1

Harmonic domain modelling of PV system for theassessment of grid integration impact

Onyema S. Nduka, Student Member, IEEE, Bikash C. Pal, Fellow, IEEE

Abstract—In this paper, a comprehensive harmonic domainreference frame (HDRF) model of a voltage source converter(VSC) grid interactive photovoltaic (PV) system is presented.The model is useful for assessing the harmonic coupling betweenthe PV system and the network. Different components of the PVsystem such as inverter, LCL filter and interconnecting trans-former have been incorporated in the model. Using this model,harmonic currents from PV system connected to both distortedand undistorted networks have been quantified. Also, the modelhas been deployed in investigating resonance occurrence in amedium-voltage distribution network (MVDN) where the resultsprovide interesting technical insight and understanding.

Index Terms—voltage source converter, grid interactive pho-tovoltaic system, resonance, harmonic domain, medium-voltagedistribution network

NOMENCLATUREVdc Inverter input dc voltage (V)Gωt 3-level inverter switching functionVinv Unfiltered inverter output voltage (V)t Time (s)x, y, z Indexing of line-line parametersq Switching function pulse indexαq;βq Switching instant and pulse widthipcc; [Ipcc] TD and HD PCC currentRd;R2 Filter damping and grid side resistorL2 Sum of grid side and filter inductorvcap; icap TD Filter capacitor voltage and current[vcap]; [icap] HD Filter capacitor voltage and currentL1;C Filter inductor and capacitor[U] HD Identity matrix[Ψ] HD differentiation matrixyg Sum of grid and Tx leakage impedance

La;Lb;Lc 3-phase LCL source-side filter inductorLaa;Lbb;Lcc 3-phase LCL filter grid side inductorC12;C13;C23 3-phase LCL filter capacitorλ Transformer effective turns ratioP Time derivative operator[V123]; [V123] Filter node voltages in TD and HD[Vabc]; [Vabc] Transformer primary side node voltages[Eabcg]; [Eabcg] Grid voltages in TD and HD[Vinvt]; [Vinvt] Inverter voltages in TD and HDh,H hth or Hth harmonic[Ypv]; [Zpv] HD PV admittance and impedance

This work was supported by the Engineering and Physical SciencesResearch Council (EPSRC) UK through the Reliable and Efficient Systemfor Community Energy Solution RESCUES grant. EP/K03619X/1: Thestudentship was funded by the Federal Government of Nigeria through thePetroleum Technology Development Fund and Presidential Special Scholar-ship for Innovation and Development. Data supporting this publication can beobtained on request from cap-publications@imperial.ac.uk.

I. INTRODUCTION

RECENT trends have shown an explosive growth in thenumber of renewable distributed generators (RDGs)

connected to power networks and this trend will continue.This has been motivated by the quest for reduction of carbonemissions, national energy independence, and cheaper energysources especially renewables such as solar and wind. Otherbenefits derived from RDG integration into power systemsinclude provision of ancillary services like frequency, voltageand reactive power support, transformer and feeder loadingstress reduction, loss minimisation etc. Among the variousRDGs, wind and PV RDG are very dominant and rapidlygrowing [1], [2].

Despite the benefits of RDG integration into the distributionsystem, it is known that operational and technical challengessuch as power quality issues, reverse power flow, powerloss increase, harmonic pollution etc are very significant andrequire technological innovations [3], [4] and [5]. Harmonicpollution often challenges the system operations by wayof device maloperation, equipment overheating, resonanceetc. Resonance investigation has been the commonest studyrelating to harmonic pollution in the literature [6], [7]. Suchanalysis is usually done using frequency or impedance scanin determining the resonant frequency. In [8], resonanceproblem due to PV system integration has been reported.The PV system was modelled as a current source in parallelwith an impedance. This impedance was modelled as a singlecapacitor. The authors argued that although a particular PVsystem may satisfy the regulatory requirements for gridintegration as stipulated in IEEE 1547 and IEC 61727, thecollective harmonic contributions of several such systemscould pose some harmonic pollution challenges to thenetwork. On the contrary, [9] suggests that such resonanceexcited by the filter capacitor of the PV system connectedto the secondary side of the distribution system is oftendampened by the passive loads and filter damping resistor.This paper also stated that the PV-DGs connected at thelow-voltage side of the network do not pose resonanceproblems to the HV or MV side of the distribution system.However, actual modelling and simulation of MV and HVnetworks for resonance study were not presented. In addition,both papers focused mainly on rooftop PVs connected to thelow voltage side of distribution network.In [10] and [11], impedance models of PV systems havebeen developed from measurement. While [10] performs aniteration (using Newton Raphsons method) to obtain values

2

of impedances that satisfy traditional circuit theorems at thePCC with measured data as initial iterates, [11] performeddata fitting of measurements. Indeed, the iterative methodmay present numerical instabilities and thus give rise toconvergence related problems. This is because four unknownvariables are to be determined from one nonlinear equationin [10] with several measured data (possibly having somemeasurement noise) been used as iterates. Also, the abovemethods can result in multiple solutions of impedances forthe same PV inverter. Again, the use of discrete Fouriertransforms (DFT) or fast Fourier transforms (FFT) in bothmethods may result in discretization errors.

Consequently, this paper seeks to achieve two main ob-jectives viz develop an improved PV model (both singleand 3-phase) and then investigate resonance occurrence in aMVDN using the proposed model. The harmonic domain (HD)technique proposed in [12] and [13] has been deployed. Theproposed model can be referred to as an HD Norton equiv-alent circuit model of a VSC-PV system. The HD approachdeployed is useful in characterizing the interaction between thePV system harmonics and the network that is operating underdistorted and undistorted conditions. Also, the proposed modelis suitable for harmonic power flow (HPF) studies formulatedusing the HD technique (taking into account both harmonicsand inter-harmonics).To perform the resonance study for the MVDN, networkcomponents such as transformer, loads, shunt capacitors etchave been modelled in the HD. The standard IEEE 13-busdistribution system has been modified and used for simulation.Insightful technical results are presented and discussed.

