Optimum placement of shunt capacitors in a radial distribution system for substation power factor improvement using fuzzy GA method

Electrical Power and Energy Systems 77 (2016) 314–326

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Optimum placement of shunt capacitors in a radial distribution system for substation power factor improvement using fuzzy GA method Srinivasa Rao Gampa, D. Das ⇑ Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, India

a r t i c l e

i n f o

Article history: Received 20 May 2014 Received in revised form 21 July 2015 Accepted 17 November 2015

Keywords: Shunt capacitors Fuzzy multiobjective Genetic algorithm Power factor improvement Energy loss reduction

a b s t r a c t In this work a combination of fuzzy multiobjective and genetic algorithm (GA) based approach is proposed for optimal shunt capacitor placement to improve the substation power factor near unity, reduce the real power loss, and reduce the burden on the substation and to improve the voltage profile of the distribution network. In order to obtain best nodes for capacitor placement, a sensitivity index based on real power loss reduction and voltage profile improvement is considered. In the present work, an attempt is made to make reactive current component drawn by distribution network through substation is nearly zero such that power factor at the substation will be near unity. A fuzzy multiobjective function is formed considering substation reactive current component reduction, real power loss reduction, branch current constraint limit, minimum and maximum voltage limit satisfaction. The fuzzy multiobjective function is maximized using GA for obtaining the optimum sizing of fixed and switched shunt capacitors. Simulation results are shown to demonstrate the advantage of the proposed method compared to optimal shunt capacitor placement based on annual energy savings method. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction Shunt capacitors usage is a common practice to supply reactive power in the distribution networks. Installation of shunt capacitors reduces power losses, improves the power factor and feeder voltage profile. Therefore it is essential to find optimal location and sizes of capacitors for gaining maximum benefits by the shunt capacitor installation. Ng et al. [1] classified the capacitor placement techniques available in the literature into four categories. They are analytical, numerical programming, heuristic and artificial intelligence based techniques. In the early literature many researchers proposed calculus based analytical techniques for capacitor placement solution [2–6]. Initially, methods are developed based on the assumption of uniform conductor size and loading throughout the feeder and later they are extended for different conductor sizes and nonuniform loading. With the development of powerful computational techniques numerical methods were developed for the selection of optimal nodes and sizing of capacitors considering all the operational constraints [7–13]. In order to reduce the large search space required by the numerical methods many authors have proposed heuristic methods to obtain nearly optimal solution for capacitor placement problem [14–18]. For ⇑ Corresponding author. E-mail address: [email protected] (D. Das). http://dx.doi.org/10.1016/j.ijepes.2015.11.056 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

the last two decades artificial intelligence techniques are widely used for identifying optimal locations and sizing of capacitors, out of them fuzzy logic, genetic algorithms are more popular. Ng et al. [19], Mekhamer et al. [20], Masoum et al. [21], Shi and Liu [22] and Bhattacharya and Goswami [23] used fuzzy set theory based on human experience and intuition for identification of optimal nodes for capacitor placement and for optimal capacitor sizing, they have used numerical methods and heuristic methods. Boone and Chiang [24], Iba [25], Sundharajan and Pahwa [26], Delfanti et al. [27], Das [28], Malik et al. [29] and Swarnakar et al. [30] applied genetic algorithms for optimal allocation of capacitors. Gallego et al. [31], De Souza et al. [32], Hsiao et al. [33], Das [34], and Abul’Wafa [35] have used combination of fuzzy logic and genetic algorithms for optimal allocation of capacitors in distribution systems. In the recent years many nature inspired based artificial intelligence techniques are developed and used for optimal capacitor allocation problem. Huang et al. [36] have proposed two stage immune algorithm based multi objective optimization approach for solving shunt capacitor placement problem. Rao [37] has used fuzzy and plant growth simulation algorithm to determine optimal locations and size of the capacitor to improve the voltage profile and reduce the active power losses. Singh and Rao [38] have proposed a particle swarm optimization based algorithm for obtaining the optimal size and locations of the capacitors utilizing the daily load curve. Ziari et al. [39] have proposed a

