Aalborg Universitet Harmonic Susceptibility Study of DC ......HVDC export cable ±150– 320kV DC Offshore substation Onshore substation MVDC array cable ±50–100kV DC MMC Line-frequency

Aalborg Universitet

Harmonic Susceptibility Study of DC collection Network Based on Frequency Scan andDiscrete Time-Domain Modelling Approach

Imbaquingo Carlos Sarra Eduard Isernia Nicola Tonellotto Alberto Chen Yu-HsingDincan Catalin Gabriel Kjaeligr Philip Carne Bak Claus Leth Wang XiongfeiPublished inJournal of Electrical and Computer Engineering

DOI (link to publication from Publisher)10115520181328736

Creative Commons LicenseCC BY 40

Publication date2018

Document VersionPublishers PDF also known as Version of record

Link to publication from Aalborg University

Citation for published version (APA)Imbaquingo C Sarra E Isernia N Tonellotto A Chen Y-H Dincan C G Kjaeligr P C Bak C L amp WangX (2018) Harmonic Susceptibility Study of DC collection Network Based on Frequency Scan and DiscreteTime-Domain Modelling Approach Journal of Electrical and Computer Engineering 2018 1-25httpsdoiorg10115520181328736

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Research ArticleHarmonic Susceptibility Study of DC Collection NetworkBased on Frequency Scan and Discrete Time-DomainModelling Approach

Carlos Enrique Imbaquingo Eduard Sarra Nicola Isernia Alberto TonellottoYu-HsingChenCatalinGabrielDincan PhilipKjaeligrClausLethBakandXiongfeiWang

Department of Energy Technology Aalborg University Aalborg Denmark

Correspondence should be addressed to Catalin Gabriel Dincan cgdetaaudk

Received 19 July 2018 Revised 31 October 2018 Accepted 6 November 2018 Published 16 December 2018

Guest Editor Luigi P Di Noia

Copyrightcopy 2018Carlos Enrique Imbaquingo et alis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e equivalent model of offshore DC power collection network for the harmonic susceptibility study is proposed based on thediscrete time-domain modelling technique and frequency scan approach in the frequency domaine proposedmethodology formodelling a power converter and a DC collection system in the frequency domain can satisfy harmonic studies of any con-figuration of wind farm network and thereby find suitable design of power components and array network e methodology isintended to allow studies on any configuration of the wind power collection regardless of choice of converter topology array cableconfiguration and control design To facilitate harmonic susceptibility study modelling DC collection network includes creatingthe harmonic model of the DC turbine converter and modelling the array network e current harmonics within the DCcollection network are obtained in the frequency domain to identify the resonance frequency of the array network and potentialvoltage amplification issues where the harmonic model of the turbine converter is verified by the comparison of the converterswitching model in the PLECStrade circuit simulation tool and laboratory test bench and show a good agreement

1 Introduction

Offshore wind power is becoming an important energyresource in Europe However electrical power losses arealways a concern in the operation of wind farm with long-range power transmission To reduce losses researchers lookto replace AC with DC in the entire path from the windturbine through power collection and transmission to shoreMedium-voltage DC (MVDC) collection of wind power isan attractive solution to reduce overall losses and installationcost [1 2] e article [2] addresses cost-efficient solution toform the multiterminals of DC grid with the diode rectifierunit (DRU) However the solution of DRU can produce aconsiderable harmonic current within the array network ifthe 6-pulse DRU topology is used

Figure 1 illustrates a DC collection network of radial typeof offshore wind farm with the HVDC transmission isgeneral configuration in Figure 1 supports any choice ofMVDC collector voltage level as well as converter topology

and the number of wind power plant (WPP) clusters Forexample the turbine and offshore substation can employ anyDCDC converter topology with multiple degrees of free-dom uni- vs bidirectional power flow modular (parallelseries connection) vs monolithic topology (single activebridge dual active bridge and more [3ndash5]) Figures 1 and 2illustrate the details of wind farm configuration includingwind turbine DCDC converters offshore substation con-verters and onshore substation converters e offshorewind farm is designed to generated and deliver 700MWpower from offshore to onshore [1]

e series resonant converter (SRC) is selected as theturbine DCDC converter based on the converter efficiencyhigh-voltage transfer ratio scale of converter and galvanicisolation property Considering the existing converter to-pology and its power rating in high-power applications themodular multilevel converter (MMC) with the galvanicisolation and unidirectional power flow control is selected toserve as the offshore substation converter to deliver the

HindawiJournal of Electrical and Computer EngineeringVolume 2018 Article ID 1328736 25 pageshttpsdoiorg10115520181328736

HVDCexport cable

plusmn150ndash320kVDC

Offshoresubstation

Onshoresubstation

MVDC array cable plusmn50ndash100kVDC

MMC

Line-frequency transformer

DCAC

ACDC

MF transformerDC wind turbinegenerator

DC wind turbine generator (generator andrectifier set + isolated DCDC converter)

MVDC power collection

Figure 1 General configuration of the wind power plant with MVDC power collection

700MW plusmn 320kVDC

HVDC export cable

plusmn50kVDC

DCDC

250 MWplusmn320kVDCplusmn50kVDC

plusmn50kVDC

DCDC

250MWplusmn320kVDCplusmn50kVDC

plusmn50kVDC

DCDC


G7 times 8MW

WT

G7 times 8MW

WT

G7 times 8MW

WT

G8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT

G G G G G G G G

2400A

Offshore substation

MVDC side

Turbine DCDC converterand MVDC connection

LVDC side

11

WTGWTGWTGWTGWTG

12131415

1A1B1C1D1E

21

WTGWTGWTGWTGWTG

22232425

2A2B2C2D2E

WTGWTGWTGWTGWTG

41

WTGWTGWTGWTG

4243444546

4A4B4C4D

313233343536

3A3B3C3D3E3F

WTG

16

1F

WTG

26

2F

WTG

WTG

17

1G

WTG

27

2G

WTG

37

3G

WTGWTG

4E4F

WTGWTG

4G4H

4748

300mm2 082kmcable100mm2 082kmcable

300 mm2 082kmcable100 mm2 082kmcable

800mm2 100km

DCAC

Onshore AC transmission

network

400kV

Onshore substation

WPP cluster 1

MVDC side HVDC side

Offshore substation converter

iHVDC onshore

VHVDC onshore

VMVDC1iMVDC1 offshore

VHVDC offshoreiHVDC1 offshore

Vturb

iturb

2320A

560A 640A

iHVDC offshore

iFeeder1

iFeeder2

iFeeder3

iFeeder4

plusmn320kVDCplusmn50kVDC

plusmn50kVDC

Onshore substation converter

SM SMSM SM

SM SMSM SM

SM SM

SM SM

SMSM

SMSM

SM

SM

HVDC side

plusmn320kVDC

5060Hztransformer

Utility

Medium-frequencytransformer

Resonant tank

SM SM SM

SM SM SMSM SM SM

SM SM SMSM SM SM

SM SM SM

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

Figure 2 Single line diagram of example 700MW offshore DC wind farm [1]

2 Journal of Electrical and Computer Engineering

offshore wind power to the onshore grid via the HVDCtransmission line e onshore substation captures thepower from the offshore wind farm by the onshore MMCwith a grid-connected 5060Hz transformer e onshoresubstation is designed to match with onshore AC gridvoltage (utility) and deliver the offshore power to the on-shore AC transmission network and alleviate the influencefrom the grid fault

In this study a series resonant converter topology(named SRC) shown in Figure 3 is selected as the candidatefor DC power conversion via plusmn50 kVDC MVDC array net-work With the series resonant converter the DC turbineconverter can take advantages of high efficiency high-voltage transformation ratio and galvanic fault isolationfor different ratings of turbine generator [6ndash9] A seriesresonant converter SRC with the resonant tank on thesecondary side and governed by the phase shift and afrequency-depended power flow control technique is hereconsidered [10 11]

To synthesize the DC-connected wind turbine converterand the MVDC gird the wind turbine converter has to meetthe following requirements

(1) Control of the LVDC bus(2) High-voltage transformation from the LVDC to

MVDC(3) Galvanic isolation(4) Roust and compact design

Large amount of possible topology of converter topo-logies for a DC turbine is investigated However all ofexisting converter designs are immature technology thusselecting an optimal topology is not a straightforward so-lution for converter design [12] e turbine converter isdesigned to deliver the captured wind energy produced bythe generator to the MVDC gird and then control the LVDCbus Regarding power rating number of components andthe possible solution of the turbine converter topologies inthe early DC wind farm design the hard-switched full-bridge power converter is selected as preferred topology[4 13ndash17]erefore the proposed series resonant converter (SRC) combined with the medium-frequency trans-former (MF transformer) which serves as the DC turbineconverter is selected to boot the low-voltage DC (LVDC) tohigh-voltage DC (MVDC-side of SRC) with a high-voltagetransferring ratio and provides a galvanic isolation to makesure that the line is immune from line faults [4 18 19]

e concept of SRC is widely used in the tractionapplications which is usually operated at constant fre-quency and subresonant mode to archive a roust andcompact design target in high-voltage specifications [20ndash24] One of famous modulation techniques (or topologies)is the half-cycle discontinuous-conduction-mode seriesresonant converter (HC-DCM-SRC) Under the tractionapplications the DCDC converter is tied between two DCvoltages with a fixed voltage transfer ratio and the open-loop control Relying on the pulse removal technique thebulky transformer in the classic SRC can be voided in theproposed SRC and thus the compact convert design can

be archived [25] More details of operation principle ofSRC is addressed Section 2

To archive the study of harmonic susceptibility in off-shore wind farm with the selected turbine converter theanalysis of array network in wind farm includes modelling theselected turbine converter (SRC) array cable and substationconverter where the substation converter in this study isconsidered as an ideal voltage source to produce DC com-ponent of MVDC and harmonic components e objectiveof the study is to investigate how the harmonic affects theturbine converter and array networks In the power collectionsystem the wind turbines and substation are connected to-gether by medium-voltage DC cables However there are nostandards or guidelines for such DC array network designand thus some studies are still made by simply selectingcommercially available medium-voltage AC submarine cableswithout giving any additional consideration in the cablersquosproperties such as insulation [26]

Modelling of the array cables is an essential task forevaluating results of cable sizing by computer simulatione complexity of modelling is dependent on the phe-nomenon which will be conducted in the study case Forexample the travelling waves (voltage and current) with afinite propagation speed cause reflectionsis phenomenonhas to be represented on the cable model when study casesare intended to conduct the overvoltage or switching tran-sient issues Typically three types of cable models Pi ModelBergeron model and frequency-dependent models whichare available in the electromagnetic transient simulationprogram PSCADtrade (EMTDC) for studying AC and DCpower collection systems are used [27ndash30] To adequatelyrepresent the cable parameters for the harmonic suscepti-bility studies over different frequency ranges the frequency-dependent array cable models are generated from geometriccable data based on the method [31] and recommendationsproposed by the articles [32] and verified by the generatedcable model in PSCADtrade

Although these DC alternatives promise lower electricallosses and lower bill-of-material it is still an immaturetechnology As the many existing AC wind power plants theharmonic pollution is an important issue since wind powercollection systems serve by large amount of turbine con-verters [33ndash36] Depending on the configuration of the windpower plant every power collection system has its owninherent resonance behavior e network which is rich inthe harmonic current can increase the power losses and thestress of power components and cause unpredicted equip-ment trip e resonance behavior inside the collectionnetwork is affected by properties of the array cable filterdesigns converter topologies and PWM schemes ere-fore the objective of this paper is to develop a methodologyto identify potential harmonic resonant problems in the DCcollection network For example the phenomenon of har-monic amplification relates to grid contingencies variationof load flow cable length and uncertainties of network[33 37] e article [38] focuses on the stability issues ofDC grid related to the travel waves on the transmissionline and harmonic content of converters e paper iden-tifies the travelling wave in the long submarine array cable

