# A Novel Hybrid Chaotic Jaya and Sequential Quadratic Programming Method for Robust Design of Power System Stabilizers and Static VAR Compensator

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Background and Literature Review

#### 1.2. Paper Contribution and Layout

- The CJaya-SQP is a novel and alternative metaheuristic algorithm for continuous optimization which redesigned the conventional Jaya by combining chaos theory, chaotic local search strategy and SQP technique. To the best of our knowledge, the Jaya algorithm is combined with the SQP method for the first time.
- The performance of the proposed CJaya-SQP technique is investigated on 17 benchmark functions and it is compared with various optimization algorithms, including the conventional Jaya, DE, PSO, ABC, IWO and FA.
- The CJaya-SQP provides the highest performance in terms of solution accuracy, convergence speed and robustness, with the highest success rate.
- A new method based on the real power loss sensitivity approach is proposed to find the optimal location of SVC in electric power systems.
- A novel methodology is proposed for optimal design of PSSs and SVC controllers in order to alleviate power system stability problems and to mitigate low frequency oscillations.

## 2. The Original Jaya Algorithm

## 3. The Hybrid CJaya-SQP Method

#### 3.1. Chaotic Map

#### 3.2. Original Jaya with Chaos

#### 3.3. Chaotic Local Search Method (CLS)

_{best}. The range around X

_{best}can be the best area to proceed to the optimum global solution. The process of the CLS method based on the Tent map is presented as follows:

**Step 1:**Set $k=1$ and then generate the chaotic variables $c{x}_{j}^{(1)}$ using Equation (5). k denotes the iteration number of the CLS.

**Step 2:**Evaluate the target vector ${X}_{t,i}=({x}_{t,i,1},{x}_{t,i,2},\dots ,{x}_{t,i,D})$ using the following equation:

**Step 3:**Check if the decision variables ${x}_{t,i,j}^{(k)},j=1,2,\dots ,D$ are still be clamped to the predefined search space $[{x}_{\mathrm{min},j},{x}_{\mathrm{max},j}]$ according to Equation (9):

**Step 4:**Evaluate the decision variables ${x}_{t,i,j}^{(k)},j=1,2,\dots ,D$. If ${X}_{t,i}^{(k)}$ is better than ${X}_{i}$ in terms of the objective function value, go to the next step. Otherwise, go to step 6.

**Step 5:**Set ${x}_{i,j}={x}_{t,i,j}^{(k)}$$(j=1,2,\dots ,D)$ and stop the CLS method.

**Step 6:**Set $k=k+1$, if the iteration time $k\le {k}_{\mathrm{max}}$, where ${k}_{\mathrm{max}}$ denotes the maximum iteration number of the CLS, evaluate the chaotic variables $c{x}_{j}^{(k+1)}$ for the next iteration using the Tent equation and turn to step 2.

#### 3.4. Sequential Quadratic Programming (SQP)

- Update the Hessian of the Lagrangian function;
- Solve the QP sub-problem;
- Calculate a line search and merit function.

