# A Comparison of Metaheuristic Techniques for Solving Optimal Sitting and Sizing Problems of Capacitor Banks to Reduce the Power Loss in Radial Distribution System

Authors: Tongfei Lei; Saleem Riaz; Hira Raziq; Munira Batool; Feng Pan; Jianfeng Wang
Journal: Complexity
 (2022)

---

## Abstract

The losses in the radial distribution system are inevitable which needs to be minimized for the proper transmission of power to the end customers. This problem can be solved by the allocation of capacitor banks at proper locations with appropriate sizing. These allocations need an efficient approach for the performance enhancement of RDS. In this paper, several metaheuristic techniques such as particle swarm optimization (PSO), Harmony search, Bat, Cuckoo, and Grey-wolf (GW) algorithms are employed to find the size of capacitor banks. Loss sensitivity analysis is considered for the indication of candidate buses where a capacitor has to be installed to reduce the total system losses and ultimately increase efficiency. Cost-effectiveness, power loss minimization, and voltage enhancement can be determined and compared for these 5 techniques and are implemented on the IEEE-34 bus system to illustrate the efficacy of each of them. The results show the advantages and drawbacks of the techniques separately. The simulations are carried out in MATLAB.


---

## Body

## 1. Introduction

The voltage in a radial distribution system goes on decreasing from the substation to the end customer due to the losses generated in lines. The percentage of losses in the form ofI2R in a radial distribution system is 10–13% of generated power. Power loss minimization and voltage profile improvement are one of the most important technical issues to deal with. These issues can be reduced by employing distributed generation units at different locations of the system. Similarly, losses can also be reduced by allocating capacitor banks at suitable locations in the system. Two important scenarios are incorporating capacitor which is the basic step for the minimization of the losses and the allocation of these capacitors at the specific locations in a system which is the main objective. Locating capacitor banks in a radial distribution system is done through some heuristic techniques. Based on the approach of these techniques size and position of a capacitor can be found [1–4]. Allocation at a certain position is critical for the system so it is really important to place the CB at a certain specified location if not placed properly that will not only lead to increased losses but also results in the failure of a system. Hence classical methods have been used by researchers to solve the optimal sizing and location of a capacitor. One of the approaches is teaching strategy-based algorithm. In this, the size and position are determined by student and instructor relation-based schemes [5, 6].To allocate the position and size of capacitors for the control of active and reactive power flow in the power system some genetic algorithms have been used. These genetic algorithms with some modifications shows better performance when compared with their standard counterparts [7–9]. To reduce the power loss and to find the exact location and size of the capacitor different genetic algorithms with some improvements are found to be more promising. One of the bio-inspired algorithms is the monkey search algorithm which uses the strategy of the monkeys for the sizing and allocating the capacitor banks [10]. The PSO optimization technique improves the result by repetitively improving a solution concerning the task given to it. The adopted PSO algorithm shows some improvements in the conventional PSO [11].A direct search algorithm was implemented on standard bus system and exhibit better power loss reduction, voltage characterization with less cost when compared with PSO. The algorithm searches for all possible locations in the system by a specific capacitor size and places the capacitor on the bus providing a significant reduction in operating power loss. The correct sizes are chosen to be the standard sizes available on the market i.e., different capacitor sizes are considered. The integration of renewable energy resources i.e. wind with the distributed system leads to higher power losses and voltage reduction due to varying wind speed. The intermittent nature of the wind creates the unbalance power which has to be improved with the PSO algorithm. Modifications are carried out in the objective function of the algorithm to obtain adequate results. Hybrid searching algorithms combine two or more algorithms and use the advantages of one algorithm on the other and results in improved results than if both of them are used individually [12–14]. The recursive techniques are used for the estimation of unknown matrix for linear system. The repetitiveness leads to exact location and dimension of the unknown variable by estimating different values in a continuous loop until correct standard is achieved and putting them in the matrix. For the reliability and high efficiency of the system specific equipment and size of the material is necessary. The supercapacitor nowadays is getting popular due to its robust nature of fast charging and discharging. For integration of supercapacitor in the energy storage system require the specific size and the parameter of the supercapacitor. The capacitor allocation in the power system requires the suitable optimization technique. Some of the researchers had used plant growth optimization to minimize the power losses. Loss sensitivity can be employed for the determination of the specific location for the employment of capacitor and the optimization technique is used for the size of the capacitor. In recent times a new algorithm i.e. Bacterial Foraging Optimization algorithm was incorporated for finding the solution of the problem by minimizing the power loss [15–18]. For several years the researchers try to minimize the power losses but that results in large compensating locations that will make the system complex and costly so minimizing the number of locations at which capacitors are to be installed is challenging. So, in the following research, the compensating locations are minimized to two or three and better results are to be achieved, with less complexity and cost-effectiveness compared to other techniques available in the literature. Budget allocation for genetic algorithm improves the search efficiency for selecting the highest solution for the optimization problem. The allocation can be extended to multi-objective optimization problem for correct selection of a candidate solution [19].Iterative control method is used to locate the specific position and time of the fault by its repetitive property. Designing the iterative control strategy is a difficult task for both linear and nonlinear systems. Some error convergence techniques i.e Norm optimal, P-type,PD∝-type iterative control are implemented. The data produced in each iteration is used for the next iteration for improvement in control system. The optimal control scheme based on optimization shows improves performance of the iterative control algorithm. For system optimization Convergence speed of iterative control algorithm is important and is improved by applying optimal control scheme [20–25]. The efficiency of a system is based on the optimization approach. Different strategies based on the failure management is proposed for the maintenance management of a system. The maintenance and failure cost increase with time that urges for the optimization based on the complexity of the system [26, 27].A harmonic search algorithm has been used to solve optimal placement and sizing of the capacitor bank. Fuzzy-based GA has been used to determine the optimal size with the multi-objective functions to minimize the energy cost and enhance the voltage profile [28]. BAT algorithm is implied to calculate the optimal size of the capacitor. The loss sensitivity factor determines the best and fewer number of locations for capacitor placement. This technique is slightly different than the other existing techniques based on LSF approach for sizing of capacitor [7]. Furthermore, the Cuckoo algorithm has been used more frequently than the BAT algorithm by researchers due to better power loss minimization but one of the drawbacks of the algorithm is the number of locations cannot be minimized [29]. Similarly, the Grey wolf algorithm is introduced to compensate for all the issues associated with power loss minimization effectively and efficiently. The convergence rate is higher than the other optimization techniques available. Similarly, the cost is lower than the direct and teaching-based algorithms [30, 31].The foremost and primary aim of this work has to reduce the total power loss of the system. The proposed algorithms are verified on the IEEE 34-bus radial distribution system. The acquired outcomes are matched with the other present methods and the results are comparably shown in the table. MATLAB software is used for the Load flow analysis of IEEE standard bus system.The main contributions of the paper are as follows:(i)
Proposal of various optimization algorithms to find the accurate size of capacitor banks for power loss reduction in distribution network(ii)
Step-by step implementation of proposed algorithms on the standard IEEE bus system for finiding the most suitable technique(iii)
Selection of most suitable technique for power loss reduction and voltage improvement of distribution network by comparitive analysis of implemented optimization algorithm(iv)
Validating the effective operation of the proposed techniques using numerical analysis


