- Email: [email protected]
Direct search algorithm for capacitive compensation in radial distribution systems
Electrical Power and Energy Systems 42 (2012) 24–30
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Direct search algorithm for capacitive compensation in radial distribution systems M. Ramalinga Raju a, K.V.S. Ramachandra Murthy b,⇑, K. Ravindra a a b
Electrical and Electronics Engineering Department, University College of Engineering, JNTUK, Kakinada 533 001, Andhra Pradesh, India Electrical and Electronics Engineering Department, G. V. P. College of Engineering, Visakhapatnam 530 048, Andhra Pradesh, India
a r t i c l e
i n f o
Article history: Received 18 September 2011 Received in revised form 15 December 2011 Accepted 10 March 2012
Keywords: Capacitor placement Radial distribution systems Power flow Direct search algorithm
a b s t r a c t In this paper a new algorithm is proposed to determine the optimal sizes of fixed and switched capacitors together with their optimal locations in a radial distribution system so that net savings are maximized and improvement in the voltage profile is achieved. The algorithm searches for all possible locations in the system for a particular size of capacitor and places the capacitor at the bus which gives maximum reduction in active power loss. The optimal sizes are chosen to be standard sizes that are available in the market i.e., discrete sizes of capacitors are considered. The algorithm is tested on standard 69 bus system, 85 bus system and practical 22 bus systems. And the results are compared with results of other methods like particle swarm optimization and genetic algorithm, as available in the literature. The loss reduction obtained in this paper for the two standard test systems is highest compared to the other technique as reported in the literature. Cost analysis is also presented with and without capacitor placement on the three bus systems considering three different loading levels. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In India, all the 11KV rural distribution feeders are radial and too long. The voltages at the far end of many such feeders are very low with very poor voltage regulation. Capacitor banks are connected to radial distribution systems to improve the voltage profile and reduce the active power loss. The computational methods used in the analysis and design of distribution systems are not as robust as they are in transmission systems. In particular, the design of compensation systems for radial distribution system has become very complex because, the system does not fit into the usual optimization methods used in transmission system. Carpinelli et al. implemented non-linear programming technique for capacitor placement [1] on three phase unbalanced system. Wang et al. implemented integer programming technique [2], and Tabu search was used by Huang et al. [3] for optimal capacitor placement. Grainger implemented equal area criterion [4] and genetic algorithm applied to capacitor placement by Dlfanti [5] for determining optimal sizes of capacitors. Das applied FuzzyGA method for capacitor placement problem [6]. Sydulu and Reddy applied Index Vector to capacitor placement problem [7], Prakash and Sydulu applied particle swarm optimization for optimal capacitor placement problem [8]. Safigianni and Salis presented optimum VAr control of radial primary power distribution networks by shunt capacitor installation [9]. Das implemented genetic algorithm [10], Hsiao implemented Fuzzy-genetic algorithm for ⇑ Corresponding author. Tel.: +91 9966803153; fax: +91 891 2739605. E-mail address: [email protected] (K.V.S. Ramachandra Murthy). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.03.006
[11] for optimal capacitor placement problem. Huang applied immune multi objective algorithm for capacitor placement problem [12]. Kannana et al. applied Fuzzy-Differential Algorithm [13], Srinivasa Rao et al. applied plant growth algorithm for optimal capacitor placement problem [14]. The new algorithm proposed for capacitive compensation in this paper considered as direct search algorithms with a possible expert interaction yields optimal locations with suitable sizes of capacitors resulting in minimum active power loss and maximum net savings. 1.1. Objective function Objective function for optimization can be stated mathematically as given below. The first part of it is cost of energy loss and second part is the purchase cost of capacitor. The objective is to minimize the total cost, S:
Minimum S ¼ K e
ncap L X X T j Pj þ K c Q ci j¼1
ð1Þ
i¼1
where Ke is the energy cost per each kW h, Tj is the duration for which a jth load level operates. Three load levels are considered in this work. They are light, nominal and peak. Pj is the active power loss during jth load level. Qci is the size of the capacitor placed at ith bus. Different size capacitors would be suitable for different load levels at the optimal locations for minimizing the total cost function. Kc is the purchase cost of capacitor per kVAr. Number of candidate locations is indicated by ‘ncap’.
25
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
2. Load flow solution Traditional transmission system load flow methods Gauss-Siedel and Newton Raphson techniques, cannot be used for distribution systems as R/X ratio is high. Kresting and Mendive [15] have presented a load flow technique based on the ladder network theory. Teng and Lin proposed solution for meshed topology. Das and Kothari have proposed a load flow solution method by writing a algebraic equation for bus voltages [16]. Network topology based technique for three phase un-balanced systems is developed by Teng [17]. Shirmohammadi et al. [18] have presented a compensation based power flow for weakly meshed transmission and distribution systems. Baran and Wu [19] have obtained the load flow solution in a distribution system by iterative solution of three fundamental equations representing real power, reactive power and voltage magnitude. In this paper, a simple algebraic expression of voltage magnitude [20] is evaluated to obtain voltages and no trigonometric terms are used. Topology based approach is used for evaluating equivalent load at every node. This eliminates the complex process of identifying nodes connected beyond a particular node as described in [16]. The two developed matrices, ‘bus injection to branch current matrix’ and ‘line loss to node power matrix’ are very easy to form. The features of this method are robustness and computer economy. Convergence is always guaranteed. The assumption is that shunt capacitance is negligible at the distribution voltage level. The Fig 1 shows a two bus system where sending end voltage is known and receiving end voltage is to be obtained. P2 and Q2 are effective active and reactive powers at node 2 in Fig 1. The magnitude of voltage and loss calculation performed with the formulae presented by Das et al. [16]. The same equations are used along with the matrices developed from network topology method [17]. This model requires less memory and faster in execution. The load flow equations used are as follows:
V2 A½j ¼ P2 R½j þ Q 2 X½j 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B½j ¼ ½A½j2 ðP22 þ Q 22 ÞðR½j2 þ X½j2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ¼ ½B½j A½j " # P22 þ Q 22 Ploss ½j ¼ R½j V 22 " # P22 þ Q 22 Q loss ½j ¼ X½j V 22
ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
The set of equations can be written by applying Kirchhoff’s current law (KCL) to the distribution network. The KCL equations will equally hold good for powers also. For example a simple six bus system shown in Fig 2. Consider N2, N3, N4, N5 and N6 as the equivalent powers at each node.
N2 ¼ S3 þ S4 þ S5 þ S6 N3 ¼ S3 þ S5 N4 ¼ S4 þ S6
where S3, S4, S5 and S6 are complex load powers respectively at buses 3, 4, 5 and 6. Effective load at node [N] can be obtained by,
½N ¼ ½BIBC½S
For distribution systems, the models which are based on the effective load at every node is very convenient. At each bus ‘i’ the complex power S is specified by:
Si ¼ Pi þ jQ i
ð7Þ
ð9Þ
The constant BIBC matrix has entries of 1 and 0 only. For a distribution system with m-branch sections and n-load buses, the dimension of the BIBC is m n. The bus where there is no load need not enter into matrix [S].
3 N2 2 1 6 7 6 N3 7 6 6 7 61 6 N4 7 ¼ 6 6 7 40 6 7 4 N5 5 0 N6 2
3 2
3 S3 7 6 0 1 0 7 6 S4 7 7 76 7 1 0 1 5 4 S5 5 1 1 1
0 0
1
S6
The relation between the line losses and node power can be obtained by following equations.
