impact of vsg parameters on transient performance of solar ... · dg are inverter terminal voltage...

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Impact of VSG Parameters on TransientPerformance of Solar PV based System

Rakesh Kumar Panda, Abheejeet Mohapatra and S. C. SrivastavaDepartment of Electrical Engineering, Indian Institute of Technology Kanpur, India 208016

Email: [email protected]; [email protected]; [email protected]

Abstract—The impact of Virtual Synchronous Generator(VSG) parameters on the transient performance of Solar Photo-Voltaic (SPV) based system has been analyzed in this paper. Inaddition, the variations in Kinetic Energy (KE) and PotentialEnergy (PE) components of a Structure Preserving EnergyFunction (SPEF) for the system under consideration have alsobeen analyzed for the same purpose. In order to do this, statespace model of the SPV system with VSG controller has beendeveloped. SPEF has then been derived with this model for aSPV source connected to a Synchronous Generator (SG) by acable with the nonlinear load. In order to analyze the transientbehavior of the system under 3φ bolted fault and induction motorswitching, a nonlinear Runge-Kutta 4th order method is used fornumerical simulation. Results from Real Time Digital Simulator(RTDS) show that the variations in KE and PE of SPEF can aidin assessing the transient behavior of SPV based system.

Index Terms—Solar PV, Virtual Synchronous Generator, En-ergy Function, Transient performance.

I. INTRODUCTIONIntegration of inverter interfaced SPV sources in the con-

ventional system has undesirable impact on system transientstability as compared to a conventional generator [1]. In atraditional generator, system transient stability is governed bythe sizable rotating mass of generators. The same is not true forSPV sources. To enhance system inertia of SPV based systemsfor improved transient stability, control methods like VSG [2],synchronverter [3], etc. have been proposed in the literature.Of these, VSG is more popular due to its easy implementation[2].

It is thus evident that the assessment of system transientstability of SPV based system with VSG is of paramount im-portance. Majority of literature deal with the issue of stability[4] and inertia contribution through VSG [2] for SPV basedsystems. However, to the best of the authors’ knowledge, noattempt has been made to analyze the impact of VSG param-eters on the transient behavior of SPV source based system inthe presence of system disturbance using the energy functionconcept. The benefit of doing so by the energy function methodis that it takes less time to give solution (degree of stability)and also provides additional information to design predictivecontrol for the system [5]. Energy function has been often usedto assess the transient stability of conventional power system[5]. However, to overcome the conservativeness of its resultsand consideration of a simple generator model without itscontrol circuit, Structure Preserving Energy Function (SPEF)of the system was first introduced in [6]. SPEF includes

detail modeling of a generator with its control circuit andalso considers models of load and transmission system. Useof SPEF for transient stability evaluation in the conventionalsystem can be seen in [7].

Thus, the major contributions of this paper are: first, toanalyze the impact of VSG parameters on the transient perfor-mance of SPV based system in the event of disturbances withvariations in SPEF. Second, the detailed state space modelof SPV source with VSG is derived for this purpose. SPVsource is assumed to be delivering maximum power to Pointof Common Coupling (PCC), regardless of system condition.Third, the derived state space model of VSG is used to obtainSPEF for transient behavior analysis. For robust transientperformance analysis under 3φ bolted fault and inductionmotor switching, a nonlinear load, which is a combinationof 3φ Induction Motor (IM) and ZIP load, is used.

II. STATE SPACE MODEL OF SPV SYSTEM

The focal point of the paper is to study the impact ofVSG parameters on the transient performance of SPV basedsystem during 3φ bolted fault and nonlinear load switching.Generally, a power source near the fault and load point (theSPV source with VSG in fig. 1) is modeled in detail for tran-sient stability analysis. Hence, detailed modeling of the VSGcontroller based SPV source is proposed. For simplicity, SG ismodeled using classical model. This section first reviews theoverall system structure, which is used for transient stabilityanalysis and, then, mathematical modeling of each componentis discussed.

