Multi-objective VAr planning for large-scale powersystems using projection-based two-layer simulatedannealing algorithms

Y.L. Chen and Y.L. Ke

Abstract: A projection-based two-layer simulated annealing method for solving general multi-objective optimisation problems is presented. The power of the simulated annealing techniqueenables the solution method to find a desirable, globally efficient solution to the problem, evenwhen the solution space is nonconvex and the objective functions are nondifferentiable. The multi-objective VAr planning problem in large-scale power systems can be solved by the proposedmethod and very encouraging results have been obtained.

1 Introduction

Multi-objective optimisation (MO) aims to find satisfactorycompromise solutions in multi-objective decision-making(DM) problems when the objectives are usually noncom-mensurable. A bound solution is usually obtained, insteadof a truly optimal solution, using the e-constraint methodto solve the MO problem. One of the objective functionsis selected as the primary objective function and all theother objective functions are regarded as constraints totreat the MO as a single objective optimisationproblem [1, 2].

The traditional method in general MO problems,which assumes that the objective function is continuous isdifficult to apply to a constrained, multi-objective andnondifferentiable optimisation problem as many of theobjective functions are nondifferentiable and the solutionspace is nonconvex. A modified projection-based methodbased on the simulated annealing (SA) technique isproposed to solve these problems. The proposed approachuses a two-level scheme, involving a decision level concernedwith the decision strategy and an analysis level whichgenerates efficient solutions to meet the DM’s preferences(goals).

The aim of reactive power (VAr) planning is to providethe system with sufficient VAr sources to enable the systemto be operated under economically conditions. The goals ofVAr planning problems include power loss reduction,investment cost minimisation and voltage deviationreduction. Consequently, the VAr planning is a constrained,multi-objective and nondifferentiable optimisationproblem.

Many of these formulations for reactive power planningare based on linear programming with a 0–1 integer torepresent whether or not the new device should be installed

[3], and the objective function is treated as continuous [4–6].An optimisation technique based on a two-layersimulated annealing (TLSA) method has been developedto efficiently find a desirable, global optimal solutionin the MO problem. The projection-based SA methodis applied to the multi-objective VAr planning problemsince the SA technique [7, 8] is a powerful tool forsolving combinatorial optimisation problems and cansearch for the global optimal solution even if the solutionspace is nonconvex and the objective functions arenondifferentiable.

2 Multi-objective VAr planning problem

This study presents a projection-based SA method forsolving the optimal multi-objective VAr planning problem.The formulations are described as follows by treating theproblem as a multi-objective mathematical programmingproblem concerned with minimising each objective simulta-neously, while satisfying the constraints.

2.1 Objective functionsThe objective function for the reactive power planningproblem includes active power loss, the investment cost ofthe VAr sources and voltage deviation.

2.2 Active power lossThe total power loss is minimised by (1).

f 1ðzÞ ¼Xnb

j¼1

Xnb

j¼1V2

kgkj � VkV jðgkj cos ykj þ bkj sin ykjÞ� �

ð1Þ

where nb the total number of buses in the systemVk, yk voltage magnitude and voltage angle at bus k,

respectivelygkj, bkj conductance and susceptance of transmission line

between bus k and bus j, respectivelyykj¼ yk�yj

z operating variable vector

Y.-L. Chen is with the Institute of Electro-Mechanical Engineering, MingchiInstitute of Technology, Taipei, Taiwan

Y.-L. Ke is with the Department of Electrical Engineering, Kun ShanUniversity of Technology, Tainan, Taiwan

r IEE, 2004

IEE Proceedings online no. 20040645

doi:10.1049/ip-gtd:20040645

Paper first received 4th September 2003 and in revised form 17th March 2004

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 555

2.3 Investment cost for VAr sourcesThe investment cost is determined by (2).

f2ðoÞ ¼X

i2E

ðdi þ sciqci þ sriqriÞri ð2Þ

whereE the set of potential buses required to expand reactive

sourcesdi the installment cost of bus iri a 0–1 variable; ri¼ 1 when bus i is selected for VAr

expansion, and ri¼ 0 otherwiseqci, qri added capacitive and inductive compensation,

respectively; both are integerssci, sri, unit cost of capacitor and reactor, respectively The

term o is called the expansion variable vector and specifieswhether or not VAr sources are installed, and identifies thetypes and sizes of the VAr sources.