II. PV SYSTEM MODELLING

A utility scale PV system essentially comprises of a PV ar-ray, power conditioning unit, filter, control circuits, protectingdevices (e.g. isolating switches), interconnecting transformeretc. The VSC type PV system is considered in this workalthough the method is also applicable to current sourceinverter PV systems. Schematic diagram of Fig. 1 and Fig.2 are used for development of circuit equations for singleand 3-phase PV systems respectively. DC voltage from PVmodules/arrays which serve as the input to the inverter isestimated using equations provided in [14], [15]. The inverteris modelled using the switching function technique. Suchmethod treats the inverter unit as a black box and is thereforebased on its operation rather than circuit element based model.The input (dc voltage) and the output are related by (1).

Vinv (t) = G (ωt)Vdc (1)

where G(wt) is the switching function and can be written as

G(wt) =

∞∑h=−∞

Cxh exp (jhwt) (2)

Indeed, switching function expressions are not unique [16] asseveral expressions could be derived for the same switchingoperation. This modelling approach was deployed in [16]though without giving explicit expressions for the switching

L1 Lfg

C

Zg

ipv

Vdc

ipccic

Rd

Eg

Fig. 1. Schematic diagram of single phase PV system

functions used. Also, that paper assumed bipolar switchingoperation which is inferior to the unipolar strategy deployedhere. This is because the former generates more harmonicsthan the latter [17]. The double Fourier series expression hasalso been used in [18] but this is computationally demandingand difficult to incorporate with models of other componentsof the PV system e.g. LCL filter. Therefore, in this paper, themethod adopted in [19] have been used to derive the followingswitching function HD coefficients for the 3-phase PV system:

Cxh =

−j4π

[ z∑q=1

1

hsin

(hγq

)sin

(h

2βq

)](3)

Cyh =

4

π

[ z∑q=1

1

hsin

(hγq

)sin

(h

2βq

)][− sin

(ζ

)−j cos

(ζ

)](4)

Czh =

4

π

[ z∑q=1

1

hsin

(hγq

)sin

(h

2βq

)][sin

(ζ

)−j cos

(ζ

)](5)

where x, y, zεab, bc, ca and q is the switching pulse index.Next, applying classical circuit theorems, the result in themathematical models are presented. Note that models are firstpresented in time domain (TD) and thereafter transformedinto HD. This is to ensure improved readability. We beginby deriving the equations for currents injected by the singleand 3-phase PV systems respectively.

A. Single-Phase PV harmonic model

The circuit equations for single phase PV generator can besummarised as shown below.

dipcc (t)

dt+R2

L2ipcc (t) =

1

L2[Vcap (t)− Eg (t)] (6)

dVcap (t)

dt= Rd

dicap (t)

dt+

1

Cicap(t) (7)

dicap (t)

dt= − 1

L1Vcap (t)− dipcc (t)

dt+

1

L1G(ωt)Vdc (8)

where icap and Vcap represent the filter capacitor currentand voltage respectively while ipcc is the PCC current. Theequations can be transformed into HD namely:

[Icap] = [Q]−1[

1

L1[Ψ]−1

[G]Vdc + [B] [Eg −Vcap]

](9)

[Vcap] = [Ψ]−1(

1

C[U] +Rd [Ψ]

)[Icap] (10)

3

Cab

La

Lga

TX

Ia

IgC

Vdc

LbLc

Ic

Lcc

Ib

Lbb

Laa

Cbc

Cca

Lgb

rgb

rga

rgc

Lgc Ig

B

IgA

Fig. 2. Schematic diagram of 3-phase grid interactive system

[Ipcc] = [Ψ]−1[− [D] [Icap] +

1

L1[G]Vdc

](11)

where[Q] = [Ψ]

−1[D] (12)

[D] = [Ψ] +1

L1[Ψ]−1(

1

C[U] +Rd [Ψ]

)(13)

[B] = (L2 [Ψ] +R2 [U])−1 (14)

[U] and [Ψ] are identity and differentiation matrices in HDrespectively. By applying simple algebra, we have

[Icap] =

[(1

L1[Θ]−1

[Q]−1

[Ψ]−1

[G]

)Vdc

+(

[Θ]−1

[Q]−1

[B])

[Eg]]

(15)

[Ipcc] = [Ψ]−1(

[Υ]Vdc −(

[D] [Θ]−1

[Q]−1

[B])

[Eg]

)(16)

where

[Υ] =

(− 1

L1[D] [Θ]

−1[Q]−1

[Ψ]−1

[G] +1

L1[G]

)(17)

[Θ] = [U] + [Q]−1

[B]

([Ψ]−1(

1

C[U] + Rd [Ψ]

))(18)

The PV current can simply be computed from KCL:

[Ipv] = [Ipcc] + [Icap] (19)

EgIs Ypv

Fig. 3. HD Norton model of single phase PV system

By substitution and rearrangement of terms, we write:

[Ipv] = [Is] + [Ypv] [Eg] (20)

where

[Is] =1

L1

[[Ψ]−1(

[U]− [D] [Θ]−1

[Q]−1

[Ψ]−1)

[G]

+(

[Θ]−1

[Q]−1

[Ψ]−1

[G])]

[Vdc] (21)

[Ypv] =(

[U]− [Ψ]−1

[D])

[Θ]−1

[Q]−1

[B] (22)

