reconductorizar_plem (1)

10 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY 2013

Optimal Conductor Size Selectionand Reconductoring in Radial DistributionSystems Using a Mixed-Integer LP Approach

John F. Franco, Student Member, IEEE, Marcos J. Rider, Member, IEEE, Marina Lavorato, Member, IEEE, andRubn Romero, Senior Member, IEEE

AbstractThis paper presents a mixed-integer linear program-ming model to solve the conductor size selection and reconduc-toring problem in radial distribution systems. In the proposedmodel, the steady-state operation of the radial distribution systemis modeled through linear expressions. The use of a mixed-integerlinear model guarantees convergence to optimality using existingoptimization software. The proposed model and a heuristic areused to obtain the Pareto front of the conductor size selectionand reconductoring problem considering two different objectivefunctions. The results of one test system and two real distributionsystems are presented in order to show the accuracy as well as theefficiency of the proposed solution technique.

Index TermsDistribution system optimization, mixed-integerlinear programming, optimal conductor size selection.

NOTATION

The notation used throughout this paper is reproduced belowfor quick reference.

Sets:

Sets of nodes.

Sets of substation nodes.

Sets of branches.

Sets of conductor type.

Sets of load levels.

Constants:

Reconductoring cost from conductor type toconductor type .

Peak power losses cost .

Minimum voltage magnitude.

Maximum voltage magnitude.

Circuit length of branch in kilometers.

Manuscript received February 10, 2011; revisedApril 27, 2011, July 11, 2011,September 07, 2011, and November 28, 2011; accepted May 18, 2012. Dateof publication June 20, 2012; date of current version January 17, 2013. Thiswork was supported by the Brazilian institutions CNPq grant 306760/2010-0,FAPESP and FEPISA. Paper no. TPWRS-00112-2011.The authors are with the Faculdade de Engenharia de Ilha Solteira,

UNESPUniversidade Estadual Paulista, Departamento de EngenhariaEltrica, Ilha SolteiraSP, Brazil (e-mail: [email protected];[email protected]; [email protected]; [email protected]).Digital Object Identifier 10.1109/TPWRS.2012.2201263

Maximum apparent power limit of substation atnode .

Nominal voltage magnitude.

Maximum current magnitude of conductortype.

Real power demand at node .

Reactive power demand at node .

Existent conductor type of branch . If ,no conductor is in branch .

Resistance of conductor type per kilometer.

Reactance of conductor type per kilometer.

Impedance of conductor type per kilometer.

Number of blocks of the piecewise linearization.

Parameter used in the calculation of the currentflow magnitude of the circuits.

Slope of the block of deviation voltagemagnitude at node .

Upper bound of the deviation voltage magnitudeblocks at node .

Slope of the block of current flow magnitudeof conductor type.

Upper bound of the current flow magnitudeblocks of conductor type.

Number of hours in a year for load level .

Variables:

Circuit that can be added on branch ofconductor type.

Real power flow that leaves node toward nodeof conductor type.

Reactive power flow that leaves node towardnode of conductor type.

Real power provided by substation at node .

Reactive power provided by substation at node .

Voltage magnitude at node .

Square of .

0885-8950/$31.00 2012 IEEE

FRANCO et al.: OPTIMAL CONDUCTOR SIZE SELECTION AND RECONDUCTORING 11

Current flow magnitude that leaves node towardnode of conductor type.

Main component of .

Secondary component of .

Square of .

Square of .

Square of .

Value of the block of deviation voltagemagnitude at node .

Value of the block of .

Value of the block of .

I. INTRODUCTION

T HE main objective of an electrical distribution system(EDS) is to provide a reliable and cost-effective serviceto consumers while ensuring that power quality is within stan-dard ranges. To achieve this objective, it is necessary to properlyplan the EDS and thus evaluate several aspects: new equipmentinstallation cost, equipment utilization rate, quality of service,reliability of the distribution system and loss minimization, con-sidering an increase of system loads and newly installed loadsfor the planning horizon [1].In EDS planning, the conductor size selection (CSS) problem

works to select the conductor size (from an available set) in eachbranch of the EDS, whichminimizes the investment cost and theenergy losses subject to feasible operation constraints. Severalparameters are taken into account to model the CSS problem:conductors economic life, discount rate, cable and installationcosts and type of circuit (overhead or underground) [2]. The re-conductoring problem is considered part of the EDS planningproblem and functions to change the existing circuit conductorsto others conductor types. Themain reasons to use the reconduc-toring problem are: 1) when there are excessive power losses inthe existing system, 2) when the maximum current capacity ofexisting circuits is violated or 3) when the voltage magnitudesin the EDS are lower than its minimum limit [3].In the specialized literature the CSS problem is commonly