## 2. Materials and Methods

The algorithms presented in the paper are applied to the IEEE 34-bus radial distribution system shown in Figure1. The line and bus data are given in [33]. The total real and reactive powers are 4636.5 KW and 2873.5 KW respectively. The real and the reactive power losses of the system without the incorporation of capacitors are 223.85 kW and 66.47 kVAR.Figure 1
IEEE-34 bus radial distribution system [32].The exact and precise approach for Distribution load flow is used for the determination ofP, Q losses, and voltages. The model of distribution system is given in Figure 2.Figure 2
Distribution system [7].Wherem, m+1 = buses.Vm, Vm+1  = voltages at bus m and m + 1, respectively.Jm  = branch current.From Figure2 above the expression of Vm+1 can be written as(1)Vm+1=Vm−JmRm+jXm,where V = IR and J = (BIBC) × (I), which is the Matrix representation of branch current in each line, BIBC is the Bus current Injection to branch current matrix.As the formula of complex power is given as(2)S=VI∗,I=SV∗,where S=P + jQ.Putting the value ofS in I,(3)I=Pm+jQmVm∗.


### 2.1. Expression for Active Power Loss

(4)PLossm,m+1=I2∗Rm=Sm|Vm|2Rm,whereS=P2+Q2=Pm2+Qm2Vm2∗Rm.ThePLossm,m+1 is the active power loss in the line section between buses m and m + 1.


### 2.2. Expression for Reactive Power Loss

(5)QLossm,m+1=Sm|Vm|2∗Xm,=Pm2+Qm2Vm2∗Xm,where QLossm,m+1  = Reactive power loss in the line section between buses m and m + 1.Pm and Qm  = Active and Reactive power flowing out of bus m,Xm  = Reactance of the line section between buses m and m + 1,Vm  = Magnitude of voltage at bus m.


### 2.3. Sum of Active Power Loss

The total power loss can be computed by adding up all the branch losses. The expression for total power loss is mentioned as follows:(6)Pt,loss=∑m=1nbPLossm,m+1.


### 2.4. Sum of Reactive Power Loss

The equation for total reactive power loss is(7)Qt,loss=∑m=1nbQLossm,m+1.


### 2.5. Objective Function for the Problem

The objective function is used for the minimization of power loss in the system. The mathematical formulation of the objective function is given by(8)MinimizeF=minPt,loss.subject to the following operating constraints.


#### 2.5.1. Power Balance

(9)Pss=∑m=2nPLm+∑m=1nbPLossm,m+1−∑m=1nbPcap,m,where PLm  = real power load at bus m.Pcap,m  = reactive power supplied by capacitor.m = bus numbern = total number of busesnb = total number of the branches.


#### 2.5.2. Voltage Deviation Limit

(10)Vm,min≤Vm≤Vm,max,where Vm,minandVm,max  = minimum and maximum voltage limits at bus m.


#### 2.5.3. Reactive Power Compensation

(11)Qcm,min≤Qcm≤Qcm,max,m=1,…,NB,where Qcm,min and Qcm,max  = minimum and maximum reactive power limits of compensated bus m.


### 2.6. Loss Sensitivity Factor for Node Selection

LSF approach is employed to assess the sensitivity of the buses and is useful for the selection of candidate buses for locating capacitor. The LSF values for all lines and buses are calculated by sorting these values in descending order. Those nodes which have the largest value of LSF are considered to be the candidate buses and capacitors will be only installed on those nodes. Following are the advantages of LSF:(i)
It helps to figure out the buses on which capacitors are to be installed(ii)
It reduces the search space for optimization(iii)
It is easy and simple to implementAssume there are two nodesm and n connected through a branch k. LSF can be calculated by partial derivative of the system losses with respect to reactive power consumption at the destination node n as follows:(12)∂Plineloss∂Qm=2QmRkVm2,where Plossm,n Power loss in nodes m and n,Qmn is reactive power in nodes m and n(13)∂Qlineloss∂Qm=2QmXkVm2,where Qm is reactive power beyond node m, Rk  = resistance of a branch k, Xk is the reactance of the branch k.


#### 2.6.1. Unit of LSF

As the formula of LSF is given as(14)∂Ploss∂Qm=2QmRkVm2=Var.ΩV.V,Var.A−1VasVΩ=Amp&ΩV=A−1=VarAV=AVAV=no unit.Hence, it is concluded from this derivation that LSF has no unit.


