Capacitor Placement

A Review of Capacitor Placement Techniques on Distribution Feeders

J.C. Carlisle A.A. El-Keib Department of Electrical Engineering University of Alabama Tuscaloosa, A1 35487-0286

D. Boyd K. Nolan Alabama Power Company Birmingham, A1 35291-0715

Abstract discrete capacitor sizes, ( 5 ) a radial feeder with no laterals. The most commonly used assumption considers capacitors to affect only the reactive current component. Some of the proposed methods base the cost savings only on reduction in peak power losses, while others consider both peak power and energy loss reductions. Some methods consider only fixed capacitors while others consider both fmed and switched capacitors. Of the methods which apply switched capacitors, there may be limitations as to the switching times.

Optimal sizing and placement of shunt capacitors on distribution feeders has received considerable attention from researchers for many years. This paper presents an extensive review of the different solution methods found in the literature and is intended as a guide for those interested in the problem or intending to do additional research in the area. The assumptions made and a brief description of the solution methods are presented.

Introduction

Review of existing solution approaches

Transfer of electric energy from the source of generation to the customer via the transmission and distribution networks is accompanied by losses. The majority of these losses occur on the distribution system. It is widely recognized that placement of shunt capacitors on the distribution system can lead to a reduction in power losses. Increased competition in the industry has created a renewed interest in improving efficiency by reducing these losses. An overview of methods previously considered for placement of capacitors is presented in this paper. No attempt is made here to verify any of the claims made by the authors, or to determine the effectiveness of the methods described. A companion paper [l] describes the general techniques in more detail and provides additional insight into the problem.

Neagle and Samson [2] assume the load is uniformly distributed along the feeder. They consider only peak kilowatt loss savings with fixed capacitors and ignore the cost of capacitors. Curves are presented which show the reduction in losses as a function of capacitor size and distance of capacitor from the substation. Where two banks are to be installed, they consider equally sized banks or one bank to be twice the size of the other. For installation of three or four banks, equally sized capacitors were assumed. Cook [3] also addresses application of fixed capacitors to a uniformly distributed load. However, instead of considering reduction of peak power losses, savings are based on energy loss reduction considering a time-varying load. This analysis is extended in [4]to include switched capacitors. Savings are calculated based on reduction of both peak power losses and energy losses. An algorithm is presented to calculate capacitor locations and savings as total compensation is varied from 0.05 to 1.0 per unit of feeder reactive load. The number of capacitors is varied from one to four, assuming equally sized banks. A second algorithm sizes and locates a fixed bank and a switched

Problem formulation The problem, in general, is to determine the optimal number, location, sizes, and switching times for capacitors to be installed on a distribution feeder to maximize cost savings subject to operating constraints. Due to the complexity of the problem, the proposed solution methods introduce one or more of the following simplifying assumptions: (1) a uniform feeder, (2) constant voltage profile, (3) linear capacitor cost, (4)non-

bank as total compensation is varied from 0.05 to 1.0 per

unit. The optimal switching time is determined by considering several switching times. In [5], Cook

