usf smart grid power systems - stability analysis of two parallel...

1

Stability Analysis of Two Parallel Converters withVoltage-Current Droop Control

Yin Li, Student Member, IEEE and Lingling Fan, Senior Member, IEEE

Abstract—Voltage-current (V-I) droop control has been pro-posed for real and reactive power sharing. Compared withthe conventional droop control using real power/frequency andreactive power/voltage, it enhances stability since adding a V-Idroop is equivalent to providing a resistor. However, for a systemwith parallel converters with V-I droops, oscillations are observedwhen small droop coefficients are adopted. In this paper, multi-input multi-output (MIMO) model of the system is derived infrequency domain. This model is based on dq-reference frame andconsists of converter control and power network model, whichis represented by an admittance matrix. Linear system analysisis carried out to identify the root causes of oscillations in bothgrid-connected mode and autonomous mode. Analysis results arevalidated by simulation results of detailed model-based systemsbuilt in Matlab/Simpowersystems.

Index Terms—V-I droop, current sharing, MIMO model, droopcoefficient, oscillations

I. INTRODUCTION

MORE and more distributed energy resources (DERs)are integrated in microgrids. The conventional droop

control achieves real and reactive power sharing by introducingadditional feedback loops, e.g., power-frequency and VAr-voltage droop. Such type of droops (ω = ω∗−m(P−P ∗), E =E∗ − n(Q − Q∗)) is adopted in [1], [2] where the converterworks as a current source. For converters operating as avoltage source (in voltage/frequency control mode), frequency-power droop (P = P ∗ − m(ω − ω∗)) is adopted [3], [4].As a variation of power-frequency (P − f ) droop, power-angle (P − δ) droop can eliminate the frequency deviation[5], [6]. For the pure resistive network, P − V droop is usednormally [7]. Nonetheless, the power control block in theconventional droop control has a low bandwidth to cause theslow dynamic problem regardless increasing or decreasing thedroop coefficients [8]–[11].

While power-frequency droop can achieve power sharingaccording to the droop coefficients, it is known that E − Qdroop based reactive power sharing is inaccurate due to lineimpedance [12].

To achieve fast dynamics and accurate real power andreactive power sharing, V-I droop is proposed in [8], which issuitable for microgrids with small inertia DERs and frequentload variation. It utilizes the function of output current (E∗d =E0−miLd, E∗q = 0−niLq) to adjust a DER’s output voltagereferences. In a microgrid consisting of multiple convertersequipped with V-I droops, real and reactive power sharing is

Y. Li and L. Fan are with Dept. of Electrical Engineering, Univer-sity of South Florida, Tampa FL 33620. Emails: [email protected],[email protected].

achieved through dq-axis current sharing. The current sharingis inversely proportional to the ratio of droop coefficients.Larger droop coefficients, mk or nk, may lead DERk’s voltageblow the range under heavy loading conditions. On the otherhand, small droop coefficients may lead to inaccurate powersharing according to [8].

Although V-I droop control achieves faster dynamics, for asystem with multiple converters, there are possible oscillationissues, which have not been identified in the literature. [8] hasmultiple converters with V-I droop, but the oscillation issueis not observed. In this paper, stability investigation will beconducted to identify scenarios when oscillations may occur.

Stability issue is very common in multiple parallel-connected converters with droop control and have been studiedin the previous literature [11], [13]–[16]. Eigenvalue analysisbased on linearized systems is a popular approach to analyzethe stability issues in multiple converters systems [14]–[19].In such system models, converters are modeled as voltagesources with control loop dynamics and output filter dynamicswhile the switching dynamics are ignored [20]–[23]. Basedon the stability analysis, several methods have been proposedin the previous literature to enhance system stability suchas a controller based on the second derivative of the out-put capacitor voltage [16], arctan gradient algorithm concept[19], an adaptive decentralized droop controller [11], virtualcomplex impedance [14], and high gain angle droop control[15]. However, for parallel converters with V-I droops, stabilityissues have not been identified and studied. In our research,oscillation issues are observed for a system with parallelconverters with different V-I droop coefficients. Different fromthe conventional droop control, the stability issue of V-I droopis caused by the smaller droop coefficients.

