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

Department of Power Systems, Ho Chi Minh City University of Technology (HCMUT), 268 Ly %uong Kiet Street, District 10, Ho Chi Minh City, Vietnam Southern Electrical Testing Company, Southern Power Corporation, Electricity of Vietnam, 22 Bis Phan Dang Luu Street, District Binh %anh, Ho Chi Minh City, Vietnam Southern Region Load Dispatch Center, National Load Dispatch Center, Electricity of Vietnam, 5 Su %ien Chieu Street, District 3, Ho Chi Minh City, Vietnam Vietnam National University Ho Chi Minh City, Linh Trung Ward, %u Duc District, Ho Chi Minh City, Vietnam


Introduction
e 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. erefore, 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. e obtained results would need to be rounded to achieve practical values. erefore, 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. erefore, numerical programming methods may not effectively deal with the OCP problem for such cases. On the contrary, the heuristic methods as in [8][9][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].
Symbiotic organisms search (SOS) is new a populationbased algorithm proposed by Cheng and Prayogo [31] in 2014. is algorithm is inspired from symbiotic relationships among species in nature. e 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. erefore, 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. e proposed SQI-SOS has been tested on 33-bus, 69-bus, and 119-bus radial distribution networks with different load models. e 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. e 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) irdly, 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. e 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. e numerical results are reported in Section 5. Finally, the paper is concluded in Section 6.

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]: 2 Complexity 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.

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:

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 α and β 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. e exponent values pertaining to the load types are given in Table 1 [33,34]. {ψ p1 , ψ p2 , ψ p3 , ψ p4 } and {ψ q1 , ψ q2 , ψ q3 , ψ q4 } 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. e different types of the load model and the practical mixed load model can be defined as follows: 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 ψ p1 and ψ q1 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 ψ p2 � 0.4 and ψ q2 � 0.4, residential load ψ p3 � 0.45 and ψ q3 � 0.45, and commercial load ψ p4 � 0.15 and ψ q4 � 0.15.

Symbiosis Organisms Search
Algorithm. e SOS proposed by Cheng and Prayogo in 2014 [31] is a novel metaheuristic algorithm. is 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. e new organisms are generated via these phases which are described as follows.

Mutualism Phase.
Mutualism is a symbiotic relationship where both organisms get benefits from each other. In the SOS, an organism X j is randomly selected from the ecosystem, which is used to interact with an organism X i . e organism X i is the ith member of the ecosystem. Both organisms increase their mutual survival abilities based on a mutualism in the ecosystem. e new organisms for X i and X j are generated based on the mutualistic symbiosis relationship between organisms X i and X j by the following equations [31]: where rand(0, 1) is a vector of random numbers between 0 and 1; X best represents the best organism in an ecosystem; MV denotes a mutual vector that represents the mutualistic symbiosis relationship between organisms X i and X j ; bf 1 and bf 2 are the benefit factors which describe the level of benefit to each organism. ese factors are stochastically selected as either 1 or 2 (1 is for partial benefit while 2 is for full benefit). e new organisms are accepted only if they give a better fitness value compared to the previous organisms.

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 X j is randomly selected from the ecosystem to interact with the organism X i . As a result from the interaction, organism X i benefits while the organism X j is neither harmed nor benefited. e new organism of X i 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.

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

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. e SQI is used to get a set of new organisms. It is executed after the completion of the parasitism phase at the present iteration. e formulation of the SQI can be described as follows [32].
Considering two organism X j and X k (j ≠ k) where X j � x j,1 , x j,2 , . . . , x j,D and X k � x k,1 , x k,2 , . . . , x k,D in the ecosystem, the organism X i is updated by the three-point SQI.
e mth dimension of the new organism is calculated by the following equation: where m � 1, 2, . . ., D and f i , f j , and f k are the fitness values of the ith, jth, and kth organisms, respectively. e 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 LB i and UB i are the lower and upper bounds of the ith organism, respectively. e pseudocode of the SQI-SOS algorithm is depicted in Algorithm 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). e first n elements are buses chosen for capacitors installation. Each element of this part is a natural number. e remaining elements are the sizes of capacitors In the SQI-SOS, each organism of the population is randomly initialized. e solution for the number of buses and sizes of capacitors for each load level is initialized as follows: Q C,ij � Q C min ,ij + rand 1 × Q C max ,ij − Q C min ,ij ; i � 1, . . . , n; j � 1, . . . , L. (18) 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 [0 · · · Q C max ] with the increment step of ΔQ C . As a result, there will be N max � ((Q C max /ΔQ C ) + 1) 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 Ind Cmin to Ind Cmax corresponding to the values from 1 to N max . Obviously, each discrete size value will be identified by an index Ind C . Also, a randomization in 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 X best in an ecosystem; for i � 1: Eco_size % Mutualism Phase % Randomly select one organism X j , where X j ≠ X i ; Modify organisms X i and X j 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 X j , where X j ≠ X i ; Modify organism X i with the help of organism X j using equation (13); Check the new organism for boundary violation and repair according to equation ( Complexity 5 the initial discrete size initialization process can be fulfilled thanks to the manipulation of equation (21). us, capacitor size with the actual discrete value can be obtained by using equation (22). is procedure can be demonstrated as follows: where rand 1 is a uniformly distributed random number in [0, 1] for each population of organisms; Ind C,ij is an index representing the ith capacitor's capacity with a discrete value in the jth load level; Ind Cmin,ij and Ind Cmax,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. e fitness function is formulated based on two components of the objective function and dependent variables. ese dependent variables are the inclusion of bus voltages, line flows, power factor, and maximum allowable reactive power. e fitness function is calculated as follows: where Cost is the objective function in equation (1) and K v , K f , K p , and K q 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. e limit values of the dependent variables in equation (23) are expressed as the following equation: where x represents the calculated value of V i , PF k , PF overall , and Q TC and x lim represents the limits of V i , PF k , PF, and Q TC .

