Optimal Conductor Size Selection in Radial

Power Distribution Systems Using EvolutionaryStrategies

Franklin Mendoza, Durlym Requena, Jose L. Bemal-Agustin, Member, IEEE, and Jose A.Dominguez-Navarro, Member, IEEE

Abstract- This paper presents the application of a heuristicmethod, called Evolutionary Strategy (ES), to select optimalsize of feeders in radial Power Distribution Systems. ES incor-porates biologically inspired structures and operators such asrecombination, mutation and fitness based selection. ES proveto be successful when compared with other iterative methodson most problems. In this paper, the posed optimization problemconsists in select a conductor type for each feeder of radial PowerDistribution Systems. The optimization procedure is subject tosome technical constraints, which are the Kirchhoff's currentlaw constraints for all the nodes, the capacity constraints forthe feeders and substations, and the voltage drop constraints. Asa case study, the proposed method is applied to radial PowerDistribution Systems with satisfactory results.

Index Terms-- Evolutionary Strategies, Optimal ConductorSize, Power Distribution Systems.

I. INTRODUCTIONM/[OST electric distribution networks work in a radial

configuration. A radial Power Distribution System hasseveral feeders; however, it is characterized by having onlyone path for power to flow from the source to each customer.For this configuration, there is an associated loss of power inthe feeders [1]. The cost of this loss can be evaluated and usedas objective function to select the optimum size of conductors.Determining the dimensions of the conductor is acommonplace task. A more difficult but important task is theconsideration of a single circuit section as a link in the wholenetwork. To determine the dimensions of the conductor isnecessary to meet the requirements of the overall general planfor the network. For example, the nature of the load connectedand the growth rate of the load [2].

Revising the available technical literature, we can affirmthat only a few researchers focus his research work finding thebest sizes of conductors in radial distribution systems. Mandaland Pahwa [3] present a mathematical method for selection ofoptimal set of conductor. Several financial and engineeringfactors are considered in the solution. Wantg et al. [4] presents

F. Mendoza and D. Requena are with the Electrical EngineeringDepartment, Universidad Nacional Experimental Politecnica Antonio Jose deSucre, UNEXPO Puerto Ordaz, Venezuela. (e-mail: fmen@cantv.net).

J. L. Bernal-Agustin and J. A. Dominguez-Navarro are with the ElectricalEngineering Department, Zaragoza University, Spain (e-mail:jlbemaalgunizar.es, jadonagunizar.es).

1-4244-0288-3/06/$20.00 (©2006 IEEE

a new approach to the optimization problem of conductor sizein planning radial Power Distribution Systems formulated asan integer programming problem. Tram and Wall [5]proposed an improved algorithm based on realisticassumptions and reliable optimization techniques.

Solving the exact problem using classical optimizationtechniques can be difficult because of the combinatorial natureof the problem. The Evolution Strategies are heuristicalgorithms to search for optimal solution. They incorporatebiologically inspired structures and processes as operatorssuch as recombination, mutation and fitness based selection[6], [7].

This paper presents an application of the EvolutionaryStrategies for the selection of optimal size of feeders in radialPower Distribution Systems which minimize sum ofcapitalized energy loss costs. Optimal conductor type isdetermined for each feeder, subject to technical constraints,which are the Kirchhoff s current law constraints for all thenodes, the capacity constraints for the feeders and substationsand the voltage drop constraints. The voltage drop is animportant constraint which has to be considered, and it may bethe critical factor in determining conductor dimensions. If theconductor is selected purely on the basics of economics, thevoltage drop per unit length will be almost constantthroughout the network.

The example of application showed in this paper,demonstrates that the Evolutionary Strategies can be usedeffectively to solve the problem of optimal selection ofconductor size in radial Power Distribution Systems.

II. PROBLEM FORMULATION

The basic problem is to select the optimum size feeders inradial Power Distribution Systems. It can be expressed like anoptimization problem where the objective function tominimize is the sum of the total costs for the network and canbe mathematically formulated by:

n

Objective = min E Cp (X)i,J

(1)

where, C. is the investment cost of branch, per unit oflength.

In general, in the method proposed we assume know

2

substation and type of conductor cost. We consider thecomplete network and only investment cost.

Obtaining the minimal cost implies determining the totallength for each branch [8] and the total length of the network,subject to technical constraints, which are the Kirchhoff'scurrent law constraints for all the nodes, the capacityconstraints for the feeders and substations, the voltage dropconstraints, and radial constraints in operation.

