Optimal Power Flow Solution of Two-terminal HVDC Systems Using Genetic Algorithm

8/9/2019 Optimal Power Flow Solution of Two-terminal HVDC Systems Using Genetic Algorithm

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Electr Eng (2014) 96:6577DOI 10.1007/s00202-013-0277-7

ORIGINAL PAPER

Optimal power ow solution of two-terminal HVDC systemsusing genetic algorithm

Ula s Kl Krsat Ayan

Received: 11 March 2012 / Accepted: 12 January 2013 / Published online: 31 January 2013 Springer-Verlag Berlin Heidelberg 2013

Abstract The usage extensively of high voltage direct cur-rent (HVDC) transmission links in recent years makes itnecessary further work in this area. Therefore, two-terminalHVDC transmission link is one of most important elementsin electrical power systems. HVDC link representation ismostly ignored and simplied for optimal power ow (OPF)studies in power systems. OPF problem of purely alternat-ing current (AC) power systems is dened as minimizationof the fuel cost to subjected equality and inequality con-straints.Hence,OPF software of purelyAC power systems isextended by taking into consideration power transfer controlcharacteristicsof HVDC links. In this paper, OPFproblem of integrated ACDC power systems is rst solved by geneticalgorithm that is a heuristic algorithm based on evolutionary.The proposed method is tested on two test systems which arethe modied 5-node test system and the modied WSCC 9-bus test system. In order to show effectiveness and efciencyof the proposed method, the obtained results are comparedto that reported in the literature.

Keywords Optimal power ow HVDC system Integrated ACDC system Heuristic method Genetic algorithm Evolutionary

U. Kl (B )Department of Electrical, Mehmet Akif Ersoy University,Burdur, Turkeye-mail: [email protected]

K. AyanDepartment of Computer Engineering,Sakarya University, Sakarya, Turkeye-mail: [email protected]

1 Introduction

In power systems, optimal power ow (OPF) problem isdened as a nonlinear, non-convexand large-scale static opti-mization problem having continuous variables together withdiscretecontrol variables [ 1].Thepresence ofdiscretecontrolvariables, such as transformer tap positions, phase shiftersand switchable shunt devices complicates to converge to theglobal optimum of the general non-convex OPF problem [ 2].Here, total generation cost that is an objective is minimizedto subject equality and inequality constraints for obtaininggeneration power outputs, bus voltage magnitudesand trans-former taps [3]. This optimization problem has nonlinearpower ow equality constraints. Currently, OPF problem isvery signicant issue for power system operation and analy-sis. In a deregulated electricity conditions, OPF problem hasbeen used to assess the variation of electricity prices andtransmission congestion studies, in recent years [ 2].

OPF problem was rst introduced in Ref. [ 4] and thisproblem has been solved using methods such as Gradi-ent base, Linear programming method [ 5] and quadraticprogramming [ 6]. For equality constrained optimizationproblems,nonlinear equationscan be solved using a Newton-type algorithm. In Newton OPF [ 7], the inequality con-straints have been included to the objective functions such asquadratic penalty terms. In interior point (IP) method [ 8,9],the inequality constraints are converted to equalities con-straints.In theunlimitedpointalgorithm[ 10], transformationof the slack and dual variables of the inequality constraintsis used. Recently, to overcome the limitations of these math-ematical programming approaches, algorithms such as GAwhichis an evolutionary-based heuristicalgorithmhave beendeveloped [ 11,12].

Effectiveness of OPF problem is limited by size of powersystems. Numerical optimization techniques are based on

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successive linearization using the rst and the second deriv-atives of objective functions and their constraints [13]. Theadvantages of such methods are that they have mathematicalbackground. However, they have disadvantages such as thesensitivity to problem formulation, algorithm selection andusually converge to a local minimum.

HVDC transmission link is an application of the powerelectronics technology to integrated ACDC power systems.The economics of bulk power transmission by undergroundDC links are increasingly moving in recent years. The one of the advantages of power transmission by HVDC links is theability to control the instantaneous power in neighbouringAC systems. Furthermore, HVDC links are used to stabilizeelectric power systems [ 14]. Recently, much researchon real-ization of HVDC models is performedforpowerow studies[1519]. The formulation for the basic model of the HVDClink is given in Ref. [ 20].