The objective of this paper is to derive the linear modelof a system with parallel converters with V-I droop and findthe root causes of oscillations. The paper is organized asfollows. The topology of the microgrid system with V-I droopand the steady-state analysis will be introduced in Section II.Section III presents the MIMO model in frequency domainand linear analysis of the MIMO model. Section IV validatesthe linear analysis using detailed model simulation resultsvia MATLAB/Simpowersystem. Section V gives simulationresults for the CIGRE microgrid benchmark test system.Section VI concludes this paper.

II. THE MICROGRID WITH V-I DROOP

A. Topology of the circuitIn the microgrid with V-I droop, the global positioning sys-

tem (GPS) signals are required to keep the system frequency

This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.The final version of record is available at http://dx.doi.org/10.1109/TPWRD.2017.2656062

Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

2

VSC

1

VSC

2

PCCR=100mΩ

RL

C=500uF

i1

i2

+

-

+

-

iL1

iL2 R

Closed at 0.7s

Open at 1.5s

1.61+j0.81Ω

4.03Ω

VPCC (rated)=110V

L-L

VDC=400V

VDC=400V

m2=0.4

n2=3

iPCC

LL=50uH

LL=50uH

L=10mH

E1 (rated)=110Vm1=0.2

n1=1.5

E1 (rated)=110V

R=100mΩ L=10mH

C=500uF

Fig. 1: Two DERs support one load through parallel VSCs.

as constant. The constant frequency, ω0, is used to synchronizeall DER voltages and currents in the same dq-frame (d+ jq).It is assumed that for the initial condition, VPCC is alignedat the d-axis (VPCCd = VPCC , VPCCq = 0). In this paper,a microgrid consisting of two DERs and one load is used toanalyze the characteristics of V-I droop control. Each DERcontains a voltage source converter (VSC) and an RLC filter.Two VSCs are controlled in coordination by V-I droop control.There is a pure inductive transmission line (LL) between DERand the point of common coupling (PCC). The reactance XL

of the line is negligible compared with droop coefficients.

B. Steady State Analysis

Assume that the two converters (including the RLC filter)are connected to the PCC bus through an RL impedance Rk+jXk, k = 1, 2. The complex power injected to the load or thegrid through the PCC bus is notated as Sk, k = 1, 2. Let thed-axis be aligned with the PCC voltage space vector (vd =|−→v PCC |). Then the complex power expression is

Sk =3

2vd(iLdk − jiLqk) =

3

2(vdiLdk − jvdiLqk) (1)

Therefore,

Pk =3

2vdiLdk (2a)

Qk = −3

2vdiLqk (2b)

Therefore, the d-axis current sharing determines the real powersharing while the q-axis current sharing determines the reactivepower sharing.At steady-state, the converter voltage, PCC voltage and thecurrent have the following relationship.[

Rk −Xk

Xk Rk

] [iLdkiLqk

]=

[EkdEkq

]−[vd0

](3)

If we apply the V-I droop and assume the droop coefficients aremk for d-axis and nk for q-axis, then we have the followingrelationship.[

EkdEkq

]−[vd0

]=

[E0

0

]−[mk 00 nk

] [iLdkiLqk

]−[vd0

](4)

⇒[Rk +mk −Xk

Xk Rk + nk

] [iLdkiLqk

]=

[E0

0

]−[vd0

](5)

If we assume that mk Rk, nk Rk, mk Xk, andnk Xk, then we will have the following relationship:

m1iLd1 = m2iLd2 (6a)n1iLq1 = n2iLq2 (6b)

Therefore, the real power sharing is proportional to 1/mk

and the reactive power sharing is proportional to 1/nk. Theassumptions are that mk Rk, nk Rk, mk Xk, andnk Xk. If we have a resistive network or Rk Xk, thenthe real power sharing is according to 1/(Rk + mk) and thereactive power sharing is according to 1/(Rk + nk).

If the droop coefficients mk and nk are comparable withRk or Xk, then we cannot obtain accurate real/reactive powersharing. On the other hand, large droop coefficients may leadconverter voltages drop below the range during heavy loadconditions.