SPV Source with VSG Controller

Transmission Line

SG

ZIP Load

Step Up transformer

11 V

rtf,xtf

22 V 33 V

CB1

Nonlinear Load

Bus 1 Bus 2 Bus 3

Fig. 1: Single line diagram of simulated system

A. System Overview

The single line diagram of the simulated system is shownin fig. 1. VSG based SPV source, and SG is supplying an IMand a ZIP load. IEEE type 1 governor/ turbine and IEEE typeST1 excitation system are used for SG to control active poweroutput and terminal voltage, respectively [8]. VSG controller978-1-5386-6159-8/18/$31.00 c© 2018 IEEE

Proceedings of the National Power Systems Conference (NPSC) - 2018, December 14-16, NIT Tiruchirappalli, India

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works for governor and excitation system action for SPVsource. The detailed control operation of VSG with associatedmathematical modeling is discussed next.

B. SPV based Source with VSG Controller [2]

The implementation structure of the SPV source is shown infig. 2. The energy storage unit is used primarily to imitate theKinetic Energy (KE) of the SG. During a disturbance, it aidsto store additional power supplied by SPV source. The inverterand its associated filter unit act as an interface between SPVsource and PCC.

SPV Source

Energy Storage

Inverter

rf , Xf

Idg

11 VdgdgE

Bus 1(PCC)

Switching Signal from

PWM

Fig. 2: Basic SPV integration with PCCThe inverter switching signal is decided by the VSG con-

troller, which is shown in fig. 3. To mimic the SG char-

VSG 1/s3-Φ Sine

PWMθ

Pm_dg

Pe_dg

f1

PIFrequency Detector

Power Calculator

V1

Idg

f1

QrefQ-V Droop

Q0V1 Qe_dg

ωdg

Qe_dg

Pe_dg

Switching signal to

the Inverter

Fig. 3: Basic VSG control Structure

acteristic, VSG block solves the swing equation using 4th

order Runga-Kutta method to decide the inverter power output.The mechanical power reference to the VSG block, Pm dg iscalculated by the MPPT algorithm, i.e., Pm dg = PMPPT .The real and reactive power injected at PCC by SPV sourceare given by

Pe dg =EdgV1xdg

sin(δdg − θ1) (1)

Qe dg =E2dgxdg

− EdgV1xdg

cos(δdg − θ1) (2)

where, Edg and δdg are inverter terminal voltage and as-sociated phase angle, respectively. V1 and θ1 are voltagemagnitude and associated phase at PCC. The xdg = xfis transfer reactance between SPV source and PCC. Edgis determined by reactive power control loop, which givesQref = Q0 − kq(V1 − V10) signal by taking V1 as an inputto the Q− V drop control. Q0 is the reactive power suppliedby SPV source at nominal voltage V10 and kq is the Q − Vdroop coefficient.

C. Proposed State Space Model of VSG ControllerDynamics of real power loop for VSG block can be written

in the form of swing equation as

δ̇dg = ωdg (3)Mdgω̇dg = Pm dg − Pe dg −Ddg(ωdg − ω0) (4)

where, ωdg is output frequency of VSG and ω0 is the nominalsystem frequency. Mdg and Ddg are virtual inertia and damp-ing coefficient of SPV source, respectively. On similar lines,the dynamics of reactive control loop can be derived as

Edg = (Q0 − kq(V1 − V10)−Qe dg)(kpdg +

kidgs

)(5)

where, kpdg and kidg are the PI parameters of the reactivecontrol loop. Since, SPV source is considered, the reactivepower support is zero, i.e. Q0 = 0. Dynamics of PCC voltageis ignored i.e. V̇1 = 0. Laplace inverse of (5) yields

Ėdg = −kpdgQ̇e dg − kqkidg (V1 − V10)− kidgQe dg (6)