There are twobasic approaches for solving thepowerowequations of integrated ACDC power systems in the litera-ture. The rst is the sequential approach [ 19,21,22]. In thismethod, the AC and DC equations are solved separately bysuccessive iterations. Although the implementation of thesequential method is easy, it has convergence problems asso-ciated with certain situations and the state vector does notcontain explicitly the DC variables. The second approach isknown as the unied approach [ 23].

GA approach was only applied to OPF and ORPF prob-lems of purely AC power systems until today [ 2426]. Thepurpose of this paper is to solve the optimization problemof two-terminal HVDC systems using GA approach. GA isessentially a search algorithm that is based on natural genet-ics. It evaluates the optimal solution with randomized, struc-tured exchanges of information among exist solutions. Theconstraints on solution space are not implemented duringthe process in GA approach. Therefore, GA approach is arobust method. The power of GA comes from its ability toexploit historical information from previoussolution guessesto increase performance of future solutions [27]. Recently,GA has been applying extensively to solve global optimiza-tion searching problem when the closed form optimizationtechnique cannot be applied. GA approach takes into consid-eration simultaneously many points in the parameter spaceand has not to assume that the search space is differentiableor continuous [ 28]. Therefore, GA approachconvergeseasilyto the global solution. In [ 29], the solution of OPF problemby using GA is based on the usage of genetic algorithm. Themethod is not sensitive to the starting points and is able todetermine the global optimum solution of OPF for a rangeof objective functions and constraints.

In this study, the factual model for the transformers isused. In the factual model, the impedance values of the trans-formers change as the tap ratios of the transformers change,and thus the bus admittance matrix of the power system also

changes. The power system calculations are accomplishedwithout inclusion the impact of the transformers to the Jaco-bian matrix. Thus, the tap ratios of the transformers can beselected as the control variables. In heuristic method, eachindividual select a different tap ratio for each transformerand the bus admittance matrix of the power system is cal-culated uniquely for each individual. This process increasescomputational time of the software.

After this introduction, the modeling of DC transmis-sion link is represented in Sect. 2. The GA methodologyis explained in Sect. 3. GA-based optimal power ow solu-tion of two-termial HVDC system is explained in Sect. 4.In Sect. 5, to demonstrate validity, the efciency and theeffectiveness of the proposed method, simulation results of the modied 5-node test system and the modied WSCC9-bus test system with an HVDC link are given and theobtained results are extensively evaluated and compared tothat reported in the literature. Finally, the conclusions anddiscussions are represented in Sect. 6.

2 The modeling of DC transmission system

Before analyzing DC transmission system, it is necessary tomodel DC transmission link and the converters. The model-ing is based on widelyaccepted assumptions in the literature.The assumptions are as follows [23]:

The main harmonic values of current and voltage in ACsystem are balanced.

The other harmonics except the main harmonic areignored. The ripples in the current and voltage waveforms of

HVDC link are ignored.

The thyristors used in theconverters areaccepted as idealswitch.

No load current of the converter transformers and thelosses are ignored.

A basic schematic diagram of a two-terminal HVDC trans-mission link interconnecting buses r (rectier) and i(inverter) is illustrated in Fig. 1. The basic converter equa-tions describing the relationship between the AC and DCvariables were expressed in Ref. [30].

The symbols appearing in Fig. 1 are dened as follows:V r is the primary line-to-line AC voltage (rms) of the recti-er side, V i the primary line-to-line AC voltage (rms) of theinverter side, r the phase angle of the rectier side, i thephase angle of the inverter side, I r the alternating current of therectier side, I i thealternating current of the inverter side,V dr the direct voltage of the rectier side, V di the direct volt-age of the inverter side, I d the direct current, t the effective

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Electr Eng (2014) 96:6577 67

Fig. 1 A basic schematicdiagram of a two-terminalHVDC transmission link [ 30]

dr V diV

dc Rd I r r V iiV

i I r I 1:it r t :1

dr dr QP ,

didi QP ,

ACSystem

ACSystem

Rectifier Inverter

transformer tap ratio and P , Q is active and reactive power,respectively.

