Application ofAveraging Technique to thePower System Optimum Placement and Sizing

of Static CompensatorsM. Tavakoli Bina, Member, IEEE, J. Rezaei Siahbidi, and K. Kanzi

Abstract--The application of static compensators (STATCOM)has been growing in transmission and distribution systems. Thisis basically followed for succeeding various objectives such as

voltage stability, loss reduction, or reactive power control.However, making a decision on the best possible locations forSTATCOMs can be achieved by arranging a combinatorialoptimization problem, and solving it by heuristic methods such as

genetic algorithm. It is also necessary to involve a suitable modelof STATCOM for the power flow analysis, which takes intoaccount the effects of the DC-link as well as the power losses andthe modulation index of the converter. This paper applies theaverage model of STATCOM to the optimal placement ofcompensators for the IEEE 14-bus benchmark network. Thegenetic algorithm is programmed with MATLAB to seek theoptimal locations of STATCOMs and their nominal values. Thisprovides the possibility of simulating and analyzing a variety ofcases in which power system issues such as losses, reactive power,

and its loadability are investigated.

Index Terms--Averaging, genetic algorithms, losses, optimalplacement, load flow analysis, reactive power, static compensator

I. INTRODUCTION

INCREASING the loading capability as well as enhancingthe controllability of power systems can potentially be

achieved by flexible compensators. These devices are alsocapable of improving the power quality and reducing thepower system losses due to uncompensated reactive power. Atthe same time, privatization of electricity market encourages

the power industries to take advantages of the power

electronics devices, managing the deregulation objectives.However, flexible compensators need high construction andinstallation cost followed by variable operating cost. Hence, itis vital to choose analytically the most favorable busbars forinstalling the static controllers along with determining theirrequired nominal powers.

It is also essential to model the power electronic devices, to

This work was supported in part by the Laboratory of reactive power andFACTS within the K. N. Toosi University of Technology.

M. Tavakoli Bina (corresponding author) is with the Faculty of ElectricalEngineering, K. N. Toosi University of Technology, Tehran--Iran (e-mail:tavakoli 0kntu.ac.ir).

J. Rezaei Siahbidi is currently a Master student at the K. N. ToosiUniversity of Technology.

K. Kanzi is with the ACECR research institute in Tehran--Iran.

be adapted for the power flow program. Different approacheshave already been taken into account. In [1-4], the proposedmodel either absorbs or injects reactive power to a bus,ignoring the power losses. It is also taken no notice of the DC-link effect along with the modulation technique on theconverter behavior, assuming an ideal positive sequencecomponent at 50/60 HZ. In [5], a separate bus is consideredfor the STATCOM's converter output. Then, an inductance inseries with a resistance connects the added bus to the powersystem busbar, including some sort of power losses for thecompensator. However, the DC-link voltage variation,containing the switching losses, is still paid no attention.

An average model for STATCOM is presented in [6],embracing the power losses as well as the modulatingparameters of the DC/AC converter. This paper adapts theaverage model of STATCOM and applies it to the power flowanalysis. In fact, the DC voltage and modulation index arelinked into the reactive power generation and active powerabsorptions from the power system. Meanwhile, we still usethe idea of having a separate bus for the compensator. Then, acombinatorial optimization is arranged in which an objectivefunction is established based on the power systemrequirements. Genetic algorithm is employed to seek the bestplacement as well as the nominal power of the compensators.The whole process was programmed with MATLAB tosimulate the proposed optimal placement for the 14-bus IEEEnetwork.

II. APPROXIMATE MODELAveraging technique is a common approach to the

modeling of power converters. Switch-mode converters have adiscontinuous behavior which is analytically very complex.During every switching period, there exists a set of state spaceequations (SSE) that mathematically models the exact system.Thus, the number of switching periods during the synchronousperiod establishes the number of SSE to be analyzed.

Averaging technique approximates the modulation of theconverter from a periodic discontinuous waveform to aperiodic continuous one. This would simplify the mentionedcomplex system to a single continuous SSE for everysynchronous period. Note that the high order harmoniccurrents of the compensator are trapped by a low-pass filter,

2id

I

cI0

-1

-2-1

P. U Aw~~~~~~~~~ 5 n i4lpha [Degee

Fig. 1. The approximate model of angle-controlled STATCOM.

introducing a similarity between the operation of the exactSTATCOM with the approximating approach. Hence, theaverage model can be usefully employed for the power flowanalysis.An average model is presented in [6], shown here by Fig.