From (2), Q̇e dg can be written as

Q̇e dg =2Edgxdg

Ėdg −V1 cos(δdg − θ1)

xdgĖdg

+EdgV1 sin(δdg − θ1)

xdgδ̇dg

(7)

On substituting δ̇dg from (3) and Q̇e dg from (7) in (6), thedynamics of reactive control loop can be expressed as

TdgĖdg=−kpdgV1sin(δdg−θ1)

xdgωdg+kidg

V1xdg

cos(δdg−θ1)

−kidgEdgxdg

−kidgkq(V1−V10)

Edg

(8)

where, Tdg = (1 + kpdg2Edgxdg

− kpdgV1xdg

cos(δdg − θ1)) 1Edg .

D. SG Model [9]

Classical model of SG is considered for transient sta-bility analysis. Hence, the quadrature axis voltage behindthe transient reactance, E

′

qsg is assumed constant and directaxis and quadrature axis reactances are considered equal, i.e.xdsg = xqsg . Dynamics of SG can be described by thefollowing swing equation

δ̇sg = ωsg (9)Msgω̇sg = Pm sg − Pe sg −Dsg(ωsg − ω0) (10)

where, Msg and Dsg are the inertia and damping coefficientof SG, respectively. δsg is load angle of SG and ωsg is SGfrequency. Pm sg and Pe sg are the mechanical input andelectrical output of SG, respectively.

E. Nonlinear Load Model [9]

The nonlinear load in fig. 1 is a combination of a static ZIPload and an IM. Hence, total real and reactive power loads arePL = PZIP + PIM and QL = QZIP + QIM , respectively.The ZIP load model can be expressed as

PZIP=P0

[aZ

(V2V20

)2+aI

(V2V20

)+aP

]=fp(V2)

QZIP=Q0

[bZ

(V2V20

)2+bI

(V2V20

)+bP

]=fq(V2)

(11)

where, aZ , aI , aP are constant impedance, current and powercoefficients, respectively of real power load. bZ , bI , bP are


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similar coefficients for reactive power load. P0 and Q0 arerespective nominal values at nominal voltage V20.

IM dynamics [10], considering both electrical and mechan-ical transients can be written as

Ė′

dim=1

T′0im

[E′

dim+(Xim−X′

im)Iqim]+(ωrim−ω0)E′

qim (12a)

Ė′

qim=1

T′0im

[E′

qim+(Xim−X′

im)Idim]−(ωrim−ω0)E′

dim (12b)

ω̇rim=ω0

2Him(Te im−Tm im) (12c)

The standard notations defined in [10] are used in (12).The real and reactive powers consumed by the IM can beexpressed as PIM = 1.5 (Vd2Idim + Vq2Iqim) and QIM =1.5 (Vq2Idim − Vd2Iqim).

F. Transmission Line and Transformer Model

The transmission line is modeled using lumped parameterπ representation and the transformer is replaced by its leakageimpedance i.e ztf = rtf+jxtf . Real and reactive power flowsin a line between bus i and bus j are

Pij = ViVj(Gij cos θij +Bij sin θij)

Qij = ViVj(Gij sin θij −Bij cos θij)(13)

where, Gij and Bij are real and imaginary parts of (i, j)th

element of system admittance matrix. θij = θi−θj and Vi andVj are the ith and jth bus voltage magnitudes, respectively.

III. STRUCTURE PRESERVING ENERGY FUNCTION [9]

The objective of this section is to derive a SPEF forthe impact of VSG parameters on transient stability analysisof the system in fig. 1. The assumptions, considered forderivation of the energy function, are a) SG is replaced byclassical model, b) governor of SG does not respond duringfault period, c) for SPV source, insolation and temperatureremains unchanged during the transient period, d) post-faultand pre-fault operating points are same. The derived SPEFincludes the load dynamics, the effect of VSG controller aswell as network dynamics. The energy function defined forthe system in fig. 1 is

WTot(δ, ω,E, V, θ, t) =WKET +WPET (14)

where, WTot is the total energy of the system and WKET ,WPET are the total KE and total Potential Energy (PE),respectively.