2.4 Voltage deviationThe voltage deviation of each load bus must be as small aspossible to obtain a performance voltage index. The voltagedeviation index is defined as follows:

f3ðzÞ ¼ maxk2OjVkr � Vkj ð3Þ

where X is the set of all load buses, and Vkr is the idealspecific voltage magnitude at bus k, which is usually set to 1p.u.

2.5 Load constraintsThe load constraints in a compact form are the real andreactive power balance in load flow equations.

LðzÞ ¼ 0 ð4Þ

2.6 Operating constraintsThe vector of the operating variable z is defined as

z ¼ðPGi : QGi : V i : yi : tkÞi ¼1; 2; . . . ; nb; k ¼ 1; 2; . . . ; nt

ð5Þ

where tk is the tap ratio of the kth controllabletransformer; nt is the total number of tap changes ofthe controllable transformer. PGi and QGi are the activeand reactive power generation at bus i, respectively.These constraints are summarised in a compact form asshown in (6).

Tmink � Tk � Tmax

k k ¼ 1; 2; . . . ; nt

PminGi � PGi � Pmax

Gi i ¼ 1; 2; . . . ; nb

QminGi � QGi � Qmax

Gi i ¼ 1; 2; . . . ; nb

Sij � jSmaxi j i ¼ 1; 2; . . . ; nl

GðzÞ � 0j

ð6Þ

where nl is the total number of transmission lines and Si isthe line flow of the ith line.

2.7 Expansion constraintsThe expansion constraints are the limits on the new VArsources to be installed. The vector of the expansionvariables, o, is constrained

0 � qri � qmaxri

0 � qci � qmaxci ; i 2 E

For convenience, the above constraints are expressed in acompact form as

RðoÞ � 0 ð7Þ

2.8 Overall problemThree objective functions: power loss, investment costand voltage deviation, are considered in this paper. Thepower loss can be converted into an expense by a factor,$/kW-h. The investment cost and voltage deviation can belinearly combined into a single objective function expressedby (8).

KeDif1ðzÞ þ f2ðoÞ ð8Þwhere Di is the duration of system operation. Since thevoltage deviation and the other objective functions areincommensurable they cannot be combined into a singleobjective function. An optimal VAr planning configurationW* is sought among all possible configurations W, such thatall of the objective functions are optimised while the loadand operating constraints are satisfied. The multi-objectiveproblem in mathematical formulations is demonstratedby (9).

minz2c

KeDif1ðzÞ þ f2ðoÞ

minz2c

f3ðzÞð9Þ

3 Solution algorithm

3.1 Projection-based SAThe VAr planning problem is a constrained, multi-objectivenondifferentiable optimisation problem. A is a powerful,general-purpose technique for solving combinatorialoptimisation problems. It is a randomisation algorithmthat can asymptotically search for a global optimalsolution with probable one [7, 8]. The proposed algorithmfor solving the above problem can be described asfollows [9, 10]:

Step 1: randomly set an initial condition (solution)

Step 2: generate a feasible point that neighbours the currentpoint in the solution space

Step 3: evaluate the increase in cost DC

Step 4: if DCo0, then accept the new solution point and goto step 6

Step 5: choose a random number g uniformly distributed inthe interval [0,1). If exp(�DC/T)4g, then accept the newpoint; otherwise, discard the new point

Step 6: if the moves have not been completed, then go toStep 2

Step 7: reduce the temperature, T¼ a*T

Step 8: if T4Tmin, go to step 2

Step 9: output the global optimal solution

At each temperature, randomly choose a new solutionpoint S’ by perturbing the current point S. Then, evaluatethe increase in cost by DC¼ cost (S’)� cost (S). If the movethat reduces the cost is termed downhill, then the move isaccepted. If the move that increases the cost is termeduphill, then the move is accepted with a probabilityexp(�DC/T). The probabilistic selection rule allows SA toalways be able to escape from a local optimum in which itcould otherwise be trapped, and to proceed to the desiredglobal optima. This characteristic distinguishes the SA fromthe greedy search approach.

