Electrical Power and Energy Systems 46 (2013) 250–257

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Electrical Power and Energy Systems

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A new technique for optimal allocation and sizing of capacitors and setting of LTC

Iman Ziari ⇑, Gerard Ledwich, Arindam GhoshSchool of Engineering Systems, Queensland University of Technology, Brisbane, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 6 November 2010Received in revised form 7 September 2012Accepted 26 September 2012Available online 23 November 2012

Keywords:Distribution systemCapacitorOptimization

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.09.010

⇑ Corresponding author. Tel.: +61 401 573 814.E-mail address: ima_zia@yahoo.com (I. Ziari).

An iterative based strategy is proposed for finding the optimal rating and location of fixed and switchedcapacitors in distribution networks. The substation Load Tap Changer tap is also set during this proce-dure. A Modified Discrete Particle Swarm Optimization is employed in the proposed strategy. The objec-tive function is composed of the distribution line loss cost and the capacitors investment cost. The lineloss is calculated using estimation of the load duration curve to multiple levels. The constraints are thebus voltage and the feeder current which should be maintained within their standard range.

For validation of the proposed method, two case studies are tested. The first case study is the semi-urban 37-bus distribution system which is connected at bus 2 of the Roy Billinton Test System whichis located in the secondary side of a 33/11 kV distribution substation. The second case is a 33 kV distri-bution network based on the modification of the 18-bus IEEE distribution system. The results are com-pared with prior publications to illustrate the accuracy of the proposed strategy.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Capacitors are used commonly in distribution systems tominimize the reactive component of line current. This compensa-tion reduces the distribution line loss and improves the voltageprofile. Particularly in the peak load, reduction of the line loss bycapacitors can prevent additional investment for using high ratingequipment. In order to minimize the capacitors cost and the lineloss and to improve the voltage profile simultaneously, the OptimalAllocation and Sizing of Fixed and Switched Capacitors (OASFSCs)problem should be solved. During this procedure, Load Tap Chang-ers (LTCs) tap is also set to minimize the line loss and improve thevoltage profile.

Modeling loads in the capacitor planning problem is a main is-sue. Some papers allocate and size the capacitors based on theaverage load level [1–8]. Since the line loss is proportional to thesquare of the rms current, calculation of the average distributionline loss based on only the average load level is not correct. Also,it is not feasible to calculate the line loss for all levels of loads sincethis needs to do optimization for hundreds times. To avoid thisproblem, loads should be modeled using an approximation of theload duration curve in multi-level.

Selection of the optimization algorithm is another important as-pect in OASFSC problem. Due to the discrete nature of the alloca-tion problem and the discreteness of the capacitors’ size, anumber of local minima are present in the objective function. This

ll rights reserved.

creates a risk of being caught in local minima. Using the analyticalbased tools as optimization methods does not improve this prob-lem. That is why only a few papers are based on this philosophy[9–12]. To deal appropriately with this difficulty, the heuristicbased optimization methods are quite common in literatures [1–8,13–19]. The Particle Swarm Optimization (PSO) [20,21] is awell-known heuristic based method. In this paper, the discrete ver-sion of PSO called Discrete Particle Swarm Optimization (DPSO)[22] is modified and employed.

In [1–8], the capacitors are optimized using the heuristic meth-ods for one specific load level. This assumption is improved in somepapers by modeling the loads in multi-level. In [9,13,15,17,19], thecapacitors are optimized for each load level sequentially. In [10–12], the OASFSC problem is solved separately for each load leveland then, the minimum optimal capacitor size among all load levelsat a bus is supposed as the fixed capacitor size and the rest as theswitched capacitor size at the relative bus. Whereas the computa-tion time is the main advantage of these two multi-level load-basedmethodologies, the capacitors calculated for the higher load levelsare not included in the lower load levels computation while thislikely reduces the total investment cost further. Another group ofthe available literatures [4,14,16,18] proposes an algorithm inwhich the capacitors in all load levels are optimized simulta-neously. Dealing with a large number of optimizing variables isthe main drawback in these methods which leads to a high compu-tation time and low accuracy. These mentioned problems highlightthe need for an algorithm with appropriate accuracy and reason-able computation time.

