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Reconfiguration and optimal capacitor placementfor losses reduction

Diana P. Montoya, Student Member, IEEE. Juan M. Ramirez, Member, IEEE.

Abstract—This paper presents a deterministic approach fornetwork reconfiguration and a heuristic technique for optimalcapacitor placement for power-loss reduction and voltageprofile improvement in distribution networks. Related to thereconfiguration, a Minimum Spanning Tree (MST) algorithmis utilized to determine the configuration of minimum losses.After that, a Genetic Algorithm (GA) is implemented to achievethe greatest savings through the optimal capacitor placement.The objective of this study is to analyze the benefits thatthe reconfiguration and optimal placement of capacitors havein the grid. The proposed algorithms have been implementedin two IEEE test systems. Results indicate that substantialannual savings may be obtained through the application of thesetwo techniques and taking into account these two techniquescomplementarily can become quite efficient.

Index Terms—Distribution systems reconfiguration, powerloss reduction, optimal capacitor placement, voltage profileimprovement.

I. INTRODUCTION

THE Smart Grid development requires that utilities lookfor strategies for increasing their efficiency, for instance,

losses reduction. There are different approaches to reducelosses: equilibrium and symmetry of loads, reduction oflength of lines or the wire sizing increment, optimal useof transformers loading, capacitor placement, elimination ofharmonics, reconfiguration, optimal change of transformerstap, use of series capacitor for loss reduction of lines, optimalD.G (distributed generation) placement, etc. In this paper twoof such strategies are approached: (i) reconfiguration; (ii) localcapacitor placement.

Network reconfiguration is the process by which thedistribution systems topology is altered through theopen/closed status of switches. The reconfiguration algorithmsmay be classified by the solution methods that are employedto attain the goal: (i) those based upon a blend ofheuristic optimization methods; (ii) deterministic methods.In open research, several meta-heuristic techniques havebeen proposed to solve the reconfiguration problem forlosses’ minimization. Likewise, evolutionary algorithms havebeen applied for losses’ minimization. Recently, some otherstochastic-based techniques like simulated annealing, particleswarm optimization [1], genetic algorithms [2]; ant colonyoptimization [3], tabu search algorithms [4], etc., also triedthe distribution systems reconfiguration.

Juan M. Ramirez acknowledges support from CONACyT project no.167933. (e-mail:jramirez@gdl.cinvestav.mx)

Diana Montoya is a M. Sc. student in Department of Electrical PowerSystem, CINVESTAV-Guadalajara. (e-mail:dmontoya@gdl.cinvestav.mx)

Juan M. Ramirez is with CINVESTAV - Guadalajara. MEXICO.

The problem of capacitor allocation for loss reduction inelectric distribution systems has been extensively researchedover the past several decades. Reactive currents account for aportion of power losses. However, losses produced by reactivecurrents can be reduced by the installation of shunt capacitors.In addition to the reduction of energy and peak power loss,effective capacitor installation can also release additional kVAcapacity and improve the system voltage profile. Thus, theoptimal capacitor allocation problem involves determiningthe locations, sizes, and number of capacitors to install ina distribution system such that the maximum benefits areachieved while all operational constraints are satisfied atdifferent loading levels.

Several papers related to optimal capacitor placement werebased on analytical methods. These algorithms were devisedwhen powerful computing resources were unavailable orexpensive. Analytical methods involve the use of calculusto determine the maximum of a capacitor savings function.The pioneers of optimal capacitor placement are, Neagle andSamson [5], Chang [6], Miles Maxwell [7], Schmill [8], Cook[9], Kuppurajulu [10] and Bae [11]. Although simple closed-form solutions were achieved, these methods were basedon unrealistic assumptions. Constant wire size and uniformloading feeders, for instance. It was from this early researchthat the famous “two-thirds” rule became established. The“two-thirds” rule advocates that for maximum loss reduction,a capacitor rated at two-thirds of the peak reactive load shouldbe installed at a position two-thirds of the distance along thetotal feeder length.

For optimal capacitor allocation, the savings functioncould be the objective function and the locations, sizes,number of capacitors, and bus voltages would be thedecision variables, which must satisfy operational constraints.Numerical programming methods allow the use of moreelaborate cost functions for solving the problem. The objectivefunctions can take the voltage and line loading constraintsinto account, discrete sizes and location for capacitors. Usingnumerical programming methods, the capacitor allocationproblem can be formulated as follows [12]

max S = KLΔL−KCC (1)

where S is the savings, KLΔL is the cost savings which mayinclude energy and peak power loss reductions, and releasedcapacity, KCC is the capacitors’ installation costs.