The temperature is lowered according to the rule,Tk+1¼ a(Tk)*Tk, where a(Tk) is a constant smaller than,but close to unity. Typical values are between 0.8 and 0.99.

556 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

When the acceptance ratio is low or the sample mean andvariance of cost values at the current temperature dropdramatically, a(Tk) is increased to prevent it from becomingstuck at a local optimal point. Otherwise, a(Tk) is reduced(say to 0.85) to increase the rate of convergence.

The SA technique can also easily be applied to themaximisation problem by replacing DC by � DC in steps 4and 5 in the above SA algorithm.

The analysis level in the projection algorithm is aminimax problem, which is generally solved by the gradientprojection method. In updating l and performing theminimisation, the general gradient method assumes that theobjective function is differentiable.

A TLSA is presented herein for solving a generalminimax problem, even when in the real world, an MOVAr planning has objective functions that are nondiffer-entiable and the solutions space is nonconvex. Figure 1,illustrates the solution algorithm of the projection-based SAmethod for solving an MO problem. The preference yk ofthe DM is given by the decision level, and the analysis levelefficiently finds a solution. In the analysis level, the sub-problems of maximisation and minimisation are executedalternatively. During minimisation, l is constant and theminimal value of h(.) is determined using the SA technique.According to the result of the minimisation, l will beupdated during the solution of the maximum subproblemand then sent to the minimisation process. The subproblemswill be solved alternatively until an efficient solution hasbeen obtained. The TLSA approach can completely solvethe minimax problem as follows.

3.2 TLSAIn this paper, a TLSA algorithm is proposed to solve aminimax problem. The pseudocode of the TLSA algorithmfor the analysis level is described as follows.

Two_layer_simulated_annealing( )

{

Initialisation( );

/*SA_outer_layer( )*/

{repeat

{repeat

{lkk+1¼ generate(lkk); /* lkk+1ALe*/

new_cost¼ SA_inner_layer(lkk+1);

move_criterion(new_cost); /*accept/discard*/

kk¼ kk+1;

} until move_length_reach( );

} until frozen( );

}

output( ); /*an efficient optimal solution output*/

}

The inner layer of SA solves the minimisation problemwith constant lkk+1, determined from Ke in the outer layerof SA, and a solution (new_cost) is returned by the innerlayer to enable the outer layer to determine the maximumsolution. It should be that the outer layer solves themaximisation subproblem and the inner layer solves theminimisation subproblem.

The main disadvantage of the SA technique is theassociated computational burden. However, only the firstminimisation process requires a high-temperature coolingschedule and the other process only uses low-temperaturecooling schedule because the solution has already convergedto a near-optimal solution point when the first minimisationhas been completed.

This Section proposes a solution algorithm for multi-objective VAr source planning to determine: first, thelocation at which to install the VAr sources; secondly, thetypes and sizes of VAr sources to be installed; and finally,the setting of the VAr sources under various loadingconditions.

Solution algorithm for MO VAr sources planning

/* Initialisation */

Step 0(a): Input system data and the decision point y. Thedecision point y is specified by the DM (then go to step 1) orautomatically generated by step 0(b)

Step 0(b): Determine a decision point automatically. Use asingle objective optimisation method to determine everydecision component of the objective functions

/* Outer Layer SA (for maximization search) */

Step 1(a): Input outer layer SA control parameters. Inputcontrol parameters such as the initial temperature, therandom number seed and the cooling rate

Step 1(b): Generate a new vector of l and perform steps 3–9 (inner layer SA)

Step 2(a): Accept or reject the new configuration. Usethe cost function to determine the configuration of thesystem. Retain the new configuration or restore theprevious configuration by applying the acceptationcriterion

Step 2(b): Check the stop criterion. Use the outer layercooling schedule to perform steps 1(b)–2(b) until theannealing is completed

Step 2(c): Print out the optimal configuration

/* Inner layer SA (for minimisation search) */

Step 3: Input the inner layer SA control parameters. Inputthe control parameters such as the initial temperature, therandom number seed, the cooling rate, the step size of thecontrol setting (e.g. 3 MVAR) and the number of moves ateach temperature. If the inner layer SA is to be executed

updatekk

SA technique

kk

θkk

(.)