#### 3.5. Hybrid CJaya-SQP Method

Algorithm 1: The procedure of CJaya-SQP algorithm. |

1: Set the number of design variables (D), population size (NP), maximum function evaluations (MaxFEs), the maximum iteration number $MaxIter=MaxFEs/NP$ and the maximum search length of CLS ${k}_{\mathrm{max}}$; 2: Initialize: the iteration count l (l = 1), the initial conditions $c{x}_{0},c{y}_{0},c{z}_{0}$ for the Lorenz system and $c{x}_{0}$ for the Tent map; 3: Initialize the NP candidate solutions $\left({X}_{i},i=1,2,\dots NP\right)$ chaotically within their search boundaries using Equation (6); 4: Calculate the value of the objective functions ${F}_{i}=F({X}_{i})$ of the initial solutions $\left({X}_{i},i=1,2,\dots NP\right)$ during iteration $l=1$; 5: Repeat 6: Identify the indexes assigned to the best candidate ${X}_{best}$, $best=Arg(Min({F}_{i},i=1,2,\dots NP))$ and the worst candidate ${X}_{worst}$, $worst=Arg(Max({F}_{i},i=1,2,\dots NP))$ and evaluate ${F}_{best}=F({X}_{best})$ and ${F}_{worst}=F({X}_{worst})$; 7: For $i=1$ To NP Do 8: Produce the new solution ${X}_{i}^{\prime}$ by updating the ${x}_{i,j}$ variables for the ${X}_{i}$ candidate solution using Equation (7); 9: Check the variables’ boundaries; {Compare the new solution ${X}_{i}^{\prime}$ with the old one ${X}_{i}$}; 10: If $F({X}_{i}^{\prime})<F({X}_{i})$ Then ${X}_{i}={X}_{i}^{\prime}$, $F({X}_{i})=F({X}_{i}^{\prime})$ End If End For {Update the best candidate solution X _{best} and its function value F_{best} = F(X_{best}) }; 11: For $i=1$ To NP Do 12: Initialize the iteration count of the CLS k (k = 1); 13: Repeat 14: Generate the chaotic variables $c{x}_{j}^{(k)}$ for the Tent map using Equation (5); 15: Evaluate the ${x}_{t,i,j}$ components of the target vector X _{t,i} using Equation (8); 16: Check if the design variables are in the search space using Equation (9); {Compare the target vector ${X}_{t,i}$ with the candidate solution ${X}_{i}$}; 17: If $F({X}_{t,i})<F({X}_{i})$ Then ${X}_{i}={X}_{t,i}$, $F({X}_{i})=F({X}_{t,i})$, $k={k}_{\mathrm{max}}$ End If 18: $k=k+1$; {increment the iteration count of the CLS} 19: Until $k\le {k}_{\mathrm{max}}$ (End Repeat) {Update the best solution X _{best} and the function value F_{best} = F(X_{best}) }; 20: If $F({X}_{i})<F({X}_{best})$ Then ${X}_{best}={X}_{i}$, ${F}_{best}=F({X}_{i})$ End If End For 21: $l=l+1$ {increment the iteration count} 22: Until $l\le MaxIter$ (End Repeat) 23: Let the best solution ${X}_{best}$ extracted by CJaya be the initial condition for SQP; 24: Use the SQP method for searching around ${X}_{best}$ which is derived by CJaya; 25: Output the global optimum solution found by SQP. |

## 4. Problem Statement

#### 4.1. Power System Model

#### 4.2. PSS Modeling and Damping Controller Structure

#### 4.3. SVC Modeling and Damping Controller Structure

#### 4.4. Optimal SVC Location Method

**Criteria for SVC optimal placement**

- The sensitivity factor ${R}_{k}=\frac{\partial {P}_{Loss}}{\partial {Q}_{k}}$ which involves changes in RPL with respect to the change in the injection of reactive power is computed for all load buses. The bus having the largest absolute value of the sensitivity is considered as the best location for connecting an SVC controller;
- The SVC device should not be installed in the transmission lines containing the generation buses and transformers, even if the sensitivity factor is the largest there. Indeed, all generators are generally equipped with a PSS whose the main function consists in improving power system stability by producing a supplementary stabilizing signal through the excitation system. Moreover, the SVC device should not be placed in buses where there is no injected power.

## 5. Coordinated Design of PSSs and SVC via Hybrid CJaya-SQP

_{1}and J

_{2}described by Equations (24) and (25). This will provide some degree of relative stability and limits the maximum overshoot of the system.

_{1}and J

_{2}are combined into a single objective function J, defined as follows:

_{1}and J

_{2}. Its value is derived from various small signal and dynamic stability analyses conducted on the studied power system.

_{PSS}, the time constants of the lead-lag blocks T

_{1p}–T

_{4p}, the SVC gain K

_{svc}and the lead-lag time constants T

_{1s}–T

_{4s}. In this research study, there are 10 bound constraints and 20 parameters to be optimized, which are selected as continuous decision variables. It is worth mentioning that the initial values of controllers’ parameters are chaotically generated using the Lorenz system within their search boundaries using Equation (6). The washout time constants for both PSS and SVC are kept fixed during the design process. In this study, ${T}_{wp}={T}_{ws}=10\mathrm{s}$ is used. The flowchart of the proposed simultaneous coordination design of the PSSs and SVC damping controllers by the proposed hybrid CJaya-SQP algorithm is provided in Figure 10.

## 6. Numerical Experiments

#### 6.1. Benchmark Functions and Parameter Settings

_{1}–f

_{5}) and multimodal (f

_{6}–f

_{9}) functions, whereas the rotated problems (f

_{10}–f

_{17}) are the rotated versions of f

_{2}–f

_{9}. The latter type of problem is more complex than conventional test functions. For minimizing the statistical errors, the results including the mean, the standard deviation (SD) and the standard error of means (SEM) of each test function for the seven algorithms are obtained for 30 independent runs. All the simulation experiments in this paper are performed on an Intel (R) Core (TM) i7-5500 U CPU 2.40 GHz personal computer and programs are implemented using a Matlab R2014a environment. For a fair comparison, all common parameters of the studied algorithms have the same values. In this study, the population size is 40 and the number of function evaluations (FEs) is 80,000.