1. In this model, L introduces the equivalent couplinginductance between the converter and the power system. Theresistance R is part of the compensator losses concerned withthe interconnection of the converter to the power system. Theother part of the power losses corresponds to the converterlosses that are absorbed by the proper modulation of theconverter switches. Fig. 2 shows typical STATCOM power

losses in P.U. against the phase shift between the converteroutput and the power system voltage (a) that is obtained bythe average model.

In Fig. 1, functions fi and f2 represent two linearcombination of the three-phase power system phase-neutralvoltages (va(t), vb(t) and v,(t)) as follows:

f1 (U(t)) 2Va (t)- Vb (t) VC (t)

f2 (U(t)) 2Vb (t) -Va (t) VC (t)

Also, two dependent voltage sources g1 and g2 are defined by

jMg, (D(t)) =Vc (2Da (t)-Db (t)-Dc (t)) (2)g2 (D(t)) Vc (2Db (t) -Da (t) -Dc (t))

and two dependent current sources h1 and h2 as

h1 (D(t), ia (t)) = ia (t)(Dc (t) -Da (t))

1h2 (D(t), ib (t)) = ib (t)(Dc (t) - Db (t))

Where Da(t), Db(t) and DC(t) are the three-phase duty ratiofunctions. Let m be the modulation index (belonging to [0, 1]),and M is the number of switching periods within a

synchronous period. Then, the three duty ratios for threephases can be approximated by:

Fig. 2. A typical power losses ofSTATCOM against ac obtained bythe average model.

[Da (t);:I-[I + m sin(o)t2

D :t(t) -[I + m sin(o)t2

1

2

/T--+ a)]

M

/T 2 /T-+a--)]

MT 23T_+x 3 )

(4)

A. Adapting the modelfor thepowerflowWhile the average model presents a time-dependent circuit,

a phasor PQ or PV model is essential for the power flowanalysis. Hence, here it is performed adaptive analysis to getthe supplied active and reactive powers of STATCOM (PCONand QcON). A new bus is added for every STATCOM as theconverter AC voltage, which is connected to an existing bus n

through the commutation reactance (XcON) and the ACresistance (R). Ignoring R for big XCON/R ratios, the active andreactive power of the compensator can be obtained by thewell-known power relationships for two busbars that are

connected through a connecting reactance:

PCON

QCON

VtVCON s Tsin(a )

XCON M

' (VCON -Vt cos(aXCON

(5)

Where V, and VCON are the magnitude of the fundamentalvoltages of bus n and the converter AC bus respectively. Fig.3 describes the power flow model, including the aboveparameters. Pn and Qn are active and reactive powers of thepower system at bus n.

Note that PCON (STATCOM losses) and QCON have to obey theaverage model obligations. Thus, the ratio a=PCONlQCON isdefined and set to an initial value. Then, the optimizationprogram is run by this initial condition for each STATCOM.

2

-.0-I. -1 _~.S V .J

3

CONO

V60R%

V

A%+jQ6 R1;^

Fig. 3. The adapted average model of STATCOM for power flow analysis.

Fig. 4. Flow chart of the combinatorial optimization for finding the bestplacement and sizing of STATCOMs.

Thus, the defined parameter a is updated, and once againthe optimization program is called. This leads to adaptation ofoperating points of the power system bus and the STATCOMaverage model. It should be emphasized that the adaptingprocedure is performed for all STATCOMs.

III. OPTIMIZATION ALGORITHM

Here an objective function along with some constraints isdefined for the optimization problem. In this paper threeobjectives are considered; A parameter for increasing powersystem voltage stability (b), reduction of power system losses(PI,,,) and transmission lines reactive power (Q). These arethen expressed by a single function as follows:

Maximize G(V, Y) b

P/OssQWhere G introduces the objective function, V is the phasors ofthe network voltages, and Y is the power system admittancematrix. The total power losses and reactive power are obtainedby conventional power system phasor relationships [7]. Theparameter b is subjected to a tolerable range imposed by thestandard practice rule. For example, when the voltage

magnitude of every power system bus lies within [0.95, 1.05]P.U., then b=1, otherwise the parameter b is penalized by:

b e-5(J1-V+0.05) (7)

Genetic algorithm is employed here to seek the bestsolution for the combinatorial optimization. An initialpopulation of strings is randomly generated based on thepower system busbars nominated for reactive powercompensation (e.g. (m, n) implies that busbars m and n aredetermined for compensator installation). Furthermore, theneeded nominal power of each STATCOM is anotherparameter to be sought. For the previous example, a ratedvalue is assigned to each installation (Sm and Sn), leading tothe form of (m, Sm, n, Sn) for a typical chromosome within thepopulation. A four digit precision is applied to the nominalpower of a compensator in P.U., while it is generally less than2 P.U.