modeled as a mixed integer nonlinear programming (MINLP)problem, and various approaches have been used to solve it.Reference [4] is one of the first works to formulate the CSSproblem. The study presents models to represent feeder cost,energy loss and voltage regulation as a function of a conductorcross-section. The dynamic programming approach was thenused to solve the CSS problem. In [5], financial and engineeringcriteria to choose the conductor size in a feeder were proposed;the study found that a conductor is most economical when bothcapital and operating costs are considered in the CSS problem.A heuristic method to solve the CSS problem is presented

in [6]; this method uses a selection phase by means of eco-nomic criteria, followed by a technical selection using a sensi-tivity index that seeks to ensure a feasible operation of the EDS.The heuristic methods are robust, easily applied and normallyconverge to a local optimum solution. In [7] the CSS problemis solved using systematic enumeration through logical rules.In [8] and [9] the optimal CSS and capacitor placement are

solved using two different approaches. Reference [8] presentsa heuristic method using a novel sensitivity index for the reac-tive power injections, whereas [9] uses a genetic algorithm.Several studies have used evolutive techniques to solve the

CSS problem [10][12]. Although these techniques are easy andsimple methods that provide good results, they present variousproblems such as high processing demand and their incapacityto guarantee the optimum solution. In [13] and [14], the re-conductoring problem of the existent circuits was modeled inthe EDS planning problem, considering also the conductor sizeselection for the new circuits. This problem is modeled as aMINLP problem and can be solved using a genetic algorithm[13] or dynamic programming [14]. Therefore, the previouslymentioned techniques, as well as the use of solvers that directlysolve the MINLP problem also represent alternatives to solvingthe CSS problem.Some of the methods mentioned above use linear approxima-

tions in the calculation of power losses or voltage regulation.Another approximation is to assume that the loads are modeledas constant current [4] or apparent power [15]. If the linear ap-proximations are not used, the mathematical model for the con-ductor size selection and reconductoring (CSSR) problem be-comes nonlinear, complicating its solution. Therefore heuristicmethods and meta-heuristics are commonly used [6], [10][12].The present study proposes a mixed integer linear model for

the problem of conductor selection size and reconductoring ofprimary feeders in radial distribution systems. Linearizationswere made to adequately represent the steady-state operationof an EDS considering the behavior of the constant power typeload. The proposed model was tested in systems of 50, 200, and600 nodes. In order to validate the approximations performed,the steady-state operation point was compared to that obtainedusing the load flow sweep method. In contrast with other worksthat use mixed integer linear models, the proposed model rep-resents the constant power type load with added precision.The main contributions of this paper are as follows:1) A novel model for the steady-state operation of a radialdistribution system through the use of linear expressions.

2) A mixed integer linear programming (MILP) model for theconductor size selection and reconductoring problem thatpresents an efficient computational behavior with conven-tional MILP solvers.

3) A heuristic to obtain the Pareto front of the CSSR problemconsidering two different objective functions (powerlosses and investment costs).

II. OPTIMAL CONDUCTOR SIZE SELECTIONAND RECONDUCTORING PROBLEM

The conductor size selection problem involves determiningthe optimal conductor configuration for a radial distributionsystem, using a set of types of conductors. Each type ofconductor has the following characteristics: 1) resistance perlength, 2) reactance per length, 3) maximum current capacityand 4) building cost per length. The reconductoring of existingcircuits is determined by the investment cost , where theinvestment cost depends on the initial conductor type andthe final conductor type . Table I shows an example offor four types of conductor. If (case without existingcircuit), then represents the building cost of a new circuitfor conductor type . is big number greater than the other


TABLE IRECONDUCTORING COST FROM CONDUCTORTYPE TO CONDUCTOR TYPE

Fig. 1. Illustrative example.

costs and is used to indicate that the reconductoring is notattractive because involves a conductor of less capacity.

A. Assumptions

In order to represent the steady-state operation of an EDS, thefollowing assumptions are made:1) The load is represented as constant real and reactive power.2) The flows of real power, reactive power and current onbranch are in the same direction, leaving node towardnode .

3) The real and reactive power losses on branch are con-centrated in destination node .

The three considerations are shown in Fig. 1, where andare the phasors of the voltage at node and the current flow

on branch , respectively. and are the real and reac-tive power flow that leaves node toward node , respectively., , and are the resistance, reactance, and impedance

of branch , respectively. and are the real andreactive power losses of branch , respectively.

