This article was downloaded by: [University of Chicago]On: 18 March 2013, At: 03:32Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Electric Power Componentsand SystemsPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/uemp20

Optimal Capacitor Switchingin a Distribution SystemUsing Functional LinkNetworkBiswarup Das & Shiv Prakash VelpulaVersion of record first published: 30 Nov 2010.

To cite this article: Biswarup Das & Shiv Prakash Velpula (2002): OptimalCapacitor Switching in a Distribution System Using Functional Link Network,Electric Power Components and Systems, 30:8, 833-847

To link to this article: http://dx.doi.org/10.1080/15325000290085055

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make anyrepresentation that the contents will be complete or accurate or up todate. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall notbe liable for any loss, actions, claims, proceedings, demand, or costs ordamages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

Electric Power Components and Systems, 30:833–847, 2002Copyright c® 2002 Taylor & Francis1532-5008/ 02 $12.00 + .00DOI: 10.1080/ 15325000290085055

Optimal Capacitor Switching in a Distribution

System Using Functional Link Network

BISWARUP DAS

Department of Electrical EngineeringUniversity of RoorkeeRoorkee - 247667, India

SHIV PRAKASH VELPULA

Wipro Technologies Ltd.Bangalore, India

One of the most important control decision functions in a modern distributionautomation system is the volt-var control. The objective of the volt-var controlis to supply reactive power by optimally switching the capacitors installed inthe distribution system so that the voltage drop and real power loss are mini-mum. Traditionally, this problem of optimal capacitor switching has been solvedthrough various optimization techniques. However, as the time taken by thesetraditional optimization methods is quite signi�cant, these methods may not bemuch suitable for on-line application. To reduce the time required to solve theoptimal capacitor-switching problem, a Functional Link Network (FLN) basedapproach has been developed in this article. For a practical number of capac-itors in the system it has been found that the FLN based technique is at least100 times faster than the traditional optimization methods. Moreover, as thenumber of capacitors in the system increases, the eŒectiveness of the FLN overthe traditional approach (in terms of the solution time) increases.

Keywords power distribution system, optimal capacitor switching, ANN,FLN

1. Introduction

To ensure safe, reliable, and un-interruptible power supply, supervisory control anddata acquisition (SCADA) systems have been employed worldwide, to continuouslymonitor and control the transmission system. Similarly, to continuously monitor andcontrol the distribution system, a few years ago [1] the concept of a \distributionautomation" system emerged. In a distribution automation system a lot of sensorsand remote terminal units (RTUs) are placed in the power distribution system.

Manuscript received in �nal form on 13 August 2001.The computational and �nancial help provided by the Department of Electrical

Engineering, University of Roorkee, Roorkee - 247 667, for carrying out this work isgratefully acknowledged.

Address correspondence to Biswarup Das. E-mail: biswafee@isc.iitr.ernet.in

833

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

834 B. Das and S. P. Velpula

The data collected by these sensors and RTUs are sent to a central computerstation over a dedicated communication channel. The data sent are then analyzedby appropriate analysis software, to assess the present health of the system. If anyanomaly in the operation of the distribution system is detected, proper remedialand control decisions are taken at the central computer station by the help of propersoftware analysis packages. For proper �eld implementation these control decisionsare then sent to the �eld through the same or other dedicated communicationchannels.

Obviously, even with the presence of accurate sensors, RTUs, and robust, reli-able communication channels, the eŒectiveness of a distribution automation systemlargely depends upon the quality of the analysis software at the central computerstation. As all the decisions regarding the present health and remedial control actionare to be taken by the analysis software packages, the algorithms of the softwarepackages must be such that the results produced are highly accurate. Since thehealth monitoring functions and control decision functions are to be carried outquickly to minimize the adverse eŒects of any abnormal conditions in a power dis-tribution system, the algorithms of the analysis software packages need also to bevery fast.

Among the various control decision functions, the volt-var control is one ofthe most important functions in any modern distribution automation system. Theobjective of the volt-var control is to locally supply the reactive load demand inthe system such that the currents corresponding to these loads do not have to �owover the feeders from the sub-station to the loads. Consequently, the voltage dropand the power loss in the distribution system become minimum.