In this paper, a new strategy is proposed to determine theoptimal placement and rating of fixed and switched capacitors in

Load Level = Average

Solve OASFSC

Load Level = 1

Solve OASFSC

Results Changed?

Print Results

Increase Load Level

Modify OF

Increase Load Level

Modify OF

No

Yes

Yes

No

Increase Load Level

Modify OF

No

Initi

aliz

atio

n

LastLoad Level?

LastLoad Level?

Yes

Fig. 1. Flowchart of the proposed algorithm.

Table 1Characteristics of the test system.

No. of loads Customer type Load points Average load level (MW)

9 Residential 1–3,10–12,17–19 0.505 Commercial 6–7,15–16,22 0.456 Government 4–5,13–14,20–21 0.572 Industrial 8–9 1.10

I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257 251

distribution networks. In this iterative-based strategy, the capaci-tors calculated for higher load levels are re-used to modify the sizeand location of the capacitors obtained for the lower load levels ineach iteration. During this procedure, the tap of LTC located at thesubstation is also set. The objective is to minimize the distributionline loss cost and the capacitors investment cost. The bus voltageand the feeder current as constraints are maintained within theirstandard range.

This paper is organized as follows. The problem formulation ispresented in Section 2. Sections 3 and 4 explain the methodologyas well as the optimization method and its implementation tothe problem. The results and conclusions are expressed in Sections5 and 6.

2. Problem formulation

The main objective of OASFSC problem is to minimize the totalcost of capacitors as well as the distribution line loss. The bus volt-age and the feeder current are also limited as the constraints andadded to the objective function with a penalty factor. Since all of

the objective function elements are simply converted into the com-posite equivalent cost, this problem can be solved using a single-objective optimization method. The objective function is definedas follows:

OF ¼ CCAPITAL þXT

t¼1

CO&MþCLOSS

ð1þ rÞtþ k ð1Þ

where CCAPITAL and CO&M are the capital cost and the operation andmaintenance cost of fixed and switched capacitors, CLoss is the lineloss cost, r is the discount rate, T is the number of years in the studytimeframe which is assumed 20 years here, and k is the constraintpenalty factor.

The line loss cost is expressed in (2):

CLOSS ¼ kL

XLL

l¼1

Tl � PLossl þ kPL � PLossLL ð2Þ

where kL is the cost per MWh, Tl is the duration of load level l, PLossl

is the line loss in load level l, LL is the number of load levels, PLossLL

is the line loss in the peak load level, and kPL is the saving per MWreduction in the peak load.

The constraints include the bus voltage and the feeder current.The bus voltage as the first constraint should be maintained withinthe standard level as defined in (3):

0:95 pu 6 Vbus 6 1:05 ð3Þ

where Vbus is the actual bus voltage. The feeder current as the sec-ond constraint should be less than the feeder rated current as ex-pressed in (4):

If 6 Iratedf ð4Þ

where If and Iratedf are the current and the rated current of a feeder,

respectively.

3. Methodology

The algorithms presented for solving the OASFSC problem canbe categorized into four main groups. Group1 [1–8] is related tothe algorithms in which the capacitors are optimized for one spe-cific load level. Consideration of one load level probably does notguarantee the accurate results. In group2 [9,13,15,17,19], thecapacitors are allocated and sized from the lowest load level tothe peak load level. The capacitors optimized in a load level are as-sumed as the fixed capacitors in optimization procedure of thenext load level. Finally, the capacitors calculated in the lowest loadlevel will be the fixed capacitors and those added to the fixedcapacitors in other load levels are assumed as the switched capac-itors. This strategy is called Building Strategy in this paper. In thisstrategy, the capacitors obtained for higher load levels are not usedfor lower load levels while some of them probably minimize theline loss much more without applying any extra cost. Group3[10–12] is related to the methods in which the capacitors are ob-tained in all load levels separately and then, the minimum capac-itor rating at a bus is assumed as the fixed capacitor, and the restas the switched capacitors at the corresponding bus. This strategyis called Separating Strategy in this paper. This strategy causesusing a large number of capacitors so high investment cost.

1 2 3

. . . . . . . .

NB

Capacitors

C1 C2 C3 CNB LTC

Fig. 2. The structure of a particle.