N02 ¼ S11 þ S12 þ S13 þ S14 þ Sl5 similarly; N03 ¼ S14
ð11Þ
N04 ¼ S15 Complex power losses associated with lines 1, 2, 3, 4, 5 are given by Sl1, Sl2, Sl3, Sl4 and Sl5 respectively. Sum of the line losses appearing at nodes 2, 3, 4, 5 and 6 are given by N02 ; N 03 ; N04 ; N 05 ; N 06 . Total line loss supplied by node2 is N02 . End nodes will not have to supply any line loss component. So, they will have all zeros in their corresponding rows. Node 2 will have to supply all the line losses except branch 1 loss. SL is the column matrix containing all line losses.
½N0 ¼ ½LLNP½SL 3 2 0 N02 6 0 7 6 6 N3 7 6 0 6 0 7 6 6N 7 ¼ 60 6 47 6 6 0 7 6 4 N5 5 4 0
3 2
3 Sl1 7 6 7 0 7 6 Sl2 7 7 6 7 6 7 17 7 6 Sl3 7 7 6 7 0 5 4 Sl4 5 0 Sl5
ð12Þ
Effective load at each node ¼ N þ N0
ð13Þ
N06
1 1 1 1 0
0 1
0
0
0
0
0
0
0 0
0
0
Pi and Qi are the real power and imaginary powers at ith node.
Fig. 1. Two bus system.
ð10Þ
2.2. Line loss to node power matrix (LLNP)
2 2.1. Bus-injection to branch current matrix (BIBC)
ð8Þ
Fig. 2. Sample distribution system.
26
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
After calculating the effective load at each node, recalculate the receiving end voltages using the above Eqs. (2)–(4). And calculate power losses using Eqs. (5) and (6). The voltage values at all nodes change after adding losses. 2.3. Algorithm for the load flow solution 1. Read the system data, V1 = 1.0 pu. Line losses are assumed to be zero in the first iteration. 2. Build BIBC matrix and LLNP matrix. 3. Obtain P effectiv e þ j Q effectiv e , at each node using Eq. (9). N represents the part of load powers in the effective load at various nodes.
½N ¼ ½BIBC½S where, S is the column matrix of all loads. 4. Initialize iteration count = 1. 5. Obtain receiving end voltages using simple formulae mentioned above.
V2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½B½j A½j
6. Calculate power loss on all lines using the formulae:
" Ploss ½j ¼ R½j
Q loss ½j ¼ X½j
P22 þ Q 22
#
V 22 " # P22 þ Q 22 V 22
Sloss ¼ Ploss þ jQ loss 7. Multiply the power loss column matrix, SL with LLNP matrix to get N0 matrix. N0 represents the part of line losses in the effective load at various nodes.
½N 0 ¼ ½LLNP½SL 8. Calculate the total effective load at various nodes by adding the N and N0 matrix.
Effective load at each node ¼ N þ N0 9. Increment the iteration count repeat the steps from step (5) using new effective loads at every node. 10. If the difference in the voltages between present iteration and previous iteration is greater than 0.001 pu, then increment the iteration count and repeat from step (5), otherwise, print the result. Number of iterations taken in this method is 3. The algorithms is efficient, robust and faster. This requires less memory and convergence is faster. For the first time, topology is fully exploited with this load flow solution model. 3. The proposed direct search algorithm for capacitor placement The algorithm proposed is for radial distribution system with source bus as slack bus and all other load buses as PQ buses. The algorithm proposed is described in following steps for deciding the optimal sizes of the capacitors in terms of standard sizes available in the market and their locations (only load buses). The algorithm is proposed in the following steps:
2. All the load buses are fully compensated with all reactive powers set to zeros and load flow study is conducted and total line loss is determined. This is considered as minimum possible loss to be aimed at for determining optimal sizes and locations. 3. To determine the optimal sizes of capacitors, a number of options having group of various capacitor sizes are to be tried. A tolerance index is chosen i.e., modulus of difference between losses under any option and minimum loss should be a very small value. All possible options may be enlisted. 4. Let m(k) be the number of capacitors in the kth option, k ranging from 1 to n where ‘n’ is the total number of options. m(1), the first option is with single capacitor, the Q of which is nearest to the total KVAR placed at all load buses, in turn, and load flow study is conducted. The line losses are determined. If the lowest loss satisfies the tolerance criterion, the process can be terminated. The size and location are considered as the optimal solution. 5. In one set of capacitors m(k), the first capacitor is kept at all load buses in turn, and the location for which losses are the lowest is considered as the optimal location for that capacitor. Placing this capacitor at that load bus, the procedure is repeated for placing the second capacitor at all load buses in turn and deciding the optimal location for the second capacitor. This procedure is repeated for all capacitors. 6. The options m(2) to m(n) are sequenced taking more and more number of capacitors of smaller size such that the total compensation is nearest to the total KVAR of the system. System losses are found out for each combination and checked for tolerance. If the tolerance is acceptable, process can be terminated. In any practical system, the authors’ experience is that, all possible options need not be exhausted. Process termination will be obtained after a finite number of options with acceptable tolerance. The sizing and sequencing of capacitors depend on load pattern and can be adjudged by the expert analyst. Even after trying all possible options, if the tolerance is not acceptable then that option for which distribution loss is minimum, is considered as the optimal solution. In this paper two standard test systems are considered for analysis and demonstrating the above algorithm with 69 bus system and 85 bus system. Standard capacitor sizes available in the literature (in KVAR): 150, 300, 450, 600, 750, 900, 1050, 1200, 1350, 1500, 1650, 1800, 1950, 2100, 2250, 2400, 2550, 2700, 2850, 3000, 3150, 3300, 3450, 3600, 3750, 3900, 4050. 4. Results 4.1. Example 1: 69 bus system Total Reactive power load in the system is 2694 kVAr and active power loss without compensation is 225 kW. By compensating 100% on all load buses (i.e., at all load buses, Qload = 0) the loss obtained is 143.52 kW. This is the minimum possible loss that should be aimed at. Table 1 shows the capacitive compensation on 69 bus system. The Table 1 is organized in such a way that column 4 gives active power loss after placing the capacitor size which is given in column2 at optimal location given in column3. Minimum loss obtained by the proposed algorithm is 147.0 kW. Data for the standard 69 system data is available in [19]. There is a reduction of 34.66% in active power loss. 4.2. Example 2: 85 bus system
1. Base case load flow study is conducted and distribution line losses are determined. This uncompensated loss is considered to be maximum loss in the system.
Total reactive power load in the system is 2622.2 kVAr and active power loss without compensation is 316.11 kW. By compen-
27
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30 Table 1 Capacitive compensation on 69 bus system using direct search algorithm.
Table 5 Load level and load duration time.
S. no.
Q kVAr compensated
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3
900 450 450
61 15 60
159.42 151.92 147.00
Total
1800 kVAr
Load level
0.5 (Light)
1.0 (Normal)
1.6 (Peak)
Duration (h)
2000
5260
1500
4.3. Example 3: Practical 22 bus agricultural distribution system
Table 2 Capacitive compensation on 85 bus system using direct search algorithm. S. no.