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Nomenclature Iac i IC i NB F P Si PQC Si QC j

current magnitude through ith branch after capacitor placement maximum current carrying capacity of ith branch total number of nodes of the distribution system fuzzy multiobjective function power injected at substation before capacitor placement at ith load level power injected at substation after capacitor placement at ith load level total capacity of shunt capacitors installed at jth rank node

modified discrete particle swarm optimization technique for the determination of rating and locations of fixed and switched capacitors. El-Fergany et al. [40] have proposed artificial bee colony based approach for allocation of static capacitors along the radial feeders of the distribution system. Hung et al. [41] classified devices capable of supplying reactive power only as type-2 kind of distributed generation. Aman et al. [42] presented a literature survey on optimal shunt capacitor placement. From the above literature survey, it can be seen that the capacitor placement problem is aimed to obtain optimal locations and sizing of capacitors to reduce the active power losses, to increase feeder capacity and to improve the voltage profile of the radial distribution system. In the present work a new sensitivity analysis is proposed to find suitable nodes for the capacitor placement problem. For obtaining optimum sizing of capacitors at the optimum locations identified the objectives of minimizing reactive current drawn by the distribution network from the grid through substation, real power loss reduction, feeder capacity improvement and voltage profile improvement are considered to improve the substation power factor, nearer to unity. Since all the objectives considered are non-commensurable in nature, the conventional approaches that optimizes a single objective function are not suitable for this problem. Therefore the fuzzy approach is adopted for considering all the multiple objectives simultaneously. The conventional calculus based techniques mainly depends upon the existence of derivatives of the single objective function and may lead toward the local optimum solution. The actual search space in majority of the practical cases is associated with many discontinuous functions and hence the conventional optimization methods are not suitable for finding optimal solutions for non differentiable multiobjective functions. Genetic Algorithms (GA) are probabilistic search techniques based on natural selection and genetics. The advantage of GA compared to other conventional techniques is it works with binary coding of parameters rather than parameters themselves and hence it is more efficient optimization technique for non differentiable multiobjective functions consisting both discrete and continuous variables. GA proceeds in the direction of maximization of fitness function in the selected multi dimensional search space and hence the solution proceeds toward global optimum. Hence for the present capacitor placement problem a genetic algorithm based fuzzy multiobjective approach is used for optimum sizing of fixed and switched capacitors. The main motivation of this work is that the distribution network should not draw reactive power from the grid. Shunt capacitors must meet the total reactive power load demand and reactive power loss of the distribution network. Therefore, in this paper, the approach is to make the substation power factor near unity, i.e. drawing negligible reactive power from the grid.

Ke CC CI CRF NL NCL Ti

energy cost in $/kW h cost of capacitor in $/kVAr capacitor installation cost in $/location capital recovery factor total number of load levels total number of capacitor installation locations total number of hours of ith load level duration in one year annual economical savings due to shunt capacitor placement

SCP

Sensitivity analysis for the placement of capacitors In the present analysis, a new sensitivity index is used for the placement of capacitors in a distribution network. In the sensitivity analysis the load flow solutions are obtained by compensating the total reactive load at every node and from the load flow solutions the active power loss reduction and maximum node voltage increment are calculated for all nodes of distribution system. Let us define a sensitivity index,

Sk ¼ LSIk  VSIk

ð1Þ

where Sk = Sensitivity index of kth node due to total reactive load compensation. LSIk = Loss sensitivity index of kth node due to total reactive load compensation. VSIk = Voltage sensitivity index of kth node due to total reactive load compensation. The loss sensitivity index can be expressed mathematically as follows:

LSIk ¼

PLRk  PLRmin PLRmax  PLRmin

PLRk ¼ PLB  PLk

ð2Þ ð3Þ

where PLB = Real power loss without shunt capacitor compensation. PLk = Real power loss due to total reactive load compensation at kth node. PLRk = Real power loss reduction due to total reactive load compensation at kth node. PLRmin = Minimum Real power loss reduction. PLRmax = Maximum Real power loss reduction. The voltage sensitivity index can be expressed mathematically as follows:

DVMk  DVMmin DVMmax  DVMmin e ac  V e bc j DVMk ¼ max j V i;k i;k

VSIk ¼

ð4Þ ð5Þ

where DVMk = Maximum node voltage increment due to total reactive load compensation at kth node. DVMmin = Minimum of Maximum node voltage improvement. DVMmax = Maximum of Maximum node voltage improvement. e bc = Voltage at ith node before capacitor placement at kth V i;k

node. e ac = Voltage at ith node after capacitor placement at kth node. V i;k

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Significance of Eq. (1) is that due to reactive power compensation in the distribution network it is expected that the voltage profile will improve and the loss will decrease. The loss sensitivity values are normalized in between 0 and 1 considering the highest loss reduction value as 1 and smallest loss reduction value to zero [35,37]. Similarly the maximum node voltage increment values are also normalized in between zero and one. The node corresponding to the highest product of loss sensitivity index and voltage sensitivity index will have highest sensitivity index and it is best suitable for capacitor placement. In the present work two case studies, i.e., an 11 kV, 51 node radial distribution network and a 12.66 kV, 69 node distribution network are considered and the single line diagrams are shown in Figs. 1 and 2 respectively. The line data and load data for the 51 node distribution system are given in Appendix A and for 69 distribution network the data are taken from [43]. The substation e 1 ¼ ð1 þ j0:0Þ pu. Ranking of nodes is made (s/s) voltage is V

