Electrical Power and Energy Systems 32 (2010) 368–374

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Electrical Power and Energy Systems

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Improved particle swarm optimization applied to reactive powerreserve maximization

L.D. Arya a,*, L.S. Titare b,1, D.P. Kothari c

a Department of Electrical Engg., SGSITS, 23-Park Road, Indore, MP 452 003, Indiab Department of Electrical Engg., Govt. Engineering College, Jabalpur, MP 456 010, Indiac Centre for Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 June 2006Received in revised form 5 October 2009Accepted 6 November 2009

Keywords:Voltage stabilityParticle swarm optimization (PSO)Reactive power reserveSchur’s inequalityProximity indicator

0142-0615/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijepes.2009.11.007

* Corresponding author. Tel.: +91 0731 2321016; faE-mail addresses: ldarya@rediffmail.com (L.D.

(L.S. Titare), dpk0710@yahoo.com (D.P. Kothari).1 Tel.: +91 0734 2514246.

This paper presents a new approach for scheduling of reactive power control variables for voltage stabil-ity enhancement using particle swarm optimization (PSO). Cost function selected is maximization ofreactive reserves of the system. To get desired stability margin a Schur’s inequality based proximity indi-cator has been selected whose threshold value along with reactive power reserve maximization assuresdesired static voltage stability margin. PSO has been selected because not only it gives global optimalsolution but also its mechanization is very simple and computationally efficient. Reactive generation par-ticipation factors have been used to decide weights for reactive power reserve for each of generating bus.Developed algorithm has been implemented on 6-bus, 7-line and 25-bus 35-line standard test systems.Results have been compared with those obtained using Devidon–Fletcher–Powell’s (DFP) method.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Power system optimization problems including reactive poweroptimization (RPO) have complex and non-linear characteristicswith large number of inequality constraints. Recently, as an alter-native to the classical mathematical approaches, the non-tradi-tional techniques [16] such as genetic algorithm, Tabu search,simulated annealing, and PSO are considered practical and power-ful solution schemes to obtain the global or quasi-global optimumsolution to engineering optimization problems. At times suchschemes are termed as heuristic optimization techniques [1]. PSOhas been selected as an optimization methodology in this paperbecause its mechanization is extremely simple, robustness to con-trol parameters and computational efficiency when compared withmathematical programs and other non-traditional algorithms.Reactive power and voltage control [2], power system stabilizerdesign [3], and dynamic security [4] studies are the areas to whichPSO has been successfully applied. Yoshida et al. [2] suggested amodified PSO to control reactive power flow and alleviating volt-age limit violations. The problem was a mixed-integer, non-linearoptimization problem with inequality constraints.

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x: +91 07312432540.Arya), lstitare@yahoo.co.in

Availability of reactive power at sources and network transfercapability are two important aspects, which should be consideredwhile rescheduling of reactive power control variables. Nedwicket al. [5] have presented a reactive management program for apractical power system. They have discussed a planning goal ofsupplying system reactive demand by installation of adequatelysized and adequately located capacitor banks which will permitthe generating unit near to unity power factor. Vaahedi et al. [6]developed a hierarchical optimization scheme, which optimized aset of control of variables such that the solution satisfied a speci-fied voltage stability margin. Menezes et al. [7] introduced a meth-odology for rescheduling reactive power generation of plants andsynchronous condenser for maintaining desired level of stabilitymargin. In Ref. [8] an algorithm for voltage stability enhancementhas been presented for rescheduling of reactive power control vari-ables for voltage stability margin improvement using linearizedincremental model. Dong et al. [9] developed an optimized reactivereserve management scheme using Bender’s decompositiontechnique. Ant colony system algorithm is applied to the reactivecontrol problem in order to minimize real power losses, subjectto operating constraints over the whole planning period byVlachogiannis et al. [10]. Yang et al. [23] presented a techniquefor reactive power planning based on chance constrained program-ming accounting uncertain factors. Generator outputs and loaddemands modeled as specified probability distribution. MonteCarlo simulation along with genetic algorithm has been used forsolving the optimization problem. Wu et al. [24] described an