DM (decision level)

θk*

(.)

f k*(.)

yk

analysis level

maximum subproblem

(outer layer SA)

minimisation

iteration index

subproblem

(inner layer SA)

* k, kk =

λ

λ

Fig. 1 Solution algorithm of projection method with interactionscheme for MO problems

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 557

first, then set a high initial temperature; otherwise set a lowinitial temperature

Step 4: Obtain a feasible configuration.

(1) Generate a new configuration by applying aperturbation

(2) Run a load flow to check the feasibility of the newconfiguration. If any constraint is violated, go to (1),otherwise proceed to (3)

(3) Calculate h(.)

Step 5: Design a proper global cooling schedule.At eachtemperature Tk, perform some Lk moves. For move¼ 1, 2,y , Lk, perform steps 5–8. Otherwise, proceed to step 8

Step 6: Generate a new feasible configuration.

(1) Generate a new configuration by applying aperturbation

(2) Run a load flow to check the feasibility of the newconfiguration. If any constraint is violated, go to (1)Otherwise, proceed to (3)

(3) Calculate the cost function, h(.)

Step 7: Update the configuration of the system. Use the costfunction to determine the system configuration. Retain thenew configuration or restore the previous one according tothe acceptation criterion

Step 8: Apply the stop criterion. If the stop criterion isnot satisfied, then the system is not yet frozen and theprocess is continued by returning to step 4. Otherwise,proceed to step 9

Step 9: Return the optimal value h(.) and the optimalconfiguration to the outer layer SA. The proposed solutionalgorithm determines the optimal configuration of theinstalled VAr sources. The solution point matches thedecision point as closely as possible.

The projection method is introduced in the Appendix(Section 7).

4 Numerical studies

A computer program to implement the proposed solutionalgorithm is developed for testing on a modified IEEE 30-bus system [6] and a real 358-bus system. In all of them, onebank of the VAr source is set to 3 MVAR; the specificvoltage Vkr of all the load buses is set to 1 p.u., and thefollowing parameters are used: power loss cost weighting(KeDi¼ 1 unit/MW-h) and VAr source cost weighting(sci¼ 0.001 unit/bank, sri¼ 0.0013 unit/bank and di¼ 0.02unit/location).

The modified IEEE 30-bus system is considered in study1. The objectives include both power loss reduction andvoltage deviation minimisation. The set of potential buses isE¼ {7,10,17,19,21,24,30}. The initialisation process acts on

a decision point y0 ¼ y01 ; y

02

� �T¼ ½0:083; 0:033�T . The y1 isa power loss in per unit based on 100MW and y2 is avoltage deviation objective. Table 1 lists the results andTable 2 shows the configuration of the VAr sources. l isvaried over Ke (contains seven sample points) to guaranteethat the solution methodology can obtain a really optimalpoint that is not only a minimum distance from the decisionpoint y but also belongs to the Pareto-optimal solutions.Clearly, the solution point presents an efficient solution and

is the closest point to the decision point. After the first innerSA has been applied, the solution trajectory converges to aPareto-optimal solution and then quickly converges to anoptimal solution.

In study 2, which involves the test system in study 1, thedecision specifies an inferior decision point, y0¼ [0.094,0.05]T on the decision level. Table 3 lists the result that thedecision point is attainable and can be improved.

It should be noted that the sign of h(.) indicates whetheror not the decision point y is attainable. It is unattainable ifh(.) is positive (Table 1). It is attainable and can beimproved if h(.) is negative (Table 3).

Study 3 concerns both cost reduction (including energyloss cost and VAr source investment cost) and improvementto the voltage profile of the practical Tai-Power 358-bussystem. The system is operated under peak load. The newVAr sources are installed in only selected 57 buses. Thedecision specifies a decision point y0¼ [0.0255,0.035]T. Thepower loss objective is expressed on a 104 MW base.Table 4 presents encouraging results: the cost is reduced by

Table 1: Result of study 1(IEEE 30-bus system)

Power loss,100MW

Voltagedeviation, p.u.

