Optimal placement of capacitor in distribution networks according to the proposed method based on gradient search

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1 Applied mathematics in Engineering, Management and Technology 2 (6) 2014: Optimal placement of capacitor in distribution networks according to the proposed method based on gradient search Gholamreza Sarlak*Reza Dashti**Amirhossen Etemadi*** *Azad Islamic University of Jasb, Engineering Department **Iran University of Science and Technology ***University of Tehran Abstract: Keywords: capacitor placement, losses, optimization, gradient Proper installation of capacitors in a distribution network reduces the energy and power loss, increases the lines' capacity, and enhances the voltage profile. Optimal placement of capacitor includes determination of location, quantity, and timing of the parallel capacitor switching on distribution feeders in order to maximize financial savings. In this paper, using the idea of search gradient approach, a method for network's critical nodes' determination is offered so that reduces the problem dimension (therefore, this method is proper for large-scale distribution systems). In the next section, after a comprehensive examination of all capacitor installation types, the optimal approach has been found. The advantage of the proposed approach is to use standard capacitance values with respect to the real price. The excellence of the proposed approach is proved after assessment of the three existent feeders in published articles. In the last part, the algorithm is performed on an actual feeder and desired results have been achieved. 1- Introduction Studies have shown that about 13 percent of total electrical energy produced in form of RI 2 power, perishes in distribution systems [1]. Reactive component of the current is responsible of a part of this loss which could be reduced by proper installation of parallel capacitors. In general, advantages of parallel capacitors' installation in distribution networks with lagging power factor could be summarized as follows [2]: 1- Reduction of reactive component of circuit current 2- Increase of voltage level in load location 3- Improvement of voltage regulation in case of proper switching 4- Reduction of RI 2 losses in system due to current reduction 5- Reduction of XI2 losses in system due to current reduction 6- Increase of network power factor 7- Reduction of loading on network equipments including generators, lines, transformers, and consequently reduction of equipments' overloading conditions and capacity release in order to confrontation with load growth 8- More loading capability on generators with respect to real power as a result of produced reactive power reduction 9- Reduction of KVA demand where the apparent power is sold 10- Reduction or delay of investment on system development Generally, the solution of optimal placement of capacitor is to find out the number, location, size, and optimum timing of switching for parallel capacitors in order to achieve the maximum financial saving with respect to the system restrictions. This problem is a combination of a variety of parameters which make it difficult to solve and raise the dimensions. At the same time, with the enlargement of the distribution system dimensions, the problem's complexity increases exponentially. On the other hand, since the capacitors are available only in standard sizes and their prices do not linearly vary; and due to the fact that system has various charges levels and is unpredictable with changes, objective function of the problem which is the same financial saving, is a non-differentiable function; therefore its optimization is not possible with normal analysis methods and its optimization entails application of more advanced algorithms. The proposed techniques resolving this problem could be divided into the following 4 groups [3]: 1- Analytical methods 2- Numerical programming methods 570