L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374 369

OPF based approach for assessing the minimal reactive power sup-port for generators in deregulated power systems. He et al. [25]proposed a method to optimize reactive power flow (ORPF) withrespects to multiple objectives while maintaining voltage security.Varadarajan and Swarup [26] applied differential evolutionaryalgorithm for optimal reactive power dispatch and voltage control.Zhang and Liu [27] developed an algorithm for reactive power andvoltage control using fuzzy adaptive particle swarm optimization(FAPSO). Zhang et al. [28] developed a computational method forreactive power market clearing.

Reactive power reserve available at source is an important andnecessary requirement for maintaining a desired level of voltagestability margin. Power network may have the transfer capabilityof reactive power but if reserve is not available and reactive powerlimit violation occurs than the static voltage stability limit may beinadequate. Further reactive reserves available at sources will notbe of much help in maintaining desired level of stability margin,if network transfer capability is limited. This paper proposes amethodology for voltage stability enhancement accounting net-work loading constraint as well as optimizing reactive power re-serves at various sources in proportion to their participationfactors decided based on incremental load model. Voltage depen-dent reactive power model has been used for determining reactivepower reserves, which utilizes field heating as well as armatureheating limit. Section 2 explains problem formulation. Section 3presents an overview of PSO technique. Section 4 presents imple-mentation of the algorithm for optimizing reactive reserves. Sec-tion 5 gives results and discussions. Section 6 gives conclusionsand highlights of main contributions of the paper.

2. Problem formulation

2.1. Reactive reserve

The voltage stability enhancement problem is formulated as anoptimal search problem whose objective is two fold: (i) maximizethe reactive reserves based on the participation of reactive sourcesfor increased loading condition and (ii) maintaining the desiredstability margin with respect to current operating point. Reactivepower reserve is the ability of the generators to support bus volt-ages under increased load or disturbance condition. Amount ofreactive power, which can be fed to network, depends on presentoperating condition, location of the source, field and armatureheating of the alternators. Nature of the change in load scenarioalso has impact on reactive reserves. Availability of reactive powerreserve of a generator is calculated using capability curves. For a gi-ven real power output the reactive power generation is limited byboth armature and field heating limit [11]. Maximum reactivepower output with respect to field current limit is expressed as:

Q g;max ¼ �ðV2g=XdÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððV2

g � E2maxÞ=X2

dÞ � P2g

qð1Þ

Maximum reactive power output Qg,max of the generator isdetermined by internal maximum voltage Emax corresponding tothe maximum field current. Thus maximum reactive power outputis determined not only on real power output Pg but also on termi-nal voltage Vg. Maximum reactive power output due to armaturecurrent limitation as follows:

Q g;max ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

g � I2g;max � P2

g

qð2Þ

Ig,maxis maximum armature current of the generator. The reac-tive power reserve of the gth generator is then represented as:

Q g;max;res ¼ Q g;max � Q g ð3Þ

where, Qg,max is the smaller of the two values obtained from Eqs. (1)and (2). Reactive reserve is calculated using relation (3) if Qg is less

than Qg,max. However if Qg reaches its limit the reactive reserve is setto zero. The bus is treated as variable voltage and internal voltage ofthe generator behind synchronous reactance is assumed constant.This way voltage dependent reactive power limits have been ac-counted in a realistic way. In such situation ‘Qg’ moves on the capa-bility curve governed by Eqs. (1) and (2). Hence if ‘Qg’ reaches theboundary the reactive reserve is set to zero and then ‘Qg’ varies asa function of terminal (bus) voltage. Such modeling (voltage depen-dent reactive power limit) has been adopted by many researchers[9,17–19].