Several authors have developed the optimal location ofcapacitors using a GA. Das et al [13], Zhang et al [14],Farahani et al [15], Srinivasas Rao et al [16], Tabatabaei etal [17], used a GA to solve the optimization problem.

978-1-4673-2673-5/12/ $31.00 c©2012 IEEE

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Das et al [13], develops a GA based fuzzy multi-objectiveapproach for determining the optimum values of fixed andswitched shunt capacitors. Moreover, Zhang et al [14], usedan improved adaptive GA (IAGA) to optimize capacitorswitching, and it used a simplified branch exchange algorithm,to find the optimal network structure for each genetic instanceat each iteration of the capacitor optimization algorithm.

Frahani et al [15], applies a discrete GA to optimize thelocation and size of capacitors and the sequence of loopsselection. In fact, the capacitor sizes have been consideredas discrete variables.

Srinivasas Rao et al [16], proposed a methodologydecomposed in two steps: (i) the loss sensitivity factorswas used to select the candidate locations for the capacitorplacement; (ii) an algorithm that employs the Plant GrowthSimulation Algorithm (PGSA) was used to estimate theoptimal size of capacitors at the optimal buses previouslydetermined.

Tabatabaei et al [17], develops a method based on a fuzzydecision making, which uses an evolutionary method. Theinstallation bus was selected by the fuzzy reasoning supportedby the fuzzy set theory in a step by step procedure.

In this paper, the reconfiguration problem is solved via adeterministic method (Minimal Spanning tree algorithm) [18],while the optimal capacitor placement is based on a geneticalgorithm. The proposed method is applied to a 33-bus systemand 69-bus system. The problem formulation, description ofsolution methods and examples are presented in Sections II,III,IV and VI, respectively. Finally, conclusions are exposed inSection VII.

II. FORMULATION

This paper proposes the capacitor placement problemfor loss reduction in distribution systems. Additionally, thetopology reconfiguration is analyzed. The main goal isrelated to improve technical and economical issues. Thus, thedistribution system’s improvement is based on two strategies:(i) the topology’s reconfiguration, and (ii) optimal capacitorplacement.

A. Reconfiguration formulation

The reconfiguration problem is formulated as a real powerloss’ minimization as follows:

minE∑

n=1

RnP 2n +Q2

n

|V 2n |

(2)

where E is the total number of lines; Rn, Pn and Qn are theresistance, active power flow and reactive power flow throughline n, respectively; Vn is the n-th bus voltage.

Eq. (2) corresponds to the objective function and representsthe total grid’s active power loss. The objective function (2)is subject to:

V mini ≤ Vi ≤ V max

i , i = 1, . . . , N (3)

φ = 0 (4)

Eq. (3) refers to voltage constraints in each bus. Eq. (4),deals with the radial topology constraint.

B. Optimal capacitor placement formulation

The proposed approach formulates the capacitor placementby a mathematical and logical problem through two objectives:

• the reduction of energy losses, and• the improvement of bus voltage.The purpose is twofold: maximizing the saving by

minimizing the energy loss due to the shunt capacitors’placement after the distribution system reconfiguration. Energyloss reductions are considered as objective function, whiletaking the voltage profile into account as constraint.

The objective function related to savings [12], NS becomes

NS = KeTP0 −KeTPC −ncap∑i=1

KC,iQC,i (5)

where Ke is the energy loss cost, T is the duration of load;P is the power loss before compensation; PC is the powerlosses after compensation; QC is the value of capacitor at theith node; and ncap is the number of candidate locations forcapacitor placement.

The security constraints may be posed as follows

Vmini ≤ Vi ≤ Vmax

i , i = 1, . . . , N (6)

QTOTALC ≤ QL (7)

where Vi is voltage at bus i, QTOTALC is total connected KVAr

in capacitors’ banks and QL is total load kVAr.