Q kVAr compensation
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
450 300 150 150 150 150 150 150 150 150 150 150 150 150
36 69 30 80 66 61 14 26 57 8 20 6 18 17
247.24 212.31 188.52 178.05 168.91 161.80 155.96 151.68 148.68 146.83 145.30 144.62 144.17 144.01
Total
2550 kVAr
Data for the practical 22 bus system data is available in [A1]. 22 Bus system represents a small portion of agricultural distribution of Eastern Power Distribution system in India having base voltage as 11 kV. Total reactive power load in the system is 657.4 kVAr and active power loss without compensation is 17.69 kW. By compensating 100% on all load buses, (i.e., at all load buses, Qload = 0) the loss obtained is 9.14 kW. This is the minimum possible loss that should be aimed at. The Table 3 shows the best combination of capacitors with location and active power loss after placing the capacitors in turn. The Table 3 is organized in same way as that of Table 1. Minimum active power loss obtained by the proposed algorithm is 9.66 kW. There is a reduction of 45.39% in active power loss. 4.4. Comparison of results with other techniques
Table 3 Capacitive compensation on 22 bus system using direct search algorithm. S. no.
Q kVAr compensated
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3 4 Total
300 150 150 150 750 kVAr
13 16 4 17
12.21 10.22 9.95 9.66
It is found that distribution losses obtained by implementing the proposed algorithm is significantly less compared to other methods. Comparison of results with different techniques is given in Table 4 along with the location and size of the capacitor banks. The table contains active power loss obtained by GA, PSO and direct search algorithm. The minimum loss obtained in 69 bus system with direct search algorithm is 147 kW which is less when compared to 156.62 kW of active power loss obtained by implementing genetic algorithm [6] and 152.48 kW obtained by implementing particle swarm optimization [8]. The minimum loss obtained on 85 bus system with direct search algorithm is 144.01 kW which is less when compared to 146.061 kW of active power loss obtained by implementing genetic algorithm [7] and 163.32 kW obtained by implementing particle swarm optimization [8]. 4.5. Cost analysis
sating 100% on all load buses, (i.e., at all load buses, Qload = 0) the loss obtained is 141.02 kW. This is the minimum possible loss that should be aimed at. Table 2 shows the best combination of capacitors with location and active power loss after placing the capacitors in turn. Table 2 is organized in the same as that of Table 1. Minimum loss obtained by the proposed algorithm is 144.01 kW. Data for the standard 85 system data is available in [16]. There is a reduction of 54% in active power loss which is considerably large.
4.5.1. Cost analysis on 69 bus system Detailed cost analysis is presented for 69 bus system. Energy cost is assumed as US $0.06 per kW h and purchase cost of capacitor is assumed as US $3.0 per kVAr. Three load levels and load duration time data for the system is given in Table 5. Table 6 presents capacitor placement locations and sizes. Three load levels are considered. They are 160%, 100% and 50% load levels. Switched
Table 4 Comparison of active power loss without compensation and with compensation by different techniques. Standard bus system
Active power loss without compensation (kW)
Active power loss with capacitor placement by genetic algorithm (kW)
Active power loss with capacitor placement by PSO (kW)
Active power loss with capacitor placement by direct search algorithm (kW)
69 Bus system
225
156.62 61 64 59
[6] 700 800 100
152.48 46 47 50
[8] 241 365 1015
147.0 61 15 60
900 450 450
146.061 19 buses
[7] 2206.25 (Qtotal)
163.32 8 58 7 27
[8] 796 453 314 90
144.01 36 69 12 Buses
450 300 150 12 = 1800
85 Bus system
316.11
28
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
Table 6 Optimal capacitor placement location and size. Optimal location
61 15 60
Table 11 Comparison of results with and without losses on 85 bus system.
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
1800 900 900
900 450 450
450 300 300
Total cost Total losses cost Total capacitor cost
1800 900 900
Total annual savings = US $95,475 Load level 1.6
Table 7 Cost of energy loss and minimum system voltage before compensation. Load level
0.50 (Light)
1.0 (Nominal)
1.60 (Peak)
Energy loss cost Minimum voltage
$6192 0.95668
$70,997 0.90919
$58,716 0.84449
Maximum voltage = V1 = 1.0 pu.
Load level 1.0
V 54 min Losses
Load level 0.5
V 54 min Losses
Optimal location
Load level
0.50 (Light)
1.0 (Nominal)
1.60 (Peak)
Energy loss cost Minimum voltage
$4262.4 0.9683
$46,393.2 0.9318
$38,457 0.8936
Table 9 Comparison of results with and without losses on 69 bus test system.
Total cost Total losses cost Total capacitor cost
V 65 min Losses V 65 min
Load level 1.0
Losses Load level 0.5
With capacitors
$196,011 $196,011 0.0
$100,536 $86586 $13,950
0.7722
0.8770
975.93 0.8713
410.69 0.9224
316.11 0.9397
144.01 0.9629
70.11
34.76
Table 12 Optimal capacitor placement location and size on 22 bus system.
Table 8 Cost of energy loss and minimum system voltage after compensation.
Total annual savings = US $35,992.4 Load level 1.6
V 54 min Losses
Without capacitors
V 65 min Losses
13 16 4 17
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
450 300 150 150
300 150 150 150
150 150 0 0
450 300 150 150
Table 13 Comparison of results with and without losses on 22 bus practical distribution system.
Without capacitors
With capacitors
$135,905 $135,905 0.0
$99,912.6 $89,112.6 $10,800
0.84449
0.8936
652.40 0.90919
427.3 0.9318
Load level 1.0
224.96 0.95668
147.0 0.9683
V 22 min Losses
Load level 0.5
51.60
35.52
V 22 min Losses
Total cost Total losses cost Total capacitor cost Total annual savings = US $1576.41 Load level 1.6
V 22 min Losses
Without capacitors
With capacitors
$10,302 $10,302 0.0
$8725.59 $5575.59 $3150
0.9560
0.9701
46.68 0.9729
24.89 0.9824
17.69 0.9866
9.66 0.9909
4.32
2.39
Maximum voltage = V1 = 1.0 pu.
Table 10 Optimal capacitor placement location and size in 85 bus system. Optimal location
36 69 30 80 66 61 14 26 57 8 20 6 18 17
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
900 600 450 450 300 300 150 150 300 150 300 150 300 150
450 300 300 150 150 150 150 150 150 150 150 150 150 150
300 150 150 0 150 0 150 0 150 0 0 0 0 150
900 600 450 450 300 300 150 150 300 150 300 150 300 150
capacitors need to be installed at nodes 61, 15 and 60 nodes. At 61 bus, 450 fixed capacitor and 450 kVAr, 900 switched capacitors need to be installed. At 15th bus, 300 kVAr fixed capacitor, one
150kVAr, one 450kVAr switched capacitors need to be installed. At 60th bus, 300 kVAr fixed capacitor, one 150 kVAr, 450 kVAr switched capacitor need to be installed to cater to the needs of varying load levels. Table 7 presents cost of energy loss without placing capacitors, i.e., bare system at different load levels. Table 8 presents cost of energy loss with capacitors placed in optimal locations at different load levels. Table 9 shows comparison of results with and without considering capacitor placements. The total cost for the 69 bus system without any compensation is US $135,905. After compensation, the total cost is US $99,912.6. The annual saving is US $35,992.4 where as total annual savings is $30,878 in [6]. It is observed that after capacitor placement, minimum voltage level of the system has improved and the system losses are reduced in each load level. Only at the peak load level, voltage is observed to be 0.8936 pu which is below 0.9 pu. In all remaining cases, voltage is above 0.90 pu. Table 9 gives the comparison of cost with and without capacitor placement. 4.5.2. Cost analysis on 85 bus system Total cost of energy loss in 85 bus system is US $196,011 without compensation, where as it is US $86,586 after capacitor placement. The total purchase cost of fixed and switched capacitors is US $13,950. Capacitor requirement indicated in the 0.5 load level
29
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30 Table A1 Data of the 22 bus agricultural distribution system. Line no.