B. Steady-State Operation of a Radial Distribution System

In Fig. 1, the voltage drop in a circuit is defined by (1):

(1)

where can be calculated using (2):

(2)

Equation (2) is then replaced in (1) to obtain (3):

(3)

Considering that , and ,where is the phase angle at node , (3) can be written as shownin (4):

(4)

Identifying the real and imaginary parts (4), we get

(5)(6)

Summing the squares of (5) and (6), we get

(7)

where the current flow magnitude is shown in (8):

(8)

In (7) the angular difference between voltages is eliminated; itis possible to obtain the voltage magnitude of the final nodein terms of the voltage magnitude of the initial node , thereal power flow , the reactive power flow , the currentmagnitude and the electrical parameters of branch . Theconventional equations of load balance are shown in (9) and(10); see Fig. 1. Equations (7)(10) represent the steady-stateoperation and are frequently used in the load flow sweepmethod[16], [17] and optimal load flow [18] of a radial distributionsystem:

(9)

(10)

C. MINLP Model of the CSSR Problem

The CSSR problem can be modeled like a mixed integer non-linear programming problem as follows:

(11)


The objective function (11a) is the total investment and op-eration cost based on [13]. The first part represents the invest-ment cost (construction/reconductoring of circuits); the secondpart represents the cost of power losses in the planning horizon,where is a factor to calculate the cost of the peak power losses;it is a function of the energy cost, loss factor, interest rate, plan-ning horizon and load growth ratio as shown in [4]. Equations(11b)(11e) represent the steady-state operation and are a nat-ural extension of (7)(10) considering different conductor types.For the CSSR problem, the and values are the demandsat the moment of maximum loading of the feeder, which is theworst case to evaluate the minimum voltage magnitude, max-imum power losses and maximum current magnitude. The limitof the flows of current in branch of conductor type is rep-resented by (11f). Equation (11g) represents the constraints ofthe voltage magnitude of the nodes, while (11h) represents themaximum capacity of apparent power at substation . Equation(11i) stipulates no superposition in the conductor type, so it ispossible to install only one conductor type per circuit.Equation (11j) represents the binary nature of conductor type

thatcanbeselected inbranch .Aconductor type isselected if thecorrespondingvalue is equal tooneand isnot selected if it is equalto zero. The binary investment variables are the decisionvariables (control variables), and a feasible operation solution forthe distribution system depends on their value. The remainingvariables represent the operating state of a feasible solution. Fora feasible investment proposal, defined through specified valuesof , several feasible operation states are possible.Given that , and arepositivevalues, theobjective func-

tion (11a) is a convex quadratic function.Constraints (11g)(11i)are linear, and constraints (11b)(11f) contain square terms.Withthe aim of using a commercial solver, it is desirable to obtain alinear equivalent for constraints (11.b)(11.f).

D. Linearization

Note that the quadratic terms and appears in(11a)(11f). The objective of this subsection is to find linearexpressions for both terms using a piecewise linear modeling.1) Square of the Voltage Magnitude: From (11g), the voltage

magnitude has a minimum value of and a maximum valueof . Let be the variable that represents the square voltagemagnitude, as shown in (12):

(12)

where has a minimum value of 0 and a maximumvalue of . From (12), the quadratic term is linearizedas described in [19] and shown in Fig. 2. Thus, the square ofvoltage magnitude is defined in (13):

(13)where

Fig. 2. Modeling the piecewise linear function.

Note that (13) is a set of linear expressions and andare constant parameters. Constraints (13a) are the linear ap-

proximations of square voltage magnitude at node . Constraints(13b) state that the voltage magnitude at node is equal to theminimum voltage magnitude plus the sum of the values in eachblock of the discretization. Constraints (13c) set the upper andlower limits of the contribution of each block of the differencebetween the voltage magnitude and the minimum voltage mag-nitude at node .2) Square of the Current FlowMagnitude: Note that the divi-

sion of two operation variables appears in (11e). Therefore, thisequation cannot be used to linearize . An alternative formto calculate the square of the current flow magnitude is shownin (14) based on (1):

(14)

Equation (14) can be separated into two terms as shown in(15):

(15)

In a radial distribution system it is possible to assume that theangular difference is small; thus, the second term (15) is neg-ligible and is normally eliminated: see [4], [6], [20], and [21].Therefore, the current flowmagnitude would depend only on thefirst term. However, it is possible to estimate the second term of(15) considering an approximation of usingand (6) for different types of conductors:


The previous approximation for cosine function is used in(15) to obtain (16):

(16)

Let be the variable that represents the square current flowmagnitude in branch of conductor type, then (16) can besubstituted with (17)(19). The above separation can be donebecause (11f) and (11i) guarantee that only one conductor typeis selected:

(17)where

(18)

(19)