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describes an incorrect method for calculating loss reduction often used in the literature. The correct method is also presented. Maxwell [6] addresses the effects of capacitors on reduction in kVA input, kW demand, and energy losses. The algorithm presented is not based on an optimization procedure, but is an aid in calculating savings due to placement of capacitors. Schmill [7] extends the work of Cook. Equations are given for sizing and placement of n capacitors on a uniform feeder with a uniformly distributed load. The necessary conditions for optimal sizing and placement of one or two capacitors on a feeder with discrete loads and non-uniform resistance are presented. An iterative approach is suggested to solve the problem. Chang [8-111 assumes a feeder with a uniform load and a concentrated end load. Accounting for both peak power losses and energy losses, he determines the optimal location of a fixed capacitor and the resulting savings, given the capacitor size. The optimal solution is determined by considering each of the available capacitor sizes. Duran [12] proposes a dynamic programming approach to find the number, locations, and sizes of fixed capacitor banks on a feeder with discrete loads. Algorithms are presented for the special cases of no capacitor cost, capacitor cost proportional to installed capacity, and cost proportional to installed capacity plus a fixed cost per capacitor bank. Alvarez and Molina [13] also use dynamic programming. However, they propose an approach to reduce the number of possible states to be considered in order to reduce the solution time. Szabados and Burgess [14] address the problem of inductive interference with communication circuits caused by the addition of shunt capacitors to an unbalanced distribution system. Procedures for applying capacitors while minimizing this interference are given. Bae [15] assumes a uniformly distributed load along the feeder, a constant voltage profile, and capacitors of equal size. Capacitors are first optimally located for a fixed load level. The methodology is extended to determine the optimal capacitor locations for all load levels up to the fixed level, without accounting for the time duration of each load level. Finally, an algorithm is presented for determination of the optimum compensation level for a load with typical yearly characteristics. In all cases, energy loss reduction as a result of fixed capacitors only is considered and capacitor cost is ignored. Grainger and Lee [16] propose a methodology which considers non-uniform feeders and non-uniform loads. It transforms a feeder with different wire sizes into an equivalent uniform feeder (or the normalized equivalent feeder). It also accounts for non-uniform reactive load

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distribution by introducing the concept of the normalized reactive current distribution function. The objective function used consists of the peak power loss reduction, energy loss reduction, and a linear cost of capacitors. Constant voltage is assumed and only fixed capacitors are considered. The necessary conditions for optimality are used to determine optimal capacitor locations and sizes. This results in the decomposition of the problem into two subproblems. The first subproblem solves for capacitor locations given sizes, while the second subproblem solves for sizes given locations. The overall solution process begins by choosing the number of capacitors to be installed. The two problems are solved sequentially using their “Equal Area Criterion” to guide the interaction between them. This process continues until convergence is achieved. In [17], Grainger, Lee, Byrd, and Clinard modify [16] to determine capacitor sizes by solving a set of linear equations representing the necessary conditions corresponding to the optimal sizes. Again, to determine the optimal locations and sizes, the two subproblems are solved sequentially in an iterative manner until convergence is reached. The paper also presents results of a sensitivity analysis to determine the effect on savings of choosing the nearest standard size capacitors to those calculated. In addition, a sensitivity analysis of cost with respect to small changes in location was also included. In [ 181, Lee and Grainger extended the methodology in [16] to consider both switched and fixed capacitors. The proposed approach uses the load duration curve to account for the time variation of the reactive load. Optimal sizes, locations, and switching time are determined for a given number of fixed and switched capacitors, assuming capacitors are switched simultaneously. The problem is decomposed into 3 subproblems: (1) given locations and switching time, determine optimal sizes, (2) given sizes and locations, determine optimal switching time, and (3) given sizes and switching time, determine locations. The three subproblems are solved alternatively until convergence is obtained. As in the case of finding sizes and locations, the switching time is determined using the necessary conditions for an optimal solution. It is suggested by the authors that several initial conditions could be chosen, and the global solution selected from among the local optimals. Different capacitor costs are used for fixed and switched banks, but both costs are linear functions of size with no fixed charge. When using this algorithm, it is necessary to specify the number of fixed and switched capacitors to be used and their relative placement with respect to each other along the feeder. In [ 191, Grainger and Lee extended [ 161 to incorporate the effect of voltage variation along the feeder. An outer loop is added in which the load and capacitor currents are