according to the descending order values of the sensitivity index Sk calculated using Eq. (1) for optimal allocation of capacitors. The ranking of the nodes for 51 node distribution system is given in Table 1, and for 69 node distribution system the order of ranking is given in Table 2. The plots for sensitivity index (Sk) values vs node number are shown in Figs. 3 and 4 for 51 node distribution system and for 69 node distribution system respectively. The nodes with top five ranks are selected as best locations for capacitor placement for the both the examples considered for capacitor placement. Formation of fuzzy multiobjective function In the present section fuzzy membership functions [44] are developed for substation reactive current component, real power loss reduction, branch current capacity, minimum voltage limit and maximum voltage limit of the distribution system.

Table 1 Ranking of nodes of 51 node distribution system for capacitor placement.

Fig. 1. 51 node distribution system network.

Order of the ranking of nodes

Node number

Order of the ranking of nodes

Node number

Order of the ranking of nodes

Node number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

15 14 11 49 40 44 48 16 51 10 09 34 43 42 38 45 26

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

41 39 08 46 23 50 19 33 37 13 05 22 29 47 20 21 30

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

35 36 32 24 18 25 31 06 07 17 03 12 28 27 02 04 01

Table 2 Ranking of nodes of 69 node distribution system for capacitor placement.

Fig. 2. 69 node distribution system network.

Order of the ranking of nodes

Node number

Order of the ranking of nodes

Node number

Order of the ranking of nodes

Node number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

61 64 59 21 65 12 11 50 62 49 18 17 16 24 08 69 68 27 26 51 55 10 07

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

54 09 67 66 14 13 22 46 45 53 48 52 34 20 06 33 40 39 35 43 37 29 36

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

28 41 63 60 58 57 56 47 44 42 38 32 31 30 25 23 19 15 05 04 03 02 01

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value is assigned. If q is less than or equal to qmin then unity membership value is assigned and if q is greater than qmax, zero membership value is assigned. From Fig. 5, lIQ can be written as:

lIQ ¼

8 > <1 > :

for q 6 qmin

ðqmax qÞ ðqmax qmin Þ

for qmin < q 6 qmax

0

for q > qmax

ð7Þ

In this work with the proposed method for shunt capacitors placement it is aimed at reducing the reactive current component at the substation to 5% of the base case value. For the values of q less than or equal to 0.05 the fuzzy membership values are given a value of unity and for the case of reactive current component greater than or equal to base case value the fuzzy membership values are considered as zero. Hence the values for qmin and qmax are taken as 0.05 and 1.0 respectively. When the value of q is less than qmax and greater than qmin a fuzzy membership value in between zero and one is assigned.

Fig. 3. Plot of sensitivity index for capacitor placement of 51 bus system.

Membership function for real power loss reduction (lPL) The objective of this membership function developed is to reduce the real power loss due to the compensation of reactive power using shunt capacitors. Let us define the real power loss index ‘x’ as, ac

x¼

Membership function for substation reactive current component reduction (lIQ) The main purpose of this membership function is to reduce substation reactive current component and to improve the power factor at the substation of the distribution system. Let us define,

Iac q Ibc q

ð6Þ

where q = Substation Reactive current component index. Ibc q = Reactive current component at substation before capacitor placement. Iac q = Reactive current component at substation after capacitor placement. Fig. 5 shows the fuzzy membership function for substation reactive current component reduction (lIQ). When the value of q exceeds qmin and less than or equals to qmax, lower membership

Fig. 5. Membership function for substation reactive current component.

ð8Þ

bc

Ploss

where Plossbc = Real power loss of the distribution network before capacitor placement. Plossac = Real power loss of the distribution network after capacitor placement.

Fig. 4. Plot of sensitivity index for capacitor placement of 69 bus system.

q¼

Ploss

Fig. 6 shows the membership function for real power loss reduction (lPL). Eq. (8) indicates that for higher the values of x the membership value will be low because real power loss reduction is low and if x has a lower value then the membership value will be high because real power loss reduction is high. From Fig. 6, lPL can be written as:

lPL ¼

8 > <1 > :

for x 6 xmin

ðxmax xÞ ðxmax xmin Þ

for xmin < x 6 xmax

0

for x > xmax

ð9Þ

In the present work with the proposed method for shunt capacitors placement it is aimed at reducing the real power loss to 40% of the base case value. For the values of x less than or equal to 0.4 the fuzzy membership values are given a value of unity and for the case of zero loss reduction and losses greater than base case value the fuzzy membership values for real power loss reduction are considered as zero. Hence the values for xmin and xmax are taken as 0.4

Fig. 6. Membership function for active power loss reduction.