From bus
To bus
Resistance (X)
Reactance (X)
Real power load (kW)
Reactive power load (kVAr)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 2 2 4 4 5 6 6 9 9 11 11 13 14 14 16 17 17 19 20 20
2 3 4 5 9 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22
0.3664 0.0547 0.5416 0.193 0.7431 1.3110 0.0598 0.2905 0.0547 0.675 0.0547 0.3942 1.0460 0.022 0.0547 0.3212 0.0949 0.574 0.1292 0.0871 0.5329
0.1807 0.0282 0.2789 0.099 0.3827 0.6752 0.0308 0.1496 0.0282 0.3481 0.0282 0.203 0.5388 0.0116 0.0282 0.1654 0.0488 0.2959 0.066 0.045 0.2744
16.78 16.78 33.80 14.56 19.31 10.49 8.821 14.35 14.35 16.27 16.27 82.13 34.71 34.71 80.31 49.62 49.62 43.77 37.32 37.32 31.02
20.91 20.91 37.32 12.52 25.87 14.21 11.66 18.59 18.59 19.48 19.48 71.65 30.12 30.12 70.12 47.82 47.82 38.93 35.96 35.96 29.36
is fixed capacitor. Table 10 presents capacitor placement locations and sizes for different loading levels. Table 11 gives the comparison of cost with and without capacitor placement on 85 bus system. Total annual savings are found to be US $95,475. The reason for having large savings is loss reduction at different loading levels are 57.9%, 54.43% and 50.42% respectively at 1.6, 1 and 0.5 load levels of the nominal level. This is much higher than 69 bus system loss reduction. 4.5.3. Cost analysis on 22 bus system Fixed capacitors of 150 kVAr size are required at bus nos. 13 and 16 one at each bus. Total cost of energy loss in 22 bus system is US $10,302 per year without compensation, where as it is US $5575.59 per year after capacitor placement. The total purchase cost of capacitors is US $3150. Total annual savings are found to be US $1576 for the agricultural distribution system. Sizes of capacitors for various load levels is presented in Table 12 and cost analysis presented in Table 13. Minimum voltages are also indicated at each loading level. 5. Conclusions The new algorithm proposed for designing the compensation in distribution system, resulted in improving the voltage profile and maximizing the net savings. Cost analysis is demonstrated on 69 bus radial distribution system. This algorithm can be classified under direct search algorithm as the approach is methodic and sequential. The process generally terminates fast and faster if implemented by an expert. Standard 69 bus and 85 bus systems have been considered for which known results are there with implementation of PSO and genetic algorithm. It has been clearly observed for the given loading conditions, the total distribution loss is significantly less than the one obtained in the other two methods. The algorithm is also tested on 22 bus practical agricultural distribution system. However, the future work can be carried out considering the installation and maintenance costs. Acknowledgements The authors would like to extend their whole hearted gratefulness to Dr. Govinda Rao Gade, Senior Professor at G.V.P. College of
Engineering, Visakhapatnam, India for providing language help, assisting in writing and for proof reading.
Appendix A See Table A1.
References [1] Carpinelli G, Varilone P, Di Vito V, Abur A. Capacitor placement in three-phase distribution systems with nonlinear and unbalanced loads. IEE Proc Generat Transm Distribut 2005;152(1):47–52. [2] Wang J, Chiang H, Nan minu K, Darling G. Capacitor placement and real time control in large-scale un-balanced distribution systems: Part I and II. IEEE Trans Power Deliv 1997;12:953–64. [3] Huang YC, Yang HT, Huang CL. Solving the capacitor placement problem in a radial distribution system using tabu search application. IEEE Trans Power Syst 1996;11:1868–73. [4] Grainger JJ, Lee SH. Optimum size and location of shunt capacitors for reduction of losses on distribution feeders. IEEE Trans Power App Syst 1981;100:1105–18. [5] Dlfanti M, Granelli GP, Maranninio P. Optimal capacitor placement using deterministic and genetic algorithm. IEEE Trans Power Syst 2000;15: 1041–6. [6] Das D. Optimal placement of capacitors in radial distribution system using Fuzzy-GA method. J Electr Power Energy Syst 2008;30: 361–7. [7] Sydulu M, Reddy VVK. Index and GA based optimal location and sizing of distribution system capacitors. In: Power engineering society general meeting, Tampa, FL; 2007. p. 1–4. [8] Prakash K, Sydulu M. Particle swarm optimization based capacitor placement on radial distribution systems. Power Eng Soc Gen Meet 2007: 1–5. [9] Safigianni AS, Salis GJ. Optimum VAr control of radial primary power distribution networks by shunt capacitor installation. Electr Power Energy Syst 2001;23:389–401. [10] Das D. Reactive power compensation for radial distribution networks using genetic algorithm. Electr Power Energy Syst 2002;24:573–81. [11] Hsiao Ying-Tung, Chen Chia-Hong, Chien Cheng-Chih. Optimal capacitor placement in distribution systems using a combination Fuzzy-GA method. Electr Power Energy Syst 2004;26(7):501–8. [12] Huang Tsong-Liang, Hsiao Ying-Tung, Chang Chih-Han, Jiang Joe-Air. Optimal placement of capacitors in distribution systems using an immune multiobjective algorithm. Electr Power Energy Syst 2008;30:184–92. [13] Kannana SM, Renugab P, Kalyania S, Muthukumarana E. Optimal capacitor placement and sizing using Fuzzy-DE and Fuzzy-MAPSO methods. Appl Soft Comput 2011. [14] Srinivasa Rao R, Narasimham SVL, Ramalingaraju M. Optimal capacitor placement in a radial distribution system using plant growth simulation algorithm. Electr Power Energy Syst 2011;33:1133–9.
30
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
[15] Kresting WH, Mendive DL. An application of ladder network theory to the solution of three phase radial load flow problem. in: IEEE PES winter meeting, NY; 1976. p. A 76044–8. [16] Das D, Kothari DP, Kalam A. Simple and efficient method for load flow solution of radial distribution networks. J Electr Power Energy Syst 1995;17(5):335–46. [17] Teng Jen-Hao. A network – topology based three-phase load flow for distribution systems. In: Proceedings of the national science council, ROC(A), vol. 24(4); 2000. p. 259–64.