The most common approach for supplying reactive power adequately, withchanging load demand, is to install switchable capacitors in the system. Moreover,because of the cost consideration, the capacitors are installed at some strategiclocations, not at each load point in the system. The location and sizes (rating ofthe capacitors in terms of KVAR supplied) of the capacitors are decided basedon forecasts of load demands (both active and reactive) in the system over a cer-tain period (called the study period). This is known as the \capacitor placement"problem, and this is an involved optimization problem. Once the capacitors areplaced in the system based on the capacitor placement study, it is necessary todetermine their optimal switching pattern (which capacitor to be switched `ON’and which capacitor to be switched `OFF’) at any loading condition, so that thepower loss and the voltage drop in the system are minimum. This is known as the\capacitor switching" problem and, indeed, this is also an involved optimizationproblem.

Thus, the capacitor-switching algorithm presupposes the locations of the in-stalled capacitors, and it strives to �nd the optimal switching pattern of thesecapacitors for the minimum loss in the system at any given loading condition.Although, a signi�cant amount of work has been reported in the literature for ca-pacitor placement problem [2{4], not much work has been published in the area ofthe optimal capacitor switching problem. An approach for determining the optimalcapacitor switching pattern, based on the sensitivity approach, has been reported in[5]. This method calculates the sensitivity factors based on the repeated load-�owof the system.

However, as mentioned earlier, for this information of optimal switching pat-tern to be of any use in the modern distribution automation system, it has to be

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 835

obtained very quickly. Now, by the approach in [5], for a practical power distri-bution system involving thousands of feeders, it takes a comparatively long time(a few minutes depending upon the system size) to perform the capacitor switchinganalysis and compute the optimal switching pattern. In this case, it may be desir-able to reduce the computation time by some alternative technique, such that theoptimal capacitor-switching pattern can be obtained quickly. Arti�cial intelligence(AI) techniques, such as arti�cial neural network (ANN), oŒer such a possibility ofcomputation of optimal switching pattern within a very short time.

Arti�cial neural networks are computation tools which try to mimic the op-eration of a human brain. Analogous to the operation of the human brain, ANNsoperate on the principle of parallel processing and, consequently, they are quite fast,especially while dealing with large volume of data without any known mathemati-cal correlation among the data. Clearly, the optimal capacitor switching algorithmbased on ANN technique is also expected to be quite fast and, hence, it is ex-pected that it will be quite suitable for on-line application in a modern distributionautomation system.

In this paper, a methodology for computing the optimal capacitor switchingpattern in a distribution system based on ANN technique is described. Essentially,the development of the methodology is comprised of two steps:

1. A suitably chosen ANN structure is to be trained �rst with a set of trainingdata (input : loading pattern, output : optimal switching pattern), with thehelp of a suitable training algorithm. The training data would be obtainedfrom an optimal capacitor switching software package based on a conven-tional capacitor switching algorithm.

2. Once the ANN is trained with a su� cient amount of training data, theANN \learns" the implicit correlation between the loading patterns and theoptimal switching patterns. Next, new loading patterns (which have not beenused to train the ANN) are fed to the network, and the network providesthe optimal switching pattern at its output within a very short span.

In this work, an FLN [6] with the modi�ed Perceptron training rule has beenused to solve quickly the optimal capacitor problem. The switching algorithm re-ported in [5] has been used for optimal capacitor switching analysis.

This paper is organized as follows: Section 2 discusses the arti�cial neural net-work algorithm used in this work and, in detail, the FLN and the modi�ed Percep-tron learning rule used. In Section 3, the conventional optimal capacitor switchingalgorithm is discussed brie�y. Section 4 presents the important results of this work,and Section 5 delineates the main conclusions.

2. Functional Link Network (FLN)

In the case of a multi-layer-mapping network, the hidden layers of neurons providethe appropriate transformations and the output layer yields the �nal mapping.Instead of carrying out a multi-stage transformation, the input/output mapping canalso be obtained through an arti�cially augmented single layer network. Since thenetwork has only one layer, instead of the generalized delta-learning rule normallyused for multi-layer networks, the mapping can be learned by using the simpledelta-learning rule. The concept of training an augmented and expanded network

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

836 B. Das and S. P. Velpula

leads to the so-called Functional Link Network (FLN), as introduced by Pao [6].FLNs are single layer networks that are able to handle linearly non-separable tasks,due to the appropriately enhanced input representation.

