Optimal Placement of Capacitors in Radial Distribution Grids via Enhanced Modified Particle Swarm Optimization

Abstract

This paper presents the integration of shunt capacitors in the radial distribution grids (RDG) with constant and time-varying load consideration for the reduction of power losses and total annual cost, which turns to enhance the voltage profile and annual net savings. To gather the stated goals, three objective functions are formulated with system constraints. To solve this identified problem, a novel optimization technique based on the modification of particle swarm optimization is proposed. The solution methodology is divided into two phases. In phase one, potential candidate buses are nominated using the loss sensitivity factor method and in phase two the proposed technique first selects the optimal buses for the capacitor placement among the potential buses then it decides the optimal sizing of the capacitors as well. To demonstrate the performance in terms of efficiency and strength, the proposed technique is tested on IEEE 15, 33, and 69 bus system for the optimal placement and sizing of capacitors (OPSC) problem. The results are achieved in terms of annual net savings for 15 bus (47.66${\%}_{case-1}$, 32.76${\%}_{case-2}$, 26.46${\%}_{case-3})$, 33 bus (33.09% ${}_{case-1}$, 27.06${\%}_{case-2}$, 24.15${\%}_{case-3}$), and 69 bus (34.51% ${}_{case-1}$, 29.43${\%}_{case-2}$, 25.83${\%}_{case-3}$) which are comparable to other state of the art methods, and it also indicates the success of the proposed technique.

1. Introduction

Power demand is increasing globally with time, so it is required to control the power flow efficiently. With the increase of the power demand, power losses are also increased due to which RDG operates at a low power factor and faces low voltage. These power losses can be minimized using conventional methods. Capacitor placement is one of the methods used to minimize the power losses of the system at the distribution level. Capacitor injects leading current in the system, which opposes the lagging current of the system. As a result, system power losses are minimized along with the improvement of voltage profile, power factor, and system stability [1]. This leads to a significant enhancement of annual net savings and total annual cost reduction. Thus, optimal placement and sizing of capacitors (OPSC) have a dynamic role in the RDG.

The complexity of the OPSC problem depends upon the size of the power system. To solve this problem the possible number of solutions depends upon the ${(idx,cq)}^n$ combinations whereas “n” is the length of “idx” and “idx” is the index of the potential candidate buses with “cq” capacitors. Several heuristic approaches have been recommended to resolve the OPSC problem in the literature as it is time-consuming to evaluate the set of solutions. A comprehensive review of reported literature, concentrating on numerous heuristic methods demonstrated in the recent years presented in [2] to discuss the OPSC problem. Various heuristic techniques are applied in the last few years such as fuzzy genetic algorithm [3], grey wolf optimization [4], particle swarm optimization [5], ant lion optimization [6], dolphin algorithm [7], salp swarm algorithm [8], two-stage method [9], Fuzzy-DE and Fuzzy-MAPSO [10], fire fly algorithm [11], crow search algorithm [12], whale optimization algorithm [13], the locust search algorithm [14], vortex search algorithm [15], to resolve OPSC related problem.

Similarly, different analytical methods i-e, loss sensitivity factor (LSF), voltage stability index (VSI), and power loss index (PLI) reported in the literature are used to nominate the potential candidate buses for the optimal capacitor placement and their combination with different optimization techniques are considered to decide the sizing of capacitors in the RDG. In this sequence, several traditional mathematical methods and their combinational schemes with optimization techniques are practiced in the last few years as in [16] LSF and artificial bee colony is presented, LSF and mine blast algorithm in [17], LSF and salp swarm algorithm [18], LSF and Flower pollination algorithm used in [19], LSF and discrete particle swarm optimization [20], VSI and genetic algorithm [21,22] offered VSI and shark smell optimization algorithm. LSF and VSI along with cuckoo search algorithm [23], VSI and LSF combined with bacterial foraging optimization algorithm in [24], LSF and VSI with hybrid algorithm [25]. PLI and crow search algorithm [26], PLI and Improved harmony algorithm in [27] to obtain the solution for OPSC problem.

In many cases, the recently referenced algorithms radiate an impression of being convincing, yet they may not guarantee to show up at ideal cost regard and are challenging to get away from the local minima. Yet PSO algorithm is very much popular among the other algorithms for solving complex optimization problems due to its better exploration and exploitation process. This enhances its ability to handle nonlinear optimization problems effectively. It is also observed in the prior mentioned work that traditional cost functions were used as an objective function for the OPSC problem and most of the researchers used just single or maximum of two cost functions to evaluate the performance of the proposed algorithm for the OPSC problem. But in this paper, three different cost functions, i.e., $Cos{t}_{TI}^{FL},Cos{t}_{TD}^{FL}$ and $Cos{t}_{TD}^{VL}$ (see Equations 11–13) are formulated to resolve the OPSC problem. Besides, a modernistic predominant method reliant upon modification of particle swarm optimization (MPSO) is proposed to unwind the OPSC issue. The alteration is finished by presenting an original inertia term in the velocity equation $(2w-w^2\times v_{iD}^k)$ which improves the capacity of the algorithm to discover the optimal solutions by staying away from the redundancy of particle velocity in each iteration. This approach helps to improve the findings of local best for each particle and global best at the end of each iteration. So as a result, the proposed algorithm has achieved better efficiency and strength. In, addition a simple recursive load flow algorithm is developed to perform the power flow, and the LSF method is used to nominate potential candidate buses for the capacitor placement. Then LSF method is integrated with the MPSO technique for the OPSC to achieve, maximization of annual net savings, reduction of total annual cost, and active power losses. To highlight the performance, the proposed technique is tested on three IEEE bus systems for the OPSC problem. Additionally, a comprehensive analysis is also demonstrated by considering the fixed and switched capacitors with constant and time-varying load for each stated IEEE bus system. The findings indicate that the proposed technique competitively address the OPSC problem in term of efficiency and strength.

Precisely, the following are the vital contributions of this research work:

• Review of analytical, heuristic and their combinational methods reported in the literature for tackling optimal placement and sizing of the capacitor problem.
• A novel modified evolutionary algorithm is executed for reactive power planning in a radial distribution system.
• Analytical and heuristic combinational algorithms are applied on three IEEE distribution systems.
• Three objectives for intensification of annual net savings are achieved while considering system constraints.

