Laplace Transform Application to Model and … · Equivalent circuit of the power network compensated. ... Laplace Transform Application to Model and Analysis of D-STATCOM 28th Power

No. E-13-AAA-0000

Laplace Transform Application to Model andAnalysis of D-STATCOM

Ehsan AfzalanRamin Power Plant

Khuzestan Regional Electric Power CompanyAhvaz, Iran

[email protected]

Mahmood JoorabianDepartment of Electrical EngineeringShahid Chamran University of Ahvaz

Ahvaz, [email protected]

Abstract—This paper proposes mathematical modeling ofD-STATCOM based on Laplace transform technique in whichthe proposed model is in the complex frequency domain. Thepower circuit of D-STATCOM is modeled by a controlledcurrent source. All state variables in the model are transformedand therefore analyzed in dq reference frame. The compensatingcurrent from the D-STATCOM is assumed to be purelysinusoidal. Therefore, the transformed signals in dq axes are DC.For performance evaluation, loads of the D-STATCOM aredivided into two types: i) constant impedance and ii) constantpower. The proposed model can be linearised and then Laplacetransform is applicable to transform the model into thefrequency domain. Together with DC-link capacitor voltagecontrol, the proposed D-STATCOM model of reactive powercompensation can be evaluated. A simplified 11-kV, 2-bus testpower system is employed for simulation. Satisfactory resultsobtained by simulating the proposed model are compared tothose obtained by simulating the switching control ofD-STATCOM power circuit created in MATLAB’s PowerSystem Blockset. As a result, the effectiveness of proposed modelis verified. It gave satisfactory responses to estimate nonlinearcharacteristics of the D-STATCOM operation.

Keywords—Static synchronous compensator, Laplacetransform, Reactive power control, Voltage control, Constant powermodel, D-STATCOM.

I. INTRODUCTION

Since the last decade, a large number of customer’sequipment has been increasingly installed in electric powerdistribution systems. These are highly sensitive to poor qualityof electric power supply (Hingorani et al., 1999; Ghosh et al.,2002). Recently, custom power technology (CPT), the low-voltage counterpart of the more widely-known flexible actransmission systems (FACTS) technology, aimed at high-voltage power transmission application, has emerged as acredible solution to solve many of the problems relating tocontinuity of supply at the end-user level. Both FACTS andCPT are directly credited to EPRI (Hingorani et al., 1999).The distribution static compensator (D-STATCOM) is avoltage source converter (VSC) based CPT which can performas a reactive source in power systems. D-STATCOM canregulate magnitude of voltage and adjust phase angle at ACbus at the point where it is connected, via generating or

absorbing reactive power from the system. Among theseapplications, control system design of D-STATCOM is vital. Itrequires an accurate model to synthesize its controller. InSTATCOM and D-STATCOM study, models in the dqreference frame are commonly used (Singh et al., 2008). Saoet al. (2002), Park et al. (2001), Deng et al. (2005), Saeedifardet al. (2009), and Zhou et al. (2011) presented a basicmodeling of D-STATCOM for controller design in the dqreference frame. Illustration of dynamic responses acquired byusing this model is also reported. Dinavahi et al. (2004)proposed a controller design algorithm of the D-STATCOM.All variables of this design process are in the dq referenceframe. Proportional plus integral (PI) control is employed.However, this research reveals complexity of using the dqmodel due to nonlinearity of D-STATCOM and the powerdistribution system. Later, Ye et al. (2010) described amodification for linearising the dq model. Therefore, itscontroller can be designed according to classical feedbackcontrol theories. Furthermore, the dq model is extended to beused in unbalanced operations (Blazic et al., 2004; Blazic etal., 2006). Positive, negative and zero sequence componentsare introduced and applied to analyse the system. [1-5]

In this paper, the dq model of D-STATCOM is developedfrom a simple feeding configuration. This model is used inreactive power compensation problems where a reactivepower source is installed to regulate the load. The overallsystem model in the dq frame is further simplified. This can becarried out by assuming that the reactive power source iseither an ideal controlled current source or even a capacitorbank. Since DC signals can be expected from the simplified dqmodel, Laplace transform is employed to enable transientanalysis in frequency-domain. Simulation through controlsystem block diagram can be implemented in MATLAB forperformance evaluation. [6-8]