A. Total Kinetic Energy WKETUsually, KE is associated with the rotating mass of the

machine. System in fig. 1 has two rotational masses i.e. SGand IM. However, to improve the KE of the system, VSGcontroller is used for SPV source, which introduces virtualKE into the system. Therefore, WKET can be written as

WKET (ω) =WKE sg +WKE dg +WKE im (15)

where, WKE sg = KE associated with SG = 12Msgω2sg ,

WKE dg = KE associated with DG = 12Mdgω2dg and

WKE im = KE associated with IM = 12Mimω2imS. Value of

WKE sg can be obtained by integrating left hand side of (10),after multiplying with δ̇sg . By adopting similar procedure to(4) and (12c), corresponding values of WKE dg and WKE imcan be obtained. S = 1, when CB1 is closed in fig. 1 andS = 0, when CB1 is opened. Energy associated with dampingis not considered here as it is not significant and appropriatejustification of not considering this is given in [11].

B. Total Potential Energy, WPETPE of a system is sum of integrals of the accelerating

power associated with the power sources and sum of energyassociated with network element. PE of the SPV source, SGand IM can be obtained by integrating right hand sides ofthe (4), (10) and (12c), respectively. By integrating (11) and(13), PE of nonlinear load and transmission network can beobtained. More details can be seen in [9]. The total PE, WPETcan be written as

WPET (δ, E, V, θ, t) =WPE sg +WPE dg +WPE im

+WPE ZIP +WPE tl(16)

The PE associated with SG, i.e. WPE sg can be stated as

WPE sg =WPE sg1(δ) +WPE sg2(δ, E, V, θ) (17)

where, WPE sg1 = PE due to Pm sg = −Pm sg(δsg − δsg0)and WPE sg2 = PE due to xsg . Appropriate expression ofWPE sg2 can be seen in [9]. The PE associated with DG,i.e. WPE dg can be expressed as

WPE dg=WPE dg1(δ)+WPE dg2(δ,E,V,θ)+WPE dg3(E) (18)

where,

WPE dg1=PE due to Pm dg=−Pm dg(δdg−δdg0) (19a)WPE dg2=PE due to xdg

=1

2xdg

[(E2dg+V

21 −2EdgV1cos(δdg−θ1))

−(E2dg0+V 210−2Edg0V10cos(δdg0−θ10))]

(19b)

WPE dg3=PE due to Edg=[−kpdgks1sin(δdg−θ1)ωdg+kidgks1cos(δdg−θ1)−

ks2E2dg+ks3ln(Edg)

]−[−kpdgks10sin(δdg0−θ10)ωdg

+kidgks1cos(δdg−θ1)−ks2E2dg+ks3ln(Edg0)]

(19c)

and ks1 =V1Edgxdg

, ks2 =kidgxdg

, ks3 = kidgkq(V1−V10), ks10 =V10Edg0xdg

. The PE associated with ZIP load, i.e. WPE ZIP is

WPE ZIP =WPE ZIPP (t) +WPE ZIPQ(V ) (20)

where, WPE ZIPP = PE due toPZIP =∫ tt0fp(V2)

(dθ2dt

)dt

and WPE ZIPQ = PE due toQZIP =∫ V2V20

fq(V2)(

dV2V2

).

On similar lines, PE due to IM [11] can be written as

WPE IM=

(Tm imδim+

1

2XimIdim+

1

2X

′

imIqim

)∣∣∣∣δim,Idim,Iqimδim0,Idim0,Iqim0

(21)

while the same for transmission line and transformer are

WPE tl =WPE tlB +WPE tlG (22)


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where,

WPE tlB=PE due to Bij

=−12

n∑i=1

n∑j=1

Bij(ViVjcosθij−Vi0Vj0cosθij0)

WPE tlG=PE due to Gij(Path dependent)

=

n∑i=1

n∑j=1

Gij

∫ViVjsinθijd(θi+θj)