In brief, various procedures are provided for the geneticalgorithm. These are selection of parents among the lastpopulation based on the objective function value, selectivebreeding such as mutation and random crossover to generate anew population of elements, and testing each element forfitness. The selective breeding is done in a way that thealgorithm avoids stopping at a local optimum. This ismanaged by generating a random number to make a decisionon selecting either mutation or crossover.

IV. SIMULATION PRINCIPLES

The IEEE benchmark 14-bus network ([8]) is used forexamination of the proposed optimized placement and sizingof the compensators. Consider the flow chart of Fig. 4 inwhich a power flow analysis is performed for everychromosome within the generated population. Results of thepower flow for each element are used to evaluate thecorresponding objective function as well as the parameter b.Then, the whole population chromosomes are ranked, andsuitable parents are chosen for an objective breeding tomanage the new population. The developed program stopswhen the objective function of the best chromosome remainsunchanged after a number of iterations.

Note that the number of STATCOMs can be detefmined atthe beginning of the program. Once this optimizationalgorithm converges, the obtained results are passed to theaverage model of STATCOM. This is followed by running theoptimization program using the adapted solutions. Thisprocedure is repeated until the best results are achieved(usually after a few iterations).

(6) A. Simulation resultsHere various simulation results are presented, based on the

developed combinatorial optimization and the approximatemodel. These are then examined to assess the proposedoptimal placement of compensators across the IEEEbenchmark 14 bus network. It is assumed that the load is fixedwith a factor of one for all cases except the last simulationTable in which the load is varied accordingly.

Fig. 5. Bus voltage magnitude profile for the uncompensated case. Fig. 6. Bus voltage magnitude profile when coupling a STATCOM to a

randomly chosen bus.bus volt;age magnittude profile

Fig. 7. Bus voltage magnitude profile when a STATCOM optimally Fig.placed and sized based on b and Q for the IEEE benchmark network. anm

1) Uncompensated case

An uncompensated power system provides useful informationconcerned with voltage regulation, reactive power and losses.This is done by running the power flow program, having no

compensator across the power system. The voltage magnitudefor 14 buses is shown by Fig. 5. It can be seen that the highestand the lowest voltage magnitudes are situated at bus 1 (1.07P.U.) and bus 14 (0.8 P.U.), respectively. Both are out of theallowable voltage stability range for b=1. Also, the magnitudedifference is equal to 0.27 P.U., describing an unacceptablevoltage regulation case. Hence, it is essential to use reactivecompensation for this power system.

2) Unplanned compensated case

Simulation results of Fig. 5 demonstrate the necessity ofvoltage regulation for the IEEE network. Here we employ a

reactive compensator with nominal power of 0.8 P.U., but it isconnected to an arbitrary bus (let say bus 4) without seekingits optimal placement. Fig. 6 shows the resulting voltagemagnitudes (bus 15 is assigned to the compensator's convertervoltage) by running the load flow program. Examining thesevoltages locates the highest and the lowest magnitudes at bus1 (1.05 P.U.) and bus 6 (0.8 P.U.), respectively. Thesemagnitudes give 0.25 P.U. difference, yet again showing a

8. Bus voltage magnitude profile when a STATCOM optimally placedd sized based on b, Q and PI,,, across the IEEE benchmark network.

poor voltage regulation. As a result, to employ STATCOM forthe power system, its nominal power as well as its bestcoupling bus has to be optimally sought.

3) Combinatorial optimization based on voltagestability and reactive powerThis simulation study is like the previous one, but the

coupling bus of STATCOM and its size are sought by thecombinatorial optimization. Variables b and Q are included inthe objective function, while Ploss is excluded. The program

suggests bus 14 as the optimal placement for STATCOM, and0.7625 P.U. as its nominal power. Fig. 7 illustrates the voltagemagnitudes of the IEEE benchmark network, showing a goodvoltage regulation within the tolerable range (b=l). However,the total power losses of the network are worked out to be0.2029 P.U.

4) Combinatorial optimization based on voltagestability, power losses and reactive powerThis case includes Ploss in the objective function in addition

to b and Q that is previously studied. Here bus 6 is suggestedfor installing a STATCOM with nominal power of 0.8772P.U. Fig. 8 depicts all 14 voltage magnitudes, confirming a

desirable voltage stability within the allowable region (b=l).Further, the analyzed power losses Ploss are 0.1658 P.U.,

4

4

5

TABLE ISIMULATION RESULTS SHOWING THE EFFECT OF C ON THE IEEE BENCHMARK

CHARACTERISTIC, WHILE THE LOAD IS FIXED BY A FACTOR OF ONE.

c Bus Si b Q PlossNo. (P. U.) (P. U.) (P. U.)

