# A Simplified Sine Cosine Algorithm for the Solution of Optimal Reactive Power Dispatch

Authors: Sushil Kumar Gupta; Manoj Kumar Kar; Lalit Kumar; Sanjay Kumar
Journal: International Transactions on Electrical Energy Systems
 (2022)

---

## Abstract

In this article, a simplified sine cosine algorithm (SSCA) is applied to solve the optimal reactive power dispatch (ORPD) issues by estimating the control variables. This algorithm uses sine cosine functions and generates number of random solutions to obtain the best solution by fluctuating inwards or outwards. The SSCA is implemented in the ORPD problem to find the best control variables to achieve minimum power loss and maximum net savings. Furthermore, the efficacy of SSCA is validated with other recently used algorithms considering three case studies, i.e., IEEE-30, -57, and -118 bus test system. The results show that the SSCA approach finds more precise and superior ORPD solutions. A comparison among SSCA and other methods proves the robustness of SSCA to attain the solution with faster convergence. The statistical analysis is performed to justify the effectiveness of SSCA by yielding minimum operating cost and maximum net savings as compared to other techniques considered in this study.


---

## Body

## 1. Introduction

With the increased growth of industrial and residential load demand, the transmission network has become stressed. The construction of new generating stations is not reccommended because of environmental and economic factors. Hence, the power transfer ability of the transmission lines has to be enhanced by the effective use of the existing networks. Since the last few decades, FACTS devices emerged to improve the performance of the power system by changing the AC transmission parameters [1].The steady state network condition is determined by using power flow analysis [2]. The loading condition can be easily predicted from the power flow in the lines [3]. Nonlinear algebraic equations are used to express the steady state and reactive power supplied by a bus in a power network. As a result, iterative approaches are utilized to solve these equations [4]. FACTS devices, on the other hand, are used in the power system to improve and maintain many characteristics such as generation cost, transmission losses, system load ability, and voltage stability in both the power and distribution systems. The location, type, and rating of the devices all have an impact on how the system operates and responds. The metaheuristic algorithms draw the attention of researchers because of their efficient computational capabilities. The purpose of this article is to optimize the device’s position or placement, as well as its type and size using efficient optimization algorithms.The importance of optimal reactive power dispatch (ORPD) in modern energy management systems has piqued the interest of the research community in the power sector, with the goal of reducing real power losses and improving bus voltage while preserving load demand and operating constraints [5]. The ORPD is a nonlinear optimization problem that involves continuous and discrete control variables satisfying both equality and inequality constraints. ORPD plays a major role in economic and reliable power supply. FACTS devices mitigate the network overloading and ORPD problems when it is optimally placed [6]. The ORPD problem was solved by using PSO [7], DE [8], BBO [9], and MFO [10,11]. Some researchers used distributed generation for minimizing loss [12,13]. The solution of optimal power flow (OPF) problem and voltage profile enhancement has been achieved incorporating some renewable sources [14–16]. However, the renewable generations are not predictable. Hence, the use of existing system to fulfill the load demand becomes the main challenge for power engineers. To examine the impact of static VAR compensator (SVC) and thyristor-controlled series capacitor (TCSC), the ORPD issue with a UPFC model was addressed utilizing lightning attachment procedure optimization [17,18]. The optimal solution was obtained in [19] with less computation time using the sine cosine algorithm. The OPF problem has been solved using an improved MFO algorithm in [20]. This algorithm was applied to different standard test bus systems with various single and multiobjective functions considered as case studies. In [21], the efficacy of MFO algorithm is justified considering various active and reactive load conditions for an IEEE-30 bus system. A fuzzy-based PSO is applied to solve the OPF problem considering different practical constraints [22]. The voltage profile was found to be enhanced by applying GSA in [23], considering nominal and contingency conditions. The maximum load ability was achieved by incorporating FACTS controllers with an objective to minimize cost in [24]. To achieve the same, the authors used a hybrid AL-MF-SS method and tested on the standard IEEE-6 and IEEE-30 bus system. The self-adaptive DE approach is utilized to provide high-quality solutions to single-objective and multiobjective optimal power flow problems [25]. To handle the current stochastic OPF problems with renewable generators, a new form of the DE method is suggested in [26]. A PSO-TS hybrid technique [27] and diversity-enhanced PSO [28] is utilized to find the control variable settings that minimize transmission active power losses and load bus voltage variations. However, no literature discussed the ORPD problems considering both active and reactive loading. A modified DE method is utilized to find the solutions to ORPD problem considering three standard test systems [29].


