Complexity

Complexity / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 5493958 | https://doi.org/10.1155/2020/5493958

Tri Phuoc Nguyen, Tuan Trong Nguyen, Trung Hieu Quang, Dieu Ngoc Vo, "Simple Quadratic Interpolation-Inspired Symbiosis Organisms Search Algorithm for Optimal Placement of Capacitors in Radial Distribution Networks with Different Loading Models", Complexity, vol. 2020, Article ID 5493958, 28 pages, 2020. https://doi.org/10.1155/2020/5493958

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

Academic Editor: Mohammad Hassan Khooban
Received04 Mar 2020
Revised18 Jul 2020
Accepted30 Jul 2020
Published30 Sep 2020

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 [810] 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) [1113], 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


Load typeConstant loadIndustrial loadResidential loadCommercial 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 XjXi;
  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 XjXi;
  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 XjXi;
  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 XjXkXi;
  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

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.

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.


ItemRate

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 [1618]. 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.


Item33-bus test system69-bus test system119-bus test system

Population size9090150
Maximum number of iterations100100200
Maximum power flow—MPF (kW)3925.99 (#L1-2)4027.1 (#L1-2)10677.9 (#L1-2)
Maximum limit of power flow—MLPF (kW)4000420011000
Bus voltage constraint
Power factor constraint
Number of capacitors to be installed328
Allowable capacitor range0–1500 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.


Load levelMethodOptimal siting and sizing in kVArTotal kVArTotal kVAr demand

Light 50%SOSc(13, 191.51); (25, 200.72); (30, 516.28)908.511150
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)1285.991725
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)1426.842300
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 approachSOSFixed: (14, 200); (25, 200); (30, 500)
Switched: (14, 100); (30, 400)
SQI-SOSFixed: (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.


Load levelItemUncompensatedCompensated
SOScSQI-SOScSOSSQI-SOS

Light 50%, p.u.0.95400.96670.96670.96780.9678
Ploss, kW48.787032.884832.884832.91632.916
PFoverall0.84970.99040.99040.98970.9897
MPF, kW1906.281890.381890.381890.411890.41
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Medium 75%, p.u.0.92950.94950.94950.95120.9512
Ploss, kW113.986976.124976.124376.184976.1849
PFoverall0.84940.98560.98560.98640.9864
MPF, kW2900.232862.372862.372862.432862.43
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Full 100%, p.u.0.90380.92710.92710.92750.9275
Ploss, kW210.9875141.1855141.1840141.5439141.5439
PFoverall0.84900.96980.96980.96820.9682
MPF, kW3925.993856.183856.183856.543856.54
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Total annual cost ($/year)71,737.756,188.3656,188.3656,198.3356,198.33
Annual net savings ($/year)15,549.3315,549.3315,539.3615,539.36
Worst cost ($/year)56,449.5056,449.5056,507.0256,458.73
Best cost ($/year)56,188.3656,188.3656,198.3356,198.33
Mean cost ($/year)56,252.5956,228.7856,265.0656,227.01
Standard deviation ()97.3778.96107.4162.58


LineReal power loss (kW)
UncompensatedCompensated by SQI-SOS

1-212.3009.127
2-352.07737.285
3-420.05313.143
4-518.85012.187
5-638.56624.693
6-71.9461.588
7-811.8739.488
8-94.2663.366
9-103.6202.826
10-110.5650.437
11-120.8990.710
12-132.7212.234
13-140.7440.658
14-150.3640.346
15-160.2870.273
16-170.2570.244
17-180.0540.051
2–190.1610.161
19-200.8320.831
20-210.1010.101
21-220.0440.044
3–233.1822.740
23-245.1444.389
24-251.2881.038
6–262.6021.199
26-273.3301.466
27-2811.3064.728
28-297.8373.077
29-303.8971.392
30-311.5941.509
31-320.2130.202
32-330.0130.012

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.