Journal of Electrical and Computer Engineering 3

and resonant problem between the cable length and theswitching action of the substation converter According tothe practical cases the DC wind farm topologies for the casestudy mainly focus on the offshore wind farm with a shortDC power transmission path

Reviewing the common techniques for wind farmanalysis the frequency scan techniques are usually used foridentifying harmonic voltage distortion of specific har-monic component in wind farms [34 39 40] In this casethe wind turbines are represented for a specific frequencywith an admittance equivalent circuit e array cablemodel is selected according to the cable length and in-teresting frequency ranges of harmonic analysis To ade-quately represent the cable parameters in the study bothequivalent PI section and frequency-dependent phasemodel are generally used to consider the frequency de-pendence of the cable parameters [28 31 41] Additionallythe wind farm configuration could be changed via theoperation of disconnectors or circuit breakers due tomaintenance or forced outages of fault erefore theharmonic study of wind farm must include all agreedswitching configuration [39]

2 Review of Operation Principle of SeriesResonant Converter

e circuit topology of series resonant converter (SRC)with the medium-frequency transformer is depicted inFigure 3 and the topology comprises a full-bridge inverter amedium-frequency transformer resonant LC tank andmedium-voltage output diode rectifier e switch pairs T1(leading leg) and T2 and T3 (lagging leg) and T4 operate at a50 duty cycle as in Figure 4(e) Commutation of switcheson the leading leg (T1 and T2) is shifted by an angle of δwhich is respect to the conduction of the switches on thelagging leg (T3 and T4)e duration of phase shift equals tohalf cycle of the LrCr tank resonant period resulting in aquasisquare excitation voltage as seen in Figure 4(a) andthen the voltage passes through the medium-frequencytransformer and excites LrCr resonant tank where the res-onant tank current ir is given in Figure 4(b) Via the dioderectifier and the output filter the current (or power) isdelivered to the medium-voltage network VMVDC Up to

this point there is no difference in the operation comparedto a constant frequency and phase-shift control of classicalSRC which is usually operated in the super resonant modeto achieve ZVS at turn on

Considering the switching losses with IGTB applica-tions the subresonant mode is preferred in SRC whichallows ZCS or a low current at turnoff (Figure 5(a)) re-gardless of switching frequency a full-resonant currentpulse is delivered to the load [12] e control law of SRC isalso allowed ZVS at turnon and ZCS at turnoff for the dioderectifier as shown in Figure 5(b) It should be noted that theoutput power has a positive relation with the switchingfrequency which means that the output power depends howmuch energy (current pulse) transfers to the output stageSRC operated in a low frequency means lower outputpower while high-frequency operation will deliver a higheroutput power

3 LinearizedModel andClosed-LoopControl ofSeries Resonant Converter

e objective of the study is to understand the harmonicsdistribution of offshore DC wind farm and how the DCwind turbines are affected by harmonics from the MVDCgird Figure 6 summarizes the derivation of the plantmodel of the DC wind turbine based on the discrete time-domain modelling approach (Steps 1ndash8) [42] the transferfunction of the output LC filter (Step 9) and SRCrsquoscontrol design (Steps 10ndash12) e details of the derivation(or equations) are address in Appendices A B and Cwhich can help reader to reach the harmonic model of theDC wind turbine and then conduct harmonic suscepti-bility study of the DC wind farm e following discussionincluding the flow chart only shows most importantconclusions (or equations) of the derivation due to thelimitation of page length

e ideal of the harmonic model of the DC windturbine converter is developed based on the harmonicmodel of the AC wind turbine and linearized model of theseries resonant converter (SRC) with discrete time-domain modelling approach [42 43] A linearized state-space model of SRC in subresonant mode is reported in (1)and in (A37) in Appendix

VLVDC

vturb

+

+ndash

ndash12 Lf

iturb

2Cf

2Cf

voRec

+

ndash

12 LfioRec

VMVDC

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime+ndash

Figure 3 Circuit topology of the DC wind turbine converter (SRC)


_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

where 1113957x1 1113957Ir and 1113957x2 1113957vcrIt is enough to recall here that the duty cycle is fixed to

50 and one event is defined as a whole time interval in

which the IGBT T1 is pulsed or not lasting half of theswitching period [10] e state variables are meaningful ofthe tank current and resonant capacitor voltage at the be-ginning of one event while the rectified output current isaveraged over one event e average output current isregulated by the proper action of the switching frequencyTransfer functions between converter output current andinput variables are

1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC

12 power14 power Full power

0

01

01

01

01

ndashN2N1 times VLVDC

T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0

Figure 4 Operation waveform of SRC (a) inverter output voltage Vg (b) resonant tank current ir (c) rectifier output current ioRec andoutput averaged current iaverage (d) transformer magnetizing current im (e) switching patterns (T1 T2 T3 and T4) [12]

T1

t

Vce

ZVS

Low-current during turn-offIce

t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)

Figure 5 Current waveform during switching of SRC [12] (a) Example of ZCS or a small current at turnoff switching waveforms T1 and T4IGBTs (b) Example of ZCS at turn-off for the rectifier diodes switching waveforms D5 and D8 diodes


e transfer function between output current and inputvariables (ie g1(s) g2(s) and g3(s)) can be obtained via

g1(s) g2(s) g3(s)1113858 1113859 C(sIminusA)minus1

B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)

ose transfer functions describe how the averaged outputcurrent of the converter before the output LC filter isinfluenced by input disturbances Regarding the voltage har-monics in the array network (MVDC network) the transferfunction g3(s) can be used to evaluate the effect of the voltageharmonics on the converter output current Detailed deriva-tion of the linearized state-space model for SRC from (1) to(3) and control design including the expression of elements in[A] [B] [C] and [D] matrixes are revealed in Appendix

e output LC filter is designed with the natural fre-quency of 100Hz (ie ff 1(2π

(LfCf )

1113968) asymp 100Hz) in

order to have constant power delivered to the MVDC gridand attenuate high frequency harmonics around twice theswitching frequency Based on the plant model of SRC thetransfer function of the compensator (or controller) gc(s) inthe control block shown in Figure 7 for a specific operatingpoint (OP) is set as

gc K

s middot 1minus sωp1113872 11138731113872 1113873 (5)

e transfer function gc in (5) is designed based onpredetermined specifications such as the ability to reach itsreference output current with least ringing overshoot andfastest dynamic during perturbations in input variables ecompensator is composed by an integrator to minimize thesteady-state error a pole around the resonant frequency ofthe filter to lower the resonant magnitude peak influence(ωp) and finally a gain K necessary to set a proper crossoverfrequency which in this case needs to be lower than theresonant frequency of the filter to avoid instability Controldeign of SRC is revealed in Appendix and the parametersof the controller are described in Section 6 that operates inthe working conditions specified in Table 1 are shown inTable 2

Figure 8 demonstrates the transfer function gc(s) at theoperation point (ωp minus250 rads K 508 dB and Pout

127MW) e result shows that the gain margin GM is

Step 12Step 10 Step 11

Step 5 Step 6 Step 7 Step 8


Circuittopologyof the DC

wind turbineconverter

Clarify theresonant tank

waveformaccording to

the modeof operation

Draw theequivalent

circuit

Derive thelarge signal

model

Linearizationand derivation

of the smallsignal model

Define thestate variables

Create thestate-spacemodel of

the converter

Createinteresting

transferfunctions

Create SRCcontroller

Specify thecontrol

performance

Draw thecomplete

control block ofthe DC wind

turbineconverter

Derivation of SRC plant model

Derivation of SRC control design

Step 9

Derivethe outputLC filter

Figure 6 Flow chart of mathematical derivation of the converter plant model and the digital controller


infinite (gt0 dB) and the phase marginΦM is around 787 degat 498 rads which represents that the controller is stablee design criterion has been implemented to select theparameters for the different operation points (differentoutput powers of the DC wind turbine) which have enoughgainmargin and the phasemargin to deal with the parameteruncertainty and variation

4 Modelling of DCWind Turbine Converter forHarmonic Susceptibility Study

To identify the harmonic model of the DC wind turbineconverter for the harmonic susceptibility study the voltageharmonics within the MVDC network are considered as aperturbation on the steady-state operation point Accordingto the control block diagram of the SRC given in Figure 7the harmonic components (disturbance component 1113957Iturb) onthe output current are dominated by the disturbance on thepower reference signal PREF the disturbance in the low-voltage DC network Vg and the disturbance in the outputterminal voltage of diode rectifier Vturb

Without the disturbance component in the power ref-erence signal PREF and in the low voltage DC network Vg(1113957PREF 0 and 1113957Vg 0) therefore the harmonic model ofWTG (wind turbine generator) considering the output LCfilter (Lf and Cf ) can be represented by an admittanceequivalent Yeq in (6) as given in Figure 9 where the har-monic admittance Yeq is affected by the circuit parameters of

DC wind turbine converter topology (plant model) thecontrol design and the output LC filter

Yeq minus1113957Iturb1113957Vturb

sCf minusg3(s)

1 + s2LfCf + gc(s)g1(s)minus sLfg3(s) (6)

where the admittance of the converter without consideringthe output filter can be represented by the followingequation

Yc minus1113957IoRec1113957VoRec

minusg3(s)

1 + gc(s) middot g1(s) (7)

Equation (6) describes satisfactorily the variation of theconverter current delivered to the grid in response to gridvoltage disturbances where the transfer functions g1 and g3 in(6) and (7) (the details of expression is in (3)) describe how theoutput current 1113957IoRec is influenced by the disturbances in thecontrol input signal 1113957fs and the output voltage 1113957Vo (prop voltageVMVDC) respectively In practical applications the output LCfilter contains a tiny amount of the parasitic resistances whichcan alleviate the peak value of admittance magnitude Yeq atthe resonances frequency as done in the following sections forthe study case described in Section 6

Figure 10(a) shows the variation of admittance magni-tude |Yeq| of the converter withwithout the output LC filterIt also represents the magnitude |Yc| of the converterwithout the output LC filter to observe the filterrsquos influenceon the equivalent admittance of the converter Admittancemagnitude of the converter with the output filter is high as itapproaches the natural frequency of the output LC filter (ff asymp100Hz) while the magnitude of admittance of the converteritself (without output filter) is low in the whole spectrum andreaches a constant value at low frequencies Regarding thephase angle of the admittance equivalent the behavior of theconverter with output filter changes from inductive at lowfrequencies to capacitive at the natural frequency of thefilter as shown in Figure 10(b)

5 Harmonic Susceptibility Study of DCCollection Network

In this section the harmonic susceptibility study of DCcollection network is revealed based on the harmonic modelof the turbine converter developed previously and compared

g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs

CompensatorSRC plant model

sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim

Vturb (connected to the MVDCnetwork where the converter finds

disturbances)

sim

sim

Iturb (output of theconverter connected

to array cable)

sim

sim

simsimsim

++

+ndash

Figure 7 Block diagram of the the DC wind turbine converter (SRC) in the closed-loop control with the output filter

Table 1 Parameters of the plant model of SRCTurns ratio of transformer (N1 N2) 1 25Resonant inductor Lr 781 (mH)Resonant capacitor Cr 025 (μF)Output filter inductor Lf 250 (mH)Output filter capacitor Cf 10 (μF)

Table 2 Specifications of operating point and control parametersLow-voltage DC VLVDC 404 (kVDC)Medium-voltage DC VMVDC 1000 (kVDC)Switching frequency fs 800 (Hz)Rated output power Pout 82 (MW)Crossover frequency 1591 (Hz)Gain K 586 (dB)ωp minus400 (rads)


WTG11WTG12WTG13WTG14WTG15

1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable

100mm2 082kmcable 300mm2 082kmcable

Admittance equivalent

Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime

Figure 9 Study case of DC collection network for one single cluster of wind turbines with the proposed harmonic model (admittanceequivalent) of series resonant converter SRC (DC wind turbine generator WTG) with the output filter included (Lf and Cf ) for theharmonic susceptibility study

