OPTIMAL CAPACITOR PLACEMENT USING FUZZY LOGIC

Transcription

1 CHAPTER - 5 OPTIMAL CAPACITOR PLACEMENT USING FUZZY LOGIC 5.1 INTRODUCTION The power supplied from electrical distribution system is composed of both active and reactive components. Overhead lines, transformers and loads consume the reactive power. So voltage/var control is an essential measure to reduce the power losses through the switching operations of capacitors and load tap changing transformers. Reactive power compensation plays an important role in the planning of an electrical system. A proper control of the reactive power will improve the voltage profile, reduces the system losses and improves the system efficiency. Proper generation and control of reactive power is important for maintaining the network voltages under normal and abnormal conditions and to reduce system losses. The system voltage collapse due to lack of global control of reactive power flow during crucial contingencies is emerging as a serious problem. The aim is principally to provide an appropriate placement and sizing of the compensation devices to ensure a satisfactory voltage profile while minimizing the cost of compensation. Mainly capacitors are used to develop reactive power near the point of consumption. Series and shunt capacitors in a power system generate reactive power to improve power factor and voltage, thereby enhancing the system capacity and reducing the losses. Due to various limitations in the use of series capacitors, shunt capacitors are widely used in distribution systems. The general capacitor placement problem is formulated as an optimization problem to determine the location of capacitors, the types and size of capacitors to be installed and the control scheme for the capacitors at the buses of radial distribution networks. Several methods of loss reduction by placing capacitors in distribution systems have been reported over the years. The early approaches to this problem include those using

2 analytical methods, heuristic methods, artificial intelligence methods and those using dynamic programming technique to include the discrete nature of the capacitor size. Baran and Wu [18] have proposed an analytical method based on mixed integer programming to find the optimal size of capacitor to reduce the losses of a radial distribution system. Haque [51] has proposed a method of minimizing the loss associated with the reactive component of branch currents by placing capacitors at proper locations. The method first finds the location of the capacitor in a sequential manner. Once the capacitor locations are determined, the optimal capacitor size at each selected location is determined by optimizing the loss saving equation. More recently, the use of various non-deterministic methods like tabu search, genetic algorithms, fuzzy expert system and simulated annealing to determine the location and size of capacitor to improve the voltage profile of the system have been reported. Mekhamer et al. [73] have proposed a method to select an optimal location of capacitors using fuzzy logic and its allocation by analytical method. Ng et al. [40] presents a methodology to convert the analytical method stated in Salama and Chikani [28] from crisp solution into fuzzy solution by modeling the parameters using possibility distribution function, thus accounting for the uncertainties in these parameters. Prasad et al. [102] have presented a genetic approach to determine optimal size of capacitor. The optimal location to install capacitor is determined by taking values of Power Loss Indices randomly. Power loss indices are calculated by compensating the self-reactive power at each bus and run the load flows to determine the total active power losses in each case. Most of the previous studies [30, 31, 45, 57, 64, 124] have presented a method to find the location and size of capacitors using heuristic, genetic, simulated annealing techniques. In this chapter, a method is proposed to determine the optimal location of capacitors using fuzzy expert system by considering power loss indices and voltage at each bus

3 simultaneously and size of the capacitor by an index based method to obtain good results without violating the voltage constraints. This method has the versatility of being applied to the large distribution systems and having any uncertain data. The proposed method is tested with different radial distribution systems. The mathematical formulation of the proposed method is explained in Section 5.2. In this Section, the objective function and its constraints are defined. The identification of sensitive bus for capacitor placement using fuzzy logic is described in Section 5.3. Also this Section explains the calculation of Power Loss Index and implementation aspects of Fuzzy Expert System (FES) to identify sensitive bus to place capacitor. The effectiveness of the proposed FES is tested with one example and the results are presented in Section 5.4. The size of the capacitor using Index based method is explained in Section 5.5. In Section 5.6, algorithm to be followed to obtain optimal location and size of capacitor are presented. The effectiveness of the proposed method is tested with different examples of distribution system and the results obtained are compared with the results of existing methods. In Section 5.7, conclusions of the proposed method are presented. 5.2 MATHEMATICAL FORMULATION The objective function is to maximize the net savings function (F) by placing the proper size of capacitors at suitable locations is formulated as: Max. F e lr K P 8760 k N K Q (5.1) I c c c where F P lr = net savings (`.) = Reduction in power losses due to installation of capacitor = (Power loss before installation of capacitor - Power loss after installation of capacitor)