II. MATHEMATICAL MODELLING OF D-STATCOM

It assumes that a D-STATCOM is installed to performpower compensation in a medium-voltage distribution feeder.For simplification, a model of an ideal current source isemployed as shown in Figure 1. (1)and(2) describe dqgoverning equations derived by applying KVL and KCL to theequivalent circuit given in Figure 2b. By applying further dq

Laplace Transform Application to Model and Analysis of D-STATCOM

28th Power System Conference - 2013 Tehran, Iran

2

transformation in stationary reference, (1) and (2) can berewritten in (3) and (4), respectively. [9-12]

sV

sItV

lIfI

a) Model of an ideal compensator

thV

thRthL

fI

tV

sI lI

lR

lL

Loop 1 Loop 2b) Per-phase equivalent circuit

Figure 1. Equivalent circuit of the power network compensated.

abctabcthabcsthabcsth vviRidt

dL ,,,, (1)

abctabcllabcll viRidt

dL ,,, (2)

0,0,0,0, dqtdqthdqsththdqsth VVIRLIdt

dL (3)

0,0,0, dqtdqllldqll VIRLIdt

dL (4)

Where:

000

00

00

L

L

L

L q

d

th, MLLL thqd , MLL th 20 ,

000

00

00

, fs 2

The Laplace transform is applied to (3) and (4), fourequations in s domain are obtained as in (5) – (8). In addition,by re-arranging these equations, the dq component of sourcecurrents in s domain are as shown in (9).

tdsdsqdsdthsdd VVILIRIsL (5)

tqsqsdqsqthsqq VVILIRIsL (6)

tdlqlldlldl VILIRIsL (7)

tqldllqllql VILIRIsL (8)

qdthqdthqd

qtdthdthdtqthq

qdthqdthqd

dtqthqthqtdthd

sq

sd

LLRsLLRsLL

LVVRsLVVLLRsLLRsLL

LVVRsLVV

I

I

222

222

(9)

Where:

fdsdld III , fqsqlq III

By neglecting the mutual inductance M, thqd LLL ,

voltage at the point of coupling connection tV can be

calculated by using the following matrix equation.

fq

fd

thq

thd

tq

td

I

I

V

V

ccbb

ccbb

V

V

aa

aa

1212

2121

1112

1211(10)

Where:

)(

))2(()(222

2211

ththlththl

thlththlththl

LLLRRR

sRRLRLsLLLa

)(12 thllth RLRLa

lthlthlthlthlth LLRRsRLLRsLLb 221 )(

)(2 thllth RLRLb ,

lthth

lththlthth

lththlthlth

RLR

sLLRRRL

sLRLRLsLLc

)(

))(2(

)2(

222

222

22321

lththlththlth LLRsLRLsLLc )(2 222222

These expressions are used for a case of the impedance loadmodel as shown in Figure 1b. In practice, loads in powerdistribution systems are commonly used in the power model.To simplify this calculation, a linearised load impedancederived from the load power and its bus corresponding voltagecan be written as in (11) – (12).

)( 22

2

ll

lTl QP

PVR (11)

)(2 22

2

ll

lTl QPf

QVL

(12)

III. CALCULATION OF REACTIVE POWERCOMPENSATION BY USING D-STATCOM

An amount of the reactive power required to supportvoltage sags in power distribution systems can be derivedfrom (10). In a particular steady-state operating point, voltageacross the load can be rewritten as in (13) and (14).



3

fqlfdlthqthdtd ILIRCVBVAV (13)

fdlfqlthqthdtq IRILBVCVDV (14)

Where

222

)()(

thth

ththdththqlththqththdl

LR

LVRVLLVRVRA

222

2

thth

thlthl

LR

LLRRB

,222thth

thlthl

LR

RLLRC

222

)()(

thth

ththdththqlththqththdl

LR

LVRVRLVRVLD

In this operation, no real power exchange between the systemand D-STATCOM is involved. In addition, given that

22tdTtq VVV and

22tdT

fdtdfq

VV

IVI

, therefore (13) and

(14) can be reduced to (15) as follows.