With the output filter |Yeq|

Without the output filter |Yc|

0 50 150100 200 250 300Disturbance frequency (Hz)

002

004

003

001

0

Adm

ittan

ce (S

)

(a)

With the output filter angYeq

Without the output filter angYc


0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)

Figure 10 Characteristic of equivalent admittance (harmonic model) of the converter for the study case of Tables 1 and 2 (a) Magnitude ofadmittance (b) Phase angle of admittance

Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200

ndash20ndash40ndash60

Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram

ndash180Phase margin ΦM asymp 787deg

Figure 8 Bode plot of gc(s)


to a switchingmodel implemented in PLECStrade According toFigure 9 a radial-type cluster of seven wind turbines of anoffshore wind power plant (OWPP) is given It will representthe study case for the comparison of the harmonic modeland switching model e parameters for the turbine con-verter and operating point of the plant model with controllerare given in Section 6 Since different configurations ofcollection network and wind turbine converter topologiescould generate different harmonic spectra backgroundharmonics of the DC grid (correctionarray network) fromthe offshore substation DCDC converter and DC windturbine generator (DC WTG) can cause undesirable reso-nant phenomena and raise stability issues One of the majortasks of the harmonic susceptibility study is to identifyresonance frequencies of network and investigate potentialvoltage amplification issues

51 Cable Model Array cables in the following study casesare modelled as a pi-equivalent model with longitudinaladmittance parallel branches in order to consider the fre-quency dependency of distributed cable parameters andrepresent correctly the behavior in the frequency rangingfrom 0 to 10 kHz [31 44] In the study the electromagnetictransient program PSCAD is usefully employed to extract thelongitudinal admittance and the shunt capacitance from thegeometric data of array cable [41] e values of resistanceinductance and capacitance to use at each frequency arecalculated by the software individually

According to the cable model in PSCAD the capacitanceis not varying within the range of interesting frequencyInstead as the frequency rises the resistance keeps in-creasing due to the skin effect and the inductance decreasesits value due to the eddy currents in the outer conductorsFor the considered study case eight RL series branches areused to approximate the longitudinal impedance

e PSCAD generates the admittance date (ie matrix[ZPSCAD] and [YPSCAD]) from the geometrical data of themarine cable (in frequency-dependent (phase) model) incertain operational frequency to form ldquopi-equivalent cablemodelrdquo [45] An equivalent pi-equivalent model is builtbased on the cable model which can fit the admittance date(matrix [ZPSCAD] and [YPSCAD]) from the geometricaldata of the submarine cable (implemented in frequency-dependent (phase) model in PSCAD) to make sure that theadequate cable model is implemented in the array cablenetwork Actually the frequency-dependent phase model inPSCAD is also regarded as a most numerically accurate cablemodel thus far e formulation of models has been provedthat the bandwidths of the model can be up to 1MHz whichcan satisfy most of transient studies and harmonics studies[41 45]

52 Frequency Scan Approach e WPP cluster consists ofseven turbines with seven MVDC cable segments where theoffshore substation converter is represented by an idealharmonic voltage source According to the frequency scanapproach for network analysis addressed in Figure 11 theharmonic voltage source (disturbance component) in the

output DC voltage of the offshore substation converter isactivated and injects harmonic voltage component into theMVDC grid with 05 (0005 pu) of voltage magnitude(|VMVDChn| 500V) and the frequency of injected har-monic voltage raises until 300Hz by steps of 20Hz eoutput voltage and output current traces of each windturbine model in each frequency step are extracted for FFTanalysis after the steady-state condition is reached isapproach allows an arbitrary choice of frequency andmagnitude of the disturbance in the offshore substationerefore it does not intend to represent only a specificrange of harmonics in the DC collection network Even-tually models of the DC collection system (or network)could be established for different frequencies of the injecteddisturbances As an example for the harmonic model thewind turbine converter can be described by a particular RLor RC series branch for disturbance frequency

Based on the harmonic model of DC collection networkin Figure 9 Figure 11 gives a flow chart of frequency scanapproach for the DC collection system in the harmonicsusceptibility study e process includes the identificationof the harmonic model of the DC wind turbine for certainfrequency of the injected disturbance (harmonic) and theestablishment of the harmonic model of DC collectionnetwork e DC collection network in Figure 9 is set as anexample to explain the sequence of frequency scan illustratedin Figure 11 Step 1 the harmonic voltage source in theoffshore substation (1113957VMVDC) is connected to the networkand injects a 20Hz harmonic disturbance into the MVDCnetwork with a specific voltage magnitude |VMVDC| eharmonic model of the DC wind turbine generator (WTG)under 20Hz perturbation (harmonic frequency) in MVDCnetwork is identified by connecting to theMVDC network att t1 Step 2 the voltage and current within network areextracted for FFT analysis after the voltage and current havereached the steady-state condition Step 3 at t t2 thefrequency for which the harmonic model of the DC windturbine is evaluated and is increased by 20Hz as the fre-quency of the injected disturbancee algorithm turns backto Step 2 Eventually the frequency sweep continues until300Hz (at t tnminus1) which depends on the control bandwidthof the DC wind turbine converter and interesting frequencyrange in the harmonic susceptibility study

6 Verification

Based on the DC collection network in Figure 9 and theconverter control design addressed in (5) Tables 1 and 2 givethe parameters used in the study of the DC collection gridwhere the geometric parameters of cables are shown inFigure 12 which is implemented in the created equivalentPI section according to the frequency-dependent (phase)model in PSCAD For the turbine harmonic model all theturbines are considered operating in the same operatingpoint described in Table 2 As far for the switching modelwind turbine converters are operated with a switchingfrequency of 800Hz that guarantees around an output powerof 82MW when the output voltage of the converter is100 kV e LVDC source (input voltage) of each converter


Identification of the harmonic model

of the DC windturbine

Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz

t1 t2 tntnndash1tnndash2t0

~

fMVDC = f~ ~

fVo = ffVo = f~ ~

DC collectionnetwork

|VMVDC|~

Figure 11 Frequency scan approach in the harmonic susceptibility study

0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)

ConductorInsulator 1

SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)

Figure 12 MVDC submarine array cables spacing and dimensions (XLPE pairs of single core conductor cross section 100mm2 (a) and300mm2 (b)) [41]


is 404 kV while the voltage drops across cables are re-sponsible for different voltages at the output terminal of thewind turbines e controller of each turbine regulates theswitching frequency in order to control the output power tofollow its reference In particular the frequencies are higheras wind turbines move further away from the offshoresubstation to compensate the lower voltage difference onthe two sides of the converter According to the frequencyscan approach illustrated in Figure 11 additionally a per-turbation with a specific sequence of frequency in the rangeof 20 to 300Hz is applied to the corresponding harmonicmodel of turbine converter and the results are given inFigure 13 for a single wind turbine

Figure 13 gives a verification of the turbine harmonicmodel (equivalent admittance) by comparing the harmonicspectrum of output current with the results generated by thecorresponding switching model of the turbine converter inPLECStrade e current spectrum shows that the derivedharmonic model and converter switching model are veryclose up to 300Hz e magnitude of output currentspectrum is reduced as the frequency of the harmoniccomponent is far from the natural frequency of the outputLC filter where the mismatch between the harmonic modeland its switching model in PLECStrade becomes higher Fur-thermore the turbine output current trace shows that theharmonic component of output current is amplified whilethe frequency of MVDC harmonic voltage (VMVDChn)approaches to the natural frequency of output LC filter of theturbine converter

e results of network (array network) simulation inFigure 14 between both models (harmonic model and thecorresponding switching model) show a good agreementHowever somemismatch coming from the harmonic modeldeveloped can be seen in all turbines in cluster Furthermoreas observed along the cluster last wind turbines (shown inFigures 14(f) and 14(g)) have second- and third-ordercurrent harmonics due to nonlinear behavior consideredby the turbine converter switchingmodelis is because theproposed harmonic model cannot predict these higher orderharmonics since the modelling approach for the turbineconverter in MVDC network are linear Figure 15 gives acomparison of the harmonic spectrum of output current forwind turbines at different locations to study the error of theharmonic model While in the closest WTG to the offshoresubstation a sinusoidal wave is observed (WTG11) inthe last ones sinusoidal waves have a considerable amountof harmonic components at multiples of the injected dis-turbance frequency (WTG17) Furthermore the amplitudesof the fundamental current harmonics decrease along thecluster approaching the farthest wind turbine from theoffshore substationis phenomenon is related to the risingamplitude of higher order harmonics as depicted in Fig-ure 15 Additionally it is worth mentioning that in theswitching model there are current harmonics around1600Hz twice of the switching frequency of converter asthe filter at the output of each converter cannot cancelcompletely these current harmonics

e harmonic current spectrum at natural frequencyof the output LC filter (asymp100Hz) was not shown in the grid

simulation in Figure 14 because it was in the equivalentturbine model (admittance equivalent) since it was ldquoidealrdquocurrent source (linear model) the parameters werereally high and were not matching the switching modelat al Probably the filter damping is too small which resultsin very high current (approx 500 A) at resonance of theoutput LC filter for a single turbine in grid simulation(equivalent model) However the results of grid simula-tion still shows that the admittance equivalent followsresults from the switching model for a single turbine inall frequencies at natural frequency of the turbine outputLC filter the admittance equivalent is clearly not wellrepresented

In the verification of modelling of the array cable withequivalent PI-cable section based on the method proposedby Beerten et al [31] the values of resistance and inductanceof each branch are calculated in order to fit the behavior ofthe longitudinal admittance in the frequency domain cal-culated by PSCADrsquos cable model (frequency-dependent(phase) model) Figure 16 shows a good matching of thefitted values of the admittance by the eight parallel branches(N 8) and the values calculated by software (PSCADrsquoscable) e error on the magnitude of the admittance be-tween two models is less than 10 within the range of0001Hzndash10000Hz e error in phase is always less than17deg in the range of interesting frequency

Since the calculation of the admittance date from thefrequency-dependent (phase) model in PSCAD is notstraightforward and is extremely complicate there are somearticles which are useful to conduct the calculation of ad-mittance date from the geometry date of the cable model[28 30 45]

7 Laboratory Test

A laboratory test is performed to prove the validity of theproposed harmonic model of the turbine converter Aschematic of the setup of the DC wind turbine converter forSRC is depicted in Figure 17 e down-scale prototypecircuit with 216VDC (VLVDC) to 400VDC (VMVDC) DC windturbine converter is built to verify the harmonic distributionof the turbine converter e wind turbine converterswhatever AC or DC turbines are the boost structures inorder to increase the efficiency of power transmissionerefore the down-scale prototype circuit is designed tomatch with this character (boost the low-voltage DCVLVDCfrom the generator) to match with the DC grid voltage(medium-voltage DC VMVDC) from the offshore substationAlthough only a single turbine converter is implemented inthe test bench the harmonic signatures of the wind turbineconverter (in the test bench) in the interesting frequencyrange have been confirmed which match with the simulationmodel as in the following results

A programmable AC source is used to emulate theharmonic voltage source (VMVDCh) with sinusoidal waves atdifferent frequencies on the output terminal of the converterA module of resistors Rload and a diode DAux are placed onthe output side of the converter to simulate the character ofunidirectional power flow of MVDC network Table 3


20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0

Disturbance frequency (Hz)

Am

plitu

de (A

)

Generated by a correspondingswitching model

Generated by theadmittance equivalent

(harmonic model) Ĩturb

(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)

Figure 13 Spectrum of turbine output current with a 0005 pu harmonic disturbance on the output voltage terminal of the single windturbine converter (VMVDC plusmn50 kV) (a) Magnitude of current harmonic (b) Phase angle of current harmonic