4 K e = Cost of energy in `./kwh = Installation cost in `. = Total number of capacitor buses Q C K C λ = total size of capacitor = Capital cost of each capacitor = rate of annual depreciation and interest charges of capacitor Constraints The objective function is subjected to the following constraints The voltage at each bus should lie within the voltage limits. V min. V i V max. i=1,2,..no. of buses The size of the capacitor to be installed at suitable bus is less than the total reactive load of the system. Q nbus c Q i i 1 where nbus= total number of buses 5.3 IDENTIFICATION OF SENSITIVE BUS FOR CAPACITOR PLACEMENT USING FUZZY LOGIC The fuzzy logic is used to identify the optimal location to place the capacitor in a radial distribution system so as to minimize the losses while keeping the voltage at buses within the limit and also by taking the cost of the capacitors in to account. The Fuzzy Expert System (FES) contains a set of rules, which are developed from qualitative descriptions. In a FES, rules may be fired with some degree using fuzzy inference, where as in a conventional Expert System, a rule is either fired or not fired. For the capacitor placement problem, rules are defined to determine the suitability of a bus for capacitor placement. Such rules are expressed in the following general form: If premise (antecedent), THEN conclusion (Consequent)

5 For determining the suitability of a particular bus for capacitor placement at a particular bus, sets of multiple-antecedent fuzzy rules have been established. The inputs to the rules are the bus voltages in p.u., power loss indices, and the output consequent is the suitability of a bus for capacitor placement Procedure to calculate power loss index The Power Loss Index at i th bus, PLI (i) is the variable which is given to fuzzy expert system to identify suitable location for the capacitor. Step 1 : Read radial distribution system data Step 2 : Perform the load flows and calculate the base case active power loss Step 3 : By compensating the reactive power injections (Q c ) at each bus (except source bus)and run the load flows, and calculate the active power loss in each case. Step 4 : Calculate the power loss reduction and power loss indices using the following equation where PLI i X i Y (5.2) Z Y X(i) Y Z = loss reduction at i th bus = minimum loss reduction = maximum loss reduction nbus = number of buses Step 5 : Stop Implementation aspects of Fuzzy expert system to identify the sensitive bus The power loss indices and bus voltages are used as the inputs to the fuzzy expert system, which determines the buses which are more suitable for capacitor installation. The

6 Degree of Membership Low High Degree of membership Low Lo-Norm Hi-Norm Norm High Degree of Membership Low High power loss indices range varies from 0 to 1, the voltage range varies from 0.9 to 1.1 and the output [Capacitor Suitability Index (CSI)] range varies from 0 to 1. These variables are described by five membership functions of high, high-medium/normal, medium/normal, lowmedium/normal and low. The membership functions of power loss indices and CSI are triangular in shape, the voltage is combination of triangular and trapezoidal membership functions. These are graphically shown in Figs. 5.1 to Low-Med Med Hi-Med Power Loss Index Fig. 5.1 Power loss index membership function Voltage (p.u.) Fig. 5.2 Voltage membership function Low-Med Med Hi-Med Capacitor Suitability Index

7 Fig. 5.3 Capacitor suitability index membership function For the capacitor placement problem, rules are defined to determine the suitability of a bus for capacitor installation. For determining the suitability for capacitor placement at a particular bus, a set of multiple antecedent fuzzy rules have been established. The rules are summarized in the fuzzy decision matrix in Table 5.1. The consequent of the rules are in the shaded part of the matrix. Table 5.1 Decision matrix for determining suitable capacitor locations Power Loss Index (PLI) Low- Med. Med. Low-Med. Low-Med. Low Low Med. High- Med. Med. Low-Med. Low Low And Voltage Low- High- Low Normal High Normal Normal Low Low-Med. Low-Med. Low Low Low High- Med. High-Med. High-Med. Med. Low-Med. Low High High High-Med. Med. Low-Med. Low-Med. After the FES receives inputs from the load flow program, several rules may fire with some degree of membership. The fuzzy inference methods such as Mamdani max-min and max-prod implication methods [34] are used to determine the aggregated output from a set of triggered rules. A final aggregated membership function is achieved by taking the union of all the truncated consequent membership functions of the fired rules. For the capacitor placement problem, resulting capacitor suitability index membership function, s, of bus i for m fired rules is s i max min i, i (5.3) m PLI v Where PLI and v are the membership functions of the power loss index and p.u. voltage level respectively.