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and 1.0 respectively. When the value of x is less than xmax and greater than xmin a fuzzy membership value in between zero and one is assigned. Membership function for current carrying capacity of branch (lR) The main purpose of this membership function is to avoid the branch current constraint violation. Let us define,

Branch current capacity index ¼

Iac i ; IC i

for i ¼ 1; 2; . . . ; NB  1

ð10Þ

Now we define,

 r ¼ max

 Iac i ; IC i

for i ¼ 1; 2; 3; . . . ; NB  1

ð11Þ

Fig. 7 shows the fuzzy membership function for branch current carrying capacity (lR). When the value of r exceeds rmin and less than or equals to rmax, lower membership value is assigned. If r is less than or equal to rmin then unity membership value is assigned and if r is greater than rmax, zero membership value is assigned. From Fig. 7, lR can be written as:

lR ¼

8 > <1

for r 6 rmin

ðr max rÞ for r min < r 6 r max ðr max r min Þ > : 0 for r > r max

ð12Þ

In the present work with the proposed method for shunt capacitor placement it is aimed at reducing maximum branch current to 40% of the base case value. For the values of r less than or equals to 0.4 the fuzzy membership values are given a value of unity and for the case of maximum branch current values greater than or equal to base case values the fuzzy membership values are considered as zero. Hence in the present work rmin and rmax are taken as 0.4 and

1.0 respectively. When the value of r is less than rmax and greater than rmin a fuzzy membership value in between zero and one is assigned. Membership functions for minimum and maximum voltage limits The placement of shunt capacitors is aimed to maintain all the node voltages to be in between specified minimum and maximum voltage limits. For this purpose two membership functions are used: Membership function for minimum voltage limit Fig. 8 shows the membership function for minimum voltage limit (lV min ). If the minimum system voltage is less than ymin or greater than ymax then zero membership value is assigned. If it is in between ymin and yp lower membership value is assigned and if it is lying in between yp and ymax, unity membership value is assigned. From Fig. 8, we can write,

lV min ¼

8 0 > > > ðyy Þ > min <

ðyp ymin Þ

> 1 > > > : 0

for y 6 ymin for ymin < y < yp

ð13Þ

for yp 6 y 6 ymax for y > ymax

In the present work with the proposed method for shunt capacitors placement it is aimed at improving the minimum voltage to 0.95 pu value. For y values lies in between 0.95 and 1.0 the fuzzy membership values are given a value of unity and for the case of y values less than 0.925 and greater than 1.0 the fuzzy membership values are considered as zero. For y values greater than 0.925 and less than 0.95, membership values in between zero and one are assigned. Hence in the present work ymin ¼ 0:925; yp ¼ 0:95 and ymax = 1.0 are considered for the fuzzy membership function developed for minimum voltage limit. Membership function for maximum voltage limit Fig. 9 shows the membership function for maximum voltage limit (lV max ). If the maximum system voltage is less than zmin or greater than zmax then zero membership value is assigned. If it is in between zmin and zp unity membership value is assigned and if

Fig. 7. Membership function for maximum branch current carrying capacity index.

Fig. 9. Membership function for maximum voltage limit.

Table 3 Annual load duration schedule.

Fig. 8. Membership function for minimum voltage limit.

Load level

No of hrs/yr

0.4 0.5 0.6 0.7 0.8 0.9 1.0

1000 1000 1000 1000 1000 1000 2760

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319

it is lying in between zp and zmax lower membership value is assigned. From Fig. 9, we can write,

lV max ¼

Fig. 10. Load duration curve for 51 node distribution system.

8 0 > > > > <1

for z < zmin for zmin 6 z 6 zp

ðzmax zÞ > for zp < z < zmax > ðzmax zp Þ > > : 0 for z P zmax

ð14Þ

In the present work with the proposed method for shunt capacitor placement it is aimed at limiting the maximum voltage to 1.025 pu value. For z values lies in between 1.0 and 1.025 the fuzzy membership values are given a value of unity and for the case of

Fig. 11. Flow chart for optimal shunt capacitor sizing using GA.