C. Control loop design

V-I droop control is based on voltage/frequency (V/F )control where the output voltage references are sent to theouter loops and frequency is fixed as shown in Fig. 2.

1) Inner current loop: For the voltage source converter, theblock diagram shown in Fig. 3 is normally used to design thePI controllers in the inner current loop [24].

The plant model is represented by 1Ls+R where ud = Vtd−

Ed − ωLiq and uq = Vtq − Eq + ωLid (Vtd and Vtq arethe converter output voltages. Ed and Eq are the capacitorvoltage.). Note that in Fig. 2, the feedforward components areadded back to uid and uiq to generate the references for theconverter voltages. Kip + Kii

s is the controller. Kip and Kii

are the proportional gain and integrator gain which can bedesigned using (7) [24].

Kip =L

τi,Kii =

R

τi(7)

where τi is the inner loop time constant.Based on Fig. 3 and Eq. (7), the closed-loop transfer

function is as follows.

Gic =idi∗d

=Gio

1 +Gio=

1

τis+ 1(8)

2) Outer voltage loop: Fig. 4 shows the block diagram forthe outer loop to design the gains of PI controller in the outerloop. 1

τis+1 , the closed-loop transfer function from the currentreference to the current measurement, is included in the plantmodel.

The gains Kvp and Kvi can be calculated using the follow-ing equations [24], [25] .

Kvp = Cωc =C√τiτv

, Kvi =Kvp

τv(9)

where ωc is the cut-off frequency and τv is the outer loop timeconstant.



3

+

-iq

*

iq

+

-

id*

id

PI

PI

+

-

++

ud

uq

Vtd*

Vtq*

dq

abc

Vta*

Vtb*

Vtc*

iLq

ωL

ωL

+

-

+

-

Ed*

PI

PI

+-

++

Ed

Eq

Eq*

iLd

θ=ω0t

(GPS)

ωC

ωC

+

+

E0

+

-m

-n

iLd

iLq

+

+

0

+

+

Ed

+

Eq

Fig. 2: Control block diagram for V-I droop.

id*(A)

+

-

Kip+Kii/siq*(A)

+

-

Kip+Kii/s1

Ls+Rid (A)

iq (A)

ud

uq

a)

b)

(V)

(V)

1

Ls+R

Fig. 3: The block diagram for the inner loop transfer function.

Ed*(V)

+

-

Kvp+Kvi/sEq*(V)

+

-

Kvp+Kvi/s1

τi s+1Ed (V)

Eq (V)

uvd*

uvq*

a)

b)

(A)

(A)

1

τi s+1

1

C s

1

C s

Fig. 4: The block diagram for the outer loop transfer function.

3) Droop control loop: The voltage/current droop generatesa voltage reference based on current feedback. After com-bining three layers of control loops, the whole control blockdiagram of V-I droop can be presented by Fig. 2. The droopcontrol is different from the one implemented in [8] in twoaspects. 1) Transformers in the system are not consideredand in turn the droop formulation (XT and RT ) are also notconsidered; 2) the considered droop functions are linear, while[8] considers piecewise linear ones.

III. STABILITY ANALYSIS

A. Linear Model Derivation

1) Linear model for converter control: To conduct linearanalysis, a linear model of the study system needs to bederived. The system can be separated into three parts, VSCwith V/F control, the transmission network, and the droopcontrol.

The block diagram of the controlled-frequency VSC shownin Fig. 5 can be used to represent the dynamics of the VSC

with V/F control. Compared with Fig. 2, Fig. 5 includes theRLC filter between VSC and the transmission line as a partof dynamics of VSC. The dynamics of RCL filter is presentedin both the inner loop and outer loop plant models, 1

Ls+R

and 1Cs . 1

Ls+R has been combined with the current feedbackcontrol and feedforward elements. The resulting block fromi∗d to id is a simple first-order unit (8).

There are totally four inputs (voltage references: E∗d , E∗q ,and load current measurements: iLd, iLq) and two outputs (thecapacitor voltages: Ed and Eq) for each VSC with an RLCfilter. For the system with two converters, there are 8 inputsand 4 outputs.