Load typeItemUncompensatedCompensated
SOSSQI-SOS

Industrial load (bus)0.9152 (18)0.9305 (18)0.9305 (18)
Ploss, kW167.7916135.9869135.9869
PFoverall0.90420.98090.9809
Total cost, $/year88,191.2480,024.6980,024.69
Annual savings ($/year)8,166.558,166.55
MPF, kW3,851.373,825.863,825.86
(#L1-2)(#L1-2)(#L1-2)

Residential load (bus)0.9160 (18)0.9314 (18)0.9314 (18)
Ploss, kW164.5408127.1548127.1548
PFoverall0.88220.97660.9766
Total cost, $/year86,482.6575,232.5475,232.54
Annual savings ($/year)11,250.1011,250.10
MPF, kW3,723.143,718.493,718.49
(#L1-2)(#L1-2)(#L1-2)

Commercial load (bus)0.9175 (18)0.9236 (18)0.9236 (18)
Ploss, kW159.5012127.1272127.1272
PFoverall0.87070.95800.9580
Total cost, $/year83,833.8574,168.0774,168.07
Annual savings ($/year)9,665.779,665.77
MPF, kW3,625.883,625.813,625.81
(#L1-2)(#L1-2)(#L1-2)

Mixed load (bus)0.9159 (18)0.9307 (18)0.9307 (18)
Ploss, kW164.9165130.0845130.0845
PFoverall0.89080.97760.9776
Total cost, $/year86,680.0976,772.4376,772.43
Annual savings ($/year)9,907.669,907.66
MPF, kW3,765.533,753.633,753.63
(#L1-2)(#L1-2)(#L1-2)


Load typeLocation and injected kVArTotal kVArMethod

Industrial load(13, 350); (25, 250); (30, 950)1550SOS
(13, 350); (25, 250); (30, 950)1550SQI-SOS
Residential load(13, 300); (25, 250); (30, 950)1500SOS
(13, 300); (25, 250); (30, 950)1500SQI-SOS
Commercial load(5, 350); (25, 100); (30, 700)1150SOS
(5, 350); (25, 100); (30, 700)1150SQI-SOS
Mixed load(13, 300); (25, 250); (30, 950)1500SOS
(13, 300); (25, 250); (30, 950)1500SQI-SOS

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.


Load levelMethodOptimal siting and sizing in kVArTotal kVArTotal kVAr demand

Light 50%CSA [16](21, 0); (62, 600)6001347.3
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)9502020.95
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)14502694.6
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 approachSOSFixed: (18, 200); (61, 600)
Switched: (18, 50); (61, 500)
SQI-SOSFixed: (18, 200); (61, 600)
Switched: (18, 50); (61, 500)


LineReal power loss (kW)
UncompensatedCompensated by SQI-SOS

1-20.0750.055
2-30.0750.055
3-40.1950.140
4-51.9371.306
5-628.24419.036
6-729.35219.780
7-86.8954.633
8-93.3752.247
9-104.7783.621
10-111.0150.763
11-122.1931.559
12-131.2870.885
13-141.2470.858
14-151.2060.833
15-160.2240.155
16-170.3210.228
17-180.0030.002
18-190.1040.103
19-200.0670.066
20-210.1080.106
21-220.0010.001
22-230.0050.005
23-240.0110.011
24-250.0060.006
25-260.0020.002
26-273.50E − 043.44E − 04
3–283.47E − 043.47E − 04
28-290.0030.003
29-300.0060.006
30-310.0010.001
31-320.0050.005
32-330.0120.012
33-340.0100.010
34-354.79E − 044.79E − 04
3–360.0010.001
36-370.0150.015
37-380.0170.017
38-390.0050.005
39-401.98E − 041.98E − 04
40-410.0490.049
41-420.0200.020
42-430.0030.003
43-440.0010.001
44-450.0060.006
45-461.26E − 051.26E − 05
4–470.0230.023
47-480.5830.583
48-491.6341.633
49-500.1160.116
8–510.0020.002
51-524.38E − 054.34E − 05
9–535.7813.701
53-546.7114.296
54-559.1255.830
55-568.7905.607
56-5749.68531.694
57-5824.48915.622
58-599.5066.064
59-6010.6716.787
60-6114.0268.921
61-620.1120.108
62-630.1350.129
63-640.6610.634
64-650.0410.040
11–660.0030.003
66-671.53E − 051.52E − 05
12–680.0230.023
68-693.71E − 053.66E − 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.