#### 2.6.2. Net Savings

Parameters utilized for the net savings are energy rateCe = $0.06/kWh, the cost of capacitor that has to be incorporated Ccl = $1000/each location, cost of buying capacitor Cc = $3.0/kVAR, T = 8760. The total savings can be given by(15)Netsavings=Ce∗PLossCap∗T−Ccl∗NBC−Cc∗Tc,where NBC  = number of buses that are compensated.(1)
Tc  = total rating of a capacitor(2)
PLossCap  = total power loss of the system with capacitor [7]


### 2.7. Algorithms

Five algorithms namely PSO, Harmony search, BAT, Cuckoo, and Grey-wolf are defined and discussed. The properties of each of them make them unique in solving complex problems. Flowcharts show the working hierarchy of the techniques in tackling difficult tasks.


#### 2.7.1. Particle Swarm Optimization Technique

PSO technique is originated from basic two concepts:(i)
The Examination of swarming modes of animals like bird or fish(ii)
The area of mutative computation is like a genetic algorithm [11]As the name suggests it works on having a swarm of particles. The particles are considered as the set of possible solutions in the population of a swarm. The particles will roam around the space based on mathematical formulas. The particle’s motion is governed by their own best positions and that of the swarms. Once the best position is located it will lead the other swarm to achieve the best location. The process is repetitive and expects to find suitable results. The flowchart shown below in Figure3 demonstrates the hierarchy of the technique.Figure 3
PSO algorithm flow chart.


#### 2.7.2. Harmony Search Algorithm

The harmony search algorithm is based on the creation of pleasant harmony by using different musical instruments and their pitches by the musician. The musician searching for new harmonies for the uniqueness of their sound made the basis for this algorithm. In this searching strategy, the algorithm finds the best solution to the problem with less computational time and rapid convergence. Figure4 shows the flowchart of the scheme.Figure 4
Harmony search algorithm flowchart [34].


#### 2.7.3. BAT Algorithm

From the nature of environmental systems, intelligent-based algorithms have been derived. Bat Algorithm (BA) was first introduced by Xin-She-Xang to optimize the engineering problem [35], is built on the microbat’s property of echolocation concerning varying pulse rates of diffusion and noise shown in Figure 5.Figure 5
Bat algorithm flowchart.BAT algorithm is based on the following:(i)
Echolocation behavior of microbats(ii)
Variation in frequency(iii)
Variation in loudness


#### 2.7.4. Cuckoo Algorithm

The cuckoo algorithm is built based on the cuckoo bird’s behavior of laying eggs in the nests of other birds. It is used for the optimization of complex problems i.e. Global optimization problems and NP-hard problems that can be efficiently resolved. The important property of the algorithm is its operation of flight for updating the search space for the new solution generation. One of its properties is the modification of the solution during the iterations by adding small changes [36–38]. The flowchart of the algorithm is shown below in Figure 6.Figure 6
Cuckoo algorithm flow chart [39].


#### 2.7.5. Grey-Wolf Algorithm

The grey-Wolf algorithm is based on the leadership hierarchy and the behavior of grey wolves as they hunt their prey in a group form. Hierarchy has four levels alpha, beta, delta, and omega with each of the levels designated with different responsibilities. The topmost alphas are the organizers and decision-makers for the entire group, which each of them has to follow. Beta advises the alpha in the decision-making process and manages the group according to the orders given by the alpha. The bot-tom-most omega is the one to solve the problems among the different levels of the group. Delta is assigned to warn the wolves in case of danger, provide food to the group, caring for them if any one of them gets sick.Another stimulating feature of the grey wolves is the hunting of their prey in the form of groups i.e. they search for their prey, encircle them and attack them collectively. This hierarchy scheme with the hunting mechanism made the basis for the optimization problem. The hierarchy scheme and hunting behavior are shown via pyramid and flowchart below in Figures7 and 8 respectively.Figure 7
GWO pyramid [40].Figure 8
Grey wolf algorithm flow chart [41].


## 2.1. Expression for Active Power Loss

(4)PLossm,m+1=I2∗Rm=Sm|Vm|2Rm,whereS=P2+Q2=Pm2+Qm2Vm2∗Rm.ThePLossm,m+1 is the active power loss in the line section between buses m and m + 1.


## 2.2. Expression for Reactive Power Loss

(5)QLossm,m+1=Sm|Vm|2∗Xm,=Pm2+Qm2Vm2∗Xm,where QLossm,m+1  = Reactive power loss in the line section between buses m and m + 1.Pm and Qm  = Active and Reactive power flowing out of bus m,Xm  = Reactance of the line section between buses m and m + 1,Vm  = Magnitude of voltage at bus m.


## 2.3. Sum of Active Power Loss

The total power loss can be computed by adding up all the branch losses. The expression for total power loss is mentioned as follows:(6)Pt,loss=∑m=1nbPLossm,m+1.


## 2.4. Sum of Reactive Power Loss

The equation for total reactive power loss is(7)Qt,loss=∑m=1nbQLossm,m+1.


## 2.5. Objective Function for the Problem

The objective function is used for the minimization of power loss in the system. The mathematical formulation of the objective function is given by(8)MinimizeF=minPt,loss.subject to the following operating constraints.


### 2.5.1. Power Balance

(9)Pss=∑m=2nPLm+∑m=1nbPLossm,m+1−∑m=1nbPcap,m,where PLm  = real power load at bus m.Pcap,m  = reactive power supplied by capacitor.m = bus numbern = total number of busesnb = total number of the branches.


### 2.5.2. Voltage Deviation Limit

(10)Vm,min≤Vm≤Vm,max,where Vm,minandVm,max  = minimum and maximum voltage limits at bus m.


### 2.5.3. Reactive Power Compensation

(11)Qcm,min≤Qcm≤Qcm,max,m=1,…,NB,where Qcm,min and Qcm,max  = minimum and maximum reactive power limits of compensated bus m.


## 2.5.1. Power Balance

(9)Pss=∑m=2nPLm+∑m=1nbPLossm,m+1−∑m=1nbPcap,m,where PLm  = real power load at bus m.Pcap,m  = reactive power supplied by capacitor.m = bus numbern = total number of busesnb = total number of the branches.