The rest of this paper is ordered as follows: in Section 2, we demonstrated three mathematical formulations of the OPSC problem (power flow, objective function, system constraints). Then loss sensitivity factor method, fixed and switched capacitors, proposed (MPSO) technique step by step approach are expressed in Section 3 (Methodology); our experimental setup with the effective outcome results presented in Section 4. Lastly, the paper concludes in Section 5.

2. OPSC Problem Formulation

The key precedence of employing capacitors in the RDG is minimization of power losses, improvement of voltage profile, power factor, and system stability. So, better results can be achieved if the sizing and location of the capacitors are pointed accurately. In this paper, minimization of total annual cost is the main focus while considering the system constraints. Several researchers used just single or, maximum, two cost functions to evaluate the performance of their proposed algorithm for the OPSC problem. However, in this research work, three different cost functions are formulated and an innovative MPSO technique is deployed to solve these cost functions.

2.1. Power Flow Formulation

Before formulating the cost function it is necessary to understand the power flow of a radial type distribution system while minimizing the power losses by placing the capacitors optimally. The flow of active and reactive power in each branch of the system illustrated in Figure 1 is determined using (1) and (2), respectively, [28]:

$$P_p=P_{eff/q}+P_{loss/k}$$

(1)

$$Q_p=Q_{(eff/q)}+Q_{(loss/k)}$$

(2)

Current is an important factor, to find out active and reactive power losses of the system. So, the current flow in each branch of the power system is evaluated using (3).

$$I_k=\frac{P_p-j{Q}_p}{U_{p<-{\delta }_p}}$$

(3)

$$I_k=\frac{U_{p\angle {\delta }_p}-U_{q\angle {\delta }_q}}{R_k+j{X}_k}$$

(4)

Similarly, using the conventional method, active and reactive power losses in each branch expressed in (5) and (6).

$$P_{loss/k}=I_k^2\times R_k$$

(5)

$$Q_{loss/k}=I_k^2\times X_k$$

(6)

$$P_{loss/k}=\left(\frac{P_{eff/q}^2+Q_{eff/q}^2}{{\left|U_q\right|}^2}\right)\times R_k$$

(7)

$$Q_{loss/k}=\left(\frac{P_{eff/q}^2+Q_{eff/q}^2}{{\left|U_q\right|}^2}\right)\times Q_k$$

(8)

By summing up, all the branch losses the resultant can be stated as total active and reactive power losses that occurred in the system are demonstrated in (9) and (10).

$$Total{P}_{loss}=\sum _{q=2}^{N_b}\sum _{k=1}^{N_{b-1}}\left(\frac{P_{eff/q}^2-Q_{eff/q}^2}{{\left|U_q\right|}^2}\right)\times R_k$$

(9)

$$Total{Q}_{loss}=\sum _{q=2}^{N_b}\sum _{k=1}^{N_{b-1}}\left(\frac{P_{eff/q}^2-Q_{eff/q}^2}{{\left|U_q\right|}^2}\right)\times Q_k$$

(10)

where $N_b$ refers to the total number of buses and $N_{b-1}$ are the total branches integer.

2.2. Objective Function

Several kinds of cost functions are reported in the literature to evaluate the performance of the algorithms. In this paper, three kinds of cost functions are considered based on total power loss cost and its combination with capacitor cost. Capacitor cost can be categorized as: fixed capacitor cost, variable capacitor cost, and capacitor installation cost. The total annual cost of the RDG reported in [17] is considered as first objective function, which is gathered by optimal placement and sizing of the capacitors and its relation is formulated as follows:

$$min{\left(Cos{t}_{TI}^{FL}\right)}_{case-1}=K_p\times Total{P}_{loss/cp}+\sum _{q=1}^n{K}_c\times Q_{qc}$$

(11)

where $FL$ is the fix load, $TI$ is time independent, $K_p$ is the cost of power losses in $/kWh, $Total{P}_{loss/cp}$ are the power losses with capacitor placement, $K_c$ is the cost in $/kVAr of $Q_{qc}$ capacitor rating in (11). For details refer to

Table 1. Parameters of objective functions.

Parameters	Study Case-1	Study Case-2	Study Case-3
$K_p$	168$	0.06$	0.06$
$K_{ic}$	- - -	1000$	1000$
T	- - -	8760 h	8760 h
$K_c^{fixed}$	Refer to Appendix A Table A1	3$/kVAr	3$/kVAr
$K_c^{switched}$	- - -	- - -	3.2$/kVAr

and

Table A1. Parameters of Objective Function (Case-1).

Capacitor Index (qc)	Capacitor Size (kVAr) Qc=Q(qc)	Reactive Power Cost ($/KVAr)	Capacitor Index (qc)	Capacitor Size (kVAr) Qc=Q(qc)	Reactive Power Cost ($/KVAr)
1	150	0.500	11	1650	0.193
2	300	0.350	12	1800	0.187
3	450	0.253	13	1950	0.211
4	600	0.220	14	2100	0.176
5	750	0.276	15	2250	0.197
6	900	0.183	16	2400	0.170
7	1050	0.228	17	2550	0.189
8	1200	0.170	18	2700	0.187
9	1350	0.207	19	2850	0.183
10	1500	0.201	20	3000	0.180

(Appendix A). In case-1 cost function is independent of time and it is minimized while considering fix load.