Clearly, (22) is the harmonic equivalent circuit of a singlephase PV system as illustrated in Fig. 3. For several paralleledsingle phase PV systems, the equivalent circuit representationis shown in Fig. 4. The total current injected by the paralleln-connected single-phase PV systems is simply the algebraicsum of the currents from each unit. Therefore, if all the single-phase units arranged in parallel are of the same type, possesingthe same inverter switching operation and are operating underthe same solar irradiation and network operating conditions,the following total current [IpvT ] and total admittance [YpvT ]expressions are valid:

[IpvT] = [Ipv1] + [Ipv2] + · · ·+ [Ipvn] = n [Ipv] (23)

[YpvT] = [Ypv1] + [Ypv2] + · · ·+ [Ypvn] = n [Ypv] (24)

B. 3-phase PV harmonic modelSimilar to the single-phase model, the 3-phase PV system

equations are developed using classical circuit theorems. Re-sulting equations are summarised below:

1

La

(vag(t)− v1g(t)

)+ C31

d2

dt2

(v3g(t)− v1g(t)

)−

C12d2

dt2

(v1g(t)− v2g(t)

)=

1

Laa

(v1g(t)− vaa(t)

) (25)

1

Lb

(vbg(t)− v2g(t)

)+ C12

d2

dt2

(v1g(t)− v2g(t)

)−

C23d2

dt2

(v2g(t)− v3g(t)

)=

1

Lbb

(v2g(t)− vbb(t)

) (26)

1

Lc

(vcg(t)− v3g(t)

)+ C23

d2

dt2

(v2g(t)− v3g(t)

)−

C31d2

dt2

(v3g(t)− v1g(t)

)=

1

Lcc

(v3g(t)− vcc(t)

) (27)

4

Eg

Is1

Ypv1

Ypv2

Ypvn

Is2

Isn Ipv

Fig. 4. HD Norton model of paralleled single-phase PV system

In compact matrix/vector form, the above equations can berewritten as follows:

[AA] [V123] = [f] [Vinvt] + [ξ] [Vabc] (28)

where

[f] =

1La

1Lb

1Lc

; [ξ] =

1Laa

1Lbb

1Lcc

;

[V123] =

∣∣∣∣∣∣v1gv2gv3g

∣∣∣∣∣∣ ; [Vinvt] =

∣∣∣∣∣∣vagvbgvcg

∣∣∣∣∣∣ ; [Vabc] =

∣∣∣∣∣∣vaavbbvcc

∣∣∣∣∣∣[AA] =

[A11] [A12] [A13][A21] [A22] [A23][A31] [A32] [A33]

[A11] =

(1

Laa+

1

La+ (C31 + C12)P 2

); [A12] = −C12P

2

[A13] = −C31P2; [A21] = [A12] ; [A31] = [A13]

[A22] =

(1

Lbb+

1

Lb+ (C12 + C23)P 2

); [A23] = −C23P

2

[A33] =

(1

Lcc+

1

Lc+ (C23 + C31)P 2

); [A32] = [A23]

Notice we have dropped the dependence of variables on timefor convenience.

Furthermore,

Laadiaadt

=

(v1g − vaa

);Lbb

dibbdt

=

(v2g − vbb

);

Lccdiccdt

=

(v3g − vcc

);

diadt

=1

La

(vag − v1g

);dibdt

=1

Lb

(vbg − v2g

);

dicdt

=1

Lc

(vcg − v3g

);

[Iabc] =1

λ[K]

([S][Vabc]− λ[Eabcg]

)(29)

where:

[K] =1

3λyg[N ]; [S] =

1 −1 00 1 −1−1 0 1

[N ] =

3 0 −3−1 −1 2−2 1 1

; [Eabcg] =

∣∣∣∣∣∣Eag

Ebg

Ecg

∣∣∣∣∣∣λ denotes the effective turns ratio of the transformer.Using the above equations, the HD expression for the 3-phasePV current injection can be written as shown below.

[Vabc] = [OSN]−1

([K][Eabcg] + [Ξ][Vinvt]

)(30)

[V123] = [AA]−1

([f][Vinvt] + [ξ][Vabc]

)(31)

where

[OSN] =1

λ[K][S]− [W]−1

([α][AA]−1[ξ] + [β]

)

[W] =

ΨΨ

Ψ

[Ξ] = [W]−1

([α][AA]−1[f]

)

[α] =

1Laa

[U]1

Lbb[U]

1Lcc

[U]

; [β] = −[α];

The expression for currents in HD are as follows.PCC current:

[Iabc] = [W]−1

([α][AA]−1[f][Vinvt]+α][AA]−1[ξ]+[β][Vabc]

)(32)

PV system current injection:

[Iabcs] = [W]−1[f]

([Γ][Vinvt] + [Σ][Eabcg]

)(33)

where

[Γ] =

([U]− [AA]−1

([f] + [ξ][OSN]−1[Ξ]

))

[Σ] = −(

[AA]−1[ξ][OSN]−1[K]

)Next, to obtain the impedance of the PV system whichis required for the resonance study, nodes k, f, m and n

5

are introduced in the Fig. 2. Then applying the well-knownmodified nodal analysis, the following expression is obtained.

[Zpv] = [Ypv]−1 (34)

where:

[Ypv] =

[Ykk] [Ykf ] [Ykm] [Ykn][Yfk] [Yff ] [Yfm] [Yfn][Ymk] [Ymf ] [Ymm] [Ymn][Ynk] [Ynf ] [Ynm] [Ynn]

εC(24H+12)×(24H+12)

Each submatrix in [Ypv] has terms arranged in the HD sense.For instance,

[Yx] =

[Y aax ]

[Y abx

][Y ac

x ][Y bax

] [Y bbx

] [Y bcx

][Y ca

x ][Y cbx

][Y cc

x ]

εC(6H+3)×(6H+3)

where

[Y aax ] =

[Y aax (−H,−H)] · · · [Y aa

x (−H,H)]...