2.1 Rectier side equations of DC link

Theequivalent circuit of a two-terminal HVDC link is shownin Fig. 2 [31].

The equations related to the rectier operation of a con-

verter can be expressed as follows:

V dor =3 2

V r t r (1)

V dr = V dor cos 3 X cr

I d (2)

where V dor is the ideal no-load direct voltage, is the igni-tion delay angle. X cr is the so called equivalent commutat-ing reactance, which accounts for the voltage drop due tocommutation overlap. The active power of rectier side isdetermined by:

Pdr =V dr I d (3)Since losses at the converter and transformer can be ignored( Pdr = Pac) , the reactive power on the rectier side is deter-mined as follows:

Q dr =|Pdr tan r| (4)where r is the phase angle between the AC voltage and thefundamental AC current and is calculated by neglecting thecommutation overlap as follows:

r =cos1 (V dr/V dor ) (5)

2.2 Inverter side equations of DC link

The equations related to the inverter operation of a convertercan be expressed as follows:

V doi =3 2

V it i (6)

V di = V doi cos 3 X ci

I d (7)

Pdi = V di I d (8)

Q di =|Pdi tan i| (9) i =cos

1 (V di / V doi ) (10)

where is the extinction advance angle.

2.3 DC link equations

The interdependent of the two DC voltages can be expressed

by Eq. (11):V dr =V di + Rdc I d (11)where Rdc is the resistance of DC transmission link.

2.4 Per unit system

In this study, the actual variables and per unit variables arerepresented by capital letters and small letters, respectively.

Pacbase = Pbase (12) I acbase

=

Pacbase

3V acbase(13)

Z acbase =V acbase

3 I acbase (14)Per unit variables:

v =V

V acbase(15)

vd =V d

V acbase(16)

id = I d

I acbase(17)

x c

= X c

Z acbase(18)

pd =Pd

Pacbase(19)

The voltage of DC link is calculated as follows:

V d =Vnkt 2

sink

cos nk X c

2 I d (20)

where n is the number of the series connected bridges, k represents peak number per period in the voltage variationon the load. For k =6, n =1:

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Fig. 2 The equivalent circuit of two-terminal HVDC link [ 31]

cr Rci Rdc

R

diV dr V d I

cos23

r r t V cos23

ii t V

V d =3 2

V t cos

3 X c

I d (21)

By dividing to V acbase either side of Eq. (21);

V dV acbase =

3 2 V t cos

V acbase 3 X c

I dV acbase

(22)

V dV acbase =

3 2 V t cos

V acbase 3 X c

I d 3 Z acbase I acbase (23)

Per unit value of DC link voltage is calculated as follows:

vd =3 2

v t cos

3

x cid (24)

pd =vdid (25)Opencircuit voltageforunload caseis representedas follows:

vdo =3 2

vt (26)

The phase difference between DC link voltage and open cir-cuit voltage for unload case is represented as follows:

d

=cos1 (vd /v do ) (27)

The reactive power given to the converters is represented asfollows:

qd =| pd tan d| (28)

3 Illustration of GA

GA is heuristic algorithm based on natural selection. GAswere rst utilized by Holland in 1975 for solving optimiza-tion problems [ 32]. The base logic of the algorithm is thatgens of powerful individualsarebased to be carried over nextgeneration and others are based to be detached in next gen-eration. In natural selection, a human born, grows and dies.These stages of human life correspond to the different oper-ators in the algorithm. GA operators related these stages canbe explained as the following.

3.1 Initial population

Initial population is determined in two ways. One of them isto form random individuals as initial population size within

their limits. The second way is to form initial populationof the certain individuals. The tness values of individualswithin population are obtained to be put into objective func-tion of the formed individuals. The individual number withininitial population is randomly determined by Eq. (29).

w i j =wmin , j +rand (0, 1) wmax , j wmin , j (29)where the parameters wmin , j and wmax , j show the minimumand maximum of the variable w j .

3.2 Fitness scaling

The scaling stage is one of GA operators. The scaling pre-vents algorithm to get stuck on a local point. There are dif-ferent scaling methods such as rank scales, top scales andshift linear scales. In this study, the better individuals thanindividual having average tness value are selected and canbe formulated as follows:

F ave

=

N k

i=1F i

N k (30)

where F ave , N k and F i show the average tness value withinpopulation, the number of individuals within population andthe tness value of i th individual, respectively.