## 2.5.2. Voltage Deviation Limit

(10)Vm,min≤Vm≤Vm,max,where Vm,minandVm,max  = minimum and maximum voltage limits at bus m.


## 2.5.3. Reactive Power Compensation

(11)Qcm,min≤Qcm≤Qcm,max,m=1,…,NB,where Qcm,min and Qcm,max  = minimum and maximum reactive power limits of compensated bus m.


## 2.6. Loss Sensitivity Factor for Node Selection

LSF approach is employed to assess the sensitivity of the buses and is useful for the selection of candidate buses for locating capacitor. The LSF values for all lines and buses are calculated by sorting these values in descending order. Those nodes which have the largest value of LSF are considered to be the candidate buses and capacitors will be only installed on those nodes. Following are the advantages of LSF:(i)
It helps to figure out the buses on which capacitors are to be installed(ii)
It reduces the search space for optimization(iii)
It is easy and simple to implementAssume there are two nodesm and n connected through a branch k. LSF can be calculated by partial derivative of the system losses with respect to reactive power consumption at the destination node n as follows:(12)∂Plineloss∂Qm=2QmRkVm2,where Plossm,n Power loss in nodes m and n,Qmn is reactive power in nodes m and n(13)∂Qlineloss∂Qm=2QmXkVm2,where Qm is reactive power beyond node m, Rk  = resistance of a branch k, Xk is the reactance of the branch k.


### 2.6.1. Unit of LSF

As the formula of LSF is given as(14)∂Ploss∂Qm=2QmRkVm2=Var.ΩV.V,Var.A−1VasVΩ=Amp&ΩV=A−1=VarAV=AVAV=no unit.Hence, it is concluded from this derivation that LSF has no unit.


### 2.6.2. Net Savings

Parameters utilized for the net savings are energy rateCe = $0.06/kWh, the cost of capacitor that has to be incorporated Ccl = $1000/each location, cost of buying capacitor Cc = $3.0/kVAR, T = 8760. The total savings can be given by(15)Netsavings=Ce∗PLossCap∗T−Ccl∗NBC−Cc∗Tc,where NBC  = number of buses that are compensated.(1)
Tc  = total rating of a capacitor(2)
PLossCap  = total power loss of the system with capacitor [7]


## 2.6.1. Unit of LSF

As the formula of LSF is given as(14)∂Ploss∂Qm=2QmRkVm2=Var.ΩV.V,Var.A−1VasVΩ=Amp&ΩV=A−1=VarAV=AVAV=no unit.Hence, it is concluded from this derivation that LSF has no unit.


## 2.6.2. Net Savings

Parameters utilized for the net savings are energy rateCe = $0.06/kWh, the cost of capacitor that has to be incorporated Ccl = $1000/each location, cost of buying capacitor Cc = $3.0/kVAR, T = 8760. The total savings can be given by(15)Netsavings=Ce∗PLossCap∗T−Ccl∗NBC−Cc∗Tc,where NBC  = number of buses that are compensated.(1)
Tc  = total rating of a capacitor(2)
PLossCap  = total power loss of the system with capacitor [7]


## 2.7. Algorithms

Five algorithms namely PSO, Harmony search, BAT, Cuckoo, and Grey-wolf are defined and discussed. The properties of each of them make them unique in solving complex problems. Flowcharts show the working hierarchy of the techniques in tackling difficult tasks.


### 2.7.1. Particle Swarm Optimization Technique

PSO technique is originated from basic two concepts:(i)
The Examination of swarming modes of animals like bird or fish(ii)
The area of mutative computation is like a genetic algorithm [11]As the name suggests it works on having a swarm of particles. The particles are considered as the set of possible solutions in the population of a swarm. The particles will roam around the space based on mathematical formulas. The particle’s motion is governed by their own best positions and that of the swarms. Once the best position is located it will lead the other swarm to achieve the best location. The process is repetitive and expects to find suitable results. The flowchart shown below in Figure3 demonstrates the hierarchy of the technique.Figure 3
PSO algorithm flow chart.


### 2.7.2. Harmony Search Algorithm

The harmony search algorithm is based on the creation of pleasant harmony by using different musical instruments and their pitches by the musician. The musician searching for new harmonies for the uniqueness of their sound made the basis for this algorithm. In this searching strategy, the algorithm finds the best solution to the problem with less computational time and rapid convergence. Figure4 shows the flowchart of the scheme.Figure 4
Harmony search algorithm flowchart [34].


### 2.7.3. BAT Algorithm

From the nature of environmental systems, intelligent-based algorithms have been derived. Bat Algorithm (BA) was first introduced by Xin-She-Xang to optimize the engineering problem [35], is built on the microbat’s property of echolocation concerning varying pulse rates of diffusion and noise shown in Figure 5.Figure 5
Bat algorithm flowchart.BAT algorithm is based on the following:(i)
Echolocation behavior of microbats(ii)
Variation in frequency(iii)
Variation in loudness


### 2.7.4. Cuckoo Algorithm

The cuckoo algorithm is built based on the cuckoo bird’s behavior of laying eggs in the nests of other birds. It is used for the optimization of complex problems i.e. Global optimization problems and NP-hard problems that can be efficiently resolved. The important property of the algorithm is its operation of flight for updating the search space for the new solution generation. One of its properties is the modification of the solution during the iterations by adding small changes [36–38]. The flowchart of the algorithm is shown below in Figure 6.Figure 6
Cuckoo algorithm flow chart [39].


