Complexity
Volume 2020 (2020), Article ID 5493958, 28 pages
https://doi.org/10.1155/2020/5493958

Research Article

Simple Quadratic Interpolation-Inspired Symbiosis Organisms Search Algorithm for Optimal Placement of Capacitors in Radial Distribution Networks with Different Loading Models

Tri Phuoc Nguyen,¹ Tuan Trong Nguyen,² Trung Hieu Quang,³ and Dieu Ngoc Vo1,4

¹Department of Power Systems, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam
²Southern Electrical Testing Company, Southern Power Corporation, Electricity of Vietnam, 22 Bis Phan Dang Luu Street, District Binh Thanh, Ho Chi Minh City, Vietnam
³Southern Region Load Dispatch Center, National Load Dispatch Center, Electricity of Vietnam, 5 Su Thien Chieu Street, District 3, Ho Chi Minh City, Vietnam
⁴Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Vietnam

Correspondence should be addressed to Dieu Ngoc Vo; vndieu@gmail.com

Received 4 March 2020; Revised 18 July 2020; Accepted 30 July 2020; Published 30 September 2020

Academic Editor: Mohammad Hassan Khooban

Copyright © 2020 Tri Phuoc Nguyen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proposes a novel hybrid algorithm based on a combination of the simple quadratic interpolation and the symbiosis organisms search algorithm (SQI-SOS) for finding the optimal location and size of capacitors in radial distribution networks. The objective of the problem is to minimize the system operating cost so that the net yearly savings of the system are increased. The effectiveness of the SQI-SOS has been tested on 33-, 69-, and 119-bus radial distribution networks with different load models. The obtained results from the test system by the proposed SQI-SOS are compared with those from the conventional SOS and other mature optimization methods in the literature. The result comparison has shown that the proposed SQI-SOS algorithm can provide a better solution than the other methods. Accordingly, the proposed SQI-SOS can be a very effective and efficient method for dealing with the optimal capacitor placement problem in distribution networks.

1. Introduction

The reactive power compensation plays an important role in the operation of power distribution networks since it brings many technical benefits such as decreasing power losses, enhancing voltage profiles, correcting power factor, and releasing system capacity. To acquire such benefits, shunt capacitors are widely employed in distribution networks to inject the reactive power. However, the inappropriate placement of capacitors may result in reducing the benefits of the system and even risking the whole system operation. Therefore, it is very important to determine the optimal location and size of capacitors to be installed in distribution networks so that the maximum benefits of the system can be acquired while satisfying all the operating constraints of the system.

In the recent past decades, the optimal capacitor placement (OCP) problem has been formulated with many objective functions such as power loss reduction, voltage profile improvement, capacitor installation cost minimization, network stability maximization, and reduction of burden on existing lines [1]. Many techniques have also been developed for dealing with such problems. In general, these techniques can be classified into four main categories including analytical, numerical programming, heuristic, and artificial intelligence methods [2]. Analytical methods have been early applied for the OCP problem, and they are easy to be understood and implemented. However, the analytical methods may provide inaccurate solutions since they use continuous variables to model the location and size of capacitors. The obtained results would need to be rounded to achieve practical values. Therefore, the designed system may suffer from overvoltage situation and has more or less power losses than the calculated one [2]. In addition to the analytical approach, numerical programming methods such as dynamic programming (DP) [3], mixed integer linear programming approach (MILP) [4], integer quadratic programming (IQP) [5], clustering-based optimization (CBO) [6], and interior point method (IPM) [7] have been also applied for solving the OCP problem. Although these methods consider the location and size of capacitors as discrete variables, their computational time might take longer than the analytical approaches. In addition, the OCP problem is considered as a nonlinear optimization problem with a nondifferentiable objective function. Therefore, numerical programming methods may not effectively deal with the OCP problem for such cases. On the contrary, the heuristic methods as in [8–10] are intuitive, easy to understand, and simple for implementation. Based on the fast and practical strategies of these methods, they may help to reduce the search space in finding the final solution. However, heuristic methods do not always guarantee to obtain the optimal solution for the OCP problem [2].

In recent years, artificial intelligence- (AI-) based methods have been developed and effectively implemented for dealing with the OCP problem. A survey of studies from the literature has shown that the OCP problem has been successfully solved by the genetic algorithm (GA) [11–13], tabu search (TS) [14], particle swarm optimization (PSO) [15], cuckoo search algorithm (CSA) [16], improved harmony algorithm (IHA) [17], flower pollination algorithm (FPA) [18], artificial bee colony (ABC) [19], teaching learning-based optimization (TLBO) [20], hybrid differential evolution and pattern search (DE-PS) [21], plant growth simulation algorithm (PGSA) [22], crow search algorithm (CSA) [23], whale optimization algorithm (WOA) [24], Limaçon inspired spider monkey optimization (LSMO) [25], multiswarm particle swarm optimization (MSPSO) [26], penalty-free genetic algorithm (PFGA) [27], competitive swarm optimizer (CSO) [28], grasshopper optimization algorithm (GOA) [29], and modified Gbest-guided Artificial Bee Colony (MGABC) [30]. Although AI-based methods show that they have a performance improvement in solving the OCP problem, they may not guarantee to obtain the optimal solution for complex and large-scale problems. Moreover, some of them may suffer from local optima and require a long computational time. Therefore, there is always a need to develop and introduce new techniques for effectively solving the OCP problem.

Symbiotic organisms search (SOS) is new a population-based algorithm proposed by Cheng and Prayogo [31] in 2014. This algorithm is inspired from symbiotic relationships among species in nature. The SOS method was first applied for various highly nonlinear benchmark functions and some other engineering optimization problems, and it is reported that the SOS has outperformed some other popular optimization methods such as GA, PSO, and DE in solving such problems [31]. Since the SOS is relatively a new algorithm, it needs to be further explored and investigated. To improve the performance of the SOS, Nama et al. [32] have proposed a hybrid symbiosis organisms search algorithm by embedding the simple quadratic interpolation (SQI) to enhance exploration ability of SOS in the search space; thus, it can efficiently improve the convergence characteristic of the algorithm. Therefore, the robustness of the SQI-SOS is increased by the combination of SQI and SOS in the exploration capability. In this paper, the SQI-SOS is proposed to determine the optimal location and size of capacitors in radial distribution networks with the objective of total operating cost reduction. The proposed SQI-SOS has been tested on 33-bus, 69-bus, and 119-bus radial distribution networks with different load models. The results obtained by the SQI-SOS are compared with the results obtained from the conventional SOS and other methods such as CSA, IHA, FPA, ABC, and MGABC.

The current paper contains the main contributions as follows:

(i)Firstly, we propose a novel hybrid method via the combination of SQI and SOS (SQI-SOS) to estimate the optimal siting and sizing of capacitors for the first time, in which the overall cost objective function is minimized.(ii)Secondly, we introduce a new initialization process for treating discrete capacitor sizing variables. Based on this initialization method, only a few adjustments are required in the original solution method.(iii)Thirdly, this is the first time in this study where both continuous and discrete capacitor sizes have been investigated for a result comparison.(iv)Fourthly, the actual voltage-dependent nonlinear load models have been utilized for investigations.(v)Finally, the proposed SQI-SOS method offers the solutions with a better quality than those acquired by the previously reported methods in the comparative cases.

The remaining of this paper is organized as follows: Section 2 provides the mathematical formulation of the OCP problem. Sections 3 and 4 represent the SQI-SOS algorithm and its implementation to the OCP problem. The numerical results are reported in Section 5. Finally, the paper is concluded in Section 6.

2. Problem Formulation

2.1. Objective Function

In this study, the optimization problem of the OCP is formulated with an objective of minimizing the system operating cost which is described by the following equation [16]:

In order to analyze the OCP problem with an actual voltage-dependent load model, the original objective function can be modified as follows (MC):

subject to following constraints.

2.2. Constraint

(i)Real and reactive power balance constraint:

(ii)Voltage limits at buses:

(iii)Maximum real power flow constraint:

(iv)Reactive power compensation limits:

(v)Total reactive power compensation limits:

(vi)Overall system power factor limits:

2.3. Load Models

In the previous load flow studies, the load models with constant active and reactive powers, i.e., constant loads, were commonly utilized. However, power demands of practical loads highly depend on the network voltage. So, these loads can be modeled as voltage-dependent loads which include residential, industrial, and commercial loads. Mathematically, the voltage-dependent load models can be formulated as [33]

where and are the active and reactive power exponents for the different types of load in which subscripts o, i, r, and c are, respectively, the constant, industrial, residential, and commercial loads. The exponent values pertaining to the load types are given in Table 1 [33, 34]. {} and {} are active and reactive power weight coefficients, respectively, and these coefficients are chosen based on the weight of the consumed active and reactive power by practical load. The different types of the load model and the practical mixed load model can be defined as follows:

(i)Constant load: (ii)Industrial load: (iii)Residential load: (iv)Commercial load: (v)Mixed or practical load: the aggregation of the different load types is implemented by

Table 1: Load types and exponent values.