3.3 Selection

In this stage, the parents to be crossed for producing a childare selected. There are different selection methods such asstochastic uniform, remainder, uniform, shift linear, rouletteand tournament. In this study, the tournament method is pre-ferred and can be formulated as follows:

i =F i

N k j=1 F j

(31)

where i shows the weight of i th individual within popula-tion. Furthermore, the sum of the elective probabilities of allthe individuals within population is 1 as given in Eq. ( 32).

N k

i=1 i =1 (32)

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Electr Eng (2014) 96:6577 69

Fig. 3 The simple ow scheme of GA [34]

The twice individual of the children number determined inthe beginning of the algorithm is selected from individualswithin population for crossover. The percentage distributionof the individuals within population is proportional to its t-ness value. The best individuals are shown by the high-levelpercentage distributions in roulette selection. After all theindividuals are shown as the percentage distribution between0 and 1, the roulette selection in the denite number is per-formed. Thus, the individuals are selected for crossover.

3.4 Crossover

In this stage, a child is produced to be crossed the parents.New individuals who are the same as the determined num-ber are produced in order to be used in the crossing methodwith the scattered parameters from parents selected via thetournament method as explained in the selection stage.

1 and 0 values same as gen number of individual are ran-domly produced. If the value is 1, then gen is taken frommother, the value is 0, then gen is taken from father and thusthe child is produced.

Cross: 1 0 1 1 0Mother: a b c d eFather: x y z u wChild: a y c d w

3.5 Mutation

In mutation stage, new individuals are produced to bechanged all or some gens of the selected individuals withinpopulation. The number of individuals undergoing mutationhas to be determined in the beginning of the algorithm. Theindividuals undergoingmutationare reproduced to be formedall the gens of the selected individuals within algorithm.Thus, new individuals same as the number determined byEq. (29) are randomly produced.

3.6 Stopping algorithm

There are many criterions for the stopping algorithm. Someof these are the tness value, time and iteration number. Inthis study, iteration number is preferred as the stopping crite-rion. More information related to GA operators is availablein Ref. [33]. Final population is formed to be included thereproduced individuals in stages above to initial population.After the individuals within nal population are classiedaccording to tness value, the individual same as initial pop-ulation is carried over the next iteration. The simple owscheme of GA is shown in Fig. 3 [34].

4 GA-based optimal power ow solution of two-terminal HVDC system

It is signicant todeterminethe stateandthe control variablesto solve the OPF problem of two-terminal HVDC system.Furthermore, these control variables should be the same as

those of the problem to be optimized. The control variablesof this optimization problem are selected as follows:

u =[uAC, u DC] (33)u AC = pg2 , . . . , pg N g , v g1 , . . . , v g N g , t 1 , . . . , t N T (34)u DC =[ pdr , pdi , qdr , qdi , id] (35)where pgi except the slack bus pgslack is the generator activepower outputs, vgi is the generator voltage magnitudes, N gis the number of generator buses and N T is the number of transformers.

The state variables of this optimization problem are

selected as follows: x = [ x AC, x DC] (36)

x AC = pgslack , qg1 , . . . , qg N g , v L1 , . . . , v L N l (37) x DC = t r , t i, , , v dr , v di (38)where pgslack is the slack bus active power output, qgi is thereactive power outputs, v Li is the load bus voltage magni-tudes and N l is the number of load buses.

Equations ( 39) and (40) are used for updating active andreactive power in rectier and inverter sides.,

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70 Electr Eng (2014) 96:6577

For rectier bus:

pupdateload = pload + pdrq updateload =qload +qdr (39)For inverter bus:

pupdateload

= pload

pdi

q updateload =qload +qdi (40)The tness value for each individual is obtained by theEq. (41) as follows:

F i =K 1 F cos t +K 2 pgslack plimgslack

+K 3 N g

i=1qgi q limgi +K 4

N l

i=1v Li v lim Li

+K 5 t r t limr +K 6 t i t

limi +K 7

lim

+K 8 lim

+K 9 vdr vlimdr +K 10 vdi v

limdi

(41)

where plimgslack , qlimgi , v

lim Li , t

limr , t

limi ,

lim , lim , v limdr and vlimdi

show the limits of the related variables, respectively; K 1 , K 2 ,K 3 , K 4, K 5, K 6, K 7 , K 8 , K 9 and K 10 are penalty weights of total fuel cost, real power output of slack bus, reactive poweroutputs of generator buses, load bus voltage magnitudes, theeffective transformer ratiosof rectier and invertersides, theanglesof rectierandinverter sides,the directvoltages of rec-tier and inverter sides respectively. Note that the value of penalty function grows with a quadratic form when the con-

straints are violated and is 0 in the region where constraintsare not violated.Thepower ow calculation foreach individual in heuristic

methods is performed and a tness value F i for each individ-ual is calculated to evaluate its quality as follows:

Step 1: Update pload and qload in rectier and invertersides using the Eqs. ( 39) and (40)Step 2: Run NewtonRaphsonStep 3: Calculate the phase angle in rectier and invertersides using Eq. (42)

=arctan qd pd (42)

Step 4: Calculate the direct voltage at either side usingEq. (43)

vd = pdid

(43)

Step 5: Calculate the ideal no-load directvoltage at eitherside using Eq. (44)

vdo =vd

cos ( )(44)

Step6: Calculatethe effective transformer ratiosat eitherside using Eq. ( 45)

t

=vdo

3 2 v(45)

Step 7: Calculate the ignition delay angle and the extinc-tion advance angle at either side using Eq. (46)

, =arccosvd +

3 x cid

vdo (46)

Step 8: Calculate F cos t using Eq. (47)

F cos t = N g

i=1a i +bi pgi +ci p

2gi (47)

where F cos t represents the total fuel cost, N g the totalnumber of generators, pgi the active power generation of i th generator and nally a i , bi and ci represent the fuelcost coefcients of i th generator.Step 9: Calculate the tness value F i using Eq. (41)

The application procedure of GA to OPF problem is illus-

trated step by step as follows:

Step 1. Read system data and GA parametersStep 2. Generate initial population of m individuals viacontrol variable uStep 3.Calculatethe tnessvalue F i ofeachchromosomein the populationStep 4. Create a new population by repeating the follow-ing steps until the new population is completedStep 5. Select the parents by tournament selectionStep 6. Crossover theparent chromosomes with scatteredto form a new childStep 7.Mutate new individualwith a mutation probabilityStep 8. Calculate the tness value F i of each new childStep 9. Include new individuals to initial populationStep 10. Classify all the individuals according to the t-ness valueStep 11. Carry over the individual same as initial popu-lation the next iterationStep 12. If the stopping criterion is satised, stop thealgorithm and show the best solution in current popula-tion else, go to Step 4.

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Fig. 4 The modied 5-node test system with an HVDC link [ 35]

5 Simulation results

To show the validity, efciency and effectiveness of GA inoptimal power ow solution of two-terminal HVDC system,the proposed method is tested on two test systems which arethemodied 5-nodetest system[ 35] and the modied WSCC9-bus test system [ 36] in this study. The data for either testsystem are given in Appendix [ 3537].

In thestudy, thestoppingcriterion of GAis selectedas100iterations. This value is the same as that in the previous stud-ies in the literature. The needed time to determine the globaloptimum also increase as the iteration number is increased

(> 1,000 iterations). Therefore, in selection of the iterationnumber, several experiments are performed for differentpop-ulation sizes. As a result of the performed experiments, thebest individual number is determined as 50, the best childnumber is determined as 25 and the best individual numberundergo mutation is determined as 10 for the GA. The run-ning times of software for two test systems are obtained as12.58 and 47.44 s, respectively.

5.1 The modied 5-node test system

This modied system having the ve buses and two gener-

ators is extended with two-terminal HVDC link connectedbetween Lake and Main buses. In this contrived example,the AC network and HVDC converters are assumed to work under three-phase balanced conditions. The modied 5-nodetest system with two-terminal HVDC link is shown in Fig. 4.

The study is performed for two different cases accordingto power and current of DC link.