### 2.7.5. Grey-Wolf Algorithm

The grey-Wolf algorithm is based on the leadership hierarchy and the behavior of grey wolves as they hunt their prey in a group form. Hierarchy has four levels alpha, beta, delta, and omega with each of the levels designated with different responsibilities. The topmost alphas are the organizers and decision-makers for the entire group, which each of them has to follow. Beta advises the alpha in the decision-making process and manages the group according to the orders given by the alpha. The bot-tom-most omega is the one to solve the problems among the different levels of the group. Delta is assigned to warn the wolves in case of danger, provide food to the group, caring for them if any one of them gets sick.Another stimulating feature of the grey wolves is the hunting of their prey in the form of groups i.e. they search for their prey, encircle them and attack them collectively. This hierarchy scheme with the hunting mechanism made the basis for the optimization problem. The hierarchy scheme and hunting behavior are shown via pyramid and flowchart below in Figures7 and 8 respectively.Figure 7
GWO pyramid [40].Figure 8
Grey wolf algorithm flow chart [41].


## 2.7.1. Particle Swarm Optimization Technique

PSO technique is originated from basic two concepts:(i)
The Examination of swarming modes of animals like bird or fish(ii)
The area of mutative computation is like a genetic algorithm [11]As the name suggests it works on having a swarm of particles. The particles are considered as the set of possible solutions in the population of a swarm. The particles will roam around the space based on mathematical formulas. The particle’s motion is governed by their own best positions and that of the swarms. Once the best position is located it will lead the other swarm to achieve the best location. The process is repetitive and expects to find suitable results. The flowchart shown below in Figure3 demonstrates the hierarchy of the technique.Figure 3
PSO algorithm flow chart.


## 2.7.2. Harmony Search Algorithm

The harmony search algorithm is based on the creation of pleasant harmony by using different musical instruments and their pitches by the musician. The musician searching for new harmonies for the uniqueness of their sound made the basis for this algorithm. In this searching strategy, the algorithm finds the best solution to the problem with less computational time and rapid convergence. Figure4 shows the flowchart of the scheme.Figure 4
Harmony search algorithm flowchart [34].


## 2.7.3. BAT Algorithm

From the nature of environmental systems, intelligent-based algorithms have been derived. Bat Algorithm (BA) was first introduced by Xin-She-Xang to optimize the engineering problem [35], is built on the microbat’s property of echolocation concerning varying pulse rates of diffusion and noise shown in Figure 5.Figure 5
Bat algorithm flowchart.BAT algorithm is based on the following:(i)
Echolocation behavior of microbats(ii)
Variation in frequency(iii)
Variation in loudness


## 2.7.4. Cuckoo Algorithm

The cuckoo algorithm is built based on the cuckoo bird’s behavior of laying eggs in the nests of other birds. It is used for the optimization of complex problems i.e. Global optimization problems and NP-hard problems that can be efficiently resolved. The important property of the algorithm is its operation of flight for updating the search space for the new solution generation. One of its properties is the modification of the solution during the iterations by adding small changes [36–38]. The flowchart of the algorithm is shown below in Figure 6.Figure 6
Cuckoo algorithm flow chart [39].


## 2.7.5. Grey-Wolf Algorithm

The grey-Wolf algorithm is based on the leadership hierarchy and the behavior of grey wolves as they hunt their prey in a group form. Hierarchy has four levels alpha, beta, delta, and omega with each of the levels designated with different responsibilities. The topmost alphas are the organizers and decision-makers for the entire group, which each of them has to follow. Beta advises the alpha in the decision-making process and manages the group according to the orders given by the alpha. The bot-tom-most omega is the one to solve the problems among the different levels of the group. Delta is assigned to warn the wolves in case of danger, provide food to the group, caring for them if any one of them gets sick.Another stimulating feature of the grey wolves is the hunting of their prey in the form of groups i.e. they search for their prey, encircle them and attack them collectively. This hierarchy scheme with the hunting mechanism made the basis for the optimization problem. The hierarchy scheme and hunting behavior are shown via pyramid and flowchart below in Figures7 and 8 respectively.Figure 7
GWO pyramid [40].Figure 8
Grey wolf algorithm flow chart [41].


## 3. Results and Discussion

Loss sensitivity analysis is carried out on IEEE-34 bus system Voltage profile improvement and power loss minimization graphs [7] can be seen for the base case and the proposed schemes as discussed below.


### 3.1. IEEE Radial 34 Bus System LSF

Loss sensitivity analysis at each node of the IEEE-34 bus system is calculated for the indication of candidate buses where capacitor has to be installed to reduce the total system losses which in turn increases the efficiency of the RDS. The loss sensitivity factor is calculated using Microsoft excel and the results are shown in Figure9.Figure 9
LSF at each node IEEE-34 bus system.


### 3.2. Voltage Profile and Power Loss in an IEEE-34 Bus System

Voltage and power loss in the 34-bus system have been determined with and without capacitors. The voltage is enhanced and power loss minimization occurs.


#### 3.2.1. Base Case

Voltage and power loss without capacitor incorporation are shown in Figure10. The maximum voltage is 1 pu and the minimum voltage is 0.9423 pu. These values are taken as the base values for the other results in which the capacitor is allocated in the system. Real Power loss is 223.85 kW while reactive power loss in the base case is 66.4783 kVAR.Figure 10
(a) Voltage profile IEEE-34 bus system. (b) Power loss (kW) at each node IEEE-34 bus system.
(a)(b)