The key idea of the method is to �nd a suitably enhanced representation of theinput data. Additional input data that are used in the scheme incorporate higherorder eŒects and arti�cially increase the dimension of the input space. The extendedinput data are then used for training instead of the actual input data. The blockdiagram of an FLN is shown in Fig. 1.

The network in Fig. 1 uses an enhanced representation of input patterns com-posed now of J +H components, where J is the total number of original inputs. Thehigher order patterns are applied at the bottom H inputs of the FLN. These are tobe produced in linearly independent manner from the original patterns. Althoughno new information is explicitly presented to the network, the input representationhas been enhanced and the proper mapping can be achieved in the extended space.The following discussion presents two methods of extending the input patterns thathave been used in this work.

In the �rst method, called the tensor method, the additional input patterns areobtained for each J dimensional input vector pattern, as a subset of the set of theproducts XiXj for all 1 µ i µ J and j = i + 1, with an additional term XJ X1.

In the second method, called the functional method, the additional terms aregenerated using the orthogonal basis functions. Here the function tanh(Xi) for all1 µ i µ J has been used.

It is to be noticed that the network with a single layer or the so-called �atneural network, based on the concept of a functional link, does not strictly belongto the class of layered networks. It has only one layer of neurons. However, due toits intrinsic mapping properties, the FLN performs similarly to the multi-layer net-work. Because of the enhanced input representation, the mapping can be performedthrough a single layer.

In the next two subsections the Perceptron learning rule used in this work totrain the FLN is described in detail.

Figure 1. Functional link network.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 837

2.1. Perceptron Learning Rule

Before discussing the Perceptron learning rule, a brief discussion on the concept ofgeneral supervised learning rule [6] would be in order.

A neuron is considered to be an adaptive element. Its weights are modi�able,depending on the input signal it receives, its calculated output value, and the de-sired output value. Let the learning of the weight vector wi , or its components wij

connecting the jth input with the ith neuron, be considered. The trained networkis shown in Fig. 2.

The following general learning rule is adopted in neural network studies. Theweight vector wi = [wi1wi2 : : : win ]t increases in proportion to the product of inputx and the learning signal r. The learning signal r is a general function of wi , x, andsometimes of the desired signal di . Thus, for the network shown in Fig. 2, we have:

r = r(wi ; x; di) (1)

The increment of the weight vector wi , produced by the learning step at time t,according to the general learning rule is,

¢wi(t) = cr[wi(t); x(t); di(t)]x(t) (2)

where c is a positive number called the learning constant that determines the rateof learning. The weight vector adapted at time t becomes at the next instant, orlearning step,

wi(t + 1) = wi(t) + crbwi(t); x(t); di(t)cx(t) (3)

The learning in equation (3) assumes the form of a sequence of discrete-timeweight modi�cations.

Figure 2. Illustration for general learning rule.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

838 B. Das and S. P. Velpula

For the Perceptron learning rule, the learning signal is the diŒerence betweenthe desired and actual responses of the neuron. Thus, learning is supervised, andthe learning signal is equal to

r = di ¡ oi (4)

where the actual response is oi = sgn(wti x), and di is the desired response. Weight

adjustments in this method, ¢wi and ¢wij , are obtained as follows:

¢wi = cbdi ¡ sgn(wti x)cx (5)

¢wij = cbdi ¡ sgn(wti x)cxj ; for j = 1; 2; : : : ; n (6)

where

sgn(X) = +1; for X ¶ 0

= 0; for X < 0

Note that this rule is applicable for only the binary neuron response, and therelationships in equations (5) and (6) express the rule for the case of uni-polar binaryoutput. Under this rule, the weights are adjusted if, and only if, oi is diŒerent fromdi , and the error, as a necessary condition of learning, is inherently included in thistraining rule. It is to be noted that the weight adjustment is inherently zero whenthe desired and actual responses agree.

It is a matter of common observation that in the most of the popular learningrules, throughout the process of training the learning constant is �xed at a particularvalue. This approach gives rise to a steady decrease in the learning error. However,during this work it has been observed that if the learning constant can be variedin a proper manner during the process of training, the learning error decreases ata very brisk rate. Now, there are many ways in which we can change the learningconstant. Here, in this work, we use the observation that if the learning constantis changed abruptly to a value much below the initial value, the learning errordecreases too steeply. This point is explained more clearly through the trainingalgorithm presented in the next sub-section.