[18] Shirmohammadi D, Hong HW, Semlyen A, Luo GX. A compensation based power flow method for weakly meshed distribution and transmission networks. IEEE Trans Power Syst 1988;3:753–62. [19] Baran ME, Wu FF. Optimal sizing of capacitors placed on a radial distribution system. IEEE Trans Power Deliv 1989;2:735–43. [20] Ramachandra Murthy KVS et al. Topology based approach for efficient load flow solution for radial distribution networks. In: National power system conference; 2010.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Direct search algorithm for capacitive compensation in radial distribution systems M. Ramalinga Raju a, K.V.S. Ramachandra Murthy b,⇑, K. Ravindra a a b
Electrical and Electronics Engineering Department, University College of Engineering, JNTUK, Kakinada 533 001, Andhra Pradesh, India Electrical and Electronics Engineering Department, G. V. P. College of Engineering, Visakhapatnam 530 048, Andhra Pradesh, India
a r t i c l e
i n f o
Article history: Received 18 September 2011 Received in revised form 15 December 2011 Accepted 10 March 2012
Keywords: Capacitor placement Radial distribution systems Power flow Direct search algorithm
a b s t r a c t In this paper a new algorithm is proposed to determine the optimal sizes of fixed and switched capacitors together with their optimal locations in a radial distribution system so that net savings are maximized and improvement in the voltage profile is achieved. The algorithm searches for all possible locations in the system for a particular size of capacitor and places the capacitor at the bus which gives maximum reduction in active power loss. The optimal sizes are chosen to be standard sizes that are available in the market i.e., discrete sizes of capacitors are considered. The algorithm is tested on standard 69 bus system, 85 bus system and practical 22 bus systems. And the results are compared with results of other methods like particle swarm optimization and genetic algorithm, as available in the literature. The loss reduction obtained in this paper for the two standard test systems is highest compared to the other technique as reported in the literature. Cost analysis is also presented with and without capacitor placement on the three bus systems considering three different loading levels. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In India, all the 11KV rural distribution feeders are radial and too long. The voltages at the far end of many such feeders are very low with very poor voltage regulation. Capacitor banks are connected to radial distribution systems to improve the voltage profile and reduce the active power loss. The computational methods used in the analysis and design of distribution systems are not as robust as they are in transmission systems. In particular, the design of compensation systems for radial distribution system has become very complex because, the system does not fit into the usual optimization methods used in transmission system. Carpinelli et al. implemented non-linear programming technique for capacitor placement [1] on three phase unbalanced system. Wang et al. implemented integer programming technique [2], and Tabu search was used by Huang et al. [3] for optimal capacitor placement. Grainger implemented equal area criterion [4] and genetic algorithm applied to capacitor placement by Dlfanti [5] for determining optimal sizes of capacitors. Das applied FuzzyGA method for capacitor placement problem [6]. Sydulu and Reddy applied Index Vector to capacitor placement problem [7], Prakash and Sydulu applied particle swarm optimization for optimal capacitor placement problem [8]. Safigianni and Salis presented optimum VAr control of radial primary power distribution networks by shunt capacitor installation [9]. Das implemented genetic algorithm [10], Hsiao implemented Fuzzy-genetic algorithm for ⇑ Corresponding author. Tel.: +91 9966803153; fax: +91 891 2739605. E-mail address: [email protected] (K.V.S. Ramachandra Murthy). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.03.006
[11] for optimal capacitor placement problem. Huang applied immune multi objective algorithm for capacitor placement problem [12]. Kannana et al. applied Fuzzy-Differential Algorithm [13], Srinivasa Rao et al. applied plant growth algorithm for optimal capacitor placement problem [14]. The new algorithm proposed for capacitive compensation in this paper considered as direct search algorithms with a possible expert interaction yields optimal locations with suitable sizes of capacitors resulting in minimum active power loss and maximum net savings. 1.1. Objective function Objective function for optimization can be stated mathematically as given below. The first part of it is cost of energy loss and second part is the purchase cost of capacitor. The objective is to minimize the total cost, S:
Minimum S ¼ K e
ncap L X X T j Pj þ K c Q ci j¼1
ð1Þ
i¼1
where Ke is the energy cost per each kW h, Tj is the duration for which a jth load level operates. Three load levels are considered in this work. They are light, nominal and peak. Pj is the active power loss during jth load level. Qci is the size of the capacitor placed at ith bus. Different size capacitors would be suitable for different load levels at the optimal locations for minimizing the total cost function. Kc is the purchase cost of capacitor per kVAr. Number of candidate locations is indicated by ‘ncap’.
25
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
2. Load flow solution Traditional transmission system load flow methods Gauss-Siedel and Newton Raphson techniques, cannot be used for distribution systems as R/X ratio is high. Kresting and Mendive [15] have presented a load flow technique based on the ladder network theory. Teng and Lin proposed solution for meshed topology. Das and Kothari have proposed a load flow solution method by writing a algebraic equation for bus voltages [16]. Network topology based technique for three phase un-balanced systems is developed by Teng [17]. Shirmohammadi et al. [18] have presented a compensation based power flow for weakly meshed transmission and distribution systems. Baran and Wu [19] have obtained the load flow solution in a distribution system by iterative solution of three fundamental equations representing real power, reactive power and voltage magnitude. In this paper, a simple algebraic expression of voltage magnitude [20] is evaluated to obtain voltages and no trigonometric terms are used. Topology based approach is used for evaluating equivalent load at every node. This eliminates the complex process of identifying nodes connected beyond a particular node as described in [16]. The two developed matrices, ‘bus injection to branch current matrix’ and ‘line loss to node power matrix’ are very easy to form. The features of this method are robustness and computer economy. Convergence is always guaranteed. The assumption is that shunt capacitance is negligible at the distribution voltage level. The Fig 1 shows a two bus system where sending end voltage is known and receiving end voltage is to be obtained. P2 and Q2 are effective active and reactive powers at node 2 in Fig 1. The magnitude of voltage and loss calculation performed with the formulae presented by Das et al. [16]. The same equations are used along with the matrices developed from network topology method [17]. This model requires less memory and faster in execution. The load flow equations used are as follows:
V2 A½j ¼ P2 R½j þ Q 2 X½j 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi B½j ¼ ½A½j2 ðP22 þ Q 22 ÞðR½j2 þ X½j2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ¼ ½B½j A½j " # P22 þ Q 22 Ploss ½j ¼ R½j V 22 " # P22 þ Q 22 Q loss ½j ¼ X½j V 22
ð2Þ ð3Þ ð4Þ ð5Þ ð6Þ
The set of equations can be written by applying Kirchhoff’s current law (KCL) to the distribution network. The KCL equations will equally hold good for powers also. For example a simple six bus system shown in Fig 2. Consider N2, N3, N4, N5 and N6 as the equivalent powers at each node.
N2 ¼ S3 þ S4 þ S5 þ S6 N3 ¼ S3 þ S5 N4 ¼ S4 þ S6
where S3, S4, S5 and S6 are complex load powers respectively at buses 3, 4, 5 and 6. Effective load at node [N] can be obtained by,
½N ¼ ½BIBC½S
For distribution systems, the models which are based on the effective load at every node is very convenient. At each bus ‘i’ the complex power S is specified by:
Si ¼ Pi þ jQ i
ð7Þ
ð9Þ
The constant BIBC matrix has entries of 1 and 0 only. For a distribution system with m-branch sections and n-load buses, the dimension of the BIBC is m n. The bus where there is no load need not enter into matrix [S].
3 N2 2 1 6 7 6 N3 7 6 6 7 61 6 N4 7 ¼ 6 6 7 40 6 7 4 N5 5 0 N6 2
3 2
3 S3 7 6 0 1 0 7 6 S4 7 7 76 7 1 0 1 5 4 S5 5 1 1 1
0 0
1
S6
The relation between the line losses and node power can be obtained by following equations.