Load levelItemUncompensatedCompensated
CSA [16]IHA [17]FPA [18]SOScSQI-SOScSOSSQI-SOS

Light 50%, p.u.0.95670.9660.96520.9650.96660.96660.96630.9663
Ploss, kW51.606435.8935.945136.523534.353634.353634.393834.3938
PFoverall0.81840.930.92180.92180.96080.96080.96010.9601
MPF, kW1952.651935.401935.401935.441935.44
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Medium 75%, p.u.0.93350.94860.9490.95060.94910.94910.94910.9491
Ploss, kW121.030183.1982.5782.570679.754879.75579.739979.7399
PFoverall0.81980.940.930.92990.95920.95920.95970.9597
MPF, kW2972.602931.322931.322931.312931.31
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Full 100%, p.u.0.90920.9300.9370.950.92820.92820.92810.9281
Ploss, kW225.0006147.95145.3236145.14148.323148.323148.4248148.4248
PFoverall0.82130.950.96560.95590.94190.94190.94150.9415
MPF, kW4027.103950.423950.423950.523950.52
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Total annual cost ($/year)76,349.958,074.5958,165.6057,753.1157,035.5257,035.5257,043.1457,043.14
Annual net savings ($/year)18,275.3118,184.3018,596.7919,314.3719,314.3719,306.7519,306.75
Worst cost ($/year)57,035.6457,035.5257,043.6157,043.14
Best cost ($/year)57,035.5257,035.5257,043.1457,043.14
Mean cost ($/year)57,035.5357035.5257,043.1557,043.14
Standard deviation ()0.01670.00.0657

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.


Load typeItemUncompensatedCompensated
SOSSQI-SOS

Industrial load (bus)0.9187 (65)0.9304 (65)0.9304 (65)
Ploss, kW175.0872143.121143.121
PFoverall0.87520.95380.9538
Total cost, $/year92,025.8282,174.3982,174.39
Annual savings ($/year)9,851.429,851.42
MPF, kW3,946.633,919.183,919.18
(#L1-2)(#L1-2)(#L1-2)

Residential load (bus)0.9203 (65)0.9335 (65)0.9335 (65)
Ploss, kW170.8264131.1693131.1693
PFoverall0.85160.94970.9497
Total cost, $/year89,786.3575,892.5675,892.56
Annual savings ($/year)13,893.7913,893.79
MPF, kW3,823.353,808.353,808.35
(#L1-2)(#L1-2)(#L1-2)

Commercial load (bus)0.9222 (65)0.9351 (65)0.9351 (65)
Ploss, kW165.0464123.2511123.2511
PFoverall0.83930.94420.9442
Total cost, $/year86,748.3771,580.7571,580.75
Annual savings ($/year)15,167.6115,167.61
MPF, kW3,731.563,728.193,728.19
(#L1-2)(#L1-2)(#L1-2)

Mixed load (bus)0.9199 (65)0.9325 (65)0.9325 (65)
Ploss, kW171.3912134.9734134.9734
PFoverall0.86060.95120.9512
Total cost, $/year90,083.2177,892.0277,892.02
Annual savings ($/year)12,191.1812,191.18
MPF, kW3,863.603,844.383,844.38
(#L1-2)(#L1-2)(#L1-2)


Load typeLocations and injected kVArsTotal kVArMethod

Industrial load(18, 300); (61, 1150)1450SOS
(18, 300); (61, 1150)1450SQI-SOS
Residential load(18, 300); (61, 1150)1450SOS
(18, 300); (61, 1150)1450SQI-SOS
Commercial load(18, 300); (61, 1100)1400SOS
(18, 300); (61, 1100)1400SQI-SOS
Mixed load(18, 300); (61, 1150)1450SOS
(18, 300); (61, 1150)1450SQI-SOS

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.