+

-

1

τis+1

uvd*

(A)

Kvp s+Kvi

s

id

(A)

1

Cs

Ed (V)

+ 1

τis+1

uvq*

(A)

Kvp s+Kvi

s

iq

(A)

1

Cs Eq (V)

-Eq

*

(V)

Ed*

(V)

Ed (V)

Eq (V)

+

+

-

+

+

iLd (A)

ωC

ωC

+

iLq (A)

id*

(A)

iq*

(A)

-+

-

+

ωC

ωC

-

+

Fig. 5: The block diagram of a VSC with an RLC filter.

2) Transmission network: The dynamics of the transmis-sion network can be represented by a multiple-input-multiple-output (MIMO) matrix. VSCs are considered as two voltagesources. The transmission network is shown in Fig. 6. A pureresistive load is assumed for the load. Based on the circuitshown in Fig. 6, the relations between the voltages E1, E2 andthe currents in abc frame can be derived using superposition:iL1 = E1

1Rlo‖LLs+LLs

− E2

(1

Rlo‖LLs+LLsRlo

Rlo+LLs

)iL2 = E2

1Rlo‖LLs+LLs

− E1

(1

Rlo‖LLs+LLsRlo

Rlo+LLs

)(10)

Since the control loop is modeled in dq frame, (10) shouldalso be converted to the dq frame by replacing s using s+jω,where ω = 377 rad/s. This is due to the fact that a spacevector in the abc-frame is related to a vector in the dq-frame



4

RloE1

iL1 iL2

AC E2 AC

LL LL

Fig. 6: The system without the VSC dynamics.

as follows.−→f abc = fdqe

jωt. (11)

After the conversion, the transfer function will be separatedinto real term and imaginary terms.

iL1diL1qiL2diL2q

=

G21r −G21i G22r −G22i

G21i G21r G22i G22r

G22r −G22i G21r −G21i

G22i G22r G21i G21r

︸︷︷︸

G2

E1d

E1q

E2d

E2q

(12)

where ω is the nominal frequency, 377rad/s. The transferfunctions, G21r,i and G22r,i, are expressed in (13).

(12) illustrates that the inputs are four voltages, Eid,q , andoutput are four currents, iLid,q , which will be sent back to thefirst part, VSC with control loop.

3) Overall Block Diagram: When the output currents areobtained, the references of the output voltages can be calcu-lated using the outmost droop control loop shown in Fig. 2.For two VSCs, the no load voltage, E0, is the same while thedroop coefficients are different. Therefore, the inputs of powersharing part are four currents and outputs are four referencesof voltages. Finally, Fig. 7 shows the combination of threedynamic parts of the microgrid system with V-I droop. G1

presents the dynamic of VSCs with control loop, G2 is thematrix to show the dynamic of load and transmission lines,and G3 illustrates the power sharing or the droop control loop.

B. Linear Analysis

1) Autonomous mode: Based on Fig. 7, the map of polesand zeros of the microgrid with V-I droop can be plotted usingControl Design Toolbox in MATLAB. The locations of thedominant poles can be found if specific droop coefficients areprovided. The m and n values are all based on physical unit(V/A). If we keep m1 = 0.4 and m2 = 0.8 as constantsand increase n1 (from 0.01 to 1) and n2 (from 0.02 to 2), themovement of dominant poles can be found easily after mergingthem in one plot (shown in Fig. 8). With an increasing nk,the dominant poles are further away from the imaginary-axis,so the dynamic response becomes faster and damping ratiobecomes larger.

Fig. 9 shows the movement of dominant poles with increas-ing m1 (from 0.004 to 0.4) and m2 (from 0.008 to 0.8) whilen1 = 1 and n2 = 2.

In the above cases, although the dominant poles will bemore closed to the imaginary-axis with decreasing mk or nk,they are still in the left half plane (LHP). In another word,either of small mk or nk may cause the oscillation issue, but

the system is always stable. However, if both of mk and nkare selected as very small values, the system will be unstablebecause of the locations of the dominant poles as shown in Fig.10a. The only exception is that when m1 = m2 and n1 = n2,the system is stable as shown in Fig. 10b.