......

[Y aax (H,−H)] · · · [Y aa

x (H,H)]

x = kk, fk,mm, · · · .The interconnecting transformer leakage admittance is modi-fied for harmonic frequencies according to [20].

C. Model implementation algorithm for harmonic currentprediction

In order to deploy the proposed model in quantifyingand characterising harmonics from PV system connected toeither distorted or undistorted distribution system, a systematicprocess was followed which is described through flowchart inFig. 5.

III. NETWORK MODEL FOR RESONANCE INVESTIGATION

A. Distribution lines

Distribution lines are modelled using the classical Carson’sequation [20]. The lines are comprised of 3-, 2- and single-phase overhead lines and underground cables. Lines with 1 or2-phases are padded with zeros. Consider a distribution lineconnected between two nodes viz k and r. The admittancematrix in HD [21] can be written as presented below.

[Y ]node =

[[Ykk] [Ykr][Yrk] [Yrr]

](35)

where:

[Y ]kk =

[Y aakk ]

[Y abkk

][Y ac

kk ][Y bakk

] [Y bbkk

] [Y bckk

][Y ca

kk ][Y cbkk

][Y cc

kk ]

(36)

and

[Y ]stkk =

[Y stkk(−H,−H)] · · · [Y st

kk(−H,H)]...

......

[Y stkk(H,−H)] · · · [Y st

kk(H,H)]

(37)

where s, t ∈ a, b, c. Similarly,

[Y ]strr =

[Y strr (−H,−H)] · · · [Y st

rr (−H,H)]...

......

[Y strr (H,−H)] · · · [Y st

rr (H,H)]

(38)

Data

collection

Start

Network

distorted?

Run HPF

Run

traditional

PF

Converged?

Stop

No

Yes

No

Stop

1) Simulate PV module

2) Send results to HD

model

Read PCC voltage

variables

Yes

1-phase model

3-phase model

Data

visualization

Solar irradiance

Temperature

Manufacturer

datasheet

Fig. 5. Harmonic prediction algorithm using proposed model

and

[Y ]stkr =

[Y stkr (−H,−H)] · · · [Y st

kr (−H,H)]...

......

[Y stkr (H,−H)] · · · [Y st

kr (H,H)]

(39)

The node voltages are

[V ]sk =[[V s

k (−H)] , · · · , [V sk (H)]

]T(40)

[V ]tk =[[V t

k (−H)] , · · · , [V tk (H)]

]T(41)

B. Network transformer modelling

The HD technique allows for modelling of transformercore saturation as can be seen in [18], [22] and [23]. Suchmodel requires more data beyond what is usually provided formost distribution systems. An example of such data requiredis the current-flux curve of the transformer. Nevertheless, ashave been verified in [20], the transformer model used doesnot change very much the frequency response of the system.However, for HPF studies, the impact of core saturation mustbe included for improved accuracy. Due to non-availability ofuseful information relating to the current-flux relationship ofthe transformer in the test system used for this study, we haveadopted the transformer harmonic model used in [20] thoughin the HD form.

6

1

43

8

11

256

10

12

7

9

TX

Fig. 6. Modified IEEE 13 bus distribution network

C. Shunt capacitors

Capacitors are considered to be balanced for 3-phase caseand hence inter-phase capacitances are neglected. In addition,all shunt capacitors have been assumed to be linear. Again,model is written in HD form.

D. Loads

Distribution system loads can be single, 2- or 3-phasehaving wye or delta connection. The loads can be distributedor spot loads or a combination of both. Loads are usuallyunbalanced in most distribution systems. Both linear andnonlinear spot loads have been modelled based on methodproposed in [24]. Nonlinear loads are modelled as harmoniccurrent sources using the current harmonic spectra provided in[25] while linear loads are modelled as passive RL elements.For purpose of this study, we have classified possible loadcombinations into four groups viz: G1, G2, G3 and G4. G1load category comprises 15% and 85% induction motor andpassive loads respectively whereas G2 has 20%, 20%, 40%,and 20% of induction motor, fluorescent lamps, passive andunspecified loads respectively.Similarly, G3 (G4) possesses 20%(15%) induction mo-tor, 60%(55%) passive loads, 10%(15%) fluorescent, and10%(15%) ASD. Indeed, load model has a significant impacton the frequency response of the system.

IV. TEST SYSTEM

A modified version of standard IEEE 13 bus distributionsystem has been used for the simulation. Network compo-nents were modelled as explained in previous section. Shuntcapacitors are located at nodes 8 and 10. There also exists adelta-grounded wye transformer between nodes 3 and 4 whichis used to supply single phase loads including an inductionmotor. The PV system is connected to node 7. Its data areas provided in Table I. The modified network diagram ispresented in Fig. 6. Two simulation cases are considerednamely distorted and undistorted scenarios.