252 I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257

Furthermore, these methods do not use the capacitors found inhigher load levels for lower load levels like group2. In group4[4,14,16,18], the capacitors in all load levels are optimized simulta-neously. However these methods can converge in a global mini-mum, they suffer from high number of optimizing variables. Thisseverely increases the computation time and decreases the methodaccuracy.

In this paper, an iterative based strategy is proposed to find theoptimal placement and size of fixed and switched capacitors andtap setting of the LTC. Fig. 1 shows the flowchart of the proposedalgorithm. As illustrated in this flowchart, the program starts fromthe average load level since the majority part of the load durationcurve is for the average load level (100%). Starting from this levelhelps the program to converge with more accurate results. Thecapacitors are allocated and sized optimally for the average load le-vel without taking notice to the fixed or switched type. After that,the program starts optimizing the next load level. For optimizingthe capacitors in this load level, the objective function is modifiedusing the following equations:

OF ¼ C0CAP þXT

t¼1

C0O&M þ CLOSS

ð1þ rÞtþ k ð5Þ

C 0CAPj¼

0 if ClCAPj6 Cl�1

CAPj

ClCAPj� Cl�1

CAPjif Cl

CAPj> Cl�1

CAPj

8<: ð6Þ

C 0O&Mj¼

0 if ClO&Mj

6 Cl�1O&Mj

ClO&Mj

� Cl�1O&Mj

if ClO&Mj

> Cl�1O&Mj

8<: ð7Þ

where ClCAPj

and ClO&amp;Mj

are the capital cost and the operation andmaintenance cost of the capacitor installed at bus j for the load levell, respectively. As observed in these equations, lower investmentcost is applied if the capacitor installed at a bus in the current loadlevel has been already selected in the previous load levels. This in-creases the probability of selecting the optimal capacitor locationsin the previous load level as the optimal locations in the currentload level. It also does not force the program to select the capacitorsobtained in the previous load level as the fixed capacitors for thecurrent load level.

This strategy leads the program to results with lower invest-ment cost and line loss. Furthermore, the computation time is sig-nificantly reduced compared with the methods in which all loadlevels are simultaneously optimized.

4. Implementation of PSO

The discrete nature of the allocation and sizing problem leadsthe optimization problem to have a number of local minima. Todeal appropriately with this issue, employing a reliable optimiza-tion method is required. The analytical methods can find the globalsolution in a short time but only if the initial values are selectedaccurately, which is not easy to do. The heuristic-based methodscan play a remarkable role in this case. They are based on the ran-dom values and if only one of these random values is located closeto the global minimum, they can find an acceptable solution.Among the heuristic-based methods, PSO is employed in thispaper.

4.1. Overview of PSO

PSO is a population-based and self adaptive technique intro-duced originally by Kennedy and Eberhart [22]. This algorithmhandles a population of individuals in parallel to probe searchareas of a multi-dimensional space where the optimal solution is

searched. The individuals are called particles and the populationis called a swarm. Each particle in the swarm moves towards theoptimal point with an adaptive velocity [21].

Mathematically, the position of particle i in an n-dimensionalvector is represented as Xi = (xi,1, xi,2, . . . , xi,n). The velocity of thisparticle is also an n-dimensional vector as Vi = (vi,1, vi,2, . . . , vi,n).Alternatively, the best position related to the lowest value of objec-tive function for each particle is represented as Pbesti = (pbesti,1,pbesti,2, . . . , pbesti,n) and the global best position among all particlesor best pbest is denoted as Gbesti = (gbesti,1, gbesti,2, . . . , gbesti,n).During the iteration procedure, the velocity and position of parti-cles are updated [21].

The DPSO, the discrete version of PSO, is an optimization meth-od which is applied to the discrete problems like OASFSC where theparticles are optimizing variables such as the capacitor size whichare analyzed as integer values. In this situation, the optimal solu-tion can be achieved by rounding off the actual particle value tothe nearest integer value during the iterations. In [21], it is men-tioned that the performance of the DPSO is not influenced by thisrounding off process. Note that the continuous methods performthe rounding after the convergence of the algorithm, while inDPSO, it is applied to all particles during each iteration of the opti-mizing procedure.