Load type
Constant load
Industrial load
Residential load
Commercial load

Exponents

To examine the standard radial distribution systems with a practical mixed load model, we assume that these systems only include industrial, residential, and commercial loads as in [33, 34]. As the constant loads are not included, the coefficients are equal to 0. In addition, this study assumes that the active and reactive power consumptions of industrial, residential, and commercial loads are, respectively, 40%, 45%, and 15% of the total load demand, resulting in the setting values of weight factors related to industrial load , residential load , and commercial load .

3. A Hybrid Symbiosis Organisms Search Algorithm

3.1. Symbiosis Organisms Search Algorithm

The SOS proposed by Cheng and Prayogo in 2014 [31] is a novel meta-heuristic algorithm. This algorithm portrays the symbiotic relationships between two different organisms in an ecosystem. Similar to most population-based algorithms, the SOS starts the search process with a population of organisms (i.e., ecosystem) which is randomly generated. Each organism represents a solution to the considered problem. Based on the population of organisms, the fitness function corresponding to each organism is calculated. In the SOS, there are three phases (i.e., mutualism phase, commensalism phase, and parasitism phase) that mimic symbiotic relationships between two organisms in nature. The new organisms are generated via these phases which are described as follows.

3.1.1. Mutualism Phase

Mutualism is a symbiotic relationship where both organisms get benefits from each other. In the SOS, an organism Xj is randomly selected from the ecosystem, which is used to interact with an organism Xi. The organism Xi is the ith member of the ecosystem. Both organisms increase their mutual survival abilities based on a mutualism in the ecosystem. The new organisms for Xi and Xj are generated based on the mutualistic symbiosis relationship between organisms Xi and Xj by the following equations [31]:

where is a vector of random numbers between 0 and 1; Xbest represents the best organism in an ecosystem; MV denotes a mutual vector that represents the mutualistic symbiosis relationship between organisms Xi and Xj; bf₁ and bf₂ are the benefit factors which describe the level of benefit to each organism. These factors are stochastically selected as either 1 or 2 (1 is for partial benefit while 2 is for full benefit).

The new organisms are accepted only if they give a better fitness value compared to the previous organisms.

3.1.2. Commensalism Phase

Commensalism is a symbiotic relationship where one organism is benefited and the other is neither harmed nor benefited. In this phase, an organism Xj is randomly selected from the ecosystem to interact with the organism Xi. As a result from the interaction, organism Xi benefits while the organism Xj is neither harmed nor benefited. The new organism of Xi produced by this interaction is calculated as follows [31]:

According to the rules, the new organism is only updated if it gives a better fitness value compared to the prior organism.

3.1.3. Parasitism Phase

Parasitism is a symbiotic relationship between two different organisms where one benefits and the other is harmed. In this phase, organism Xi is offered a role player of a parasite through a vector called “Parasite_Vector.” The organism Xi is duplicated and modified itself by using a random number to create a Parasite_Vector in the search space. Another organism Xj is randomly chosen and serves as a host to the Parasite_Vector. If the Parasite_Vector has a better fitness value, then it will kill organism Xj and replace it in the ecosystem. Otherwise, organism Xj will have immunity from the parasite, and the Parasite_Vector will disappear from that ecosystem [31].

3.2. Hybrid Symbiosis Organisms Search Algorithm

In 2016, Nama et al. [32] proposed the hybrid symbiosis organisms search algorithm by embedding the SQI to the original SOS for improving the entire search capability of the algorithm. The SQI is used to get a set of new organisms. It is executed after the completion of the parasitism phase at the present iteration. The formulation of the SQI can be described as follows [32].

Considering two organism Xj and where and in the ecosystem, the organism Xi is updated by the three-point SQI. The mth dimension of the new organism is calculated by the following equation:

where m = 1, 2, …, D and fi, fj, and fk are the fitness values of the ith, jth, and kth organisms, respectively.

The new organism is set to the ecosystem if its fitness value is better than that of the corresponding organism in the ecosystem.

When a new organism is created, it is further checked for boundary violation. If any organism violates the boundary, that organism will be repaired by the following strategy:

where LBi and UBi are the lower and upper bounds of the ith organism, respectively.

The pseudocode of the SQI-SOS algorithm is depicted in Algorithm 1.

Set control parameters (dimension of problem D, ecosystem size Eco_size, and maximum number of iteration itermax)

Initialize the population of organisms randomly

Set iter = 1

while (iter < itermax)

    Identify the best organism Xbest in an ecosystem;

    for i = 1: Eco_size

        % Mutualism Phase %

        Randomly select one organism Xj, where Xj  Xi;

        Modify organisms Xi and Xj using equations (10)–(12);

        Check the new organisms for boundary violation and repair according to equation (15);

        Calculate fitness value of the new organisms;

        Update the new organisms with better fitness;

        % Commensalism Phase %

        Randomly select one organism Xj, where Xj  Xi;

        Modify organism Xi with the help of organism Xj using equation (13);

        Check the new organism for boundary violation and repair according to equation (15);

        Calculate fitness value of the new organism;

        Update the new organism with better fitness;

        % Parasitism Phase %

        Randomly select one organism Xj, where Xj  Xi;

        Modify organism Xi according to Section 3.1.3;

        Check the new organism for boundary violation and repair according to equation (15);

        Calculate fitness value of the new organism;

        Update the new organism with better fitness;

    end

    for i = 1: Eco_size

        Randomly select two organisms Xj and Xk, where Xj  Xk  Xi;

        Modify organism Xi according to equation (14);

        Check the new organism for boundary violation and repair according to equation (15);

        Calculate the fitness value of the new organism;

        Update the new organism with better fitness;

    end

    iter = iter + 1;

end

Algorithm 1: Pseudocode of the SQI-SOS algorithm.

4. Implementation of SQI-SOS to OCP Problem

4.1. Initialization

A population of organisms is created by a matrix with Eco_size rows and D columns. Each row represents a solution vector with 4n elements (n is the number of capacitors). The first n elements are buses chosen for capacitors installation. Each element of this part is a natural number. The remaining elements are the sizes of capacitors at the corresponding load levels. Thus, the solution vector for the OCP problem is formed as follows:

In the SQI-SOS, each organism of the population is randomly initialized. The solution for the number of buses and sizes of capacitors for each load level is initialized as follows:

For the size of capacitors, there are two different ways to initialize before starting the optimization process depending on the study cases. In the case with continuous capacitor size, the capacitor sizes corresponding to each load level are initialized using equation (20). Meanwhile, for the actual discrete capacitor size, a novel initialization process is proposed as follows.

Assuming that the actual capacitor size has a range of with the increment step of . As a result, there will be discrete values of capacitor size in the search space. Firstly, these discrete size values are sorted in the ascending order, and then they will be numbered from IndCmin to IndCmax corresponding to the values from 1 to Nmax. Obviously, each discrete size value will be identified by an index IndC. Also, a randomization in the initial discrete size initialization process can be fulfilled thanks to the manipulation of equation (21). Thus, capacitor size with the actual discrete value can be obtained by using equation (22). This procedure can be demonstrated as follows:

where rand₁ is a uniformly distributed random number in [0, 1] for each population of organisms; IndC,ij is an index representing the ith capacitor’s capacity with a discrete value in the jth load level; IndCmin,ij and IndCmax,ij are the lower and upper limits of the index that deputize the discrete size value of the ith capacitor in the jth load level, respectively.

After initialization, each organism needs to be evaluated by calculating its fitness function. The fitness function is formulated based on two components of the objective function and dependent variables. These dependent variables are the inclusion of bus voltages, line flows, power factor, and maximum allowable reactive power. The fitness function is calculated as follows:

where Cost is the objective function in equation (1) and , Kf, Kp, and Kq are the penalty factors for bus voltages, line flows, power factor, and maximum allowable reactive power, respectively. In this study, the used penalty factors are set to 100000.

The limit values of the dependent variables in equation (23) are expressed as the following equation:

where x represents the calculated value of , PFk, PFoverall, and QTC and xlim represents the limits of , PFk, PF, and QTC.

4.2. Stopping Criteria

This study uses the maximum number of iterations (itermax) as the stopping condition. The optimization process of the algorithm is stopped when the number of iterations (iter) is equal to the maximum number of iterations (itermax).

4.3. Overall Procedure

The flowchart of the proposed SQI-SOS method for solving the OCP problem is given in Figure 1.

Figure 1: Flowchart of the SQI-SOS for solving the OCP problem.

5. Numerical Results

To validate the effectiveness of the proposed SQI-SOS algorithm, it is tested on several test systems including 33-bus, 69-bus, and 119-bus radial distribution networks to find the optimal locations and sizes of capacitors for minimizing the objective function as formulated in Section 2. In addition, two scenarios including Scenario 1: constant power (CP) load model with different load levels and Scenario 2: voltage-dependent load model with maximum load demand have been adopted for investigations. Moreover, in Scenario 1 of this study, the proposed SQI-SOS is adapted to deal with both continuous and discrete sizing variables of the OCP problem. More specifically, in the case of the continuous size variable, the results are theoretically significant while the results in the case of the discrete size variable suit for practical applications. Note that the solution methods are applied to solve the problem with the continuous capacitor size variable named as SOSc and SQI-SOSc.