2 3- Creative methods 4- Methods based on artificial intelligence Selection of the proper solution could be based on the following 4 parameters: 1- Dimensions and requirements of the problem 2- Complexity of the problem 3- Necessary accuracy for report of final results 4- Feasibility of the proposed method Efforts to resolve the problem of optimal placement of capacitor initiated since 1960s, which were more based on conjectures, engineering judgments, and simplifying assumptions making the problem unrealistic. These struggles continued to make the problem more realistic until 1980s; but due to the lack of access to digital computers of proper memory and high computational speed, most of the proposed methods were based on analytical methods and mathematical calculations. Using analytical approach in [4] and [5], a concept was introduced as normalized feeder; and differentiable equations were acquired in order to optimize the compensation for reactive systems and maximize the achieved financial revenues. In [6] and [7] the problem is divided in to two parts. In the first part, a programming problem is solved in order to obtain the number and location of capacitors, then in the next part of the problem is introduced and solved in order to attain the type and size of them. In 1990s, with the development of digital computers' speed and efficiency, the aforementioned problem was studied with artificial intelligence approach. These approaches could be divided into the following 9 groups: 1- Genetic algorithm [8] 2- Expert systems [9] 3- Steel- plating [10] 4- Nervous system [11] 5- Fuzzy logic [12] 6- Graph search [13] 7- Tabular search [25] 8- Optimization inspired by the human immune system [14] 9- Ant colony system [19] Furthermore, creative methods which were based on experience, intuitive rules, conjectures, and engineering judgments that obviously could not find the main optimum point, were used. Using creative methods in [15], first the critical nodes of distribution network having the most impact on loss reduction were determined. Then by maximizing the economical gain due to energy loss and peak power reduction, optimum capacitance was obtained. By the way, various charge levels are considered in the above study. These efforts are still going to continue in the early years of the third millennium. In [16], a combination of artificial intelligence method and creative method are used to solve the problem. In this study, non-linear charges are also considered. In [17] a comparison between fuzzy and creative methods that were used theretofore is performed; and efficiency of mentioned methods on five sample and standard networks is examined and the results were compared. In [18], the problem is solved by using genetic algorithm so that the problem is divided into two internal and external parts. In external part, efforts are done to minimize the energy loss and maximize the achieved gain. In internal part, it is attempted to reduce the peak power and it loss. In [21], by using a method based on ant colony system, called Hybrid Differential Evolution (HDE) method, considering an appropriate mutation operator, the problem has been improved, so that it moves more rapidly towards the optimal point. In this paper, we have tried to provide a simple and practical method with inspiration of gradient search. The mentioned method determines the number, amount, and optimal installation location of parallel capacitors in distribution network so as to maximize the saving function. The offered method has the following advantages: 1- There is no need to accurately present cost function in terms of network parameters and capacitor placement variables. 2- Obtained values for capacitors are completely standard and real costs are applied. 3- The maximum number and amount of capacitor bank switch are installable in network are determinable if necessary. 4- There is no need for additional and non-realistic assumptions for value and cost of continuous capacitor or feeder without lateral branches. In this method, in order to reduce the dimension and volume of calculations, the creative approach is initially applied so as to find the critical points in network. Critical points are nodes that in case of installation of capacitors on them, the maximum loss reduction occurs and consequently maximum saving is achieved. 571

3 2. Problem formulation Applied mathematics in Engineering, Management and Technology 2014 Objective function could be presented as follows: max S K p LP K e LE Nc i1 K Q c ci In which: S: Savings in Rial LP: Reduction of peak power losses LE: Reduction of energy losses K p : Conversion factor for peak power losses in Rial K e : Conversion factor for energy losses in Rial K c : Conversion factor for peak power losses in Rial N c : Number of capacitor banks Conversion factors could be determined based on market information. The following assumptions are considered for the problem solution: 1- All loads are considered as three balanced phases. 2- Only one level of load is considered. 3- Only fixed capacitors are used. For different load levels the algorithm can be implemented proportionally, then by using a combination of fixed and switching capacitor, an appropriate mode can be obtained. In case of capacitor placement, there is no need to distribute the load. Forward-Backward sweep method is a very appropriate method for radial networks which is applied in this study using [20]. In this method, reference voltage bus is kept in a fixed value; then the following equations can be solved recursively and iteratively, and this will continue until the difference between current of two consecutive phases are much less than a certain value. I k i Pi jq i k V i k1 k V [ Z] B * (1) (2) In which: I i k : Bus current of node i, at k phase Q i and P i : Active and reactive power at bus i V i k :Bus voltage of node i, at k phase B k : Current matrix of network branches at k phase Z: A matrix including impedances of system branches V k+1 : Difference between system bus voltage with reference bus at "k+1" stage In this study, it is assumed that the size of installed capacitors on each bus is either less than or equal to sum of its own reactive load with all subsequent branch buses. If we consider the size of standard capacitors as integer multiples of the fundamental values of Q 0, then we can express the above sentence as follows: Q c i Q j i Lj L jq 0 (3) In which: α i : A set including i th bus and subsequent branch buses up to the end of network. Q Lj : Reactive load for j th bus Q i c : Value of capacitor placed on i th bus L i : An integer factor assigned to each bus to determine the maximum value. For instance, consider the network in figure 1. If we want to place the capacitor on bus 3, then we have: 572