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z values greater than 1.05 and less than 1.0 the fuzzy membership values are considered as zero. For z values greater than 1.025 and less than 1.05, membership values in between zero and one are assigned. For z values greater than 1.025 and less than 1.05, membership values in between one and zero are assigned. Hence in the present work zmin = 1.0, zp = 1.025 and zmax = 1.05 are considered for the fuzzy membership function developed for maximum voltage limit. The above membership functions developed for each objective function above are added to form a single objective function through weighting factors and is given by Maximize

F ¼ w1 lIQ þ w2 lPL þ w3 lR þ w4 lV min þ w5 lV max

ð15Þ

In Eq. (15), we have assumed that all the objectives have equal importance and w1 = w2 = w3 = w4 = w5 = 1.0 is assumed. Optimal sizing of shunt capacitors using GA In the present work binary coded Genetic algorithm (GA) [45] is used for optimum sizing of shunt capacitors. In GA fitness function is needed to be maximized for obtaining the optimum values of parameters of the objective function. Here the fuzzy multiobjective function ‘F’ described by Eq. (15) is considered as fitness function for optimum sizing of capacitors using GA. Optimum shunt capacitor sizing is done in a step by step process starting from lowest load level. There are seven load levels varying from 0.4 to 1.0 in a step of 0.1 are considered for the present analysis. The load duration time for the corresponding load levels considered for the 51 node distribution system is shown in Table 3. The approximated load duration curve is shown in Fig. 10. First for 0.4 load level the optimum sizing of shunt capacitors is done and they are used as fixed capacitors. The shunt capacitors obtained at 0.4 load level are incorporated into the distribution system and the additional switched capacitors required for 0.5 level are optimized again using GA. Similarly the additional switched capacitors required for the remaining load levels are

obtained for all load levels upto the peak load level successively by incorporating the optimally sized shunt capacitors upto previous load level. Fig. 11 shows the flow diagram for optimal sizing of shunt capacitors using GA. The optimum sizing of shunt capacitor values obtained using GA are real values and they are adjusted to the nearest integer values of capacitances available in the market for incorporation into the distribution system. The fixed capacitors and switched capacitors obtained for all load levels are shown in Table 4 for 51 node distribution node network and in Table 6 for 69 node distribution network. The total optimal shunt capacitors sizing obtained using GA for performance improvement of the distribution system for all load levels is shown in Table 5 for 51 node distribution system and in Table 7 for 69 node distribution system. The GA parameters, i.e., no of generations, population size, crossover rate, mutation rate used for obtaining optimum sizing of capacitor placement are shown in Table 8. The computational efficiency of proposed fuzzy GA methodology is compared with energy savings method proposed in [30]. The algorithms are developed for both the methodologies using MATLAB 2007a software and executed on a Pentium dual core CPU with 3.2 GHz speed and 2 GB RAM computer. The computational time for 51 node distribution system with energy savings method [30] is 51 seconds and with proposed method is 35 seconds. The computational time for 69 node distribution system with energy savings method [30] is 89 s and with proposed method is 39 s. Hence it can be said that the proposed method is computationally efficient compared to GA based energy savings method proposed in [30].

Table 6 Optimal capacitor sizing obtained using GA for 69 node distribution system. Type of capacitors

Load level

QC61

QC64

QC59

QC21

QC65

Fixed capacitors

0.4

350

50

300

275

25

Switched capacitors

0.5 0.6 0.7 0.8 0.9 1.0

100 50 125 100 100 75

25 25 0 0 150 75

75 150 25 100 0 75

50 75 75 75 0 0

0 0 0 25 100 75

Table 4 Optimal capacitor sizing obtained using GA for 51 node distribution network. Type of capacitors

Load level

Reactive power of the capacity of capacitors installed (kVAr) QC15

QC14

QC11

QC49

QC40

Fixed capacitors

0.4

25

50

125

125

300

Switched capacitors

0.5 0.6 0.7 0.8 0.9 1.0

25 0 25 0 0 75

0 0 0 25 0 0

0 25 25 25 50 75

50 25 25 50 25 0

100 100 75 75 100 0

Table 5 Optimal sizing of total capacitors installed for 51 node distribution network. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Total available reactive power capacity of the capacitors installed (kVAr) QC15

QC14

QC11

QC49

QC40

25 50 50 75 75 75 150

50 50 50 50 75 75 75

125 125 150 175 200 250 325

125 175 200 225 275 300 300

300 400 500 575 650 750 750

Reactive power of the capacity of capacitors installed (kVAr)

Table 7 Optimal sizing of total capacitors installed for 69 node distribution system. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Total available reactive power capacity of the capacitors installed (kVAr) QC61

QC64

QC59

QC21

QC65

350 450 500 625 725 825 900

50 75 100 100 100 250 325

300 375 525 550 650 650 725

275 325 400 475 550 550 550

25 25 25 25 50 150 225

Table 8 GA parameters used in the optimum sizing of shunt capacitors. No of generations Population size Crossover rate Mutation rate

