Small Signal Impedance Measurementin Droop Controlled AC Microgrids

Malte JohnInstitute for Drive Systems and Power Electronics

Leibniz Universitat HannoverWelfengarten 1

D-30167 Hannover, GermanyEmail: malte.john@ial.uni-hannover.de

Patricio A. Mendoza-ArayaElectrical Engineering Department

University of ChileAv. Tupper 2007Santiago, Chile

Email: pmendoza@ing.uchile.cl

Giri VenkataramananDepartment of ECE

University of Wisconsin-Madison1415 Engineering Dr

Madison, WI USAEmail: giri@engr.wisc.edu

Abstract—Stable operation of microgrids is highly influencedby characteristics of generators due to their small size andinertia and fast and variable dynamics. Recent work has shownthat microgird stability can be determined by analysis of smallsignal impedances at the point of interconnection. The purpose ofsmall signal impedance measurement is to verify the analyticalmodels of single devices and to provide measured impedanceswhere analytical models are not available, so that stability maybe established. Small signal impedance measurement typicallyinvolves in-situ injection of currents or voltages superimposedupon the operating system. In droop-controlled microgrids, sincethe frequency is a function of the power demand, the injectionhas to be independent of the frequency. In this paper an injectionapproach is proposed using a three phase buck converter. Analyt-ical models using dynamic phasors are compared to measurementresults obtained in a laboratory microgrid. The injection methodand the incremental phasor models of passive loads are verifiedand small signal impedance measurements of an islanded andgrid-connected microgrid are obtained including a voltage sourcePWM converter.

I. INTRODUCTION

Microgrids as a detachable subunit of the electric powersystem can operate in grid-connected and islanded modeand can help to improve reliability and efficiency [1]. Toallow large scale application, certain requirements have to befulfilled. New units shall be included without the necessity toredesign the existing system, known as plug and play com-patibility [2]. To determine a stable operation, stability criteriahave been developed, using impedance matching. Therefore,two subsystems, e.g. the microgrid and a new source, arerepresented by transfer functions in the form of impedancesor admittances. Choosing one subsystem to represent a newdevice and the second subsystem to be the existing microgrid,these two parts can be determined independently. Each partis represented by a single transfer function. Connection ofboth subsystems results in a closed-loop system. Its transferfunction can be derived from the two subsystems using linearcontrol theory. This turns out to be a suitable approach formicrogrids, where a high number of small devices exist andadjustment of the overall system’s analytical model should belimited [3].

An essential feature of microgrids is the wireless com-munication approach. Stable operation has to be granted

during grid-connected and islanded mode by control thatonly relies on locally measured quantities. Therefore, activepower-frequency (P-f) and reactive power-voltage (Q-V) droopcontrol was introduced to the microgrid sources to conduct thepower sharing and voltage stabilization [4]. As a consequencethe frequency during islanded operation is subject to largervariations than during grid-connected operation. Moreover,constant power loads (CPL) build an important challengefor the stability of microgrids. Their negative incrementalimpedance behavior can lead to stability problems. CPL canalso comprise droop control, resulting in an impedance thatdepends on the droop gain.

This paper aims at the development of a small signalimpedance measurement technique suitable for a microgridenvironment. This work proposes a straightforward injectionusing a three phase buck converter that does not rely onthe measurement of the phase angle. Experimental results arecompared to analytical models obtained in dynamic phasorexpressions that describe the small signal quantities.

II. STATE OF THE ART

The basic idea of the impedance matching approach orig-inates from the filter design of DC/DC converters. Stabilitycriteria for DC systems refer to the Middlebrook criterion [5].It describes the influence of the converter’s input filter on theimpedance and therefore its transfer function. The stabilityis determined by investigation of the closed-loop transferfunction using the Nyquist criterion. Middlebrook stated astability criterion [5] over this return ratio, which has beenextended to less conservative criteria in several publications, assummarized in [6]. As the modeling of the subsystems providesnon-linear transfer functions, a linearization is conducted andresults in linearized transfer functions that represent smallsignal impedances. Later stability analysis methods have beenextended to three phase AC systems [7], including CPL [8],[9].