N02 ¼ S11 þ S12 þ S13 þ S14 þ Sl5 similarly; N03 ¼ S14
ð11Þ
N04 ¼ S15 Complex power losses associated with lines 1, 2, 3, 4, 5 are given by Sl1, Sl2, Sl3, Sl4 and Sl5 respectively. Sum of the line losses appearing at nodes 2, 3, 4, 5 and 6 are given by N02 ; N 03 ; N04 ; N 05 ; N 06 . Total line loss supplied by node2 is N02 . End nodes will not have to supply any line loss component. So, they will have all zeros in their corresponding rows. Node 2 will have to supply all the line losses except branch 1 loss. SL is the column matrix containing all line losses.
½N0 ¼ ½LLNP½SL 3 2 0 N02 6 0 7 6 6 N3 7 6 0 6 0 7 6 6N 7 ¼ 60 6 47 6 6 0 7 6 4 N5 5 4 0
3 2
3 Sl1 7 6 7 0 7 6 Sl2 7 7 6 7 6 7 17 7 6 Sl3 7 7 6 7 0 5 4 Sl4 5 0 Sl5
ð12Þ
Effective load at each node ¼ N þ N0
ð13Þ
N06
1 1 1 1 0
0 1
0
0
0
0
0
0
0 0
0
0
Pi and Qi are the real power and imaginary powers at ith node.
Fig. 1. Two bus system.
ð10Þ
2.2. Line loss to node power matrix (LLNP)
2 2.1. Bus-injection to branch current matrix (BIBC)
ð8Þ
Fig. 2. Sample distribution system.
26
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
After calculating the effective load at each node, recalculate the receiving end voltages using the above Eqs. (2)–(4). And calculate power losses using Eqs. (5) and (6). The voltage values at all nodes change after adding losses. 2.3. Algorithm for the load flow solution 1. Read the system data, V1 = 1.0 pu. Line losses are assumed to be zero in the first iteration. 2. Build BIBC matrix and LLNP matrix. 3. Obtain P effectiv e þ j Q effectiv e , at each node using Eq. (9). N represents the part of load powers in the effective load at various nodes.
½N ¼ ½BIBC½S where, S is the column matrix of all loads. 4. Initialize iteration count = 1. 5. Obtain receiving end voltages using simple formulae mentioned above.
V2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½B½j A½j
6. Calculate power loss on all lines using the formulae:
" Ploss ½j ¼ R½j
Q loss ½j ¼ X½j
P22 þ Q 22
#
V 22 " # P22 þ Q 22 V 22
Sloss ¼ Ploss þ jQ loss 7. Multiply the power loss column matrix, SL with LLNP matrix to get N0 matrix. N0 represents the part of line losses in the effective load at various nodes.
½N 0 ¼ ½LLNP½SL 8. Calculate the total effective load at various nodes by adding the N and N0 matrix.
Effective load at each node ¼ N þ N0 9. Increment the iteration count repeat the steps from step (5) using new effective loads at every node. 10. If the difference in the voltages between present iteration and previous iteration is greater than 0.001 pu, then increment the iteration count and repeat from step (5), otherwise, print the result. Number of iterations taken in this method is 3. The algorithms is efficient, robust and faster. This requires less memory and convergence is faster. For the first time, topology is fully exploited with this load flow solution model. 3. The proposed direct search algorithm for capacitor placement The algorithm proposed is for radial distribution system with source bus as slack bus and all other load buses as PQ buses. The algorithm proposed is described in following steps for deciding the optimal sizes of the capacitors in terms of standard sizes available in the market and their locations (only load buses). The algorithm is proposed in the following steps:
2. All the load buses are fully compensated with all reactive powers set to zeros and load flow study is conducted and total line loss is determined. This is considered as minimum possible loss to be aimed at for determining optimal sizes and locations. 3. To determine the optimal sizes of capacitors, a number of options having group of various capacitor sizes are to be tried. A tolerance index is chosen i.e., modulus of difference between losses under any option and minimum loss should be a very small value. All possible options may be enlisted. 4. Let m(k) be the number of capacitors in the kth option, k ranging from 1 to n where ‘n’ is the total number of options. m(1), the first option is with single capacitor, the Q of which is nearest to the total KVAR placed at all load buses, in turn, and load flow study is conducted. The line losses are determined. If the lowest loss satisfies the tolerance criterion, the process can be terminated. The size and location are considered as the optimal solution. 5. In one set of capacitors m(k), the first capacitor is kept at all load buses in turn, and the location for which losses are the lowest is considered as the optimal location for that capacitor. Placing this capacitor at that load bus, the procedure is repeated for placing the second capacitor at all load buses in turn and deciding the optimal location for the second capacitor. This procedure is repeated for all capacitors. 6. The options m(2) to m(n) are sequenced taking more and more number of capacitors of smaller size such that the total compensation is nearest to the total KVAR of the system. System losses are found out for each combination and checked for tolerance. If the tolerance is acceptable, process can be terminated. In any practical system, the authors’ experience is that, all possible options need not be exhausted. Process termination will be obtained after a finite number of options with acceptable tolerance. The sizing and sequencing of capacitors depend on load pattern and can be adjudged by the expert analyst. Even after trying all possible options, if the tolerance is not acceptable then that option for which distribution loss is minimum, is considered as the optimal solution. In this paper two standard test systems are considered for analysis and demonstrating the above algorithm with 69 bus system and 85 bus system. Standard capacitor sizes available in the literature (in KVAR): 150, 300, 450, 600, 750, 900, 1050, 1200, 1350, 1500, 1650, 1800, 1950, 2100, 2250, 2400, 2550, 2700, 2850, 3000, 3150, 3300, 3450, 3600, 3750, 3900, 4050. 4. Results 4.1. Example 1: 69 bus system Total Reactive power load in the system is 2694 kVAr and active power loss without compensation is 225 kW. By compensating 100% on all load buses (i.e., at all load buses, Qload = 0) the loss obtained is 143.52 kW. This is the minimum possible loss that should be aimed at. Table 1 shows the capacitive compensation on 69 bus system. The Table 1 is organized in such a way that column 4 gives active power loss after placing the capacitor size which is given in column2 at optimal location given in column3. Minimum loss obtained by the proposed algorithm is 147.0 kW. Data for the standard 69 system data is available in [19]. There is a reduction of 34.66% in active power loss. 4.2. Example 2: 85 bus system
1. Base case load flow study is conducted and distribution line losses are determined. This uncompensated loss is considered to be maximum loss in the system.
Total reactive power load in the system is 2622.2 kVAr and active power loss without compensation is 316.11 kW. By compen-
27
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30 Table 1 Capacitive compensation on 69 bus system using direct search algorithm.
Table 5 Load level and load duration time.
S. no.
Q kVAr compensated
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3
900 450 450
61 15 60
159.42 151.92 147.00
Total
1800 kVAr
Load level
0.5 (Light)
1.0 (Normal)
1.6 (Peak)
Duration (h)
2000
5260
1500
4.3. Example 3: Practical 22 bus agricultural distribution system
Table 2 Capacitive compensation on 85 bus system using direct search algorithm. S. no.