Note that has two components and, from the assumptionsshown in Section II-A, is always positive and representsthe main component in the calculation of . However,can be positive or negative and has as its objective improvingthe precision of the calculation of . As the voltage magni-tudes of the nodes of the EDS are limited, it is possible to obtaina linear expression for (19), approximating by a constantparameter for all circuits, as shown in (20). The term iscalculated before solving the CSSR problem, using the solutionof a load flow problem, as shown in Section III. This consider-ation causes an error in the calculation of , but, as will beshown in Section IV, it is negligible:

(20)

In the same way, for the square of voltage magnitude shownin Section II-D1, the square of and the square of

from (17) are linearized as shown in (21):

(21)where

As in (13), note that (21) is a set of linear expressions andand are constant parameters. Constraint (21a) replace

constraint (17) and is the linear approximation of square currentflow magnitude on branch of conductor type. and

are non-negative auxiliary variables to obtain asis shown in (21b). Constraints (21c) and (21d) are the linearapproximations of and , respectively. Constraints(21e) and (21f) state that and are equal to the sumof the values in each block of the discretization, respectively.Constraints (21g) and (21h) set the upper and lower limits of thecontribution of each block of and , respectively.

E. MILP Model for the CSSR ProblemThe CSSR problem could be modeled like a mixed integer

linear programming problem, as follows:

(22)where (22a), (22b), (22c), (22d) and (22e) replace (11a), (11b),(11c), (11d) and (11f), respectively. The limits of the flows ofreal and reactive power in branch of conductor type arerepresented by (22f) and (22g), respectively, and are auxiliaryconstraints used to make feasible the MILP model of the CSSRproblem. In the MINLP model (see Section II-C) if( conductor type on branch is not selected), then the respec-tive flows of current, real power and reactive power are equalto zero. In the MILP model these conditions are guaranteed by(18), (20)(21) and (22e)(22g), where is themaximum apparent power limit of conductor type and pro-vides a sufficient degree of freedom to the flows of real and re-active power in branch of conductor type when .Note that (22) is a piecewise linear model and the number of op-eration variables has increased with the linearization, while thenumber of investment variables does not change and, as will beillustrated later in Section IV, this kind of optimization problemcan be solved with the help of standard commercial solvers, ashas been done in other work in this area (see [20] and [21]).Note that (13), (18), (20)(21) and (22b)(22d) represent the

steady-state operation of the radial distribution system and arelinear expressions. Considering the assumptions in Section II-A,these expressions can be used to analyze a EDS with distributedgenerators or to model other optimization problems of the radialdistribution systems through the use of linear expressions andsolve it using classical optimization techniques.


TABLE IICOMPUTATIONAL COMPLEXITY

F. Comments on the Use of MINLP and MILP Models

The MINLP model for the CSSR problem can be solved byusing heuristic methods, meta-heuristics, or solvers that directlysolve a MINLP problem. However, these techniques cannotguarantee the optimal solution. On the other hand, if accuratelinear approximation of the quadratic terms and inthe MINLP model is used, a recast of the MINLP model intoa mixed-integer linear model is obtained. This MILP modelguarantees convergence to optimality while using existingoptimization software.Table II summarizes the computational complexity for the

CSSR problem using both models. The following observationscan be made: The number of binary variables for both models is thesame, ; which means that, for both models, thesearch space is solutions.

The number of variables of the MINLP model is propor-tional to the number of branches and the number ofconductor types . In addition to that, the number of vari-ables of the MILP model is also proportional to the numberof blocks of the piecewise linearization, . Asand , the size of both models is essentially depen-dent on the size of the system.

Note that the magnitude order of the constraints for bothmodels is ; it is not dependent on .

Taking into account the above comments, we can conclude thatthe linearization does not contribute to increasing the searchspace or to the complexity order of the constraints of the CSSRproblem. If the size of the system increases, then the processingtime to achieve convergence may increase prohibitively. Thisis a common drawback of using exact techniques to solveMILP and MINLP problems. However, the results presented inSection IV show that the computational time do not increaseexponentially with the system dimensions and that the method-ology can be used to solve real systems.