AC systems lack a constant operating point, presentinginstead a sinusoidal steady state. Consequently linearizationis not possible without appropriate methods. Several modelingand analysis tools have been published, including Park dq-transformation, phasor-based approaches and harmonic lin-earization. A review is given in [10]. The approach to trans-

978-1-4799-5776-7/14/$31.00 ©2014 IEEE 702

form into rectangular dq-coordinates provides DC values forthe steady-state and therefore an operation point for the lin-earization. Measurement methods for these impedances havebeen published in [11]–[13]. However, the injection devicesuse phase-locked loops (PLLs) to align the injected smallsignal perturbation to the reference frame. Changes to thesystem frequency should not be allowed and the perturbationsignal should not have a disturbing effect on the frequency.Therefore several publications also deal with the frequencybehavior of the PLL-device [14], [15].

Alternate modulation and injection techniques that do notuse the d-q transformations have been developed and pub-lished. Using a grid connected inverter to generate a sharp cur-rent sag, the grid’s impulse response is measured to obtain itstransfer function [16]. Measurements have been implementedin a passive grid. Ref. [17], [18] are using a pseudorandombinary sequence voltage perturbation, which is a form ofinjecting white noise, and digital network analyzer and cross-correlation techniques to process the system identification.

As mentioned before, the frequency in microgrids with adroop controlled power sharing is a time-varying value as afunction of the power demand, particularly during islandedoperation. This work proposes a straightforward injectionmethod in Sec. IV, that does not rely on the measurementof the phase angle. The measured small signal impedances arepart of the subsystem’s transfer function, expressed in polarcoordinates. The centerpiece of this paper is the measurementof small signal impedances to verify analytical models ob-tained in dynamic phasor expression. These models have beendeveloped in [3] and a short summary is given in the followingsection.

III. ANALYTICAL MODEL

The analytical models are built around dynamic phasors.They are a generalization of the complex-valued steady-statephasor quantity to the quasi-steady state, resulting in a mag-nitude and phase angle of the phasor as functions of time. Adynamic phasor represents the time-domain function

V1(t) = V a1 cos(ωot+ V θ

1 ), (1)

where V a1 and V θ

1 are both functions of time. The corre-sponding dynamic phasor in the complex notation would beV1 = V a

16 V θ

1 . The dynamic phasor models presented here arestrongly influenced by previous developments shown in [19],and an example of the derivation of phasor dynamic equationsis shown there. The method can be regarded as close to”harmonic linearization” [20], or ”method of averaging” [21],in the sense that it corresponds to the 1-phasor approximationof the non-linear system [22].

To study small-signal perturbations, a linearization of thoseequations around an operating point provides a linear time-invariant system that can be used for stability assessment. Thedynamics of this linearized model, which we call incrementalphasor model, describe the interactions between perturbationsin phasor currents and voltages (v, i) around the static phasoroperating point (V , I), as shown for reference in Fig. 1.

In general, the state space representation of a set of

V

I

v

i

(a) (b)

Fig. 1. Dynamic phasor diagrams for (a) large and (b) small signals

Area 1 Area 2I

V

PoCMicrogrid

Newsource National

Grid

NeighbourMicrogrid

Fig. 2. System division into two subsystems for impedance matchingapproach

Y1

−Z2

i1

−i2v2

v1

Area 1 Area 2PoC

Y1 Z2

Fig. 3. Impedances in a conceptual Microgrid

nonlinear differential equations is given by

x(t) = f(x(t),u(t))

y(t) = g(x(t),u(t)).(2)

where x(t) is the state variable, u(t) the input variable andy(t) the output variable. The functions f and g representthe non-linearity. To obtain an LTI-model, the state spaceequations are linearized at an operating point:

x(t0) = f(x(t0),u(t0)) (3)

The linearized state space representation contains fourmatrices A,B,C and D with constant complex values. Thesystem variables are small signal variables, representing thedeviation from the operating point:

˙x = Ax + Bu (4)y = Cx + Du (5)

The system’s transfer function using the incremental phasormodel H(s) can be obtained using the following equation:

H(s) =y(s)

u(s)= C · (sI −A)

−1 ·B + D (6)

It represents the magnitude and phase components of theinput phasor u(s) and the output phasor y(s). By analogyto Cartesian dq-coordinates, where the d- and q-componentsare cross-coupled, the phasor representation contains a cross-coupling of the polar components magnitude and phase.