8 Once the suitability membership function of a bus is calculated, it must be defuzzified in order to determine the buses suitability ranking. The centroid method of defuzzification is used; this finds the center of area of the membership function. Thus, the capacitor suitability index is determined by: s CSI z z s.zdz dz (5.4) Illustration of FES for a sample system The proposed method is explained with a sample system. Consider a 15 bus system whose single line diagram is shown in Fig The line and load data of this system is given in Appendix A (Table A.1). After performing the load flows for base case, the total active power loss and minimum voltage is given as kw and p.u. Considering one bus at a time, every bus is compensated with reactive power injection equivalent to that of self reactive load. Now perform the load flows to determine the active power loss, power loss index and the voltage in each case. These are given in Table 5.2. Table 5.2 Power Loss Index and voltage Bus No. Voltage Voltage PLI Bus No. (p.u.) (p.u.) PLI

9 The Capacitor Suitability Indices (CSI) of 15 bus system from FES is given in Table 5.3. The most suitable buses for capacitor placement are selected based on the maximum value of CSI of the system and they are 3, 4, 6, 11 and 15. Table 5.3 Capacitor suitability indices of 15 bus system Bus No. CSI Bus No. CSI PROCEDURE TO CALCULATE CAPACITOR SIZE USING INDEX BASED METHOD After knowing the optimal locations to place the capacitor, the size of the capacitor can be calculated by using index based method. Where Index i 1 V V k k total q effectiveload,i (5.5) i i I I p Q Q V i Ip[k], Iq[k] Q effectiveload,i Q total = Voltage at i th bus. = real and reactive component of current in k th branch. = total reactive load beyond i th bus (including Qload at i th bus) = total reactive load of the given distribution system capacitors ize i index i Qload i (5.6) where Qload[i] = local reactive load at i th bus 5.5 ALGORITHM FOR CAPACITOR PLACEMENT AND SIZING USING FES AND INDEX BASED METHOD

10 Step 1: Read the system input data a) Number of buses, number of branches, resistance and reactance of each branch, from bus and to bus of each branch, active and reactive power of each bus. b) Base kv, base kva, tolerance, etc. Step 2: Run load flow program and calculate the voltage at each bus and calculate the active power loss before compensation. Step 3: Run the load flow program by compensating the reactive load at each bus, considering one bus at a time, and calculate the loss reduction at each bus. Step 4: The power-loss reduction indices and the bus voltages are the inputs to the fuzzy expert system. Step 5: The outputs of FES, the capacitor suitability index, CSI are obtained from which the optimal location for the capacitor placement is selected by considering the maximum value of it. Step 6: The index vector is determined at selected buses using Eqn.(5.5). Step 7: Calculate the size of capacitor at selected buses by multiplying the reactive load at that bus with index vector at that bus (Eqn. (5.6)). Step 8: Then placing the calculated size of capacitors at best locations conduct a load flow study. Step 9: Print the results. Step 10: Stop

11 5.6 FLOW CHART FOR OPTIMAL CAPACITOR PLACEMENT USING FES Start Read Distribution System line and load data, base kv and kva, iteration count (IC) =1and tolerance (ε) = Perform load flows and calculate voltage at each bus, real and reactive power losses Calculate the loss reduction by running load flow by compensating the reactive load at each bus, considering one bus at a time Calculate power loss reduction indices, PLI using Eqn. (5.2) Obtain Capacitor Suitability Index (CSI) from the FES by

12 Fig. 5.4 Flow chart for optimal capacitor placement using FES 5.7 ILLUSTRATIVE EXAMPLES The proposed method is tested with four different radial distribution systems having of 15, 33, 34 and 69 buses Example 1 Consider a 15 bus system whose single line diagram is shown in Fig The line and load data of this system is given in Appendix A (Table A.1). The total real power loss and minimum bus voltage before compensation are kw and p.u.