The second objective function considered in this paper, suggested recently in [14]. It is worthy to be mention that this objective function also considers capacitor installation cost to find out annual net savings of the RDG. The second objective function is stated in (12):

$$min{\left(Cos{t}_{TD}^{FL}\right)}_{case-2}=K_p\times T\times Total{P}_{loss/cp}+K_{ic}\times N_q+\sum _{q=1}^n{K}_c\times Q_{qc}$$

(12)

where $FL$ is the fix load, $TD$ is time dependent, $K_p$ is the cost of power losses in $/kWh, T is total annual operation time in hours, $Total{P}_{loss/cp}$ are the power losses with capacitor placement, $K_{ic}$ is the installation cost of $N_q$ number of capacitors, $K_c$ are the capacitor cost in $/kVAr of $Q_{qc}$ capacitor rating in (12). For details refer to Table 1. In case-2 cost function is dependent on time and it is also minimized while considering fix load. The cost function reported in [29] is considered as third objective function considered in this paper and it is expressed as:

$$\begin{array}{ll}min{\left(Cos{t}_{TD}^{VL}\right)}_{case-3}=K_p\times T\times Total{P}_{loss/cp}+K_{ic}\times N_q& +\sum _{q=1}^n{K}_c^{fixed}\times Q_{qc}^{fixed}\\ & +\sum _{q=1}^n{K}_c^{switched}\times Q_{qc}^{switched}\end{array}$$

(13)

where $VL$ is the variable load, $TD$ is time dependent,$K_p$ is the cost of power losses in USD/kWh,T is total annual operation time in hours, $Total{P}_{loss/cp}$ are the power losses with capacitor placement, $K_{ic}$ is the installation cost of $N_q$ number of capacitors, $K_c^{fixed}$ and $K_c^{switched}$ are the fixed and switched capacitor cost in USD/kVAr of $Q_{qc}$ capacitor rating in (13). For details refer to Table 1. It is entrusting to see that this objective function considers the fixed and switched capacitors ratings along with capacitor installation cost to find out annual net savings of the RDG. In case-3 the cost function is also dependent on time and it is minimized while considering time-varying load.

2.3. System Constraints

While achieving the objective functions following constraints are needed to be considered:

2.3.1. Voltage Limitation

While performing the capacitor placement it is necessary to consider the upper and lower voltage constraints i-e, $U_i^{max}$=1.05 and $U_i^{min}$=0.95.

$$U_i^{min}\le U_i\le U_i^{max}$$

(14)

2.3.2. Reactive Injection Limitation

It is necessary to control reactive power injection within the constraints while doing capacitor placement.

$$Q_{cj}^{min}\le Q_{cj}\le Q_{cj}^{max}$$

(15)

where $Q_c{j}^{min}$ is the lower constraint and $Q_{cj}^{max}$ is the upper constraint and $Q_{cj}$ is the rating of the capacitor which is placed at bus j. Here $Q_{cj}^{min}$=50 kVAr and $Q_{cj}^{max}$= 3000 kVAr are considered as constraints.

2.3.3. Maximum Reactive Power Limitation

The total compensated capacitor rating needed to be monitor so that it does not go beyond the total reactive power of the load at each bus.

$$\sum _{i=1}^n{Q}_c\left(i\right)\le \sum _{j=1}^n{Q}_L\left(j\right)$$

(16)

Here ${\sum }_{i=1}^n{Q}_c\left(i\right)$ is the sum of total compensated power of capacitors and ${\sum }_{j=1}^n{Q}_L\left(j\right)$ is the sum of the total reactive power demand of the loads.

3. Methodology

To find the initial losses of the network it is necessary to perform load flow analysis. Through which power system current state can be expressed, in terms of the voltage level at each bus, the overall power factor of the system and power losses in each branch. Several methods are reported in the literature to perform load flow analysis of the system. In this paper, the method reported in [28], is adapted to perform load flow analysis of the system, to find the state of the network before and after optimal capacitor placement. Equations (9) and (10) are used to find the total active and reactive power losses of the system before and after capacitor placement.

3.1. Loss Sensitivity Factor

To minimize the power losses in the system, after performing load flow and evaluating the system initial conditions, it is necessary to nominate the candidate busses for the capacitor placement. In this paper, the loss sensitivity factor method [30] is adopted, for the nomination of busses to optimally place the capacitors. The loss sensitivity factor method helps to predict the candidate buses, which have more capability to reduce the power losses of the system.

$$\frac{\partial P_{loss/k}}{\partial Q_{eff/q}}=\frac{\partial }{\partial Q_{eff/q}}\left[\left(\frac{P_{eff/q}^2+Q_{eff/q}^2}{{\left|U_q\right|}^2}\right)\times R_k\right]$$

(17)

To nominate the candidate buses four simple steps are involved in this method.

3.1.1. Power Loss Indexing

In step one loss sensitivity index (LSI) is achieved for all busses by using (18), which is the resultant derivative of (7).

$$LSF=\left[\left(\frac{2\times Q_{eff/q}}{{\left|U_q\right|}^2}\right)\times R_k\right]$$

(18)

3.1.2. LSF Ranking

The resultant LSF of each bus placed in descending order in step 2 and its position index also stored.

3.1.3. Operating Voltage Normalization

In step three operating voltage of all busses which are obtained from the load flow method without OPSC, converted to a normalized voltage using (19).

$$U_{norm}\left(i\right)=\frac{U\left[i\right]}{0.95}$$

(19)

3.1.4. Selection of Candidate Buses

The last step is very crucial and important because it is compulsory to make sure that the resultant normalized bus voltage, indexing order is the same as the LSF bus indexing order. In the last, only those buses are selected as candidate buses, which are having normalized voltage less than 1.01p.u, whether the power system is large or small.

3.2. Fixed and Switched Capacitors

In the RDG power demand varies with the time as load varies and this may cause enhancement of power losses and instability in the system. To minimize these losses and stabilize the RDG fixed and switched capacitors are used. The capacitors which are permanently installed are called fixed capacitors and with the load demand which can be switched ON/OFF are known as switched capacitors. In this paper, optimal placement and sizing of the fixed and switched capacitors with respect to load variation is also considered to enhance the annual net saving and reduce the total annual cost, power losses. For this purpose, three levels of load are considered with a specific duration of time (see

Table 2. Annual load levels with respect to time duration.

Load Level	Off-Peak Load	Mid-Peak Load	Peak Load
Time Duration	2000	5260	1500

).