100 200 0.95 0.001

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Results and discussions The effect of optimum capacitor placement on the performance of the distribution system with the proposed method is analyzed considering two examples of 51 node distribution network and 69 node distribution system. The proposed methodology is compared with the methodology considering the objective of maximizing net annual savings proposed in [30]. The effect of optimum capacitor placement on substation real power supply and reactive power supply for all the load levels is shown in Tables 9 and 10. From Tables 9 and 10, it can be noticed that the burden on the substation is reduced to the significant level with the shunt capacitor placement in the distribution system. From Table 9, it is seen that the real power supply reduction at the substation in the case of lower load levels is almost same with both methodologies considered for comparison for the both the networks considered. But as the load level increases the real power supply reduction at the substation is slightly higher in the case of methodology proposed in [30] compared to the proposed fuzzy GA methodology. From Table 10, it is seen that the reactive power sup-

ply reduction at the substation is much higher in the case of proposed methodology as compared to the methodology proposed in [30]. The effect on substation reactive current component and current drawn at substation due to optimal shunt capacitor placement is shown in Tables 11 and 12 respectively. It can be seen from Table 11 that the reactive current component is reduced greatly in the case of proposed methodology compared to the methodology based on annual energy savings proposed in [30]. From Table 12 it can be observed that the current drawn from the substation is reduced significantly in the case of proposed fuzzy GA based step by step shunt capacitor based methodology compared to the annual energy savings methodology proposed in [30]. The real power loss comparison before and after capacitor placement is shown in Figs. 12 and 13 for 51 node distribution system and 69 node distribution system respectively. From Figs. 12 and 13, it can be observed that the real power losses are reduced significantly with the placement of shunt capacitors compared to base case. It can also be observed that the real power loss is almost same at lower load levels both in the case of energy savings

Table 9 Real power drawn from substation before and after placement of shunt capacitors. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (kW)

After placement of shunt capacitors (Method in [30]) (kW)

After placement of shunt capacitors (Proposed method) (kW)

Before placement of shunt capacitors (kW)

After placement of shunt capacitors (Method in [30]) (kW)

After placement of shunt capacitors (Proposed method) (kW)

1004.30 1261.73 1521.91 1784.95 2050.98 2320.13 2592.56

1000.00 1254.22 1510.76 1769.73 2030.75 2294.54 2561.33

1001.49 1257.39 1515.14 1775.16 2038.13 2303.56 2571.86

1553.39 1952.70 2356.84 2766.06 3180.65 3600.92 4027.19

1543.59 1936.99 2335.26 2733.95 3137.43 3545.98 3959.86

1545.22 1939.50 2338.39 2739.11 3145.66 3559.88 3978.13

Table 10 Reactive power drawn from substation before and after placement of shunt capacitors. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (kVAr)

After placement of shunt capacitors (Method in [30]) (kVAr)

After placement of shunt capacitors (Proposed method) (kVAr)

Before placement of shunt capacitors (kVAr)

After placement of shunt capacitors (Method in [30]) (kVAr)

After placement of shunt capacitors (Proposed method) (kVAr)

644.11 810.62 979.49 1150.83 1324.73 1501.30 1680.68

214.81 278.83 319.40 437.18 506.61 628.57 803.18

15.46 4.90 20.85 38.63 33.63 30.54 54.62

1092.69 1370.85 1651.19 1933.83 2218.88 2506.48 2796.77

363.38 413.95 191.57 419.68 549.97 832.54 1117.51

88.99 114.90 92.83 146.67 128.05 62.86 49.50

Table 11 Substation reactive current component before and after placement of shunt capacitors. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (A)

After placement of shunt capacitors (Method in [30] (A)

After placement of shunt capacitors (Proposed method) (A)

Before placement of shunt capacitors (A)

After placement of shunt capacitors (Method in [30]) (A)

After placement of shunt capacitors (Proposed method) (A)

58.55 73.69 89.04 104.62 120.43 136.48 152.80

19.53 25.35 29.04 39.74 46.06 57.14 73.02

1.41 0.45 1.90 3.51 3.06 2.78 4.97

86.31 108.28 130.43 152.75 175.27 197.98 220.91

28.70 32.70 15.13 33.15 43.44 65.76 88.27

7.03 9.08 7.33 11.59 10.12 4.97 3.91

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Table 12 Substation current component before and after placement of shunt capacitors. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (A)

After placement of shunt capacitors (Method in [30]) (A)

After placement of shunt capacitors (Proposed method) (A)

Before placement of shunt capacitors (A)

After placement of shunt capacitors (Method in [30]) (A)

After placement of shunt capacitors (Proposed method) (A)

108.46 136.33 164.53 193.07 221.96 251.23 280.88

92.94 116.80 140.38 165.72 190.27 216.28 244.03

91.06 114.31 137.75 161.41 185.31 209.43 233.86

150.02 188.46 227.31 266.59 306.33 346.55 387.29

125.26 156.46 185.08 218.48 251.60 287.71 325.00

122.26 153.47 184.85 216.67 248.68 281.24 314.25

Fig. 12. Real power loss with and without capacitor placement of 51 node network.