### 1.1. Research Contributions

The contribution of the proposed work is summarized as follows:(1)
A simplified SCA (SSCA) algorithm is proposed that combines the exploration capability of SCA/rand-target updating schemes with the exploitation capability of SCA/best-target updating schemes through the parallel use of sine cosine operators(2)
The effectiveness of the proposed SSCA approach is assessed using 13 benchmark unimodal and multimodal functions(3)
As comparison techniques, SCA, BBO, MFO, and PSO, the four promising metaheuristic algorithms, are used(4)
Minimizing the active power loss and operating cost(5)
The IEEE-30, -57, and -118 test bus systems are used to assess the performance of the proposed SSCA under various active and reactive loading scenariosThe outlines of this paper are as follows: problem formulation is given in Section2, the description of proposed SSCA is described in Section 3, the discussion of simulation results is explained in Section 4, and the research findings of this work is concluded in Section 5.


## 1.1. Research Contributions

The contribution of the proposed work is summarized as follows:(1)
A simplified SCA (SSCA) algorithm is proposed that combines the exploration capability of SCA/rand-target updating schemes with the exploitation capability of SCA/best-target updating schemes through the parallel use of sine cosine operators(2)
The effectiveness of the proposed SSCA approach is assessed using 13 benchmark unimodal and multimodal functions(3)
As comparison techniques, SCA, BBO, MFO, and PSO, the four promising metaheuristic algorithms, are used(4)
Minimizing the active power loss and operating cost(5)
The IEEE-30, -57, and -118 test bus systems are used to assess the performance of the proposed SSCA under various active and reactive loading scenariosThe outlines of this paper are as follows: problem formulation is given in Section2, the description of proposed SSCA is described in Section 3, the discussion of simulation results is explained in Section 4, and the research findings of this work is concluded in Section 5.


## 2. Problem Formulation

The prime objective of ORPD is to reduce both the active power loss (APL) and the operational cost by maintaining the operational limits. ORPD is mathematically modeled as a mixed integer nonlinear and nonconvex optimization program. The solution of ORPD has to be optimized because of the nonlinearity behavior of both the objective function and constraints.The APL of the transmission line which has to be minimized can be expressed mathematically as(1)MinimizepLoss=∑i,j∈NGijVi2+Vj2−2ViVjcosδi−δj,where Gij represents the line conductance between the ith and jthbus, N is the number of buses, Vi and Vj represents the sending and receiving end voltages, respectively, and δi and δj represents the sending and receiving end voltages angle, respectively.The cost minimization function can be written as(2)Costtotal=Costenergy+Costfacts,where Costenergy=pLoss×Energy rate and Energy rate=0.06×100000×365×24.Cost due to energy loss is 0.06$/kWh, the capital cost of shunt capacitor is 1000$, the number of hours per day is 24, and the number of days in a year is 365. Equation (2) signifies the operating cost.(3)Costfacts=CostSVC+CostTCSC,CostSVC=0.0003SVCvalue2−0.3051SVCvalue+127.38US$kVAR,CostTCSC=0.0015TCSCvalue2−0.7130TCSCvalue+153.75US$kVAR.The following equality and inequality constraints must be satisfied in order to minimize the above objective function.


### 2.1. Equality Constraints

In most cases, the load flow balancing equations are expressed by equality constraints. (4) and (5) signifies the power balance equations satisfying the equality constraints as shown below:(4)Pgi−Pdi−Vi∑j=1NVjGijcosδi−δj+Bijsinδi−δj=0,(5)Qgi−Qdi−Vi∑j=1NVjGijsinδi−δj−Bijcosδi−δj=0,where Pgi and Pdi are the active power with respect to generation and load demand of ith bus, respectively; Qgi and Qdi are the reactive power with respect to Generation and demand of ith bus, respectively; and Gij and Bij are the conductance and susceptance of the lines connected to ith and jth bus, respectively.


### 2.2. Inequality Constraints

The following are the power system inequality constraints that correspond to the operational control variables.The real power generation, reactive power generation, and voltage limit for the generator at ith bus is represented by equation (5), (6), and (7), respectively, as(6)Pgimin≤Pgi≤Pgimax,where i=1,2,…,nPV, and(7)Qgimin≤Qgi≤Qgimax,where i=1,2,…,nPV, and(8)Vgimin≤Vgi≤Vgimax,where i=1,2,…,nPV.Equation (9) signifies the the tap setting of ith transformers; Timin and Timax are the minimum and maximum tap settings of the transformer; and nT is the number of transformers, which is given as(9)Timin≤Ti≤Timax,where i=1,2,…,nPV.Equations (10) and (11) signify the limits of SVC and TCSC. SVCiminandSVCimax are the minimum and maximum limits of SVC, whereas TCSCiminandTCSCimax are the minimum and maximum limits of TCSC, which are given as(10)SVCimin≤SVCi≤SVCimax,where i=1,2,…,nS, and(11)TCSCimin≤TCSCi≤TCSCimax,where i=1,2,…,nTC.Ti denotes the ith transformer’s tap setting. nPVnT, nS, and nTC represents the number of generator buses, number of transformers, number of SVC, and number of TCSC, respectively. The mathematical model of FACTS devices such as SVC and TCSC is considered same as [21].The reactive power generations of generator (QG), tap positions of transformer (Tap), susceptance value of SVC (0.05pu≤Bsvc≤0.15pu), and reactance value of TCSC (0.01pu≤xtcsc≤0.06pu) are considered as control variables.