#### 6.2. Results for Low-Dimensional Problems

#### 6.2.1. Comparison of Solution Accuracy

_{5}and f

_{13}test functions, the CJaya-SQP algorithm converges to a more accurate solution and significantly outperforms the original Jaya and five other algorithms in terms of mean and standard deviations of the best objective values, as well as the standard error of means for all the unimodal, multimodal and rotated problems over the 30 runs. For f

_{5}, the CJaya-SQP performs slightly worse than FA regarding the SD and SEM, but it produces the best mean for this function and it is still competitive compared to the other five algorithms. For f

_{13}, the FA algorithm attains the best mean, whereas CJaya-SQP produces the best SD and SEM values. Furthermore, the proposed algorithm provides the best results for the multimodal problems from f

_{6}to f

_{9}, which highlights its strong ability to solve complicated problems and to shun local optimums. As a whole, the CJaya-SQP algorithm achieves the most competitive results regarding solutions accuracy. It should be noted that the inferior values of the standard deviations prove the high stability and reliability of the new hybrid algorithm.

#### 6.2.2. Comparison of Convergence and Success Ratios

_{5}test function in which the success ratio is equal to 96.66%. It is also observed from the results that for f

_{7}, f

_{14}and f

_{15}test functions, only CJaya-SQP succeeds to attain the T-Value in the 30 runs. Hence, the proposed algorithm is the most efficient among all the comparative algorithms.

#### 6.2.3. Statistical Tests

#### 6.3. Results for High-Dimensional Problems

#### 6.3.1. Comparison of Solution Accuracy

_{5}and f

_{13}functions. For f

_{5}, the PSO algorithm obtains the best SD and SEM values, whereas CJaya-SQP provides the best mean result. For f

_{13}, the PSO provides the best mean result and the proposed algorithm has the best SD and SEM results. Overall, the CJaya-SQP is the most competitive algorithm regarding solutions accuracy and ability to move away from local optimums for high dimensional problems.

_{2}, f

_{3}, f

_{5}, f

_{6}, f

_{7}and f

_{8}conventional problems. These box-plots are illustrated in Figure 11. Compared to the conventional Jaya, the improvement by the proposed CJaya-SQP algorithm is obvious. Compared to other competitive algorithms, the CJaya-SQP also features an excellent performance in terms of robustness and solution quality, which is illustrated from the span of the solution distributions.

_{10}, f

_{11}, f

_{12}, f

_{14}, f

_{15}and f

_{16}rotated problems are illustrated in Figure 12. It is obvious that the CJaya-SQP provides the highest solutions quality and it is the most robust among the other competitive algorithms.

#### 6.3.2. Comparison of Convergence and Success Ratios

_{5}benchmark function in which the SR is equal to 86.66%, the proposed algorithm achieves 100% success ratio for all unimodal, multimodal and rotated problems. For f

_{13}, all algorithms failed to converge to an acceptable solution. It is also observed from the results that except for the proposed CJaya-SQP, the remaining comparative algorithms failed to converge to the threshold value for f

_{2}, f

_{5}, f

_{6}, f

_{7}, f

_{10}, f

_{11}, f

_{14}, f

_{15}and f

_{16}test functions in the 30 runs. Hence, the performance of CJaya-SQP is significantly superior to other optimization algorithms. It is worth mentioning that the conventional Jaya algorithm failed to produce a feasible solution for all 100-dimensional problems, which points out its weak global search behavior. Owing to the modifications embedded into Jaya, the CJaya-SQP presents high speed convergence and efficiency.

_{1}, f

_{4}, f

_{5}, f

_{7}, f

_{8}and f

_{9}conventional problems are shown in Figure 13. It is worth noting that the objective function values are shifted by a gap of 10

^{−3}because of some zero values, and therefore they can be plotted ona semi-log scale. It is observed from Figure 13 that CJaya-SQP algorithm is the best performing in terms of solutions quality and convergence speed.

_{10}, f

_{12}, f

_{13}, f

_{14}, f

_{16}and f

_{17}rotated problems are illustrated in Figure 14. It is obvious that the proposed algorithm has a faster convergence speed in all rotated problems than the remaining algorithms. It can be concluded that CJaya-SQP is a powerful algorithm for the design of complex optimization problems.