#### 3.2.2. Proposed Method

Allocation of capacitor in power system leads to improved voltages and power loss minimization. Different techniques are used for locating and sizing of capacitor for voltage improvement and power loss minimization as shown in Table1. Outputs of BAT and grey wolf are shown in Figures 11 and 12.Table 1
Results comparison of metaheuristic techniques on an IEEE-34 bus system.
ParameterBase casePSO [34]Harmonic searchBATCuckooGrey wolfCapacitor size and sitting (node)—781 (19)803 (22) 479 (20)1000 (19)800 (26)650 (10)1150 (25)10 (650)19 (850)25 (700)500(10)500(25)660 (19)290 (22)Minimum voltage (node)0.9423 (bus 27)—0.9501 (27)0.9500 (27)0.9503 (27)0.9507 (27)Maximum voltage (node)1.00 (bus 1)—1.00 (1)1.00 (1)1.00 (1)1.00 (1)Real power loss (kW)223.85168.6167.9468165.3482160.3159.29Reactive power loss (kVAR)66.478349.992449.042249.957948.824347.8127%P loss reduction—24.681724.973526.134328.389528.84%%Q loss reduction—24.798926.228224.850826.55628.0777kVAR injected—20631800180022001950Computation time (s)0.90311717.8094939.8920122.0050970.146820.43199Net savings ($)—1839820635217111919420115Figure 11
BAT (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)Figure 12
GWO (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)For the Bat algorithm, the maximum voltage is 1 pu but the minimum voltage is 0.95 pu which is a considerable improvement from the base case. Similarly, real power loss is 165.34 kW, and reactive power loss is 49.95 kVAr which is minimized then the base case.Grey-wolf gives the most satisfactory result among the 5 optimization problems. Voltages and power losses show the best results among the proposed techniques. The minimum voltage is 0.9507 pu which is the highest among the proposed algorithms similarly active and reactive powers are 159.29 kW and 47.812 kVAR, respectively.Table1 compare the metaheuristic techniques and the base case. Capacitor sizes and their location at nodes in the 34-bus system are mentioned. Different parameters such as minimum and maximum voltages, kVAR injected, active and reactive power losses, percentage power loss reduction and cost analysis have been compared based on different algorithms. The results show satisfactory outputs after the incorporation of CBs.The comparison of the real power losses for different algorithms is shown in Figure13 Without capacitors the losses are the highest among them. The grey-wolf have the lowest losses hence the highest optimization.Figure 13
comparison of Real power losses for different techniques.The net savings for the PSO algorithm is the lowest i.e. $18398 and are highest for Bat and GWO algorithms $21711 and $20115 respectively shown in Figure14. This is one of the advantages of the Bat and GWO in terms of cost.Figure 14
Net savings in different techniques.Power loss reduction is one of the most important features of the power system. For the proper functioning of the system, the losses should be the least otherwise it will lead to other system problems. The power loss reduction is the lowest in PSO and highest in the Grey-wolf optimization technique as shown below in Figure15.Figure 15
Power loss reduction for different techniques.Table2 compares the metaheuristic techniques based on their different approach towards optimization of a problem. It can be seen from Figure 10(a) that the minimum voltage that appeared without compensation kVAR is 0.9423 p.u but as soon as kVAR is injected the voltage profile gets improved and the minimum voltage hence appears is 0.9507 p.u shown in Figure 10(b). If one looks at the power loss graph in Figure 11(a) it can be seen that the power loss has also been reduced power loss at node 1 is about 30.1754 kW Figure 10(a)) but after the capacitor placement, it reduces to about 25.4049 kW shown in Figure 11(b). Using GWO nearly 28.84% real power loss and 28.0777% reactive power loss have been reduced. Moreover, comparison Tables 1 and 2 predicts the pros and cons of each of the techniques. PSO technique converges very slowly and does not have appreciable power loss minimization as far as other techniques are concerned GWO can better minimize the power losses as compared to other techniques and also converges quickly but BAT algorithm can minimize the locations where capacitors have to be installed. The cuckoo algorithm gives better power loss reduction whereas the harmony search algorithm can have minimum kVAR injected and ultimately the number of candidate buses. So a hybrid algorithm could be developed using this research that could achieve superiority over all the mentioned techniques.Table 2
Comparison of metaheuristic techniques.
PSOHarmonic searchBatCuckooGrey-wolfBased on the concept of swarm intelligence for solving complex problemsInspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony.Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm.Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species, e.g., off-spring parasites.The hierarchy scheme and hunting mechanism of grey wolves made the basis of this optimization problemA. Future work.(1)
Using system reconfiguration techniques and then applying the algorithm.(2)
System power loss minimization could be accomplished by optimal location of DG (distributed generation).(3)
An improved Bat algorithm based on the novel initialization technique for global optimization problem could be developed.(4)
Another technique is used that is the hybrid between different techniques, for example, Bat algorithm (BA) and Artificial Bee Colony (ABC) which enables a communication strategy. In this technique, the drawbacks or the bad individuals or bats are covered by good agents of the artificial bee colony and vice versa. At every iteration, the advantages got increased and accuracy and convergence is achieved in a better way. BA is up to 78% and original ABC is at 11% on finding the near best solution improvement.


## 3.1. IEEE Radial 34 Bus System LSF

Loss sensitivity analysis at each node of the IEEE-34 bus system is calculated for the indication of candidate buses where capacitor has to be installed to reduce the total system losses which in turn increases the efficiency of the RDS. The loss sensitivity factor is calculated using Microsoft excel and the results are shown in Figure9.Figure 9
LSF at each node IEEE-34 bus system.


## 3.2. Voltage Profile and Power Loss in an IEEE-34 Bus System

Voltage and power loss in the 34-bus system have been determined with and without capacitors. The voltage is enhanced and power loss minimization occurs.


### 3.2.1. Base Case

Voltage and power loss without capacitor incorporation are shown in Figure10. The maximum voltage is 1 pu and the minimum voltage is 0.9423 pu. These values are taken as the base values for the other results in which the capacitor is allocated in the system. Real Power loss is 223.85 kW while reactive power loss in the base case is 66.4783 kVAR.Figure 10
(a) Voltage profile IEEE-34 bus system. (b) Power loss (kW) at each node IEEE-34 bus system.
(a)(b)