## 2.1. Equality Constraints

In most cases, the load flow balancing equations are expressed by equality constraints. (4) and (5) signifies the power balance equations satisfying the equality constraints as shown below:(4)Pgi−Pdi−Vi∑j=1NVjGijcosδi−δj+Bijsinδi−δj=0,(5)Qgi−Qdi−Vi∑j=1NVjGijsinδi−δj−Bijcosδi−δj=0,where Pgi and Pdi are the active power with respect to generation and load demand of ith bus, respectively; Qgi and Qdi are the reactive power with respect to Generation and demand of ith bus, respectively; and Gij and Bij are the conductance and susceptance of the lines connected to ith and jth bus, respectively.


## 2.2. Inequality Constraints

The following are the power system inequality constraints that correspond to the operational control variables.The real power generation, reactive power generation, and voltage limit for the generator at ith bus is represented by equation (5), (6), and (7), respectively, as(6)Pgimin≤Pgi≤Pgimax,where i=1,2,…,nPV, and(7)Qgimin≤Qgi≤Qgimax,where i=1,2,…,nPV, and(8)Vgimin≤Vgi≤Vgimax,where i=1,2,…,nPV.Equation (9) signifies the the tap setting of ith transformers; Timin and Timax are the minimum and maximum tap settings of the transformer; and nT is the number of transformers, which is given as(9)Timin≤Ti≤Timax,where i=1,2,…,nPV.Equations (10) and (11) signify the limits of SVC and TCSC. SVCiminandSVCimax are the minimum and maximum limits of SVC, whereas TCSCiminandTCSCimax are the minimum and maximum limits of TCSC, which are given as(10)SVCimin≤SVCi≤SVCimax,where i=1,2,…,nS, and(11)TCSCimin≤TCSCi≤TCSCimax,where i=1,2,…,nTC.Ti denotes the ith transformer’s tap setting. nPVnT, nS, and nTC represents the number of generator buses, number of transformers, number of SVC, and number of TCSC, respectively. The mathematical model of FACTS devices such as SVC and TCSC is considered same as [21].The reactive power generations of generator (QG), tap positions of transformer (Tap), susceptance value of SVC (0.05pu≤Bsvc≤0.15pu), and reactance value of TCSC (0.01pu≤xtcsc≤0.06pu) are considered as control variables.