Case A:The current ofDClink is takeninto considerationas 0.10 p.u.

Table 1 Upper and lower limitsof thecontrol variables of the modied5-node test system

Variables Case A/Case B Variables Case A Case B

u AC uminAC umaxAC uDC u

minDC u

maxDC u

minDC u

maxDC

pg2 0.1 2.0 pdr 0.1000 0.1500 0.1500 0.2250

vg1 0.90 1.10 pdi 0.1000 0.1500 0.1500 0.2250

vg2 0.90 1.10 qdr 0.0 0.1 0.0 0.1

qdi 0.0 0.1 0.0 0.1

id 0.1000 0.1000 0.1500 0.1500

Table 2 Upper and lower limits of the state variables of the modied5-node test system

Variables Case A/Case B Variables Case A/CaseB

x AC x minAC x minAC x DC x

minDC x

maxDC

pgslack 0.1 2.0 t r 0.90 1.10

qg1 3.0 3.0 t i 0.90 1.10qg2 3.0 3.0 10 20v L1 0.90 1.10 15 25

v L2 0.90 1.10 vdr 1.00 1.50

v L3 0.90 1.10 vdi 1.00 1.50

Case B:Thecurrent of DClink is taken into considerationas 0.15 p.u.

Upper and lower limits of the control variables and the statevariables related to two-terminal HVDC system are given inTables 1 and 2, respectively.

Thegenerator active andreactive poweroutputs andnodalvoltages of the modied 5-node test system for cases A andB are shown in Table 3. It can be seen that the minimumgeneration costs for cases A and B are obtained as 748.0335and 748.1335 $/h. These generation costs are obtained byrunning 20 times of the algorithm. The voltage magnitudeobtained for North bus in Ref. [ 35] was 1.109 p.u. This valueis higher than its upper limit. Furthermore, the limit overowin the voltage of North bus is overcome by GA.

OPF results obtained by method in Ref. [ 35] and GA aregiven in Table 4. It can be seen clearly from this that theminimum generation cost obtained by GA is lesser than thatreported in Ref. [35].

DC link parameter results obtained using GA of the mod-ied 5-node system for cases A and B are given in Table 5.It can be seen clearly that all the parameters of DC link arewithin their upper and lower limits.

The tness value variation against iteration for cases Aand B is shown in Fig. 5. The optimum points for casesA and B are obtained by 12 and 30 iterations, respectively.

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Table 3 The generator activeand reactive power outputs andnodal voltages of the modied5-node test system for cases Aand B

Case A Case B

Bus name: v pg qg v pg qg

North 1.099632 0.794997 0.140369 1.099498 0.810151 0.188901South 1.094199 0.885547 0.064935 1.096783 0.870860 0.142612

Lake 1.068605 1.067430

Main 1.077238 1.079512

Elm 1.071843 1.074457

Cost ($/h) 748.0335 748.1335

Losses (MW) 3.0544 3.1011

Table 4 Comparative results of OPF for 5-node system

Case A Case B

GA Ref. [35] GA Ref. [35]

Cost ($/h) 748 .0335 748 .156 748 .1335 748 .451

Losses (MW) 3 .0544 3.094 3.1011 3.168

Again, the optimum points for cases A and B are obtainedwithin 1.5096 and 3.774 s, respectively.

The reactive power variations in rectier and invertersidesagainst iteration number for cases A and B are shown in Figs.

Table 5 DC link parameter results obtained using GA for the modied5-node system

Control Effective Active Reactive DCangles transformer power power current

(degree) ratio (p.u.) (p.u.) (p.u.)

Case ARec. 10.39955 1.052508 0.149325 0.027793 0.1000

Inv. 15.26741 1.064044 0.149292 0.040908

Case BRec. 10.55394 0.913851 0.194103 0.037022 0.1500

Inv. 17.40562 0.930288 0.194028 0.061141

6 and 7, respectively. The transformer ratio variations at bothside against iteration number for cases A and B are shown inFigs. 8 and 9, respectively.