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

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

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: voltagedependent 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 SOS c and SQI-SOS c . e 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. e Matpower 6.0 toolbox [35] is used for power flow calculation.

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. ere 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. e ecosystem size and the maximum number of iterations can be fixed depending on the complexity and dimension of optimization problems. ey can be set to a small value for a smallscale problem and to a higher value for a large-scale problem. e 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. e 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][17][18]. e 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. e 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. e 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.

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

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-SOS c and SOS c 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 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  Complexity 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. is 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 (#L 1-2 ) 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. e 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. e 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.

Scenario 2: Voltage-Dependent Load Model with Maximum Load Demand.
e test results of the 33-bus system acquired by the applied SOS and proposed SQI-SOS   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. e 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. e 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 L 1-2 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  Complexity 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.

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

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   10 Complexity 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.
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-SOS c , SOS c , 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. e 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. is 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 L 1-2 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. erefore, the SQI-SOS method exposes the ability not only to improve the convergence speed, but also to  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  Figure 13 shows that the proposed SQI-SOS outperforms the SOS in terms of convergence speed when examining this load model. e 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 L 1-2 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 voltagedependent load models.

119-Bus Test
System. e performance of the proposed SQI-SOS is finally tested on a large-scale radial distribution network with 119 buses. e total load demand of this system is 22.709 MW and 17.041 MVAr with a rated voltage of 11 kV. e system data are referred from [38].  Table 14 provides the optimal siting and sizing of capacitors obtained by the SQI-SOS as well as other methods such as SOS, SQI-SOS c , SOS c , 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. e 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. e result comparison between the proposed SQI-SOS and other methods is represented in Table 15. e 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. e 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     SOS SQI-SOS Figure 9: Convergence curves of the total cost of the 69-bus system with the CP load model at 100% loading. 16 Complexity 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 L 1-2 . Furthermore, the power flow on most lines after compensation is lower than that before compensation. e 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.
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.

Scenario 2: Voltage-Dependent Load Model with Maximum Load Demand.
e 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. e detailed information on the optimal capacitor locations and sizes for each load model is given in Table 18. e clustered column graph in Figure 19 Figure 12: Annual cost of the 69-bus system before and after compensation with different load models.     (1500) 43 (1000) 40 (1500) 35 (1050) 35 (1500)    u., and 0.8464 lagging for the uncompensated case. e 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 L 1-2 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.  e 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.

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. e proposed SQI-SOS is an improvement of the SOS method to enhance its searchability in terms of the solution quality and convergence speed. e 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. e proposed SQI-SOS has been tested on different large-scale distribution systems with 33, 69, and 119 buses. e original SOS method has also been implemented to validate the exploitation capacity of the SQI-SOS. e 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. erefore, the proposed SQI-SOS can be a favorable method for solving the OCP problem in distribution systems.

K P :
Energy cost per each kWh P loss,i : Real power loss at any load level i T i : Duration of load level i K C : Purchase rate of the capacitor per kVAr Q C,j : Size of the capacitor placed at the bus j K ci : Installation cost K o : Operating cost L: Number of load levels T: Designed period CB: Number of capacitor locations P slack : Active power of the slack bus Q slack : Reactive power of the slack bus P D,i : Active power demand at the ith bus P Do,i : Active power demand operating point at the ith bus Q D,i : Reactive power demand at the ith bus Q Do,i : Reactive power demand operating point at the ith bus P L,j : Real power loss at the jth branch Q L,j : Reactive power loss at the jth branch Q C,j : Amount of reactive power at the jth bus Q min C,j : Minimum reactive power limit of the compensated bus j Q max C,j : Maximum reactive power limit of the compensated bus j Q TC : Total reactive power injected by capacitors n l : Number of load buses n br : Total number of branches CB: Number of buses where to install the capacitors V i, min : Minimum voltage level at bus i V i, max : Maximum voltage level at bus i V i : Voltage at the ith bus V o : Operating point voltage N b : Total number of buses of the network PF k : Power flow through the kth line PF max k : e maximum limit of power flow through the kth line PF overall : System overall power factor PF min : Minimum system power factor limit at the slack bus PF max : Maximum system power factor limit at the slack bus.