The implementation of the SQI-SOS to the OCP problem is coded in the Matlab R2016a platform, and 50 independent trials are run on a computer with Intel core i5-3337U CPU of 1.80 GHz speed and 8 GB RAM. The Matpower 6.0 toolbox [35] is used for power flow calculation.

5.1. Assumptions, Control Parameters, and Constraints

In order to examine variable load conditions in Scenario 1, it is assumed that the networks are operated at three load levels: 0.5 (light), 0.75 (medium), and 1.0 (full) for time percentages of 25, 35, and 40%, respectively. To calculate the total cost, the rates stated in Table 2 are applied. In addition, the net yearly savings are calculated as follows:

where NYS is termed as the net yearly savings; A is the yearly energy cost savings; C is the cost of capacitors calculated by the sum of capacitor purchase, operation, and installation costs.

Table 2: Rates for energy, purchase, operation, and maintenance costs.

Item
Rate

Average energy cost (KP)
0.06k$/kWh

Purchase cost (KC)
3k$/kVAr

Installation cost (Kci)
1000l$/location

Operating cost (Kco)
300y$/year/location

Hours per year (T)
8760.0000

There are two control parameters of the SQI-SOS (i.e., ecosystem size Eco_size and maximum number of iteration itermax) that have to be predetermined. The ecosystem size and the maximum number of iterations can be fixed depending on the complexity and dimension of optimization problems. They can be set to a small value for a small-scale problem and to a higher value for a large-scale problem. The parameter setting for the SQI-SOS algorithm for all test systems is given in Table 3. Table 3 also provides the information for the setting of inequality constraints for the OCP problem. Regarding the inequality constraint setting, the maximum limit of power flow is proposed to control the power flow in each line after reactive power compensation. The upper limit of the overall power factor is set to 1 for all test systems, whereas its lower limit is set to 0.95 for the 33-bus system based on the repeated trials and 0.9 for both the 69-bus and 119-bus systems according to the previous studies [16–18]. The upper and lower limits of system voltage (0.9 and 1.05, respectively) are applied to all the tested systems. In addition, for the test distribution systems, the numbers of capacitors to be installed are different. The reason is that these numbers of capacitors are chosen based on the basis of the effect assessment of the different numbers of capacitors on the system performance, i.e., the improvement in the total annual cost with compensation. The experimented results reveal that the penetration of the chosen number of capacitors into each system leads to a significant reduction in the total cost as compared to that of the remaining numbers of capacitors, but this penetration still fulfills in all relevant operational constraints.

Table 3: Parameter setting for the SQI-SOS algorithm and inequality constraint setting.

Item
33-bus test system
69-bus test system
119-bus test system

Population size
90
90
150

Maximum number of iterations
100
100
200

Maximum power flow—MPF (kW)
3925.99 (#L₁₋₂)
4027.1 (#L₁₋₂)
10677.9 (#L₁₋₂)

Maximum limit of power flow—MLPF (kW)
4000
4200
11000

Bus voltage constraint

Power factor constraint

Number of capacitors to be installed
3
2
8

Allowable capacitor range
0–1500 kVAr for continuous size variables; 0–1500 kVAr with the fixed step of 50 kVAr for discrete size variables

5.2. 33-Bus Test System

The first test system is a 33-bus radial distribution network with the line and load data from [36]. This system supplies a total power load demand of 3.72 MW and 2.3 MVAr at a voltage level of 12.66 kV.

5.2.1. Scenario 1: CP Load Model with Different Load Levels

In the first scenario, the OCP problem was examined with a CP load model to make a performance comparison with previously reported methods.

Table 4 shows the optimal siting and sizing of capacitors obtained by the tested SQI-SOSc and SOSc as well as the proposed SQI-SOS and SOS at different load levels. Fixed and switched capacitors on the load levels by the proposed SQI-SOS and SOS are provided in Table 4 as well. As seen from Table 4, for the case of full load level, the SQI-SOS provides the optimal locations to install capacitors which are buses 14, 25, and 30 with the optimal capacitor sizes of 300 kVAr, 200 kVAr, and 900 kVAr, respectively.

Table 4: Optimal siting and sizing of capacitors for the 33-bus system with the CP load model at different load levels.

Load level
Method
Optimal siting and sizing in kVAr
Total kVAr
Total kVAr demand

Light 50%
SOSc
(13, 191.51); (25, 200.72); (30, 516.28)
908.51
1150

SQI-SOSc
(13, 191.48); (25, 200.85); (30, 516.33)
908.67

SOS
(14, 200); (25, 200); (30, 500)
900

SQI-SOS
(14, 200); (25, 200); (30, 500)
900

Medium 75%
SOSc
(13, 295.44); (25, 200.72); (30, 789.82)
1285.99
1725

SQI-SOSc
(13, 295.42); (25, 200.85); (30, 789.84)
1286.11

SOS
(14, 300); (25, 200); (30, 800)
1300

SQI-SOS
(14, 300); (25, 200); (30, 800)
1300

Full 100%
SOSc
(13, 325.37); (25, 200.72); (30, 900.73)
1426.84
2300

SQI-SOSc
(13, 325.37); (25, 200.85); (30, 900.75)
1426.97

SOS
(14, 300); (25, 200); (30, 900)
1400

SQI-SOS
(14, 300); (25, 200); (30, 900)
1400

Final optimal ratings (location and size) by proposed approach
SOS
Fixed: (14, 200); (25, 200); (30, 500)
Switched: (14, 100); (30, 400)

SQI-SOS
Fixed: (14, 200); (25, 200); (30, 500)
Switched: (14, 100); (30, 400)

Table 5 shows the system conditions before and after the installation of capacitors at different load levels. For the case of 100% load level, before the installation of capacitors, the system has a real power loss of 210.9875 kW, minimum voltage of 0.9038 p.u., overall power factor of 0.8490 lagging, and total annual cost of 71,737.7 $. After the capacitors are installed according to the proposed SQI-SOS method, the power loss is reduced from 210.9875 kW to 141.5439 kW, the minimum voltage is improved from 0.9038 p.u. to 0.9275 p.u., the system overall power factor is corrected from 0.849 lagging to 0.9682 lagging, and the total annual cost is reduced from 71,737.7 $ to 56,198.33 $. Obviously, with the objective of system operating cost achieved, it leads to an annual net savings of 15,539.36 $. Although the proposed SQI-SOS and SOS methods provide the same total annual cost, the proposed SQI-SOS obtains the results with a smaller number of iterations than the SOS. It clearly shows that the convergence speed of the proposed SQI-SOS is faster than the original SOS, and this is confirmed as can be seen from Figure 2. Note that this system was first experimented when considering the OCP problem, so a comparison study for calculation performance with the previous approaches cannot be made. In addition, the statistical performance of the proposed SQI-SOS including the worst, best, mean, and standard deviation of the total annual cost is tabulated in Table 5. As shown in Table 5, the standard deviation of solutions yielded by the SQI-SOS is lower than that obtained by the SOS algorithm. This proves that the SQI-SOS method is more robust than the SOS algorithm when applied to this test system. It is worth mentioning here that the simulated actual capacitor sizing results are similar to the simulated theoretical ones, showing the reliability of computation. Moreover, in 100% load level, the connection of capacitors into the system results in a remarkable improvement in the bus voltage profile, as shown in Figure 3. In addition, the comparison of lines power flow before and after compensation with the CP load model can be observed in Figure 4. Clearly, the maximum power flow (MPF) on lines 1-2 (#L₁₋₂) is decreased from 3925.99 kW to 3856.54 kW due to optimal compensation by the proposed SQI-SOS, and this value is much lower than the maximum limit of power flow (MLPF) of 4000 kW. Furthermore, the connection of capacitors to the system leads to the reduction of power flow in most lines. The detail of power loss comparison before and after compensation for this system corresponding to the full load level using the SQI-SOS method in each line is given in Table 6. In general, the proposed SQI-SOS method is more suitable than the original SOS algorithm for dealing with the OCP problem in the CP load model of the 33-bus system. The reason is that the exploration ability of the SQI-SOS method is significantly better than that of the original SOS algorithm. Furthermore, the robustness of SQI-SOS is validated compared with the SOS algorithm.

Table 5: Comparison results of the 33-bus system before and after compensation with the CP load model at different load levels.