4.2. Applying hybrid PSO to problem

Identification of the optimizing variables is the first step in anoptimization procedure. In OASFSC problem, the placement andrating of capacitors along with the tap setting of the LTC are theoptimizing variables, particles, as shown in Fig. 2.

As illustrated for the single LTC problem in Fig. 2, the particle iscomposed of NB cells with the value of Ci and one cell with the va-lue of LTC tap setting. Each candidate bus for capacitors is assignedby a value entry in a cell. The notation used is Ci which is the ratingof the capacitor installed at bus i. If the value of a cell, the capacitorsize, is more than a specific threshold, it indicates that a capacitoris installed at the relative bus. Otherwise, no capacitor is placed atthe corresponding bus. This specific threshold is the minimum sizeof available set of capacitors. For the LTC, the value of this cell is thetap setting of LTC. The number of cells assigned for LTCs is equal tothe number of LTCs. Therefore, this problem can be extended formore than one LTC. NB is the number of buses when all busesare available as candidates for installing capacitors.

It was observed that the results obtained by pure DPSO are im-proved when GA operators are applied to DPSO procedure. This ismainly because these operators increase the diversity of optimiz-ing variables. Fig. 3 shows the flowchart of the proposed method.The description and comments of steps are presented as follows.

Step 1. (Input system data and initialization)

In this step, the distribution network configuration and dataand the available capacitors are input. The maximum allowed volt-age drop and the characteristics of feeders, impedance and ratedcurrent, are also specified. The DPSO parameters, number of popu-lation members and iterations, as well as the PSO weight factors,

Input DataInitialize

Iteration Number = 1

Calculate OFEquation (1)

LastIteration Number?

Print Results

IncreaseIteration Number

Crossover

No

Yes

Do Not ChangeHalf of Population

Calculate PbestEquation (8)

Calculate GbestEquation (9)

Update VelocityEquation (10)

Update PositionEquation (11)

Mutation

Fig. 3. Algorithm of proposed PSO-based approach.

I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257 253

are also identified. The random-based initial population of particlesXj (size of capacitors and tap setting of LTC) and the particles veloc-ity Vj in the search space are also initialized.

Step 2. (Calculate the objective function)

Given the capacitors size determined in the previous step, theadmittance matrix is reconstructed. Using the new admittance ma-trix, a load flow is run and the buses voltage and the feeders cur-rent are calculated. After that, the distribution line loss of allfeeders is calculated.

The objective function is now constituted by (1). The constraintsare also computed using (3) and (4) in this step and included in theobjective function with penalty factors. It means that if a constraintis not satisfied, a large number as a penalty factor is added to theobjective function to exclude the relevant solution from the searchspace.

Step 3. (Calculate pbest)

The component of the objective function value associated withthe position of the particles is compared with the corresponding

value in previous iteration and the position with lower objectivefunction is recorded as pbest for the current iteration.

pbestkþ1j ¼

pbestkj ifOFkþ1

j P OFkj

xkþ1j if OFkþ1

j � OFkj

8<: ð8Þ

where k is the number of iterations, and OFj is the objective functioncomponent evaluated for particle j.

Step 4. (Calculate gbest)

In this step, the lowest objective function among the pbests asso-ciated with all particles in the current iteration is compared with it inthe previous iteration and the lower one is labeled as gbest.

gbestkþ1 ¼gbestk if OFkþ1 P OFk

pbestkþ1j if OFkþ1 � OFk

(ð9Þ

Step 5. (Update position)

The position of particles for the next iteration can be calculatedusing the current pbest and gbest as follows:

Vkþ1j ¼ xVk

j þ c1rand pbestkj � Xk

j

� �þ c2rand gbestk

j � Xkj

� �ð10Þ

where Vkj is the velocity of particle j at iteration k, x is the inertia

weight factor, ci is the acceleration coefficients, Xkj is the position

of particle j at iteration k, pbestkj is the best position of particle j

at iteration k, and gbestk is the best position among all particles atiteration k.

As mentioned before, using the available data, x as inertiaweight factor, and c1 and c2 as acceleration coefficients, the veloc-ity of particles is updated. It should be noticed that the accelerationcoefficients, c1 and c2, are different random values in the interval[0, 1].