### 3.2.2. Proposed Method

Allocation of capacitor in power system leads to improved voltages and power loss minimization. Different techniques are used for locating and sizing of capacitor for voltage improvement and power loss minimization as shown in Table1. Outputs of BAT and grey wolf are shown in Figures 11 and 12.Table 1
Results comparison of metaheuristic techniques on an IEEE-34 bus system.
ParameterBase casePSO [34]Harmonic searchBATCuckooGrey wolfCapacitor size and sitting (node)—781 (19)803 (22) 479 (20)1000 (19)800 (26)650 (10)1150 (25)10 (650)19 (850)25 (700)500(10)500(25)660 (19)290 (22)Minimum voltage (node)0.9423 (bus 27)—0.9501 (27)0.9500 (27)0.9503 (27)0.9507 (27)Maximum voltage (node)1.00 (bus 1)—1.00 (1)1.00 (1)1.00 (1)1.00 (1)Real power loss (kW)223.85168.6167.9468165.3482160.3159.29Reactive power loss (kVAR)66.478349.992449.042249.957948.824347.8127%P loss reduction—24.681724.973526.134328.389528.84%%Q loss reduction—24.798926.228224.850826.55628.0777kVAR injected—20631800180022001950Computation time (s)0.90311717.8094939.8920122.0050970.146820.43199Net savings ($)—1839820635217111919420115Figure 11
BAT (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)Figure 12
GWO (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)For the Bat algorithm, the maximum voltage is 1 pu but the minimum voltage is 0.95 pu which is a considerable improvement from the base case. Similarly, real power loss is 165.34 kW, and reactive power loss is 49.95 kVAr which is minimized then the base case.Grey-wolf gives the most satisfactory result among the 5 optimization problems. Voltages and power losses show the best results among the proposed techniques. The minimum voltage is 0.9507 pu which is the highest among the proposed algorithms similarly active and reactive powers are 159.29 kW and 47.812 kVAR, respectively.Table1 compare the metaheuristic techniques and the base case. Capacitor sizes and their location at nodes in the 34-bus system are mentioned. Different parameters such as minimum and maximum voltages, kVAR injected, active and reactive power losses, percentage power loss reduction and cost analysis have been compared based on different algorithms. The results show satisfactory outputs after the incorporation of CBs.The comparison of the real power losses for different algorithms is shown in Figure13 Without capacitors the losses are the highest among them. The grey-wolf have the lowest losses hence the highest optimization.Figure 13
comparison of Real power losses for different techniques.The net savings for the PSO algorithm is the lowest i.e. $18398 and are highest for Bat and GWO algorithms $21711 and $20115 respectively shown in Figure14. This is one of the advantages of the Bat and GWO in terms of cost.Figure 14
Net savings in different techniques.Power loss reduction is one of the most important features of the power system. For the proper functioning of the system, the losses should be the least otherwise it will lead to other system problems. The power loss reduction is the lowest in PSO and highest in the Grey-wolf optimization technique as shown below in Figure15.Figure 15
Power loss reduction for different techniques.Table2 compares the metaheuristic techniques based on their different approach towards optimization of a problem. It can be seen from Figure 10(a) that the minimum voltage that appeared without compensation kVAR is 0.9423 p.u but as soon as kVAR is injected the voltage profile gets improved and the minimum voltage hence appears is 0.9507 p.u shown in Figure 10(b). If one looks at the power loss graph in Figure 11(a) it can be seen that the power loss has also been reduced power loss at node 1 is about 30.1754 kW Figure 10(a)) but after the capacitor placement, it reduces to about 25.4049 kW shown in Figure 11(b). Using GWO nearly 28.84% real power loss and 28.0777% reactive power loss have been reduced. Moreover, comparison Tables 1 and 2 predicts the pros and cons of each of the techniques. PSO technique converges very slowly and does not have appreciable power loss minimization as far as other techniques are concerned GWO can better minimize the power losses as compared to other techniques and also converges quickly but BAT algorithm can minimize the locations where capacitors have to be installed. The cuckoo algorithm gives better power loss reduction whereas the harmony search algorithm can have minimum kVAR injected and ultimately the number of candidate buses. So a hybrid algorithm could be developed using this research that could achieve superiority over all the mentioned techniques.Table 2
Comparison of metaheuristic techniques.
PSOHarmonic searchBatCuckooGrey-wolfBased on the concept of swarm intelligence for solving complex problemsInspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony.Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm.Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species, e.g., off-spring parasites.The hierarchy scheme and hunting mechanism of grey wolves made the basis of this optimization problemA. Future work.(1)
Using system reconfiguration techniques and then applying the algorithm.(2)
System power loss minimization could be accomplished by optimal location of DG (distributed generation).(3)
An improved Bat algorithm based on the novel initialization technique for global optimization problem could be developed.(4)
Another technique is used that is the hybrid between different techniques, for example, Bat algorithm (BA) and Artificial Bee Colony (ABC) which enables a communication strategy. In this technique, the drawbacks or the bad individuals or bats are covered by good agents of the artificial bee colony and vice versa. At every iteration, the advantages got increased and accuracy and convergence is achieved in a better way. BA is up to 78% and original ABC is at 11% on finding the near best solution improvement.


## 3.2.1. Base Case

Voltage and power loss without capacitor incorporation are shown in Figure10. The maximum voltage is 1 pu and the minimum voltage is 0.9423 pu. These values are taken as the base values for the other results in which the capacitor is allocated in the system. Real Power loss is 223.85 kW while reactive power loss in the base case is 66.4783 kVAR.Figure 10
(a) Voltage profile IEEE-34 bus system. (b) Power loss (kW) at each node IEEE-34 bus system.
(a)(b)