Load level
Item
Uncompensated
Compensated

SOSc
SQI-SOSc
SOS
SQI-SOS

Light 50%
, p.u.
0.9540
0.9667
0.9667
0.9678
0.9678

Ploss, kW
48.7870
32.8848
32.8848
32.916
32.916

PFoverall
0.8497
0.9904
0.9904
0.9897
0.9897

MPF, kW
1906.28
1890.38
1890.38
1890.41
1890.41

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Medium 75%
, p.u.
0.9295
0.9495
0.9495
0.9512
0.9512

Ploss, kW
113.9869
76.1249
76.1243
76.1849
76.1849

PFoverall
0.8494
0.9856
0.9856
0.9864
0.9864

MPF, kW
2900.23
2862.37
2862.37
2862.43
2862.43

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Full 100%
, p.u.
0.9038
0.9271
0.9271
0.9275
0.9275

Ploss, kW
210.9875
141.1855
141.1840
141.5439
141.5439

PFoverall
0.8490
0.9698
0.9698
0.9682
0.9682

MPF, kW
3925.99
3856.18
3856.18
3856.54
3856.54

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Total annual cost ($/year)
71,737.7
56,188.36
56,188.36
56,198.33
56,198.33

Annual net savings ($/year)
—
15,549.33
15,549.33
15,539.36
15,539.36

Worst cost ($/year)
—
56,449.50
56,449.50
56,507.02
56,458.73

Best cost ($/year)
—
56,188.36
56,188.36
56,198.33
56,198.33

Mean cost ($/year)
—
56,252.59
56,228.78
56,265.06
56,227.01

Standard deviation ()
—
97.37
78.96
107.41
62.58

Figure 2: Convergence curves of the total cost of the 33-bus system under the CP load model with 100% loading.

Figure 3: Voltage profile of the 33-bus system before and after compensation with the CP load model at 100% loading.

Figure 4: Comparison of lines power flow of the 33-bus system before and after compensation with the CP load model at 100% loading.

Table 6: Results of real power loss in all lines of the 33-bus system before and after the installation of capacitors.

Line
Real power loss (kW)

Uncompensated
Compensated by SQI-SOS

1-2
12.300
9.127

2-3
52.077
37.285

3-4
20.053
13.143

4-5
18.850
12.187

5-6
38.566
24.693

6-7
1.946
1.588

7-8
11.873
9.488

8-9
4.266
3.366

9-10
3.620
2.826

10-11
0.565
0.437

11-12
0.899
0.710

12-13
2.721
2.234

13-14
0.744
0.658

14-15
0.364
0.346

15-16
0.287
0.273

16-17
0.257
0.244

17-18
0.054
0.051

2–19
0.161
0.161

19-20
0.832
0.831

20-21
0.101
0.101

21-22
0.044
0.044

3–23
3.182
2.740

23-24
5.144
4.389

24-25
1.288
1.038

6–26
2.602
1.199

26-27
3.330
1.466

27-28
11.306
4.728

28-29
7.837
3.077

29-30
3.897
1.392

30-31
1.594
1.509

31-32
0.213
0.202

32-33
0.013
0.012

5.2.2. Scenario 2: Voltage-Dependent Load Model with Maximum Load Demand

The test results of the 33-bus system acquired by the applied SOS and proposed SQI-SOS methods with different voltage-dependent load models are represented in Table 7. It can be seen that in all the experimented load models, the system performance parameters have been significantly improved with capacitor installation. Table 8 reveals the detailed information on the optimal placements and reactive powers of capacitors for each load model. The reduction in the total annual cost of the system after compensated by the SOS and SQI-SOS can be observed in all the load types as depicted in Figure 5. In the actual mixed load case, the power loss before compensation is 164.9165 kW, and this loss is enhanced after optimal compensation to be 130.0845 kW. The minimum system voltage without compensation is 0.9159 p.u. and is increased to 0.9307 p.u. due to capacitor allocations. Also, the system overall power factor has been appreciably enhanced from 0.8908 lagging to 0.9776 lagging. With the installation of capacitors at the optimal locations and sizes in kVAr of {(13, 300); (25, 250); (30, 950)}, the total system annual cost is reduced to 76772.43 $/year, i.e., the net saving is 9907.66 $/year, compared to 86680.09 $/year in the uncompensated case. Although the objective values of total costs found are the same, the SOS-SQI requires less number of iterations than the SOS in the searching process as shown in Figure 6. Moreover, the enhancements in the voltage magnitudes in all buses can be perceived as seen in Figure 7. In addition, Figure 8 illustrates the comparison of lines power flow before and after compensation. It is worthy to mention that through the SQI-SOS, the MPF on L₁₋₂ is reduced to 3753.63 kW, which is lower than a maximum limit of power flow (MLPF) of 4000 kW. Moreover, the power flow in most lines for the compensated case is lower than that for the uncompensated case. Based on the experiment results, it can be concluded that the proposed SQI-SOS is capable of dealing with the practical voltage-dependent load model scenarios for this system.

Table 7: Comparative results of the 33-bus system after compensation with different load models.

Load type
Item
Uncompensated
Compensated

SOS
SQI-SOS

Industrial load
(bus)
0.9152 (18)
0.9305 (18)
0.9305 (18)

Ploss, kW
167.7916
135.9869
135.9869

PFoverall
0.9042
0.9809
0.9809

Total cost, $/year
88,191.24
80,024.69
80,024.69

Annual savings ($/year)
—
8,166.55
8,166.55

MPF, kW
3,851.37
3,825.86
3,825.86

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Residential load
(bus)
0.9160 (18)
0.9314 (18)
0.9314 (18)

Ploss, kW
164.5408
127.1548
127.1548

PFoverall
0.8822
0.9766
0.9766

Total cost, $/year
86,482.65
75,232.54
75,232.54

Annual savings ($/year)
—
11,250.10
11,250.10

MPF, kW
3,723.14
3,718.49
3,718.49

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Commercial load
(bus)
0.9175 (18)
0.9236 (18)
0.9236 (18)

Ploss, kW
159.5012
127.1272
127.1272

PFoverall
0.8707
0.9580
0.9580

Total cost, $/year
83,833.85
74,168.07
74,168.07

Annual savings ($/year)
—
9,665.77
9,665.77

MPF, kW
3,625.88
3,625.81
3,625.81

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Mixed load
(bus)
0.9159 (18)
0.9307 (18)
0.9307 (18)

Ploss, kW
164.9165
130.0845
130.0845

PFoverall
0.8908
0.9776
0.9776

Total cost, $/year
86,680.09
76,772.43
76,772.43

Annual savings ($/year)
—
9,907.66
9,907.66

MPF, kW
3,765.53
3,753.63
3,753.63

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Table 8: Optimal locations and kVArs of capacitors for the 33-bus system with different load models.

Load type
Location and injected kVAr
Total kVAr
Method

Industrial load
(13, 350); (25, 250); (30, 950)
1550
SOS

(13, 350); (25, 250); (30, 950)
1550
SQI-SOS

Residential load
(13, 300); (25, 250); (30, 950)
1500
SOS

(13, 300); (25, 250); (30, 950)
1500
SQI-SOS

Commercial load
(5, 350); (25, 100); (30, 700)
1150
SOS

(5, 350); (25, 100); (30, 700)
1150
SQI-SOS

Mixed load
(13, 300); (25, 250); (30, 950)
1500
SOS

(13, 300); (25, 250); (30, 950)
1500
SQI-SOS

Figure 5: Annual cost of the 33-bus system before and after compensation with different load models.

Figure 6: Convergence curves of the total cost of the 33-bus system with the mixed load model.

Figure 7: Voltage profile of the 33-bus system before and after compensation with the mixed load model.

Figure 8: Comparison of lines power flow of the 33-bus system before and after compensation with the mixed load model.

5.3. 69-Bus Test System

The second test system is a 69-bus radial distribution network with the total load demand of 3.8 MW and 2.69 MVAr. The data for branch and load of this system are taken from [37] with a base voltage of 12.66 kV.

5.3.1. Scenario 1: CP Load Model with Different Load Levels

In this scenario, the siting and sizing of capacitors obtained by the proposed SQI-SOS, applied SOS, and other methods are shown in Table 9 with different load levels. Table 9 also provides fixed and switched capacitors on the load levels. At the full level of loads, the SQI-SOS provides the optimal locations which are buses 18 and 61 corresponding to optimal capacitor sizes of 250 kVAr and 1100 kVAr to install the capacitors. The detail of power loss comparison before and after compensation for this system corresponding to the full load level using the SQI-SOS method in each line is given in Table 10.

Table 9: Optimal siting and sizing of capacitors for the 69-bus system with the CP load model at different load levels.

Load level
Method
Optimal siting and sizing in kVAr
Total kVAr
Total kVAr demand

Light 50%
CSA [16]
(21, 0); (62, 600)
600
1347.3

IHA [17]
(21, 0); (61, 550)
550

FPA [18]
(17, 0); (61, 550)
550

SOSc
(18, 178.14); (61, 627.28)
805.42

SQI-SOSc
(18, 178.09); (61, 627.43)
805.52

SOS
(18, 200); (61, 600)
800

SQI-SOS
(18, 200); (61, 600)
800

Medium 75%
CSA [16]
(21, 0); (62, 950)
950
2020.95

IHA [17]
(21, 0); (61, 900)
900

FPA
(17, 0); (61, 900)
900

SOSc
(18, 242.09); (61, 952.60)
1194.69

SQI-SOSc
(18, 242.01); (61, 952.60)
1194.62

SOS
(18, 250); (61, 950)
1200

SQI-SOS
(18, 250); (61, 950)
1200

Full 100%
CSA [16]
(21, 250); (62, 1200)
1450
2694.6

IHA [17]
(21, 350); (61, 1350)
1700

FPA [18]
(17, 300); (61, 1250)
1550

SOSc
(18, 242.09); (61, 1113.35)
1355.44

SQI-SOSc
(18, 242.01); (61, 1113.41)
1355.43

SOS
(18, 250); (61, 1100)
1350

SQI-SOS
(18, 250); (61, 1100)
1350

Final optimal ratings (location and size) by proposed approach
SOS
Fixed: (18, 200); (61, 600)

Switched: (18, 50); (61, 500)

SQI-SOS
Fixed: (18, 200); (61, 600)

Switched: (18, 50); (61, 500)

Table 10: Results of real power loss in all lines of the 69-bus system before and after the installation of capacitors.