Fig. 13. Real power loss with and without capacitor placement for 69 node network.

Table 13 Substation power factor before and after placement of shunt capacitors. Load level

0.4 0.5 0.6 0.7 0.8 0.9 1.0

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (lag)

After placement of shunt capacitors (Method in [30] (lag)

After placement of shunt capacitors (Proposed method) (lag)

Before placement of shunt capacitors (lag)

After placement of shunt capacitors (Method in [30]) (lag)

After placement of shunt capacitors (Proposed method) (lag)

0.8418 0.8413 0.8409 0.8405 0.8400 0.8396 0.8391

0.9777 0.9762 0.9784 0.9708 0.9703 0.9645 0.9542

0.9999 0.9999 0.9999 0.9998 0.9999 0.9999 0.9998

0.8179 0.8185 0.8190 0.8196 0.8201 0.8207 0.8214

0.9734 0.9779 0.9967 0.9884 0.9850 0.9735 0.9624

0.9983 0.9982 0.9992 0.9986 0.9992 0.9998 0.9999

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S.R. Gampa, D. Das / Electrical Power and Energy Systems 77 (2016) 314–326 Table 14 Minimum voltage magnitude before and after placement of shunt capacitors. Load level

51 node distribution system

69 node distribution system

Before placement of shunt capacitors (pu)

After placement of shunt capacitors (Method in [30]) (pu)

After placement of shunt capacitors (Proposed method) (pu)

Before placement of shunt capacitors (pu)

After placement of shunt capacitors (Method in [30]) (pu)

After placement of shunt capacitors (Proposed method) (pu)

0.4

V 16 min = 0.9650

V 16 min = 0.9740

V 16 min = 0.9803

V 65 min = 0.9656

V 65 min = 0.9731

V 65 min = 0.9767

0.5

V 16 min = 0.9559

V 16 min = 0. 9664

V 16 min = 0.9753

V 65 min = 0.9567

V 65 min = 0.9662

V 65 min = 0.9708

0.6

V 16 min V 16 min V 16 min V 16 min V 16 min

V 16 min V 16 min V 16 min V 16 min V 16 min

V 16 min V 16 min V 16 min V 16 min V 16 min

V 65 min V 65 min V 65 min V 65 min V 65 min

V 65 min V 65 min V 65 min V 65 min V 65 min

= 0.9634

V 65 min = 0.9652

= 0. 9545

V 65 min = 0.9586

= 0. 9454

V 65 min = 0.9529

= 0. 9361

V 64 min = 0.9498

= 0. 9266

V 64 min = 0.9451

0.7 0.8 0.9 1.0

= 0.9466 = 0.9373 = 0.9277 = 0.9180 = 0.9081

= 0. 9597 = 0.9520 = 0. 9445 = 0. 9363 = 0. 9277

= 0.9693 = 0.9640 = 0.9592 = 0.9538 = 0.9506

= 0.9476 = 0.9383 = 0.9288 = 0.9191 = 0.9092

Fig. 14. Voltage profile comparison at 40% load level for 51 node network.

Fig. 15. Voltage profile comparison at peak load level for 51 node network.

method [30] and proposed method but the real power losses are slightly high in the case of proposed method compared to energy savings method. In the case of proposed method eventhough the real power loss of the distribution system is slightly high, from Table 12, it can be observed that the total current drawn at the substation is significantly less and hence the real power loss will be greatly reduced beyond substation and the grid voltage profile will be improved. The substation power factor improvement due to shunt capacitor placement is shown in Table 13. From Table 13 it can be

observed that the power factor improvement is much better in the case of proposed method as compared to the energy savings method [30] and in the case of proposed fuzzy GA method the power factor is almost improved to unity. The operation of substation at unity power factor using shunt capacitors will be helpful to reduce the on-peak operating cost, deferral of network upgradation, reduction of real power losses, and lower transmission and distribution costs. This kind of operation will help the grid in much better way because the network is almost not drawing any reactive power from the grid.

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Fig. 16. Voltage profile comparison at 40% load level for 69 node network.

Fig. 17. Voltage profile comparison at peak load level for 69 node network.

The voltage profile improvement is shown in Table 14 and from Table 14, it can be observed that in the case of proposed method the minimum voltage improvement is much better compared to all other cases considered. The voltage profile improvement due to shunt capacitor placement at 40% lower load level is shown in Figs. 14 and 16 and the voltage profile improvement at peak load level is shown in Figs. 15 and 17 for both 51 node and 69 node distribution systems respectively. From Figs. 14–17 it can be observed that the proposed fuzzy GA method gives the best voltage profile improvement compared to energy savings method [30] at lower and higher load levels for both the examples of the distribution systems considered.