1 6 0.8772 1 2.1220 0.16582 6 0.4081 1 1.1470 0.1589

9 0.36983 6 0.3726 1 0.9505 0.1614

9 0.35823 0.2184

4 6 0.3380 1 0.9300 0.16189 0.40003 0.17191 0.0014

5 6 0.2747 1 0.9300 0.16189 0.36503 0.18821 0.001213 .10 13

which is 17% less than the previous case. Note that inclusionof PF,0s in the analysis not only changes the coupling point ofSTATCOM but also total network losses are significantlyreduced.

5) Increasing the number ofcompensatorsDespite the last three simulation cases, the developed program

is capable of seeking optimal places and corresponding sizesfor several compensators as well as adapting the obligations oftheir approximate model. Assume the optimal locations of twocompensators are needed to be suggested by the program. Theresultant locations are obtained buses 6 and 9 with nominalpowers of 0.4081 P.U. and 0.3698 P.U., respectively.Coupling these two STATCOM with the suggested buseswould reduce the power system losses from 0.1658 P.U. forthe previous case to 0.1589 P.U. Moreover, the reactive powerof transmission lines decreases from 2.1220 P.U. for theprevious case to 1.1470 P.U. At the same time, the total ratedpower of the two STATCOM is less than that of one

compensator for the previous case.

Additionally, assume c is the number of STATCOMs to beinstalled across the IEEE benchmark. Various simulationswere performed by increasing c. The results are gathered inTable I, where Si denotes to the rated power of STATCOMcoupled with bus i. It can be observed that the bigger thenumber of STATCOM coupled with the IEEE network thelower the reactive power of transmission lines. Also,increasing c from one to two decreases the power systemlosses, while any further increments would gradually add tothe power system losses (the parameter b is equal to one). As a

result, there exists an optimal c needed for every network thatsatisfies reduction of both power system losses and reactivepower of lines. Fig. 9 illustrates all simulation cases describedby Table I, showing the IEEE benchmark power losses as wellas reactive power changes due to increasing the number of theinstalled STATCOMs.

6) Loading capability oftransmission linesFormer simulation results demonstrate the need for seeking

optimal placement and sizing of compensators for the power

TABLE II

SIMULATION RESULTS EXAMINING THE EFFECT OF BOTH LOAD INCREASE ANDC ON THE IEEE BENCHMARK CHARACTERISTIC.

system. However, when the network is overloaded, theloading capability of transmission lines can be improved byraising the rated power of compensators. Different simulationswere carried out by increasing the network load, and theresults are collected in Table II. For example, when the load ischanged by a factor of 1.29, then a STATCOM with ratedpower of 1.1676 P.U. coupling with bus 6 can potentiallymanage the required loading capability. Comparing this withthe first simulated case of Table 1, the rated power ofSTATCOM has to be raised by about 3300. Nevertheless, thepower system losses worsen to 0.2937 P.U., showing about7700 growth.

Furthermore, it is unavoidable to increase c when thenetwork is keeping overloaded; Otherwise, the voltagestability parameter gets worse as shown by the second row ofTable II (b=0.6047). Then, in the third row, c is added up totwo, resulting in b=1, but the power system losses P,oss are

raised to 0.3937 P.U. As another example, rows four and fivecan be compared, where reactive power is decreased by1.1191 P.U. and Ploss is slightly increased. In fact, for a fixedc, there should be an optimal load factor that any furtheroverload may result in abnormal growth of power systemlosses and reactive power along with unacceptable b. At thispoint, the parameter c has to be varied to prevent any sharpgrowth of losses. Fig. 10 summarizes all simulation cases ofTable II, describing the effect of load increase factor on powerlosses and reactive power. Here, two pictures illustrate theeffect of load factor and number of STATCOMs on the IEEEbenchmark reactive power and losses for seven differentsimulation cases.