C. Significance of Variations in SPEF

It can be seen from (15), (18) and (19c) that VSG param-eters, i.e. virtual inertia Mdg , reactive power loop parameterskidg and kpdg (and virtual active power damping coefficientDdg when considered) have direct impact on KE and PEfunctions. Conventionally, the impact of the control circuitparameters of a generator has not been considered in theenergy function evaluations [9]. However, here, this has beenconsidered and is one of the contributions of this paper. Also,it is to be noted that the transient behavior of a systemduring a disturbance can be analyzed by observing variationsin the system states’. However, the reliability of such ananalysis depends on which state is being observed. On theother hand, variation in the total KE and/ or PE of a systemduring a disturbance can directly aid in analyzing the system’stransient behavior as a large variation in KE and PE impliesthe system is prone to instability if the disturbance is allowedto persist. Also, if the time taken for KE or PE to settle at post-disturbance value is large, then it indicates that the system hasless damping. SPEF thus provides an accurate energy profileconsidering the effect of all system components along withthe control parameters of VSG.

IV. SIMULATION RESULTS

System data for the considered system in fig. 1 is givenin table I. A 1km long, 70mm2 three core XLPE insulatedcable is used as the transmission line (associated parametersare in [12]). At steady state, SPV power output at PCC is375kW . SG accounts for the mismatch power. For SPEFevaluations, bus voltage phasors obtained by simulating thesystem in RTDS. The measured bus voltages for a specifictest scenario are used in numerical simulations. Runge-Kutta4th order method is used to solve the differential equations fornumerical simulation. To obtain the initial conditions, Newton-Raphson power flow is solved. The transient behavior of thesystem is analyzed under the following conditions.

A. Case 1: 3φ Bolted Fault

A 3φ bolted fault at bus 1 is simulated at 2s for 2 cycles,i.e. 40ms. Also, as can be seen from (15), Mdg has thedirect impact on system’s KE, and, hence, the effect of anychange in KE is quite straightforward (higher is Mdg , largeis system’s KE and thus improved system inertia) and hasnot been discussed further. What is rather interesting to see isthe effect of change in kidg , kpdg and Ddg on SPEF. Table II

TABLE I: Data for System in fig. 1

SGParameter Values Parameter ValuesVb sg 6.6kV Sb sg 1MVAP0 sg 1.0pu ω0 314.159rad/sHsg 2.775s Xd 1.90pu

X′d 0.314pu X

′′d 0.280pu

Xq = X′q 0.770pu X

′′q 0.375pu

T′′d 0.0348s T

′′q 0.0.0346s

SPV Filter detailsEdg 0.415kV rf 0.005Ω

PMPPT 0.375MW Lf 0.4mHJ0 56.3kgm2 Cf 300µFD0 17pu Transformerθpcc 0 Turns 415V/6.6kVVb pcc 6.6kV xtf 0.1pu

DC side LoadVDC 0.85kV P0 0.4MWCDC 5mF Q0 0.2MVAR

TABLE II: Test Conditions

Scenario Ddg(pu) kidg kpdg

S10.17

20 0.0251.717

S2 1710

0.0252050

S3 17 200.050.025

0.0025

Fig. 4: SPV states in Case 1 (S1) (a) δdg (b) Edg (c) ωdg (d)state plane

(a) PE due to electri-cal power

(b) PE due to me-chanical power

(c) PE due to Edg

Fig. 5: PE of VSG based SPV in Case 1 (S1)


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Fig. 6: (a) Total KE (b) Total energy of system in Case 1 (S1)

shows the various test scenarios in which the system’s transientbehavior is analyzed.

Ddg has not been explicitly considered in SPEF. However,it has direct impact on damping out the virtual rotor angleoscillation δdg , which improves the transient stability boundary[13]. To observe the impact of Ddg , system is tested in S1 forthe stated 3φ bolted fault. Associated variations are shown inFigs. 4-6. As Ddg increases, the states of VSG based SPVreach the pre-fault operating point quickly after the 3φ faultas in fig. 4. Fig. 5 shows that the oscillation in PE is higherfor lower values of Ddg . Fig. 6 shows that KE oscillations dieout quickly as Ddg increases.