4 Q 3 c 3 Q 3,4,5, 6 3 Q 4 Q 5 Applied mathematics in Engineering, Management and Technology 2014 Q c L3 Q3 / Q0 In which [ ] is the symbol of integer component. 6 Figure 1: 6 bus system The first solution for capacitor placement problem that comes to mind is to examine all possible states of capacitor's installation and select the state with the most financial revenue. For example, first we set the value of Q 0 for i th bus and after the load distribution, we calculate S from equation 1 and gradually increase the value of capacitor until we reach L i Q 0, at each phase we calculate the achieved gain. We iterate this operation for all buses in one-bus state and then we continue with two-bus and three-bus modes, and finally we calculate for all modes of buses. The corresponding location and value with the most saving are the answers of the problem. If we want to use the above method, the required number of iteration for each assumed network is (n is the number of network buses): n (4) ( 1) i2 L i If we consider the above equation for the following three standard networks assuming Q 0 = 25 kvar, number of iteration is shown in table 1: Table 1: number of iterative calculations for three sample feeders 10 bus [21] 6.3* bus [23] 2.3* bus [22] 2.8*10 60 According to the above values, it is observed that this method is practical by no means. In order to make it practical and feasible, it is necessary to limit the number of candidate buses for capacitor placement. For this purpose, using the idea of gradient search approach, with a creative but totally numerical method, first the candidate buses for capacitor placement are identified, and then the aforementioned method is implemented on them. 3. Gradient search method Two following conditions are usually fulfilled in the optimization problems [24]: i) Objective function has a smooth procedure which guarantees the existence of partial derivatives. ii) The resultant equations of objective function can be analytically solved. But in practice, these two conditions are not usually fulfilled. Therefore, it is possible to apply numerical and iterative methods to achieve the optimum point. Imagine a mountaineer aims to reach the foggy summit of a mountain. Since he cannot see the summit of the mountain, he tries to climb an uphill with a greater slope to reach the summit. Gradient search approach issues from on this fact. If C is an objective function dependent on variables from x 1 to x n (C=f (x 1,,x n )), then partial differential vector dx with n variables can be defined as follows: 573

5 dx dx1 dx2.. dx n (5) And gradient vector is defined as: C x 1 C.. C x n (6) Then the total differential of objective function is: dc C T dx (7) Since the inner product of two vectors is maximized when they are two parallel, we conclude that if the changes are in the direction of the gradient vector, we will have the most changes in the objective function. So we can offer the following algorithm for optimization of gradient search: 1) Select a desired starting point like X 0. 2) Calculate the value of gradient at that point and take a differential step in that direction until you reach the next point. 3) Repeat the previous step up to the point that an appropriate benchmark assures that you are sufficiently close to the optimal point. 4. Proposed method Inspired by the gradient search approach, a creative method based on computer numerical calculations and distribution of various loads can be presented for solving the optimal placement of capacitance. First we want to find the sensitive buses according to their sensitivity. Critical node is a node on which installation of a certain value of capacitor result in maximum loss reduction of system and consequently the most financial profit is gained. Sensitivity factor could be defined as follows: P S Q Loss ci (8) In which S is described as sensitivity factor based on which the system buses are classified and should be calculated individually for each bus. Each bus having the most sensitivity is introduced as the first candidate of capacitor placement. Since the accurate function of P Loss from Q c is not clear, in this study we try to find the system critical buses by using numerical methods. We act in the following order: First step: Set the minimum value of standard capacitor on i th bus. Second step: After load distribution, calculate financial saving according to equation 1 and save this value. Third step: Reiterate the previous step until reaching the maximum value distinguished in equation 2 (L i Q 0 ). 574