## 3. Methodology

Mirjalili [20] proposed a population-based algorithm utilizing sine and cosine function and named it as the sine cosine algorithm. The simplicity and ease of implementation makes this algorithm popular. However, it suffers from slow convergence rate and fails to maintain balance between exploitation and exploration resulting in premature convergence. Hence, to overcome the demerits of SCA, we have proposed a simplified version of SCA (SSCA) using fewer parameters. Two search agent updating schemes, namely SCA/best-target and SCA/rand-target, are proposed. The SCA/best-target assists in the exploitation of search space, whereas SCA/rand-target assists in the exploration of search space. Finally, the exploitation and exploration property are combined by taking their mean and thus generating a new search agent.In original SCA, each search agent represents a candidate solution to the problem. The solutions vary far or near the best solution. The best solution after the completion of final iteration is utilized as the solution to the problem as(12)Xit+1=Xit+r1×sinr2×r3×Pit−Xit,ifr4<0.5,Xit+r1×cosr2×r3×Pit−Xit,ifr4≥0.5,where Xit, and Pit represents the ith position of the current solution and destination point in tth iteration, respectively, Xit+1 is the ith position of the solution in t+1 iteration, and r1,r2,r3andr4 are the parameters of the algorithm.Figure1 shows the flowchart of the proposed SSCA method. The operation includes three steps: initialization of parameters, iteration step, and selection of best solution. The best search agent of the final iteration is selected as the candidate solution to the problem.Figure 1
Flowchart for the SSCA method.In SSCA, two proposed search agent schemes are ensembled to generate a new search agent with the parallel use of sine and cosine functions. SCA/best-target moves the search agent toward the best search agent which is shown in (13) and (14) Again, SCA/rand-target moves the search agent towards a random search agent as shown in (15) and (16) The balance between exploitation and exploration is maintained by taking the mean of these four search agents and given by equation. (16). The original SCA has four parameters, whereas the SSCA has three parameters, which helps in improving the robustness of SSCA method. The three parameters r1, r2, and r3 used in SSCA are determined using equations (17), (18), and (19), respectively. The equations are shown as follows:(13)Y1=Xbestt+r1×sinr2×r3×Xrandt−Xit,(14)Y2=Xbestt+r1×cosr2×r3×Xrandt−Xit,(15)Y3=Xrandt+r1×sinr2×r3×Xbestt−Xit,(16)Y4=Xrandt+r1×cosr2×r3×Xbestt−Xit,(17)Xit+1=MeanY1,Y2,Y3,Y4,(18)r1=2×1−tTmax,(19)r2=2×pi×rand0,1,(20)r3=2×rand0,1,where t and Tmax denote current iteration and maximum iteration respectively, and rand0,1 represents the generation of a random number between 0 and 1.The mean and standard deviation of the fitness value for different methods considering 500 iterations is evaluated and shown in Table1. The bold face represents the best fitness value among all the methods. The proposed SSCA method is tested on various unimodal (F1–F7) and multimodal (F8–F13) functions as shown in Table 2 to justify its robustness. The proposed method gives better result in terms of fitness value for 10 functions (F1, F2, F3, F4, F5, F7, F9, F10, F11, and F13), whereas PSO performs better for F6 and F12 and BBO performs better for F8.Table 1
Statistical analysis of different methods (best values are presented in bold).
SSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SDF19.77E-29 ± 1.64E-281.058E-10 ± 2.048E-109.318E-20 ± 2.266E-196.217E-12 ± 1.058E-111.063E-18 ± 2.243E-18F21.65E-17 ± 1.73E-173.93E-09 ± 6.24E-095.6E-12 ± 8.95E-121.78E-08 ± 2.59E-089.1E-13 ± 1.84E-12F39.87E-17 ± 2.02E-160.02295 ± 0.0417310.04077 ± 0.041970.657616 ± 0.4416650.001023 ± 0.001323F45.56E-10 ± 9.22E-100.007677 ± 0.0191620.961993 ± 2.0177565.546965 ± 5.0854240.002221 ± 0.001851F54.809174 ± 0.6378077.542727 ± 0.4423416.309724 ± 1.613368119.3389 ± 231.18946.072164 ± 2.416001F60.184376 ± 0.1608250.516629 ± 0.2024893.884666 ± 2.8610713.89E-11 ± 1.1E-101.92E-20 ± 3.32E-20F70.00149 ± 0.0008510.005295 ± 0.0063490.005182 ± 0.0018620.019673 ± 0.0147210.002788 ± 0.001546F8-2798.49 ± 172.9513-2104.07 ± 155.2715-3915.95 ± 189.4259-3238.98 ± 310.8421-2298.53 ± 346.1811F90.519919 ± 1.6441280.554686 ± 1.7485457.987651 ± 3.17009822.884 ± 10.1465613.43193 ± 5.185881F103E-14 ± 8E-150.000324 ± 0.0005837.04E-11 ± 4.4E-113.51E-07 ± 1.99E-077.48E-11 ± 1.89E-10F110.042491 ± 0.0434160.218004 ± 0.2724180.054998 ± 0.0478940.139545 ± 0.0876550.083155 ± 0.072638F120.031975 ± 0.0269420.133127 ± 0.0911791.68E-13 ± 5.33E-130.311232 ± 0.5871182.14E-20 ± 6.59E-20F130.131962 ± 0.082030.373384 ± 0.0948151.097407 ± 0.3143393.731186 ± 4.6562495.692137 ± 4.747002Table 2
Statistical analysis of APL for IEEE 30 bus system.
Loading (Pd, Qd)in %MethodsSSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SD1000.0687 ± 0.000140.0697 ± 0.000370.0694 ± 0.000460.0689 ± 0.000210.0696 ± 0.000311150.1049 ± 0.008280.1090 ± 0.000330.1086 ± 0.000340.1084 ± 0.00070.1094 ± 0.000411250.1059 ± 0.016570.1285 ± 0.021960.1409 ± 0.000320.1407 ± 0.000020.1420 ± 0.00055


## 4. Results and Discussions

The performance of proposed SSCA method is demonstrated and validated considering standard IEEE-30, -57, and -118 bus test systems with different (i.e.,100%, 115%, and 125%) active and reactive loading conditions. The study is performed by using MATLAB 2019a software environment.