703

TABLE I. INCREMENTAL PHASOR IMPEDANCES FOR SEVERAL COMPONENTS

Component Large-signal impedance Incremental phasor impedanceResistor ZR(s) = R ZR(s) = R

Capacitor ZC(s) = 1sC

ZC(s) = |ZC(jωo)| · ω2o

s2+ω2o

R-C load ZR−C(s) = 1C· 1s+ 1

RC

ZR−C(s) = |ZR−C(jωo)| ·1

RCs+ 1

(RC)2+ω2

o

s2+ 2RC

s+ 1(RC)2

+ω2o

R||L load ZR||L(s) = R s

s+RL

ZR||L(s) = |ZR||L(jωo)| ·s2+3R

Ls+

(R2

L2 +ω2o

)s2+2R

Ls+

(R2

L2 +ω2o

)

H(s) =

[Haa(s) Hθa(s)Haθ(s) Hθθ(s)

](7)

Each component will be modeled in such a way that theresulting state space representation yields either an admittanceor an impedance. This way, the incremental phasor modelscan be connected together to form an impedance-admittanceequivalent that resembles the block diagram of Fig. 3. In thisdiagram, Y1 and Z2 represent the small signal dynamics ofArea 1 and Area 2 in the figure at the operating point that isestablished at its interface.

The matrix element Haa(s) represents the ratio of voltagemagnitude V and current magnitude I . In terms of small signalstability this element is dominant and will be addressed asincremental phasor impedance Z(s) or incremental phasoradmittance Y (s) depending on the state space formulation.

Impedance matching criteria can be used to predict stabilityof two connected subsystems. Fig. 2 shows a possible divisionof an example system. It consists of a source connectedto a microgrid, which in turn can be connected to othermicrogrids or the utility grid. The division is arbitrary andcan be conducted for every point in the system. At the pointof connection (PoC) two subsystems can be defined for bothareas and electrical values like currents and voltages can beconsidered as inputs and outputs. These subsystems are shownin Fig. 3.

To determine the system stability, the open loop transferfunction L(s) = Yl(s) · Zs(s) is examined. The admittanceYl(s) and impedance Zs(s) traditionally correspond to loadand source, but in our case they represent the Areas 1 and 2,in form of the transfer function element Haa(s).

A. Example Models

In the case of a symmetrical load containing a capacitor inparallel to a resistor, the state space representation is given by

x =

(V ac

V θc

), u =

(IacIθc

), y = x

A =

(− 1

RCIAC

C

− Iac

C(V ai )2 − 1

RC

),

(8)

B =1

C

(IaR

Iai

IacIac

V ai Ia

i

IaR

V ai

),

C =

(1 00 1

), D = 0 .

The complex valued transfer function can be obtained using(6). The incremental phasor impedance as an interpretation ofthe magnitude-magnitude channel of this transfer function islabeled as ZR||C and is compared to the large signal impedanceZR||C as its corresponding steady-state phasor:

ZR||C(s) =1

C· 1

s+ 1RC

(9)

ZR||C(s) =∣∣ZR||C(j ω0)

∣∣ · 1RC s+ 1

(RC)2 + ω20

s2 + 2RC s+ 1

(RC)2 + ω20

(10)

The analytical models for different passive loads can bederived similarly. Refer to Tab. I for selected examples. Theincremental phasor model of a droop controlled microsourceis given in (11) in the form of a small signal admittance. Thenumerator and denominator of this admittance are functions ofthe droop gain Mp. More example models of components canalso be found in [19].