For pmaxdr = pmindr = 0.1371 p.u. and pmaxdr = pmindr =0.1945 p.u. which correspond id = 0.1 p.u. and id =0.15 p.u., the fuel costs are obtained as 748.0708 and748.2054 $/h, respectively. For the same pdr and id , the fuelcosts were obtained as 748.156 and 748.451 $/h, respec-tively, in Ref. [ 35]. According to these results, it can beseen that GA is more effective and superior than method inRef. [35].

5.2 The modied WSCC 9-bus test system

The modied WSCC 9-bus test system extended with two-terminal HVDC link is shown in Fig. 10. AC transmission

between buses4 and5 in theoriginalWSCC9-bustestsystemis replaced with two-terminal HVDC link (Fig. 10) [36].

The upper and lower limits of the control and state vari-ables related to the test system are given in Tables 6 and 7,respectively.

According to the obtained results, the variations of thebest, the worst and the average tness value for each individ-ual against iteration number are shown in Fig. 11.

Fig. 5 Fitness value variationsagainst iteration number for

cases A and B

10 20 30 40 50 60 70 80 90 100740

750

760

770

780

790

800

810

Iteration number

F i t n e s s v a

l u e

Case A Case B

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Electr Eng (2014) 96:6577 73

Fig. 6 The reactive powervariations at either side againstiteration number for case A

10 20 30 40 50 60 70 80 90 1000.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

Iteration number

R e a c

t i v e p o w e r

( p . u . )

Rectifier Side Inverter Side

Fig. 7 The reactive powervariations at either side againstiteration number for case B

10 20 30 40 50 60 70 80 90 100

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Iteration number

R e a c

t i v e p o w e r

( p . u . )

Recti fier Side Inverter Side

Fig. 8 Transformer ratiovariations at either side againstiteration number for case A

10 20 30 40 50 60 70 80 90 1000.85

0.9

0.95

1

1.05

1.1

Iteration number

T r a n s f o r m e r r a

t i o


Fig. 9 Transformer ratiovariations at either side againstiteration number for case B

10 20 30 40 50 60 70 80 90 1000.85

0.9

0.95

1

1.05

1.1

Iteration number


t i o


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74 Electr Eng (2014) 96:6577

Fig. 10 The modied WSCC 9-bus and 3-machine test system [ 36]

Table6 Upper andlower limitsof thecontrol variables forthe modiedWSCC 9-bus system

u AC uminAC umaxAC uDC u

minDC u

maxDC

pg2 0.10 3.0 pdr 0.1000 1.500

pg3 0.10 2.7 pdi 0.1000 1.500

vg1 0.90 1.10 qdr 0.0 1.0

vg2 0.90 1.10 qdi 0.0 1.0

vg3 0.90 1.10 id 0.1 1.0

t 14 0.85 1.15t 27 0.85 1.15t 39 0.85 1.15

The best individual within these three individuals reachesto the optimum point in almost 15 iterations. Again, the com-putational times are obtained in 7.116 s. In Table 8, thefuel costs obtained via the best, the worst and the averageindividual are given as 1,145.9525, 1,147.9315 and1,146.942 $/h, respectively. The reactive power variationsof rectier and inverter sides to iteration number are shownin Fig. 12.

Table 7 Upper and lower limits of the state variables for the modiedWSCC 9-bus system

x AC x minAC x minAC x DC x

minDC x

maxDC

pgslack 0.1 2.5 t r 0.85 1.15

qg1 3.0 3.0 t i 0.85 1.15qg2 3.0 3.0 7.00 10.00qg3 3.0 3.0 10.00 15.00v L1 0.90 1.10 vdr 1.00 1.50v L2 0.90 1.10 vdi 1.00 1.50

v L3 0.90 1.10

v L4 0.90 1.10

v L5 0.90 1.10

v L6 0.90 1.10

Table 8 The obtained fuel costs for the modied WSCC 9-bus system

Best Worst Average

Cost ($/h) 1,145.9525 1,147.9315 1,146.942Time (s) 7.116

The transformer ratio variations at either side against iter-ation number are shown in Fig. 13.

DC and AC load ow results of the modied WSCC 9-bussystem are given in Tables 9 and 10, respectively. As seenfrom Tables 6, 7, 9, and 10, all the results are kept withintheir upper and lower limits.