In the proposed algorithm, it is necessary to calculatevoltage drop of each node and the current flowing in eachsection. Therefore a load flow method must be used in theprocess. In [9] a simple and efficient load flow method forradial networks based in current injection is proposed, whichwe have been used in our optimization approach. This loadflow method considers all aspects of modeling of sections anddetailed load modeling. The solution technique is based onforward and backward propagation to calculate current in eachsection and voltage at each node. The most favorable situationis given when the upper and lower voltage at every node valueis narrow. The voltage requirement at node n [10], [1 ], beingthis the function aptitude:

I + sgn(fV -V max ) (TVn TV,,mx)+2 V (2)n,nom(2

1+sgn(V1mi, -LVn) (Vn,mi- Vn)2 V

n,nom

where,

OAn function aptitude (fitness)sgn sign functionVn voltage at node n.Vn max maximum allowed voltage valueVn,min minimum allowed voltage valueVn,nom nominal voltage.

The algorithm proposedat node is zero.

to voltage requirement

III. DEVELOPED EVOLUTION STRATEGYFrom the various possible types of ES, in this research

work the ([t+X)-ES [12] has been applied. This technique isbased on evolution principles: selection, reproduction andmutation. They were first reported by Rechenberg [7] andSchwefel [6].

The Evolutionary Strategies work directly with the realrepresentation of the decision variables. Other EvolutionaryAlgorithms, like the Genetic Algorithms, work with a codingof the solutions. The key element of the evolution strategy isan individual (possible solution) that includes:

X =(X x...... .x (3)

07 (07c,07 * --- ) (4)

a = (a1,a2, ...... an ) (5)where, (Xl,x2......X) represents the objective variables(diameters of the feeders), (07,q2...... --) and (a1,a2.a,)represent their associated standard deviations and preferableaddress vector that guides the search [6].that means:

a ((xl,x2......X) ( 2, n), (a1, a2,).).)an (6)

The aptitude (fitness) of an individual represents that sogood it is the vector, to satisfy a function objective. Therefore,the value of the aptitude of an individual is to evaluate thefunction objective, this is: P(a) = F(xi).

The performance of the Evolution Strategy techniquedepends on a suitable choice of internal strategy controlparameter. The first vector X represents a point in the searchspace. The second vector a is a vector of standard deviations.This perturbation vector provides instructions on how mutateX and is itself subject to mutation. In other words, bothcomponents, X and o, are submitted to evolution process byapplication of the operator of mutation and alsorecombination. Thus, a suitable adjustment and diversity ofparameter mutation should be provided under arbitrarycircumstances. The Evolution Strategy technique facilitates anadjustment of such parameters within a self-adaptationprocedure.

The most common strategies are described as (1+1)-ES,([t+X)-ES called Plus Strategy, and ([t,2)-ES called CommaStrategy. The strategy used in this paper is (pt+X)-ES. In theformer, [t parents generate X offspring and all solutionscomplete for survival with the best [t individuals beingselected as parents of the next generations. In the latter, only Xoffspring complete for survival and the [t parents arecompletely replaced in each generation. In the ([t,X)-ES case ateach generation the [t individuals are selected only among theX children of the generation, whereas in the (pt+X)-ES case theparents are also included in the evaluation process.

The ES starts from an initial set of specific network anddifferent type of conductors, randomly generated, thus of [iindividuals. The best individuals (lower cost) of thispopulation are selected; these are the parents of the nextgeneration. If one parent with a value of the fitness functionequal to zero exists, the algorithm ends because an optimalconductor is found. Else, a new generation of X children isgenerated by recombination of the parents. In the next step thevariables (x, u and a) are slightly modified by mutation. Themutation operator is the most important operator of thealgorithm, and the modification implies changes of conductordiameter. This operator is applied each individual generatedby the operator of recombination; this it introduces newinformation to the process of optimization, avoiding this way,the stagnation during the search of the good solution.

A. Mutation operatorThe mutation consists on correlated variations of the

objective variables, this means that the deviations applied to

3

the objective variables are correlated, by means of an angle uithat indicates the preferable address for the search of the goodsolution. To carry out this process, first the mutation processis applied to the vectors of deviations and angles, like it isindicated next:

z=vrN(0,1)+±iWN (0,1)

a-.= a1. + ,8-N (0,1)(7)(8)

where; Ni(O, 1) and Nj(O, 1), they are random numbers ofnormal distribution, calculated for each variable i, whileN(0, 1), it is only calculated once, that is to say, it remainsconstant during the whole mutation process; the constants r', rand ,6, they represent factors of adaptation.