V. SIMULATION AND RESULTS

MATLAB software has been used for simulation. Fig. 7and Fig. 8 are the capacitor voltage and current injection forsingle-phase PV system operating in an undistorted network.Phase current is smooth. Similarly, the current injected

0 2 4 6 8 10 12

−250

−200

−150

−100

−50

0

50

100

150

200

250

time (s)

Vol

tage

am

plitu

de (

V)

Waveform of Capacitor voltage, Vcap

Fig. 7. Single phase PV system capacitor voltage for undistorted scenario

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time(s)

Nor

mal

ised

Ipcc

am

plitu

de

Waveform of PCC Injected current Ipcc

Fig. 8. Current flowing into PCC for undistorted scenario with single phasePV system

0 2 4 6 8 10 12−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time(s)

Nor

mal

ised

am

plitu

de o

f cur

rent

s

Waveform of line currents Injected into PCC

a−phaseb−phasec−phase

Fig. 9. Line currents for 3-phase PV system for distorted scenario

0 2 4 6 8 10 12

−500

−400

−300

−200

−100

0

100

200

300

400

500

Time t(s)

|AB

Vca

p|

Voltage across line−to−line ab Capacitor

Fig. 10. Line-to-line (AB) capacitor voltage (Vcap) for 3-phase PV systemfor distorted scenario

7

TABLE IPV SYSTEM DATA

Inverter switching frequency 3120 HzFundamental frequency 60HzPV rated output power 1MWFilter inductors L1,L2 2.1mH, 1.3mHFilter capacitor 10.6µFTransformer leakage impedance 0.1 puTransformer MVA 1.5Effective turns ratio of ratio 0.48/4.16KV

0 2 4 6 8 10 12

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time(s)

Ipcc

Mag

nitu

de

Waveform of line currents Injected into PCC

a−phaseb−phasec−phase

Fig. 11. Line currents for 3-phase PV system for undistorted scenario

into the grid and the capacitor voltage for the 3-phase PVsystem for distorted network are shown in Fig. 9 and Fig. 10respectively. Here, the line currents contain some harmoniccontents. The corresponding plots for 3-phase currents andcapacitor voltages for undistorted network scenario are shownin Fig. 11 and Fig. 12.

Furthermore, for resonance studies, the system’s nodal ad-mittance matrix is obtained. This matrix is very sparse ascan be seen from Fig.13. This is usually the case for mostpower systems. In order to investigate resonance, driving pointimpedances for all buses have been determined for upto the300th harmonic frequency (i.e. 18kHz) but we have presentedonly those of selected nodes and at selected frequencies. Reso-nance can be series or parallel. A series resonance occurs whenthe reactive capacitance is equal to the reactive inductancethus the equivalent impedance becomes equal to the seriesresistance. This situation leads to over-current in the system

0 2 4 6 8 10 12

−400

−300

−200

−100

0

100

200

300

400

Time t(s)

|AB

Vca

p|

Voltage across line−to−line ab Capacitor

Fig. 12. Line-to-line (AB) capacitor voltage for 3-phase PV system forundistorted scenario

0 2000 4000 6000 8000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

nz = 49100

Fig. 13. Sparsity of system nodal admittance

0 20 40 60 80 1000

50

100

150

Harmonic order h

Nor

mal

ised

|Z|

a−phaseb−phasec−phase

Fig. 14. Node 1 frequency response

and is often responsible for capacitor and fuse failures. On theother hand, a parallel resonance occurs when the impedanceat the point of scan overshoots to an unreasonably high value(theoretically infinite). It is one of the causes of over-voltagesin a power system. It can result in insulation breakdownof transformer windings and insulation of cables [26]. Inthe resonance study presented here, nodes 8 and 10 are ofparticular interest due to the presence of shunt capacitors atthose nodes. Similarly, detailed attention is given to node 7ie the PCC where the PV system is connected. A plot ofdriving point impedance for the three phases of node 1 isshown in Fig.14. The response reveals a positive impedancecharacteristics for this node since the impedance magnitudeincreases with the frequency. Resonance does not occur atthis node. Also, the plot of self impedance for node 8 behavesdifferently from node 1 in some way. Although, like node 1,approximately linear relationship exist between the impedancemagnitude and the frequency prior to the 20th harmonicfrequency as can be seen in the Fig. 15; however, beyond thispoint such linear relationship is not sustained. Nonetheless, theplot does not show any case of sharp resonance in the system.It is reasonable to infer that the passive RL load connectedto this bus must have contributed significantly in the responseabove resulting in possible avoidance of resonance excitationby the capacitor connected to this node.Also, node 10 response is slightly similar to that of node 8with the exception that only a single phase (phase C) exist atthe former. This node also has passive RL loads. Its responseis shown in Fig. 16.Unlike the response of previous nodes presented, resonance(both series and parallel) occur at the PV interconnecting

8

0 20 40 60 80 1000

2

4

6

8

10

12

Harmonic order h

Nor

mal

ised

|Z|

a−phaseb−phasec−phase

Fig. 15. Node 8 frequency response

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

Harmonic order h

Nor

mal

ised

|Z|

c−phase

Fig. 16. Node 10 frequency response

transformer side as shown in Fig. 17. This resonance is dueto the filter capacitor interaction with the impedance at theaffected node. Actually such resonance could lead to damagingof the interconnecting transformer windings. Nevertheless,from the above plot, the resonant frequencies (for both seriesand parallel resonance cases) are beyond the 80th harmonicfrequency for the test system. Indeed, significant harmonic

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

Harmonic order h

Nor

mal

ised

|Z|

a−phaseb−phasec−phase

Fig. 17. PV system (without Rd) node frequency response

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

Harmonic order h

Nor

mal

ised

|Z|

a−phaseb−phasec−phase

Fig. 18. PV system (with Rd = 2.2Ω) node frequency response

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

Harmonic order h

Nor

mal

ised

|Z|

a−phaseb−phasec−phase

Fig. 19. PV system (with Rd = 6.0Ω) node frequency response

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

Harmonic order hN

orm

alis

ed |Z

| (Ω

)

a−phaseb−phasec−phase

Fig. 20. PV system (with no Rd) response when loads at PCC are modified

currents cannot emanate from PVs at such high frequencyexcept the filter circuit is faulty or not in operation. Ideally,an LCL filter eliminates the high order harmonics from theinverter output voltages. Also, it is very practically rare tohave nonlinear loads with significant harmonic currents at suchhigh frequency connected to the distribution network withoutfiltering. It can also be observed that prior to the resonanceoccurrence, the impedance at this node exhibited a negativecharacteristics.Next, the well known solution of connecting a dampingresistor in series with the filter capacitor is incorporated inthe simulation. Fig. 18 and Fig. 19 show the impact of thedamping resistor on damping the resonance. Two values ofresistances (i.e. 2.2Ω and 6.0Ω ) have been used and compared.From the two frequency response plots above, it is inferredthat the higher the value of the damping resistor, the betterthe damping.