As observed in (10), x is to adjust the effect of the velocity inthe previous iteration on the new velocity for each particle.Regarding the obtained velocity of each particle by (10), the posi-tion of particles can be updated for the next iteration using (11).

Xkþ1j ¼ Xk

j þ Vkþ1j ð11Þ

The inertia weight factor is set as 0.9 and both the accelerationcoefficients as 0.5 in this paper.

After this step, half of the population continues DPSO procedureand other half goes through the genetic algorithm operators. Thefirst half continues their route at Step 7 and the second half goesthrough step 6.

Step 6. (Apply GA operators)

In this step, the crossover and mutation operators are applied tothe half of the population. This is done to increase the diversity theoptimizing variables to improve the local minimum problem.Figs. 4 and 5 show the operation of crossover and mutationoperators.

Step 7. (Check convergence criterion)

If Iter = Itermax or if the output does not change for a specificnumber of iterations the program is terminated and the resultsare printed, else the program goes to step 2.

5. Results

To validate the proposed method, two different test systems arestudied. The first test system is the 11 kV semi-urban distribution

0.60.3 0.61.5 0.30.3 1.20.9 0.9

0.90.6 0.91.2 0.90.3 1.20.6 0.6...........

...........pop i

pop j

0.60.3 0.91.5 0.90.3 1.20.6 0.6

0.90.6 0.61.2 0.30.3 1.20.9 0.9...........

...........pop i

pop j

Crossover

Fig. 4. A sample crossover operation.

1.20.3 0.61.5 0.30.9 1.20.9 1.2...........pop i

0.30.3 0.61.5 1.50.9 1.20.9 1.2...........pop i

Mutation

Fig. 5. A sample mutation operation.

L2

L3

L1

L5

L6

L7

L9

L10

L4

L8

L11

L12

L13

L14L15

L16

L17

L18

L19

L20 L21

L22

33 kV

11 kVF1 F2

F3F4

1

2

3

4

56

7

8

9

10

11 12

1314

1516

17

18

19

20

21

2223

24

25

26

27 28

29

30

31

32

3334

35

36

37

Fig. 6. Distribution System for RBTS Bus 2.

160%

50%

100%

8760450050

Load(%)

Time(hour)

Fig. 7. Load duration curve used in the testing distribution system.

254 I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257

system connected to bus 2 of the Roy Billinton Test System (RBTS)as shown in Fig. 6. This 37-bus test system has 22 loads located inthe secondary side of a 33/11 kV distribution substation. The char-acteristics of this test system are given in Table 1. The second testsystem is based on the modification of the 18-bus IEEE distributionsystem. This test system is also employed in some references

[15,17,19], related to the allocation and sizing of capacitorsproblem.

The load duration curve of the test systems is shown in Fig. 7.Investigation of every point of the load duration curve leads theprogram to very slow computation time. Consideration of only2–3 levels is probably inaccurate. In this paper, to implement acompromise between accuracy and computation time, this curveis approximated with five load levels as shown in Fig. 8. However,using sensitivity analysis to find the optimal load level number canbe included in the future.

160%

50%

100%

8760

Load(%)

Time(hour)

80%

120%

Fig. 8. Approximation of load duration curve.

3 4 6 7 9 10 12 14 16 20 21 23 24 26 28 29 31 32 34 36 370

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Bus Number

Cap

acio

Rat

ing

(Mva

r)

Switched Capacitor Fixed Capacitor

Fig. 9. Fixed and Switched capacitors location and rating.

50% 80% 100% 120% 160%0

100

200

300

400

500

600

700

800

900

Load Level

Line

Los

s (k

W)

Before Installation After Installation

Fig. 10. Line loss before and after installation of capacitors.

I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257 255

As shown in Fig. 8, the load is peak for 2% of a time domain andlowest for 3% of a time domain. The average load is drawn from thenetwork for 40% of a time domain. For 30% and 25% of a timedomain, the load level is 120% and 80% of the average load,respectively.