Line
Real power loss (kW)

Uncompensated
Compensated by SQI-SOS

1-2
0.075
0.055

2-3
0.075
0.055

3-4
0.195
0.140

4-5
1.937
1.306

5-6
28.244
19.036

6-7
29.352
19.780

7-8
6.895
4.633

8-9
3.375
2.247

9-10
4.778
3.621

10-11
1.015
0.763

11-12
2.193
1.559

12-13
1.287
0.885

13-14
1.247
0.858

14-15
1.206
0.833

15-16
0.224
0.155

16-17
0.321
0.228

17-18
0.003
0.002

18-19
0.104
0.103

19-20
0.067
0.066

20-21
0.108
0.106

21-22
0.001
0.001

22-23
0.005
0.005

23-24
0.011
0.011

24-25
0.006
0.006

25-26
0.002
0.002

26-27
3.50E − 04
3.44E − 04

3–28
3.47E − 04
3.47E − 04

28-29
0.003
0.003

29-30
0.006
0.006

30-31
0.001
0.001

31-32
0.005
0.005

32-33
0.012
0.012

33-34
0.010
0.010

34-35
4.79E − 04
4.79E − 04

3–36
0.001
0.001

36-37
0.015
0.015

37-38
0.017
0.017

38-39
0.005
0.005

39-40
1.98E − 04
1.98E − 04

40-41
0.049
0.049

41-42
0.020
0.020

42-43
0.003
0.003

43-44
0.001
0.001

44-45
0.006
0.006

45-46
1.26E − 05
1.26E − 05

4–47
0.023
0.023

47-48
0.583
0.583

48-49
1.634
1.633

49-50
0.116
0.116

8–51
0.002
0.002

51-52
4.38E − 05
4.34E − 05

9–53
5.781
3.701

53-54
6.711
4.296

54-55
9.125
5.830

55-56
8.790
5.607

56-57
49.685
31.694

57-58
24.489
15.622

58-59
9.506
6.064

59-60
10.671
6.787

60-61
14.026
8.921

61-62
0.112
0.108

62-63
0.135
0.129

63-64
0.661
0.634

64-65
0.041
0.040

11–66
0.003
0.003

66-67
1.53E − 05
1.52E − 05

12–68
0.023
0.023

68-69
3.71E − 05
3.66E − 05

To verify the efficacy of the proposed SQI-SOS, the resulting solutions acquired by the SQI-SOS are compared with those from the other well-established methods such as SOS, SQI-SOSc, SOSc, CSA [16], IHA [17], and FPA [18], as shown in Table 11. For the case of 100% load level, before reactive power compensation, the system has a real power loss of 225.0006 kW with a minimum voltage of 0.9092 p.u. The overall power factor of the system is 0.8213 lagging. After reactive power compensation by the SQI-SOS method, the real power loss of the system is decreased to 148.4248 kW, the minimum system voltage is enhanced to 0.9281 p.u., and the system overall power factor is corrected to 0.9415. Regarding the total annual cost, it is reduced from 76,349.9 $ to 57,043.14 $ by the SQI-SOS for the case of full load level. As observed from Table 11, both the proposed SQI-SOS and original SOS find the same total annual cost. However, the SQI-SOS converges at the optimum result faster than the SOS as manifested in Figure 9. Moreover, the statistical performance parameters such as the worst, best, mean, and standard deviation values of the total annual cost by the SQI-SOS method are better than the ones by the SOS algorithm. Also, the SQI-SOS method yields the lowest total annual cost compared to that from previously reported approaches. Specifically, the best total annual cost attained by the SQI-SOS method is 57,043.14 $, whereas the best cost by CSA [16], IHA [17], and FPA [18] is 58,074.59 $, 58,165.60 $, and 57,753.11 $, respectively. Besides, the results pertaining to the actual capacitor sizing case are also similar to the results from the theoretical capacitor sizing case. This again confirms the reliability of computation. Furthermore, in Figure 10, the improvement of the bus voltage profile of the system in the 100% load level with the presence of capacitors can be verified. Figure 11 depicts the lines power flow change before and after capacitor allocation. It is realized that MPF on L₁₋₂ is reduced to 3,950.52 kW by the proposed SQI-SOS, and it is still within permissible limit. In addition, the power flow on most of the lines is diminished due to reactive power compensation. Generally, the proposed SQI-SOS method outperforms the previously published methods regarding the gain of the lowest total annual cost for the test system in this scenario. On the other hand, the SQI-SOS approach introduces a fairly new solution to the OCP problem for the 69-bus system with the CP load model. Therefore, the SQI-SOS method exposes the ability not only to improve the convergence speed, but also to enhance the robustness of the original SOS algorithm when solving the problem under the CP load model.

Table 11: Comparison results of the 69-bus system before and after compensation with the CP load model at different load levels.

Load level
Item
Uncompensated
Compensated

CSA [16]
IHA [17]
FPA [18]
SOSc
SQI-SOSc
SOS
SQI-SOS

Light 50%
, p.u.
0.9567
0.966
0.9652
0.965
0.9666
0.9666
0.9663
0.9663

Ploss, kW
51.6064
35.89
35.9451
36.5235
34.3536
34.3536
34.3938
34.3938

PFoverall
0.8184
0.93
0.9218
0.9218
0.9608
0.9608
0.9601
0.9601

MPF, kW
1952.65
—
—
—
1935.40
1935.40
1935.44
1935.44

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Medium 75%
, p.u.
0.9335
0.9486
0.949
0.9506
0.9491
0.9491
0.9491
0.9491

Ploss, kW
121.0301
83.19
82.57
82.5706
79.7548
79.755
79.7399
79.7399

PFoverall
0.8198
0.94
0.93
0.9299
0.9592
0.9592
0.9597
0.9597

MPF, kW
2972.60
—
—
—
2931.32
2931.32
2931.31
2931.31

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Full 100%
, p.u.
0.9092
0.930
0.937
0.95
0.9282
0.9282
0.9281
0.9281

Ploss, kW
225.0006
147.95
145.3236
145.14
148.323
148.323
148.4248
148.4248

PFoverall
0.8213
0.95
0.9656
0.9559
0.9419
0.9419
0.9415
0.9415

MPF, kW
4027.10
—
—
—
3950.42
3950.42
3950.52
3950.52

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Total annual cost ($/year)
76,349.9
58,074.59
58,165.60
57,753.11
57,035.52
57,035.52
57,043.14
57,043.14

Annual net savings ($/year)
—
18,275.31
18,184.30
18,596.79
19,314.37
19,314.37
19,306.75
19,306.75

Worst cost ($/year)
—
—
—
—
57,035.64
57,035.52
57,043.61
57,043.14

Best cost ($/year)
—
—
—
—
57,035.52
57,035.52
57,043.14
57,043.14

Mean cost ($/year)
—
—
—
—
57,035.53
57035.52
57,043.15
57,043.14

Standard deviation ()
—
—
—
—
0.0167
0.0
0.0657

Figure 9: Convergence curves of the total cost of the 69-bus system with the CP load model at 100% loading.

Figure 10: Voltage profile of the 69-bus system before and after compensation with the CP load model at 100% loading.

Figure 11: Comparison of lines power flow of the 69-bus system before and after compensation with the CP load model at 100% loading.

5.3.2. Scenario 2: Voltage-Dependent Load Model with Maximum Load Demand

The OCP results for the 69-bus system with voltage-dependent load models are tabulated in Table 12. From Table 12, it is observed that after optimal compensation by the proposed SQI-SOS, the system conditions are remarkably improved in all the studied load models. Also, the optimal locations and sizes of capacitors obtained by the methods for the different load types are given in Table 13. Figure 12 shows the comparison of the total system annual cost with and without capacitor allocations in different load models. For the practical mix load model, the proposed approach offers a solution with the connection of only two capacitor placements of {18, 61} with respective sizes in kVAr of {300, 1150}. As a result, the total system annual cost is reduced to 77,892.02 $/year corresponding to a net savings of 12,191.18 $/year. In addition, the real power loss is decreased from 171.3912 kW (uncompensated case) to 134.9734 kW (compensated case). Furthermore, the minimum voltage and overall power factor of the system are corrected from 0.9199 p.u. to 0.9325 p.u. and from 0.8606 lagging to 0.9512 lagging, respectively. Moreover, Figure 13 shows that the proposed SQI-SOS outperforms the SOS in terms of convergence speed when examining this load model. The improvements in the system voltage profile can be observed in Figure 14. Finally, the power flow in lines before and after compensation can be compared through a clustered column graph as depicted in Figure 15. It can be observed from the figure that the MPF on L₁₋₂ is decreased from 3,863.60 kW (in the uncompensated case) to 3,844.38 kW (in the compensated case). Besides, the power flow in most lines is improved after compensation. From the simulated results, it can be realized that the proposed SQI-SOS method can effectively deal with the problem in the voltage-dependent load models.