2) Grid-connected mode: Fig. 8-Fig. 10a show the move-ment of the eigenvalues for the system in the autonomousmode. To check if there are stability issues for a system ingrid-connected mode, the following test system shown in Fig.11 is examined, where the two converters not only serve a loadbut also connected to the grid through a line with inductanceLG.

The same method is used to derive a new G2, the dynamicsof the grid and load. Because the oscillation only happens withvery small droop coefficients, n1 and n2 are selected as 0.01and 0.02 to make the system have stability issue. Short circuitratio (SCR) is a good factor to decide if the grid is strong orweak. The SCR can be changed by changing the inductance ofthe transmission line connected to the grid, LG. Fig. 12 showsthe movement of the dominant eigenvalues corresponding tothe variation of SCR (from 0.5 to 5). According to Fig. 12,the damping ratio of the dominant eigenvalue becomes largerwith larger SCR.

Remarks: Based on the above analysis, we identified theroot causes of oscillations are due to the following aspects.Small values for m and n, generally, cause poles to be locatedcloser to the imaginary axis, therefore decreasing the stability,and showing poorer damping ratios (see Figs. 8 and 9); 2) Ifthe mk and/or nk values are different from each other, somepoles are in the RHP (see Fig. 10); 3) Weaker grids also showpoorer damping ratios; stronger grids show better dampingratios (see Fig. 12).

IV. SIMULATION VALIDATION: THE TWOPARALLEL-CONVERTER CASE

In this section, a detailed model was designed to verify thecapability of V-I droop control’s power sharing as well as theeffect of parameters on stability.

The test system is the autonomous system in Fig. 1. Thedetailed model including power electronic switching detailsand controls shown in Fig. 2 was built and simulated in MAT-LAB/Simpowersystems. The parameters are listed in Table I.Note that the reactance from the converter to the PCC voltageis 0.0038 Ω. Therefore, m and n should be 0.0038 to haveaccurate power sharing. The gains of the PI controllers arelisted in Table II. The gains are based on physical units.

In Section III, the effect of droop coefficients on themovement of the dominant poles has been presented in Fig. 9and Fig. 8. Both mk and nk can affect the system stability. Asmentioned in Section III, the load was considered as a pureresistive load (Rlo = 1.34Ω) when studying the stability issue.

Three cases with different sets of droop parameters weresimulated to verify this stability issue based on the detailedmodel. The values of mk and nk are shown in the caption ofFig. 13.

The dynamic event is designed as follows. At the initialcondition, two DERs support one resistive load (rated power:



5

G21r =L3Ls

3 + 3L2Rlos2 + (L3

L + ω2 + 2LLR2lo)s+ L2

LRloω2

G21D

G21i =L3Lωs

2 + 2L2LRloωs+ (L3

Lω3 + 2LLR

2loω)

G21D

G22r =L4LRlos

4 + 4L3R2los

3 + 5L2LR

3los

2 + 2LR4los− (L2

LR3loω

2 + L4LRloω

4)

G22D

G22i =L4LRlos

4 + 4L3LR

2los

3 + 5L2LR

3los

2 + 2LR4los− (L2

LR3loω

2 + L4LRloω

4)

G22D

where

G21D = L4Ls

4 + 4L3Rlos3 + (2L4ω2 + 4L2

LR2lo)s

2 + 4L3LRloω

2s+ (L4Lω

4 + 4L2LR

2loω

2)

G22D = L6Ls

6 + 6L5LRlos

5 + (3L6Lω

2 + 13L4LR

2lo)s

4 + (12L5LRloω

2 + 12L3LR

3lo)s

3

+ (3L6Lω

4 + 18L4LR

2loω

2 + 4L2LR

4lo)s

2 + (6L5LRloω

4 + 12L3LR

3loω

2)s+ (L6Lω

6 + 5L4LR

2loω

4 + 4L2LR

4loω

2) (13)

G1

8x4

G2

4x4

E1d

E1q

E2d

E2q

iL1d

iL1q

iL2d

iL2q

E1d*

E1q*

E2d*

E2q*

E1d*

E1q*

E2d*

E2q*

-m1 0 0 0

0 -n1 0 0

0 0 -m2 0

0 0 0 -n2

G3

Fig. 7: The block diagram of the linear model of the systemwith two VSCs with V-I droop.