200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


20 40 60 80 100 120 140 160 180 200 220 240 260 280 300Disturbance frequency (Hz)

Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)

20 40 60 80 100 120 160 180 200 220 240 260 280 300Disturbance frequency (Hz)

140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)

Figure 14 Output current spectrum of individual turbines in the operating point described in Tables 1 and 2 being WTG11 the closestto the offshore substation and WTG17 the furthest Harmonic current amplitude and phase angle are represented for a switching modelof the DC wind turbine converter (|IturbswWTGxx|angθturbswWTGxx) and for the admittance equivalent model (harmonic model)(|IturbeqWTGxx|angθturbeqWTGxx) (a) WTG11 (b) WTG12 (c) WTG13 (d) WTG14 (e) WTG15 (f ) WTG16 (g) WTG17

Fundamentalcomponent

2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Fundamental frequency disturbance (Hz)

(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)

Fundamental frequency disturbance (Hz)20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

(b)

Figure 15 A comparison of output current magnitudes for the switching model in PLECStrade for the WTG11 (first WTG) (a) and WTG17(seventh WTG) (b) (fundamental component injected harmonic component)


describes the power rating and circuit parameters of thelaboratory test bench of SRC where the average of outputvoltage is around 400VDC (VMVDC Vturb DC component

of output voltage of the turbine) and the output current (DCcomponent of output current of the turbine) is around14ADC

Bode plot of longitudinal admittance (magnitude) (Ωndash1)

Angular frequency (rads)10ndash3 10ndash2 10ndash1 100 102101 103 105104

ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10

Optimum solution of admittancefor PI section |YOptimum|

Admittance generated by geometricalcable data in PSACD |YPSCAD|

(a)

10ndash3 10ndash2 10ndash1 100 102101 103 105104

Bode plot of longitudinal admittance (phase) (deg)

Angular frequency (rads)

ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30

Optimum solution of admittancefor PI-section angYOptimum

Admittance generated by geometricalcable data in PSACD angYPSCAD

(b)

Figure 16 Results of the optimum vector fitting for the longitudinal admittance of the submarine array cable with the parallel of eight RLseries branchese longitudinal admittance values found in PSCAD are highlighted in red star (points) while the admittance of the parallelof the eight series RL branches is depicted in blue curve (line) (a) Magnitude of calculated admittance (b) Phase angle of calculatedadmittance


A harmonic model and switching model in PLECStrade arealso developed with the parameters of the laboratory setupfor the comparison e operating switching of SRC is setto 800Hz with 50 duty cycle and an output power of550W Input voltage (VLVDC) is set to 216 V which is 11 Vslightly higher than the input voltage (205 V) in theswitching model developed in PLECStrade to take into accountvoltage drops in the practical setup e DC component ofconverterrsquos output voltage (VMVDC0) is set to 400V Aharmonic voltage source (VMVDCh) of 35VRMS with fre-quency varying in the range of 20 to 300Hz is used forharmonic susceptibility tests Harmonics current in theoutput are recorded and the results from the laboratoryexperiment PLECStrade simulations and the admittanceequivalent developed for the laboratory setup are com-pared in Figure 18 e results show that there is a goodagreement with the proposed DC turbine harmonicmodel However there is a mismatch in the amplitudes(Figure 18(a)) when the injected frequencies approach thenatural frequency of the output LC filter As given inSection 4 the admittance of the DC turbine harmonic modelis heavily influenced by the output LC filter e behavior ofthe converter changes from inductive to capacitive at thenatural frequency of the filter as in Figure 18(b)

It should be noted that there is no standard for selectingthe medium-voltage DC (MVDC) grid e MVDC of400VDC is chosen as output voltage of the down-scaleprototype circuit of SRC from the available test equip-ment in the laboratory for simulating the MVDC gird(controllable DC power source) In practical applicationsthe selection of MVDC is based on the electromagnetictransient (EMT) simulation of the grid and existing off-shore wind farm technologies such as power convertersprotection devices available offshore platform and arraycables [1]

8 Conclusion

A model-based methodology in conducting the harmonicsusceptibility study is revealed based on the discrete time-domain modelling technique and the frequency scan ap-proach in the frequency domain e developed harmonicmodel in the paper can be used to perform a harmonicsusceptibility study of power collection network in an earlystage of a wind farm project because of its simplicity andlow computational cost compared to a switching model Inthe later stage the proposed method is recommended toconsider the switching model of the converters in order tobe closer to reality e process of harmonic susceptibilitystudy contains identification of the harmonic model of theturbine converter and the establishment of array networkin the frequency domain e validation of the harmonicmodel of the turbine converter is given in Sections 6 and 7via computer simulation implemented in PLECStrade and thelaboratory tests respectively Additionally the distributionof current harmonics within DC collection network isinvestigated including the identification of possible reso-nance frequency within array network Moreover ana-lytical data from simulation and the laboratory tests giveindications to set a proper design and natural frequency forthe output LC filter of the DC turbine converter and amatched switching frequency for the converter placed inthe offshore substation

Appendix

According to the flow chart of mathematical derivation inFigure 6 in Section 3 firstly the circuit topology of theturbine converter and mode of operations are decided asshown in Figures 3 and 19 and then the equivalent circuitbased on the switching sequence of transistors is created asshown in Figure 20 In Figure 20 the large-signal model ofthe converter is generated (Step 4) and then the interestingstate variables (Step 5) are defined to create the small-signalequations Steps 6 and 7 establish the space model of theplant Finally the converter plant mode is established basedon the discrete time-domain modelling approach proposed in(Step 8) [42 46] Step 9 gives a set of interesting transferfunction of the output LC filter Eventually Steps 10 to 12focus on the control design for SRC

A Derivation of the Plant Model

Steps 1ndash3 Decide the circuit topology of DC turbineconverter resonant tank waveform and equivalent circuit

Steps 1ndash3 describe how to obtain the large-signal modeland the corresponding equivalent circuit in subresonantCCM in Figure 20 from the circuit topology of SRC whichis operated in subresonant CCM as in Figures 3 and 20respectively e waveform in Figure 19 is divided by dif-ferent time zones (different switching sequences) based onthe discrete time-domain modelling approache figures areeventually used to generate the large-signal model of SRC

SRC (Figure 3)Turns ratio of transformer N = 2

Lr = 20mH Cr = 1microFLf = 25mH Cf = 10mF

+

ndash

VMVDC

iturb

VMVDC0

VMVDChVLVDC = 216VDC

= 400VDC

Perturbation

DC power source Rload =125Ω

DAux

+

ndashVturb

Figure 17 Schematic of the laboratory setup for experiment

Table 3 Specifications of the laboratory test benchLow-voltage DC source VLVDC 216 (VDC)DC component of medium-voltage source VMVDC0 400 (VDC)Transformer winding voltage ratio N1 N2 1 2Output power Pout 550 (W)Resonant inductor Lr 200 (mH)Resonant capacitor Cr 10 (μF)Output filter inductor Lf 25 (mH)Output filter capacitor Cf 10 (mF)Resistive load Rload 125 (Ω)Duty cycle 50


Step 4 Large-signal model

In the large-signal model the final value of interestingstate variables in each switching interval is represented by theinitial values of states According to discrete time-domainmodelling approach the procedure is only valid when thevariations in MVDC grid voltage vo(t) (output terminalvoltage) and LVDC voltage vg(t) (input terminal voltage) inthe event (switching) are relatively small than their initial andfinal values [42] According to Figures 19 and 20 Equations(A1) (A3) (A4) (A6)ndash(A10) (A12) and (A13) show thedetails of derivation of the large-signal model of resonantinductor current ir(t) capacitor voltage vCr(t) and their endvalues at kth event in terms of initial values of kth event

For t0(k) le t le t1(k) (T1 T4 ON)

vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)

e resonant inductor current ir(t) and resonant ca-pacitor voltage vCr(t) can be obtained by (A1)

20 40 60 80 120 140 160 180 220 240 260 280200100 300Disturbance frequency (Hz)

2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel

(admittanceequivalent)

(a)

Laboratorytest Switching model (PLECSTM)

Harmonic model (admittance equivalent) 300

250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)

Figure 18 Harmonic spectrum of output currents for the scaled-down experiment with 50V perturbation in amplitude in the range of 20to 300Hz laboratory test result result generated by the switching model in PLECStrade simulation and result generated by the proposedharmonic model (a) Magnitude of current harmonics (b) Phase angle of current harmonics


ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)

ωst2(k) =ωst0(k+1)

ωst2(k+1) =ωst0(k+2)

Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)

Figure 19 Resonant inductor current and resonant capacitor voltage waveforms of SRC in subresonant CCM

+ndash

ndash

+

ndash

+

irprime(t)

vprimeg(t) vg(t) = Vg0(k) = VprimeLVDC= N2N1 times VLVDC

vo(t) = Vo0(k) = VMVDC

+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)

vprimeg(t) vg(t) = Vg1(k) = 0

vo(t) = Vo1(k) = ndashVMVDC

+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)

vprimeg(t) vg(t) = Vg0(k+1)= ndashVprimeLVDC

vo(t) = Vo0(k+1) = ndashVMVDC

+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)

vprimeg(t) vg(t) = Vg1(k+1) = 0


+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)

Figure 20 Equivalent circuit of SRC for large-signal analysis of conduction intervals in subresonant CCM (a) t0(k) le t le t1(k) (T1 T4 ON)(b) t1(k) le t le t2(k) (D1 T3 ON) (c) t0(k+1) le t le t1(k+1) (T2 T3 ON) (d) t1(k+1) le t le t2(k+1) (D2 T4 ON)


ir 1

ZrVg0(k) minusVo0(k) minusVCr0(k)1113872 1113873sin ωrt( 1113857

+ Ir0(k) cos ωrt( 1113857

(A3)

vCr Vg0(k) minus Vg0(k) minusVo0(k) minusVCr0(k)1113872 1113873cos ωrt( 1113857

+ Ir0(k)Zr sin ωrt( 1113857minusVo0(k)(A4)

where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)

At time t t1(k) the tank current ir reaches the point ofzero crossing commutating the IGBT switches T1 and T4turnoff and turning on body-diode D1 and IGBT switch T3erefore the resonant inductor current ir(t) can beexpressed byIr1(k) ir t1(k)1113872 1113873

1Zr

Vg0(k) minusVo0(k) minusVCr0(k)1113872 1113873sin ωrt1(k)1113872 1113873

+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr

ωsβK ωrs middot βK for t0(k) 0

tan ωrsβK( 1113857 minusIr0(k)Zr

Vg0(k) minusVo0(k) minusVCr0(k)1113872 1113873

(A7)

0lt ωrsβK( 1113857le π t1(k) βKωs

VCr1(k) vCr t1(k)1113872 1113873 Vg0(k) minus Vg0(k) minusVo0(k) minusVCr0(k)1113872 1113873

middot cos ωrsβK( 1113857 + Ir0(k)Zr middot sin ωrsβK( 1113857minusVo0(k)

(A8)

For t1(k) le t le t2(k) (D1 T3 ON)

ir tprime( 1113857 minus1Zr

Vo1(k) + VCr1(k)1113872 1113873sin ωrtprime( 1113857 (A9)

where

tprime tminus t1(k)

vCr tprime( 1113857 Vo1(k) + VCr1(k)1113872 1113873cos ωrtprime( 1113857minusVo1(k)(A10)

where

Ir1(k) 0

Vg1(k) 0

Vo1(k) minusVMVDC minusVo0(k)

(A11)

Eventually the end values of inductor current ir(t)

and capacitor voltage vCr(t) (at time t t2(k)) can bewritten as

Ir2(k) minussin ωrsβK( 1113857 middot sin ωrsαK( 11138571113858 1113859 middot Ir0(k)

+ minus1

Zrmiddot cos ωrsβK( 1113857 middot sin ωrsαK( 11138571113890 1113891 middot VCr0(k)