#### 6.3.3. Statistical Tests

## 7. Practical Application

#### 7.1. PSSs Locations

#### 7.2. SVC Location and Input Signal

#### 7.3. Damping Controllers Design and Robustness Analysis

**Scenario I:**Nominal loading, a three phase fault of six cycle duration occurs at bus 7 at $t=0.4\mathrm{s}$ and it is removed by tripping the faulted line with successful reclosure after 100 ms.

**Scenario II:**Nominal loading, a load curtailment contingency of twelve cycle duration is applied at bus 7 at $t=1\mathrm{s}$

**.**

**Scenario III:**Nominal loading, a ten cycle three phase fault occurs at bus 7 at $t=0.4\mathrm{s}$ and it is removed by an appropriate circuit breaker (outage of one of the tie lines connecting buses 7 and 8).

**Scenario IV:**Heavy loading (143% of nominal loading), a load curtailment contingency of twelve cycle duration occurs at bus 7 at $t=1\mathrm{s}$.

**Scenario I:**A three phase fault of six cycle duration at bus 7 at $t=0.4\mathrm{s}$ is applied in one of the tie lines. The fault is cleared by opening the faulted line with successful reclosure after 100 ms. The behavior of the test system is evaluated for 10 s. The local and inter-area mode responses with the coordinated and uncoordinated tuning of PSSs and SVC via CJaya-SQP are shown in Figure 17a–d. These figures show the speed deviations of G1–G2, G3–G4, G1–G3 and G1–G4. The rotor angle responses of generators G1 and G2 with respect to the swing generator G3 are provided by Figure 17e,f. It is clear from the results that both the CJaya-SQP-PSSs and CJaya-SQP-PSSs&SVC controllers are able to mitigate the EMOs and stabilize the power system. Moreover, the proposed coordinated CJaya-SQP-PSSs&SVC provides better damping characteristics than the CJaya-SQP-PSSs controller. Hence, the coordinated design method has an excellent capability to improve the dynamic stability performance and damping the low frequency oscillations quickly and efficiently.

_{PSS-G1}, U

_{PSS-G2}, U

_{PSS-G4}and B

_{SVC}are shown in Figure 17g–j. It can be concluded that the coordinated tuning of PSSs and SVC controllers has a strong ability to reduce the control effort more than CJaya-SQP-PSSs and CJaya-SQP-SVC, helping the power system to damp out the EMOs with a high speed. This confirms the potential of the suggested CJaya-SQP-based coordinated controller to boost the system damping performances.

**Scenario II**: In this scenario, the performance of the CJaya-SQP-based tuned controllers is also tested under load curtailment. At t = 1 s, the load at bus 7 is disconnected for 200 ms. Figure 18 illustrates the system dynamic response for the considered load curtailment contingency. Figure 18a,b show the speed deviation response of G1-G2 and G3-G4 and depict the local mode responses of area 1 and area 2 using the three system design methodologies. Figure 18c,d show the speed deviation response of G1-G3 and G1-G4 that illustrate the inter-area mode response using the coordinated and uncoordinated tuning of SVC and PSSs via the CJaya-SQP method. The rotor angle responses of generators G1 and G2 with respect to G3 are shown in Figure 18e,f. It can be observed from these figures that the CJaya-SQP-SVC controller is unable to mitigate both local and inter-area mode of oscillations. Moreover, it is evident that both the CJaya-SQP-based PSSs and the CJaya-SQP-based coordinated controller have the ability to damp out the local and inter-area oscillations and stabilize the test system. Moreover, it can be seen that the proposed CJaya-SQP-PSSs&SVC controller for damping the EMOs is robust and provides the best damping performances to the local and inter-area oscillations.

**Scenario III:**To highlight the robustness of the coordinated design approach, a three phase fault and transmission line outage are simultaneously considered. In this scenario, a ten cycle three phase fault occurs at bus 7 at $t=0.4\mathrm{s}$ in one of the tie lines. The fault is removed by an appropriate circuit breaker. Therefore, the power system operates with one of the tie lines between two areas. The speed deviations and the rotor angle responses under the considered fault are shown in Figure 19a–f. Assessment of these figures reveals that both the CJaya-SQP-PSSs and CJaya-SQP-SVC controllers have an inferior effect on the local oscillations and they are insufficient to dampen the inter-area oscillations. It is also observed from Figure 19a–d that the proposed CJaya-SQP-based coordinated controller has the potential to grant superior damping performances of the inter-area and local oscillations, when compared to the uncoordinated tuning of PSSs and SVC controllers. Furthermore, by the use of the coordinated CJaya-SQP-PSSs&SVC controller, the EMOs are quickly damped and the system overshoot and undershoot are greatly improved.