## 3.2.2. Proposed Method

Allocation of capacitor in power system leads to improved voltages and power loss minimization. Different techniques are used for locating and sizing of capacitor for voltage improvement and power loss minimization as shown in Table1. Outputs of BAT and grey wolf are shown in Figures 11 and 12.Table 1
Results comparison of metaheuristic techniques on an IEEE-34 bus system.
ParameterBase casePSO [34]Harmonic searchBATCuckooGrey wolfCapacitor size and sitting (node)—781 (19)803 (22) 479 (20)1000 (19)800 (26)650 (10)1150 (25)10 (650)19 (850)25 (700)500(10)500(25)660 (19)290 (22)Minimum voltage (node)0.9423 (bus 27)—0.9501 (27)0.9500 (27)0.9503 (27)0.9507 (27)Maximum voltage (node)1.00 (bus 1)—1.00 (1)1.00 (1)1.00 (1)1.00 (1)Real power loss (kW)223.85168.6167.9468165.3482160.3159.29Reactive power loss (kVAR)66.478349.992449.042249.957948.824347.8127%P loss reduction—24.681724.973526.134328.389528.84%%Q loss reduction—24.798926.228224.850826.55628.0777kVAR injected—20631800180022001950Computation time (s)0.90311717.8094939.8920122.0050970.146820.43199Net savings ($)—1839820635217111919420115Figure 11
BAT (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)Figure 12
GWO (a) Voltage profile after kVAR compensation IEEE-34 bus system. (b) Power loss after kVAR compensation IEEE- 34 bus system.
(a)(b)For the Bat algorithm, the maximum voltage is 1 pu but the minimum voltage is 0.95 pu which is a considerable improvement from the base case. Similarly, real power loss is 165.34 kW, and reactive power loss is 49.95 kVAr which is minimized then the base case.Grey-wolf gives the most satisfactory result among the 5 optimization problems. Voltages and power losses show the best results among the proposed techniques. The minimum voltage is 0.9507 pu which is the highest among the proposed algorithms similarly active and reactive powers are 159.29 kW and 47.812 kVAR, respectively.Table1 compare the metaheuristic techniques and the base case. Capacitor sizes and their location at nodes in the 34-bus system are mentioned. Different parameters such as minimum and maximum voltages, kVAR injected, active and reactive power losses, percentage power loss reduction and cost analysis have been compared based on different algorithms. The results show satisfactory outputs after the incorporation of CBs.The comparison of the real power losses for different algorithms is shown in Figure13 Without capacitors the losses are the highest among them. The grey-wolf have the lowest losses hence the highest optimization.Figure 13
comparison of Real power losses for different techniques.The net savings for the PSO algorithm is the lowest i.e. $18398 and are highest for Bat and GWO algorithms $21711 and $20115 respectively shown in Figure14. This is one of the advantages of the Bat and GWO in terms of cost.Figure 14
Net savings in different techniques.Power loss reduction is one of the most important features of the power system. For the proper functioning of the system, the losses should be the least otherwise it will lead to other system problems. The power loss reduction is the lowest in PSO and highest in the Grey-wolf optimization technique as shown below in Figure15.Figure 15
Power loss reduction for different techniques.Table2 compares the metaheuristic techniques based on their different approach towards optimization of a problem. It can be seen from Figure 10(a) that the minimum voltage that appeared without compensation kVAR is 0.9423 p.u but as soon as kVAR is injected the voltage profile gets improved and the minimum voltage hence appears is 0.9507 p.u shown in Figure 10(b). If one looks at the power loss graph in Figure 11(a) it can be seen that the power loss has also been reduced power loss at node 1 is about 30.1754 kW Figure 10(a)) but after the capacitor placement, it reduces to about 25.4049 kW shown in Figure 11(b). Using GWO nearly 28.84% real power loss and 28.0777% reactive power loss have been reduced. Moreover, comparison Tables 1 and 2 predicts the pros and cons of each of the techniques. PSO technique converges very slowly and does not have appreciable power loss minimization as far as other techniques are concerned GWO can better minimize the power losses as compared to other techniques and also converges quickly but BAT algorithm can minimize the locations where capacitors have to be installed. The cuckoo algorithm gives better power loss reduction whereas the harmony search algorithm can have minimum kVAR injected and ultimately the number of candidate buses. So a hybrid algorithm could be developed using this research that could achieve superiority over all the mentioned techniques.Table 2
Comparison of metaheuristic techniques.
PSOHarmonic searchBatCuckooGrey-wolfBased on the concept of swarm intelligence for solving complex problemsInspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony. Inspired by the musical process of searching for a perfect state of harmony.Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm. Inspired by the echolocation behavior of bats. It is a bio-inspired algorithm.Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species e.g. off-spring parasites. Inspired by the behavior of cuckoo species, e.g., off-spring parasites.The hierarchy scheme and hunting mechanism of grey wolves made the basis of this optimization problemA. Future work.(1)
Using system reconfiguration techniques and then applying the algorithm.(2)
System power loss minimization could be accomplished by optimal location of DG (distributed generation).(3)
An improved Bat algorithm based on the novel initialization technique for global optimization problem could be developed.(4)
Another technique is used that is the hybrid between different techniques, for example, Bat algorithm (BA) and Artificial Bee Colony (ABC) which enables a communication strategy. In this technique, the drawbacks or the bad individuals or bats are covered by good agents of the artificial bee colony and vice versa. At every iteration, the advantages got increased and accuracy and convergence is achieved in a better way. BA is up to 78% and original ABC is at 11% on finding the near best solution improvement.


## 4. Conclusion

In this paper, five techniques have been applied to the IEEE-34 bus test system for optimally locating the capacitor banks. The size of the CBs is another issue that has been optimized. The importance of these algorithms in RDS has been pointed out by comparing the results with the base case. Sizing and sitting of the capacitors can also be done without the algorithmic approach but that will increase the chances of risk and in turn reliability of a system becomes less which makes the use of these techniques necessary. Comparison between different algorithms can be carried out to find the best technique for the given problem. PSO is applied for the allocation of CBs but the results do not assure a reliable solution to the problem. Bat algorithm shows satisfactory results by determining the best and minimum optimal locations at which capacitor should be placed. Harmonic search and cuckoo algorithm has been used in the search for better results than the previous ones. The cuckoo algorithm shows better power loss minimization than Bat but the locations of the CBs cannot be minimized which is one of its drawbacks. Similarly, the Grey wolf algorithm is introduced to compensate for the issue of power loss minimization and shows better performance, converges rapidly in an effective and efficient manner.



---

*Source: 4547212-2022-03-25.xml*