### 4.1. IEEE-30 Bus System

This system has six generators, forty-one numbers of transmission lines, four numbers of tap changing transformers, four numbers of SVC, and four numbers of TCSC [30]. The total demand of real and reactive powers is 283.4 MW and 126.2 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered, and APL and the corresponding operating cost are calculated. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times, and the statistical analysis consisting mean and standard deviation values are represented in Table 3. It is noticed from Table 2 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions. The statistical verification is justified by using WSRT analysis, as shown in Table 4. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods. A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table 5. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 2, Figures 3 and 4, respectively. The optimal control variables obtained for IEEE-30 bus system is given in Table 6. The variation of bus voltage at different loadings is shown in Figure 5. Figure 5 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 3
WSRT results on IEEE 30 bus system.
Loading\MethodSCABBOMFOPSO100−−−−115≈≈≈≈125−−−−Table 4
Cost analysis without and with FACTS controllers for IEEE 30 bus system (best values are presented in bold).
Loading (Pd, Qd) in %Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1000.07193779064SSCA3610872168192SCA3665534113530BBO3647664131400MFO3620858158206PSO36634321156321150.11205886720SSCA5513018373702SCA5729565157155BBO5708542178178MFO5698030188690PSO57490131377071250.14767757856SSCA55645272193329SCA67518581005998BBO7405178352678MFO7396769361087PSO7461418296438Table 5
Statistical analysis of APL for IEEE 57 bus system.
Loading (Pd, Qd) in %MethodsSSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SD1000.1764 ± 0.049970.1848 ± 0.049980.2503 ± 0.003160.2500 ± 0.003920.2560 ± 0.020121150.3684 ± 0.105130.3802 ± 0.087590.4596 ± 0.042770.4256 ± 0.072310.4741 ± 0.057161250.5241 ± 0.126980.6422 ± 0.097910.5823 ± 0.119850.6667 ± 0.025020.6013 ± 0.14315Figure 2
Loss variation in the IEEE-30 bus system at 100% loading.Figure 3
Loss variation in the IEEE-30 bus system at 115% loading.Figure 4
Loss variation in the IEEE 30 bus system at 125% loading.Table 6
WSRT results on IEEE 57 bus system.
Loading\MethodSCABBOMFOPSO100≈−−−115≈−≈−125−≈−≈Figure 5
Bus voltage variation at different loadings in the IEEE-30 bus system.This system has seven generators, eighty numbers of transmission lines, fifteen numbers of tap changing transformers, four numbers of SVC, and four numbers of TCSC [30]. The total demand of real and reactive powers is 1250.8 MW and 336.4 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered to calculate the APL and the corresponding operating cost. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times and the statistical analysis consisting mean and standard deviation values are represented in Table 7. It is noticed from Table 7 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions.Table 7
Cost analysis without and with FACTS controllers for IEEE 57 bus system (best values are presented in bold).
Loading (Pd, Qd) in %Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1000.295515531480SSCA92715846259896SCA97120375819443BBO131557682375712MFO131410512390429PSO1345325820782221150.687636140256SSCA1936520616775050SCA1998226016157996BBO2415815311982103MFO2237111313769143PSO24916594112236621251.087957180024SSCA2754511929634905SCA3375403223425992BBO3060621326573811MFO3504332822136696PSO3160275125577273The statistical verification is justified by using WSRT analysis, as shown in Table8. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods.Table 8
Statistical analysis of APL for IEEE 118 bus system.