+2

Zrmiddot sin ωrsαK( 1113857 +

minus1Zr

middot cos ωrsβK( 1113857 middot sin ωrsαK( 11138571113890 1113891

middot Vo0(k) + 1113890minus1Zr

middot sin ωrsαK( 1113857 +1

Zrmiddot cos ωrsβK( 1113857

middot sin ωrsαK( 11138571113891 middot Vg0(k)

(A12)

VCr2(k) Zr middot sin ωrsβK( 1113857 middot cos ωrsαK( 11138571113858 1113859 middot Ir0(k)

+ cos ωrsβK( 1113857 middot cos ωrsαK( 11138571113858 1113859 middot VCr0(k) + 1113876minus2

middot cos ωrsαK( 1113857 + cos ωrsβK( 1113857 middot cos ωrsαK( 1113857 + 11113877

middot Vo0(k) + 1113890cos ωrsαK( 1113857minus cos ωrsβK( 1113857

middot cos ωrsαK( 11138571113891 middot Vg0(k)

(A13)

where

ωs middot t2(k) minus t1(k)1113872 1113873 αK (A14)

e same process of derivation as given in (A1) (A3)(A4) (A6)ndash(A10) (A12) and (A13) can be applied forobtaining the large-signal expression of resonant inductorcurrent ir(t) and resonant capacitor voltage vCr(t) in (k +1)th event (t0(k+1) le t le t1(k+1) and t1(k+1) le t le t2(k+1))

A1 Steady-State Solution of Large-Signal Model e fol-lowing equation expresses the conditions for obtainingsteady-state solution (at certain operating points of CCM) ofthe discrete state equation of the large-signal model

Ir2(k) minusIr0(k)

VCr2(k) minusVCr0(k)(A15)

By substituting (A15) into (A12) and (A13) the steady-state solution of Ir0(k) and VCr0(k) can be written in term ofVo0(k) Vg0(k) βk and αk

Ir Ir0(k) f Vo0(k) Vg0(k) βK αk1113872 1113873

VCr VCr0(k) f Vo0(k) Vg0(k) βK αk1113872 1113873(A16)

where the overbar used in equations is to indicate the steady-state value of the interesting state variables For simplifying thederivation of the plant model the output LC filter of SRC(ie Lf and Cf ) is neglected due to a very slow dynamics involtage and current comparing with the dynamics in theresonant inductor current ir(t) and resonant capacitor voltage


vCr(t)erefore only theDC component of the output currentdiode rectifier ioutRec is considered as an output variable whichis expressed by io Eventually the output current equationdelivered by the SRC during the Kth event is express as

io 1ck

1113946βk

0ioutRec θs( 1113857dθs +

1ck

1113946ck

βk

ioutRec θs( 1113857dθs

1ck

middot1ωrs

sin ωrsβk( 1113857 +1ωrs

sin ωrsβK( 1113857 middot 1minus cos ωrsαk( 11138571113858 11138591113896 1113897

middot Ir0(k) +1ck

middot 1113896minus1ωrs

1Zr

1minus cos ωrsβk( 11138571113858 1113859 +1ωrs

1Zr

cos

middot ωrsβK( 1113857 middot 1minus cos ωrsαk( 11138571113858 11138591113897 middot VCr0(k) +1ck


1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857

middot 1minus cos ωrsαk( 11138571113858 11138591113897 middot Vo0(k) +1ck

middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857

middot 1minus cos ωrsαk( 11138571113858 11138591113897 middot Vg0(k)

(A17)

whereθs ωst

αk ck minus βk(A18)

It should be noted that the initial value of inductor currentis represented by Ir0(k) VCr0(k) is the initial value of capacitorvoltage the initial value of rectifier output voltage is written asVo0(k) and Vg0(k) is then the initial value of input voltage ofSRC Characteristic impedance Zr is defined by the pa-rameter of resonant tanks (

LrCr

1113968) αk is the transistor and

diode conduction angle during the switching interval event kand the switching frequency of the converter is represented byθs (ωsmiddott) Finally the steady-state solution of the discrete-state equation for the output variable io can be obtained bysubstituting the steady-state condition into (A17) as

Io io1113868111386811138681113868 βkβαkαckcIr0(k)IrVCr0(k)VcrVo0(k)Vo Vg0(k)Vg( 1113857

(A19)

Step 5 Define state variable

Control design technique based on the linear controltheory cannot directly be applied for mainly because of thehigh nonlinearity in the discrete large-signal state equationsin (A12) (A13) and (A17) erefore the linearization ofthe above large-signal equations is necessary Equation(A20) shows the definitions of interesting state variables inboth kth switching event (t0(k) le t le t2(k)) and (k + 1)thswitching event (t2(k) le t le t2(k+1)) which is used in the

linearization process Eventually the following equations ofapproximation of derivative are applied for converting thediscrete large-signal model into the continuous time

x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)

VCr2(k) minusx2(k+1)

(A20)

_xi tk( 1113857 xi(k+1) minus xi(k)

t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where

ck ωs t2(k) minus t0(k)1113872 1113873 ωs t0(k+1) minus t0(k)1113872 1113873 (A22)

Instead of using the state variables in (A12) and (A13) thestate variables in (A12) and (A13) are replaced by the definedstate given in (A20) With the approximation of derivation in(A21) the nonlinear state-space model is given by

_x1(k) ωs

ck

middot sin ωrsβK( 1113857 middot sin ωrsαK( 1113857minus 11113858 1113859 middot x1(k)

+ωs

ck

middot1

Zrmiddot cos ωrsβK( 1113857 middot sin ωrsαK( 11138571113890 1113891 middot x2(k) +

ωs

ck

middotminus2Zr


Zrmiddot cos ωrsβK( 1113857 middot sin ωrsαK( 11138571113890 1113891

middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr

middot cos ωrsβK( 1113857


f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck

middot flowast1 x1 x2 vo vg α1113966 1113967

_x2(k) ωs

ck

middot minusZr middot sin ωrsβK( 1113857 middot cos ωrsαK( 11138571113858 1113859 middot x1(k)

+ωs

ck

middot minuscos ωrsβK( 1113857 middot cos ωrsαK( 1113857minus 11113858 1113859 middot x2(k) +ωs

ck

middot 2 middot cos ωrsαK( 1113857minus cos ωrsβK( 1113857 middot cos ωrsαK( 1113857minus 11113858 1113859

middot Vo0(k) +ωs

ck

middot 1113890minuscos ωrsαK( 1113857 + cos ωrsβK( 1113857


f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)


where the output equation is

io fout x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

fout x1 x2 vo vg α1113966 1113967

1ck

middot flowastout x1 x2 vo vg α1113966 1113967

(A24)

Step 6 Linearization and small-signal model

Inject a small perturbation in all the interesting statevariables in Step 5 during the steady state (near the certainoperating point OP) e nonlinear state equations can berewritten with Taylor series expansion in terms of the op-erating point (OP) and the perturbations

(i) Taylor series expansion of resonant inductorcurrent

x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck

middot flowast1 1113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21

11138681113868111386811138681113868111386811138681113868OP1113957x21 + middot middot middot

(A27)

where the subscript OP in equations indicate thatthe equation is evaluated at that steady-state point

OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)

(ii) Taylor series expansion of resonant capacitorvoltage

x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)

(iii) Taylor series expansion of the output currentequation

Io + 1113957Io fout x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg + 1113957Vg α + 1113957α1113966 1113967

1

(c + 1113957c)middot flowastout1113882x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then


Io + 1113957Io fout x1 x2 Vo Vg α1113966 1113967 +zfout

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c

middot 1minus1113957c

c1113888 1113889 +

1113957c

c1113888 1113889

2

minus middot middot middot + middot middot middot⎧⎨

⎩

⎫⎬

⎭

middot 1113896flowastout x1 x2 Vo Vg α1113966 1113967 +

zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP

middot 1113957x2 +zflowastoutzvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo +zflowastout

zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1

11138681113868111386811138681113868111386811138681113868OP1113957x21 + middot middot middot 1113897

(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c

middot 1113896zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OPminus Io middot

zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891 middot 1113957α1113897

(A36)

e linearized equations can be obtained by neglectingthe higher order terms of perturbation signals and retainingonly the linear terms in the Taylor series expansion for(A27) (A31) and (A35)

Step 7 State-space model

A set of the linearized state-space model (in (A37))of SRC in the subresonant mode can be generated from(A27) (A31) and (A35) Additionally the expressionof transfer functions between the input state variables and theinteresting state variables are given in (A44) and (A46)

_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c

middotzflowastoutzx1


zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c

middotzflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)

For derivative of equations f1 and f2

zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)

With the steady-state operating conditions

ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)

e same derivation process which are used tothe derivatives of f1 and f2 with respect to input statesx1 and x2 can be used to calculate the derivative of flowastoutin the output equation and the derivatives of f1 and f2with respect to input states α vg and vo Additionallythe angle β and its steady-state solution can be calculatedby the derivation of the large-signal model in (A7) as

tan ωrsβ( 1113857 minusx1Zr

vg minus vo minusx21113872 1113873 tan ωrsβminus π( 1113857

β|OP πωrs

+1ωrs

middot tanminus1minusIrZr

Vg minusVo minusVCr1113872 1113873⎡⎢⎣ ⎤⎥⎦

(A42)

e derivative of β with respect to input states x1 x2 vgand vo at the given operating points is expressed by thefollowing equation


zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusZr middot vg minus vo minus x21113872 1113873

vg minus vo minus x21113872 11138732

+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusZr middot Vg minusVo minusVCr1113872 1113873

Vg minusVo minusVCr1113872 11138732

+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)

where ωrs (ωrωs) is defined by the ratio between thenatural frequency of the resonant tank (ωr) and theswitching frequency of SRC (ωs)

Step 8 Transfer functions

According to (A37) the transfer functions between theconverter output current (output rectifier current) and inputstate variables can be obtained

1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

1113957fs 1113957α(s) middotfr

minusπ

(A45)

and transfer functions between defined internal state vari-ables and input state are

1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)

e details of derivation of the linearized plant modeland the expression of elements in matrixes [A] [B] [C] and[D] are shown in (A37) where the expressions of transferfunctions g1(s) g2(s) and g3(s) in (A44) can be ob-tained by (3)

B Derivation of Output LC Filter

Step 9 Derivation of transfer function of output LC filter

e output current of SRC delivered from the outputdiode rectifier to the MVDC grid has a high-harmoniccontent at twice the switching frequency and its multi-ples and at the moment only the mean value of the outputcurrent in one event is considered as the real output currentAn output LC filter has to be placed on the output terminalof the converter in order to decrease as much as possible thecurrent harmonics and mitigate the inrush current duringfault e following equations give the transfer functions ofthe output LC filter used to derive the harmonic model of theDC turbine converter


gf1 1113957Iturb1113957IoRec

RC RC + RL( 1113857

1 + s middot Lf + RCRLCf( 1113857 RC + RL( 1113857( 1113857 + s2 middot LfCfRC( 1113857 RC + RL( 1113857( 1113857

gf2 1113957VoRec1113957IoRec

RCRC + RL 1 + s Lf RL( 1113857( 1113857


gf3 1113957Iturb1113957Vturb

RC RC + RL( 1113857( 1113857 middot 1 + sRCCf( 1113857


gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)

where RC and RL are the parasitic resistances of filtersand filter capacitance and filter inductance are representedbyCf and Lf respectively 1113957Vturb (1113957VMVDC) is theMVDC-sidevoltage 1113957VoRec is the output diode rectifier voltage of SRCand 1113957Vg (1113957VLVDC) is the LVDC-side voltage of SRC edynamic model of the output LC filter in the control blockis given by Figure 7