From the R+t individuals (solutions) the best t individualsare chosen. In the next iteration these j individuals will be theparents. The approach iterates till the fitness function value isequal to zero or the maximal number of generations isreached.

The flow chart of the developed ES is given in Fig. 1.

Indices s and r denote two parent individuals selected atrandom from P(t), and k E [0,1 ] is a uniform random variable.The steps are:

a) To select by random, two best members of population of[t parents and X offspring.

b) A new offspring is created with recombination globalintermediate, thus a new network with news conductor.

C. Selection operatorWhen the mutation and recombination operators are

finished, selecting the [t best of the union of parents andoffspring. Although the (pt+X)-selection is elitist and thereforeguarantees a monotonously improving performance.

The selection operator criterion is: To choose from amongthe [t+- population, the best [t members according to costfunction.

IV. RESULTS AND DISCUSSION

In Fig. 2 the test system is shown. The distribution systemconsists of 13.8 kV line-to-line voltage, 8 demand nodes, 1substation and 7 feeders in radial configuration. The length ofconductor for each branch is fixed in 1000 m. The voltagedrop is 1%.That network is used to illustrate the performanceof the ES for select the best size for each feeder of de RadialPower Distribution System. In our approach to solve theproblem, in each branch we have fixed the smallest sizes ofthe conductors initially. For the sake of simplicity, thefollowing conditions apply in this paper: i) only a peak load isconsidered and ii) the feeder configuration is known.

Table I shows the power demands of the distributionnetwork nodes.

Fig. 1. Flow chart of the ES appliedproblem.

to optimal conductor size selection

B. Recombination operatorDifferent mechanisms of recombination used in

Evolutionary Strategies any of these in their usual form a newindividual exist he takes place starting from two parentsselected randomly, where they are generated descendingstarting from a population of parents. The most common typesof recombination are:

x . no recombinationS,lX V discreteS,l r, 1

x . + k (x .x .) intermediateS,l r, S,lX . V Xri global discrete

x . + k. (x ., .-x ., .) global intermediateSl,l Sl,

(9)

N04 )

L04

S/E L01 NCN01

L05

L06 N06N05

IF I F~~~

L02 N03

L07

N08 EJ

Fig. 2. Test Power Distribution System.

TABLE IDEMAND REQUIREMENTS, IN KW

Node DemandNOI 0.0N02 1054.2N03 806.5N04 2632.5N05 609.0N06 2034.5N07 932.8N08 1731.4

L03 N07

xi

D2

Table II shows types of conductor proposed.The values of the parameters employed by the ES are

indicated in table III. The optimal cost is obtained in 10generations, therefore in this paper we consider that 30generations are sufficient for demonstrate the effectiveness ofthe proposed algorithm.

TABLE IIPARAMETERS FOR CONDUCTOR TYPE

Type Max. Current (A) R(Q/km) X(Q/km)2 180 0.8763 0.41331 200 0.6960 0.4133

1/0 230 0.5518 0.40772/0 270 0.4387 0.39833/0 300 0.3480 0.38994/0 340 0.2765 0.3610

TABLE IIIPARAMETERS FOR THE ES

Gen max 3020

X. 10Recombination Global intermediateSelection From t + X

TABLE VVOLTAGE OBTAINED IN EACH NODE

Node Voltage [%] Voltage Drop[%]NOI 100.00 0.00N02 99.67 0.25N03 99.35 0.45N04 99.52 0.16N05 99.39 0.17N06 99.01 0.29N07 99.06 0.53N08 99.09 0.56

Fitness0.25-

0.

0.15-

0.

0.05 -

0

0 5 1 1 2 2 3 3

Fig. 4. Fitness evolution in optimization process. g

Using the data mentioned above, was demonstrated thevalidity and effectiveness of the proposed algorithm. The mostfavorable situations voltage range between the upper andlower value is narrow. Table IV shows the type of conductorobtained using the ES with lowest overall cost by taking intoaccount size of conductor and voltage drop constrain. Fig 3shows the distribution system obtained. For the optimalsolution, the voltages in each node are indicated in Table V,and Fig 4 shows fitness evolution of the best individual foreach generation.

TABLE IVTYPE OF CONDUCTOR OBTAINED

Feeder Current Conductor[A] Selecting

LOI 95.4 4/0L02 73.2 3/0L03 22.7 2L04 55.3 1L05 55.8 2L06 43.0 1L07 36.6 1/0

N04

1

S/ENo1

1/0

N08

Fig. 3. Optimal solution (Cost: 5.94 106 MU).