III. MINIMAL SPANNING TREE ALGORITHM

The minimal spanning tree problem is one of the oldest andmost basic graph algorithms in theoretical computer science.Its history dates back to Boruvka’s algorithm in 1926 [19],[20]. Nowadays, there are two algorithms commonly used: (i)Prim, and (ii) Kruskal. Both are greedy algorithms that run inpolynomial time.

To solve the reconfiguration problem, in this paper the usedMST algorithm is based on the greedy Kruskal’s strategy,because it finds a minimum spanning tree by a global searchin the network [18]. If the weight of edges are integers, thendeterministic algorithms are known to solve the problem inO (m+ n) integer operations [21], where m and n refer toedges and vertices, respectively.

The proposed weights for use in the MST are the absolutevalue of the active power flows through branches, calculatedby a load flow run using the configuration of the entirenetwork (lines normally open and closed). After the load flowconvergence, all active power flows are calculated througheach branch, Fig. 1.

Vk∠θk Vm∠θmIk

R + jX

Yk

2Yk

2

Im

Fig. 1. Tie-lines power flow

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Thus, in this paper the branches’ weight is proposed asfollows,

W =1

|Pe| =1∣∣∣�

(|Vk|2

[1−VkV ∗

m

R−jX +Y ∗k

2

])∣∣∣(8)

where � (x) is the real part of a complex number x.The implemented algorithm was developed using Matlab,

and its performance is assessed by two distribution systems(33-bus system and 69-bus system) [22], [23].

IV. GENETIC ALGORITHM

The development of GA is largely credited to the workof Holland [24] and Goldberg [25]. Fig. 2 summarizes thisalgorithm.

N

Mutation

No. of Crossover<CR?

Crossover

Selection of Parents

Initialization...

Read Data:System data, switches state before reconfigurationGA parameters: population, generations max,tolerances

Start

Y

No. of Mutations<MR?

Y

Stop

Is TerminationCriteria Satisfied?

Elitism

Perform Power Flow. Apply Constraints.Evaluate Fitness Function

N

Y

N

wheel

Crossover

Mutation

Operator

Operator

Roulette

Fig. 2. Flow chart of the GA for capacitor placement.

V. OPTIMAL CAPACITOR PLACEMENT CRITERIA

After reconfiguration, the proposed method, Fig. 2, isapplied to the distribution systems described in Section VI.The cost constants and conditions are as follows [12]: (i)energy cost Ke = US$0, 06/kWh, (ii) cost of capacitor Kc =US$3, 0/kVAr, and (iii) load duration time T = 8760hours.

The decision variables in the proposed strategy are: x1 = Q(Size Capacitor) and x2 = a (location), which were searched

within the decision space described below:

0 ≤x1 ≤ 1000 kVAr (9)

x2 ∈ [1, NBUS] (10)

where NBUS is the total number of buses in the studiedsystem.

VI. EXAMPLES

A. 33-Bus System

The first test system is a hypothetical 12.66 kV system with33 nodes, 37 edges, and 5 looping branches (tie lines). Thetotal real and reactive power loads on the system are 3715 kWand 2300 kVAr, respectively. The total loss is about 8% ofthe total load. A lossy system is selected because the lossreduction is expected to be appreciable. The system data isgiven in [22]. The single-line diagram of the system is depictedin Fig. 3.

Fig. 3. Single line diagram of 33-bus system

To carry out the power flow study, the selected base powerand base voltage are 100 MVA and 12.66 kV, respectively. Theslack bus is the main substation (bus 1), and its voltage angleand voltage magnitude are set to 1.0∠0◦ p.u. The remainingbuses are load buses (PQ type).

The active power loss before reconfiguration (initialconditions of test systems, Fig. 3) become 197.379 kW, whilethe minimum voltage is Vmin = V18 = 0.914 p.u. Brancheswith greater losses are the closest to the most loaded buses.Higher loads (greater than 200 kW) are connected to nodes 8,9, 24, 25, 30, and 21; in the neighboring branches respectto these nodes are the highest proportion of losses. Whenapplying the MST algorithm, the configuration that presentsfewer losses is that where branches 7, 10, 14, 28, and 36 areopen.

According to the cost data and load duration timegiven in Section V, the cost and savings are calculated.Before compensation, the cost of losses becomes US$72.966.Simulations were performed for different numbers ofcapacitors, i.e, the optimal number of capacitors wereinvestigated. It is worth nothing that the saving has a

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maximum. That is, after some number of capacitors, savingdoes not grows anymore, Fig. 4 (Table I).