Nonetheless, a closer examination of Fig. 19 reveals thateven with a damping resistor as high as 6Ω, the parallelresonance in phase b could not be completely averted. Thisis of major concern especially for networks where resonanceexcitation occurs at lower harmonic frequencies. Indeed, phaseb differs from phases a and c in terms of loads connected tothe PCC to which the PV is also connected as shown in the testsystem in Fig.6. Next, the load was modified and equalizedfor all phases and the effect of the damping resistance re-examined as shown in Fig. 20 and Fig. 21. All three-phasesare now dampened as shown in Fig. 21. This suggests that thepassive loads connected to the PCC contribute significantly indamping of the resonance.

9

10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

Harmonic order h

Nor

mal

ised

|Z| (

Ω)

a−phaseb−phasec−phase

Fig. 21. PV system (with Rd = 6.0Ω) response when loads at PCC aremodified

VI. PROPOSED MODEL VALIDATION

Some works have earlier investigated the impact of networkvoltage distortion on the output performance (in terms ofharmonic generation) of PV systems. In order to validatethe proposed model and simulation results, we shall relateour findings to those of existing models. In [27], [28], and[29], both simulations and field tests results showed that theharmonic currents generation associated with grid connectedPV systems are strongly correlated with the grid voltageharmonics present. In [27], SPICE simulator was used forsimulations and experiments were also conducted to verifythe results. Both ideal grid and distorted (real) networkvoltages impacts were examined. Authors concluded that thenature of harmonic currents from grid integrated PV systemswere dependent on the grid voltage condition. Similarly, [28],and [29] performed measurements and thereafter deployedsome modelling and simulations (using the phasor domainapproach) to verify their results. The notion of ’phase anglediversities and summation exponent’ were also suggested by[29] in order to assess the harmonic interactions between thegrid and the PV system in their model.

Comparing the 3-phase PV currents generation from oursimulation results for undistorted and distorted case as shownin Fig. 11 and Fig. 9 respectively, it is seen that the waveformof the latter is distorted in relation to the former which issmooth. This shows an agreement between our simulationresults and those of other researchers cited above.

Apart from agreements in results between our proposedmodel and those that already exist in the literature, our modelshows some strength in some ways. First, the model proposedin this paper can accurately capture both characteristic anduncharacteristic harmonics thus eliminating the assumptionof presence of characteristic harmonics alone. Secondly,harmonic cross-modulation between a PV system anda nonlinear component (e.g. a transformer operating insaturated state) or load (e.g. asynchronous speed drive) can befully accounted for in our proposed analytical model unlikethe phasor-domain approach that decouples such interaction.Premised upon this, possible cancellations between harmonics

from PVs and the grid if any can also be captured easily usingour model. Thus, an assumption on phase angle diversities(as in [29]) of harmonic components becomes unnecessary.

Next, the resonance investigation conducted with theproposed model also confirms the impact of damping resistorand passive loads on damping of the resonance excited by theinteraction between the inverter LCL filter and the network(see Fig. 21). This also agrees with [9].

Having validated our model, it is important to make clearthat the model proposed in this paper is the first HDRF modelof VSC-PV system (to the best of knowledge of authors) andit has useful applications beyond resonance investigation as itcan be deployed also in HPF studies formulated in the HDRF.

VII. CONCLUSION

In this work, a complete HDRF model of a VSC PV systemis proposed for both single and 3-phase grid interactive PVs.These HD Norton equivalent models developed have beenused to assess and quantify the interaction between the PVsystem harmonics performance and the network. Simulationresults show an insightful interaction between the PV systemharmonic currents produced and the network backgrounddistortions. In addition, an investigation into resonanceoccurrence in a MVDN with 3-phase utility scale PV systemusing the above model has been presented. Results confirmthat resonance excitations (both series and parallel) arepossible due to the system impedance interaction with the PVfilter capacitor. While the popular solution of series connecteddamping resistor was able to dampen the resonance of phasesa and c, the parallel resonance of phase b could not becompletely averted. This is not unconnected with the differentlevel of passive loads connected to the different phases. Withequalization of loads, all 3-phases were dampened; henceshowing that the passive loads at the PCC help significantlytowards damping of the resonance.

In summary, simulation results obtained using the modelproposed show that a strong correlation exists between PVsystem harmonic performance and network backgroundvoltage distortion. Also, passive load components connectedto PCC contribute significantly in damping of resonance thatmay arise from inverter filter capacitor interaction with thenetwork impedance.

We therefore, recommend that detailed assessment andperhaps measurement of harmonic performance of a networkbe conducted pre- and post-installation of PVs in order to dif-ferentiate actual harmonic impacts caused by PV integration.

REFERENCES

[1] E. Romero-Cadaval, G. Spagnuolo, L. Garcia Franquelo, C.-A. Ramos-Paja, T. Suntio, and W. M. Xiao, “Grid-connected photovoltaic gener-ation plants: Components and operation,” Industrial Electronics Maga-zine, IEEE, vol. 7, no. 3, pp. 6–20, 2013.