**Scenario IV:**To further validate the robustness of the CJaya-SQP-based coordinated controller in mitigating the power system oscillations, another severe contingency is applied by changing the loading condition of the studied system from nominal to heavy loading (143% of nominal loading). In this scenario, a load curtailment contingency of twelve cycle duration occurs at bus 7 at $t=1\mathrm{s}$. Figure 20 shows the system dynamic response under this type of contingency. The inter-area and local oscillation modes with the CJaya-SQP-based tuned controllers are shown in Figure 20a–d. The rotor angle responses of generators G1 and G2 with respect to G3 are given by Figure 20e,f. From these figures, it is clear that the CJaya-SQP-PSSs and CJaya-SQP-SVC controllers fail to damp out both local and inter-area oscillations and thus the power system is unstable under this contingency. Moreover, only the proposed coordinated CJaya-SQP-PSSs&SVC controller succeeds to stabilize the power system and mitigates both the local and inter-area oscillations quickly and efficiently under this load curtailment contingency and change in system operating condition.

#### 7.4. Quantify the Enhancement of Proposed Approach

_{1}and ITAE

_{2}are two performance indices used to provide a clear perspective of the inter-area and local mode responses, respectively. The numerical results of these indices for all four suggested scenarios are given in Table 15 and Table 16. It is worth noting that the lower values of the ITAE performance index point out the better system dynamic responses regarding damping performances and time-domain characteristics.

_{1}for CJaya-SQP-PSSs and CJaya-SQP-SVC controllers are respectively decreased by 98.51% and 99.61% to 0.1937 for the CJaya-SQP-PSSs&SVC controller. This confirms that the coordinated tuning of PSSs and SVC controllers via the CJaya-SQP method has the best system responses in terms of damping performances and time-domain characteristics of the inter-area oscillations. Additionally, it is evident from Table 16 that the total values of ITAE

_{2}for the individual design of PSSs and SVC controllers are respectively decreased by 95.69% and 98.04% to 0.0228 for the CJaya-SQP-PSSs&SVC controller. This proves that the CJaya-SQP-based coordinated controller provides superior damping performances to the local oscillations when compared to the uncoordinated tuning of PSSs and SVC controllers.