G. Modeling Load Levels for the CSSR Problem

The CSSR problem usually has as an objective function theminimization of the cost of peak power losses. However, dueof the varying yearly loss patterns, it is advisable the minimiza-tion of the energy losses cost considering the time variation ofthe loads. The proposed model for the CSSR problem can beextended in order to represent several load levels taking as ob-jective function the minimization of the energy losses cost, aspresented in (23), and using the additional index for all

the operational variables of the proposed model and for the ac-tive and reactive power demand:

(23)

III. METHODOLOGY OF THE SOLUTION

This section will show an expression for the calculation ofthe constant parameter and proposes constraint (24), whichtakes into account a minimum value for the current magnitude inevery circuit, with the aim of reducing the computational effortneeded to solve the CSSR problem.Since the loads are modeled as constant power, it is possible

to demonstrate that the minimum values for the current magni-tudes in the circuits appear when the voltage magnitude dropsare lowest. For the CSSR problem, the lowest voltage magni-tude drops appear when all the circuits are built with the con-ductor type of lowest impedance . Thus, we solve a load flowproblem assuming that the conductor type for all circuits is .When solving this load flow problem, the obtained current mag-nitudes for every circuit give a lower bound for the currentmagnitudes in the CSSR problem. Using this lower bound, (24)is defined, which is added to the proposed model:

(24)

Additionally, using the information obtained with the solu-tion of the load flow problem aforementioned, a value forcan be estimated in accordance with (25), thereby taking a con-stant factor for the EDS. Equation (25) is designed to find avalue to reduce the error in the calculation of real power lossesassociated with the component . It isobtained by equating the real power losses calculated usingunder (19) with the approximated real power losses calculatedusing according (20):

(25)

where , and are the real and reactive power flowof circuit and the voltage magnitude at node , respectively,obtained with the solution of the load flow problem. The stepsof the proposed methodology to solve the CSSR problem arepresented in the flowchart in Fig. 3.

A. Approximation of the Pareto FrontThe proposed model for the CSSR problem can be used to

solve a multiobjective problem considering the power lossesand the investment cost as two different objective functions.Those objectives are two conflicting functions because to re-duce the power losses it is necessary to build circuits with lowerresistance, which implies an increase in the investments; on theother hand, if one wishes to reduce the investment cost, then a


Fig. 3. Flowchart of the proposed methodology.

rise in the power losses is to be expected. The multiobjectiveCSSR problem can be stated as shown in (26):

(26)

where represents the investment cost, are the power losses,and is a feasible solution for the conductors of the circuits.The multi-objective optimization allows to obtain a set of Paretosolutions, which is known as a Pareto front. A solution isPareto if it is not dominated by another solution, that is, thereis no another solution that satisfies both and

[22].In order to obtain the Pareto front for the multiobjective

CSSR problem, a constraint limiting the investment cost, asshown in (27), is included in the CSSR model. Each th solu-tion of the Pareto front is found by establishing an appropriatevalue for the maximum limit of investments and solvingthe resulting CSSR model. The first solution is found withan arbitrarily high limit of investments. The next solution isobtained by setting the limit of investments as the investmentcost of the previous solution; the process is repeated until theproblem becomes unfeasible:

(27)

Knowing the Pareto front brings flexibility to the decisionprocess and allows for better adaptation to the policies of eachelectrical distribution company. Thus, a set of solutions is avail-able that ranges between one that minimizes real power lossesand another one that minimizes investment costs to satisfy op-erational constraints. In order to support the decision process,several multi-criteria decision analysis methods can be used, asshown in [23].We suggest a simple way to choose the best solution given

the Pareto Front. If the electrical distribution company wantsto reduce their real power losses under a goal value of ,the best solution is given by

, which represents the solution with minimum invest-ment cost that has a power losses lower than the specified goal.A similar analysis can be done for the investment cost, if theelectrical distribution company has an investment cost limit of

, the best solution is given by, which represents the solution with minimum

power losses that has a investment cost lower than the specifiedlimit.

TABLE IIITECHNICAL CHARACTERISTICS OF THE CONDUCTOR TYPES

IV. TESTS AND RESULTSA test system of 50 nodes and two real distribution systems

of 200 and 600 nodes were used to show the performanceand robustness of the proposed methodology. For all tests,the maximum and minimum voltage magnitude was 1.00 puand 0.95 pu, respectively, the voltage magnitude of the sub-station was fixed to 1.00 pu, the peak power losses cost was

, the reconductoring costs are shown inTable I and the technical characteristics of the four conductortypes are shown in Table III. The number of blocks of thepiecewise linearization is equal to 40. The CSSR model wasimplemented in AMPL [24] and solved with CPLEX [25](called with default options, with a maximum gap ofas optimality criterion) using a workstation with an Intel XEONW3520 processor.