Real Axis (seconds −1)

Imag

inar

y A

xis

(sec

onds

−1)

Pole−Zero Map

−400 −300 −200 −100 0

−300

−200

−100

0

100

200

300

n increase

Fig. 8: Increasing n1 and n2 leads to the dominant polesmoving to the left half plane (LHP).

TABLE I: Circuit Parameters

Parameter ValuesDC voltage 400 VNo-load voltage, E0 102 VRated voltage on load 110 V (rms, L-L)Nominal frequency, ω0 377 rad/sSwitching frequency 3060 HzR of RLC filter 100 mΩL of RLC filter 10 mHC of RLC filter 0.5 mFLL of transmission line 0.05 mH (0.0038 Ω)

Pole−Zero Map


Imag

inar

y A

xis

(sec

onds

−1)

−400 −300 −200 −100 0

−300

−200

−100

0

100

200

300

mincrease

Fig. 9: The values of m1 and m2 affect the continuousmovement of dominant eigenvalues.

TABLE II: Gains of PI Controllers

Loop Time constant Kp Ki

Inner 0.5 ms 20 200Outer 5 ms 0.316 63.25

6 kW). At 1.5 s, a pure resistive load (rated power: 3 kW)was added.

In the case study, load is modeled as impedances. Theimpedance of the load will be reduced to have a powerincrease. Due to the effect of the current controller, the currentsare kept constant at the moment when the step change isenforced. This will cause a sudden reduction in the PCCvoltage or load bus voltage.

A. Effect of the droop coefficients

The effect of droop coefficients are shown in Fig. 13. Incase (a), there is no oscillation issue with the chosen droopcoefficients. The voltages are within the ±4% range. If m1

and m2 are reduced to 0.004 and 0.008, real power showsoscillations as shown in (b). If n1 and n2 are reduced to0.01 and 0.02, then the reactive power shows oscillations.



6

Pole−Zero Map


Imag

inar

y A

xis

(sec

onds

−1)

−350 −300 −250 −200 −150 −100 −50 0 50

−300

−200

−100

0

100

200

300

(a)Pole−Zero Map


Imag

inar

y A

xis

(sec

onds

−1)

−350 −300 −250 −200 −150 −100 −50 0 50

−300

−200

−100

0

100

200

300

(b)

Fig. 10: (a) shows poles and zeros when m1 = 0.004, m2 =0.008, n1 = 0.01, and n2 = 0.02; (b) shows poles and zeroswhen m1 = m2 = 0.04, n1 = n2 = 0.04.

LG

E1

iL1 iL2

AC E2

LL LL

ACEgrid

ACRlo

Fig. 11: The grid-connected system without the VSC dynam-ics.

The oscillations in reactive power are caused by circulatingcurrents which are basically q-axis currents going back andforth between the DER units, and therefore increasing thestress on and losses in transmission lines.

B. Effect of the network resistance

Sensitivity analysis of the network parameters is conducted.The network parameters are varied. For the same droopcoefficients employed in case (c), instead of using a pureinductor LL to connect the converter after the RLC filter tothe PCC bus, the inductor was replace by an RL circuit wherer = 0.1Ω or r = 0.01Ω. It can be found from Fig. 14 that

Pole−Zero Map


Imag

inar

y A

xis

(sec

onds

−1)

−400 −300 −200 −100 0

−300

−200

−100

0

100

200

300

SCRincrease

Fig. 12: When a grid is connected, the value of SCR affectsthe continuous movement of dominant eigenvalues.

the network resistance plays a big role in oscillations. With ahigh R/X ratio at 26, oscillations will be suppressed. For a R/Xratio at 2.6, oscillations still exist. For a resistive network, evenwith very small droop coefficients, oscillations are not issues.

C. Sensitivity of SCR

As mentioned above, a strong grid can enhance the systemstability. This case was designed to check how the strengthof the grid affects the system stability. The strength of thegrid is dependent on SCR. Fig. 15(a) and Fig. 15(b) showthe effect of SCR on oscillations. A grid with SCR=1 isnormally considered as a weak grid while SCR=5 means thevery strong grid. It can be concluded that the strong grid canreduce oscillations even with small droop coefficients, whilethe weak grid case still shows obvious oscillations.