Table 12: Comparative results of the 69-bus system after compensation with different load models.

Load type
Item
Uncompensated
Compensated

SOS
SQI-SOS

Industrial load
(bus)
0.9187 (65)
0.9304 (65)
0.9304 (65)

Ploss, kW
175.0872
143.121
143.121

PFoverall
0.8752
0.9538
0.9538

Total cost, $/year
92,025.82
82,174.39
82,174.39

Annual savings ($/year)
—
9,851.42
9,851.42

MPF, kW
3,946.63
3,919.18
3,919.18

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Residential load
(bus)
0.9203 (65)
0.9335 (65)
0.9335 (65)

Ploss, kW
170.8264
131.1693
131.1693

PFoverall
0.8516
0.9497
0.9497

Total cost, $/year
89,786.35
75,892.56
75,892.56

Annual savings ($/year)
—
13,893.79
13,893.79

MPF, kW
3,823.35
3,808.35
3,808.35

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Commercial load
(bus)
0.9222 (65)
0.9351 (65)
0.9351 (65)

Ploss, kW
165.0464
123.2511
123.2511

PFoverall
0.8393
0.9442
0.9442

Total cost, $/year
86,748.37
71,580.75
71,580.75

Annual savings ($/year)
—
15,167.61
15,167.61

MPF, kW
3,731.56
3,728.19
3,728.19

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Mixed load
(bus)
0.9199 (65)
0.9325 (65)
0.9325 (65)

Ploss, kW
171.3912
134.9734
134.9734

PFoverall
0.8606
0.9512
0.9512

Total cost, $/year
90,083.21
77,892.02
77,892.02

Annual savings ($/year)
—
12,191.18
12,191.18

MPF, kW
3,863.60
3,844.38
3,844.38

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Table 13: Optimal locations and sizes of capacitors for the 69-bus system with different load models.

Load type
Locations and injected kVArs
Total kVAr
Method

Industrial load
(18, 300); (61, 1150)
1450
SOS

(18, 300); (61, 1150)
1450
SQI-SOS

Residential load
(18, 300); (61, 1150)
1450
SOS

(18, 300); (61, 1150)
1450
SQI-SOS

Commercial load
(18, 300); (61, 1100)
1400
SOS

(18, 300); (61, 1100)
1400
SQI-SOS

Mixed load
(18, 300); (61, 1150)
1450
SOS

(18, 300); (61, 1150)
1450
SQI-SOS

Figure 12: Annual cost of the 69-bus system before and after compensation with different load models.

Figure 13: Convergence curves of the total cost of the 69-bus system with the mixed load model.

Figure 14: Voltage profile of the 69-bus system before and after compensation with the mixed load model.

Figure 15: Comparison of lines power flow of the 69-bus system before and after compensation with the mixed load model.

5.4. 119-Bus Test System

The performance of the proposed SQI-SOS is finally tested on a large-scale radial distribution network with 119 buses. The total load demand of this system is 22.709 MW and 17.041 MVAr with a rated voltage of 11 kV. The system data are referred from [38].

5.4.1. Scenario 1: CP Load Model with Different Load Levels

Table 14 provides the optimal siting and sizing of capacitors obtained by the SQI-SOS as well as other methods such as SOS, SQI-SOSc, SOSc, CSA [16], IHA [17], FPA [18], ABC [19], and MGABC [30]. Fixed and switched capacitors on the load levels are also provided in Table 14. At the load level of 100%, the optimal locations obtained by the SQI-SOS are buses 32, 42, 50, 74, 80, 96, 107, and 111. The capacitors installed at those buses have optimal sizes of 1050 kVAr, 600 kVAr, 1500 kVAr, 1400 kVAr, 1100 kVAr, 800 kVAr, 900 kVAr, and 1450 kVAr, respectively.

Table 14: Optimal siting and sizing of capacitors for the 119-bus system with the CP load model at different load levels.

Item
Load level—light 50%

Method
CSA [16]
IHA [17]
FPA [18]
ABC [19]
MGABC [30]
SOSc
SQI-SOSc
SOS
SQI-SOS

Locations (kVArs)
32 (0)
39 (1200)
32 (0)
32 (500)
33 (300)
96 (561.52)
50 (977.53)
42 (400)
32 (850)

39 (1100)
43 (0)
40 (1100)
35 (0)
35 (750)
107 (636.82)
96 (548.25)
111 (750)
80 (800)

40 (0)
70 (900)
70 (750)
40 (0)
45 (300)
80 (807.44)
32 (899.34)
80 (800)
74 (750)

70 (350)
74 (0)
74 (1100)
50 (0)
46 (150)
50 (982.00)
80 (811.47)
74 (750)
111 (800)

74 (500)
86 (0)
89 (500)
70 (0)
49 (900)
74 (753.84)
107 (626.67)
96 (550)
96 (550)

86 (450)
91 (600)
104 (0)
73 (750)
54 (450)
42 (393.18)
42 (392.52)
107 (600)
107 (550)

108 (0)
107 (0)
109 (0)
79 (500)
71 (750)
32 (904.33)
111 (733.85)
50 (950)
50 (1000)

118 (1000)
109 (0)
112 (0)
105 (300)
76 (150)
111 (753.84)
74 (751.84)
33 (750)
42 (400)

118 (1000)
118 (1000)
106 (0)
86 (750)

109 (0)
94 (300)

110 (950)
101 (750)

110 (600)

111 (450)

114 (150)

115 (150)

Total kVAr
3400
3700
3350
3900
6900
5777.11
5741.51
5550
5700

Total kVAr demand
8520.55

Load level—medium 75%

Locations (kVArs)
32 (900)
39 (1500)
32 (0)
32 (0)
33 (600)
96 (804.70)
50 (1499.95)
42 (550)
32 (1050)

39 (1500)
43 (0)
40 (1500)
35 (800)
35 (1200)
107 (903.13)
96 (820.33)
111 (1100)
80 (1100)

40 (0)
70 (900)
70 (750)
40 (1200)
45 (300)
80 (1107.43)
32 (1137.43)
80 (1100)
74 (1150)

70 (600)
74 (600)
74 (600)
50 (450)
46 (150)
50 (1497.64)
80 (1087.39)
74 (1150)
111 (1150)

74 (750)
86 (0)
89 (1500)
70 (550)
49 (1350)
74 (1146.86)
107 (904.62)
96 (800)
96 (800)

86 (700)
91 (1500)
104 (0)
73 (750)
54 (750)
42 (553.04)
42 (556.16)
107 (950)
107 (900)

108 (750)
107 (0)
109 (0)
79 (850)
71 (1050)
32 (1122.45)
111 (1132.41)
50 (1500)
50 (1500)

118 (1100)
109 (0)
112 (0)
105 (0)
76 (300)
111 (1133.50)
74 (1145.73)
33 (1000)
42 (600)

118 (1200)
118 (1000)
106 (0)
86 (1050)

109 (800)
94 (450)

110 (1000)
101 (1200)

110 (900)

111 (600)

114 (150)

115 (150)

Total kVAr
6300
5700
5350
6400
10200
8268.79
8284.06
8150
8250

Total kVAr demand
12780.83

Load level—full 100%

Locations (kVArs)
32 (1500)
39 (1500)
32 (1500)
32 (850)
33 (900)
96 (804.85)
50 (1499.95)
42 (550)
32 (1050)

39 (1500)
43 (1000)
40 (1500)
35 (1050)
35 (1500)
107 (903.63)
96 (820.46)
111 (1400)
80 (1100)

40 (550)
70 (1000)
70 (850)
40 (1300)
45 (450)
80 (1107.66)
32 (1137.43)
80 (1100)
74 (1400)

70 (950)
74 (1000)
74 (1100)
50 (800)
46 (450)
50 (1499.53)
80 (1087.45)
74 (1400)
111 (1450)

74 (750)
86 (900)
89 (1500)
70 (550)
49 (1500)
74 (1395.53)
107 (904.64)
96 (800)
96 (800)

86 (1050)
91 (1500)
104 (500)
73 (1300)
54 (1050)
42 (553.16)
42 (556.20)
107 (950)
107 (900)

108 (1500)
107 (850)
109 (900)
79 (1200)
71 (1200)
32 (1122.81)
111 (1435.61)
50 (1500)
50 (1500)

118 (1200)
109 (850)
112 (250)
105 (700)
76 (450)
111 (1436.75)
74 (1408.13)
33 (1000)
42 (600)

118 (1200)
118 (1150)
106 (250)
86 (1350)

109 (800)
94 (750)

110 (1200)
101 (1500)

110 (1350)

111 (750)

114 (150)

115 (150)