## 8. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Su, Q.; Khan, H.U.; Khan, I.; Choi, B.J.; Wu, F.; Aly, A.A. An optimized algorithm for optimal power flow based on deep learning. Energy Rep.
**2021**, 7, 2113–2124. [Google Scholar] [CrossRef] - Farah, A.; Guesmi, T.; Hadj Abdallah, H.; Ouali, A. A novel chaotic teaching–learning-based optimization algorithm for multi-machine power system stabilizers design problem. Int. J. Electr. Power Energy Syst.
**2016**, 77, 197–209. [Google Scholar] [CrossRef] - Welhazi, Y.; Guesmi, T.; Hadj Abdallah, H. Eigenvalue Assignments in Multimachine Power Systems using Multi-Objective PSO Algorithm. Int. J. Energy Optim. Eng.
**2015**, 4, 33–48. [Google Scholar] [CrossRef][Green Version] - Hu, W.; Liang, J.; Jin, Y.; Wu, F. Model of Power System Stabilizer Adapting to Multi-Operating Conditions of Local Power Grid and Parameter Tuning. Sustainability
**2018**, 10, 2089. [Google Scholar] [CrossRef][Green Version] - Jolfaei, M.G.; Sharaf, A.M.; Shariatmadar, S.M.; Poudeh, M.B. A hybrid PSS-SSSC GA-stabilization scheme for damping power system small signal oscillations. Int. J. Electr. Power Energy Syst.
**2016**, 75, 337–344. [Google Scholar] [CrossRef] - Singh, B.; Kumar, R. A comprehensive survey on enhancement of system performances by using different types of FACTS controllers in power systems with static and realistic load models. Energy Rep.
**2020**, 6, 55–79. [Google Scholar] [CrossRef] - Bruno, S.; De Carne, G.; La Scala, M. Distributed FACTS for Power System Transient Stability Control. Energies
**2020**, 13, 2901. [Google Scholar] [CrossRef] - Farah, A.; Guesmi, T.; Hadj Abdallah, H. A new method for the coordinated design of power system damping controllers. Eng. Appl. Artif. Intell.
**2017**, 64, 325–339. [Google Scholar] [CrossRef] - Hasanvand, H.; Arvan, M.R.; Mozafari, B.; Amraee, T. Coordinated design of PSS and TCSC to mitigate interarea oscillations. Int. J. Electr. Power Energy Syst.
**2016**, 78, 194–206. [Google Scholar] [CrossRef] - Movahedi, A.; Niasar, A.H.; Gharehpetian, G.B. Designing SSSC, TCSC, and STATCOM controllers using AVURPSO, GSA, and GA for transient stability improvement of a multi-machine power system with PV and wind farms. Int. J. Electr. Power Energy Syst.
**2019**, 106, 455–466. [Google Scholar] [CrossRef] - Singh, B.; Agrawal, G. Enhancement of voltage profile by incorporation of SVC in power system networks by using optimal load flow method in MATLAB/Simulink environments. Energy Rep.
**2018**, 4, 418–434. [Google Scholar] [CrossRef] - Yorino, N.; El-Araby, E.E.; Sasaki, H.; Harada, S. A new formulation for FACTS allocation for security enhancement against voltage collapse. IEEE Trans. Power Syst.
**2003**, 18, 3–10. [Google Scholar] [CrossRef] - Minguez, R.; Milano, F.; Zarate-Minano, R.; Conejo, A.J. Optimal network placement of SVC devices. IEEE Trans. Power Syst.
**2007**, 22, 1851–1860. [Google Scholar] [CrossRef] - Chang, R.W.; Saha, T.K. Maximizing Power System Loadability by OPTIMAL allocation of SVC Using Mixed Integer Linear Programming. In Proceedings of the IEEE Power & Energy Society General Meeting, Minneapolis, MN, USA, 25–29 July 2010. [Google Scholar]
- Chang, R.W.; Saha, T.K. A novel MIQCP method for FACTS allocation in complex real-world grids. Int. J. Electr. Power Energy Syst.
**2014**, 62, 735–743. [Google Scholar] [CrossRef] - Jordehi, A.R. Brainstorm optimization algorithm (BSOA): An efficient algorithm for finding optimal location and setting of FACTS devices in electric power systems. Int. J. Electr. Power Energy Syst.
**2015**, 69, 48–57. [Google Scholar] [CrossRef] - Raj, S.; Bhattacharyya, B. Optimal placement of TCSC and SVC for reactive power planning using Whale optimization algorithm. Swarm Evol. Comput.
**2018**, 40, 131–143. [Google Scholar] [CrossRef] - Weiss, M.; Abu-Jaradeh, B.N.; Chakrabortty, A.; Jamehbozorg, A.; Habibi-Ashrafi, F.; Salazar, A. A wide-area SVC controller design for inter-area oscillation damping in WECC based on a structured dynamic equivalent model. Electr. Power Syst. Res.