V. CONCLUSIONS

This study has presented an application of ES to solve theoptimal conductor selection problem in a radial PowerDistribution network. The objective function was optimizedconsidering several conductors. The cost for the obtainedoptimal solution has been calculated. The test case shows thatthe ES possess high robustness to find the optimal conductorsizes. From this reason, the evolution strategy can be usedeffectively to solve the optimization problem stated in thispaper.

VI. REFERENCES[1] M. A. Yehia, E. E. Matar, N. Y. Hobelia, A. D. Avedikian, "A Heuristic

Algorithm for Electric Distribution Networks Optimal FeederConfiguration Using Geographic Information System", IEEETransactions on Power Systems, vol. 17, no 4, pp. 1232-1237. Nov.2002.

[2] E. Lakervy, E. J. Holmes, "Electricity Distribution Network Design".IEE Power Series 21, 2nd Edition, London, 1995.

[3] S. Mandal, A. Pahwa, "Optimal Selection of Conductors for DistributionFeeders". IEEE Transactions on Power Systems, vol. 17, no 1, pp. 192-197. Feb. 2002.

[4] Z. Wang, H. Lui, D. Yu, X. Xiaohui and H. Song. "A PracticalApproach to the Conductor Size Selection in Planning Radil DistributionSystems". IEEE Transactions on Power Systems, vol. 15, no 1, pp. 350-354. Jan. 2000.

[5] H. N. Tram and D. L. Wall. "Optimal Conductor Selection in PlanningRadial Distribution System". IEEE Transactions on Power Systems, vol.3, no 1, pp. 200-206. Feb. 1988.

[6] H. P. Schwefel, "Evolution and Optimum Seeking". Wiley, New York,1995.

[7] I. Rechenberg. "Evolutionsstrategie - Optimierung technischer Systemenach Prinzipien der biologischen Evolution". Frommann-Holzboog.1973.

[8] E. Diaz-Dorado, E. Miguez,, "Application of Evolutionary Algorithmsfor the Planning of Urban Distribution Networks of Medium Voltage,"IEEE Transactions on Power Systems, vol.170, no. 3, pp. 879-884, Aug.2002.

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[9] J. L. Bernal-Agustin, "Application of genetic algorithms to the optimaldesign of power distribution systems," Ph.D. dissertation (in Spanish),Dept. Elect. Eng., University of Zaragoza. Spain. 1998.

[10] M. Doll, J.F. Verstege, "An Evolution Strategy based Approach for aCongestion Management System", IEEE PES Winter Meeting, 2001,Paper 2001-WM-022.

[11] R. Ranjan, B. Venkatesh, D. Das, "A New Algorithm for PowerDistribution Systems Planning", Journal of Electrical Power SystemsResearch, vol 62, no 1, pp. 55-65, May 2002.

[12] T. Back, H. P. Schwefel, "Evolutionary Computation: an overview",Proceedings of IEEE. International Conference on EvolutionaryComputation, pp 20-29, May 1996

VII. BIOGRAPHIESFranklin Mendoza received his MSc degree in Electrical Engineering fromthe Universidad Nacional Politecnica Antonio Jose de Sucre -UNEXPO-(Venezuela) in 1994. In 1990 he joined the UNEXPO where he is ProfesorAgregado. Currently he is completing doctoral studies in the University ofZaragoza. His investigations are focused towards the area of optimization ofpower distribution systems by applying evolutionary algorithms.

Durlym Requena received his Lic degree in Physics in the UniversidadPedagogica Experimental Liberatdor in 1979 In 1987 he joined the UNEXPOwhere she is Profesor Asociado. Currently he is completing doctoral studies inthe University of Zaragoza. His investigations are focused towards the area ofoptimization of distributed generation by applying evolution strategy.

Jose L. Bernal-Agustin (M'04) received his Ph.D. degree in ElectricalEngineering from the University of Zaragoza (Spain) in 1998.In 1991, he joined the University of Zaragoza, where he is, currently, ProfesorTitular. He has been involved in research projects referred to design of powerdistribution network and renewable energy sources.His current interest areas are applications of evolutionary algorithms inelectrical power systems, power distribution networks design, electric marketsand renewable energy sources.

Jose A. Dominguez-Navarro (M'04) received the Ph.D. degree in electricalengineering from the University of Zaragoza, Zaragoza, Spain, in 2000. Since1992, he has been with the University of Zaragoza, where he is now aProfesor Titular de Universidad. His current interest areas are applications ofsoft computing in electric power systems, power distribution networksplanning, electric markets, and metaheuristic optimization.