[2] M. M. Begovic, I. Kim, D. Novosel, J. R. Aguero, and A. Rohatgi,“Integration of photovoltaic distributed generation in the power distribu-tion grid,” in 45th Hawaii International Conference on System Sciences.IEEE Computer Society, Conference Proceedings, pp. 1977–1986.

10

[3] E. Caamao-Martn, H. Laukamp, M. Jantsch, T. Erge, J. Thornycroft,H. De Moor, S. Cobben, D. Suna, and B. Gaiddon, “Interaction betweenphotovoltaic distributed generation and electricity networks,” Progress inPhotovoltaics: Research and Applications, vol. 16, no. 7, pp. 629–643,2008.

[4] M. Thomson and D. G. Infield, “Impact of widespread photovoltaicsgeneration on distribution systems,” IET Renewable Power Generation,vol. 1, no. 1, p. 33, 2007.

[5] G. Chicco, J. Schlabbach, and F. Spertino, “Characterisation and assess-ment of the harmonic emission of grid-connected photovoltaic systems,”in Power Tech, 2005 IEEE Russia. IEEE, Conference Proceedings, pp.1–7.

[6] N. Eghtedarpour, M. Karimi, and M. Tavakoli, “Harmonic resonancein power systems-a documented case,” in Harmonics and Quality ofPower (ICHQP), 2014 IEEE 16th International Conference on. IEEE,Conference Proceedings, pp. 857–861.

[7] L. Sainz, M. Caro, and J. Pedra, “Study of electric system harmonicresponse,” IEEE transactions on power delivery, vol. 19, no. 2, pp. 868–874, 2004.

[8] J. H. Enslin and P. J. Heskes, “Harmonic interaction between a largenumber of distributed power inverters and the distribution network,”IEEE transactions on power electronics, vol. 19, no. 6, pp. 1586–1593,2004.

[9] H. Hu, Q. Shi, Z. He, J. He, and S. Gao, “Potential harmonic resonanceimpacts of pv inverter filters on distribution systems,” SustainableEnergy, IEEE Transactions on, vol. 6, no. 1, pp. 151–161, 2015.

[10] C. Limsakul, A. Sangswang, D. Chenvidhya, M. Seapan, B. Meunpinij,N. Chayavanich, and C. Jivacate, “An impedance model of a pvgrid-connected system,” in Photovoltaic Specialists Conference, 2008.PVSC’08. 33rd IEEE. IEEE, Conference Proceedings, pp. 1–4.

[11] E. C. Aprilia, V. Cuk, J. Cobben, P. F. Ribeiro, and W. L. Kling,“Modeling the frequency response of photovoltaic inverters,” in In-novative Smart Grid Technologies (ISGT Europe), 2012 3rd IEEEPES International Conference and Exhibition on. IEEE, ConferenceProceedings, pp. 1–5.

[12] J. Arrillaga and N. Watson, “The harmonic domain revisited,” inHarmonics and Quality of Power, 2008. ICHQP 2008. 13th InternationalConference on. IEEE, Conference Proceedings, pp. 1–9.

[13] J. Arrillaga, A. Medina, M. Lisboa, M. Cavia, and P. Sanchez, “Theharmonic domain. a frame of reference for power system harmonicanalysis,” Power Systems, IEEE Transactions on, vol. 10, no. 1, pp.433–440, 1995.

[14] Y. Mahmoud, W. Xiao, and H. Zeineldin, “A simple approach tomodeling and simulation of photovoltaic modules,” Sustainable Energy,IEEE Transactions on, vol. 3, no. 1, pp. 185–186, 2012.

[15] Y. Mahmoud and E. El-Saadany, “Accuracy improvement of the idealpv model,” IEEE Transactions on Sustainable Energy, vol. 6, 2015.

[16] B.-K. Lee and M. Ehsami, “A simplified functional simulation model forthree-phase voltage-source inverter using switching function concept,”Industrial Electronics, IEEE Transactions on, vol. 48, no. 2, pp. 309–321, 2001.

[17] N. Mohan and T. M. Undeland, Power electronics: converters, applica-tions, and design. John Wiley and Sons, 2007.

[18] G. W. Chang, H.-W. Lin, and S.-K. Chen, “Modeling characteristicsof harmonic currents generated by high-speed railway traction driveconverters,” IEEE Transactions on Power Delivery, vol. 19, no. 2, pp.766–773, 2004.

[19] C. Marouchos and M. Darwish, “Teaching power electronics using theswitching function approach,” in 2012 47th International UniversitiesPower Engineering Conference (UPEC). IEEE, Conference Proceed-ings, pp. 1–7.

[20] T. Densem, P. Bodger, and J. Arrillaga, “Three phase transmisssionsystem modelling for harmonic penetration studies,” IEEE transactionson power apparatus and systems, no. 2, pp. 310–317, 1984.

[21] M. Madrigal and E. Acha, “Modelling of custom power equipmentusing harmonic domain techniques,” in Harmonics and Quality of Power,2000. Proceedings. Ninth International Conference on, vol. 1. IEEE,Conference Proceedings, pp. 264–269.

[22] A. Semlyen, E. Acha, and J. Arrillaga, “Harmonic norton equivalentfor the magnetising branch of a transformer,” in IEE Proceedings C-Generation, Transmission and Distribution, vol. 134. IET, ConferenceProceedings, pp. 162–169.

[23] E. Acha and M. Madrigal, Power systems harmonics. John Wiley andSons, Inc., 2001.