C Control Design of SRC

Steps 10 and 12 Control design of DC wind turbineconverter

Main specifications for the design of the controller areaddressed in articles [47 48] Since this is the first paper toaddress the closed-loop control of the DC wind turbine andthen conductor to the susceptibility study of offshore DCwind farm the paper did not provide specific numbers(range) for control design of the DC wind turbine in theoffshore wind farme following descriptions mainly focuson the identification of the coefficient of transfer functiongc(s) of the SRC turbine converter and thus it can give thereaders a full freedom of control design related to the DCturbine in the offshore wind farms to meet the performancerequirements such as stability zero steady-state error set-tling time overshoot and ringing and disturbances rejectioncapability

According to the structure of the control block in Fig-ure 7 the control deign of SRC gc(s) is influenced by theplant model and output LC filter H(s) where the open-looptransfer function T(s) is defined as

T(s) gC middot H(s) (C1)

where the current delivered to theMVDC grid is related withthe switching frequency of the converter by the transferfunction H(s) is represented by

H(s) 1113957iturb1113957fs

g1 middot gf1

1minusgf2 middot g3 (C2)

e design of control loop gc(s) starts by evaluating thefrequency response of H(s) At the resonant frequency(asymp100Hz) of the output LC filter a peak on the magnitudeoccurs e parasitic resistances of the capacitor Cf and

inductor Lf can provide a limited damper to attenuate theresonant peak

Following the specifications mentioned at the beginningof this subsection the design of the controller starts byadding an integrator to minimize steady-state error with astep input us the output current will reach the referencevalue with a minimized tracking error after the transient

Second a pole around the resonant frequency of theoutput LC filter is added to lower the resonant magnitudepeak influence of the filter

ird a gain K is necessary to set a proper crossoverfrequency fc which is always lower than the resonant fre-quency of the output LC filter otherwise the resonant peakwill be present in the feedback loop and the phase margin ofthe open-loop transfer function H(s) will be less than 0degleading to the instability of the closed-loop system Finallythe generic expression of the transfer function of thedesigned controller gC can be expressed as

gC K

s middot 1minus sωp1113872 11138731113872 1113873 (C3)

where K is the gain (HzA) of the controller and ωp is a pole(rads) close to the resonant frequency of the output LC filterIn order to improve the transient response of the system andto reject harmonic disturbances generally the angularcrossover frequency ωc (2π middot fc) of H(s) should be as high aspossible Typically the selected crossover frequencies fc for allthe operating points are always less than 10 of the switchingfrequency fs (fc lt 01 fs) in order to minimize the influencefrom the switching harmonics [48] e crossover frequencyof the open-loop transfer function H(s) which can be as-sumed equal to the bandwidth of the system characterizes therapidity of the feedback system erefore the higher thecrossover frequency the faster the response of the system is

Nomenclature

Lr Inductor in the resonant tankCr Capacitor in the resonant tankir Resonant inductor currentvCr Resonant capacitor voltagevg Input voltage of the resonant tank referred to the

secondary side of themedium-frequency transformervo Output voltage of the resonant tank


VLVDC Low-voltage DCVMVDC Medium-voltage DCvturb Output terminal voltage of the DC wind turbine

converteriturb Output current of the DC wind turbine converter

( iLf )ioRec Output current of the diode rectifier of the DC

wind turbine convertervoRec Output voltage of the diode rectifier of the DCwind

turbine converter ( vCf )Lf Inductor in the output filterCf Capacitor in the output filterfs Switching frequency of the series resonant

converter defined by ωs2πωr Natural resonant frequency of the tank defined by

ωr 1LrCr

1113968

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] Y-H Chen C G Dincan R J Olsen M C SchimmelmannP C Kjaeligr and C L Bak ldquoStudies for characterisation ofelectrical properties of DC collection system in offshore windfarmsrdquo in Proceedings of CIGRE General Session 2016 B4-301Paris France August 2016

[2] T Christ S Seman and R Zurowski ldquoInvestigation of DCconverter nonlinear interaction with offshore wind powerpark systemrdquo in Proceedings of EWEA Off-Shore 2015 pp 14ndash44 Copenhagen Denmark March 2015

[3] C Meyer and R W De Doncker ldquoDesign of a three-phaseseries resonant converter for offshore DC gridsrdquo in Pro-ceedings of 42nd IAS Annual Meeting Industry ApplicationsConference pp 216ndash223 September 2007

[4] C Meyer Key components for future offshore dc grids PhDdissertation Institute for Power Electronics and ElectricalDrives RWTH Aachen University Aachen Germany 2007

[5] S Vogel ldquoInvestigation of DC collection network for offshorewind farmsrdquo MS thesis Department of Wind EnergyTechnical University of Denmark Kongens Lyngby Den-mark 2014

[6] V Vorperian and S Cuk ldquoA complete DC analysis of theseries resonant converterrdquo in Proceedings of 13th AnnualPower Electronics Specialists Conference (PESCrsquo82) pp 85ndash100 Cambridge MA USA June 1982

[7] A F Wittulski and R W Erickson ldquoSteady-state analysis ofthe series resonant converterrdquo IEEE Transactions on Aero-space and Electronic Systems vol AES-21 no 6 pp 791ndash7991985

[8] R U Lenke J Hu and R W De Doncker ldquoUnified steady-state description of phase-shift-controlled ZVS-operatedseries-resonant and non-resonant single-active-bridge con-vertersrdquo in Proceedings of IEEE Energy Conversion Congressand Exposition (IEEE-ECCE) pp 796ndash803 IEEE San JoseCA USA September 2009

[9] G Ortiz H Uemura D Bortis J W Kolar andO Apeldoorn ldquoModeling of soft-switching losses of IGBTs inhigh-power high-efficiency dual-active-bridge DCDC con-vertersrdquo IEEE Transactions on Electron Devices vol 60 no 2pp 587ndash597 2013

[10] C G Dincan P Kjaeligr Y Chen et al ldquoDesign of a high powerresonant converter for DC wind turbinesrdquo IEEE Transactionson Power Electronics 2018

[11] P C Kjaeligr C L Bak P F M Da Silva Y H Chen andC G Dincan ldquoPower collection and distribution in mediumvoltage DC networksrdquo 2015 httpwwwdccetaaudk

[12] C Dincan P C Kjaeligr Y H Chen S Munk-Nielsen andC L Bak ldquoHigh power medium voltage series resonantconverter for DC wind turbinesrdquo IEEE Transactions on PowerElectronics vol 33 no 9 pp 7455ndash7465 2018

[13] D Jovcic ldquoStep-up DC-DC converter for megawatt size ap-plicationsrdquo IET Power Electronics vol 2 no 6 pp 675ndash6852009

[14] D Jovcic ldquoBidirectional high-power DC-transformerrdquo IEEETransactions on Power Delivery vol 25 no 4 pp 2276ndash22832010

[15] W Chen A Q Huan C Li G Wang and W Gu ldquoAnalysisand comparison of medium voltage high power DCDCconverters for offshore wind energy systemsrdquo IEEE Trans-actions on Power Electronics vol 28 no 4 pp 2014ndash20232013

[16] A Parastar and J K Seok ldquoHigh gain resonant switchedcapacitor cell based DCDC converter for offshore windenergy systemsrdquo IEEE Transactions on Power Electronicsvol 30 no 2 pp 644ndash656 2015

[17] L Max Design and control of A DC grid for offshore windfarms PhD dissertation Department of Energy and Envi-ronment Chalmers University of Technology GoteborgSweden 2009

[18] R Lenke A contribution to the design of isolated DC-DCconverters for utility applications PhD dissertation Institutefor Power Electronics and Electrical Drives RWTH AachenUniversity Aachen Germany 2012

[19] K Park and Z Chen ldquoAnalysis and design of a parallel-connected single active bridge DC-DC converter for high-power wind farm applicationsrdquo in Proceedings of 15thEuropean Conference on Power Electronics and Applications(EPE) vol 110 Lille France September 2013

[20] D Duji F Kieferndorf and F Canales ldquoPower electronictransformer technology for traction applicationsmdashan over-viewrdquo in Proceedings of 7th International Power Electronicsand Motion Control (PEMC) vol 16 no 1 p 5056 DaeguKorea June2012

[21] M Steiner and H Reinold ldquoMedium frequency topology inrailway applicationsrdquo in Proceedings of European Conferenceon Power Electronics and Applications (EPE) Aalborg Den-mark September 2007

[22] H Hoffmann and B Piepenbreier ldquoMedium frequencytransformer in resonant switching dcdc-converters for rail-way applicationsrdquo in Proceedings of 2011 14th EuropeanConference on Power Electronics and Applications pp 1ndash8Birmingham UK August-September 2011

[23] L Heinemann ldquoAn actively cooled high power high fre-quency transformer with high insulation capabilityrdquo inProceedings of APEC Seventeenth Annual IEEE Applied PowerElectronics Conference and Exposition vol 1 Los CabosMexico 2002

[24] J W Kolar and G I Ortiz ldquoSolid state transformer conceptsin traction and smart grid applications scheduleoutlinerdquo in


Proceedings of 15th International Power Electronics andMotion Control Conference EPE-PEMC pp 1ndash166 Novi SadSerbia September 2012

[25] C Dincan P Kjaer Y H Chen S Munk-Nielsen andC L Bak ldquoAnalysis of a high power resonant DC-DCconverter for DC wind turbinesrdquo IEEE Transactions onPower Electronics vol 33 no 9 pp 7438ndash7454 2017

[26] Report 9639ndash01ndashR0 MVDC Technology Study Market Op-portunities and Economic Impact TNEI Services Ltd Man-chester UK 2015

[27] M Barnes and A Beddard ldquoVoltage source converter HVDClinksndashthe state of the art and issues going forwardrdquo in EnergyProcedia vol 24 pp 108ndash122 2012

[28] F F Da Silva and C L Bak Electromagnetic Transients inPower Cables Springer London UK 2013

[29] Cigre WG B457 Guide for the Development of Models forHVDC Converters in a HVDC Grid CIGRE Technical Bro-chure Paris France 2014

[30] Manitoba HVDC Research CentreUSERrsquoS GUIDE on the Useof PSCAD Manitoba HVDC Research Centre Division ofManitoba Hydro International Ltd Manitoba ON Canada2018

[31] J Beerten S DrsquoArco and J A Suul ldquoCable modelorder reduction for HVDC systems interoperability anal-ysisrdquo in 11th IET International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February2015

[32] C L Bak Power System Technical Performance Issues Relatedto the Application of Long HVAC Cable WG C4502s CigreTechnical brochure Paris France 2013

[33] J B Glasdam L H Kocewiak J Hjerrild and C L BakldquoControl system interaction in the VSC-HVDC grid con-nected offshore wind power plantrdquo in Proceedings of CigreSymposium 2015 pp 142ndash149 Lund Sweden 2015

[34] L Kocewiak Harmonics in large offshore wind farms PhDdissertation Aalborg University Aalborg Denmark 2012

[35] V Preciado ldquoHarmonics in a wind power plantrdquo in Pro-ceedings of IEEE Power and Energy Society General Meetingpp 1ndash5 Denver CO USA July 2015

[36] K M Hasan K Rauma A Luna J I Candela andP Rodriguez ldquoHarmonic resonance study for wind powerplantrdquo in Proceedings of International Conference on Re-newable Energies and Power Quality (ICREPQrsquo12) Santiago deCompostela Spain March 2012

[37] Y Fillion and S Deschanvres ldquoBackground harmonic am-plifications within offshore wind farm connection projectsrdquoin Proceedings of Power Systems Transient (IPST 2015) CavtatCroatia June 2015

[38] F Mura C Meyer and R W De Doncker ldquoStability analysisof high-power dc gridsrdquo IEEE Transactions on Industry Ap-plications vol 46 no 2 pp 584ndash592 2010

[39] A Shafiu A Hernandez F Schettler J Finn andE Joslashrgensen ldquoHarmonic studies for offshore windfarmsrdquo inProceedings of 9th IET International Conference on AC andDCPower Transmission (ACDC 2010) pp 1ndash6 London UKOctober 2010