2.2. Training Algorithm for Functional Link Network (FLN)

Given are P training pairs arranged in the training set fy1; d1; y2; d2; : : : yP ; dP gwhere yi is a (JX1) input vector, di is a (KX1) output vector, and i = 1; 2; : : : ; P .The steps of the algorithm are as follows:

Step 1: c > 0, Emax > 0 are set to some initial values, where c is the learningconstant and Emax is the maximum allowable average error during training.

Step 2: Weights W are initialized at small random values, where W is the weightmatrix of dimension (KXJ ). Set q ¬ 1, p ¬ 1.

Step 3: The training step starts here. Input is presented and the output is com-puted:

y ¬ yp ; d ¬ dp ; ok ¬ f (wtk y); for k = 1; 2; : : : ; K

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 839

where wk is the kth row of W and f (net) is as de�ned below:

f (net) = 0; net < 0

f (net) = 1; net ¶ 0

Step 4: Weights are updated:

wk ¬ wk + c(dk ¡ ok )y; for k = 1; 2; : : : ; K

where wk is the kth row of W .Step 5: If p < P , then p ¬ p + 1 and go to step 3; otherwise go to step 6.Step 6: The average error is computed as follows:

Eq =PX

i=1

KX

j =1Eij (k) and Eav e = Eq =(P ¤

K)

where Eij (k) = (dk ¡ ok )2, for k = 1; 2; : : : ; K

Step 7: q ¬ q + 1.If q is a multiple of 10,

c = c=100,else

c remains at the initial value.Step 8: The training cycle is completed; Eav e < Emax terminates the training

session. The output are weights W , q, and E. If Eav e > Emax , then p ¬ 1, anda new training cycle is initiated by going to step 3.

3. Optimal Capacitor Switching Algorithm

From a mathematical point of view the optimal capacitor switching problem is aconstrained minimization problem, where the constraints are inequality constraints.In this work the objective is to minimize the total real-power loss in the distributionsystem subjected to diŒerent inequality constraints. As the total real-power loss inthe system is the summation of I

2r losses in the feeders and the transformers, this

is a non-linear objective function. The diŒerent inequality constraints, which couldbe imposed are constraints on bus voltages, current �ow through each feeder andtransformer, power factor, and reactive power demand at the substation, etc. [5].The value of the real-power loss can be computed by using an accurate load-�owprogram, given the load-pro�le in the system and the setting of other control vari-ables (such as transformer tap settings, settings of any shunt-connected reactors,capacitors etc.). Also, it is desirable that the solution of this constrained optimiza-tion problem be obtained in a minimum number of steps.

In this work, among all the previously mentioned constraints, only the con-straint on bus voltages is considered. Hence, the statement of the constrained opti-mization problem is as follows:

\Under the current loading condition �nd the optimal switching patternof the already installed shunt capacitors in the distribution system, suchthat the real power loss in the system is minimum, and simultaneouslythe voltages at all the buses lie within their respective operating limits."

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

840 B. Das and S. P. Velpula

Since the capacitors that are to be switched are discrete, not continuous innature, the optimal capacitor switching problem is essentially a discrete (integer)programming problem with non-linear objective function and inequality constraints.To solve such discrete optimization problems, combinatorial methods are generallyused. In combinatorial method all possible discrete solutions are checked, and �nallythe optimal solution is selected. Depending upon the size and nature of the opti-mization problem, the number of possible solutions may be very large and hence,time taken to �nd the optimum solution may be large. To overcome this problem,combinatorial methods use special strategies such as only eŒective subsets of allpossible solutions are searched.

One of the most commonly used, simple, and reliable combinatorial strategiesis gradient descent method. This method can be used for any type of variables andobjective function. In this method, the control variables are moved by a reasonablestep in the direction in which the objective function decreases the most. For discretecontrol variables the chosen step size is generally equal to the discrete incrementsof the control variables. This direction (which is generally termed \the descentdirection") can be chosen by a number of methods, such as by chance (Monte Carlomethod) , by any evaluation strategy, or by the largest negative partial derivativeof the objective function with respect to the control variables. In the last case, themethod is known as the gradient or oriented discrete co-ordinate descent method.

For a given loading condition, in this work the oriented discrete co-ordinatedescent method [5] has been used to determine the optimal capacitor switchingpattern. For a detailed discussion of this method, the interested reader is referredto [5].