2 2

2 2

td fdtd td T td l l fd

T td

V IV D V C B V V R L I

V V

(15)

From the above equation, VT is the desired magnitude ofthe load voltage. Thus, the calculated value of thecompensating reactive power is shown in the followingequation.

. .f tq fd td fqQ V I V I (16)

(16) provides the equation to calculate an amount of thereactive power compensation to support voltage sags.However, in some particular conditions, this calculation mayfail to regulate the load voltage within a given limit.

IV. CONTROL OF D-STATCOM

When D-STATCOM is connected to a point of couplingconnection, the interaction between AC side and DC side ofthe D-STATCOM’s inverter is controlled by some digitalcontroller. In this paper, this interaction must be simplified inorder to include the D-STATCOM’s controller in the analysis.Assume that the reactive power injected from D-STATCOMis controlled by using a simple block diagram as shown inFigure 2. The D-STATCOM’s control scheme can be dividedinto three parts: i) current control, ii) DC link voltage controland iii) AC voltage control. As can be seen from Figure 3,voltage and current relations in both AC and DC sides can beexpressed as (17) – (24).

tastafafa vvRiidt

dL (17)

tbstbfbfb vvRiidt

dL (18)

tcstcfcfc vvRiidt

dL (19)

fcstcfbstbfastadcdc iviviviv (20)

dcdcdc ivdt

dC (21)

*adcsta vvv (22)

*bdcstb vvv (23)

*cdcstc vvv (24)

tdstdfqfd VVLIIRLs (25)

tqstqfdfq VVLIIRLs (26)

*ddcstd VVV (27)

*qdcstq VVV (28)

By transforming (17) – (24) into the dq reference axis andtherefore applying Laplace transform, (25) – (28) are obtained

where *dV and *

qV are the reference voltages. (25) and (26)

can be used to find the D-STATCOM’s current as shown in(29) and (30), while the voltage and current of DC linkcapacitor can be calculated by using (31) and (32).

22222 LRLRssL

VVLRLsVVI tqstqtdstd

fd

(29)

22222 LRLRssL

VVLRLsVVI tdstdtqstq

fq

(30)

dc

fqstqfdstddc V

IVIVI

(31)

sC

IV

dc

dcdc (32)

tavtbvtcv

R Lfai fbi

fcistav

stbvstcv

dcV

dcI

fd fqI ,IlossPfQ

* *std stqV ,V

dc ,refV

acVac ,refV

Figure 2. D-STATCOM with control structure.



4

By adding a current control loop with *fdI and *

fqI as the

reference currents, the reference value of the voltage andcurrent control loops can be calculated by using (33) – (36).

fdfdvdi

vdpd IIs

kkV

*,

,*

(33)

fqfqvqi

vqpq IIs

kkV

*,

,*

(34)

22*

tqtd

losstdftqfd VV

PVQVI

(35)

22*

tqtd

ftdlosstqfq VV

QVPVI

(36)

Where Ploss is the output from the DC voltage controllercompensating the switching losses and Qf is the output fromthe AC voltage controller representing an amount of Desiredreactive power injection from the D-STATCOM. These can besummarized as described by (37) and (38).

dcdcdci

dcploss VVs

kkP

*,

, (37)

ttaci

acpf VVs

kkQ

*,

, (38)

V. RESULTS AND DISCUSSION

A simple 11-kV, 2-bus power distribution system asdescribed in Figure 3 is used for study. In normal operationswhere the load voltage at bus 2 is 11 kV, the voltage at bus 1can be measured as 13.2943 kV. The situation for this study isthe compensation from D-STATCOM under the voltage sags.Assume that a sudden sag occurs and thus the voltage at bus 1and 2 are reduced to 9.2943 kV and 7.6903 kV, respectively.The required reactive power of 16.7242 Mvar to support thissag can be calculated by (15) and (16). With this reactivepower calculation, the voltage at bus 2 can be resumed within100 ms as shown in Figure 4a.