2.2. Proximity indicator based on Schur’s inequality

It is assumed that load flow Jacobian at current solution point isknown. Following relation can be written based on Schur’s inequality[12]:

Skmax 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiXi;j

s2i;j

sð4Þ

where, Skmax is greatest eigenvalue of sensitivity matrix given, asthe inverse of load flow Jacobian, si,j is ijth element of sensitivitymatrix [S], which is inverse of load flow Jacobian. It is observed frommatrix theory that minimum eigenvalue magnitude of load flowJacobian is reciprocal of greatest eigenvalue of sensitivity matrix[S]. Hence following relation follows:

Jkmin P 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiX

i;j

s2i;j

sð5Þ

Jkmin is minimum eigenvalue of Jacobian.Right hand side of the above expression is lower bound on the

minimum eigenvalue and termed in further application of this pa-per as proximity indicator (s).

Under low loading condition elements of sensitivity matrixare smaller and value of proximity indicator is large. As the loadon the system increases the value of proximity indicator de-creases since element of sensitivity matrix (si,j) increases in mag-nitude. In the vicinity of collapse point the value of proximityindicator practically becomes zero. Hence magnitude of ‘s’ hasbeen used for voltage stability assessment and control in this pa-per. For secure operation a threshold value of proximity indica-tor must be maintained. Variation of proximity indicator canbe co-related with load on the system with the help of powerflow run. A magnitude of ‘s’ which provides adequate voltagestability margin (distance to voltage collapse point from currenttotal load) is selected as threshold value. This value is systemdependent.

Computation of proximity indicator requires Jacobian inver-sion, which is available directly at the end of current load flowsolution. Further computational efficiency is achieved using spar-sity and inversion using LU factorization [21,22]. Developed algo-rithm is for base point setting of reactive power controlvariables.

2.3. Mathematical formulation

The reactive reserve optimization problem is formulated assearch problem whose objective is to maximize the effective reac-tive reserve subject to various operating and stability constraints.Objective function is given as follows:

J ¼MaxX

Wi � Q i;res ð6Þ

Above objective function is optimized subject to followingconstraints:

370 L.D. Arya et al. / Electrical Power and En

(i) Power flow equations:

P ¼ f ðV ; dÞQ ¼ gðV ; dÞ

ð7Þ

(ii) Inequality constraints on load bus voltages:

Vmini 6 Vi 6 Vmax

i i 2 NL ð8Þ

(iii) Voltage stability constraint:

s P sth ð9Þ

sth is threshold value of ‘s’ proximity indicator

(iv) Reactive power generation constraint:

Qminp 6 Q p 6 Q max

p p ¼ 1;2; . . . ;NG ð10Þ

It is stressed here that Qmaxp is voltages dependent and

obtained using Eqs. (1) and (2). It is further clarified here thatQmin

p based on minimum rotor current limiter, the purpose ofwhich is to avoid very small rotor current (these may causeproblem for excitation systems) are of interest for synchro-nous compensators not for synchronous generators [17].

(v) Inequality constraint on control variables:

Umini 6 Ui 6 Umax

i i 2 NC ð11Þ

NL and NC denotes set of load buses and number of controlvariables.

In objective function (6) Qi,res denotes reactive power reserveavailable at ith generation bus and Wi is the weighting factor forith generator bus. Weighting factor Wi is selected based on the rel-ative participation of each generator to reactive load increase inspecific direction in load parameter space. The bus, which partici-pates to a smaller extent, is given higher weight and the bus par-ticipating to a greater extent should be given lesser weight suchthat reserve at such bus is reduced and increased respectively.Such weighting factors can be obtained by incremental power flowequations with some algebraic manipulation. Following sensitivityrelation is obtained using incremental power flow equation:

SQDi ¼ @Q i=@Q d ð12Þ

Qi and Qd represent reactive power injection at ith generator busand total reactive power demand of the system. This sensitivityrepresents change in reactive power injection at ith bus with re-spect to change in total reactive power demand of the system. Thenweighting factor for ith generator bus is given as:

Wi ¼X

p

SQDp

!,SQDi ð13Þ

The search optimization problem is solved using particle swarmoptimization (PSO). Next section presents an overview of PSOtechnique.