adjusted based on the voltages obtained from a load flow solution. The inner loop contains the two subproblems as before. After convergence of the inner loop, it is necessary to check for convergence of the outer loop. In [201, Grainger, Lee, and El-Keib extend 1181 to allow for different switching times for different capacitor banks. This is done by assigning the capacitors to different “groups” such that the capacitors in each group are switched together. The relative switching times of the groups and the relative placement of the individual capacitors along the feeder must be specified. A procedure is presented for determining the switching times for capacitors which have been already placed. It should be noted that [ 16-20] do not consider laterals. Ponnavaikko and Rao [21] include the effect of load growth and energy cost increase into the objective function and solve the problem using the Method of Local Variations. In this procedure, the capacitor at each bus is a state variable. The procedure starts with any solution which satisfies all voltage constraints. A check is made to determine if additional savings will result from increasing the size of the capacitor at a bus by a discrete amount, which is determined by the minimum available capacitor size. If so, the capacitor is set to the new size. Otherwise, a check is made to see if a decrease in size will result in an increase in savings. If so, the capacitor size at the bus is decreased, otherwise no change is made and the method proceeds to the next bus. An iteration is complete when all buses have been checked. Convergence is reached when all state variables remain unchanged throughout an entire iteration. Fawzi, El-Sobki, and Abdel-Halim [22] propose an approach which considers two distinct optimization problems depending on whether the kVA released by the capacitor is used to feed additional loads through the same feeder or a different feeder. They consider placing a single capacitor on a uniform feeder. First, the optimization problem is formulated for a constant feeder load. Because of the complexity of the problem, a number of special cases were considered. These are (1) neglecting cost of energy losses, (2) maximum energy loss reduction, (3) optimum solution based on released kVA with adjustment for energy losses, and (4) optimum solution based on the cost of energy losses against the cost of capacitors. Each case simplifies the problem of determining capacitor size and location. The second optimization problem is then formulated assuming an increase in feeder load due to the released kVA. Here it is assumed that the maximum saving obtained depends on the amount of load which can be added due to the released kVA. The capacitor size, therefore, is dependent on the voltage drop on the feeder at maximum load and the voltage increase along the feeder at light loads.

Grainger, Civanlar, and Lee [23] assume continuous control of capacitor output. The problem is divided into an optimal design subproblem and an optimal control subproblem. Sizes are rounded to the nearest standard size and voltage variations are not considered. The authors suggest the use of lookup tables to implement switching on a seasonal or daily load profile. The equation needed for solving for the location of capacitors is not given. In [24], Grainger, Civanlar, Clinard, and Gale are concemed only with control, assuming capacitors are already placed. The cases of equal tap sizes and tap size dependent on capacitor size are addressed. The first case is solved directly, while the second uses a branch-andbound technique. Both problems begin with the continuous control problem. In [25], they include design and control problems assuming capacitors are continuously controlled. Balanced loads which are varying conformally are assumed. The authors introduce the V-P model which they claim obviates the need for a load-flow solution while allowing inclusion of the effects of voltage variation on the design and control phases of an optimal capacitor compensation scheme. In [26], Grainger, El-Keib, and Lee extend [16,19] to include unbalanced feeders, In [27], El-Keib, Grainger, Clinard, and Gale further extend this work to include switched capacitors and laterals. Kaplan [28] presents a heuristic approach which is claimed to make none of the simplifying assumption used in much of the earlier work. The approach first determines the “best” locations and types (fixed or switched) for the smallest available standard capacitor size. After placement of these initial capacitors, an attempt is made to improve savings by the addition of larger banks or by combination of smaller banks into larger units. Iyer, Ramachandran, and Hariharan [29] formulate the problem as a mixed integer linear programming problem, which is decomposed into two smaller subproblems. The first is a pure integer programming problem with b i i variables, while the second is a linear programming problem. Salama, Chikhani, and Hackam [30] assume a fixed load condition and a uniform feeder. A concentrated load at the end of the feeder is dealt with separately. They transform a non-uniform feeder into an equivalent uniform feeder using the “base resistance” technique. The example feeder used satisfies the conditions of a uniformly distributed load. The proposed solution algorithm determines capacitor sizes and locations, given the number of capacitors and considering all capacitors to be of the same size. The objective function does not include energy loss reduction and a linear capacitor cost is