TABLE IOPTIMAL CAPACITOR PLACEMENT: LOCATION AND SIZE IN THE 33-BUS

TEST SYSTEM.

No. GA Q1 Bus Q2 Bus Q3 bus SavingsCap Gen kVAr L1 kVAr L2 kVAr L3 US$

1 22 1000 30 - - - - 56742 56 521 31 948 30 - - 64773 73 504 31 207 17 974,27 30 67254 40 507 27 475 30 250,46 14 66555 32 431 32 493 29 285,7756 31 65656 76 79 1 45 25 339,65 32 6246

No. GA Q4 Bus Q5 Bus Q6 Bus SavingsCap Gen kVAr L4 kVAr L5 kVAr L6 US$

1 22 - - - - - - 56742 56 - - - - - - 64773 73 - - - - - - 67254 40 550,8 32 - - - - 66555 32 114,5 18 305,1484 30 - - 65656 76 317,855 14 673,69 30 312,05 15 6246

Fig. 4. Performance of savings in the 33-bus system.

The biggest savings will be when 3 capacitors are installed.The first capacitor must be installed in bus 31 and must havea capacity of approximately 504 kVAr. The second capacitorat bus 17, with an approximate capacity of 207 kVAr, and thethird capacitor in bus 30, with a capacity of approximately974 kVAr.

Fig. 5 depicts the evolution of the maximum value, respectto the number of generations; the best value of GA of savingsbecomes US$6.725.

Under these conditions, Table II illustrates losses, minimumvoltages, costs and savings. It is noteworthy that savings byan amount of $45.340 may be obtained.

Fig 6 shows the voltage profile. Notice that magnitudesof most of the bus voltages have been improved afterreconfiguration embedding capacitors.

10 20 30 40 50 60 706000

6100

6200

6300

6400

6500

6600

6700

6800Performance of GA(best value)

number of generations

max

val

ue o

f bes

t sol

utio

n

Fig. 5. Best value with 3 capacitors, 33-bus system.

TABLE IICOMPARISON OF RESULTS

Initial Reconfig. Reconfig.conf. only + Capacitor

losses [kW] 197,379 138,825 101,499Min. node voltage [p.u] 0,914 0,939 0,957

Total annual cost [US$] 103.742 72.966 58.403Total losses cost [US$] 103.742 72.966 53.348

Total capacitor cost [US$] 0 0 5.055Total annual savings [US$] 0 30.776 45.340

5 10 15 20 25 300.9

0.92

0.94

0.96

0.98

1

1.02

Bus No.

Bus

Vol

tage

[p.u

.]

Before reconfigurationAfter reconfiguration without capacitorsAfter reconfiguration with capacitors

Fig. 6. Comparison results for the 33-bus test system.

B. 69-bus test system

The second test system is a hypothetical 12.66 kV systemwith 69 nodes, 73 edges, and 5 looping branches (tie lines).The total real and reactive power loads on the system are3801.5 kW and 2694.6 kVAr, respectively. The system data isgiven in [23]. The schematic diagram of the system is depictedin Fig. 7.

To carry out the power flow study, the selected base powerand base voltage are 10 kVA and 12.66 kV, respectively. Theslack bus is the main substation (bus 1), and its voltage angleand voltage magnitude are set to 1.0∠0◦ p.u. The remainingbuses are load buses (PQ type).

The active power loss before reconfiguration (initialconditions of test systems, Fig. 7) becomes 204.45 kW, whilethe minimum voltage is Vmin = V65 = 0.9178 p.u. Brancheswith greater losses are the closest to the most loaded buses.Higher loads (greater than 100 kW) are connected to buses 10,

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11, 48 and 49; in the neighboring branches respect to thesebuses are allocated the highest proportion of losses. Whenapplying the MST algorithm, the configuration that exhibitsfewer losses is that where branches 13, 20, 58, 61, and 69 areopen.

Fig. 7. Single line diagram of 69-bus system

According to cost data and load duration time givenin Section V the cost and savings are calculated. Beforecompensation, losses cost becomes US$42.504. Similarlyto the previous example, simulations were performed fordifferent number of capacitors. The results of the proposedalgorithm indicate that the saving is zero. That is, this networkdoes not require location of capacitors to reduce losses afterreconfiguration (Table III).