ItemLoad level—light 50%
MethodCSA [16]IHA [17]FPA [18]ABC [19]MGABC [30]SOScSQI-SOScSOSSQI-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 kVAr340037003350390069005777.115741.5155505700
Total kVAr demand8520.55

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 kVAr6300570053506400102008268.798284.0681508250
Total kVAr demand12780.83

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 kVAr90009800925010000135008823.948849.9187008800
Total kVAr demand17041.1

Fixed capacitorsSOS(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 capacitorsSOS(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.


Load levelItemUncomp.Compensated
CSA [16]IHA [17]FPA [18]ABC [19]MGABC [30]SOScSQI-SOScSOSSQI-SOS

Light 50%, p.u.0.93850.9550.95520.95390.95660.95670.95670.95660.9566
Ploss, kW297.15208.96206.8402209.0868207.52206.65194.8344194.8289195.4589194.8911
PFoverall0.79980.870.91880.9140.89150.98830.97010.96930.96540.9685
MPF, kW5233.935202.315202.315202.935202.32
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Medium 75%, p.u.0.90490.9330.920.93130.93360.93350.93350.93350.9335
Ploss, kW697.33471.01473.1488.83471.78482.01451.2657451.2566452.5665451.4294
PFoverall0.79980.910.92070.94190.90680.98600.96360.96380.96180.9633
MPF, kW7927.177852.327852.287853.627852.45
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Full 100%, p.u.0.86880.9060.9020.908860.90730.90470.90490.90470.9047
Ploss, kW1298.09858.89843.1459853.1543854.39874.03846.6694846.2974847.3224847.0243
PFoverall0.79980.920.94880.94190.92950.98440.93600.93630.93440.9357
MPF, kW10677.9210547.8310547.6010548.4710548.23
(#L1-2)(#L1-2)(#L1-2)(#L1-2)(#L1-2)

Total annual cost ($/year)440,236.40332,077.38332,573.27336,216.33337,983.73359,583.69323,491.68323,488.99323,578.50
Annual net savings ($/year)108,159.02107,663.13104,020.07102,252.6780,652.70116,744.71116,747.40116,657.89
Worst cost ($/year)327346.43325,776.65326,745.33
Best cost ($/year)323,491.68323,488.99323,578.50
Mean cost ($/year)324,441.73324,248.97324,445.95
Standard deviation (786.63629.26644.11

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.

LineReal power loss (kW)
UncompensatedCompensated by SQI-SOS

1-254.69340.757
2-31.07E − 041.07E − 04
2–443.48030.111
4-50.0320.032
5-60.0210.021
6-70.0080.008
7-80.0030.003
8-90.0020.002
2–109.0169.009
10-115.0605.056
11-120.2420.242
12-130.1550.155
13-140.1410.140
14-150.0180.018
15-160.0100.010
16-170.0020.002
11–185.6925.687
18-192.9992.997
19-203.2883.286
20-210.8290.829
21-220.4390.439
22-233.5963.593
23-240.5160.516
24-250.0800.080
25-260.0070.007
26-270.0020.002
4–2813.0668.854
28-298.8885.880
29-3037.66122.342
30-3140.93422.299
31-3217.8059.099
32-3318.52310.325
33-3416.3438.905
34-3512.3316.343
30–362.0832.046
36-370.3480.342
29–3811.4348.464
38-399.4566.855
39-403.6892.654
40-414.0372.855
41-427.6216.211
42-430.2630.257
43-440.0640.063
44-450.0280.028
45-460.0050.005
35–4715.0227.490
47-488.1073.970
48-499.4644.546
49-508.5803.956
50-512.0161.887
51-520.5250.492
52-531.3401.254
53-541.0861.017
29–554.1624.140
55-563.8343.815
56-572.9572.941
57-584.0744.053
58-590.3290.327
59-600.2750.274
60-610.0810.081
61-620.0140.014
1–6321.51414.882
63-6478.02053.567
64-6576.19452.927
65-6626.43118.290
66-6737.67425.135
67-6848.43932.331
68-6937.60724.970
69-7084.70255.975
70-719.0906.320
71-728.7406.211
72-737.1495.446
73-744.4993.497
74-750.3130.289
75-760.6130.565
76-770.0090.008
64–7837.40424.564
78-799.6786.266
79-803.9492.301
80-811.8221.773
81-820.4610.449
82-830.1520.148
83-840.0190.018
84-850.0050.005
79–860.4190.409
86-870.0110.011
87-880.0010.001
65–8920.67714.298
89-907.5585.200
90-917.1804.964
91-920.2910.283
92-930.1030.100
93-940.0220.021
94-950.0220.021
91–962.9742.224
96-970.9170.890
97-980.0120.012
98-990.0020.002
1-10024.07415.852
100-10145.61128.730
101-10233.41220.393
102-10343.44225.825
103-10469.35938.948
104-10522.98612.902
105-10643.60024.262
106-10719.22310.705
107-10831.59018.174
108-10911.1536.153
109-11015.0477.897
110-1114.0692.678
110-1120.4090.388
112-1130.0110.010
100-1142.1602.157
114-1150.2100.210
115-1160.2480.248
116-1170.0310.031
117-1180.0060.006