3. Particle swarm optimization technique: an overview

The PSO is a population-based optimization algorithm. Its pop-ulation is called swarm and each individual is called a particle [13].Each particle flies through the solution space to search for globaloptimal solution. The mechanization of the PSO procedure is ex-plained in following steps:

Step-1 A current position is an n-dimensional search space,which represents a potential solution (particle or agent):

UKi ¼ ðuK

i;1; uKi;2; . . . ; uK

i;nÞ

i denotes ith particle and K denotes the iteration.

Step-2 A current velocity for ith agent in Kth iteration is givenas:

ergy Systems 32 (2010) 368–374

qðKÞi ¼ ðqKi1;q

Ki2; . . . ;qK

inÞ

Step-3 At each iteration, the particle is updated by the followingrelation:

qðKþ1Þi ¼W � qK

i þ c1�r1ð�ÞðPKbestðiÞ � UK

i Þ þ c2 � r2ð�ÞðGKbest � UK

i Þð14Þ

UðKþ1Þi ¼ UK

i þ qKþ1i ð15Þ

where, PKbestðiÞ is the best previous position of ith particle,

GKbest is the global best position among all the participa-

tion in the swarm from objective function viewpoint,r1(�) and r2(�) are random digit generated from uniformdistribution [0, 1], W is an inertia weight that is typicallychosen in the range [0, 1]. A large inertia weight facili-tates global exploration and a smaller inertia weighttends to facilitate local exploration to fine-tune the cur-rent search area. Therefore the inertia weight W is animportant parameter for the PSO’s convergence behavior.A suitable value for the inertia weight usually providesbalance between global and local exploration abilitiesand consequently results in a better optimum solution.In view of this it has been suggested that iteration wisethe weight W is varied according to following relation[1]:

W ¼Wmax � ððWmax �WminÞ=NITÞ � NITmax

where NITmax is the maximum number of iteration sup-plied and NIT denotes current number of iteration. Wmax

and Wmin denote maximum and minimum values of iner-tia weights. Thus as iteration increases inertia weightvaries from Wmax say 2.0 to Wmin say 0.5, c1 and c2 areacceleration constant selected in the range 1–2.

4. Implementation of PSO for reactive reserve optimizationproblem

In this section PSO implementation to reactive reserve optimi-zation as formulated in Section 2 has been explained. The processof optimization is summarized as follows:

Step-1 Determination of initial population:

Each particle in the population consists of NC componentas reactive power control variables. Each reactive powercontrol variables for an agent is selected from a uniformdistribution between Ui,min and Ui,max satisfying theinequality constraint. ‘M’ such set of initial populationmay be selected. The following procedure is adopted forinitialization of an agent: (a) Set j = 1(b) Select a random digit from [0, 1].(c) Using the random digits create the value of Uj, satisfy-

ing the inequality constraint (11).(d) If j P NC (total number of control variables) then go

to step (e) otherwise j = j + 1 and repeat from (b)(e) Stop the initialization.

Step-2 This step consists of initialization of velocities of eachparticle selected in step-1 above. Initial velocity of eachparticle is also created at random. The velocity of ele-ment ‘j’ of individual is generated at random within theboundary given as follows:

7

L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374 371

Uminj � U0

ij 6 q0ij 6 Umax

j � U0ij

5

6

n (J

)

The initial P0bestðiÞ of individual i is set as the initial posi-

tion of individual i and the initial G0ðbestÞ is determined

as the position of an individual with maximum valueof objective function as obtained using relation (6).

0

1

2

3

4

No of iterations

Obj

ectiv

e fu

nctio

0 20 40 60

Fig. 1. Plot of objective function with respect to number of iteration for 6-bussystem.

10

Step-3 Velocity of each individual is updated using relation(14).