Loading (Pd, Qd) in %MethodsSSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SD1001.2525 ± 0.062041.3328 ± 0.208091.2982 ± 0.001131.2955 ± 0.000801.3098 ± 0.035421151.7058 ± 0.164731.9660 ± 0.105911.8123 ± 0.192341.7810 ± 0.164841.8531 ± 0.166121252.2427 ± 0.241842.6438 ± 0.177752.4435 ± 0.188642.3232 ± 0.275462.4518 ± 0.31980A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table9. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 6, Figures 7 and 8, respectively. The variation of bus voltage at different loadings is shown in Figure 9. Figure 9 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 9
WSRT results on IEEE 118 bus system.
Loading\MethodSCABBOMFOPSO100−−−−115−≈≈−125−−≈≈Figure 6
Loss variation in the IEEE-57 bus system at 100% loading.Figure 7
Loss variation in the IEEE-57 bus system at 115% loading.Figure 8
Loss variation in the IEEE-57 bus system at 125% loading.Figure 9
Bus voltage variation at different loadings in the IEEE-57 bus system.This system has fifty-three generators, one hundred eighty-six numbers of transmission lines, nine numbers of tap changing transformers, five numbers of SVC, and five numbers of TCSC [30]. The total demand of real and reactive powers is 4242 MW and 1438 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered, and the APL and the corresponding operating cost are calculated. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times, and the statistical analysis consisting mean, standard deviation, values are represented in Table 10. It is noticed from Table 10 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions.Table 10
Cost analysis without and with FACTS controllers for IEEE 118 bus system (best values are presented in bold).
Loading (Pd, Qd) in %)<Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1001.405073846800SSCA658292988017502SCA700551213791679BBO682344435612357MFO680941085752692PSO6884571650010841152.1605113555880SSCA8965474623901134SCA10333033210225548BBO9525396218301918MFO9360725819948622PSO97399462161564181253.0488160244928SSCA12210686638138062SCA13895917921285749BBO12842825831816670MFO13125598528988943PSO12886398031380948The statistical verification is justified by using WSRT analysis, as shown in Table11. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods.Table 11
Benchmark Functions used for performance evaluation.
FunctionFormulationLimitsSphere [F1]f1x=∑i=1Nxi2[−100, 100]Schwefel 2.22 [F2]f2x=∑i=1nxi+∏i=1nxi[−10,10]Schwefel 1.2 [F3]f3x=∑i=1n∑j−1ixj2[−100, 100]Schwefel 2.21 [F4]f4x=maxixi,1≤i≤n[−100, 100]Rosenbrock [F5]f5x=∑i=1n−1100xi+1−xi22+xi−12[−30, 30]Step [F6]f6x=∑i=1nxi+0.52[−100, 100]Quartic [F7]f7x=∑i=1nixi4+random0,1[−1.28, 1.28]Schwefel [F8]f8x=∑i=1n−xisinxi[−500, 500]Rastrigin [F9]f9x=∑i=1nxi2−10cos2πxi+10[−5.12, 5.12]Ackley [F10]f10x=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e[−32, 32]Griewangk [F11]f11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+1[−600, 600]Penalized [F12]f12x=π/n101sinπy1+∑i=1n−1yi−12∑1+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4whereyi=1+xi+1/4,uxi,a,k,m=kxi−amxi>a0−a<xi<ak−xi−amxi<−a[−50, 50]Penalized2 [F13]f13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,4[−50, 50]A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table12. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 10, Figures 11 and 12, respectively. The variation of bus voltage at different loadings is shown in Figure 13. Figure 13 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 12
Optimal Control Variable at various loading of IEEE 30 bus system.
Control variable\ loading100% (Pd, Qd)115% (Pd, Qd)125% (Pd, Qd)QG (2)0.210130.0238580.45941QG (5)0.0025550.021930.35855QG (8)0.301870.452270.33745QG (11)0.09482900.26904QG (13)0.183510.400490.033199Tap (11)0.987440.951920.10439Tap (12)1.0030.953080.99152Tap (15)1.0131.02681.0491Tap (36)0.963190.968221.05SVC (7)0.0555150.131580.1117SVC (15)0.0842510.0349970.14645SVC (17)0.114670.0243530.14021SVC (21)0.101430.119030TCSC (5)0.060.0522270.000107TCSC (25)0.0414780.014130.011153TCSC (28)0.0185450.0073690.06TCSC (41)0.0027780.0171830.005297Figure 10
Loss variation in the IEEE-118 bus system at 100% loading.Figure 11
Loss variation in the IEEE-118 bus system at 115% loading.Figure 12
Loss variation in the IEEE-118 bus system at 125% loading.Figure 13
Bus voltage variation at different loadings in the IEEE-118 bus system.