3.3. Novel Modified Particle Swarm Optimization

In 1995, Dr. Kennedy and Dr. Ebenhart [31] introduced the standard particle swarm optimization, which is a smart stochastic technique. This algorithm starts with the initialization of particles positions $X_i=(x_{i1},x_{i2},\dots \dots ,x_{iD})$ and their velocities “$V_i=(v_{i1},v_{i2},\dots \dots ,v_{iD})$” in “D” dimensional search space along with inertia weight coefficient, acceleration constants, random variables, personal best of each particle, “$P_{besti}=(p{b}_{i1},p{b}_{i2},\dots \dots ,p{b}_{iD})$” and global best among all particles, “$G_{besti}=(p{g}_{i1},p{g}_{i2},\dots \dots ,p{g}_{iD})$”.

$$w=w_{max}-\frac{w_{max}-w_{min}}{ite{r}_{max}}\times iter$$

(20)

where “w” is inertia, $w_{max}=0.9,w_{min}=0.4,$ “$ite{r}_{max}$” is the maximum number of iterations and “iter” is the $k^{th}$ iteration number. In this research, we have proposed an unusual modified inertia term to find the new velocity of each particle and it is formulated as:

$$v_{iD}^{k+1}=\underset{\mathrm{Novel}\mathrm{Inertia}\mathrm{Term}}{\underbrace{\left(2w-w^2\right)\times v_{iD}^k}}+\underset{\mathrm{Congnative}\mathrm{Term}}{\underbrace{c_1\times r_1\times \left(pbes{t}_{iD}^k-x_{iD}^k\right)}}+\underset{\mathrm{Social}\mathrm{Term}}{\underbrace{c_2\times r_2\times \left(gbes{t}_{iD}^k-x_{iD}^k\right)}}$$

(21)

$$x_{iD}^{k+1}=x_{iD}^k+v_{iD}^{k+1}$$

(22)

Here it is fascinating to see the trait of new velocity using modified inertia term in Figure 2, i.e., resultant vector is $(2w-w^2)\times v_{iD}^k$. Essentially, this unique inertia term assists with moderating the dull qualities for the $v_{iD}^{k+1}$ which goes to expand the quantity of opportunities for every particle to improve individual best in every iteration. This upgrades the presentation of the proposed strategy as far as effectiveness and strength. During every iteration velocity and position of every particle is revived utilizing (21) and (22) and gbest and pbest values are taken care of, while getting the best outcomes algorithm stops.

$$v_{iD}^{k+1}=\left\{\begin{array}{cc}\mathrm{rand}\times v_{iD}^{k+1}& \mathrm{if}\left|v_{iD}^{k+1}\right|=\left|v_{iD}^k\right|\\ v_{iD}^{k+1}& \mathrm{otherwise}\end{array}\right.$$

(23)

Normally (23) is used in standard PSO and other modified PSO methods reported in the literature to settle the repetitive elements for new velocity. Our proposed technique mitigates this repetitive issue; it can be seen in Figure 3, that each value of “w” either positive or negative outcomes for unfamiliar inertia term $(2w-w^2)$ is different. So there is no need to use (23) in our technique. For more accurate results, each particle velocity is restricted between the maximum and minimum values $[v_{max},v_{min}]$ using (24).

$$v_{iD}^{k+1}=\left\{\begin{array}{c}{v}_{max}=ca{p}_{max}\\ v_{min}=ca{p}_{min}\end{array}\right.$$

(24)

where $ca{p}_{max}$ is 3000kVAr and $ca{p}_{min}$ is 50 kVAr.

$$v_{iD}^{k+1}=\left\{\begin{array}{ccc}{v}_{max}& if& v_{iD}^{k+1}>v_{max}\\ v_{iD}^{k+1}& if& \left|v_{iD}^{k+1}\right|\le v_{max}\\ v_{min}& if& v_{iD}^{k+1}<v_{min}\end{array}\right.$$

(25)

While updating the velocity for kth iteration, particle velocity is lemmatized by (23), and resultant velocity gathered in the form of (25).

Figure 3. Novel inertia term response.

Figure 3. Novel inertia term response.

4. Execution of MPSO for OPSC

To understand the performance of the proposed MPSO algorithm for the OPSC problem, the pseudocode is stated below.

Step 1

Perform load flow to evaluate the initial state of the RDG in terms of power losses, total annual cost, minimum voltage.

Step 2

Use loss sensitivity method to highlight the potential buses for the optimal capacitor placement.

Step 3

Initialize random set of solutions; particle velocity, particle position, cognitive, social components, and random variables.

Step 4

Set maximum iteration limit.

Step 5

Check whether generated solution is satisfying the constraints; install capacitors on optimal buses while satisfying the (14)–(16).

Step 6

Evaluate the fitness of the solution in each iteration for (11)–(13); total annual cost.

Step 7

Generate new set of solutions using (21) and (22); update particle velocity and particle position.

Step 8

Stop the algorithm and display the result when it reached to maximum iteration or optimal solution is achieved; power losses reduction, total annual cost reduction, improvement of voltage profile, enhancement of annual net saving.

5. Results and Discussion

To show the effectiveness of the proposed algorithm for the OPSC problem, the approach was tested on IEEE 15, 33, and 69 bus RDG and the outcomes were compared with the reported results in the recent literature. A novel inertia term was added in the standard PSO algorithm. This approach helps to get away from local minima by mitigating the repetition of velocity for each particle through this possibility to get optimal result is increased. This ability enables our algorithm to perform better as compared to conventional PSO and other algorithms. All results were tabulated for the stated objective functions i-e case-1, case-2 and case-3. To show the efficiency and strength of the proposed modernistic MPSO; the technique has been executed 20 times on MaTLAB® 2015b with intel® Core™ i3-2370M CPU @ 2.4GHz and 4GB RAM, for each execution number of iteration (k) is taken as 1000 with 20 number of particles (N).

5.1. Statistical Analysis

To demonstrate the efficiency of the proposed algorithm, the statistical breakdown for the stated objective functions (11)–(13) is given in

Table 3. MPSO based statistical results for optimal capacitor placement.