Total kVAr
9000
9800
9250
10000
13500
8823.94
8849.91
8700
8800

Total kVAr demand

17041.1

Fixed capacitors
SOS
(42, 400); (111, 750); (80, 800); (74, 750); (96, 550); (107, 600); (50, 950); (33, 750)

SQI-SOS
(32, 850); (80, 800); (74, 750); (111, 800); (96, 550); (107, 550); (50, 1000); (42, 400)

Switched capacitors
SOS
(42, 150); (111, 650); (80, 300); (74, 650); (96, 250); (107, 350); (50, 550); (33, 250)

SQI-SOS
(32, 200); (80, 300); (74, 650); (111, 650); (96, 250); (107, 350); (50, 500); (42, 200)

The result comparison between the proposed SQI-SOS and other methods is represented in Table 15. The comparison shows that the SQI-SOS yields a better solution quality than SOS, CSA [16], IHA [17], FPA [18], ABC [19], and MGABC [30] in terms of the total annual cost. Regarding system conditions, the system without capacitor installation has a real power loss of 1298.09 kW, minimum voltage of 0.8688 p.u., and overall power factor of 0.7998 lagging at full load level. With the installation of capacitors by the SQI-SOS, the real power loss is reduced to 847.0243 kW, the minimum system voltage is improved to 0.9047 p.u., and the overall system power factor is enhanced to 0.9357. The total annual cost is also decreased from 440,236.40 $ (uncompensated case) to 323,532.03 $ (compensated case) by the proposed SQI-SOS at the full load level. In this case, the original SOS settles at an annual cost objective value of 323,578.50 $, which is higher than the one found by the SQI-SOS. Furthermore, the standard deviation of total annual costs for 50 trial runs obtained by the SQI-SOS method is lower than that from the SOS method, proving that the SQI-SOS algorithm is more robust than the SOS approach in terms of the statistical performance. Figure 16 illustrates the convergence curves obtained by the SQI-SOS and SOS methods. It is obvious that the SQI-SOS approach converges to the optimum solution with less number of iterations compared to the SOS method. It shows that the convergence capacity of the SQI-SOS is better than that of the SOS. In addition, computational reliability can be verified for this system because there is no significant difference between the solution with the continuous capacitor sizing variables and the solution with the discrete ones. Another notable result is that the bus voltage profile of the system in the 100% load level is improved after the connection of capacitors by the proposed method, which is illustrated in Figure 17. In addition, the effect of capacitor placement on the lines power flow can be perceived in Figure 18. From the figure, it is clear that after the connection of capacitors results in the curtailment of MPF on L₁₋₂. Furthermore, the power flow on most lines after compensation is lower than that before compensation. The detail of power loss comparison before and after compensation corresponding to the full load level using the SQI-SOS method in each line is given in Table 16.

Table 15: Comparison results of the 119-bus system before and after compensation with the CP load model at different load levels.

Load level
Item
Uncomp.
Compensated

CSA [16]
IHA [17]
FPA [18]
ABC [19]
MGABC [30]
SOSc
SQI-SOSc
SOS
SQI-SOS

Light 50%
, p.u.
0.9385
0.955
0.9552
—
0.9539
0.9566
0.9567
0.9567
0.9566
0.9566

Ploss, kW
297.15
208.96
206.8402
209.0868
207.52
206.65
194.8344
194.8289
195.4589
194.8911

PFoverall
0.7998
0.87
0.9188
0.914
0.8915
0.9883
0.9701
0.9693
0.9654
0.9685

MPF, kW
5233.93
—
—
—
—
—
5202.31
5202.31
5202.93
5202.32

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Medium 75%
, p.u.
0.9049
0.933
0.92
—
0.9313
0.9336
0.9335
0.9335
0.9335
0.9335

Ploss, kW
697.33
471.01
473.1
488.83
471.78
482.01
451.2657
451.2566
452.5665
451.4294

PFoverall
0.7998
0.91
0.9207
0.9419
0.9068
0.9860
0.9636
0.9638
0.9618
0.9633

MPF, kW
7927.17
—
—
—
—
—
7852.32
7852.28
7853.62
7852.45

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Full 100%
, p.u.
0.8688
0.906
0.902
—
0.90886
0.9073
0.9047
0.9049
0.9047
0.9047

Ploss, kW
1298.09
858.89
843.1459
853.1543
854.39
874.03
846.6694
846.2974
847.3224
847.0243

PFoverall
0.7998
0.92
0.9488
0.9419
0.9295
0.9844
0.9360
0.9363
0.9344
0.9357

MPF, kW
10677.92
—
—
—
—
—
10547.83
10547.60
10548.47
10548.23

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Total annual cost ($/year)
440,236.40
332,077.38
332,573.27
336,216.33
337,983.73
359,583.69
323,491.68
323,488.99
323,578.50

Annual net savings ($/year)
—
108,159.02
107,663.13
104,020.07
102,252.67
80,652.70
116,744.71
116,747.40
116,657.89

Worst cost ($/year)
—
—
—
—
—
—
327346.43
325,776.65
326,745.33

Best cost ($/year)
—
—
—
—
—
—
323,491.68
323,488.99
323,578.50

Mean cost ($/year)
—
—
—
—
—
—
324,441.73
324,248.97
324,445.95

Standard deviation (
—
—
—
—
—
—
786.63
629.26
644.11

The sign “” implies that for a fair comparison, the obtained load flow results are recycled with capacitor parameters given in [30] using our proposed load flow technique, and the total cost in [30] is also recalculated using equation (1) with the price data provided in Table 2.

Figure 16: Convergence curves of the total cost of the 119-bus system with the CP load model at 100% loading.

Figure 17: Voltage profile of the 119-bus system before and after compensation with the CP load model at 100% loading.

Figure 18: Comparison of lines power flow of the 119-bus system before and after compensation with the CP load model at 100% loading.

Table 16: Results of real power loss in all lines of the 119-bus system before and after the installation of capacitors.

Line
Real power loss (kW)

Uncompensated
Compensated by SQI-SOS

1-2
54.693
40.757

2-3
1.07E − 04
1.07E − 04

2–4
43.480
30.111

4-5
0.032
0.032

5-6
0.021
0.021

6-7
0.008
0.008

7-8
0.003
0.003

8-9
0.002
0.002

2–10
9.016
9.009

10-11
5.060
5.056

11-12
0.242
0.242

12-13
0.155
0.155

13-14
0.141
0.140

14-15
0.018
0.018

15-16
0.010
0.010

16-17
0.002
0.002

11–18
5.692
5.687

18-19
2.999
2.997

19-20
3.288
3.286

20-21
0.829
0.829

21-22
0.439
0.439

22-23
3.596
3.593

23-24
0.516
0.516

24-25
0.080
0.080

25-26
0.007
0.007

26-27
0.002
0.002

4–28
13.066
8.854

28-29
8.888
5.880

29-30
37.661
22.342

30-31
40.934
22.299

31-32
17.805
9.099

32-33
18.523
10.325

33-34
16.343
8.905

34-35
12.331
6.343

30–36
2.083
2.046

36-37
0.348
0.342

29–38
11.434
8.464

38-39
9.456
6.855

39-40
3.689
2.654

40-41
4.037
2.855

41-42
7.621
6.211

42-43
0.263
0.257

43-44
0.064
0.063

44-45
0.028
0.028

45-46
0.005
0.005

35–47
15.022
7.490

47-48
8.107
3.970

48-49
9.464
4.546

49-50
8.580
3.956

50-51
2.016
1.887

51-52
0.525
0.492

52-53
1.340
1.254

53-54
1.086
1.017

29–55
4.162
4.140

55-56
3.834
3.815

56-57
2.957
2.941

57-58
4.074
4.053

58-59
0.329
0.327

59-60
0.275
0.274

60-61
0.081
0.081

61-62
0.014
0.014

1–63
21.514
14.882

63-64
78.020
53.567

64-65
76.194
52.927

65-66
26.431
18.290

66-67
37.674
25.135

67-68
48.439
32.331

68-69
37.607
24.970

69-70
84.702
55.975

70-71
9.090
6.320

71-72
8.740
6.211

72-73
7.149
5.446

73-74
4.499
3.497

74-75
0.313
0.289

75-76
0.613
0.565

76-77
0.009
0.008

64–78
37.404
24.564

78-79
9.678
6.266

79-80
3.949
2.301

80-81
1.822
1.773

81-82
0.461
0.449

82-83
0.152
0.148

83-84
0.019
0.018

84-85
0.005
0.005

79–86
0.419
0.409

86-87
0.011
0.011

87-88
0.001
0.001

65–89
20.677
14.298

89-90
7.558
5.200

90-91
7.180
4.964

91-92
0.291
0.283

92-93
0.103
0.100

93-94
0.022
0.021

94-95
0.022
0.021

91–96
2.974
2.224

96-97
0.917
0.890

97-98
0.012
0.012

98-99
0.002
0.002

1-100
24.074
15.852

100-101
45.611
28.730

101-102
33.412
20.393

102-103
43.442
25.825

103-104
69.359
38.948

104-105
22.986
12.902

105-106
43.600
24.262

106-107
19.223
10.705

107-108
31.590
18.174

108-109
11.153
6.153

109-110
15.047
7.897

110-111
4.069
2.678

110-112
0.409
0.388

112-113
0.011
0.010

100-114
2.160
2.157

114-115
0.210
0.210

115-116
0.248
0.248

116-117
0.031
0.031

117-118
0.006
0.006

To sum up, the proposed SQI-SOS reveals an outstanding performance when compared with the previous methods available in the literature in terms of the obtained minimum total annual cost. Moreover, it can be realized that the proposed SQI-SOS is more efficient and robust than the original SOS algorithm in solving the OCP problem in this scenario.