V. SIMULATION VALIDATION: THE CIGRE BENCHMARKTEST CASE

Simulation case study is also conducted for the CIGREbenchmark test case [26]. The system topology is shown inFig. 16. The ratings of DERs are shown in the figure. Ratedload real power levels are also shown. All loads are assumedto have 0.7 lagging power factor. The loads are modeled asconstant impedances. The network consists of cables withlarge R/X ratio. In the simulation model, the network dynamicsare ignored. Converters are modeled in the dq reference frame.The simulation results are shown in Fig. 17. A set of smalldroop coefficients is used in Case (a) while a set of large droopcoefficients is used in case (b). At t = 1 second, the load atDER2 is doubled by increasing its admittance to twice of itsoriginal value. The simulation results show that even with verysmall droop coefficients, there is no oscillation issue. This isdue to the resistive network of the CIGRE benchmark system.The droop coefficients were designed based on the power

ratings. For example, the rating real power of DER1 10 timesof that of DER4, so m4 = 10m1. The same method wasused to designed n. Fig. 17 illustrated the real and reactive



7

1 1.5 2 2.50

5

10

P (

kW)

Load

DER1

DER2

1 1.5 2 2.5−4

−2

0

2

4

Q (

kVA

r)

1 1.5 2 2.560

65

70

VR

MS (

V)

Time (s)

1 1.5 2 2.50

5

10Load

DER1

DER2

1 1.5 2 2.5−4

−2

0

2

4

1 1.5 2 2.560

65

70

Time (s)

1 1.5 2 2.50

5

10

Load

DER1

DER2

1 1.5 2 2.5−4

−2

0

2

4

Load

DER1

DER2

1 1.5 2 2.560

65

70

Time (s)

Fig. 13: (a) m1 = 0.09,m2 = 0.18, n1 = 1, n2 = 2; (b) m1 = 0.004,m2 = 0.008, n1 = 1, n2 = 2; (c) m1 = 0.09,m2 =0.18, n1 = 0.01, n2 = 0.02. The two straight lines in voltage plots indicate ±4% range.

1 1.5 2 2.5−4

−2

0

2

4

(a)

Q (kVAr)

with 0.1 Ω resistance

1 1.5 2 2.5−4

−2

0

2

4

(b)

with 0.01 Ω resistance

1 1.5 2 2.5−4

−2

0

2

4

(c)

Time (s)

without resistance

Fig. 14: Effect of the network resistance. (a) with 0.1 Ωresistance; (b) with 0.01 Ω resistance; (c) without resistance.

power shared by five DERs based on the relative droopcoefficients. Compared to case (b), case (a) shows a moreobvious oscillation mode with sufficient damping. For thisnetwork, due to the resistive network, there is no oscillationissue concerned.We would like to point out that small droop coefficientslead to inaccurate power sharing as observed in case (a).With increased droop coefficients, the power sharing are morealigned towards the desired power sharing ratio. Note that theP and Q presented in Fig. 17 are real power and reactive

1 1.5 2 2.5−3

−2

−1

0

1

2

3

(a)

1 1.5 2 2.5−5

−4

−3

−2

−1

0

1

2

Time (s)

(b)

SCR=1

SCR=5

(a)

(b)

Fig. 15: (a) Effect of SCR = 1 on reactive power; (b) Effectof SCR = 5 on reactive power. In both cases, m1 = 0.4,m2 = 0.8, n1 = 0.01 and n2 = 0.02.

power measured at the DERs. Fig. 17(b) shows that thereactive power sharing is not accurate. This may due to theassumption of shunt constant impedances as load models. Theshunt impedance reduces the equivalent Thevenin networkimpedance and causes inaccurate sharing when the droopcoefficients are close to the network resistance/reactance.Remarks: It is possible that oscillations occur when convertersare connected to PCC through inductive devices. Oscillations



8

4.5kW

3φ,

VPCC (rated)=400V

30kW

DER1

30kW45kW10kW

18kW

DER2DER3

10kW

20kW

DER5

4.5kW

3kW

DER4

6

7

8

9

1

3

5

2

4

Fig. 16: CIGRE microgrid test case. The load power factor is0.7.

are possible when multiple batteries or DERs are connectedin parallel to a PCC with small droop coefficients.