6 Fourth step: Reiterate the previous steps for all of the network buses. Once comparing the results, find location and value of the capacitor corresponding with the maximum saving. Compensate the network with the given value in the corresponding location. Therefore, one critical node has been obtained. Fifth step: Reiterate the steps from one to four in order to obtain the next critical nodes. Repeat this task up to the point that either you reach the maximum number of capacitor banks indicated by user or the additional gain in new step becomes negative. In this way, candidate buses for capacitor placement are determined according to their sensitivity. However, are these calculated values the same desired values? We are not sure. Since capacitor values are calculated individually, it is possible that we are trapped in a local optimum point and we have lost the main optimum point. Thus, capacitor placement on sensitive buses must be performed simultaneously. In this section, the problem dimensions are extremely limited due to selection of sensitive buses. Hence in the last step, we examine all feasible compounds and all possible combinations of capacitor placement on candidate buses in order to select the best model. This approach works in this way: first we form spiral loops proportional to the number of sensitive buses, and in each loop we increase the capacitor value of each bus from zero to the maximum value indicated in equation 2; in each step with load distribution and by using equation 1, we save the gained financial benefit with corresponding values of capacitors. In the end, attaining the maximum value of financial saving, corresponding values of capacitors are calculated. Thus with a simple and practical method, without a need for calculations and application of complex algorithms we will obtain the desired response. The aforementioned steps could be presented in flowchart as follows (n is the number of network buses, Q imax is the maximum allowable value of capacitor for i th bus, Q min is the minimum standard capacitor, S max is the maximum financial benefit in each step, and N c is the maximum number of specified capacitor banks). 5. Implementation of proposed approach Due to simplicity of handling with arrays and complex numbers, load distribution program are written in MATLAB TM software based on the mentioned method in section 2,the results are compared with the commercial program of DIGSILENT TM and their validity are verified. Meanwhile, capacitor placement algorithm is also written in MATLAB TM software and is run on a Pentium IV computer. In order to prove the efficiency of proposed algorithm, we test it on four sample feeders. Among authentic papers provided about capacitor displacement, the first three networks are selected and the results of these papers are compared with the results of proposed algorithm. The fourth network is a real network which is thoroughly examined. First experimental feeder An experimental feeder includes 9 buses with a nominal voltage of 23 K volt. Single-line diagram, load information and branches of this feeder are all presented in Figure 2 and table 2. Figure 2: 9- bus feeder Total load of this feeder is j4186 kva. After load distribution, the following information is achieved. The minimum network voltage with the value of pu belongs to bus number 9; and total loss without compensation would be kW. Denoted coefficients in equation 1 are as follows: K e = 0 and K p = 168$/kW; and capacitor cost could be calculated from the following relation: C C $ = Q C kvar In which C C is the capacitor cost and Q C is its value in terms of kvar. The results of two papers are compared with proposed algorithm in Table 3. It is noted that the most financial benefits gained as a result of proposed method. 575

7 Table 2: characteristics of a 9-bus feeder Bus Load Feeder R+jX No. KW KVAr Section (Ω/Km) j j j j j j j j j Table 3: evaluation of results for capacitor placement on 9-bus feeder Method Candidate buses Value of capacitor Loss reduction Financial benefit (kw) ($) Fuzzy [28] 3,4,5,9 3300,2100,1650, Fuzzy [29] 2,3,5,9 3900,3300,2100, Proposed 4,5,8,9,7 2850,1050,300,150, Figure 3: loss changes' trend and financial benefit in terms of capacitor number for a three-bus feeder 9 It is noted that the more the number of buses including capacitor are, the slower the system improvement trend becomes and obviously after a certain point, the capacitors are not only beneficial but only they cause financial losses and deficiencies. This feeder consists of 34 buses with 11 kw nominal voltages, whose single-line diagram is shown in figure 5;in order to see its characteristics please refer to [23]. Figure 4: linear diagram for a 34-bus feeder 576