A. 50-Node Distribution SystemThe 50-node distribution system is based on [26] and the

data are shown in Table IV. It is a 15.0-kV distribution systemsupplying 56.54-MVA and feeds 50 load nodes. The 50-nodesystem had 15 existing circuits (2 circuits of conductor type 2and 13 circuits of conductor type 1) and 35 circuits to be built.The value for the parameter used to solve the CSSR problem,obtained using (25), was 0.9859 pu. The number of binary vari-ables of the CSSR problem was 200.The solution of the CSSR problem was found evaluating 293

nodes of a branch and bound algorithm with a computationaltime of 8 s and a total cost of with an invest-ment cost of . The proposed model selected11 circuits with conductor type 4, 1 circuit with conductor type2 and 23 circuits with conductor type 1 and reconductored 5circuits (4 circuit of initial conductor type 1 and 1 circuit of ini-tial conductor type 2) to conductor type 4, as shown in the fifthcolumn of Table IV. In order to compare the results found by theproposed methodology, an exhaustive enumeration was used tosolve the CSSR problem. The found solution by the exhaustiveenumeration is the same found by the proposed model. Thus, theproposed methodology found the optimal solution of the CSSRproblem for this test system.The operation point for the solution of the CSSR problemwas

compared using the load flow sweep method. The results of thereal power losses and the voltage magnitude at node 10 (whichhad the largest error) are shown in the two first rows of Table V.Note that the errors are negligible, showing the accuracy of themodel.In order to show the influence of the component in the

CSSR problem, the model was solved without considerationof this component, with a computational time of 9 s, and theconductors configuration solution did not change. Additionally,the operation point of the optimal solution can be seen in thethird row of Table V. Further, Table VI shows a comparison


TABLE IVDATA AND RESULTS FOR THE 50-NODE DISTRIBUTION SYSTEM

TABLE VPOWER LOSSES AND MINIMUM VOLTAGEMAGNITUDE FOR THE 50-NODE SYSTEM

of the calculation for the five greatest current magnitudes, withtheir relative errors indicated in parentheses. The above resultsshow that, when the component is disregarded, the voltagemagnitudes are overestimated with an error of , thereal power losses are underestimated with an error of ,and the current magnitudes are underestimated with a maximumerror of . These errors are considered acceptable in thesolution of distribution planning problems, and for this reasonthe sole use of the component is common in studies of EDSplanning [20], [21] and conductor size selection [4], [6].Using the heuristic presented in Section III-A, the Pareto front

for the 50-node system was found as seen in Fig. 4. The pointto the right is the solution found for the base case, without aninvestment limit, which has the minimum total cost. The pointto the left is the solution that presents the largest real powerlosses (1231.49 kW), but with the minimum investment neces-

TABLE VICOMPARISON OF CURRENT MAGNITUDES FOR THE 50-NODE SYSTEM [A]

Fig. 4. Pareto front for the 50-node system.

sary for the system to operate while satis-fying the operational constraints.The Pareto front allows selecting a solution according to the

needs and policies of the electrical distribution company. Forexample, if the limit for investments is , with thehelp of Fig. 4 can be determined that the best solution that sat-isfies that limit and also the operational constraints has a realpower losses of 1140 kW. On the other hand, if the electricaldistribution company wants to reduce their real power lossesunder a goal value of 1100 kW, the Pareto front provides a so-lution with an investment cost of .A test considering load levels was carried out with the

50-node distribution system. For this test three load levelswere considered, which were obtained by multiplication of thenominal loads by the factors 1.0 (heavy loading), 0.4 (mediumloading) and 0.3 (light loading), with respective durations of1000, 6760 and 1000 h. The solution of the CSSR problemwas found with a computational time of 25 s and a total cost of

with an investment cost of ,which has the same selection for conductors that the caseconsidering maximum loading. The obtained solution was thesame because the constant was calculated using a loss factorof 0.25, which represents adequately the energy losses for theload levels in terms of the maximum power losses; also theselection of conductors considering load levels must to accom-plish the minimum voltage magnitude, where the worst case isactually the heavy loading. The power losses at each load levelcalculated using the load flow sweep method and the valuesobtained from the proposed model are shown in Table VII.

B. 200-Node Real Distribution SystemThe 200-node distribution system data are based on the

system in [13]. It is a 11.5-kV distribution system supplying


TABLE VIIPOWER LOSSES FOR THE 50-NODE SYSTEM WITH LOAD LEVELS

TABLE VIIIPOWER LOSSES AND MINIMUM VOLTAGEMAGNITUDE FOR THE 200-NODE SYSTEM

18.63-MVA and feeds 200 load nodes. The 200-node systemhad 40 existing circuits (20 of conductor type 1 and 20 ofconductor type 2) and 160 circuits to be built. The data for thethis test system can be obtained upon request. The value for theparameter used to solve the CSSR problem was 0.9833 pu.