4. Results and Discussion

To illustrate the development of an FLN-based optimal switching algorithm, a sam-ple 30-bus system [7] has been chosen. The one-line diagram of the network is shownin Fig. 3. The data for this test system are given in Tables A.1 and A.2 in AppendixA. The loading pattern given in Table A.1 is termed \base operating condition".In this system there are a total of 22 load points. At each load points, both real(KW) and reactive (KVAR) loads are speci�ed. Hence, the total number of realand reactive loads in the system is 44. There are also 17 capacitors installed in thesystem. As any capacitor in the system can switch independently, irrespective ofany other capacitor in the system, without following any �xed switching pattern,there are 17 positions of switchable capacitors in the system. Hence, the FLN ar-chitecture used in this work has 44 original input nodes and 17 output nodes. Aswas discussed in Section 2, the original inputs have been enhanced from 44 to 88internally, so that we have eŒectively 88 input nodes and 17 output nodes.

To train the FLN, it is necessary to generate a number of input-output patternsat diŒerent loading conditions. The diŒerent loading conditions in the system areachieved by varying the KW and KVAR loads in the system within a certain rangewith respect to the \base operating condition". For example, the KW and KVARloads can be varied in such a way that the new loading condition always remainswithin a range of 90{110% of the base operating condition. Similarly, for any otherspeci�ed ranges, the KW and KVAR loads can be varied accordingly. In this worktwo diŒerent ranges of loading have been considered. These are, (a) 90{110% and(b) 60{140%, of the base operating condition. To vary the loading within any spec-

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 841

Figure 3. One-line diagram of the sample distribution system.

i�ed range (e.g. 90{110%), a large quantity of random numbers within a range of0.9{1.1 have been generated, and subsequently, the KW and KVAR loads at baseoperating condition have been multiplied by these random numbers. Consequently,a large number of loading conditions within a range of 90{110% of the base operat-ing condition are generated. Once these new loading conditions are generated, theoriented discrete descent method [5], using the load-�ow algorithm reported in [8],has been used to �nd out the optimal capacitor switching patterns at each of thesenew loading conditions. Thus, a number of input-output patterns are generated.Out of these generated input-output patterns, some of the patterns have been usedto train the network, and some remaining patterns have been used to test the per-formance of the trained FLN. To train the FLN, the Perceptron learning algorithmas described in Subsection 2.2 has been used.

The results of training and testing the FLN with an initial value of the learningconstant = 0.01, for both tensor and functional methods and for both the rangesof loading are tabulated in Table 1. As observed from this table, for example, forthe operating range 90{110% of the \base operating point", 5000 input-outputpatterns have been used to train the FLN, and after the FLN is trained, 5000input-output patterns, which had not been used during training, have been usedto test the performance of the FLN. Similarly for the other range, the number ofinput and output patterns are also given in the table. The FLN is trained for atotal of 201 iterations in each case. The performance of the FLN is quanti�ed by

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

842 B. Das and S. P. Velpula

Table 1

No. of No. of Testing TestingRange of training testing error 1 error 2loading patterns patterns (%) (%)

90%{110% 5,000 5000 0.441 0.28560%{140% 10,000 5000 2.957 1.556

\testing error" (percentage mismatch). During testing, continuous values (between0 and 1) are available at the ANN output nodes. To map these continuous valuesto capacitor \ON/OFF" status, any value less than or equal to 0.5 is taken to bezero (\OFF" status) and any value greater than 0.5 is taken to be 1 (\ON" status).It is shown that for the operating range 90{110%, the testing error is 0.441% formethod 1 and 0.285% for method 2. Here method 1 involves the �rst (tensor) formof augmentation of input patterns, while method 2 involves the second (functional)form of augmentation of input patterns. From this table it is also observed that incomparison with the tensor form of FLN, the functional form of FLN always givesbetter \testing error".

The meaning of testing error is as follows. As discussed earlier, each pattern has17 output values. Hence, for a total of 5000 test patterns, there are a total of 85,000output values. Upon testing, the FLN predicts wrong output values for 0.441% of85,000 output values, i.e., for 0:441¤ 850 = 375 output values, the predictions ofthe FLN do not match the output values obtained from oriented discrete descentmethod. For the rest 85; 000 ¡ 375 = 84; 625 output values, predictions of FLNmatch exactly those obtained from the oriented discrete descent method.