13 The optimal locations and the size of the capacitors obtained by the proposed method are given in Table 5.4. In addition, voltage at these buses before and after compensation, loss reduction and net savings due to compensation are also given in the same table. The effect of using the nearest standard size capacitors instead of actual size of the capacitors is presented in Table 5.5 and it is observed that the changes in loss reduction and net savings are marginal. The active power loss reduction due to compensation is from kw to kw i.e., a reduction of 47.98% of the original active power loss. The voltage profile of the system before and after compensation is given in Table 5.6. The minimum voltage is improved from p.u. to p.u. The voltage regulation is improved from 5.55% to 3. 33%. The line flows of the system is given in Table 5.7. Table 5.4 Capacitor allocation and loss reduction of 15 bus RDS for calculated size of capacitor Without capacitor With capacitor Bus No. Q-Cap Voltage (p.u.) Voltage (p.u.) Total size of capacitor 1, Without capacitor With capacitor Improvement P loss (kw) Q loss P loss (kw) Q loss P loss (kw) Q loss Net Saving (`.) Without Capacitor With Capacitor , 79, 844/-

14 Table 5.5 Capacitor allocation and loss reduction of 15 bus system for standard size of capacitor Without capacitor With capacitor Bus No. Q-Cap Voltage (p.u.) Voltage (p.u.) Total size of capacitor 1,400 Without capacitor With capacitor Improvement P loss (kw) Q loss P loss (kw) Q loss P loss (kw) Q loss Net Saving (`.) Without Capacitor With Capacitor ,74, 695/- Table 5.6 Voltage profile before and after compensation of 15 bus RDS Before compensation After compensation Bus Voltage magnitude Voltage magnitude No. Angle (deg.) Angle (deg.) (p.u.) (p.u.)

15 Real Power Loss (kw) Bus No. Table 5.7 Line flows of 15 bus system Before compensation Active power loss (kw) Reactive power loss After compensation Active power loss (kw) Reactive power loss The variations of real power loss at each branch and voltage magnitude at each bus with and without compensation are shown in Figs. 5.5 and 5.6 respectively Losses after capacitor placement Losses before capacitor placement Branch number Fig. 5.5 Real power loss at each branch of 15 bus RDS with and without

16 Voltages (p.u.) capacitor Voltages after capacitor placement Voltages before capacitor placement Bus number Fig. 5.6 Voltages at each bus of 15 bus RDS with and without capacitor Example 2 Consider a 34 bus system whose single line diagram is shown in Fig The line and load data of this system is given in Appendix A (Table A.4).The total real power loss and minimum bus voltage before compensation are kw and p.u. The capacitor Suitability Index and capacitor sizes (nearest standard size of capacitors to the actual value) at the best suitable buses are given in Table 5.8. Fig. 5.7 Single line diagram of 34 bus radial distribution system Table 5.8 CSI and size of capacitor of 34 bus RDS Bus No. CSI Capacitor size

17 Total size of capacitor 1,500 The summary of results before and after compensation is given in Table 5.9. The comparison of results with existing methods is given in Table Table 5.9 Capacitor allocation and loss reduction for 34 bus system Without capacitor With capacitor Bus No. Q-Cap Voltage (p.u.) Voltage (p.u.) Total size of capacitor required 1,500 Without capacitor With capacitor Improvement P loss (kw) Q loss P loss (kw) Q loss P loss (kw) Q loss Description Real power losses (kw) Net saving (`.) Total size of capacitor required Net Saving (`.) Without Capacitor With Capacitor , 05, 926/- Min. Voltage (p.u.) Table 5.10 Comparison of results of 34 bus system with existing methods Existing method [45] Existing method [36] Proposed method Before compensation After compensation Before compensation After compensation Before compensation After compensation ,40,563/ ,65,200/ ,05,926/ From Table 5.9 it is observed that, the minimum voltage is improved from p.u. to p.u., total real power loss reduced from kw to kw (i.e., 29.45%) and total reactive power loss reduced from kvar to kvar (i.e., 38.76%) due to reactive power compensation. Thus, voltage regulation is improved from