Annual Net Savings in $ (Case-1)
Best	Worst	Average	Standard Deviation	IEEE
$4.95\times {10}^{+3}$	$4.87\times {10}^{+3}$	$4.92\times {10}^{+3}$	$5.62\times {10}^{-2}$	15
$1.17\times {10}^{+4}$	$1.10\times {10}^{+4}$	$1.15\times {10}^{+4}$	$1.21\times {10}^{-1}$	33
$1.30\times {10}^{+4}$	$1.28\times {10}^{+4}$	$1.29\times {10}^{+4}$	$1.02\times {10}^{-1}$	69
Annual Net Savings in $ (Case-2)
Best	Worst	Average	Standard Deviation	IEEE
$1.06\times {10}^{+4}$	$1.01\times {10}^{+4}$	$1.05\times {10}^{+4}$	$1.44\times {10}^{-1}$	15
$3.00\times {10}^{+4}$	$2.95\times {10}^{+4}$	$2.99\times {10}^{+4}$	$1.44\times {10}^{-1}$	33
$3.48\times {10}^{+4}$	$3.40\times {10}^{+4}$	$3.46\times {10}^{+4}$	$2.23\times {10}^{-1}$	69
Annual Net Savings in $ (Case-3)
Best	Worst	Average	Standard Deviation	IEEE
$9.70\times {10}^{+3}$	$9.56\times {10}^{+3}$	$9.65\times {10}^{+3}$	$4.54\times {10}^{-2}$	15
$4.70\times {10}^{+4}$	$4.66\times {10}^{+4}$	$4.68\times {10}^{+4}$	$1.29\times {10}^{-1}$	33
$1.90\times {10}^{+4}$	$1.88\times {10}^{+4}$	$1.89\times {10}^{+4}$	$5.81\times {10}^{-2}$	69

in terms of best (maximum), worst (minimum), average (mean), and standard deviation values. While convergence curves in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 reflect the strength of the proposed unusual MPSO technique. Here it is necessary to mention that CPU time represents the total time to compute: load flow with and without capacitor, LSF method, number of iterations of MPSO to reach optimal solution. Furthermore, comparative results are explained in the below subsections.

Figure 4. Flow chart of MPSO for solving OPSC problem.

Figure 4. Flow chart of MPSO for solving OPSC problem.

5.2. IEEE 15 Bus RDG

In this study, initially, IEEE 15 bus system is taken to test the proposed unique technique. System data are taken from [32], it has 15 buses, 14 branches (see Figure 5a) and it works at 11kV, 100MVA base value, with 1226kW active load, and 1251kVAr reactive load. While examining the system for case-1 and case-2 (without capacitor placement) active power loss is 61.7926kW, the minimum system voltage is 0.9445p.u. at bus 13, power loss cost, and the total annual cost is 10381$ and 32478$ with zero net savings (see

Table 4. Comparative results for IEEE 15 bus RDG (case-1).

	Base Case	MAPSO [10]	TSM [9]	dPSO [21]	DE [33]	IHA [27]	FPA [33]	MPSO
Total Power Losses (kW)	61.7926	30.9534	32.4262	30.4463	32.3	31.1255	30.7112	30.1814
Loss Reduction(%)	- - -	50.04	47.51	50.72	47.86	49.76	50.30	51.16
Minimum Bus Voltage (p.u)	0.9445	0.978	0.9695	0.9712	- - -	0.9658	0.9676	0.9690
Candidate Buses	- - -	4, 6, 7, 11, 15	3, 6, 4	4, 6, 13, 15	3, 6, 11	6, 11, 15	6, 11, 15	4, 7, 11
OPSC (kVAr)	- - -	345, 264, 143, 300, 143	175, 375, 750	450, 450, 150, 150	454, 500, 178	350, 300, 300	350, 350, 300	510, 345, 325
Total kVar	- - -	1195	1300	1200	1132	950	1000	1180
Annual power loss Cost ($)	10381	5200	5448	5115	5426	5229	5160	5071
Saving Due to Power Losses Reduction ($)	- - -	5209	4933	5265	4982	5179	5221	5311
Cap kVar cost ($)	- - -	501	426	378	317	333	350	364
Total Annual Cost ($)	10381	5701	5873	5493	5743	5562	5510	5434
Annual Net Saving ($)	- - -	4708	4506	4887	4665	4847	4872	4947
Net Saving (%)	- - -	45.23	43.41	47.08	44.82	46.57	46.93	47.66

and

Table 5. Comparative results for IEEE 15 bus RDG (case-2).

	Base Case	MAPSO [10]	TSM [9]	dPSO [21]	DE [33]	IHA [27]	FPA [33]	MPSO
Total Power Losses (kW)	61.7926	30.9534	32.4262	30.4463	32.3	31.1255	30.7112	31.634
Loss Reduction(%)	- - -	50.04	47.51	50.72	47.86	49.76	50.30	48.81
Minimum Bus Voltage (p.u)	0.9445	0.978	0.9695	0.9712	- - -	0.9658	0.9676	0.9630
Candidate Buses	- - -	4, 6, 7, 11, 15	3, 6, 4	4, 6, 13, 15	3, 6, 11	6, 11, 15	6, 11, 15	4, 6
OPSC (kVAr)	- - -	345, 264, 143, 300, 143	175, 375, 750	450, 450, 150, 150	454, 500, 178	350, 300, 300	350, 350, 300	670, 400
Total kVar	- - -	1195	1300	1200	1132	950	1000	1070
Annual power loss Cost ($)	32,478	16,269	17,043	16,002	16,977	16,359	16142	16627
Saving Due to Power Losses Reduction ($)	- - -	16,200	15,428	16,472	15,586	16,203	16,337	15,851
Cap kVar cost ($)	- - -	8585	6900	7600	6396	5850	6000	5210
Total Annual Cost ($)	32,478	24,854	23,943	23,602	23,373	22,209	22,142	21,837
Annual Net Saving ($)	- - -	7615	8528	8872	9190	10,353	10,337	10,641
Net Saving (%)	- - -	23.45	26.26	27.32	28.22	31.79	31.83	32.76

). Candidate buses are nominated using the LSF method, and potential buses are ranked in descending order, as shown in Figure A2 (Appendix A). Comparative results are presented to show the effectiveness of the proposed technique with other reported techniques; MAPSO, TSM, dPSO, DE, IHA, FPA.

Our proposed unfamiliar technique suggested 510, 345, and 325kVAr rating at bus 4, 7 and 11 (case-1) and bus 4, 6 with 670 and 400kVAr (case-2) for the OPSC problem. The suggested sizing results in the power losses reduction up to 51.16%, 48.81% with 47.66%, 32.76% net saving (see Figure 6) with an enhanced minimum voltage of 0.9690, 0.9630 p.u. (observe Figure 5b). It is observed that a remarkable enhancement in annual net saving is achieved after OPSC using MPSO.