Before planning 0.11741 0.11950

DM goals 0.08300 0.03300

After planning 0.08756 0.03719

Reduce rate, % 25.4 68.9

l1¼0.716718, y(.)¼ 0.0044557

Table 2: Results of the VAr locations, sizes and types

Bus no. 7 10 17 19 21 24 30

Installedbanks

9 �3 3 3 4 3 2

one bank¼ 3 MVAr

Table 3: Result of study 2(IEEE 30-bus system)

Power loss,100MW

Voltagedeviation, p.u.

Before planning 0.11741 0.11950

DM goals 0.09400 0.05000

After planning 0.08600 0.04137

Reduce rate, % 26.8 65.4

l1¼0.741554, y(.)¼�0.0081628

Table 4: Results for Tai-Power 358-bus system

cost, 104 unit loss, 104MW Voltagedeviation, p.u.

Initial 0.0303564 0.0303564 0.107443

DM goals 0.0255000 � 0.035000

Result 0.0259216 0.0257735 0.037716

Reducerate, %

14.6 15.1 64.9

l1¼0.585460, h(.)¼ 0.00137272total locations¼ 45 (for installing new VAr sources)

558 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004

44.348 units (or 14.6%), the power loss is reduced by 45.8MW (or 15.1%) and the voltage profile is reduced by0.069727 p.u. (or 64.9%). The value of h(.), (0.0013727240)indicates that the decision point of the decision isunattainable.

5 Conclusions

This study developed a projection-based SA method forsolving multi-objective optimisation problems with non-differentiable objective functions and nonconvex solutionspace. The proposed method constitutes an effectiveapproach to finding a desirable, globally efficient solutionto an MO problem. The numerical results prove the goodperformance and the proposed methodology algorithm is apowerful tool that will enable planners to develop correctinvestment policies in response to MO problems.

6 References

1 Chiang, H.D., and Jean-Jumeau, R.: ‘Optimal network reconfigura-tions in distribution systems: Part 1: A new formulation and a solutionmethodology’, IEEE Trans. Power Deliv., 1990, 5, (4), pp. 1902–1909

2 Chiang, H.D., and Jean-Jumeau, R.: ‘Optimal network reconfigura-tions in distribution systems: Part 2: Solution algorithmsand numerical results’, IEEE Trans. Power Deliv., 1990, 5, (3),pp. 1568–1574

3 Aoki, K., Fan, M., and Nishikori, A.: ‘Optimal VAr planning byapproximation method for recursive mixed-integer linear program-ming’, IEEE Trans. Power Syst., 1988, 3, (4), pp. 1741–1747

4 Iba, K., Suzuki, H., Suzuki, K.I., and Suzuki, K.: ‘Practical reactivepower allocation/operation planning using successive linear program-ming’, IEEE Trans. Power Syst., 1988, 3, (2), pp. 558–566

5 Hong, Y.Y., Sun, D.I., Lin, S.Y., and Lin, C.J.: ‘Multi-year multi-caseoptimal VAr panning’, IEEE Trans. Power Syst., 1990, 5, (4),pp. 1294–1301

6 Deeb, N.I., and Shahidehpour, S.M.: ‘An efficient technique forreactive power dispatch using a revised linear programmingapproach’, Electr. Power Syst. Res., 1988, 15, pp. 121–134

7 Aarts, E., and Korst, J.: ‘Simulated annealing and Boltzmannmachines’, (Wiley, 1989)

8 Kirkpatrick, S., Gelatt, C.D. Jr., and Vecchi M., P.: ‘Optimization bysimulated annealing’, Science, 1983, 220, pp. 671–679

9 Chiang, H.D., Wang, J.C., Cockings, O.R., and Shin, H.D.: ‘Optimalcapacitor placements in distribution systems: part 2: solutionalgorithms and numerical results’, IEEE Trans. Power Deliv., 1990,5, (2), pp. 643–649

10 Lasdon, L.S.: ‘Optimization theory for large systems’, (Macmillan,London, 1970)

11 Chankong, V., and Haimes, Y.Y.: ‘On characterization of noniferiorsolutions of a vector optimization problem’, Automation, 1982, 18, (6),pp. 697–707