Q kVAr compensation
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3 4 5 6 7 8 9 10 11 12 13 14
450 300 150 150 150 150 150 150 150 150 150 150 150 150
36 69 30 80 66 61 14 26 57 8 20 6 18 17
247.24 212.31 188.52 178.05 168.91 161.80 155.96 151.68 148.68 146.83 145.30 144.62 144.17 144.01
Total
2550 kVAr
Data for the practical 22 bus system data is available in [A1]. 22 Bus system represents a small portion of agricultural distribution of Eastern Power Distribution system in India having base voltage as 11 kV. Total reactive power load in the system is 657.4 kVAr and active power loss without compensation is 17.69 kW. By compensating 100% on all load buses, (i.e., at all load buses, Qload = 0) the loss obtained is 9.14 kW. This is the minimum possible loss that should be aimed at. The Table 3 shows the best combination of capacitors with location and active power loss after placing the capacitors in turn. The Table 3 is organized in same way as that of Table 1. Minimum active power loss obtained by the proposed algorithm is 9.66 kW. There is a reduction of 45.39% in active power loss. 4.4. Comparison of results with other techniques
Table 3 Capacitive compensation on 22 bus system using direct search algorithm. S. no.
Q kVAr compensated
Min loss location
Active power loss after placing the capacitors in turn (kW)
1 2 3 4 Total
300 150 150 150 750 kVAr
13 16 4 17
12.21 10.22 9.95 9.66
It is found that distribution losses obtained by implementing the proposed algorithm is significantly less compared to other methods. Comparison of results with different techniques is given in Table 4 along with the location and size of the capacitor banks. The table contains active power loss obtained by GA, PSO and direct search algorithm. The minimum loss obtained in 69 bus system with direct search algorithm is 147 kW which is less when compared to 156.62 kW of active power loss obtained by implementing genetic algorithm [6] and 152.48 kW obtained by implementing particle swarm optimization [8]. The minimum loss obtained on 85 bus system with direct search algorithm is 144.01 kW which is less when compared to 146.061 kW of active power loss obtained by implementing genetic algorithm [7] and 163.32 kW obtained by implementing particle swarm optimization [8]. 4.5. Cost analysis
sating 100% on all load buses, (i.e., at all load buses, Qload = 0) the loss obtained is 141.02 kW. This is the minimum possible loss that should be aimed at. Table 2 shows the best combination of capacitors with location and active power loss after placing the capacitors in turn. Table 2 is organized in the same as that of Table 1. Minimum loss obtained by the proposed algorithm is 144.01 kW. Data for the standard 85 system data is available in [16]. There is a reduction of 54% in active power loss which is considerably large.
4.5.1. Cost analysis on 69 bus system Detailed cost analysis is presented for 69 bus system. Energy cost is assumed as US $0.06 per kW h and purchase cost of capacitor is assumed as US $3.0 per kVAr. Three load levels and load duration time data for the system is given in Table 5. Table 6 presents capacitor placement locations and sizes. Three load levels are considered. They are 160%, 100% and 50% load levels. Switched
Table 4 Comparison of active power loss without compensation and with compensation by different techniques. Standard bus system
Active power loss without compensation (kW)
Active power loss with capacitor placement by genetic algorithm (kW)
Active power loss with capacitor placement by PSO (kW)
Active power loss with capacitor placement by direct search algorithm (kW)
69 Bus system
225
156.62 61 64 59
[6] 700 800 100
152.48 46 47 50
[8] 241 365 1015
147.0 61 15 60
900 450 450
146.061 19 buses
[7] 2206.25 (Qtotal)
163.32 8 58 7 27
[8] 796 453 314 90
144.01 36 69 12 Buses
450 300 150 12 = 1800
85 Bus system
316.11
28
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
Table 6 Optimal capacitor placement location and size. Optimal location
61 15 60
Table 11 Comparison of results with and without losses on 85 bus system.
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
1800 900 900
900 450 450
450 300 300
Total cost Total losses cost Total capacitor cost
1800 900 900
Total annual savings = US $95,475 Load level 1.6
Table 7 Cost of energy loss and minimum system voltage before compensation. Load level
0.50 (Light)
1.0 (Nominal)
1.60 (Peak)
Energy loss cost Minimum voltage
$6192 0.95668
$70,997 0.90919
$58,716 0.84449
Maximum voltage = V1 = 1.0 pu.
Load level 1.0
V 54 min Losses
Load level 0.5
V 54 min Losses
Optimal location
Load level
0.50 (Light)
1.0 (Nominal)
1.60 (Peak)
Energy loss cost Minimum voltage
$4262.4 0.9683
$46,393.2 0.9318
$38,457 0.8936
Table 9 Comparison of results with and without losses on 69 bus test system.
Total cost Total losses cost Total capacitor cost
V 65 min Losses V 65 min
Load level 1.0
Losses Load level 0.5
With capacitors
$196,011 $196,011 0.0
$100,536 $86586 $13,950
0.7722
0.8770
975.93 0.8713
410.69 0.9224
316.11 0.9397
144.01 0.9629
70.11
34.76
Table 12 Optimal capacitor placement location and size on 22 bus system.
Table 8 Cost of energy loss and minimum system voltage after compensation.
Total annual savings = US $35,992.4 Load level 1.6
V 54 min Losses
Without capacitors
V 65 min Losses
13 16 4 17
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
450 300 150 150
300 150 150 150
150 150 0 0
450 300 150 150
Table 13 Comparison of results with and without losses on 22 bus practical distribution system.
Without capacitors
With capacitors
$135,905 $135,905 0.0
$99,912.6 $89,112.6 $10,800
0.84449
0.8936
652.40 0.90919
427.3 0.9318
Load level 1.0
224.96 0.95668
147.0 0.9683
V 22 min Losses
Load level 0.5
51.60
35.52
V 22 min Losses
Total cost Total losses cost Total capacitor cost Total annual savings = US $1576.41 Load level 1.6
V 22 min Losses
Without capacitors
With capacitors
$10,302 $10,302 0.0
$8725.59 $5575.59 $3150
0.9560
0.9701
46.68 0.9729
24.89 0.9824
17.69 0.9866
9.66 0.9909
4.32
2.39
Maximum voltage = V1 = 1.0 pu.
Table 10 Optimal capacitor placement location and size in 85 bus system. Optimal location
36 69 30 80 66 61 14 26 57 8 20 6 18 17
Control setting (kVAr)
Optimal size (kVAr)
1.6
1.0
0.50
900 600 450 450 300 300 150 150 300 150 300 150 300 150
450 300 300 150 150 150 150 150 150 150 150 150 150 150
300 150 150 0 150 0 150 0 150 0 0 0 0 150
900 600 450 450 300 300 150 150 300 150 300 150 300 150
capacitors need to be installed at nodes 61, 15 and 60 nodes. At 61 bus, 450 fixed capacitor and 450 kVAr, 900 switched capacitors need to be installed. At 15th bus, 300 kVAr fixed capacitor, one
150kVAr, one 450kVAr switched capacitors need to be installed. At 60th bus, 300 kVAr fixed capacitor, one 150 kVAr, 450 kVAr switched capacitor need to be installed to cater to the needs of varying load levels. Table 7 presents cost of energy loss without placing capacitors, i.e., bare system at different load levels. Table 8 presents cost of energy loss with capacitors placed in optimal locations at different load levels. Table 9 shows comparison of results with and without considering capacitor placements. The total cost for the 69 bus system without any compensation is US $135,905. After compensation, the total cost is US $99,912.6. The annual saving is US $35,992.4 where as total annual savings is $30,878 in [6]. It is observed that after capacitor placement, minimum voltage level of the system has improved and the system losses are reduced in each load level. Only at the peak load level, voltage is observed to be 0.8936 pu which is below 0.9 pu. In all remaining cases, voltage is above 0.90 pu. Table 9 gives the comparison of cost with and without capacitor placement. 4.5.2. Cost analysis on 85 bus system Total cost of energy loss in 85 bus system is US $196,011 without compensation, where as it is US $86,586 after capacitor placement. The total purchase cost of fixed and switched capacitors is US $13,950. Capacitor requirement indicated in the 0.5 load level
29
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30 Table A1 Data of the 22 bus agricultural distribution system. Line no.