Figure 5. (a) IEEE 15 bus system. (b) Voltage profile.

Figure 5. (a) IEEE 15 bus system. (b) Voltage profile.

Moreover, our proposed technique is tested for different loading conditions; off-peak, mid-peak, and peak load (case-3) for 15 bus system. Initially, (without capacitor placement) system faces 14.698, 61.7926, and 168.877kW power losses with 0.9730, 0.9445 and 0.9081 p.u. minimum voltage at bus 13, power loss cost, and the total annual cost is 1764$ and 19502$ 15199$ with zero net savings (see

Table 6. Results for IEEE 15 Bus RDG in different loading conditions (case-3).

	Load Level	Total Power Losses (kW)	Minimum Bus Voltage (p.u)	Candidate Buses	Number of Switched Capacitors/200kVAr	Capacitor (kVAr)	Cap kVAr Cost ($)	Annual Power Loss Cost($)	Total Annual Cost ($)	Annual Net Saving ($)
Base Case	Off-Peak Load	14.698	0.9730	- - -	- - -	- - -	- - -	1764	1764	- - -
	Mid-Peak Load	61.7926	0.9445	- - -	- - -	- - -	- - -	19,502	19,502	- - -
	Peak Load	168.877	0.9081	- - -	- - -	- - -	- - -	15,199	15,199	- - -
Total Without OPSC		- - -	- - -	- - -	- - -	- - -	- - -	36,465	36,465	- - -
Proposed	Off-Peak Load	14.698	0.9730	- - -	- - -	- - -	- - -	1764	1764	0
MPSO	Mid-Peak Load	32.6404	0.9615	4,6	3,2	600, 400	3200	10,301	13,501	6000
	Peak Load	97.922	0.9296	4	1	200	640	8912	9552	5647
Total With OPSC Including 2000$ Cap Installation Cost		- - -	- - -	4,6	6	1200	5840	20,977	26,817	9648

). Our proposed innovative technique suggested bus 4,6 for the optimal placement of capacitor with 0, 1000, and 200kVAr sizing. The proposed sizing results in the reduction of power losses 0%, 47.18%, and 42% along with enhancement of minimum voltage 0.9730, 0.9615, and 0.9296 p.u. Similarly, total power losses cost is reduced from 36,465$ to 20,977$ and total annual cost from 36,465$ to 26,817$ which turns to enhances the total net saving from 0$ to 9648$. So, it can be seen that significant enhancement is gathered through MPSO for the OPSC problem.

Figure 6. Convergence curve IEEE 15 bus system.

Figure 6. Convergence curve IEEE 15 bus system.

Figure 7. (a) IEEE 33 bus system (b) Voltage profile.

Figure 7. (a) IEEE 33 bus system (b) Voltage profile.

5.3. IEEE 33 Bus RDG

IEEE 33 bus system is taken as second test system for the proposed unique technique. System data are taken from [14], this system has 33 buses, 32 branches (observe Figure 7a), and it operates at 12.66kV, 100MVA base value with 3715kW active load, and 2230 kVAr reactive load. While examining the system for case-1 and case-2 (without capacitor placement) active power loss is 210.97 kW, the minimum system voltage is 0.9038p.u. at bus 18, power loss cost, and the total annual cost is 35,443$ and 110,886$ with zero net savings (see

Table 7. Comparative results for IEEE 33 bus RDG (case-1).

	Base Case	IP [34]	SA [34]	TSM [9]	MMS [32]	FPA [19]	LS [14]	MPSO
Total Power Losses (kW)	210.97	171.78	151.75	144.04	135.77	134.47	138.61	137.37
Loss Reduction(%)	- - -	18.58	28.07	31.73	33.01	33.65	34.30	34.89
Minimum Bus Voltage (p.u)	0.9038	0.9501	0.9591	0.9251	0.9416	0.9365	0.9325	0.9370
Candidate Buses	- - -	9, 29, 30	10, 30,14	7, 29, 30	9, 13, 29	6, 9, 30	5, 8, 11, 16, 24, 26, 30, 32	8, 16, 24, 30, 32
OPSC (kVAr)	- - -	450, 800, 900	450, 350, 900	850, 25, 900	300, 300, 900	250, 400, 950	150, 150, 150, 150, 450, 150, 750, 150	320, 340, 500, 660, 380
Total kVar	- - -	2150	1700	1775	1500	1600	2100	2200
Annual power loss Cost ($)	35,443	28,859	25,494	24,199	22,809	22,591	23,287	23,079
Saving Due to Power Losses Reduction ($)	- - -	6584	9949	11,244	11,290	11,458	12,157	12,364
Cap kVar cost ($)	- - -	499	401	507	375	439	771	636
Total Annual Cost ($)	- - -	29,358	25,895	24,611	23,184	23,030	24,057	23,714
Annual Net Saving ($)	35,443	6085	9548	10,832	10,915	11,019	11,807	11,729
Net Saving (%)	- - -	17.17	26.93	30.56	32.01	32.36	32.12	33.09

and

Table 8. Comparative results for IEEE 33 bus RDG (case-2).

	Base Case	IP [34]	SA [34]	TSM [9]	MMS [32]	FPA [19]	LS [14]	MPSO
Total Power Losses (kW)	210.97	171.78	151.75	144.04	135.77	134.47	139.23	141.23
Loss Reduction(%)	- - -	18.58	28.07	31.73	33.01	33.65	34.00	33.06
Minimum Bus Voltage (p.u)	0.9038	0.9501	0.9591	0.9251	0.9416	0.9365	0.9291	0.9298
Candidate Buses	- - -	9, 29, 30	10, 30, 14	7, 29, 30	9, 13, 29	6, 9, 30	12, 25, 30	13, 30
OPSC (kVAr)	- - -	450, 800, 900	450, 350, 900	850, 25, 900	300, 300, 900	250, 400, 950	450, 350, 900	425, 1125
Total kVar	- - -	2150	1700	1775	1500	1600	1700	1550
Annual power loss Cost ($)	110,886	90,288	79,760	75,707	71,361	70,677	73,179	74,231
Saving Due to Power Losses Reduction ($)	- - -	20,598	31,126	35,178	35,168	35,841	37,707	36,654
Cap kVar cost ($)	- - -	9450	8100	8325	7500	7800	8100	6650
Total Annual Cost ($)	- - -	99,738	87,860	84,032	78,861	78,477	81,279	80,881
Annual Net Saving ($)	110,886	11,148	23,026	26,853	27,668	28,041	29,607	30,004
Net Saving (%)	- - -	10.05	20.76	24.21	25.97	26.33	26.70	27.06