Fig. 7: SPV states in Case 1 (S2). (a) δdg (b) Edg (c) ωdg (d)state plane

Any change in kidg and kpdg affects the virtual inverterterminal voltage Edg , which in a way affects the VSG states.In S2, by decreasing kidg while keeping kpdg constant, VSGbased SPV states achieve their corresponding steady stateoperating values quickly as shown in fig. 7. The associatedKE and total energy variations due to different kidg values areshown in fig. 8.

In S3, an increase in kpdg leads to slower variation in VSGbased SPV states with more settling time to reach the pre-faultoperating values as shown in fig. 9. Associated variation in PEdue to different kpdg values are shown in fig. 10. A similarinference can be drawn about PE for different kpdg values.

Fig. 8: (a) Total KE (b) Total energy of system in Case 1 (S2)

Fig. 9: SPV states in Case 1 (S3). (a) δdg (b) Edg (c) ωdg (d)state plane

B. Case 2: Induction Motor Switching

The event of IM switching has been studied for all scenariosin table II. However, due to limitation of space, S1 withDdg = 17pu has been discussed here. Initially, CB1 in fig.1 is OFF. SPV and SG feed power to ZIP load in steady state.At 2s, 3φ IM (parameters in [14]) is switched ON by closingCB1. IM operates in lock free mode. Due to IM switching, alarge transient is experienced in the system. Pre-equilibriumpoint of the VSG states, i.e. δdg and Edg change due toIM switching. However, the power injected to PCC remainsunchanged by SPV source. This is because the system voltagelevel decreases, and hence VSG adjusts its states, and thus,operates at new equilibrium point, which is shown in fig. 11a.

Similarly, to balance the real power, SG provides the

(a) PE due to Elec-trical Power

(b) PE due to Me-chanical Power

(c) PE due to Edg

Fig. 10: PE of VSG based SPV in Case 1 (S3)


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(a) State plane of VSG basedSPV

(b) State plane of SG

Fig. 11: State plane of sources in Case 2

mismatch power. Therefore, the SG states i.e. δsg and ωsgalso change, which after oscillations settle at new steady statevalues as in fig. 11b. The associated variations in PE and KEduring IM switching are shown in figs. 12-13. It has beenobserved from these variations that a higher Ddg with lowerkidg and kpdg parameters in VSG lead to improved transientbehavior.

Fig. 12: PE in Case 2

Fig. 13: Total KE in Case 2

V. CONCLUSION

In this paper, the impact of the change in VSG parameterson transient stability of an SPV based system has beenanalyzed through the novel use of a SPEF. To this end, astate space model of the SPV source with VSG controller isderived. Using the derived model, an energy candidate functionin SPEF is developed considering the VSG control parameters.By including the energy function of the VSG controller withthe energy function of other system components, the energyprofile of the overall system is analyzed. The major advantage

of this analysis is that it includes the control parameters ofthe VSG controller, which further helps to study the properenergy profile of the system during the transient period in theevent of a disturbance. Simulation results from RTDS suggestthat by choosing adequate control parameters of the VSGcontroller, stability of the system can be enhanced. DuringIM switching, reactive power consumption leads to change inthe stable operating point of SPV source DG with the powerinjection of SPV source to PCC remaining unchanged. Theproper energy profile obtained from the analysis may furtheraid in finding accurate stability margin and critical clearingtime of the system in the event of a disturbance.

ACKNOWLEDGEMENT

The authors would like to thank the DST/Indo-US Scienceand Technology Forum (IUSSTF), New Delhi, India for pro-viding financial support to carry out this research work underan Indo-US collaborative project UI-ASSIST with project no.IUSSTF/EE/2017282B.

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