IV. SMALL SIGNAL IMPEDANCE MEASUREMENT

The purpose of small signal impedance measurement istwofold. The first purpose is the validation of an analyticalmodel, in this case of the incremental phasor models, derivedin Sec. III. Secondly, measurements are necessary when ananalytical model is not available or takes an unacceptable effortdue to its complexity.

Yusrc(s) = − 1

∆(s)· 1

Za·[Za

LcosφLs

2 +1

L2

(Za2 cos(φL − ϕ)− 1

2LMpV

ae U sinφe cosφL

)s

+1

2L2ZaMpV

ae (V a

o sinφL + U sin(ϕ− φe) cosφL)

]∆(s) = s3 +

2R

Ls2 +

1

L2

(Za2 +

1

2LMpV

ae V

ao sin(δ)

)s+

1

2L2ZaMpV

ae V

ao sin(δ + ϕ)

(11)

704

A. Small Signal Injection Method

To measure the small signal impedance of a system, adeviation from the operation point has to be conducted. Thequantities of the operating point are regarded as the carrier ofthe small signal, that is modulated on top of the large signal.In a three phase system with the angular frequency ωe thismodulated current, with the small signal amplitude Is and thesmall signal angular frequency ωs, can be expressed by thefollowing equation:

isa = Is cos(ωst+ φis) cos(ωet)

isb = Is cos(ωst+ φis) cos(ωet− 2π/3)

isc = Is cos(ωst+ φis) cos(ωet+ 2π/3).

(12)

Injecting a small signal current in a system results in voltagedrop across the impedances, that can be described by amagnitude and phase angle respective to the current. In returna modulated voltage injection

vsa = Vs cos(ωst+ φvs) cos(ωet)

vsb = Vs cos(ωst+ φvs) cos(ωet− 2π/3)

vsc = Vs cos(ωst+ φvs) cos(ωet+ 2π/3)

(13)

would result in a modulated small signal current. Thereforethe shunt current injection or the series voltage injection areboth possible methods for measuring small signal quantities.Both methods are depicted in Fig. 4.

The injected signal contains two frequency componentsωs ± ωe, which can be shown by trigonometric identities. Forexample, the small signal current in phase a can be written as

isa =Is2

cos ((ωs + ωe) t+ φis) +Is2

cos ((ωs − ωe) t+ φis) .

(14)It is possible to either inject both frequency componentsdirectly (14) or to disturb the system with a modulated smallsignal using a carrier, as shown in (12) and (13).

As its name implies, small signal measurement deals withrelatively small quantities, whereas the measurement deviceshave to be rated for the superimposed large signals of theoperating point. Moreover, the measurement faces severalproblems with line frequency harmonics, switching effects andother sources of noise, as it takes place for a broad frequencyrange.

B. Data Aquisition and Small Signal Filtering Software

The centerpiece of the small signal measurement is thetwelve channel oscilloscope Yokogawa DL750. Six high-voltage differential probes and six current probes, all suitablefor AC and DC values, are used to provide the measurementsignals to the oscilloscope. The obtained data is saved fora single frequency injection and the whole spectrum can beprocessed by math software such as Matlab.

To obtain the small signal values of current and voltage,a transformation to symmetrical components is conducted.Thereby, measurement errors in the three phases are averagedand the measurement results can be compared to the singlephase models. Fast Fourier Transform (FFT) or Fourier seriesdecomposition can be used to find the small signal part ofthe positive sequence component, denoted by the subscript()1. Both methods provide similar outcome and are used for

System 1 IsV

I1 I2

System 2

System 1 V1

I

System 2

~

V2

Vs

Fig. 4. Possible small signal injection methods shunt current injection (top)and series voltage injection (bottom)

time-domain simulations in [3] as well as for the experimentalresults in this paper. The small signal impedance Z or theadmittance Y is calculated by using the current and voltage inpolar coordinates, expressed by (15). The superscripts ()a and()θ denote magnitude and phase angle respectively.