8 The total load of this feeder is j kva. After load distribution we achieve the following information. The minimum network voltage belongs to bus 27 and its value is pu. The net loss before compensation is kw. The indicated coefficients in equation 1 for capacitor placement are as follows: K p = 150$/kW, K e = 0.07$/kWh, and K c = 6$/kVAR. The presented results in three articles are compared with proposed algorithm in table 4: Table 4: A comparison between results of capacitor placement on a 34-bus feeder Method Candidate nodes and kvar values Loss reduction (kw) Financial benefit ($) traditional Expert 5,22,750, systems [24] Creative method [21] 20,21,11,26, ,250,400,750 Fuzzy 7,17, Method[25] 450,750,1500 Proposed method 32,9,24,20 100,500,500, It can be noticed that the proposed method operates rather better. 6.Third experimental feeder A 69-bus feeder with V voltages, whose single-line diagram is shown in figure 6; in order to see its characteristics please refer to [6]. Figure 5: Single-line diagram for a 69-bus feeder The total load of this feeder is j kva. After load distribution we achieve the following information: The minimum network voltage belongs to bus 65 and its value is pu. The net loss before compensation is 225 kw. The indicated coefficients in equation 1 for capacitor placement are as follows: K p = 0$/kW, K e = 0.06$/kWh, and capacitor costs are calculated so that K c = 3$/kVAR. First, we can merely use the 300kVAR capacitors or their integer multiples- this issue has been observed in the compared articles-and second, the installation cost of capacitor on each bus is considered 1000$. The presented results in three articles are compared with proposed algorithm in table 5. We expect an improved voltage profile after the installation of capacitor. Hence, we draw the voltage profile before and after the compensation. Voltage improvement is entirely obvious (figure 7). Figure 6: voltage profile for 9-bus feeder 577

9 Table 5: comparison between capacitor placement results on 69-bus feeder Method Candidate nodes and kvar values Loss reduction (kw) Financial benefit ($) Numerical programming 19,63,300, [6] Steel plating [30,31] 21,61,300, Proposed method 61,17,300, Again, it can be noticed that the proposed method operates better. 7.Fourth experimental feeder Ordib feeder This feeder belongs to power network of Pakdasht area which is a subsidiary of Distribution Company insouth East Tehran and is branched from63/20 kv Khatoon Abad post. A single-line diagram of this feeder can be observed in figure 8. Figure 7: single-line diagram for a 48-bus practical feeder Information of this feeder is gathered in table 7: It can be noted that the minimum voltage of this feeder is , which belongs to bus 15. The total loss of this feeder is kw. The indicated coefficients in equation 1 for capacitor placement stage are as follows: K p = Rls/kW, K e = 55Rls/kWh, and K c = 59600Rls/kVAR. Capacitor placement results in this feeder are as shown in table 6: 578

10 Table 6: Results of capacitor placement on a 48-bus feeder Bus number Capacitor value kvar Loss reduction (kw) Financial benefit (Rials) If we sketch the trend for loss reduction and financial benefit increase in terms of parallel capacitor values' increase, we will achieve the chart in figure 9: Figure 8: trend of loss reduction and financial benefit in terms of capacitor numbers for a 48-bus feeder We expect an improved voltage profile after the installation of capacitor. Hence, we draw the voltage profile before and after the compensation. Voltage improvement is entirely obvious (figure 10). Figure 9: voltage profile for 48-bus feeder 6. Conclusion In this study, a simple practical method is presented without a need for additional and non-realistic assumptions in order to resolve the optimal capacitor placement problem in distribution networks. For this purpose, with inspiration of gradient search optimization approach, a creative method is proposed. First critical nodes are determined in order to reduce the problem dimension. Critical nodes are determined based on the achieved net 579