TABLE IIIOPTIMAL CAPACITOR PLACEMENT: LOCATION AND SIZE IN THE 69-BUS

SYSTEM.

No. GA Q1 Bus Q2 Bus SavingsCap Gen kVAr L1 kVAr L2 US$

1 70 0 63 - - 02 24 0 31 0 26 03 80 0 48 0 29 04 39 0 44 0 45 0

No. GA Q3 bus Q4 Bus SavingsCap Gen kVAr L3 kVAr L4 US$

1 70 - - - - 02 24 - - - - 03 80 0 5 - - 04 39 0 44 0 34 0

Figs. 8 depict the evolution of the maximum value withinthe genetic algorithm, respect to the number of generations.Ultimately, the value of savings with GA was US$0.

Under these conditions, Table IV illustrates losses,minimum voltage, costs, and savings. The annual savings afterreconfiguration become $30.776.

Fig. 9 illustrates the voltage profile improvement achievedby the proposed strategy (only reconfiguration). Notice that

most of the bus voltage magnitudes have been improved afterreconfiguration. The minimum bus voltage is equal to 0.9178p.u., and after reconfiguration it is raised to 0.9593 p.u.

10 20 30 40 50 60 70 80−1000

−800

−600

−400

−200

0

200Performance of GA(best value)

number of generations

max

val

ue o

f bes

t sol

utio

n

Fig. 8. Best value with 3 capacitors, 69-bus system.

TABLE IVCOMPARISON OF THE RESULTS.

Initial Reconfig. Reconfig.config. only + Capacitors

losses [kW] 197,379 138,825 138,825Minimum node voltage [p.u] 0,914 0,939 0,939

Total annual cost [US$] 103.742 72.966 72.966Total losses cost [US$] 103.742 72.966 72.966

Total capacitor cost [US$] 0 0 0Total annual savings [US$] 0 30.776 30.776

10 20 30 40 50 600.9

0.92

0.94

0.96

0.98

1

1.02

Bus No.

Bus

Vol

tage

[p.u

.]

Before reconfigurationAfter reconfiguration

Fig. 9. Voltage profile for test 69-bus system

These results demonstrate that for losses reduction indistribution network, the reconfiguration technique is a goodoption due to the reconfiguration is obtained by changing thetopology of the network, without modifying its structure (Noinstallation costs and maintenance of new equipment.).

VII. CONCLUSION

Nowadays, distribution systems are regaining attention dueto represent the key element within novel tendencies ofmanaging electrical energy, such as the smart-grid philosophy.In this context, this paper tries to contribute to the distributionsystems operation’s improvement by reducing losses. Two

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strategies are used to this purpose: (i) reconfiguration, and (ii)and capacitor allocation.

This paper presented the application of the minimalspanning tree algorithm to solve the distribution networkreconfiguration problem. Additionally, an optimal capacitorplacement strategy is proposed to increase economic savings.

The graph theory application creates only feasible radialtopologies. The MST algorithm is easy to apply and to reply. Itis deterministic, which means that, under the same conditions,the same result is guaranteed for different runs. The proposedalgorithm is tested on two distribution test systems (33-bus and69-bus) for loss minimization showing promising results. As asecondary advantage, in general, voltage magnitudes becomebetter than those obtained with other formulations.

By finding the optimal capacitors placement after thereconfiguration, it is shown that, under some conditions, itis not necessary to install capacitors for greater savings foractive power losses.

REFERENCES

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Diana P. Montoya obtained her degree inElectrical Engineering from Universidad Nacionalde Colombia, Medellı́n-Colombia in 2008; M. Sc.in Electrical Engineering from CINVESTAV-Mexicoin 2012. She is currently pursuing her Ph. D. inElectrical Engineering in CINVESTAV, Guadalajara,Mexico. Her research interest is Smart-Grid andElectric Power Systems.

Juan M. Ramirez (M1986) obtained his BSin Electrical Engineering from Universidad deGuanajuato, Mexico in 1984; M. Sc. in ElectricalEngineering from UNAM-Mexico in 1987; Ph. D. inElectrical Engineering from UANL-Mexico in 1992.He joined the department of Electrical Engineeringof CINVESTAV in 1999, where he is currently a fulltime professor. His areas of interest are in operationand control of electric power systems.