Za =V a1

Ia1Zθ = V θ

1 − Iθ1

Y a =Ia1V a1

Y θ = Iθ1 − V θ1

(15)

As can be seen, the impedance phase angles are differ-ences. An angle reference in relation to a reference voltage isunnecessary. Therefore, a PLL-device measuring the system’sfrequency is not required.

C. Injection Device

In frequency dependent systems, the injection deviceshould be independent of measuring the system’s angularfrequency ωe. The proposed topology for the small signalinjection is the three phase buck converter depicted in Fig. 5. Itcontains three on-cycle IGBTs (S1, S2, S3) that are controlledby PWM signal 1. The three off-cycle IGBTs (S4, S5, S6) takeover the task of the freewheeling diode of DC buck convertersand they are controlled by PWM signal 2. Details on theconverter topology are covered in [23].

Both PWM signals initially have a fixed duty cycle of D0

and (1 −D0), respectively. By varying this duty cycle at thefrequency fs with an amplitude D, the desired small signalperturbation can be generated. The duty cycle variation

D = D0 + D sin (ωst) (16)

is illustrated in Fig. 6. Therefore the output voltage is mod-ulated with the small signal angular frequency ωs on top ofthe system angular frequency ωe. When connecting a linearload to the converter, a small signal current that fulfilling (12)is injected back into the system without the need to measurethe system frequency. Another advantage of this topology isthe straightforward control. No current or voltage feedback isnecessary, unless it is needed for protection.

The converter input and output filters are designed to havea sufficient margin to the switching frequency fsw of 10 kHzto prevent oscillations. Measurements showed that a marginof about one decade from the highest small signal frequency

705

A

B

C

a

b

c

L1 L2

C1 C2

S2

S3

S1

S4

S5

S6

Fig. 5. Schematic of the PWM buck converter

1/fsw

t

VG

PWM1

PWM2

t

VG

D0

Fig. 6. Duty cycle variation for PWM channel 1 and 2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−500

0

500

Lo

ad

vo

lta

ge

(V

)

Time (s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−10

−5

0

5

10

Lo

ad

cu

rre

nt

(A)

Time (s)

Fig. 7. Measured output voltage and output current of the buck converter forresistive load (fs = 5 Hz, fe = 60 Hz, R = 80 Ω, D = 0.75± 0.2).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−500

0

500

Vo

lta

ge

(V

)

Time (s)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−5

0

5

Cu

rre

nt

(A)

Time (s)

Fig. 8. Measured input voltage and current phase A (blue) and output voltageand current phase a (red) of the buck converter for resistive load (fs = 5 Hz,fe = 60 Hz, R = 80 Ω, D = 0.75± 0.2).

TABLE II. CONVERTER PARAMETER

Parameter Symbol ValueInput filter C1 500 nF

L1 10 nHOutput filter C2 1 µF

L2 5 mHff2 2.25 kHz

Duty cycle D 0.75± 0.2Switching frequency fsw 10 kHz

of 200 Hz is sufficient. Refer to Tab. II for the converterparameter.

The measured output voltages and currents of the buckconverter in case of a symmetrical resistive load of 80 Ω isdepicted in Fig 7. The low frequency of the duty cycle at 5 Hzis clearly visible. Fig 8 shows the input and output values ofone phase of the buck converter for the same load case. Both,input and output currents point out of the buck converter, asit is useful for the definition of impedances at the PoC.

As an inherent behavior the buck converter operationalways reduces the voltage at its output. When connected inseries at the PoC, this results in a modification of the operatingpoint. Using a variable transformer (variac), the transmissionratio can be set inverse to the constant part of the duty cycle D0

to avoid an overall transmission ratio of the injection device.When used in shunt current connection, the low voltage sideof the converter is connected to a burden resistor Rb. Theamount of the injected large signal current is a function ofthe constant part of the duty cycle D0 and the burden resistor.Both combinations are shown in Fig. 9.