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cost is approximated by a linear function and a fixed charge. The solution method is decomposed into levels. The top level, called the “master problem”, is an integer programming problem which determines the number and location of capacitors. The bottom level, called the “slave problem”, is used to determine the sizes of the capacitors. The slave problem is further decomposed into smaller problems referred to as base problems. These base problems are solved by an algorithm developed for a special capacitor placement problem called the “sizing problem”. The sizing problem is solved using an algorithm based on a Phase I - Phase I1 feasible directions approach, the details of which are presented in [39]. Finally, a heuristic procedure determines whether the capacitors are of the fixed or switched type. Ertem [40] proposes a quadratic programming approach in which a quadratic objective function and linear bus voltage and line loading constraints are used. The programming problem is solved using Beale’s algorithm. An iterative optimization procedure is necessary, with bus voltages and angles updated between iterations by running a load flow program. Santoso and Tan [41] divide the distribution system into several smaller subsystems, each of which is optimized. “Interaction” variables are updated and the process is repeated until the absolute value in savings between iterations is small. Capacitor cost is a differentiable function of size. The subsystem problems are solved by a gradient search method. Chiang, Wang, Cockings, and Shin [42] modify the formulation of [38] to treat capacitor cost as a step-like function and capacitor sizes as discrete variables. The formulation allows the switched capacitors to be switched as a block or in several consecutive steps as load varies. The proposed solution method is based on the simulated annealing optimization technique. The method is further extended to unbalanced systems by Chiang, Wang, Tong, and Darling in [43,44]. Baghzouz and Ertem [45] present an algorithm for optimizing shunt capacitor sizes on radial distribution lines with distorted voltages, such that the R M S voltages and their total harmonic distortion lie within prescribed limits. A heuristic algorithm based on the Method of Local Variations is employed. Since only a local optimal solution is guaranteed, it is suggested that several runs with different initial solutions be made to identify other local optimal solutions. Augugliaro, Dusonchet, and Mangione [46] use nonlinear programming for optimization of the number, size, location, and switching-on times of both fixed and switched capacitors. The model incorporates constraints on the voltage rise at each bus. Implementation consists of a combination of Fortran and MINOS/Augmented (a

assumed. Capacitor sizes are rounded to the nearest available size and energy losses are then calculated. Relocation of existing capacitors after a load growth and inclusion of released thermal capacity into the savings calculation are also addressed. The paper includes a simplified equation for calculation of voltage along the equivalent uniform feeder once the capacitors have been placed. In [311, the authors extend this paper to consider a time-varying load. Grainger and Civanlar [32,33] propose a solution approach to the problem of determining voltage regulator placement and control setting as well as capacitor placement. This approach is based on decoupling the overall problem into two subproblems, assuming that the voltage regulators hold the bus voltages relatively constant. Lateral and sublateral branches are considered. A closed-form solution is given for the’problem of capacitor sizing once the locations are known. The nonlinear cost of capacitors is incorporated. The claim is made that regardless of system size, the computational effort required is a function only of the number of capacitors. The proposed algorithm initially assumes all capacitors are switched. After the sizes and locations are determined, the switching times are determined. During this procedure, it may be determined from the switching times that some of the capacitors should be fixed. In [34], Rao and Radhakrishna summarize the results of their investigation of the compensation of rural Indian feeders which have poor voltage profiles. They propose a two-level compensation scheme which incorporates both shunt and series capacitors. Bishop and Lee [35] describe an algorithm which starts with the 2/3 rule, then tries all combination of userspecified (or fewer) locations. Constraints are placed on power factor at each bus and the user is permitted to specify the minimum and maximum size of the capacitor allowed at any location. Their paper does not include details of the algorithm. Rinker and Rembert [36] claim the biggest problem in placing capacitors is a lack of data concerning the reactive current profile along the feeder. They address acquisition and treatment of data which is used to determine the size and placement of both fixed and switched capacitors. The method used to size and locate capacitors is attributed to Grainger and Lee. It is assumed switched capacitors are switched ”ON” whenever the load reaches 1/2 the capacitor rating. Ertem and Tudor [37] use the Method of Approximate Programming. This method uses the first order terms of a Taylor series expansion of both the objective function and constraints to construct a linear program. Baran and Wu [38] formulate the problem as a nonlinear, mixed integer programming problem. Capacitor