It is assumed that the cost per kWh is different for different loadlevels, 3 ¢ for 50% and 80% of the average load, 6 ¢ for 100%, 8 ¢ for120% and 10 ¢ for the peak load level. These prices are in the samerange of [3,5,9] and show a similar trend to the generating electric-ity cost from different types of technology in the UK [23]. The sav-ing per MW reduction in the peak power loss is presumed$168,000. The installation cost of capacitors is assumed to be 4 $/kvar and the annual incremental cost is selected 0.35 $/kvar. Theavailable capacitors are considered as 300 kvar banks. The LTC lo-

Table 2Scheduling of switched capacitors and LTC.

Bus number

3 4 6 9 10 14 16 20 23

Load level50% 0 0 0 0 0 0 0 0 080% 0.3 0.3 0.3 0 0.3 0.3 0.3 0.3 0.3

100% 0.3 0.3 0.3 0 0.3 0.3 0.3 0.3 0.3120% 0.3 0.3 0.3 0.3 0.3 0.3 0.6 0.3 0.6160% 0.3 0.3 0.3 0.3 0.3 0.3 0.6 0.3 0.6

cated at the substation is characterized by 32 taps which canchange its output voltage between 0.9 and 1.1 times of its inputvoltage. Therefore, each step can change the input voltage0.00625 pu.

5.1. Test system 1

The RBTS is used in this subsection as the first test system. Afterapplying the proposed technique, the optimal placement and rat-ing of fixed and switched capacitors are found as shown in Fig. 9.As observed in this figure, the total size of fixed and switchedcapacitors is 3.3 and 6 Mvar, respectively. Maximum capacitor rat-ing is installed at bus 16, 0.9 Mvar. The scheduling of switchedcapacitors along with the LTC is given in Table 2.

As expected, the LTC tap setting increases when the load in-creases. The LTC is also helpful for improving the voltage profile.Fig. 10 shows a comparison between the line loss before and afterthe scheduling of capacitors and the LTC. It should be noted thatthe line losses before installation of capacitors for the 120% load le-vel case and the peak load case are invalid because the voltage con-straint is not satisfied. However, it is shown in Fig. 10 only toillustrate the affect of capacitors.

Fig. 10 illustrates the line loss decreases in all load levels byadjusting the capacitors and LTC tap. A cost analysis is performedin Table 3 for two states, before and after planning of capacitorsand LTC. This table highlights the benefit of planning the capacitorsand LTC in distribution systems so that the total cost decreasesfrom $2,363,777 to $1,674,167, about 30%.

5.2. Test system 2

The 18-bus IEEE distribution test system is modified and stud-ied in this subsection. The optimal placement and rating of fixedand switched capacitors are shown in Fig. 11. As shown in this

LTC

24 28 29 31 32 34 36 37

0 0 0 0.3 0 0 0 0 �50 0.3 0.3 0 0 0 0.3 0 �30 0 0.3 0.3 0 0 0.6 0 �20.3 0.3 0.3 0.3 0.3 0.3 0.6 0 �10.3 0.3 0.3 0.3 0.3 0.3 0.6 0.3 +2

Table 3A comparison between total cost before and after scheduling ($).

Loss cost for load levels Capacitor cost Total cost

50% 80% 100% 120% 160%

Before installation 5245 196,930 808,476 956,029 397,097 0 2,363,777Proposed strategy 3652 126,602 539,455 643,561 289,214 71,683 1,674,167

4 5 6 7 8 9 21 23 24 25 260

1

2

3

4

5

Bus Number

Num

ber o

f Ban

ks

Fixed Capacitors Switched Capacitors

Fig. 11. Fixed and Switched capacitors location and rating.

1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

Bus NumberB

us V

olta

ge (P

U)

Before InstallationAfter InstallationBefore InstallationAfter Installation

Fig. 13. Voltage profile before and after installation of capacitors in peak load.

50% 80% 100% 120% 160%0

100

200

300

400

500

600

700

800

900

Load Level

Line

Los

s (k

W)

Before InstallatinAfter Installation

Fig. 12. Line loss before and after installation of capacitors.

256 I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257

figure, the capacitors are optimally located at buses 4–9, 21, 23–26of Fig 6. The total fixed and switched capacitors are 4.2 and4.5 Mvar, respectively. Maximum number of capacitor banks is in-stalled at bus 5, 5 fixed capacitors and 3 switched capacitor banks.The LTC tap setting is set at �4, �1, 0, +2, and +6 as the load in-creases from the lowest load level to the peak load level.