From bus
To bus
Resistance (X)
Reactance (X)
Real power load (kW)
Reactive power load (kVAr)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 2 2 4 4 5 6 6 9 9 11 11 13 14 14 16 17 17 19 20 20
2 3 4 5 9 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22
0.3664 0.0547 0.5416 0.193 0.7431 1.3110 0.0598 0.2905 0.0547 0.675 0.0547 0.3942 1.0460 0.022 0.0547 0.3212 0.0949 0.574 0.1292 0.0871 0.5329
0.1807 0.0282 0.2789 0.099 0.3827 0.6752 0.0308 0.1496 0.0282 0.3481 0.0282 0.203 0.5388 0.0116 0.0282 0.1654 0.0488 0.2959 0.066 0.045 0.2744
16.78 16.78 33.80 14.56 19.31 10.49 8.821 14.35 14.35 16.27 16.27 82.13 34.71 34.71 80.31 49.62 49.62 43.77 37.32 37.32 31.02
20.91 20.91 37.32 12.52 25.87 14.21 11.66 18.59 18.59 19.48 19.48 71.65 30.12 30.12 70.12 47.82 47.82 38.93 35.96 35.96 29.36
is fixed capacitor. Table 10 presents capacitor placement locations and sizes for different loading levels. Table 11 gives the comparison of cost with and without capacitor placement on 85 bus system. Total annual savings are found to be US $95,475. The reason for having large savings is loss reduction at different loading levels are 57.9%, 54.43% and 50.42% respectively at 1.6, 1 and 0.5 load levels of the nominal level. This is much higher than 69 bus system loss reduction. 4.5.3. Cost analysis on 22 bus system Fixed capacitors of 150 kVAr size are required at bus nos. 13 and 16 one at each bus. Total cost of energy loss in 22 bus system is US $10,302 per year without compensation, where as it is US $5575.59 per year after capacitor placement. The total purchase cost of capacitors is US $3150. Total annual savings are found to be US $1576 for the agricultural distribution system. Sizes of capacitors for various load levels is presented in Table 12 and cost analysis presented in Table 13. Minimum voltages are also indicated at each loading level. 5. Conclusions The new algorithm proposed for designing the compensation in distribution system, resulted in improving the voltage profile and maximizing the net savings. Cost analysis is demonstrated on 69 bus radial distribution system. This algorithm can be classified under direct search algorithm as the approach is methodic and sequential. The process generally terminates fast and faster if implemented by an expert. Standard 69 bus and 85 bus systems have been considered for which known results are there with implementation of PSO and genetic algorithm. It has been clearly observed for the given loading conditions, the total distribution loss is significantly less than the one obtained in the other two methods. The algorithm is also tested on 22 bus practical agricultural distribution system. However, the future work can be carried out considering the installation and maintenance costs. Acknowledgements The authors would like to extend their whole hearted gratefulness to Dr. Govinda Rao Gade, Senior Professor at G.V.P. College of
Engineering, Visakhapatnam, India for providing language help, assisting in writing and for proof reading.
Appendix A See Table A1.
References [1] Carpinelli G, Varilone P, Di Vito V, Abur A. Capacitor placement in three-phase distribution systems with nonlinear and unbalanced loads. IEE Proc Generat Transm Distribut 2005;152(1):47–52. [2] Wang J, Chiang H, Nan minu K, Darling G. Capacitor placement and real time control in large-scale un-balanced distribution systems: Part I and II. IEEE Trans Power Deliv 1997;12:953–64. [3] Huang YC, Yang HT, Huang CL. Solving the capacitor placement problem in a radial distribution system using tabu search application. IEEE Trans Power Syst 1996;11:1868–73. [4] Grainger JJ, Lee SH. Optimum size and location of shunt capacitors for reduction of losses on distribution feeders. IEEE Trans Power App Syst 1981;100:1105–18. [5] Dlfanti M, Granelli GP, Maranninio P. Optimal capacitor placement using deterministic and genetic algorithm. IEEE Trans Power Syst 2000;15: 1041–6. [6] Das D. Optimal placement of capacitors in radial distribution system using Fuzzy-GA method. J Electr Power Energy Syst 2008;30: 361–7. [7] Sydulu M, Reddy VVK. Index and GA based optimal location and sizing of distribution system capacitors. In: Power engineering society general meeting, Tampa, FL; 2007. p. 1–4. [8] Prakash K, Sydulu M. Particle swarm optimization based capacitor placement on radial distribution systems. Power Eng Soc Gen Meet 2007: 1–5. [9] Safigianni AS, Salis GJ. Optimum VAr control of radial primary power distribution networks by shunt capacitor installation. Electr Power Energy Syst 2001;23:389–401. [10] Das D. Reactive power compensation for radial distribution networks using genetic algorithm. Electr Power Energy Syst 2002;24:573–81. [11] Hsiao Ying-Tung, Chen Chia-Hong, Chien Cheng-Chih. Optimal capacitor placement in distribution systems using a combination Fuzzy-GA method. Electr Power Energy Syst 2004;26(7):501–8. [12] Huang Tsong-Liang, Hsiao Ying-Tung, Chang Chih-Han, Jiang Joe-Air. Optimal placement of capacitors in distribution systems using an immune multiobjective algorithm. Electr Power Energy Syst 2008;30:184–92. [13] Kannana SM, Renugab P, Kalyania S, Muthukumarana E. Optimal capacitor placement and sizing using Fuzzy-DE and Fuzzy-MAPSO methods. Appl Soft Comput 2011. [14] Srinivasa Rao R, Narasimham SVL, Ramalingaraju M. Optimal capacitor placement in a radial distribution system using plant growth simulation algorithm. Electr Power Energy Syst 2011;33:1133–9.
30
M. Ramalinga Raju et al. / Electrical Power and Energy Systems 42 (2012) 24–30
[15] Kresting WH, Mendive DL. An application of ladder network theory to the solution of three phase radial load flow problem. in: IEEE PES winter meeting, NY; 1976. p. A 76044–8. [16] Das D, Kothari DP, Kalam A. Simple and efficient method for load flow solution of radial distribution networks. J Electr Power Energy Syst 1995;17(5):335–46. [17] Teng Jen-Hao. A network – topology based three-phase load flow for distribution systems. In: Proceedings of the national science council, ROC(A), vol. 24(4); 2000. p. 259–64.
[18] Shirmohammadi D, Hong HW, Semlyen A, Luo GX. A compensation based power flow method for weakly meshed distribution and transmission networks. IEEE Trans Power Syst 1988;3:753–62. [19] Baran ME, Wu FF. Optimal sizing of capacitors placed on a radial distribution system. IEEE Trans Power Deliv 1989;2:735–43. [20] Ramachandra Murthy KVS et al. Topology based approach for efficient load flow solution for radial distribution networks. In: National power system conference; 2010.