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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
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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
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12 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY
2013
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)
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FRANCO et al.: OPTIMAL CONDUCTOR SIZE SELECTION AND
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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:
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14 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY
2013
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.
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FRANCO et al.: OPTIMAL CONDUCTOR SIZE SELECTION AND
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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
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16 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY
2013
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
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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
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18 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 1, FEBRUARY
2013
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]
Fig. 5. Pareto front for the 200-node system.
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
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FRANCO et al.: OPTIMAL CONDUCTOR SIZE SELECTION AND
RECONDUCTORING 19
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
Fig. 6. Pareto front for the 600-node system.
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.
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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.