Similar to test system 1, the line loss before and after planningof capacitors and LTC are compared as shown in Fig. 12. It shouldbe noted that the line loss before installation of capacitors is whenthe LTC tap is set as 0. The voltage constraints for the 100%, 120%,and peak load level cases are not satisfied which leads the resultsto be invalid. However, they are shown in Fig. 12 only to showthe affect of capacitors. By increasing the LTC tap setting, the volt-age profile improves but the line loss increases. Fig. 13 shows thevoltage profile before and after the installation of capacitors.

As shown in Fig. 13, the voltage at eight of the buses is lowerthan the specified limit level when no compensator is installed.After planning the capacitors and LTC, the minimum bus voltage

increases to 0.95 pu. That is why the LTC tap setting increases from0 to +6 for the peak load level to improve the voltage profile.

To validate the proposed strategy, the results are comparedwith those obtained by the methods described in Section 3, calledBuilding and Separating Strategies. Table 4 shows the results ob-tained by the methods associated with these strategies. Thesestrategies start from the lowest load level. By applying these strat-egies to the test system, no capacitor is found for the lowest loadlevel, so no fixed capacitor is found. An economic analysis is sum-marized in Table 5. The results for no capacitors, Building Strategy,Separating Strategy, and the proposed strategy are compared inthis table.

As shown in this table, $574,444 is saved when the proposedstrategy is applied to the test system compared with no capacitorstate. This highlights the importance of planning the capacitors andLTCs. Furthermore, the total cost is lower than those from theBuilding and Separating Strategies. This illustrates that theproposed strategy is a lower cost approach than other strategiesparticularly in large network in which the line loss is a criticalissue.

6. Conclusions

A new iterative based strategy is proposed in this paper to findthe placement and size of fixed and switched capacitors optimally.In addition, the switched capacitors and the LTC located at the sub-station are scheduled. In this strategy, the capacitors obtained forhigher load levels can be used for lower load levels. This is donewithout applying any extra cost because they have been alreadyinstalled for higher load levels. A hybrid optimization method com-posed of DPSO and GA is employed in the proposed algorithm. Thisalgorithm enjoys low computation time and high accuracy. Theobjective is to minimize the distribution line loss with minimumcapacitors investment. Two constraints are applied to this problem,the bus voltage and the feeder current. The bus voltage should bemaintained within the range of 1.05 and 0.95 pu. The feeder cur-rent should be less than the rated current of feeder.

Table 4Scheduling of switched capacitors.

Bus number

4 5 6 7 8 9 21 22 23 2425 26

Building strategyFixed capacitor – – – – – – – – – – –Switched capacitor 0.3 2.4 0.6 0.3 0.9 0.3 0.3 0.3 0.9 0.6 1.2 0.3

Separating strategyFixed capacitor – – – – – – – – – – – –Switched capacitor 0.9 2.4 0.9 0.3 1.2 0.3 0.3 0.3 1.2 0.6 1.2 0.3

Table 5A comparison among the cases ($).

Loss cost for load levels Capacitor cost Total cost

50% 80% 100% 120% 160%

Before installation 4861 181,082 739,585 870,128 357,839 0 2,153,495Proposed strategy 3103 119,847 503,199 612,347 273,496 67,059 1,579,051Building strategy 4861 130,590 511,861 611,540 273,115 64,746 1,596,713Separating strategy 4861 130,590 511,789 607,875 283,250 76,308 1,614,673

I. Ziari et al. / Electrical Power and Energy Systems 46 (2013) 250–257 257

Two test systems are used to evaluate the proposed strategy,the 11 kV semi-urban distribution system connected to bus 2 ofthe RBTS and the 18-bus IEEE test system. The results demonstratethe desirability of jointly planning the capacitors and the LTC.Additionally, compared with algorithms previously presented, theproposed technique demonstrates higher performance.

Acknowledgements

The authors thank the Australian Research Council (ARC) andErgon Energy for the financial support for this Project throughthe ARC Linkage Grant LP 0560917.

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