VI. CONCLUSION

This paper investigates the stability issues for a microgridwith parallel converters equipped with V-I droop control.Our paper’s investigation shows that oscillations are possiblewith small droop coefficients. To have accurate power shar-ing among converters, large droop coefficients are desired.However, large droop coefficients may lead voltages belowthe range under heavy load conditions. With small droopcoefficients, not only power sharing may not be accuratebut also there is possibility of oscillations depending on thenetwork characteristics. A linear frequency-domain model isderived in this paper to show the effect of droop coefficients onstability. The analysis results are validated by detailed model-based simulation conducted in Matlab/SimPowersystems.

ACKNOWLEDGEMENT

The authors wish to thank anonymous reviewers for provid-ing thorough reviews and detailed comments.

REFERENCES

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9

0.9 0.95 1 1.05 1.1 1.150

10

20

30

40

50

PD

ER

(kW

)DER2

0.9 0.95 1 1.05 1.1 1.150

10

20

30

40

50

QD

ER

(kV

Ar)

DER2

0.9 0.95 1 1.05 1.1 1.151.03

1.035

1.04

1.045

1.05

VD

ER

(pu

)

Time (s)

0.9 0.95 1 1.05 1.1 1.150

5

10

15

20

25(b)

DER1

DER2

DER3

DER4

DER5

0.9 0.95 1 1.05 1.1 1.150

10

20

30

DER1

DER2

DER3DER4

DER5

0.9 0.95 1 1.05 1.1 1.150.8

0.85

0.9

0.95

1

Time (s)

(a)

Fig. 17: (a) m = 0.001[1, 1, 3, 10, 3], n = 0.001[1, 1, 3, 10, 3]. (b)m = 0.02[1, 1, 3, 10, 3], n = 0.01[1, 1, 3, 10, 3]. The m and nvalues are in per unit (Sb = 10 kW and Vb = 400 V).

[23] T. L. Vandoorn, J. D. M. D. Kooning, B. Meersman, J. M. Guerrero, andL. Vandevelde, “Automatic power-sharing modification of p / v droopcontrollers in low-voltage resistive microgrids,” IEEE Transactions onPower Delivery, vol. 27, no. 4, pp. 2318–2325, Oct 2012.

[24] A. Yazdani and R. Iravani, Voltage-sourced converters in power systems:modeling, control, and applications. John Wiley & Sons, 2010.

[25] A. Tazay, Z. Miao, and L. Fan, “Blackstart of an induction motor inan autonomous microgrid,” in Power Energy Society General Meeting,2015 IEEE, Jul 2015, pp. 1–5.

[26] S. Papathanassiou, N. Hatziargyriou, and K. Strunz, “A benchmarklow voltage microgrid network,” in Proc. CIGRE Symp. Power Syst.Dispersed Gen., Dec 2005, pp. 1–8.

Yin Li received his B.S. in electrical engineering from University of SouthFlorida in May 2014. He is currently a Ph.D. student at USF SPS Lab. Hisresearch interests include microgrids and power electronics.

Lingling Fan received the B.S. and M.S. degrees in electrical engineeringfrom Southeast University, Nanjing, China, in 1994 and 1997, respectively,and the Ph.D. degree in electrical engineering from West Virginia University,Morgantown, in 2001. Currently, she is an Associate Professor with theUniversity of South Florida, Tampa, where she has been since 2009. She was aSenior Engineer in the Transmission Asset Management Department, MidwestISO, St. Paul, MN, form 2001 to 2007, and an Assistant Professor with NorthDakota State University, Fargo, from 2007 to 2009. Her research interestsinclude power systems and power electronics. Dr. Fan serves as a technicalprogram committee chair for IEEE Power System Dynamic PerformanceCommittee and an editor for IEEE Trans. Sustainable Energy.



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