The number of binary variables of the CSSR problem was 800.The solution of the CSSR problem was found by evaluating

978 nodes of a branch and bound algorithm with a computa-tional time of 484 s and a total cost of withan investment cost of . The proposed modelselected 9 circuits with conductor type 4, 7 circuits with con-ductor type 2 and 144 circuits with conductor type 1 and recon-ductored 13 circuits (6 circuits of initial conductor type 1 and7 circuits of initial conductor type 2) to conductor type 4. Theoperation point for the solution of the CSSR problem was eval-uated using the load flow sweep method. The results of the realpower losses and the voltage magnitude at node 74 (which hadthe largest error) are shown in the two first rows of Table VIII.As in the previous test, the errors are negligible, demonstratingthe accuracy of the model. In order to compare the results foundby the proposed methodology, the model for the CSSR problempresented in [15] was solved, and it found a solution with a totalcost of . The obtained results show that theproposed methodology found a better solution than the one in[15] for the CSSR problem.Also, for this test system, the CSSR problem was solved

without the component, and the conductors configuration so-lution did not change. The solution was found with a computa-tional time of 215 s. The operation point of the optimal solutioncan be seen in the third row of Table VIII. Furthermore, Table IXshows a comparison of the calculation for the five greatest cur-rents magnitudes, with their relative errors indicated in paren-theses. As in the previous test, the above results show that, whenonly the component is used, the voltage magnitudes are over-estimated with an error of , the real power losses areunderestimated with an error of , and the current mag-nitudes are underestimated with a maximum error of .Using the heuristic presented in Section III-A, the Pareto front

for the 200-node system was found as is seen in Fig. 5. As in theprevious test, the point to the right is the solution found for thebase case, without an investment limit, which has the minimumtotal cost. The point to the left is the solution that presents thelargest real power losses (579.33 kW), but with the minimuminvestment necessary in order for the systemto operate while satisfying its operational constraints.

TABLE IXCOMPARISON OF CURRENT MAGNITUDES FOR THE 200-NODE SYSTEM [A]


Similarly, as in the previous test, if the electrical distributioncompany wants to reduce their real power losses under a goalvalue of 540 kW, an investment cost of is nec-essary, as shown in Fig. 5. On the other hand, if the limit forinvestments is , with the Pareto front can be de-termined that the best solution has a real power losses of 535kW.

C. 600-Node Real Distribution SystemThe 600-node distribution system data are based on the

system in [13]. It is a 11.5-kV distribution system supplying18.72-MVA and feeds 600 load nodes. The 600-node systemhas 74 existing circuits (36 of conductor type 1 and 38 ofconductor type 2) and 526 circuits to be built. The value forthe parameter used to solve the CSSR problem was 0.9825pu. The number of binary variables of the CSSR problem was2400.The solution of the CSSR problemwas foundwith a computa-

tional time of 2375 seconds and a total cost ofwith an investment cost of . The proposedmodel selected 2 circuits with conductor type 4, 7 circuits withconductor type 2 and 517 circuits with conductor type 1 and re-conductored 13 circuits (5 circuits of initial conductor type 1and 8 circuits of initial conductor type 2) to conductor type 4.The operation point for the solution of the CSSR problem wasevaluated using the load flow sweep method. The results of thereal power losses and the voltage magnitude at node 74 (whichhad the largest error) are shown in the two first rows of Table X.As in the previous test, the errors are negligible, demonstratingthe accuracy of the model.In order to compare the results found by the proposed

methodology, the model for the CSSR problem presented in[15] was solved, and it found a solution with a total cost of

. The obtained results show that the proposed


TABLE XPOWER LOSSES AND MINIMUM VOLTAGEMAGNITUDE FOR THE 600-NODE SYSTEM

TABLE XICOMPARISON OF CURRENT MAGNITUDES FOR THE 600-NODE SYSTEM [A]

methodology found a better solution than the one in [15] forthe CSSR problem.Also, for this test system, the CSSR problem was solved

without the component, and the conductors configuration so-lution found had a total cost of with a com-putational time of 618 s, which was different from the solutionfor the complete model in 5 circuits.With the aim to show the accuracy of the proposed model, the

operation point of the optimal solution for the complete modelwas calculated using the equations of the model without con-sidering (as can be seen in the third row of Table X). Fur-thermore, Table XI shows a comparison of the calculation forthe five greatest currents magnitudes, with their relative errorsindicated in parentheses. As in the previous test, the above re-sults show that, when only the component is used, the voltagemagnitudes are overestimated with an error of , thereal power losses are underestimated with an error of ,and the current magnitudes are underestimated with a maximumerror of .Using the heuristic presented in Section III-A, the Pareto front

for the 600-node system was found as is seen in Fig. 6. As in theprevious test, the point to the right is the solution found for thebase case, without an investment limit, which has the minimumtotal cost. The point to the left is the solution that presents thelargest real power losses (604.93 kW), but with the minimuminvestment necessary in order for the systemto operate while satisfying its operational constraints.Similarly, as in the previous test, if the electrical distribution

company wants to reduce their real power losses under a goalvalue of 580 kW, an investment cost of is nec-essary, as shown in Fig. 6. On the other hand, if the limit forinvestments is , with the Pareto front can be de-termined that the best solution has a real power losses of 604kW.