## 4.1. IEEE-30 Bus System

This system has six generators, forty-one numbers of transmission lines, four numbers of tap changing transformers, four numbers of SVC, and four numbers of TCSC [30]. The total demand of real and reactive powers is 283.4 MW and 126.2 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered, and APL and the corresponding operating cost are calculated. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times, and the statistical analysis consisting mean and standard deviation values are represented in Table 3. It is noticed from Table 2 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions. The statistical verification is justified by using WSRT analysis, as shown in Table 4. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods. A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table 5. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 2, Figures 3 and 4, respectively. The optimal control variables obtained for IEEE-30 bus system is given in Table 6. The variation of bus voltage at different loadings is shown in Figure 5. Figure 5 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 3
WSRT results on IEEE 30 bus system.
Loading\MethodSCABBOMFOPSO100−−−−115≈≈≈≈125−−−−Table 4
Cost analysis without and with FACTS controllers for IEEE 30 bus system (best values are presented in bold).
Loading (Pd, Qd) in %Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1000.07193779064SSCA3610872168192SCA3665534113530BBO3647664131400MFO3620858158206PSO36634321156321150.11205886720SSCA5513018373702SCA5729565157155BBO5708542178178MFO5698030188690PSO57490131377071250.14767757856SSCA55645272193329SCA67518581005998BBO7405178352678MFO7396769361087PSO7461418296438Table 5
Statistical analysis of APL for IEEE 57 bus system.
Loading (Pd, Qd) in %MethodsSSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SD1000.1764 ± 0.049970.1848 ± 0.049980.2503 ± 0.003160.2500 ± 0.003920.2560 ± 0.020121150.3684 ± 0.105130.3802 ± 0.087590.4596 ± 0.042770.4256 ± 0.072310.4741 ± 0.057161250.5241 ± 0.126980.6422 ± 0.097910.5823 ± 0.119850.6667 ± 0.025020.6013 ± 0.14315Figure 2
Loss variation in the IEEE-30 bus system at 100% loading.Figure 3
Loss variation in the IEEE-30 bus system at 115% loading.Figure 4
Loss variation in the IEEE 30 bus system at 125% loading.Table 6
WSRT results on IEEE 57 bus system.
Loading\MethodSCABBOMFOPSO100≈−−−115≈−≈−125−≈−≈Figure 5
Bus voltage variation at different loadings in the IEEE-30 bus system.This system has seven generators, eighty numbers of transmission lines, fifteen numbers of tap changing transformers, four numbers of SVC, and four numbers of TCSC [30]. The total demand of real and reactive powers is 1250.8 MW and 336.4 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered to calculate the APL and the corresponding operating cost. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times and the statistical analysis consisting mean and standard deviation values are represented in Table 7. It is noticed from Table 7 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions.Table 7
Cost analysis without and with FACTS controllers for IEEE 57 bus system (best values are presented in bold).
Loading (Pd, Qd) in %Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1000.295515531480SSCA92715846259896SCA97120375819443BBO131557682375712MFO131410512390429PSO1345325820782221150.687636140256SSCA1936520616775050SCA1998226016157996BBO2415815311982103MFO2237111313769143PSO24916594112236621251.087957180024SSCA2754511929634905SCA3375403223425992BBO3060621326573811MFO3504332822136696PSO3160275125577273The statistical verification is justified by using WSRT analysis, as shown in Table8. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods.Table 8
Statistical analysis of APL for IEEE 118 bus system.
Loading (Pd, Qd) in %MethodsSSCA Mean ± SDSCA Mean ± SDBBO Mean ± SDMFO Mean ± SDPSO Mean ± SD1001.2525 ± 0.062041.3328 ± 0.208091.2982 ± 0.001131.2955 ± 0.000801.3098 ± 0.035421151.7058 ± 0.164731.9660 ± 0.105911.8123 ± 0.192341.7810 ± 0.164841.8531 ± 0.166121252.2427 ± 0.241842.6438 ± 0.177752.4435 ± 0.188642.3232 ± 0.275462.4518 ± 0.31980A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table9. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 6, Figures 7 and 8, respectively. The variation of bus voltage at different loadings is shown in Figure 9. Figure 9 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 9
WSRT results on IEEE 118 bus system.
Loading\MethodSCABBOMFOPSO100−−−−115−≈≈−125−−≈≈Figure 6
Loss variation in the IEEE-57 bus system at 100% loading.Figure 7
Loss variation in the IEEE-57 bus system at 115% loading.Figure 8
Loss variation in the IEEE-57 bus system at 125% loading.Figure 9