). Candidate buses are nominated using the LSF method, and potential buses are ranked in descending order, as shown in Figure A2 (Appendix A). Comparative results are presented to show the effectiveness of the proposed technique with other reported techniques; IP, SA, TSM, MMS, FPA, and LS. Our proposed modernistic technique suggested 320, 340, 500, 660, 380kVAr rating at bus 8, 16, 24, 30, 32 (case-1) and bus 13, 30 with 425 and 1125 kVAr (case-2) for the OPSC problem. The suggested sizing results in the power losses reduction up to 34.89%, 33.06% with 33.09%, 27.06% net saving (see Figure 8) with an enhanced minimum voltage of 0.9370, 0.9298 p.u. (observe Figure 7b). It is observed that a remarkable enhancement in annual net saving is achieved after OPSC using MPSO.

Figure 8. Convergence curve IEEE 33 bus system.

Figure 8. Convergence curve IEEE 33 bus system.

Moreover, our proposed technique is tested for different loading conditions; off-peak, mid-peak, and peak load (case-3) for 33 bus system as well. Initially, (without capacitor placement) system faces 48.783, 210.97 and 603.374kW power losses with 0.9540, 0.9038 and 0.8360 p.u. minimum voltage at bus 18, power loss cost, and the total annual cost is 5854$ and 66,582$, and 54,304$ with zero net savings (see

Table 9. Results for IEEE 33 Bus RDG in different loading conditions (case-3).

	Load Level	Total Power Losses (kW)	Minimum Bus Voltage (p.u)	Candidate Buses	Number of Switched Capacitors/200kVAr	Capacitor (kVAr)	Cap kVAr Cost ($)	Annual Power Loss Cost($)	Total Annual Cost ($)	Annual Net Saving ($)
Base Case	Off-Peak Load	48.783	0.9540	- - -	- - -	- - -	- - -	5854	5854	- - -
	Mid-Peak Load	210.97	0.9038	- - -	- - -	- - -	- - -	66,582	66,582	- - -
	Peak Load	603.374	0.8360	- - -	- - -	- - -	- - -	54,304	54,304	- - -
Total Without OPSC		- - -	- - -	- - -	- - -	- - -	- - -	126,740	126,740	- - -
Proposed	Off-Peak Load	48.783	0.9540	- - -	- - -	- - -	- - -	5854	5854	0
MPSO	Mid-Peak Load	142.726	0.9277	13, 30	7	400, 1000	4480	45,044	49,524	17,058
	Peak Load	402.095	0.8737	13, 30	4	200, 600	2560	36,189	38,749	15,555
Total With OPSC Including 2000$ Cap Installation Cost		- - -	- - -	13, 30	11	2200	9040	87,087	96,127	30,613

). Our proposed unusual technique suggested bus 13,30 for the optimal placement of capacitor with 0, 1400, and 800kVAr sizing. The proposed sizing results in the reduction of power losses 0%, 32.35%, and 33.36% along with enhancement of minimum voltage 0.9540, 0.9277, and 0.8737 p.u. Similarly, total power losses cost is reduced from 126,740$ to 87,087$ and total annual cost from 126,740$ to 96,127$ which turns to enhances the total net saving from 0$ to 30,613$. So, it can be seen that significant enhancement is gathered through MPSO for the OPSC problem.

Figure 9. (a) IEEE 69 bus system. (b) Voltage profile.

Figure 9. (a) IEEE 69 bus system. (b) Voltage profile.

5.4. IEEE 69 Bus RDG

IEEE 69 bus system is taken as third test system for the proposed unfamiliar technique. System data is taken from [14], This system has 69 buses, 68 branches (observe Figure 9a), with connected 3801.89kW, 2693.6kVAr active and reactive load. While examining the system for case-1 and case-2 (without capacitor placement) active power loss is 225kW, the minimum system voltage is 0.9092p.u. at bus 65, power loss cost, and the total annual cost is 37,800$ and 118,260$ with zero net savings (see

Table 10. Comparative results for IEEE 69 bus RDG (case-1).

	Base Case	Fuzzy-GA [3]	DE [33]	MMS [32]	TSM [9]	FPA [19]	LS [14]	MPSO
Total Power Losses (kW)	225	156.62	151.3763	146.29	148.91	150.28	144.25	144.79
Loss Reduction(%)	- - -	30.40	32.70	34.98	33.73	33.2	35.89	35.63
Minimum Bus Voltage (p.u)	0.9092	0.9369	0.9311	0.9341	0.9289	0.9333	0.9315	0.9311
Candidate Buses	- - -	59, 61, 64	57, 58, 59, 60, 61	12, 21, 61, 62, 64	19, 62, 63	61	12, 21, 50, 54, 61	21, 61, 64
OPSC (kVAr)	- - -	100, 700, 800	150, 50, 100, 150, 1000	200, 200, 600, 600, 200	225, 900, 225	1350	350, 150, 40, 150, 1200	320, 1200, 230
Total kVar	- - -	1600	1450	1800	1350	1350	2300	1750
Annual power loss Cost ($)	37,800	26,312	25,431	24,577	25,017	25,247	24,235	24,325
Saving Due to Power Losses Reduction ($)	- - -	11,479	12,359	13,214	12,774	12,544	13,557	13,475
Cap kVar cost ($)	- - -	425	408	564	390	280	590	431
Total Annual Cost ($)	37,800	26,737	25,839	25,141	25,407	25,527	24,825	24,756
Annual Net Saving ($)	- - -	11,054	11,951	12,650	12,384	12,264	12,975	13,044
Net Saving (%)	- - -	29.25	31.63	33.47	32.77	32.45	34.32	34.51

and

Table 11. Comparative results for IEEE 69 bus RDG (case-2).