5.4.2. Scenario 2: Voltage-Dependent Load Model with Maximum Load Demand

The simulation results for the 119-bus system after compensation with the nonlinear load models are given in Table 17. It is realized that the optimal compensation by the proposed method leads to a significant improvement in the system performance indices in all the examined load models. The detailed information on the optimal capacitor locations and sizes for each load model is given in Table 18. The clustered column graph in Figure 19 shows the difference in the system annual cost before and after capacitor allocations. In the actual mixed load model, according to the proposed SQI-SOS, the shunt capacitors are placed at buses {32, 41, 50, 74, 80, 96, 108, 111} with respective sizes of {1400, 900, 1500, 1350, 1200, 900, 1250, 1050} kVAr. The system parameters for the compensated case such as annual cost, real power loss, minimum system voltage, and overall power factor are remarkably enhanced to 441,643.41 $/year, 765.9692 kW, 0.9104 p.u., and 0.9522 lagging from 514,676.09 $/year, 979.2163 kW, 0.8905 p.u., and 0.8464 lagging for the uncompensated case. The optimal capacitor allocations by the SQI-SOS lead to a net savings of 73,032.68 $/year. In comparison with the SOS, the solution quality obtained by the SQI-SOS is better due to the gain of lower minimum total costs. Furthermore, the solution for OCP in this load model attained by the SQI-SOS contributes to a better improvement in the system indices as compared to the one attained by the SOS. In addition, Figure 20 discloses that the SQI-SOS has a better convergence profile than the SOS. Moreover, a marked improvement in the system voltage profile after capacitor allocation can be observed in Figure 21. Also, the change on lines power flow for the cases without capacitors and with capacitors is demonstrated in Figure 22. It is observed that after the penetration of capacitors into the system, the MPF on L₁₋₂ is reduced to 10,344.84 kW as compared to 10,360.30 kW for the case without capacitors. Besides, the power flow on most lines is also improved after compensation. In summary, the proposed SQI-SOS again shows the ability to cope effectively with the nonlinear load models when applied to the large-scale distribution system.

Table 17: Comparative results of the 119-bus system after compensation with the different load models.

Load type
Item
Uncompensated
Compensated

SOS
SQI-SOS

Industrial load
(bus)
0.8888 (77)
0.9069 (77)
0.9081 (77)

Ploss, kW
996.4081
804.7756
802.4568

PFoverall
0.8627
0.9564
0.9581

Total cost, $/year
523,712.09
462,490.07
461,721.30

Annual savings ($/year)
—
61,222.01
61,990.79

MPF, kW
10,532.80
10,483.41
10,480.68

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Residential load
Vmin (bus)
0.8910 (77)
0.9121 (77)
0.9121 (77)

Ploss, kW
977.8818
755.8888
753.3145

PFoverall
0.8361
0.9393
0.9426

Total cost, $/year
513,974.67
433,645.15
433,042.09

Annual savings ($/year)
—
80,329.51
80,932.57

MPF, kW
10,273.71
10,273.42
10,272.36

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Commercial load
(bus)
0.8939 (77)
0.9139 (77)
0.9148 (77)

Ploss, kW
948.3656
748.8654
738.5942

PFoverall
0.8222
0.9244
0.9356

Total cost, $/year
498,460.96
426,653.67
423,805.13

Annual savings ($/year)
—
71,807.28
74,655.83

MPF, kW
10,069.95
10,069.89
10,069.86

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Mixed load
(bus)
0.8905 (77)
0.9103 (77)
0.9104 (77)

Ploss, kW
979.2163
769.1398
765.9692

PFoverall
0.8464
0.9481
0.9522

Total cost, $/year
514,676.09
442,259.89
441,643.41

Annual savings ($/year)
—
72,416.20
73,032.68

MPF, kW
10,360.30
10,347.46
10,344.84

(#L₁₋₂)
(#L₁₋₂)
(#L₁₋₂)

Table 18: Optimal locations and kVArs of capacitors for the 119-bus system with different load models.

Load type
Locations and injected kVArs
Total kVAr
Method

Industrial load
(50, 1500); (41, 950); (107, 1050); (91, 950); (72, 1450); (35, 1050); (110, 1500); (80, 1250)
9700
SOS

(80, 1250); (32, 1450); (50, 1500); (96, 900); (41, 850); (107, 1050); (111, 1400); (74, 1450)
9850
SQI-SOS

Residential load
(74, 1350); (111, 1100); (52, 650); (50, 1300); (96, 900); (41, 950); (109, 1150); (80, 1250)
8650
SOS

(74, 1350); (32, 1000); (96, 1000); (50, 1400); (80, 1100); (42, 650); (111, 1400); (107, 1000)
8900
SQI-SOS

Commercial load
(111, 1450); (108, 900); (6, 1100); (81, 1050); (58, 500); (52, 700); (75, 1250); (96, 600)
7550
SOS

(107, 1000); (74, 1300); (4, 1300); (96, 850); (51, 750); (111, 1400); (58, 600); (80, 1200)
8400
SQI-SOS

Mixed load
(74, 1350); (80, 1300); (50, 1500); (41, 950); (47, 850); (91, 800); (111, 1400); (107, 1050)
9200
SOS

(74, 1350); (50, 1500); (111, 1050); (96, 900); (80, 1200); (32, 1400); (41, 900); (108, 1250)
9550
SQI-SOS

Figure 19: Annual cost of the 119-bus system before and after compensation with different load models.

Figure 20: Convergence curves of the total cost of the 119-bus system with the mixed load model.

Figure 21: Voltage profile of the 119-bus system before and after compensation with the mixed load model.

Figure 22: Comparison of lines power flow of the 119-bus system before and after compensation with the mixed load model.

6. Conclusion

In this paper, the proposed SQI-SOS has been successfully implemented for solving the OCP problem with the objective of total operating cost reduction at different load models. The proposed SQI-SOS is an improvement of the SOS method to enhance its searchability in terms of the solution quality and convergence speed. The main advantage of the SQI-SOS is that it has a simple structure with only two controllable parameters; thus, it is easy to be implemented to optimization problems. The proposed SQI-SOS has been tested on different large-scale distribution systems with 33, 69, and 119 buses. The original SOS method has also been implemented to validate the exploitation capacity of the SQI-SOS. The result simulations have confirmed that the convergence speed of the proposed SQI-SOS is faster than that of the conventional SOS. In addition, the SQI-SOS method has also offered a better solution quality than other compared methods such as SOS and many other methods in terms of the total annual cost. It proves that the SQI-SOS has a good performance to compete with other optimization methods in terms of the solution quality and convergence rate for the OCP problem as well as for other optimization problems in power systems. Therefore, the proposed SQI-SOS can be a favorable method for solving the OCP problem in distribution systems.

Appendix

Detailed results of real power loss in all lines of 33-, 69-, and 119-bus test systems are tabulated in Tables 6, 10, and 16, respectively. From the tables, it can be seen that real power losses in lines are improved thanks to the installation of capacitors by the proposed SQI-SOS method.

Nomenclature

KP: | Energy cost per each kWh
Ploss,i: | Real power loss at any load level i
Ti: | Duration of load level i
KC: | Purchase rate of the capacitor per kVAr
QC,j: | Size of the capacitor placed at the bus j
Kci: | Installation cost
Ko: | Operating cost
L: | Number of load levels
T: | Designed period
CB: | Number of capacitor locations
Pslack: | Active power of the slack bus
Qslack: | Reactive power of the slack bus
PD,i: | Active power demand at the ith bus
PDo,i: | Active power demand operating point at the ith bus
QD,i: | Reactive power demand at the ith bus
QDo,i: | Reactive power demand operating point at the ith bus
PL,j: | Real power loss at the jth branch
QL,j: | Reactive power loss at the jth branch
QC,j: | Amount of reactive power at the jth bus
: | Minimum reactive power limit of the compensated bus j
: | Maximum reactive power limit of the compensated bus j
QTC: | Total reactive power injected by capacitors
nl: | Number of load buses
nbr: | Total number of branches
CB: | Number of buses where to install the capacitors
: | Minimum voltage level at bus i
: | Maximum voltage level at bus i
: | Voltage at the ith bus
: | Operating point voltage
Nb: | Total number of buses of the network
PFk: | Power flow through the kth line
: | The maximum limit of power flow through the kth line
PFoverall: | System overall power factor
PFmin: | Minimum system power factor limit at the slack bus
PFmax: | Maximum system power factor limit at the slack bus.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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