12 Ferreira, P.A.V., and Geromel J., C.: ‘An interactive projectionmethod for multicriteria optimization problems’, IEEE Trans. Syst.Man Cybern., 1990, 20, (3), pp. 596–605

7 Appendix

7.1 General multi-objective optimizationproblemsA general MO problem is considered as follows:

minz2w

f T ðzÞ ¼ ½f1ðzÞf2ðzÞ . . . fmðzÞ�

subject toGðzÞ � 0ð10Þ

Let w, a feasible region, be a set of state vector z that isassumed to be nonempty and convex. The functionsf1(.)yfm(.) are generally conflicting and incommensurable.A decision-making (DM) is used to solve the inevitableconflicts among the objectives. The optimal solution is theone that matches the DM goals as closely as possible.

The concept of Pareto-optimality with a noninferiorsolution is fundamental to the MO problem. Compromisesmust be made to select a solution among the Pareto-optimalsolutions.

7.2 Definition of Pareto-optimal solutionsA feasible point z* is said to be a Pareto-optimalsolution if no other zAw exists such that f(z)rf(z*) andf(z)af(z*).

The Pareto optimality solution for a given MO problemis always used as the trade-off solution among infiniteefficient points. Let w*, the set of efficient solutions, is thecollection of all efficient points.

minz2w

ol; f ðzÞ4 ð11Þ

w* can thus be completely generated as l varies over

LE ¼ fl 2 Rms:t:li � e;Xm

i¼1li ¼ 1g with e ¼ 0 ½11�

The optimal solutions of (11), called the properefficient solutions of (10), are the desirable goals ofthe DM if e40 is selected. However, the improperefficient solutions are undesirable as li¼ 0 for some integersiA[1, 2y, m].

The decision point y is chosen to represent the desiredgoal and strategy of the DM. Defining a satisfactorysolution is very important and is the basis of the projectionmethod.

7.3 Review of the projection methodThe projection method [12] is that the entire satisfactorysolutions can be considered as the projections of constraintsf(z)�yr0 and zAw onto the space of the variable yARm

(implicit trade-off) represented by the convex setfG ¼ y 2 Rms:t:f ðzÞ � y � 0; z 2 Xg. The projection re-presents the complete set of satisfactory solutions through asupporting halfspace of C. The Pareto-optimal solution isgenerated to meet the required goals yi, i¼ 1, 2,ym byusing this projection for solving the following minimaxproblem.

yðy; eÞ ¼ max minl2Lz2w

e

ol; f ðzÞ � y4 ð12Þ

The DM problem is to find a solution point from the setc that minimises the distance between this point and y. Thevector l is a gradient vector of w, and the values of ol,f(z)�y4will be maximum as the vector l and f(z)�y in onedirection, such that the distance between the point f(z) and yis minimal. The value of h(.) indicates whether or not thegoals are attainable. An improved solution will be obtainedif the decision is inferior. The value of h(.) also indicates theminimal distance between the decision point and aboundary curve, w¼ f(x*) [12]. The basic algorithm isdescribed as follows.

7.4 Basic algorithmStep 1 (Initialisation): set k¼ 0 and determine

goi ¼ min

z2wfiðzÞ for i ¼ 1; 2; . . . m ð13Þ

Step 2 (analysis): solve the min-max problem

yðyk; EÞ ¼ maxl2Le

minz2w

ol; f ðzÞ � yk4 ð14Þ

subject G(z)r0.The optimal solution f*(z) of (14) is a point of the set of

Pareto-optimal solutions. If h(.)r0 or f*(z) is the DMpreference, then f*(z) is the optimal solution of (10).Otherwise, go to step 3.

Step 3 (decision): Set k¼ k+1; relax (or another chosen)the decision variables yk, and return to step 2.

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004 559

The interactive procedure with the projection methodusing a two-level scheme yields a truly global optimalsolution. The projection method efficiently solves MOproblems under the limitations that the solution space is

convex and the objective functions must be differentiable. Aprojection-based SA method has been proposed to solvethese problems and has been applied to multi-objectiveoptimal VAr planning in large-scale power systems.

560 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 4, July 2004