CRF ¼

ir ðir þ 1Þ N

N

ðir þ 1Þ  1

ð17Þ

Here the value of N is taken as 20 and ir is taken as 10%. For a value of the energy cost Ke = $0.096/kW h, the capacitor purchase cost CC = $6/kVAr and the capacitor installation cost CI = $1000/location, the calculated annual economical savings (SCP) obtained by the installation of shunt capacitor placement are 8870 dollars in the case of 51 node distribution system and 24,200 dollars in the case of 69 node distribution system. Hence it can be said that annual economical benefits can also be obtained with the installation of shunt capacitors in addition to the improvement of the performance of the distribution system.

Annual economical savings due to optimal capacitor placement The annual economical savings due to the effect of optimal shunt capacitor placement using the proposed fuzzy GA method into the distribution system can be obtained by considering the energy savings at the substation. The annual economical savings (SCP) can be given as:

SCP ¼ K e

( ) ! NL  NCL  X X  CRF C PSi  PQC QC þ ðC  NCLÞ T i C I j Si i¼1

ð16Þ

j¼1

where CRF is the capital recovery factor and is defined as the annual loan payment on the borrowed amount for N years at the rate of interest of ir and is given as:

Conclusions In the present work a GA based fuzzy multiobjective approach has been proposed for improving the power factor of the substation and voltage profile of the distribution system. A new sensitivity analysis is used for identifying optimal nodes for capacitor placement. A fuzzy multiobjective function is formed with the objectives of reduction of substation reactive power component current, the real power loss reduction, to avoid branch current constraint violation and to satisfy minimum and maximum voltage limit conditions. The optimum sizing of shunt capacitors is obtained using GA by maximizing the fuzzy multiobjective

S.R. Gampa, D. Das / Electrical Power and Energy Systems 77 (2016) 314–326

function. From the simulation results it is observed that the proposed method is advantageous in improving the substation power factor and voltage profile compared to the shunt capacitor placement based on annual energy savings method. In the case of proposed fuzzy GA method the reactive current component became negligible and the substation power factor improved almost to unity at all the load levels with the incorporation of optimum shunt capacitors into the distribution system. The voltage profile has been improved to much better level compared to other cases and real power losses are reduced to significant value. In addition to that a significant amount of annual savings has also been achieved.

Appendix A See Table A.

Table A 51 node radial distribution system data. Br no

Send. end

Recev. end

R (X)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3 17 18 19 20 21 4 23 24 25 26 27 28 6 30 31 32 33 34 35 36 37 38 7 40 41 42 43 44 9 46 47 48 49 50

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

0.2740 0.1370 0.3288 0.1096 0.5400 0.3600 0.3600 0.7200 2.7320 2.0490 2.0490 0.9562 1.0928 1.5026 3.0052 2.7320 0.8196 1.3660 1.3660 2.0490 1.5026 1.6392 1.7758 1.0928 0.8196 0.5464 1.0928 0.2732 0.7020 0.6480 0.6480 0.6480 0.5400 0.3240 0.3888 0.4320 0.5940 0.7020 1.9124 3.0052 2.4588 2.1856 2.1856 0.6830 0.9562 1.0245 1.2294 1.7758 1.6392 1.3660

X (X)

0.3560 0.1780 0.4272 0.1424 0.4356 0.2904 0.2904 0.5808 0.7792 0.5844 0.5844 0.2727 0.3117 0.4286 0.8571 0.7792 0.2338 0.3896 0.3896 0.5844 0.4286 0.4675 0.5065 0.3117 0.2338 0.1558 0.3117 0.0779 0.4774 0.4406 0.4406 0.4406 0.3672 0.2203 0.2644 0.2938 0.4039 0.4774 0.5454 0.8571 0.7013 0.6234 0.6234 0.1948 0.2727 0.2922 0.3506 0.5065 0.4675 0.3896

Recev. end PL (kW)

QL (kVAr)

40 60 20 80 38 20 60 70 60 80 10 25 55 120 40 35 60 80 60 50 50 80 45 38 78 16 18 40 40 20 30 36 50 27 33 42 55 44 80 60 45 48 68 77 60 40 45 70 30 35

30 40 10 60 18 15 40 45 35 50 5 15 45 80 25 25 30 50 35 30 30 60 25 18 48 8 10 30 30 15 20 26 40 18 16 22 30 26 70 30 30 28 38 23 30 20 45 50 20 30

IC (A) 324 324 324 324 234 234 234 234 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 115 208 208 208 208 208 208 208 208 208 208 115 115 115 115 115 115 115 115 115 115 115 115

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