sItVlI

fI

th thR 10 ,L 10mH

kVVth 2943.13

Figure 3. 11-kV, 2-bus test system

0.1 0.15 0.2 0.25 0.3 0.35 0.40.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

a) Impedance load model

0.1 0.15 0.2 0.25 0.3 0.35 0.40.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

b) Power load model

Figure 4. Responses of the compensation by using a capacitor

However, this compensation is performed by switching acapacitor of 16.7242 Mvar. Due to capacitance from thecompensator and RL characteristic in the power network, 20%overshoot of the response is expected. By using the capacitorcompensation, it can support the voltage sag as low as 6.64 kVonly. When any sag below this value occurs, the voltage at bus2 will not be resumed to 11 kV. Interestingly, when the powerload model as described in (11) and (12) is used, the capacitorcompensation can resume the voltage at bus 2 if the capacitoris switched into the system within 200 ms after the occurrenceof the voltage sag. If not, the capacitor compensation fails toresume the voltage across the load as shown in Figure 4-b.

The performance of reactive power compensation can beimproved by employing the feedback control scheme asdescribed in the previous section. The proposed controlscheme can be implemented according to the block diagramgiven in Figure 5 and 6 for the impedance load model and thepower load model, respectively. The simulation results showthat this control scheme can work satisfactorily when the loadmodel used is the impedance model as shown in Figure 7. Itassumes that the voltage sag occurs in the system at t = 0.2 s.Figure 7-a is similar to those plots in Figure 4-a, however it



5

just includes the bus voltage resulting from the proposedcontrol scheme as shown in Figure 5. The response of thisproposed control scheme shows that the bus voltage can beresumed to the desired level in less than 2 cycles. In addition,unlike the use of the capacitor, the overshoot is minimized.Figure 7-b shows real and reactive power responses. In Figure7-c, 7-d, 7-e and 7-f, the load voltage, the compensatingcurrent, the reactive power injection and the DC bus voltageare illustrated, respectively. Moreover, to verify the accuracyof the simplified model proposed in this paper, simulatedresults obtained by using MATLAB’s simulink - PowerSystem Blockset are also compared.

Figure 8-a is similar to those plots in Figure 4-b, however itjust includes the bus voltage resulting from the proposedcontrol scheme as shown in Figure 6. The response of thisproposed control scheme shows that the bus voltage can beresumed to the desired level in less than 2 cycles. By using theproposed expression for the optimal reactive powercalculation, the voltage at load bus can be restored quickly.This control situation is not disturbed significantly to the realpower responses of the system. However, compensating thevoltage sags in power distribution systems usingD-STATCOM causes a considerable amount of the activepower loss in the D-STATCOM’s inverter and theD-STATCOM’s coupling transformer devices. Unlike the useof the capacitor, the load bus voltage could be regulatedsuccessfully at the specified value if the capacitor is switchedsimultaneously when the fault occurred. Figure 8-b shows thereal and reactive power responses of theD-STATCOM. In Figure 8-c, 8-d, 8-e and 8-f, the activepower loss in the D-STATCOM, the real and reactive powertransfers at the source and the load, the DC bus voltage, andthe D-STATCOM’s current are illustrated, respectively.

2 2 2 2 22sd td s s sq tq s

sds s s s s

(V V )( L .s R ) (V V ) LI

L .s R L .s ( R L )

2 2 2 2 22sq tq s s sd td s

sqs s s s s

(V V )( L .s R ) (V V ) LI

L .s R L .s ( R L )

td ld l ld l lq lV I L .s I R I L

tq lq l lq l ld lV I L .s I R I L

2 2td tqV V

sdV

sqVtdV

tqV

sdI

sqIfdI

fqI

ldI

lqI

tdVtqV tV

tV

q

*std dc d

*stq dc

V V .V

V V .V

*fdI

*fqI

fdIfqI

stdVstqV

stdV

stqV dcV

*dV*

qV

dcV

fQ

lossP

,t refV

,dc refV

dcVstd fd stq fqdc

dc

V .I V .II

V

1.dc dcdc

V Is CdcI

2 2fdtq f td loss*

td tq

V .Q V .PI

V V

2 2fdtq loss td f*

td tq

V .P V .QI

V V

2 2 2 2 2std tq stq tq

fd(V V )( L.s R ) (V V ) L

IL .s RL.s ( R L )