	Base Case	Fuzzy-GA [3]	DE [33]	MMS [32]	TSM [9]	FPA [19]	LS [14]	MPSO
Total Power Losses (kW)	225	156.62	151.3763	146.29	148.91	150.28	146.61	145.27
Loss Reduction (%)	- - -	30.4	32.7	34.98	33.73	33.2	34.84	35.44
Minimum Bus Voltage (p.u)	0.9092	0.9369	0.9311	0.9341	0.9289	0.9333	0.9300	0.9300
Candidate Buses	- - -	59, 61, 64	57, 58, 59, 60, 61	12, 21,61, 62, 64	19, 62, 63	61	17, 61	18, 61
OPSC (kVAr)	- - -	100, 700, 800	150, 50, 100, 150, 1000	200, 200, 600, 600, 200	225, 900, 225	1350	350, 1200	300, 1400
Total kVar	- - -	1600	1450	1800	1350	1350	1550	1700
Annual power loss Cost ($)	118,260	82,319	79,563	76,890	78,267	78,987	77,058	76,354
Saving Due to Power Losses Reduction ($)	- - -	35,830	38,641	41,370	39,830	39,218	41,202	41,906
Cap kVar cost ($)	- - -	7800	9350	10400	7050	5050	6650	7100
Total Annual Cost ($)	118,260	90,119	88,913	87,290	85,317	84,037	83,708	83,454
Annual Net Saving ($)	- - -	28,030	29,291	30,970	32,780	34,168	34,552	34,806
Net Saving (%)	- - -	23.72	24.78	26.19	27.76	28.91	29.21	29.43

). Candidate buses are nominated using the LSF method, and potential buses are ranked in descending order, as shown in Figure A3 (Appendix A). Comparative results are presented to show the effectiveness of the proposed technique with other reported techniques; Fuzzy-GA, DE, MMS, TSM, FPA, LS.

Figure 10. Convergence curve IEEE 69 bus system.

Figure 10. Convergence curve IEEE 69 bus system.

Our proposed innovative technique suggested 320, 1200, 230kVAr rating at bus 21, 61, 64 (case-1) and bus 18, 61 with 300 and 1400kVAr (case-2) for the OPSC problem. The suggested sizing results in the power losses reduction up to 35.63%, 35.44% with 34.51%, 29.43% net saving (see Figure 10) with an enhanced minimum voltage of 0.9311, 0.9300 p.u. (observe Figure 9b). It is observed that remarkable enhancement in annual net saving is achieved after OPSC using MPSO.

Moreover, our proposed technique is tested for different loading conditions; off-peak, mid-peak, and peak load (case-3) for 69 bus system as well. Initially, (without capacitor placement) system faces 51.60, 225 and 652.43kW power losses with 0.9567, 0.9092 and 0.8445 p.u. minimum voltage at bus 65, power loss cost, and the total annual cost is 6192$ and 70487$ 58,719$ with zero net savings (see

Table 12. Results for IEEE 69 Bus RDG in different loading conditions (case-3).

	Load Level	Total Power Losses (kW)	Minimum Bus Voltage (p.u)	Candidate Buses	Number of Switched Capacitors/200kVAr	Capacitor (kVAr)	Cap kVAr Cost ($)	Annual Power Loss Cost($)	Total Annual Cost ($)	Annual Net Saving ($)
Base Case	Off-Peak Load	51.60	0.9567	- - -	- - -	- - -	- - -	6192	6192	- - -
	Mid-Peak Load	225	0.9092	- - -	- - -	- - -	- - -	70,487	70,487	- - -
	Peak Load	652.43	0.8445	- - -	- - -	- - -	- - -	58,719	58,719	- - -
Total Without OPSC		- - -	- - -	- - -	- - -	- - -	- - -	135,921	135,921	- - -
Proposed	Off-Peak Load	51.60	0.9567	- - -	- - -	- - -	- - -	6192	6192	0
MPSO	Mid-Peak Load	150.90	0.9291	61	7	1400	4480	47,625	52,105	18,905
	Peak Load	432.81	0.8767	61	4	800	2560	38,953	41,513	17,206
Total With OPSC Including 1000$ Cap Installation Cost		- - -	- - -	61	11	2200	8040	92,770	100,810	35,111

). Our proposed unique technique suggested bus 61 for the optimal placement of capacitor with 0, 1400, and 800 kVAr sizing. The proposed sizing results in the reduction of power losses 0%, 32.93%, and 33.66% along with enhancement of minimum voltage 0.9567, 0.9291, and 0.8767 p.u. Similarly, total power losses cost is reduced from 135,921$ to 92,770$ and total annual cost from 135,921$ to 100,810 $ which turns to enhances the total net saving from 0$ to 35,111$. So, it can be stated that significant enhancement is achieved from MPSO for OPSC problem.

6. Conclusions

In this paper, a novel particle swarm optimization technique based on the modification of particle velocity is utilized to resolve the OPSC problem effectively while considering constant and time-varying load for the reduction of power losses and total annual cost. The proposed technique is tested on IEEE 15, 33, and 69 bus system. To highlight the performance in solving the OPSC problem results are compared and evaluated for the reduction of power losses and total annual cost along with enhancement of voltage profile and total annual net saving. To compel a fair comparison, MPSO is tested on IEEE 15, 33, 69 RDG and acquired results are contrasted with other techniques reported recently in the literature.

The compared results authenticate that the MPSO is well appropriate for taking care of OPSC problem in RDG. This ability to handle the non-linear and incoherent complex formulation of MPSO is because of avoiding the repetition of particle velocity in each iteration. This modernistic characteristic enhances the exploration and exploitation process of MPSO. Due to this, the presented technique achieves an optimal solution for the OPSC problem in RDG. Furthermore, this proposed technique is not restricted to OPSC problems, it can also be utilized for other engineering and global optimization problems as well.

More efforts will be focused on the assessment of game plan strategies of reactive power compensation for the real time practical power system. More efforts will be focused on the assessment of game plan strategies of reactive power compensation for the real time practical power system. Additionally, some multi objective techniques can be taken on to get the best compromise course of action for the capacitor allocation, with various objectives such as voltage regulation and harmonic distortion.