V. CONCLUSIONSA mixed-integer linear programming model to solve the

CSSR problem in radial distribution systems was presented.The use of a MILP model guarantees convergence to optimalityusing conventional MILP solvers.In the proposed MILP model, the steady-state operation of

the radial distribution system is modeled through the use of


linear expressions. The results show that the power losses,voltage magnitude, and current flow magnitudes are calculatedwith great precision in comparison with the load flow sweepmethod. This fact, combined with the use of a branch andbound algorithm, provides a high degree of accuracy for theproposed methodology in order to solve the CSSR problem, asshown in Section IV.One test system and two real distribution systems were used

to test the proposed model. For the test system, the solutionfound by the proposed model is the same as the one found bythe exhaustive enumeration; whereas, for the two real distribu-tion systems, the proposed methodology found a better solutionwhen compared to the methodology shown in [15].The Pareto front for the conductor size selection and recon-

ductoring problem considering two different objective functionsis easily found using a heuristic, making it possible to obtain theset of non-dominated solutions according to power losses andinvestment costs.

REFERENCES[1] T. Gnen, Electric Power Distribution Systems Engineering. New

York: McGraw-Hill, 1986.[2] G. J. Anders, M. Vainberg, D. J. Horrocks, S. M. Foty, J. Motlis, and

J. Jarnicki, Parameters affecting economic selection of cable sizes,IEEE Trans. Power Del., vol. 8, no. 4, pp. 16611667, Oct. 1993.

[3] H. Lee, Power Distribution Planning Reference Book. New York:Basel: M. Dekker, 1997.

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[14] E. Daz-Dorado and J. C. Pidre, Optimal planning of unbalanced net-works using dynamic programming optimization, IEEE Trans. PowerSyst., vol. 19, no. 4, pp. 20772085, Nov. 2004.

[15] H. N. Tram and D. L. Wall, Optimal conductor selection in planningradial distribution systems, IEEE Trans. Power Syst., vol. 3, no. 1, pp.200206, Feb. 1988.

[16] M. E. Baran and F. F. Wu, Network reconfiguration in distributionsystems for loss reduction and load balancing, IEEE Trans. PowerDel., vol. 4, no. 2, pp. 14011407, Apr. 1989.

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John F. Franco (S11) received the B.Sc and M.Sc degrees in 2004 and2006, respectively, from the Universidad Tecnolgica de Pereira, Colombia.Currently, he is pursuing the Ph.D. degree in electrical engineering at theUniversidade Estadual Paulista, Ilha Solteira, Brazil.His areas of research are the development of methodologies for the optimiza-

tion and planning of distribution systems.

Marcos J. Rider (S97M06) received the B.Sc. (Hons.) and P.E. degreesin 1999 and 2000, respectively, from the National University of Engineering,Lima, Per; the M.Sc. degree in 2002 from the Federal University of Maranho,Maranho, Brazil; and the Ph.D. degree in 2006 from the University of Camp-inas, Brazil, all in electrical engineering.Currently he is a Professor in the Electrical Engineering Department at the

Universidade Estadual Paulista, Ilha Solteira, Brazil. His areas of research arethe development of methodologies for the optimization, planning and controlof electrical power systems, and applications of artificial intelligence in powersystems.

Marina Lavorato (S07M11) received the B.Sc and M.Sc degrees in 2002and 2004, respectively, from the Federal University of Juiz de Fora, Brazil, andthe Ph.D. degree in 2010 from the University of Campinas, Brazil, all in elec-trical engineering.Currently she is carrying out postdoctorate research at the Universidade Es-

tadual Paulista, Ilha Solteira, Brazil. Her areas of research are the developmentof methodologies for the optimization, planning and control of electrical powersystems.

Rubn Romero (M93SM08) received the B.Sc. and P.E. degrees in 1978and 1984, respectively, from the National University of Engineering, in Lima,Per, and the M.Sc and Ph.D degrees from the University of Campinas, Brazil,in 1990 and 1993, respectively.Currently he is a Professor in the Electrical Engineering Department at the

Universidade Estadual Paulista, Ilha Solteira, Brazil. His general research inter-ests are in the area of electrical power systems planning.

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