Bus voltage variation at different loadings in the IEEE-57 bus system.This system has fifty-three generators, one hundred eighty-six numbers of transmission lines, nine numbers of tap changing transformers, five numbers of SVC, and five numbers of TCSC [30]. The total demand of real and reactive powers is 4242 MW and 1438 MVAR, respectively, at 100 MVA. At first, three different active and reactive loadings (i.e., 100%, 115%, and 125%) are considered, and the APL and the corresponding operating cost are calculated. Next, the weak buses and weak branches are detected by using power flow analysis. The weak buses are chosen for the placement of SVC and the weak branches are chosen for the placement of TCSC. The optimization methods are used to set the control variables. The control variables are optimized using the proposed SSCA, SCA, BBO, MFO, and PSO. Each of the aforementioned metaheuristic method is simulated 20 times, and the statistical analysis consisting mean, standard deviation, values are represented in Table 10. It is noticed from Table 10 that the proposed SSCA method achieves lower loss as compared to other methods for different loading conditions.Table 10
Cost analysis without and with FACTS controllers for IEEE 118 bus system (best values are presented in bold).
Loading (Pd, Qd) in %)<Without FACTS controllersWith FACTS controllersNet savings in $ (X–Y)APL (p. u.)Operating cost in $ (X)TechniquesOperating cost in $ (Y)1001.405073846800SSCA658292988017502SCA700551213791679BBO682344435612357MFO680941085752692PSO6884571650010841152.1605113555880SSCA8965474623901134SCA10333033210225548BBO9525396218301918MFO9360725819948622PSO97399462161564181253.0488160244928SSCA12210686638138062SCA13895917921285749BBO12842825831816670MFO13125598528988943PSO12886398031380948The statistical verification is justified by using WSRT analysis, as shown in Table11. This test shows whether the proposed method is statistically inferior (−), superior (+), or equivalent (≈) when compared to other methods.Table 11
Benchmark Functions used for performance evaluation.
FunctionFormulationLimitsSphere [F1]f1x=∑i=1Nxi2[−100, 100]Schwefel 2.22 [F2]f2x=∑i=1nxi+∏i=1nxi[−10,10]Schwefel 1.2 [F3]f3x=∑i=1n∑j−1ixj2[−100, 100]Schwefel 2.21 [F4]f4x=maxixi,1≤i≤n[−100, 100]Rosenbrock [F5]f5x=∑i=1n−1100xi+1−xi22+xi−12[−30, 30]Step [F6]f6x=∑i=1nxi+0.52[−100, 100]Quartic [F7]f7x=∑i=1nixi4+random0,1[−1.28, 1.28]Schwefel [F8]f8x=∑i=1n−xisinxi[−500, 500]Rastrigin [F9]f9x=∑i=1nxi2−10cos2πxi+10[−5.12, 5.12]Ackley [F10]f10x=−20exp−0.21/n∑i=1nxi2−exp1/n∑i=1ncos2πxi+20+e[−32, 32]Griewangk [F11]f11x=1/4000∑i=1nxi2−∏i=1ncosxi/i+1[−600, 600]Penalized [F12]f12x=π/n101sinπy1+∑i=1n−1yi−12∑1+10sin2πyi+1+yn−12+∑i=1nuxi,10,100,4whereyi=1+xi+1/4,uxi,a,k,m=kxi−amxi>a0−a<xi<ak−xi−amxi<−a[−50, 50]Penalized2 [F13]f13x=0.1sin23πx1+∑i=1nxi−121+sin23πxi+1+xn−121+sin22πxn+∑i=1nuxi,5,100,4[−50, 50]A comparative analysis of APL, operating cost, and net savings without and with FACTS controllers is given in Table12. The variations of APL at 100%, 115%, and 125% active and reactive loading cases using different metaheuristic methods are shown in Figure 10, Figures 11 and 12, respectively. The variation of bus voltage at different loadings is shown in Figure 13. Figure 13 shows that the bus voltages are within the specified limits when FACTS devices are incorporated, thus satisfying the equality constraints.Table 12
Optimal Control Variable at various loading of IEEE 30 bus system.
Control variable\ loading100% (Pd, Qd)115% (Pd, Qd)125% (Pd, Qd)QG (2)0.210130.0238580.45941QG (5)0.0025550.021930.35855QG (8)0.301870.452270.33745QG (11)0.09482900.26904QG (13)0.183510.400490.033199Tap (11)0.987440.951920.10439Tap (12)1.0030.953080.99152Tap (15)1.0131.02681.0491Tap (36)0.963190.968221.05SVC (7)0.0555150.131580.1117SVC (15)0.0842510.0349970.14645SVC (17)0.114670.0243530.14021SVC (21)0.101430.119030TCSC (5)0.060.0522270.000107TCSC (25)0.0414780.014130.011153TCSC (28)0.0185450.0073690.06TCSC (41)0.0027780.0171830.005297Figure 10
Loss variation in the IEEE-118 bus system at 100% loading.Figure 11
Loss variation in the IEEE-118 bus system at 115% loading.Figure 12
Loss variation in the IEEE-118 bus system at 125% loading.Figure 13
Bus voltage variation at different loadings in the IEEE-118 bus system.


## 5. Conclusion

In this paper, the ORPD problems are solved by using a novel SSCA method. Three test bus systems such as IEEE-30, -57, and -118 bus system is considered as case studies with 100%, 115%, and 125% active and reactive loadings. The SSCA method is used to provide the optimal setting of control variable parameters. The optimal positioning of FACTS controllers significantly reduces the net savings. The efficacy of the proposed SSCA method is validated in achieving lower real power loss and hence operating cost. From the simulation results, it is clear that the SSCA method is capable to reduce losses as compared to the other methods. The WSRT result shows the statistical supremacy of proposed SSCA over other methods. Therefore, the SSCA method may be used for solving the real-world problems. [31–33]



---

*Source: 2165966-2022-11-15.xml*