It may be argued that during training the training error would probably bereduced if the training of the FLN had been continued for a greater number of cyclesand, consequently, the test error would also be reduced. To verify this, a graph ofaverage training error during training with respect to the number of training cycleshas been plotted (for the range 90{110%) as shown in Fig. 4.

From Fig. 4 it is evident that after approximately 200 iterations the averagetraining error reaches almost a plateau. Thus the training error and, consequently,the testing error would reduce at a very slow rate, even if the training is continuedfurther. Hence, in this work the number of training cycles has been limited to 201.Similarly, the corresponding plot for range 60{140% is shown in Fig. 5.

Also from the Table 1 it is found that the best performance of the FLN isachieved for the lowest range (90{110%). For the lowest range, the outputs of variouspatterns were quite close to each other. Therefore, after training, the FLN wasable to predict the correct output patterns for most of the input patterns, as itwas relatively easily able to learn the implicit relationship between the input andoutput patterns. On the other hand, for the other range, the outputs of variouspatterns were diverse in nature. Hence, to learn the implicit relation properly, thenumber of training patterns required by the FLN doubled (10,000) the numberthat was required for the lowest range (5000). Also, the test error was greater. Itis to be noted that the values of the learning constants were found by trial anderror, and that these values are not universal in nature. They are valid for only thetest system under consideration. To investigate the eŒect of the learning constant

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 843

Figure 4. Average training error (range of loading : 90–110%).

Figure 5. Average training error (range of loading : 60–140%).

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

844 B. Das and S. P. Velpula

on FLN training, the training of the FLN has been repeated for two more valuesof learning constants. Though the rate of decrease in average training error wasslightly lower initially, almost all the sets tend �nally, after a few iterations, to goto the same point. So, it can be concluded that the initial value of learning constantsselected does not have a pronounced eŒect on the learning rate. Fig. 6 shows thevariation of average learning error with learning constants.

It has been already discussed in Section 1 that the prime motive behind thedevelopment of the ANN-based algorithm is to reduce the time taken for decidingthe optimal capacitor switching pattern for a given loading condition. For a range of90{110%, it has been found that the CPU time (on a 100 MHz system), taken by theoriented discrete descent method for determining 5000 optimal switching patterns,is 2732.472 secs, whereas, the CPU time, for the same number of switching patterns,taken by the FLN (for functional method) is 9.784 sec. Hence, it is evident that theFLN-based method is at least 287 times faster. Obviously, for larger systems, thediŒerence between the time taken by the combinatorial method and the FLN-basedmethod would be much greater and, consequently, for a practical real life largesystem, the FLN-based method is expected to be much faster than the traditionalmethods and would be more suitable for on-line application. For the other ranges,the comparison of CPU time taken by the traditional capacitor switching algorithmand the FLN-based method is given in Table 2.

However, it should be noted that in a practical distribution system the num-ber of switchable capacitor positions is rarely as high as 17. Hence, it would beinformative to note the change in gain in the computational time as the number ofswitchable capacitors in the system varies. Thus, the number of switchable capaci-tors in the system has been reduced progressively and the gain in the computational

Figure 6. Variation of average learning error with learning constants.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 845

Table 2

Time taken by theFLN algorithmTime taken by

oriented discreteRange of descent method Method 1 Method 2loading (secs of CPU time) (secs of CPU time) (secs of CPU time)

90%{100% 2732.472 9.505 9.78460%{140% 2843.297 9.505 9.835

Table 3

Faster than the OCSNo. of switchable positions algorithm by

17 279 times13 243 times9 195 times5 145 times

time has been calculated. The results are shown in Table 3. It is observed from thistable that, as the number of switchable positions decreases, the e� cacy of the FLNalgorithm is reduced. However, it is also to be noted that for a realistic number ofswitchable capacitors, such as 5, the FLN algorithm is at least 145 times faster thanthe conventional algorithm. Even in the systems where the number of switchablecapacitors is low, the FLN technique is still many times faster than the conventionalalgorithm and, thus, better for on-line application.

5. Conclusion

In this work, a Functional Link neural network-based algorithm for optimal ca-pacitor switching in a power distribution system has been developed. The mainconclusions of this work are

° The FLN-based method is much faster than the traditional combinatorialmethods for solving the optimal capacitor switching problem, even for thosesystems in which the number of capacitors is realistic.

° As the range of loading patterns increases, the number of training patternsrequired to train the network also increases.