Step-4 The position of each individual is modified using relation(15). The resulting position of an individual does notguarantee the satisfaction of inequality constraints.Hence in this paper a modified constraint handlingmethod has been adopted [13]. In the proposed proce-dure the intuitive idea to maintain a feasible populationis for a particle to fly-back to its previous position, whenit is outside the feasible region. This has been termed asfly-back mechanism. Since the population is initializedin the feasible region flying back to a previous positionwill guarantee the solution to be feasible. Flying backto its previous position when a particle violates the con-straint will allow a new search closer to boundary.

Step-5 This step consists of updating the Pbest(i) and Gbest. ThePbest at (K + 1)th iteration is updated as follows:

9

PðKþ1ÞbestðiÞ ¼ UðKþ1Þ

i ; if JðKþ1Þi > JðKÞi ¼ PðKÞbestðiÞ; if JðKþ1Þ

i 6 JðKÞi

7

8n

(J)

where Ji is the objective function (6) evaluated at the po-

6

nctio

sition of individual ‘i’. GðKþ1Þ

best at iteration is set as the bestvalue in term of objective function among PðKþ1Þ

bestðiÞ .

1

2

3

4

5

Obj

ectiv

e fu

Step-6 Further a search reduction strategy [1] has been

employed to accelerate the convergence. Space reduc-tion strategy is introduced at a stage, when the enhance-ment in the objective function has not been taken placeduring a pre-specified iteration period. In such situationthe search space is dynamically reduced according tofollowing relation:

00 20 40 60

Number of iterations

Fig. 2. Plot of objective function with respect to number of iteration for 25-bussystem.

UðKþ1Þj;max ¼ UðKÞj;max � ðU

ðKÞj;max � GðKÞbestðjÞÞ � D

UðKþ1Þj;min ¼ UðKÞj;min þ ðU

ðKÞj;min � GðKÞbestðjÞÞ � D

9=; ð16Þ

where, D is known as step size, which is pre-specified. Infact magnitude of D will decide how search space isreduced.

Step-7 The computational algorithm is stopped if maximumnumbers of iteration have been executed.

PSO is a heuristic search procedure and as such no analyticalconvergence criterion exists, except that PSO is terminated whena pre-specified maximum numbers of iterations are executed [1].This maximum number of iterations may be specified based onexperience on the system. Further the convergence is seen by plot-ting a graph between objective function and number of iterations(see Figs. 1 and 2).

5. Results and discussions

The developed algorithm has been implemented on 6-bus and25-bus systems [14]. 6-bus system consists of two generator busesand four load buses. This system has in all 6-reactive power controlvariables. Two generators buses and shunt compensation at busNos. 4 and 6. OLTCs provided at line numbers 4 and 7. Maximuminternal voltages and synchronous reactances were assumed asEmax1 = 2.20 pu, Emax2 = 2.05 pu, Xd1 = 1.0 pu, and Xd2 = 1.15 pu.The limits of PV-bus voltages have been assumed as 0.95 and1.15 pu. Shunt compensation limits were assumed as between0.00 pu and 0.055 pu. OLTC limits have been assumed between

0.90 and 1.10. Total base case real and reactive power load onthe system is 1.35 pu and 0.32 pu.Value of proximity indicator atbase case condition is 0.4251. Table 1 shows PV-bus voltage andall other load bus voltages under base condition. Star marked busesare violating the load bus voltage limit. The desired range of loadbus voltage is 0.95–1.05 pu. The static voltage stability limit is2.199415 pu for base case setting of control variables. The staticvoltage stability limit is total load in pu as obtained using contin-uation power flow up to nose point of PV curve with voltage set-tings of PV-buses as specified above [20]. A threshold value of sas proximity indicator has been assumed as 0.55. Initially five par-ticles were selected satisfying all inequality constraints by proce-dure explained in Section 4 and are given in Table 2. The valuesof c1 and c2 have been selected same as one. As in most power sys-tem problems Wmax = 1.0 and Wmin = 0.5 have been selected. A typ-ical value of D selected as 0.12. Maximum numbers of iterationswere set equal to 50. Table 3 gives maximum value of objectivefunction ‘J’ has been obtained after 19 iterations. Fig. 1 shows var-iation of objective function (J) with respect to number of iteration.It is observed from this Fig. 1 that solution converged in 19 itera-tions. Maximum numbers of iterations were specified as 50. After19th iteration no improvement is found in Gbest. Table 4 givesoptimized set of control variables. In this situation all load bus

Table 1Load flow solution for 6-bus test system under base case condition.