2 2 2 2 2stq tq std td

fq(V V )( L.s R ) (V V ) L

IL .s RL.s ( R L )

fdIfqI

fdIfqI

dcV

dcV

tdV

tqV

tdV

tqV

Figure 5. Block diagram describing a power distribution system with the RL

load model

tdVtqV tV

q

*std dc d

*stq dc

V V .V

V V .V

*fdI*fqI

fdIfqI

stdVstqV

stdV

stqV dcV

*dV*

qV

dcV

fQ

lossP

,t refV

,dc refV

dcVstd fd stq fqdc

dc

V .I V .II

V

1.dc dcdc

V Is CdcI

2 2fdtq f td loss*

td tq

V .Q V .PI

V V

2 2fdtq loss td f*

td tq

V .P V .QI

V V

2 2 2 2 2std tq stq tq

fd(V V )( L.s R ) (V V ) L

IL .s RL.s ( R L )

2 2 2 2 2stq tq std td

fq(V V )( L.s R ) (V V ) L

IL .s RL.s ( R L )

fdIfqI

fdIfqI

dcV

dcV

tdV

tqV

2 2 2 2 22sd td s s sq tq s

sds s s s s

(V V )( L .s R ) (V V ) LI

L .s R L .s ( R L )

2 2 2 2 22sq tq s s sd td s

sqs s s s s

(V V )( L .s R ) (V V ) LI

L .s R L .s ( R L )

22 2 2 2 2 2

( . )2 ( ) ( ) 2 ( )

l l ltd t ld ld lq

l l l l l l

Q P QV V I s I I

f P Q P Q f P Q

22 2 2 2 2 2

( . )2 ( ) ( ) 2 ( )

l l ltq t lq lq ld

l l l l l l

Q P QV V I s I I

f P Q P Q f P Q

22tqtd VV

sdV

sqV

tdV

tqV

tdV

tqV

sdI

sqIfdI

fqI

ldI

lqI

tV

1

0.004 1s 2

tV

Figure 6. Block diagram describing a power distribution system with the PQ

load model

0.1 0.15 0.2 0.25 0.3 0.35 0.40.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

(a)

(b)

Active and Reactive Power at D-STATCOM

0.1 0.15 0.2 0.25 0.3 0.35 0.4-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

Mvar

andM

W

16.725 Mvar

0 MW

Q with Capacitor

Q with D-STATCOM

P with Capacitor

P with D-STATCOM



6

0.1 0.15 0.2 0.25 0.3 0.35 0.48.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

(c)

0.1 0.15 0.2 0.25 0.3 0.35 0.4-1400

-1200

-1000

-800

-600

-400

-200

0

200

fdI

fdI

fqIfqI

(d)

0.1 0.15 0.2 0.25 0.3 0.35 0.4-2

0

2

4

6

8

10

12

14

16

18

(e)

0.1 0.15 0.2 0.25 0.3 0.35 0.418.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0DC Voltage

Math Model

SimPowerSystems

(f)Figure 7. Simulation results of the proposed control scheme(RL load model)

0.1 0.15 0.2 0.25 0.3 0.35 0.40.0

2.0

4.0

6.0

8.0

10.0

12.0

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10

-5

0

5

10

15

20

(b)

0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

(c)



7

0.1 0.15 0.2 0.25 0.3 0.35 0.4-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0Mv

arand

MW

(d)

0.1 0.15 0.2 0.25 0.3 0.35 0.418.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

(e)

0.1 0.15 0.2 0.25 0.3 0.35 0.4-1400

-1200

-1000

-800

-600

-400

-200

0

200

fdI

fqI

(f)

Figure 8. Simulation results of the proposed control scheme

(PQ load model)