[40] B Badrzadeh M Gupta N Singh A Petersson L Max andM Hogdahl ldquoPower system harmonic analysis in wind powerplantsmdashPart I study methodology and techniquesrdquo in Pro-ceedings of Industry Applications Society Annual Meeting(IAS) Las Vegas NV USA October 2012

[41] PSCAD PSCADEMTDC On-Line Help System HVDC Re-search Centre Manitoba ON Canada 2013

[42] R J King and T A Stuart ldquoSmall-signal model for the seriesresonant converterrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-21 no 3 pp 301ndash319 1985

[43] X Wang F Blaabjerg andW Wu ldquoModeling and analysis ofharmonic stability in an AC power-electronics-based powersystemrdquo IEEE Transactions on Power Electronics vol 29no 12 pp 6421ndash6432 2014

[44] B Gustavsen and A Semlyen ldquoRational approximation offrequency domain responses by vector fittingrdquo in IEEETransactions on Power Delivery vol 14 no 3 pp 1052ndash10611999

[45] J R Marti ldquoAccuarte modelling of frequency-dependenttransmission lines in electromagnetic transient simula-tionsrdquo IEEE Transactions on Power Apparatus and Systemsvol PAS-101 no 1 pp 147ndash157 1982

[46] Y H Chen C G Dincan P C Kjaeligr C L Bak X Wang andC E Imbaquingo ldquoModel-based control design of seriesresonant converter based on the discrete time domainmodelling approach for DC wind turbinerdquo Journal of Re-newable Energy vol 2108 Article ID 7898679 18 pages 2018

[47] C Imbaquingo N Isernia E Sarra and A TonellottoldquoModelling DC collection of offshore wind farms for harmonicsusceptibility studyrdquo May 2017 httpswwwdccetaaudkdigitalAssets390390654_aau-2017-carlos-dc-harmonics-on-src-converterpdf

[48] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Science amp Business Media BerlinGermany 2007


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Submit your manuscripts atwwwhindawicom

Research ArticleHarmonic Susceptibility Study of DC Collection NetworkBased on Frequency Scan and Discrete Time-DomainModelling Approach

Carlos Enrique Imbaquingo Eduard Sarra Nicola Isernia Alberto TonellottoYu-HsingChenCatalinGabrielDincan PhilipKjaeligrClausLethBakandXiongfeiWang

Department of Energy Technology Aalborg University Aalborg Denmark

Correspondence should be addressed to Catalin Gabriel Dincan cgdetaaudk

Received 19 July 2018 Revised 31 October 2018 Accepted 6 November 2018 Published 16 December 2018

Guest Editor Luigi P Di Noia

Copyrightcopy 2018Carlos Enrique Imbaquingo et alis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e equivalent model of offshore DC power collection network for the harmonic susceptibility study is proposed based on thediscrete time-domain modelling technique and frequency scan approach in the frequency domaine proposedmethodology formodelling a power converter and a DC collection system in the frequency domain can satisfy harmonic studies of any con-figuration of wind farm network and thereby find suitable design of power components and array network e methodology isintended to allow studies on any configuration of the wind power collection regardless of choice of converter topology array cableconfiguration and control design To facilitate harmonic susceptibility study modelling DC collection network includes creatingthe harmonic model of the DC turbine converter and modelling the array network e current harmonics within the DCcollection network are obtained in the frequency domain to identify the resonance frequency of the array network and potentialvoltage amplification issues where the harmonic model of the turbine converter is verified by the comparison of the converterswitching model in the PLECStrade circuit simulation tool and laboratory test bench and show a good agreement

1 Introduction

Offshore wind power is becoming an important energyresource in Europe However electrical power losses arealways a concern in the operation of wind farm with long-range power transmission To reduce losses researchers lookto replace AC with DC in the entire path from the windturbine through power collection and transmission to shoreMedium-voltage DC (MVDC) collection of wind power isan attractive solution to reduce overall losses and installationcost [1 2] e article [2] addresses cost-efficient solution toform the multiterminals of DC grid with the diode rectifierunit (DRU) However the solution of DRU can produce aconsiderable harmonic current within the array network ifthe 6-pulse DRU topology is used

Figure 1 illustrates a DC collection network of radial typeof offshore wind farm with the HVDC transmission isgeneral configuration in Figure 1 supports any choice ofMVDC collector voltage level as well as converter topology

and the number of wind power plant (WPP) clusters Forexample the turbine and offshore substation can employ anyDCDC converter topology with multiple degrees of free-dom uni- vs bidirectional power flow modular (parallelseries connection) vs monolithic topology (single activebridge dual active bridge and more [3ndash5]) Figures 1 and 2illustrate the details of wind farm configuration includingwind turbine DCDC converters offshore substation con-verters and onshore substation converters e offshorewind farm is designed to generated and deliver 700MWpower from offshore to onshore [1]

e series resonant converter (SRC) is selected as theturbine DCDC converter based on the converter efficiencyhigh-voltage transfer ratio scale of converter and galvanicisolation property Considering the existing converter to-pology and its power rating in high-power applications themodular multilevel converter (MMC) with the galvanicisolation and unidirectional power flow control is selected toserve as the offshore substation converter to deliver the

HindawiJournal of Electrical and Computer EngineeringVolume 2018 Article ID 1328736 25 pageshttpsdoiorg10115520181328736

HVDCexport cable


Offshoresubstation

Onshoresubstation


MMC


DCAC

ACDC






HVDC export cable

plusmn50kVDC

DCDC


plusmn50kVDC

DCDC


plusmn50kVDC

DCDC


G7 times 8MW

WT

G7 times 8MW

WT

G7 times 8MW

WT

G8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT

G G G G G G G G

2400A

Offshore substation

MVDC side


LVDC side

11

WTGWTGWTGWTGWTG

12131415

1A1B1C1D1E

21

WTGWTGWTGWTGWTG

22232425

2A2B2C2D2E

WTGWTGWTGWTGWTG

41

WTGWTGWTGWTG

4243444546

4A4B4C4D

313233343536

3A3B3C3D3E3F

WTG

16

1F

WTG

26

2F

WTG

WTG

17

1G

WTG

27

2G

WTG

37

3G

WTGWTG

4E4F

WTGWTG

4G4H

4748



800mm2 100km

DCAC


network

400kV

Onshore substation

WPP cluster 1

MVDC side HVDC side


iHVDC onshore

VHVDC onshore



Vturb

iturb

2320A

560A 640A

iHVDC offshore

iFeeder1

iFeeder2

iFeeder3

iFeeder4


plusmn50kVDC


SM SMSM SM

SM SMSM SM

SM SM

SM SM

SMSM

SMSM

SM

SM

HVDC side

plusmn320kVDC

5060Hztransformer

Utility


Resonant tank

SM SM SM

SM SM SMSM SM SM

SM SM SMSM SM SM

SM SM SM

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash
























VLVDC

vturb

+

+ndash

ndash12 Lf

iturb

2Cf

2Cf

voRec

+

ndash

12 LfioRec

VMVDC

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime+ndash



_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)




1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC


0

01

01

01

01


T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0


T1

t

Vce

ZVS


t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)





B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


HVDCexport cable


Offshoresubstation

Onshoresubstation


MMC


DCAC

ACDC






HVDC export cable

plusmn50kVDC

DCDC


plusmn50kVDC

DCDC


plusmn50kVDC

DCDC


G7 times 8MW

WT

G7 times 8MW

WT

G7 times 8MW

WT

G8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT7 times 8MW

WT7 times 8MW

WT7 times 8MW

WT8 times 8MW

WT

G G G G G G G G

2400A

Offshore substation

MVDC side


LVDC side

11

WTGWTGWTGWTGWTG

12131415

1A1B1C1D1E

21

WTGWTGWTGWTGWTG

22232425

2A2B2C2D2E

WTGWTGWTGWTGWTG

41

WTGWTGWTGWTG

4243444546

4A4B4C4D

313233343536

3A3B3C3D3E3F

WTG

16

1F

WTG

26

2F

WTG

WTG

17

1G

WTG

27

2G

WTG

37

3G

WTGWTG

4E4F

WTGWTG

4G4H

4748



800mm2 100km

DCAC


network

400kV

Onshore substation

WPP cluster 1

MVDC side HVDC side


iHVDC onshore

VHVDC onshore



Vturb

iturb

2320A

560A 640A

iHVDC offshore

iFeeder1

iFeeder2

iFeeder3

iFeeder4


plusmn50kVDC


SM SMSM SM

SM SMSM SM

SM SM

SM SM

SMSM

SMSM

SM

SM

HVDC side

plusmn320kVDC

5060Hztransformer

Utility


Resonant tank

SM SM SM

SM SM SMSM SM SM

SM SM SMSM SM SM

SM SM SM

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash

+ndash
























VLVDC

vturb

+

+ndash

ndash12 Lf

iturb

2Cf

2Cf

voRec

+

ndash

12 LfioRec

VMVDC

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime+ndash



_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)




1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC


0

01

01

01

01


T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0


T1

t

Vce

ZVS


t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)





B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia























VLVDC

vturb

+

+ndash

ndash12 Lf

iturb

2Cf

2Cf

voRec

+

ndash

12 LfioRec

VMVDC

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime+ndash



_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)




1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC


0

01

01

01

01


T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0


T1

t

Vce

ZVS


t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)





B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











VLVDC

vturb

+

+ndash

ndash12 Lf

iturb

2Cf

2Cf

voRec

+

ndash

12 LfioRec

VMVDC

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime+ndash



_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)




1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC


0

01

01

01

01


T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0


T1

t

Vce

ZVS


t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)





B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


_1113957x1

_1113957x2

⎡⎣ ⎤⎦ [A]1113957x1

1113957x21113890 1113891 +[B]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io [C]1113957x1

1113957x21113890 1113891 +[D]

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)




1113957IoRec(s) g1(s) g2(s) g3(s)1113858 1113859

1113957fs

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (2)

N2N1 times VLVDC


0

01

01

01

01


T1(leading leg)

ioRec

ir

iaverage

Vg

T2

T3(lagging leg)

T4

im

δ δ δ

(a)

(b)

(c)

(d)

(e)

t

t

t

t

t

t

t

t

0

0

0


T1

t

Vce

ZVS


t

T4

Vce

ZVS


(a)

D8

Vd Id

t

D5

Vd

ZVS ZCS

Id

tZVS ZCS

(b)





B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




B + D (3)

where

g1(s) 1113957IoRec(s)

1113957fs(s)

1113868111386811138681113868111386811138681113868111386811138681113957Vo(s)1113957Vg(s)0

g2(s) 1113957IoRec(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vo(s)0

g3(s) 1113957IoRec(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957fs(s)1113957Vg(s)0

(4)



(LfCf )

1113968) asymp 100Hz) in


gc K













Draw theequivalent

circuit


model





the converter

Createinteresting

transferfunctions


Specify thecontrol

performance

Draw thecomplete


turbineconverter



Step 9










sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








sCf minusg3(s)




minusg3(s)






g3(s)g2(s)

g1(s)gc(s)

Vg (prop VLVDC)

IoRec

Vo Rec = VCf

PREF

Io REF fs


sLf

sCfsCf

ILf

ICf

++ +

+ndash

VLf

Vo Rec = VCf

1Vturb

Output filter

sim

sim

sim

sim sim

sim sim


disturbances)

sim

sim


to array cable)

sim

sim

simsimsim

++

+ndash






1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1A1B1C1D1E

WTG16

1F

WTG17

1G

I~

Sub

Offshoresubstation

Array cable



Yeq

V~

Sub = V~

MVDC

I~

turb

+

ndash

V~

turb

Switching model

VLVDC vturb

+

ndash(12)Lf

iturb

2Cf

2Cf

voRec

+

ndash

(12)LfioRec

Lr CrN1N2

T1 T3

T2 T4

D5 D7

D6 D8

ir

ndashvg

+

ndashvo

+

ndash

+

irprime

vgprime





002

004

003

001

0

Adm

ittan

ce (S

)