Sr. No. Control variables Control variables magnitudes (pu) Load bus voltages Load bus voltages magnitudes (pu)

1 V1 1.0000 V3 0.8303*

2 V2 0.9500 V4 0.8588*

3 BSH4 0.0500 V5 0.7901*

4 BSH6 0.0550 V6 0.8420*

5 TAP4 1.00006 TAP7 1.0000

Total load; Pd = 1.35 pu, Qd = 0.32pu.Proximity indicator s = 0.4251.* is the buses are violating the bus voltage limit.

Table 2Initial solutions for 6-bus test system for PSO.

Sr. No. V1 (pu) V2 (pu) BSH4 (pu) BSH6 (pu) TAP4 TAP7 J

1 1.1480 1.0933 0.0232 0.0234 0.9481 0.9554 2.338522 1.1448 1.1249 0.0397 0.0491 1.0299 1.0676 2.414683 1.1445 1.1279 0.0450 0.0362 1.0495 1.0408 2.487284 1.1421 1.1405 0.0466 0.0007 0.9142 0.9842 2.586205 1.1478 1.1228 0.0497 0.0075 1.0682 1.0289 1.44843

Table 3Various parameters in PSO for 6-bus test system.

Case Wmax Wmin C1 C2 No. of initial particles Max J No. of iterations for convergence

1 1.0 0.5 1.0 1.0 05 6.0529 19

372 L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374

voltages are also given in the same table. Total reactive reserveavailable 1.42294 pu. Continuation power flow was carried withoptimized set of reactive power control variable and static voltagestability limit was obtained as 2.31484 pu. Best initial solution(particle) selected as V1 = 1.1421 pu, V2 = 1.1405 pu, BSH4 =0.0466 pu, BSH6 = 0.0007 pu, TAP4 = 0.9142, TAP7 = 0.9842. Reactivereserves at bus Nos. 1 and 2 with 6 best initial solution were0.2937 pu and 0.7839 pu. Where as with optimized solution thesereactive reserves are obtained as 0.50874 pu and 0.9142 pu.Weighting factors are 3.1875 and 4.8472 obtained by sensitivityanalysis. Magnitude of proximity indicator with optimized solutionis s = 0.5523.

Similar results have been obtained for 25-bus test system. Thissystem consists of five generator buses and these are reactive powercontrol variables. Remaining 20 buses are load buses. Maximuminternal voltages and synchronous reactances were assumed asEmax1= 2.60 pu, Emax2 = 2.15 pu, Emax3 = 2.10 pu, Emax4 = 2.30 pu,Emax5 = 2.15 pu, Xd1 = 1.00 pu, Xd2 = 1.15 pu, Xd3 = 1.05 pu,Xd4 = 1.20 pu, and Xd5 = 1.15 pu. Total base case real and reactivepower load on the system is 12.41 pu and 3.876 pu.Value of proxim-ity indicator at base case condition is 0.30. Table 5 shows PV-busvoltage and all other load bus voltages under base condition. Star

Table 4Optimized set of control variables and all bus voltages for 6-bus system.