To Area 1 To Area 2

Injection Device

To Area 1 To Area 2In

jecti

on D

evic

e

Rba) b)Fig. 9. PWM buck converter connection: (a) shunt current injection, (b)series voltage injection

D. Laboratory Setup

A lab-scale microgrid (Fig. 10) provides the test environ-ment to verify the injection method and the incremental phasormodeling. The microgrid contains two droop controlled mi-crosources. These inverter-based distributed generators includea frequency and voltage droop control for power sharing andvoltage stability. The first microsource (MS1) includes a lead-acid battery bank and can function as a source or a (constant-power) load. The second microsource (MS2) is fed by a DCpower supply. Both sources comprise an LCL-filter and an iso-lation transformer. A static switch allows synchronization anddisconnection from the utility grid. Three power cables (PC)are integrated between the sources and the grid to simulatean electrical distance. Moreover three detachable loads grantdifferent operating points and the circuit breakers (CB) allowmechanical disconnection.

706

InverterBattery

UtilityGrid

CN1

CB1

Tgr

CB2 PC1PCgr

T1

Load1

PC2

T2

CB4 CN2CB3

MS2MS1

Load3Load2

StaticSwitch

480 V

208 V

InverterDC Source

BuckConverter

Variac208 V

480 V

208 V

480 V

208 V

208 V

Fig. 10. Topology of the lab-scale microgrid

V. EXPERIMENTAL RESULTS

The laboratory setup is used to verify the analytical modelsof different load and source combinations.

A. Passive Loads

In the first case, MS2 is connected to the buck converter inislanded mode and MS1 is disconnected. A variety of passiveloads is attached to the buck converter’s low voltage side(terminals a,b,c in Fig. 5).

Fig. 11 compares the measurement of a parallel RC-loadwith R = 38 Ω and C = 24µF to the small signal impedanceof the analytical model defined in (10). Experimental resultsand analytical model show very high accuracy. Deviationscan only be detected for measurements around the systemfrequency of 60 Hz, which can be explained by parasiticeffects and difficult measurement conditions for this frequencyarea. For frequencies above 250 Hz the signal-to-noise ratiodecreases and the data is not used.

Fig. 12 shows a comparison of the small signal admittanceof a series RL-load. For this load combination of R = 0.9 Ωand L = 30.5 mH a visible peak in the admittance at the

frequency√(

RL

)2+ ω2

0 ≈ ωe is visible. YR−L represents thelarge signal admittance and YR−L the small signal admittanceof the series RL-load:

YR−L(s) =1

Ls+R(17)

YR−L(s) = |YR−L(j ω0)| ·RL s+

(RL

)2+ ω2

0

s2 + 2RL s+

(RL

)2+ ω2

0

(18)

Except for few measurement points, the analytical model andmeasurement show high conformity.

B. Single Microsource Connected to a Resistive Load

The incremental phasor model of a droop controlled mi-crosource is compared in Fig. 13 to the analytical model given

in (11). The connected resistive load is depicted in Fig. 14as small signal impedance. The experimental results for theresistive load side closely match the analytical model. Thesame is not true for the microsource, as the experimentaladmittance magnitude departs from the analytical model forlow frequencies, although maintaning a fairly close phaseangle.

C. Single Microsource Connected to the Utility Grid

Fig. 15 and Fig. 16 show the microsource admittance ofMS2 and the grid impedance during grid-connection. Thisparticular experiment is very challenging, as the impedanceof the grid is usually low and, in most cases, unknown. Byinjecting small signal currents into the system, it is verydifficult to generate, under the presence of the grid, the smallvoltage perturbations needed to determine the incrementalphasor impedances.

D. Discussion

Small signal measurement is challenging as it deals witha low signal-to-noise ratio. The low impedance values fordistributed energy sources and above all for the utility gridrequires fairly high injected currents. These can lead to achanging operating point and therefore distortion of the results.