**2016**, 133, 1–11. [Google Scholar] [CrossRef][Green Version] - Panda, S.; Yegireddy, N.K.; Mohapatra, S.K. Hybrid BFOA–PSO approach for coordinated design of PSS and SSSC-based controller considering time delays. Int. J. Electr. Power Energy Syst.
**2013**, 49, 221–233. [Google Scholar] [CrossRef] - Bian, X.Y.; Tse, C.T.; Zhang, J.F.; Wang, K.W. Coordinated design of probabilistic PSS and SVC damping controllers. Int. J. Electr. Power Energy Syst.
**2011**, 33, 445–452. [Google Scholar] [CrossRef] - Furini, M.A.; Pereira, A.L.S.; Araujo, P.B. Pole placement by coordinated tuning of Power System Stabilizers and FACTS-POD stabilizers. Int. J. Electr. Power Energy Syst.
**2011**, 33, 615–622. [Google Scholar] [CrossRef] - Robak, S. Robust SVC controller design and analysis for uncertain power systems. Control. Eng. Pract.
**2009**, 17, 1280–1290. [Google Scholar] [CrossRef] - Panda, S.; Patidar, N.P.; Singh, R. Simultaneous Tuning of Static Var Compensator and Power System Stabilizer Employing Real-Coded Genetic Algorithm. Int. J. Electr. Comput. Eng.
**2008**, 2, 948–955. [Google Scholar] - Abd-Elazim, S.M.; Ali, E.S. Coordinated design of PSSs and SVC via bacteria foraging optimization algorithm in a multi-machine power system. Int. J. Electr. Power Energy Syst.
**2012**, 41, 44–53. [Google Scholar] [CrossRef] - Eslami, M.; Shareef, H.; Khajehzadeh, M. Optimal design of damping controllers using a new hybrid artificial bee colony algorithm. Int. J. Electr. Power Energy Syst.
**2013**, 52, 42–54. [Google Scholar] [CrossRef] - Baadji, B.; Bentarzi, H.; Bakdi, A. Comprehensive learning bat algorithm for optimal coordinated tuning of power system stabilizers and static VAR compensator in power systems. Eng. Optim. Syst.
**2019**, 52, 1761–1779. [Google Scholar] [CrossRef] - Narne, R.; Panda, P.C. PSS with multiple FACTS Controllers Coordinated Design and Real-Time Implementation Using Advanced Adaptive PSO. Int. J. of Electr. Comput. Eng.
**2014**, 8, 137–147. [Google Scholar] - Ali, E.S.; Abd-Elazim, S.M. Stability improvement of multimachine power system via new coordinated design of PSSs and SVC. Complexit
**2014**, 21, 256–266. [Google Scholar] [CrossRef] - Ali, E.S.; Abd-Elazim, S.M. Stability Enhancement of Multimachine Power System via New Coordinated Design of PSSs and SVC. WSEAS Trans. Syst.
**2014**, 13, 345–356. [Google Scholar] - Esmaili, M.R.; Hooshmand, R.A.; Parastegari, M.; Panah, P.G.; Azizkhani, S. New coordinated design of SVC and PSS for multi-machine power system using BF-PSO algorithm. Proc. Technol.
**2013**, 11, 65–74. [Google Scholar] [CrossRef][Green Version] - Shayeghi, H.; Shayanfar, H.A.; Safari, A.; Aghmasheh, R. A robust PSSs design using PSO in a multi-machine environment. Energy Convers. Manag.
**2010**, 51, 696–702. [Google Scholar] [CrossRef] - Jordehi, A.R. Time varying acceleration coefficients particle swarm optimisation (TVACPSO): A new optimisation algorithm for estimating parameters of PV cells and modules. Energy Convers. Manag.
**2016**, 129, 262–274. [Google Scholar] [CrossRef] - Rao, R.V. A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int. J. Ind. Eng. Comput.
**2016**, 7, 19–34. [Google Scholar] - Rao, R.V.; Savsani, V.J.; Vakharia, D.P. Teaching–Learning-Based Optimization: An optimization method for continuous non-linear large scale problems. Inform. Sci.
**2012**, 183, 1–15. [Google Scholar] [CrossRef] - Rao, R.V.; More, K.C. Design optimization and analysis of selected thermal devices using self-adaptive Jaya algorithm. Energy Convers. Manag.
**2017**, 140, 24–35. [Google Scholar] [CrossRef] - Rao, R.V.; Rai, D.P.; Balic, J. A multi-objective algorithm for optimization of modern machining processes. Eng. Appl. Artif. Intell.
**2017**, 61, 103–125. [Google Scholar] [CrossRef] - Rao, R.V.; Saroj, A. Economic optimization of shell-and-tube heat exchanger using Jaya algorithm with maintenance consideration. Appl. Therm. Eng.
**2017**, 116, 473–487. [Google Scholar] [CrossRef] - Singh, S.P.; Prakash, T.; Singh, V.P.; Babu, M.G. Analytic hierarchy process based automatic generation control of multi-area interconnected power system using Jaya algorithm. Eng. Appl. Artif. Intell.