Sr. No. Control variables Optimized control variables magnitudes (pu)

PSO DFP

1 V1 1.0868 1.10422 V2 1.0661 1.07113 BSH4 0.0497 0.04744 BSH6 0.0526 0.05425 TAP4 0.9455 0.95296 TAP7 0.9872 0.9870

Total load Pd = 1.35pu, Qd = 0.32 pu.Proximity indicator using (PSO) s = 0.5523.Proximity indicator using (DFP) s = 0.5590.

marked buses are violating the bus voltage limit. The desired rangeof load bus voltage is 0.95 pu to 1.05 pu.The static voltage stabilitylimit is 17.229396 pu in base case setting of control variables. Athreshold value of s as proximity indicator has been assumed as0.39. Initially five particles were selected satisfying all inequalityconstraints by procedure explained in Section 4 and are given inTable 6. The values of c1 and c2 have been selected same as 1.00.As in most power system problems Wmax = 1.0 and Wmin = 0.5 havebeen selected. A typical value of D selected as 0.12. Maximumnumbers of iterations were set equal to 50. Table 7 gives maximumvalue of objective function ‘J’ has been obtained after 27 iterations.Fig. 2 shows variation of objective function (J) with respect to num-ber of iteration. Solution gets converged in 27 iterations. Maximumnumbers of iterations specified were 50. After twenty seven itera-tions no improvement in objective function was found. Table 8 givesoptimized set of control variables. In this situation all load bus volt-ages are also given in the same table. Total reactive reserve available2.4526 pu.Continuation power flow was carried with optimized setof reactive power control variable and static voltage stability limitwas obtained as 20.132291 pu. Best initial solution (particle)selected as V1 = 1.1373 pu, V2 = 1.0740 pu, V3 = 1.0210 pu, V4 =1.0488 pu, V5 = 1.1002 pu. Reactive reserves at bus Nos. 1–5 with 5

Load bus voltages Load bus voltages magnitudes (pu)

PSO DFP

V3 0.9586 0.9608V4 0.9699 0.9714V5 0.9530 0.9500V6 0.9500 0.9509

Table 5Load flow solution for 25-bus test system under base case condition.

Sr.No.

Controlvariables

Control variablesmagnitudes (pu)

Load busvoltages

Load bus voltagesmagnitudes (pu)

1 V1 1.0000 V6 0.9467*

2 V2 1.0000 V7 0.9421*

3 V3 1.0000 V8 0.9374*

4 V4 1.0000 V9 0.9192*

5 V5 1.0000 V10 0.9398*

6 V11 0.9312*

7 V12 0.9333*

8 V13 0.9472*

9 V14 0.8845*

10 V15 0.8809*

11 V16 0.9065*

12 V17 0.9309*

13 V18 0.9305*

14 V19 0.976415 V20 0.9378*

16 V21 0.8903*

17 V22 0.8425*

18 V23 0.8991*

19 V24 0.8130*

20 V25 0.8282*

Total load Pd = 12.41 pu, Qd = 3.876 pu.Proximity indicator s = 0.30.* is the buses are violating the bus voltage limit.

Table 6Initial solutions for 25-bus test system for PSO.

Sr. No. V1 (pu) V2 (pu) V3 (pu) V4 (pu) V5 (pu) J

1 1.1396 1.0903 1.0318 1.0527 1.1043 2.938522 1.1406 1.1021 1.0195 1.0481 1.1105 3.114683 1.1373 1.0740 1.0210 1.0488 1.1002 3.295284 1.1427 1.0628 1.0363 1.0511 1.0993 3.046205 1.1362 1.1028 1.0861 1.0137 1.0987 2.87043

Table 7various parameters in PSO for 25-bus test system.

Case Wmax Wmin C1 C2 No. o

1 1.0 0.5 1.0 1.0 05

Table 8Optimized set of control variables and all bus voltages for 25-bus system.

Sr. No. Control variables Optimized control variables magnitudes (pu)

PSO DFP

1 V1 1.1354 1.13942 V2 1.0719 1.08133 V3 1.0161 1.02684 V4 1.0488 1.04755 V5 1.0954 1.09426789

1011121314151617181920

Total load Pd = 12.41 pu, Qd = 3.876 pu.Proximity indicator using (PSO) s = 0.3920.Proximity indicator using (DFP) s = 0.3997.