18 Voltages (p.u.) Real Power Loss (kw) 5.83% to 4.91%. From Table 5.10, the size of the capacitor required is 1500 kvar and the net saving is `.16, 05, 926/- which is comparable with the existing methods. The variations of real power loss at each branch and voltages at each bus for with and without compensation are shown in Figs. 5.8 and 5.9 respectively Losses after capacitor placement Losses before capacitor placement branch number Fig. 5.8 Real power loss at each branch of 34 bus RDS with and without capacitor Voltages after capacitor placement Voltages before capacitor placement Bus number Fig. 5.9 Voltages at each bus of 34 bus RDS with and without capacitor Example 3 Consider a 33 bus system whose single line diagram is shown in Fig The line and load data of this system is given in Appendix - A (Table A.2). The CSI and size of capacitor

19 is given in Table The summary of results before and after compensation is given in Table From results it is observed that, the minimum voltage is improved from p.u. to p.u. The improvement in voltage regulation is 1.06%. Also, the total real power loss reduces from kw to kw (i.e., 28.36%) and reactive power loss reduces from kvar to kvar (i.e., 28.25%) after capacitor placement. The net saving is `.14,57,428/-. The variation of real power loss at each branch and voltages at each bus with and without capacitor are shown in Figs and 5.11 respectively. Table 5.11 CSI and size of capacitor for 33 bus system Bus No. CSI Capacitor size Table 5.12 Capacitor allocation and loss reduction for 33 bus system Description Without capacitor With capacitor Min. Voltage Voltage regulation (%) Total real power loss(kw) Total reactive power loss Improvement in real power loss (kw) Improvement in reactive power loss Total capacitor size at bus Net saving (`.) ,57,428/-

20 Voltages (p.u.) Real Power Losses (kw) Losses after capacitor placement Losses before capacitor placement Branch number Fig Real power loss at each branch of 33 bus RDS with and without capacitor Voltages after capacitor placement Voltages before capacitor placement Bus number Fig Voltages at each bus of 33 bus RDS with and without capacitor Example 4 Consider a 69 bus system whose single line diagram is shown in Fig The line and load data of this system is given in Appendix - A (Table A.3). The CSI and size of capacitor (nearest standard size of capacitor to the actual value) is given in Table The summary of results before and after compensation is given in Table From results it is observed that, the minimum voltage is improved from p.u. to p.u. The improvement in voltage regulation is 2.18%. Also, the total real power loss reduces from kw to

21 Real Power Loss (kw) kw (i.e., 32.40%) and reactive power loss reduces from kvar to kvar (i.e., 30.99%) after capacitor placement. The net saving is `. 20, 65, 298/-.The variation of real power loss at each branch and voltages at each bus with and without capacitor are shown in Figs and 5.13 respectively. Table 5.13 CSI and size of capacitor for 69 bus system Bus No. CSI Capacitor size Table 5.14 Capacitor allocation and loss reduction for 69 bus system Description Without capacitor With capacitor Min. Voltage Voltage regulation (%) Total real power loss(kw) Total reactive power loss Improvement in real power loss (kw) Improvement in reactive power loss Total capacitor size at bus Net saving (`.) ,65,298/ Losses after capacitor placement Losses before capacitor placement Branch number Fig Real power loss at each branch of 69 bus RDS with and without capacitor

22 Voltages (p.u.) Voltages after capacitor placement Voltages before capacitor placement Bus number Fig Voltages at each bus of 69 bus RDS with and without capacitor 5.8 CONCLUSIONS A method has been proposed to determine most sensitive buses to place capacitors using fuzzy logic and its size is calculated using index based method in radial distribution systems. The FES considers loss reduction and voltage profile improvement simultaneously while deciding which buses are the most ideal for placement of capacitor. Hence, a good compromise of loss reduction, voltage profile improvement and net saving is achieved when compared to existing methods. The proposed method has been tested on four distribution systems consisting of 15, 33, 34 and 69 buses. It has been noticed that losses are reduced and voltage profile is improved.