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general-purpose projected augmented Lagrangian code). Once the solution is obtained, capacitor sizes are chosen as the nearest standard size. In [47], these authors, along with Morana, consider on-line control of switched capacitors in addition to the design problem. Salama and Chikhani [48] attempt to formulate the problem in a simple manner, without the use of a sophisticatedoptimization technique. Laterals are handled by first treating each lateral as a separate feeder. The shunt capacitor location and size is then determined to reduce peak power and energy losses. If the savings for the lateral is zero or negative, no capacitor is placed on that lateral. After determining whether capacitors should be placed on each lateral, the optimum size and location for all the capacitors is determined using the methods presented in [3 11. Lehtonen [49] follows a procedure similar to [18], where the problem is broken into smaller subproblems. Given the number of capacitors to be installed, initial locations are chosen. The first subproblem determines the sizes and switching times using an iterative process. The second subproblem attempts to relocate the capacitors one by one without changing the control times. The two subproblems are repeated successively until convergence is reached. Ajjarapu and Albanna [50], Boone and Chiang [51], and Sundhararajan and Pahwa [52], propose the use of a genetic algorithm for placement and sizing of fixed capacitors. Bengiamin, Swain, and Holcomb [53-551 focus on extending [16] to include laterals. The laterals are considered in a process referred to as “zoning” in which the feeder is separated into sections or “zones” based on the position of load transfer switches. Compensation of each zone is accomplished by treating the zone as a separate feeder. Compensation on the main feeder is then performed while accounting for the loads and any capacitors placed on the laterals. Roytelman, Wee, and Lugtu [56] focus on real-time control of capacitors and voltage regulators. The algorithm is based on the “oriented discrete coordinate descent method’. Emphasis is placed on optimization in a small number of steps, resulting in convergence to a local optimal solution. This paper suggests that decoupling the volt/VAR problems is undesirable. The power-flow problem is repeatedly solved within the algorithm. The effect of voltage on customer demand is also included in the objective function. Chen, Hsu, and Yan [57] formulate the problem as a nonlinear programming problem and solve using the MINOS optimization package. The formulation includes the effectsof system unbalance.

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Yang, Huang, and Huang [58] use a Tabu Search in which the search is biased toward solutions with a better objective function, while special features of the algorithm prevent the solution from being trapped at a local optimum solution. Jiang and Baldick [59] present an algorithm which combines the control of capacitors with switch reconfiguration to minimize losses. Simulated annealing is used to optimize the switch configuration while a discrete optimization algorithm is used to determine the optimal capacitor control. Shao, Rao, and Zhang in [60] propose an expert system solution approach that is based on a heuristic graph search method using an evaluation function. It uses the power loss sensitivity vector to guide the search procedure. In [61], Santoso and Tan, proposed a twostage Artificial Neural Network (ANN) to solve the problem of real-time control of multitap capacitors considering a non-conforming load profile. Gu and Rizy [62], proposed an ANN to control shunt capacitors and voltage regulators. Ng and Salama [63], have proposed a solution approach to the capacitor placement problem based on fuzzy sets theory. Using this approach, the authors attempted to account for uncertainty in the parameters of the problem. They model these parameters by possibility distribution functions. Chin [MI, uses a fuzzy dynamic programming model to express real power loss, voltage deviation, and harmonic distortion in fuzzy set notation. Bortignon and El-Hawary [65] have also presented a review of capacitor placement techniques.

Conclusion The capacitor placement problem is quite complex. Researchers have used a wide variety of methods in an attempt to solve the problem. This paper presents an overview of these methods. It is clear from the existing literature that several issues associated with both the design and control problems need to be addressed.

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