L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374 373

best initial solution were 0.2841 pu, 0.4094 pu, 0.3092 pu,0.2527 pu, and 0.1723 pu. Whereas with optimized solution thesereactive reserves are obtained as 0.5826 pu, 0.8121 pu, 0.4609 pu,0.2969 pu, and 0.3001 pu respectively. Weighting factors for fivegenerations are 3.9328, 1.2648, 5.5761, 6.2756 and 5.8215 obtainedby sensitivity analysis. Magnitude of proximity indicator with opti-mized solution is s = 0.3920.

The reactive power reserve maximization problem as formu-lated in Section 2 has been solved using one of the most popularnon-linear optimization techniques known as Devidon–Fletcher–Powell’s (DFP) method and accounting inequality constraints byan exterior penalty function method [15]. This gradient basedmethod has been extensively used for non-linear optimizationproblem solution in power system studies. Mechanization of DFPmethod is much more involved than PSO technique. Various sensi-tivities are required to evaluate the gradient vector. A penaltyparameter is selected at a low initial value say 0.5 and if violationpersists than this is increased in step size by 10% each time till allinequality constraints are satisfied.

Results obtained with this method for 6-bus test system are gi-ven in Table 4. This table gives value of control variable as obtainedby DFP method. It is seen; control variables as obtained by both themethods are in close agreement. Value of Max J obtained using DFPmethod is 5.9619 as against 6.0529 obtained using PSO. Reactivepower reserve obtained as 0.4551 pu, and 0.9307 pu. Similar re-sults have been obtained for 25-bus system. Control variables asobtained using DFP method has shown in Table 8 along with thoseobtained using PSO. Value of objective function obtained using DFPmethod is 9.1749 as against those obtained using PSO as 9.4987(Table 7). Results obtained by both the method are in close agree-ment. Reactive reserves obtained at all five buses obtained for 25-bus test system are 0.5973 pu, 0.8035 pu, 0.4816 pu, 0.3011 pu,and 0.2677 pu.

f initial particles Max J No. of iterations for convergence

9.4987 27

Load bus voltages Load bus voltages magnitudes (pu)

PSO DFP

V6 0.9897 1.0026V7 1.0326 1.0376V8 1.0306 1.0350V9 1.0171 1.0199V10 1.0402 1.0412V11 1.0345 1.0355V12 1.0309 1.0341V13 0.9823 0.9949V14 0.9820 0.9871V15 0.9940 0.9974V16 1.0297 1.0326V17 1.0368 1.0378V18 1.0226 1.0231V19 1.0488 1.0487V20 1.0017 1.0004V21 0.9751 0.9745V22 0.9539 0.9531V23 1.0358 1.0371V24 0.9500 0.9500V25 0.9741 0.9750

374 L.D. Arya et al. / Electrical Power and Energy Systems 32 (2010) 368–374

Results obtained by both the methods are in close agreementand this justifies the use of PSO technique for its ease of implemen-tation and computational efficiency, which is an establishedadvantage for such evolutionary algorithm [16].

6. Conclusions

This paper discusses the management of reactive power reservesin order to improve static voltage stability. This has been achievedvia a modified PSO algorithm. Advantage of PSO algorithm is thatits mechanization is simple without much mathematical complex-ity. Moreover global optimal solution is obtained and local optimalsolution is avoided via search procedure. Important about the meth-odology is that not only reactive reserve is optimized but inequalityconstraint on proximity indicator guarantee required static voltagestability margin. Network as well as source capabilities are impor-tant from voltage instability viewpoint. Further contribution toreactive reserve has been considered from participation viewpointby weighting factors. This is important aspect, which has been con-sidered since large reactive reserve available at a generator bus,which is not utilized in a load increased scenario, is not of great sig-nificance. Hence a generator participating to a larger extent hasbeen given lesser weight in the PSO algorithm. The algorithm hasbeen implemented on 6-bus and 25-bus sample test systems.

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