6 Conclusion and discussion

In this study, OPF problem of the modied 5-node testsystem and the modied WSSC 9-bus test system is rstsolved to show the validity and the effectiveness of GA. Theresults obtained for two different cases were compared tothat reported in Ref. [ 35]. As seen clearly from compara-tive OPF results, the results obtained by GA are better thanthat obtained by numerical method proposed in Ref. [ 35].

Fig. 11 The variations of thebest, the worst and the averagetness value for each individual

against iteration number

10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12 x 10

4

Iteration number

F i t n e s s v a

l u e

Best Worst Average

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Electr Eng (2014) 96:6577 75

Fig. 12 The reactive powervariations at either side againstiteration number

10 20 30 40 50 60 70 80 90 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Iteration number

R e a c t

i v e p o w e r

( p . u . )


Fig. 13 Transformer ratiovariations at either side againstiteration number

10 20 30 40 50 60 70 80 90 1000.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Iteration number


t i o


Table 9 DC load ow results of the modied WSCC 9-bussystem

Control angles Effective transformer Active power Reactive power DC current(degree) ratio (p.u.) (p.u.) (p.u.)

GARec. 9.417373 0.976568 0.136029 0.024080 0.1000Inv. 10.474138 1.109794 0.136019 0.026697

Table 10 AC load ow resultsof the modied WSCC 9-bussystem using GA

Bus number v pg qg Transformer tap ratio

1 1.063640 0.96905 0.01092 t 14 1.0161652 1.083900 1.22509 0.70183 t 27 0.9981103 1.061280 1.02942 -0.13526 t 39 1.0227534 1.047477

5 0.924871

6 1.034400

7 1.047932

8 1.034598

9 1.046922

Furthermore, GA has many advantages such as easy conver-gence and adaptability. GA converges reliably and rapidlyto the optimal solution and can be also adapted to the other

areas. In future, this algorithm and the other heuristic algo-rithmscanbeapplied easilyto OPFproblemof thelarge-scalemulti-terminal HVDC systems.

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76 Electr Eng (2014) 96:6577

Appendix

A. 5-node system data

See Tables 11, 12, 13.

Table 11 AC system bus data of the 5-node systemBus name pload qload a i bi ci

North 0.00 0.00 60 340 40

South 0.20 0.10 60 340 40

Lake 0.45 0.15

Main 0.40 0.05

Elm 0.60 0.10

Table 12 AC transmission line characteristics of the 5-node system

From bus To bus R (p.u.) X (p.u.) B/ 2 (p.u.)

North South 0.02 0.06 0.06

North Lake 0.08 0.24 0.05

South Lake 0.06 0.18 0.04

South Main 0.06 0.18 0.04

South Elm 0.04 0.12 0.03

Main Elm 0.08 0.24 0.05

Table 13 DC transmission link characteristics of the 5-node system

AC node Rectier Inverter

Lake Main

Commutation reactance (p.u.) 0.01260 0.00728

DC-link resistance (p.u.) 0.00334

B. WSCC 9-bus system data

See Tables 14, 15, 16.

Table 14 AC system bus data of the WSCC 9-bus test system

Bus number pload qload a i bi ci

1 0.00 0.00 140 200 60

2 0.00 0.00 120 150 75

3 0.00 0.00 80 180 70

4 0.00 0.00

5 1.25 0.50

6 0.90 0.30

7 0.00 0.00

8 1.00 0.35

9 0.00 0.00

Table 15 AC transmission line characteristics of WSCC 9-bus system

From bus To bus R (p.u.) X (p.u.) B/ 2 (p.u.)

1 4 0.0000 0.0576 0.0000

4 6 0.0170 0.0920 0.0790

5 7 0.0320 0.1610 0.1530

6 9 0.0390 0.1700 0.1790

2 7 0.0000 0.0625 0.00007 8 0.0085 0.0720 0.0745

8 9 0.0119 0.1008 0.1045

3 9 0.0000 0.0586 0.0000

Table 16 DC transmission link characteristics of WSCC 9-bus system

Rectier Inverter

Bus number 4 5

Commutation reactance (p.u.) 0.0459 0.0518

DC link resistance (p.u.) 0.001

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Optimal Power Flow Solution of Two-terminal HVDC Systems Using Genetic Algorithm

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