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.


Load typeItemUncompensatedCompensated
SOSSQI-SOS

Industrial load (bus)0.8888 (77)0.9069 (77)0.9081 (77)
Ploss, kW996.4081804.7756802.4568
PFoverall0.86270.95640.9581
Total cost, $/year523,712.09462,490.07461,721.30
Annual savings ($/year)61,222.0161,990.79
MPF, kW10,532.8010,483.4110,480.68
(#L1-2)(#L1-2)(#L1-2)

Residential loadVmin (bus)0.8910 (77)0.9121 (77)0.9121 (77)
Ploss, kW977.8818755.8888753.3145
PFoverall0.83610.93930.9426
Total cost, $/year513,974.67433,645.15433,042.09
Annual savings ($/year)80,329.5180,932.57
MPF, kW10,273.7110,273.4210,272.36
(#L1-2)(#L1-2)(#L1-2)

Commercial load (bus)0.8939 (77)0.9139 (77)0.9148 (77)
Ploss, kW948.3656748.8654738.5942
PFoverall0.82220.92440.9356
Total cost, $/year498,460.96426,653.67423,805.13
Annual savings ($/year)71,807.2874,655.83
MPF, kW10,069.9510,069.8910,069.86
(#L1-2)(#L1-2)(#L1-2)

Mixed load (bus)0.8905 (77)0.9103 (77)0.9104 (77)
Ploss, kW979.2163769.1398765.9692
PFoverall0.84640.94810.9522
Total cost, $/year514,676.09442,259.89441,643.41
Annual savings ($/year)72,416.2073,032.68
MPF, kW10,360.3010,347.4610,344.84
(#L1-2)(#L1-2)(#L1-2)


Load typeLocations and injected kVArsTotal kVArMethod

Industrial load(50, 1500); (41, 950); (107, 1050); (91, 950); (72, 1450); (35, 1050); (110, 1500); (80, 1250)9700SOS
(80, 1250); (32, 1450); (50, 1500); (96, 900); (41, 850); (107, 1050); (111, 1400); (74, 1450)9850SQI-SOS
Residential load(74, 1350); (111, 1100); (52, 650); (50, 1300); (96, 900); (41, 950); (109, 1150); (80, 1250)8650SOS
(74, 1350); (32, 1000); (96, 1000); (50, 1400); (80, 1100); (42, 650); (111, 1400); (107, 1000)8900SQI-SOS
Commercial load(111, 1450); (108, 900); (6, 1100); (81, 1050); (58, 500); (52, 700); (75, 1250); (96, 600)7550SOS
(107, 1000); (74, 1300); (4, 1300); (96, 850); (51, 750); (111, 1400); (58, 600); (80, 1200)8400SQI-SOS
Mixed load(74, 1350); (80, 1300); (50, 1500); (41, 950); (47, 850); (91, 800); (111, 1400); (107, 1050)9200SOS
(74, 1350); (50, 1500); (111, 1050); (96, 900); (80, 1200); (32, 1400); (41, 900); (108, 1250)9550SQI-SOS

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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