V. CONCLUSION

Averaging is an important modeling technique in power

electronics, which approximates a discontinuous complexsystem to a simple continuous one, while preserving theoriginal characteristics such as DC-link voltage andmodulation index. This paper first examines the average

model of STATCOM in the time domain, and then adapts it

Load c Bus Si b Q PlossFactor No. (P. U.) (P. U.) (P. U.)1.29 1 6 1.1676 1 2.6818 0.29371.30 1 6 1.1702 0.6047 2.6945 0.29871.50 2 6 0.7020 1 2.1650 0.3937

9 0.64771.90 2 4 2.2121 1 4.0107 0.7515

13 0.81082.00 3 6 0.9161 1 2.8916 0.7758

9 0.68303 0.9875

2.60 3 6 1.1862 1 4.4516 1.35019 1.06873 1.5821

6

Increrement initSTATICOM number

Hurd3er of STATOOMAs

1 2 3 4 5 6 7

.

3 4 5

Simulation case number

Fig. 9. Variations of power losses and reactive power of the IEEE benchmarkagainst the number of installed STATCOM corresponding to those of Table I.

power flow analysis. Further, a combinatorial optimization isarranged which focuses on voltage stability, reactive power

and losses of transmission lines as three power system mainobjectives. Genetic algorithm is employed to seek the optimalsolution for sizing and placing STATCOMs across the IEEE14-bus network, while a correcting power ratio is defined foradapting the optimized values with those obtained by theaverage model. This procedure continues until the program

reaches the adapted solution. Here we discuss the importanceof the proposed optimization process for loss reduction andreactive power of the lines, effects of increasing the number ofSTATCOMs as well as management of the needed loadingcapability. The whole system was programmed by MATLAB,and various simulation results are provided to show thepossibility of investigating different steady states power

system issues.

VI. REFERENCES[1] D. J. Gotham and G. T. Heydt, "Power flow control and power flow

studies for systems with FACTS devices," IEEE Transactions on PowerSystems, vol. 13, no. 1, pp. 537--544, February 1998.

[2] Y. Ni, et al., "STATCOM power frequency model with VSC chargingdynamics and its applications in power system stability analysis,"Proceedings of the APSCOM'97, pp. 119--124, November 1997.

[3] ""P. Petitclair, et al., "Average modeling and nonlinear control of an

ASVC (Advanced ststic var compensator)," Proceedings of IEEEPESC'96, vol. 1, pp. 753--758, June 1996.

[4] L. Gyugyi, et al., "Advanced static var compensator using gate-turn-offthyristors for utility applications," CIGRE records, April 1990.

[5] Y. Zhiping, et al., "An improved STATCOM model for power flowanalysis," IEEE PES winter meeting, paper no. 0-783-6420, 2000.

[6] M. Tavakoli Bina and D. C. Hamill, "Average model for angle-controlled STATCOM", IEE Proceedings Electric Power Applications,vol. 152, no. 3, May 2005.

[7] W. D. Stevenson, Elements of power system analysis, McGraw-Hill,Fourth edition, 1982.

[8] M. Mithulananthan, C. A. Canizares, and J. Keeve "Indices to detectHopf bifurcation in power systems," In Proceedings ofNAPS-2000, pp.

15-23, October 2000.

Fig. 10. Variations of power losses and reactive power of the IEEEbenchmark against the load factor of the network as well as the number of

installed STATCOM corresponding to those of Table II.

VII. BIOGRAPHIES

M. Tavakoli Bina was born in Tehran, Iran, on July

14, 1Q62. He received a B.Sc. degree from theUniversity of Teheran in 1992, and his Ph.D. degreein power electronics and power system

interconnection from the University of Surrey in the

UK in June 200p1.He is presently holding an

assistant professor position at the K. N. Toosi

University of Technology, researching in the area of

designing power electronics devices as well as power

system analysis. His latest work corresponds to thedesign and installation of a research Laboratory in the area of power

electronics multilevel converters and FACTS devices.

Javad Rezaei Siahbidi was born in Kermanshah,Iran on March 23, 1979. He received his B.Sc.degree from the Razi University in July 2001 in thefield of Electronics. He is currently a M.Sc. studentat the K. N. Toosi University of Technology in

Teheran. He is finalizing his master project to be

defended in June 2005. His Master project includesthe optimal placement of the static power electroniccompensators in power systems. He is interested instatic compensator modeling as well as power system

control, namely reactive power control and loss reduction along with voltagestability.

Khalil Kanzi was born in Yazd, Iran on

March 21, 1959. He received the B.Sc. degree fromK. N. Toosi University and M.Sc from TarbiatModaress University of Teheran in 1984 and 1992respectively. He has worked for the ACECRresearch Institute in Teheran in the area of power

system for more than 15 years. He is now a Ph.D.student at the K. N. Toosi University and hisworking area is control and modulation of activepower filters and FACTS controllers.

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