11 financial benefit, unlike other methods which are based on system's loss reduction. The proposed method includes the following advantages: 1- Since only a limited number of nodes are candidate for capacitor placement, problem dimension reduces and makes the proposed method suitable for large systems. 2- In this method, standard sizes and real values of capacitor banks are regarded and consequently we achieve the optimum practical result, not the theoretic one. Due to the reiterative nature of the creative method, results of each step are could be updated and accordingly are amended. 3- Limit violations do not occur. 4- Number and maximum value of capacitor could be determined if necessary. Meanwhile, the capacitors which are already installed in network are also included in calculations. 5- After implementation of the proposed method in three standard feeders used in published literature and comparison of results, it is noted that the financial benefit issued from this method is the largest one. The aforementioned method was applied on a practical feeder and desired results were achieved. 7. Appendix The information of a 48-bus Ordib feeder in Tehran Regional Electricity is provided in table 7. Branch current (A) V (pu) Table 7: Information of a 48-bus Ordib feeder in Tehran Regional Electricity Features of receiver bus Q(kVAR) P(kW) X(ohm) R(ohm) Receiver bus Transmitter bus Branch number Reference [1] J. B. Bunch, R. D. Miller, and J. E. Wheeler, Distribution system integrated voltage and reactive power control, IEEE Trans. Power Apparatus and Systems, vol. 101, no. 2, pp , Feb [2] Westinghouse Electric Corporation: Electric Utility Engineering Reference Book-Distribution Systems, Vol. 3, East Pittsburgh, Pa., [3] H. N. Ng, M. M. A. Salarna, A.Y.Chikhani, " Classification of Capacitor Allocation Techniques ", IEEE Trans. on Power Delivery, Jan. 2000, Vol. U, No. 1, pp [4 ] S. H. Lee, J. J. Grainger, " Optimum Placement of Fixed and Switched Capacitors on Primary Distribution Feeders ", leee Trans. on Power Apparatus and Systems, Vol. PAS-100, No.1, Jan , pp [5] J. J. Grainger, S. H. Lee, '' Optimum Size and Location of Shunt Capacitors for reduction of Losses on Distribution Feeders ", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100, No.3, March1981, pp [6] M. E. Baran, F. F.Wu, " Optimal Capacitor Placement on a Radial Distribution System ", IEEE Trans. on Power Delivery, Vol. 4, NO. 1, 1989, pp [7] M. E. Baran, F. F.Wu, " Optimal Sizing of Capacitors Placed on a Radial Distribution System ", IEEE Trans. on Power Delivery, Vol. 4, No.1, 1989, pp [8] K. N. Miu, H. D. Chiang, and G. Darling, Capacitor placement, re-placement and control in large-scale distribution systems by a GA-based two-stage algorithm, IEEE Trans. Power Systems, vol. 12, no. 3, pp , Aug [9] J. R. P. R. Laframboise, G. Ferland, A. Y. Chikhani, and M. M. A Salama, An expert system for reactive power control of a distribution system, Part 2: System implementation, IEEE Trans. Power Systems, vol. 10, no. 3, pp , Aug [10] T. Ananthapadmanabha, A. D. Kulkarni, A. S. Gopala Rao, and K. Raghavendra Rao, Knowledge-based expert system for optimal reactive power control in distribution system, Electrical Power & Energy Systems, vol. 18, no. 1, pp , [11] N. I. Santoso and O. T. Tan, Neural-net based real-time control of capacitors installed on distribution systems, IEEE Trans. Power Delivery, vol. 5, no. 1, pp , Jan [12] H. N. Ng, M. M. A. Salama, and A. Y. Chikhani, Capacitor allocation by pproximate reasoning: Fuzzy capacitor placement, IEEE Trans. Power Systems, Sept. 1998, submitted for publication. [13] J. C. Carlisle and A. A. El-Keib, " A Graph Search Algorithm for Optimal Placement of Fixed and Switched Capacitors on Radial Distribution Systems ", IEEE Transactions on Power Delivery, Jan. 2000, Vol. 15, No. 1, pp

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