[24] R. Burch, G.-k. Chang, C. Hatziadoniu, M. Grady, Y. Liu, M. Marz,T. Ortmeyer, S. Ranade, P. Ribeiro, and W. Xu, “Impact of aggregatelinear load modeling on harmonic analysis: A comparison of common

practice and analytical models,” Power Delivery, IEEE Transactions on,vol. 18, no. 2, pp. 625–630, 2003.

[25] A. Bonner, T. Grebe, E. Gunther, L. Hopkins, J. Mahseredjian, N. Miller,T. Ortmeyer, V. Rajagopalan, S. Ranade, and P. Ribeiro, “Modeling andsimulation of the propagation of harmonics in electric power networks.2. sample systems and examples,” IEEE Transactions on Power Delivery,vol. 11, no. 1, pp. 466–474, 1996.

[26] F. C. De La Rosa, Harmonics, power systems, and smart grids. CRCPress, 2015.

[27] A. Simmons and D. Infield, “Current waveform quality from gridcon-nected photovoltaic inverters and its dependence on operating condi-tions,” Progress in Photovoltaics: Research and Applications, vol. 8,no. 4, pp. 411–420, 2000.

[28] A. Bosman, J. Cobben, J. Myrzik, and W. Kling, “Harmonic modellingof solar inverters and their interaction with the distribution grid,” inProceedings of the 41st International Universities Power EngineeringConference, vol. 3. IEEE, Conference Proceedings, pp. 991–995.

[29] A. Varatharajan, S. Schoettke, J. Meyer, and A. Abart, “Harmonicemission of large pv installations case study of a 1 mw solar campus,”in International Conference on Renewable Energies and Power Quality(ICREPQ14), Cordoba, Spain, 8th-10th April, Conference Proceedings.

[30] M. Madrigal and J. Rico, “Operational matrices for the analysis ofperiodic dynamic systems,” Power Systems, IEEE Transactions on,vol. 19, no. 3, pp. 1693–1695, 2004.

[31] F. Toutounian, E. Tohidi, and A. Kilicman, “Fourier operational matricesof differentiation and transmission: introduction and applications,” inAbstract and Applied Analysis, vol. 2013. Hindawi Publishing Corpo-ration, Conference Proceedings.

VIII. APPENDIX

A. Orthogonal series function approach to circuit analysis

The use of orthogonal series functions (such as Fourierseries and transforms, Walsh, Hartley transforms etc) forsolving equations of integral-differential form that result fromelectric circuits is well known [23]. For purpose of readability

0 2 4 6 8 10 12

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time (s)

Am

plitu

de

Switching function G(θ)

Fig. 22. 3-level inverter switching pulse train

of this paper, we present a little insight into the HD method.Interested readers are advised to consult suggested references[13], [23], [30] and [31]. Here, we only discuss the Fourierseries for convenience. Consider that the function f(t) is atruncated Fourier series; then

f(t) = 〈[Φ(t)], [F]〉

11

such that 〈〉 := inner productBasis vector is [Φ(t)] and [F] is a vector of harmonic coeffi-cients with expressions thus:

[Φ(t)] =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

exp (−jHwt)exp (j(−H + 1)wt)

...1...

exp (j(H − 1)wt)exp (jHwt)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣; [F] =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

F−HF−H+1

...F0

...FH−1FH

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣It is easy to observe that

˙f(t) = [Φ(t)]T [Ψ][F] (42)

where

[Ψ] =

−jHω

−j(H − 1)ω. . .

j(H − 1)ωjHω

By induction (fx(t)) = [Φ(t)]T [Ψ]x[F] where x represents

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (s)

Am

plitu

de

Switching function G(θ)

Fig. 23. Zoomed version of a section of the switching function pulse train

the xth derivative and xth power on the left and right handside of the expression respectively. Now, consider a simpleseries RL electric circuit which is supplied by a non-sinusoidalvoltage source v(t). We assume that v(t) obeys the Jordan-Dirichlet’s conditions and thus is Fourier series expressible.Then applying KVL, we have:

v(t) = ri(t) + Ldi(t)

dt(43)

Applying the HD technique, we have solution for harmoniccoefficients of currents as follows:

[I] = [Y][V]; (44)

[Y] = (r[U] + L[Ψ])−1 (45)

Then the TD and the HD currents are related thus:

i(t) = [Φ(t)]T [I] (46)

B. Switching function plotsUsing the switching function equations generated in (2),

the plot in Fig. 22 was realized using MATLAB. For clarity, azoomed version of part of the plot is shown in Fig. 23. This isactually the switching pattern of a 3-level (unipolar) inverter.

Onyema S Nduka (S’14) received an M.Sc degreein Control Systems from Imperial College London,United Kingdom and a B.Eng in Electrical andElectronic Engineering (with specialisation in Powersystems Engineering) from Federal university ofTechnology, Owerri, Imo State, Nigeria in 2014 and2011 respectively. Currently, he is pursuing a PhDat Imperial College London. His research interestsincludes distribution system - modelling, control andoptimisation, power quality (harmonics and inter-harmonics), computer aided power systems, com-

putational power systems, small scale energy storage and renewable energyintegration into power distribution networks.

Bikash C Pal (M’00-SM’02-F’13) received theB.E.E.(with honours) degree from Jadavpur Univer-sity, Calcutta, India, the M.E. degree from the IndianInstitute of Science, Bangalore, India, and the Ph.D.degree from the Imperial College London, London,U.K, in 1990, 1992, and 1999, respectively, all inelectrical engineering.

Currently, he is a Professor in the Departmentof Electrical and Electronic Engineering, ImperialCollege London, London, U.K. His current researchinterests include state estimation, power system dy-

namics, and flexible ac transmission system controllers.Dr. Pal is a Fellow and the Editor-in-Chief of the IEEE TRANSACTIONS

ON SUSTAINABLE ENERGY.