A comparison of current injection and voltage perturbationemphasizes the need to pick a measurement method well suitedto the task. The voltage perturbation is preferred for highimpedances, when low parallel impedances are present.

Recurring problems occur for measurements during os-cillations. A variation of passive components in the system,like capacitors and inductors, creates a number of resonancefrequencies. For example the three phase buck converter con-tains an input and an output filter, which can be be excited tooscillations with itself or other system components.

707

10−1

100

101

102

103

16

18

20

22

Magnitude (

dB

)

10−1

100

101

102

103

−60

−40

−20

0

Phase (

deg)

Frequency (Hz)

Measured

Analytical

Fig. 11. Bode plots for a parallel RC-load

10−1

100

101

102

103

−40

−30

−20

−10

0

10

Magnitude (

dB

)

10−1

100

101

102

103

−200

−150

−100

−50

0

Phase (

deg)

Frequency (Hz)

Measured

Analytical

Fig. 12. Bode plots for a series RL-load

−20

−10

0

10

Mag

nitu

ders

dBF

Ia/Va,rY1

Analytical

Simulation

Experiment

101

102

103

0

45

90

135

180

225

Pha

sers

degF

Frequencyrsrad/sF

Fig. 13. Bode plots for a microsource, resistive load

27.5

28

28.5

Mag

nitu

dexs

dBF

Ia/Va,xZ2

x

Analytical

Simulation

Experiment

101

102

103

−20

−10

0

10

20

Pha

sexs

degF

Frequencyxsrad/sF

Fig. 14. Bode plots for a resistive load, fed from a microsource

−20

−10

0

10

20

Mag

nitu

ders

dBF

Ia/Va,rY1

Analytical

Simulation

Experiment

101

102

103

−180

−90

0

90

180

270

360

Pha

sers

degF

Frequencyrsrad/sF

Fig. 15. Bode plots for a microsource, grid-connected

−20

−10

0

10

Mag

nitu

dey(

dB)

Ia/Va,yZ2

101

102

103

−180

−90

0

90

180

Pha

sey(

deg)

Frequencyy(rad/s)

Fig. 16. Bode plots for the grid, connected to a microsource

708

As the small signal injection is conducted for a broadfrequency area, excitation of these resonances can occur.Moreover, excitation due to power oscillations and the IGBTswitching leave a small area for the filter design.

VI. CONCLUSION

In this paper, a practical method for small signal impedancemeasurement was successfully developed. With this method,incremental phasor impedances can be obtained in a microgridenvironment, which is prone to frequency deviations.

For this purpose a three phase PWM buck converterwas successfully implemented in a laboratory microgrid. Thefrequency independent small signal injection method of thistopology proved to be a major advantage. Measurements inseries voltage perturbation and shunt current injection wereconducted for the frequency area between 0.2 Hz and 200 Hz.In the case of two units with large differences in theirimpedance value, the series voltage method showed lower mea-surement errors. Further experiments included measurementsof an islanded microsource connected to a passive load, ingrid-connected mode as well as in islanded mode.

There is ample space for improvement of the measurementtechnique, specially in the low-impedance, grid-connectedcase. Techniques such as impulse or harmonic response couldbe adapted to the dynamic phasor polar representation. Im-proved techniques, however, need to maintain compatibilitywith frequency varying microgrids.

Besides the application of stability criteria, future researchcould include different system topologies: Other load types,i.e. machines and power electronic converters, could be im-plemented. The impedance measurement of the sources couldbe conducted in series voltage perturbation to increase themeasurement accuracy. Furthermore, other injection devices,like a wound rotor induction machine could be investigated.

ACKNOWLEDGMENT

This work was partially supported by the Wisconsin Elec-tric Machines and Power Electronics Consortium at the Uni-versity of Wisconsin-Madison. This material is also basedupon work supported by the National Science Foundationunder Grant No. SEP 1230751. Any opinions, findings, andconclusions or recommendations expressed in this material arethose of the author(s) and do not necessarily reflect the viewsof the National Science Foundation.

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