VI. CONCLUSION

This paper describes a simplified model of D-STATCOMin power distribution systems by using Laplace transform. Aset of equations in the frequency domain can be collected andtherefore used to develop the control scheme for voltage sagcompensation. For a case of impedance load model, the

capacitor compensation can be used to regulate the load busvoltage. The overshoot of the voltage response is inevitabledue to the transient characteristic of RLC under dampedcircuits. However, the response’s overshoot can be reduced byapplying the proposed control scheme as described in Figure6. When the load is characterized by the PQ model, thecapacitor compensation fails to resume the load voltage if theswitching time of the capacitor is not fast enough. With theproposed control scheme in Figure 7, the voltage response ofthe load bus can be resumed within 2 cycles after the fault. Ascan be seen, the proposed model in the frequency domain issimple and then used for voltage sag compensation in thepower distribution system. The results shows that the proposedcontrol scheme derived from this simplified model can beperformed well for all test cases. In addition, the simulationresults obtained by using the proposed model in the frequencydomain are compared with those acquired from MATLAB’sSimulink - Power System Blockset to confirm the accuracy ofthis simulation.

REFERENCES

[1] A. Arulampalam, M. Barnes, N. Jenkins, JB. Ekanayake, “Power qualityand stability improvement of a wind farm using STATCOM supportedwith hybrid battery energy storage”, IEE Proc. Gener. Transm. Distrib,Vol. 153, pp. 701-710, 2006.

[2] B. Blazic, I. Papic, “Improved D-Statcom control for operation withunbalanced currents and voltages”, IEEE Trans. Power Deliv, Vol. 21,pp. 225-233, 2006.

[3] W. Chao, Z. Yao, “Approach on nonlinear control theory for designingSTATCOM controller”, IEEE Grey Systems and Intelligent Services Int.Conf., pp. 871-875, 2007.

[4] WL. Chen, WG. Liang, HS. Gau, “Design of a mode decouplingSTATCOM for voltage control of wind-driven induction generatorsystems”, IEEE Trans. Power Deliv,Vol. 25, pp.1758-1767, 2010.

[5] T. Manmek, CP. Mudannayake, C. Grantham , “ Voltage dip detectionbased on an efficient least squares algorithm for D-STATCOMapplication”, IEEE Power Electronics and Motion Control Conf, pp. 1-60, 2006.

[6] MG. Molina, PE. Mercado, “Control of tie-line power flow of microgridincluding wind generation by DSTATCOM-SMES controller”, IEEEEnergy Conversion Congress and Expo, pp.2014-2021, 2009.

[7] M. Saeedifard, R. Iravani, J. Pou, “Control and DC-capacitor voltagebalancing of a space vector-modulated five-level STATCOM”, IETPower Electron, Vol. 2, pp. 203-215, 2009.

[8] B. Singh, R. Saha, A. Chandra, K. Al-Haddad, “Static synchronouscompensators (STATCOM)”, a review. IET Power Electron. Vol. 2,pp.297-324, 2008.

[9] Y. Xu, LM. Tolbert, JN. Chiasson, JB. Campbell, FZ. Peng , “Ageneralised instantaneous non-active power theory for STATCOM”, IETElectr. Power Appl, Vol. 1, pp.853 -861, 2007.

[10] CF. Xue, XP. Zhang, KR. Godfrey, “Design of STATCOM dampingcontrol with multiple operating points: a multimodel LMI approach”,IEE Proc. Gener. Transm. Distrib, Vol. 153, pp.375-382, 2006.

[11] Y. Ye, M. Kazerani, VH. Quintana, “Modeling, control andimplementation of three-phase PWM converters”, IEEE Trans. PowerElec, Vol. 18, pp.857-864, 2010.

[12] H. Zhou, G. Yang, J. Wang, H. Geng , “ Control of a hybrid high-voltage DC connection for large doubly fed induction generator-basedwind farms”, IET Renew. Power Gener, Vol. 5, pp.36-47, 2011.

Laplace Transform Application to Model and … · Equivalent circuit of the power network compensated. ... Laplace Transform Application to Model and Analysis of D-STATCOM 28th Power

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