(a)




0

100

150

50

ndash50

ndash100

Phas

e (deg)

(b)


Frequency (rads)

0

ndash45

ndash90

ndash135

101 102 103 104

101 102 103 104

Phas

e (deg)

200


Mag

nitu

de (d

B)

ndash80ndash100

Bode diagram











6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









6 Verification





Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Offshoresubstation

0 t

f(Hz) 300Hz

280Hz

260Hz

20Hz

40Hz

60Hz

80Hz


~

fMVDC = f~ ~

fVo = ffVo = f~ ~


|VMVDC|~


0009772050025372100276721002937210032522100358221

000564190019341900213419002284190025341900281419

0028141895835478 (m) 0035822050238058 (m)Cable 1 Cable 2

10 (m) 10 (m)


SheathInsulator 2

ArmourInsulator 3


SheathInsulator 2

ArmourInsulator 3

(a) (b)










7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









7 Laboratory Test




20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

2

4

6

8

18

16

14

12

10

20

0


Am

plitu

de (A

)




(a)

300

250

200

150

100

Phas

e (deg)

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300





(b)


200

100

0

8

46

2

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300



Am

plitu

de (A

)

Admittanceequival

Admittance equival

Switching model

Switchingmodel

Phas

e (deg)

(a)



0

8

46

2

Am

plitu

de (A

)

200

100

0

Phas

e (deg)

(b)

0

8

4

6

2


Am

plitu

de (A

)


200

100

0

Phas

e (deg)

(c)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(d)

Figure 14 Continued


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


0

8

46

2

Am

plitu

de (A

)


200

100

0

Phas

e (deg)


(e)


0

8

46

2

Am

plitu

de (A

)


140

200

100

0

Phas

e (deg)

(f )

0

8

46

2

Am

plitu

de (A

)



300

200

0

Phas

e (deg)

(g)



2nd orderharmonic

3rd orderharmonic

9

8

7

6

5

4

Curr

ent a

mpl

itude

(A)

3

2

1

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


(a)


2nd orderharmonic

3rd orderharmonic

1

2

3

4

5

6

0

Curr

ent a

mpl

itude

(A)


(b)







ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






ndash30

ndash20

0

10

20

30

Long

itudi

nal a

dmitt

ance

mag

nitu

de (d

B)

ndash10



(a)




ndash90

ndash70

ndash20

0

Long

itudi

nal a

dmitt

ance

pha

se (d

eg)

ndash60

ndash80

ndash50

ndash40

ndash10

ndash30



(b)





8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




8 Conclusion


Appendix







+

ndash

VMVDC

iturb

VMVDC0


= 400VDC

Perturbation


DAux

+

ndashVturb







vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





vg Lrdir

dt+ vCr + vo

ir CrdvCr

dt

(A1)

where

vg Vg0(k)

vo Vo0(k)(A2)



2

4

6

8

10

0

Am

plitu

de (A

)

12Laboratory test

Switchingmodel

(PLECSTM)

Harmonicmodel


(a)



250

200

150

100

50

020 40 60 80 100 120 140 160 180 200 220 240 260 280 300


Phas

e (deg)

(b)



ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ωst

ωst

ωst

ir(ωst)

|ir(ωst)|

ωst0(k) ωst1(k)



Vg(ωst)Vo(ωst)

VCr(ωst)

ioutRec(ωst)

0

0

0

βkαkγk

βkαkγk

Kthevent

(K + 1) thevent

ωst1(k+1)


+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(a)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(b)

+ndash

ndash

+

ndash

+

iprimer(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(c)

+ndash

ndash

+

ndash

+

irprime(t)



+ vTank(t)ndash

ir(t) Lr Cr

N1N2

(d)



ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ir 1


+ Ir0(k) cos ωrt( 1113857

(A3)



where

Zr

Lr

Cr

1113971

ωr 1

LrCr

1113968

(A5)


1Zr


+ Ir0(k) cos ωrt1(k)1113872 1113873 0

(A6)

whereωrt1(k)

ωr




(A7)




(A8)




where

tprime tminus t1(k)


where

Ir1(k) 0

Vg1(k) 0


(A11)




+ minus1


+2


minus1Zr






(A12)






(A13)

where




Ir2(k) minusIr0(k)








io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



io 1ck

1113946βk


1ck

1113946ck

βk


1ck

middot1ωrs



middot Ir0(k) +1ck


1Zr


1Zr

cos



1Zr


1Zr

cos ωrsβK( 1113857minus 2( 1113857


middot 11138961ωrs

1Zr


1Zr

1minus cos ωrsβK( 1113857( 1113857


(A17)

whereθs ωst



LrCr




(A19)




x1(k) Ir0(k)

x2(k) VCr0(k)

Ir2(k) minusx1(k+1)


(A20)


t0(k+1) minus t0(k)

ωs

ck

xi(k+1) minus xi(k)1113872 1113873 (A21)

where



_x1(k) ωs

ck


+ωs

ck

middot1


ωs

ck

middotminus2Zr



middot Vo0(k) +ωs

ck

middot 11138901


minus1Zr



f1 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f1 x1 x2 vo vg α1113966 1113967

ωs

ck


_x2(k) ωs

ck


+ωs

ck


ck


middot Vo0(k) +ωs

ck



f2 x1(k) x2(k) Vo0(k) Vg0(k) αk1113966 1113967

f2 x1 x2 vo vg α1113966 1113967

ωs

ck


(A23)




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




fout x1 x2 vo vg α1113966 1113967

1ck


(A24)




x1 + 1113957x11113858 1113859middot

f11113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A25)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

(A26)

then

x1 + 1113957x11113858 1113859middot

f1 x1 x2 Vo Vg α1113966 1113967 +zf1

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf1

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf1

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f

zx21


(A27)


OP Ir VCr Vo Vg α

f1 x1 x2 Vo Vg α1113966 1113967 _Ir 0(A28)


x2 + 1113957x21113858 1113859middot

f21113864x1 + 1113957x1 x2 + 1113957x2 Vo + 1113957Vo Vg

+ 1113957Vg α + 1113957α1113865

ωs

ck


+ 1113957Vg α + 1113957α1113865

(A29)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vs

α α + 1113957α

(A30)

then

x2 + 1113957x21113858 1113859middot

f2 x1 x2 Vo Vg α1113966 1113967 +zf2

zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zf2

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zf2

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2f2

zx21


(A31)

where

f2 x1 x2 Vo Vg α1113966 1113967 _VCr 0 (A32)



1


+ 1113957Vg α + 1113957α1113883

(A33)

where

x1 x1 + 1113957x1

x2 x2 + 1113957x2

vg Vg + 1113957Vg

vo Vo + 1113957Vo

α α + 1113957α

1113957c 1113957α + 1113957β

(A34)

then



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



zx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

+zfout

zx2

11138681113868111386811138681113868111386811138681113868OP1113957x2 +

zfout

zvg

111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP

1113957Vo

+zfout

zα

1113868111386811138681113868111386811138681113868OP1113957α +

12

z2fout

zx21


1c


c1113888 1113889 +

1113957c

c1113888 1113889

2


⎩

⎫⎬

⎭


zflowastoutzx1

11138681113868111386811138681113868111386811138681113868OP1113957x1

zflowastoutzx2

11138681113868111386811138681113868111386811138681113868OP


111386811138681113868111386811138681113868111386811138681113868OP

1113957Vg +zfout

zvo

11138681113868111386811138681113868111386811138681113868OP


zα

11138681113868111386811138681113868111386811138681113868OP1113957α

+12

z2flowastoutzx2

1


(A35)

where

Io fout x1 x2 Vo Vg α1113966 1113967

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x1

+zflowastoutzx2


zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957x2

+zflowastoutzvg


zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vg

+zflowastoutzvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 1113891 middot 1113957Vo

+zflowastout

zα


(A36)




_1113957x1

_1113957x2

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

zf1

zx1

11138681113868111386811138681113868111386811138681113868OP

zf1

zx2

11138681113868111386811138681113868111386811138681113868OP

zf2

zx1

11138681113868111386811138681113868111386811138681113868OP

zf2

zx2

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ +

zf1

zα

1113868111386811138681113868111386811138681113868OP

zf1

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf1

zvo

11138681113868111386811138681113868111386811138681113868OP

zf2

zα

1113868111386811138681113868111386811138681113868OP

zf2

zvg

111386811138681113868111386811138681113868111386811138681113868OP

zf2

zvo

11138681113868111386811138681113868111386811138681113868OP

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113957Io 1c



zβzx1

11138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c



zβzx2

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113890 1113891

middot

1113957x1

1113957x2

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ + 11138901c

middotzflowastout

zα

11138681113868111386811138681113868111386811138681113868OPminus Io1113890 1113891

1c



zβzvg

111386811138681113868111386811138681113868111386811138681113868OP1113890 1113891

1c

middotzfout

zvo


zβzvo

11138681113868111386811138681113868111386811138681113868OP1113890 11138911113891

middot

1113957α

1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A37)


zfi

zxj

z

zxj

ωs

c1113888 1113889 middot f

lowasti

111386811138681113868111386811138681113868111386811138681113868OP+ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A38)

where

c α + β for i 1 2 j 1 2 (A39)


ωs

c

11138681113868111386811138681113868111386811138681113868OPne 0

flowasti

1113868111386811138681113868OP 0

(A40)

erefore

zfi

zxj

ωs

c

zflowastizxj

111386811138681113868111386811138681113868111386811138681113868OP (A41)




β|OP πωrs

+1ωrs



(A42)



zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


zβzx1

11138681113868111386811138681113868111386811138681113868OP

1ωrs



+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs



+ IrZr( 11138572

zβzx2

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvo

11138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotminusIrZr


+ IrZr( 11138572

zβzvg

111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotx1Zr


+ x1Zr( 11138572

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

11138681113868111386811138681113868111386811138681113868111386811138681113868OP

1ωrs

middotIrZr


+ IrZr( 11138572

(A43)




1113957Io(s) g1(s) g2(s) g3(s)1113858 1113859

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (A44)

where

g1(s) 1113957Io(s)

1113957α(s)

111386811138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

g2(s) 1113957Io(s)

1113957Vg(s)

1113868111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

g3(s) 1113957Io(s)

1113957Vo(s)

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0


minusπ

(A45)


1113957X 1113957Ir

1113957vCr

⎡⎣ ⎤⎦ 1113957x1

1113957x21113890 1113891

gxu11 gxu12 gxu13

gxu21 gxu22 gxu231113890 1113891

1113957α1113957Vg

1113957Vo

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A46)

where

gxu11 1113957Ir1113957α

11138681113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu12 1113957Ir1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu13 1113957Ir1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)0

gxu21 1113957vCr1113957α

1113868111386811138681113868111386811138681113957Vg(s)01113957Vo(s)0

gxu22 1113957vCr1113957Vg

111386811138681113868111386811138681113868111386811138681113957α(s)01113957Vo(s)0

gxu23 1113957vCr1113957Vo

11138681113868111386811138681113868111386811138681113957α(s)01113957Vg(s)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A47)







RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



RC RC + RL( 1113857



RCRC + RL 1 + s Lf RL( 1113857( 1113857





gf4 1113957VoRec

1113957Vg

RC RC + RL( 1113857


(B1)








H(s) 1113957iturb1113957fs

g1 middot gf1







gC K



Nomenclature










ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








ωr 1LrCr

1113968

Data Availability




References






















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Documents

Aalborg Universitet Harmonic Susceptibility Study of DC ......HVDC export cable ±150– 320kV DC Offshore substation Onshore substation MVDC array cable ±50–100kV DC MMC Line-frequency