° The performance of the FLN reaches a plateau, after which the improvementin performance can be achieved only at a cost of unacceptably greater trainingtime.

° The error incurred by only this method in predicting the optimal switchingpattern is low (less than 2%).

Hence, the proposed methodology can be quite eŒective for on-line application ina modern distribution automation system.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

846 B. Das and S. P. Velpula

References

[1] K. Kato, H. Nagasaka, Akimichi-Okimato, T. Kunieda, and T. Nakamura, 1991, “Dis-tribution automation systems for high quality power supply,” IEEE Transaction onPower Delivery, Vol. 6, No. 3, pp. 1196–1203.

[2] H.-D. Chiang, J.-C. Wang, O. Cockings, and Hyoun-Duckshin, 1990, “Optimal ca-pacitor placement in distribution systems: Part 1: A new formulation and the overallproblem,” IEEE Transaction on Power Delivery, Vol. 5, No. 2, pp. 634–642.

[3] H.-D. Chiang, J.-C. Wang, O. Cockings, and Hyoun-Duckshin, 1990, “Optimal ca-pacitor placement in distribution systems: Part 2: Solution algorithms and numericalresults,” IEEE Transaction on Power Delivery, Vol. 5, No. 2, pp. 643–649.

[4] Y. Baghzoug, 1991, “EŒects of nonlinear loads on optimal capacitor placement in radialfeeders,” IEEE Transaction on Power Delivery, Vol. 6, No. 1, pp. 245–251.

[5] I. Roytelman, B. K. Wee, and R. L. Lugtu, 1995, “Volt/ var control algorithm for mod-ern distribution management system,” IEEE Transactions on Power Systems, Vol. 10,No. 3, pp. 1454–1460.

[6] J. M. Jurada, 1997, Introduction to Arti�cial Neural Systems, Jaico Publishing House.[7] N. I. Santoso and O. T. Tan, 1991, “Neural-net based real-time control of capacitors

installed on distribution system,” IEEE Transaction on Power Delivery, Vol. 5, No. 1,pp. 262–272.

[8] D. Shirmohammadi et. al., 1988, “A compensation based power �ow method forweakly meshed distribution and transmission networks,” IEEE Transactions on PowerSystems, Vol. 3, No. 2, pp. 753–762.

Appendix A

Table A.1System network and load data

Branch impedance Max. load at bus jBus Bus

i j rij («) xij («) P (kW) Q(k Var)

0 1 0.5096 1.7030 | |1 2 0.2191 0.0118 522 1742 3 0.3485 0.3446 | |3 4 1.1750 1.0214 936 3124 5 0.5530 0.4806 | |5 6 1.6625 0.9365 | |6 7 1.3506 0.7608 | |7 8 1.3506 0.7608 | |8 9 1.3259 0.7469 189 639 10 1.3259 0.7469 | |

10 11 3.9709 2.2369 336 11211 12 1.8549 1.0449 657 21912 13 0.7557 0.4257 783 26113 14 1.5389 0.8669 729 2438 15 0.4752 0.4131 477 159

15 16 0.7282 0.4102 549 183

(continued)

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013

Optimal Capacitor Switching 847

Table A.1(Continued )

Branch impedance Max. load at bus jBus Bus

i j rij («) xij («) P (kW) Q(k Var)

16 17 1.3053 0.7353 477 1596 18 0.4838 0.4206 432 144

18 19 1.5898 1.3818 672 22419 20 1.5389 0.8669 495 1656 21 0.6048 0.5257 207 693 22 0.5639 0.5575 522 174

22 23 0.3432 0.3393 1917 6323 24 0.5728 0.4979 | |24 25 1.4602 1.2692 1116 37225 26 1.0627 0.9237 549 18326 27 1.5114 0.8514 792 2641 28 0.4659 0.0251 882 294

28 29 1.6351 0.9211 882 29429 30 1.1143 0.6277 882 294

Vrated = 23 kV

Table A.2Capacitor KVARs at diŒerent tap positions

Capacitor kVar

Tap Cap. #1 Cap. #2 Cap. #3 Cap. #4 Cap. #5position at bus 13 at bus 15 at bus 19 at bus 23 at bus 25

1 875 875 500 Fixed at 750 6002 700 700 425 5253 525 525 350 4504 350 350 275 375

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go]

at 0

3:32

18

Mar

ch 2

013