**2017**, 60, 35–44. [Google Scholar] [CrossRef] - Warid, W.; Hizam, H.; Mariun, N.; Abdul-Wahab, N.I. Optimal power flow using the jaya algorithm. Energies
**2016**, 9, 678. [Google Scholar] [CrossRef] - Rao, R.V.; Waghmare, G.G. A new optimization algorithm for solving complex constrained design optimization problems. Eng Optim.
**2017**, 49, 60–83. [Google Scholar] [CrossRef] - Yu, K.; Liang, J.; Qu, B.Y.; Chen, X.; Wang, H. Parameters identification of photovoltaic models using an improved JAYA optimization algorithm. Energy Convers. Manag.
**2017**, 150, 742–753. [Google Scholar] [CrossRef] - Majumdar, M.; Mitra, T.; Nishimura, K. Optimization and Chaos; Springer: New York, NY, USA, 2000. [Google Scholar]
- Farah, A.; Belazi, A. A novel chaotic Jaya algorithm for unconstrained numerical optimization. Nonlinear Dyn.
**2018**, 93, 1451–1480. [Google Scholar] [CrossRef] - Gokhale, S.S.; Kale, V.S. An application of a tent map initiated chaotic firefly algorithm for optimal overcurrent relay coordination. Int. J. Electr. Power Energy Syst.
**2016**, 78, 336–342. [Google Scholar] [CrossRef] - Mirjalili, S.; Gandomi, A.H. Chaotic gravitational constants for the gravitational search algorithm. Appl. Soft Comput.
**2017**, 53, 407–419. [Google Scholar] [CrossRef] - Boggs, P.T.; Tolle, J.W. Sequential quadratic programming. Acta Numer.
**1995**, 4, 1–51. [Google Scholar] [CrossRef][Green Version] - Morshed, M.J.; Asgharpour, A. Hybrid imperialist competitive-sequential quadratic programming (HIC-SQP) algorithm for solving economic load dispatch with incorporating stochastic wind power: A comparative study on heuristic optimization techniques. Energy Convers. Manag.
**2014**, 84, 30–40. [Google Scholar] [CrossRef] - Elaiw, A.M.; Xia, X.; Shehata, A.M. Hybrid DE-SQP and hybrid PSO-SQP methods for solving dynamic economic emission dispatch problem with valve-point effects. Electr. Power Syst. Res.
**2013**, 103, 192–200. [Google Scholar] [CrossRef] - Krishnasamy, U.; Nanjundappan, D. Hybrid weighted probabilistic neural network and biography based optimization for dynamic economic dispatch of integrated multiple-fuel and wind power plants. Int. J. Electr. Power Energy Syst.
**2016**, 77, 385–394. [Google Scholar] [CrossRef] - Modares, H.; Naghibi Sistani, M.-B. Solving nonlinear optimal control problems using a hybrid IPSO-SQP algorithm. Eng. Appl. Artif. Intell.
**2011**, 24, 476–484. [Google Scholar] [CrossRef] - Xu, W.; Geng, Z.; Zhu, Q.; Gu, X. A piecewise linear chaotic map and sequential quadratic programming based robust hybrid particle swarm optimization. Inform. Sci.
**2013**, 218, 85–102. [Google Scholar] [CrossRef] - Muhammad, M.A.; Mokhlis, H.; Naidu, K.; Amin, A.; Franco, J.F.; Othman, M. Distribution Network Planning Enhancement via Network Reconfiguration and DG Integration Using Dataset Approach and Water Cycle Algorithm. J. Mod. Power Syst. Clean Energy
**2020**, 8, 86–93. [Google Scholar] [CrossRef] - Helmi, A.M.; Carli, R.; Dotoli, M.; Ramadan, H.S. Efficient and Sustainable Reconfiguration of Distribution Networks via Metaheuristic Optimization. IEEE Trans. Autom. Sci. Eng.
**2022**, 19, 82–98. [Google Scholar] [CrossRef] - Lorenz, E.N. Deterministic Nonperiodic Flow. J. Atmos Sci.
**1963**, 20, 130–141. [Google Scholar] [CrossRef][Green Version] - Teh, J.S.; Samsudin, A.; Akhavan, A. Parallel chaotic hash function based on the shuffle-exchange network. Nonlinear Dyn.
**2015**, 81, 1067–1079. [Google Scholar] [CrossRef] - Kundur, P. Power System Stability and Control; McGraw-Hill: New York, NY, USA, 1994. [Google Scholar]
- Alizadeh, M.; Ganjefar, S.; Alizadeh, M. Wavelet neural adaptive proportional plus conventional integral-derivative controller design of SSSC for transient stability improvement. Eng. Appl. Artif. Intell.
**2013**, 26, 2227–2242. [Google Scholar] [CrossRef]

**Figure 11.**Boxplot of final best objective values obtained by different algorithms over 30 runs on the 100-dimensional conventional problems: (

**a**) f

_{2}Schwefel 1.2; (

**b**) f

_{3}Schwefel 2.22; (

**c**) f

_{5}Rosenbrock; (

**d**) f

_{6}Rastrigin; (

**e**) f

_{7}Salomon; and (

**f**) f

_{8}Ackley.

**Figure 12.**Boxplot of final best objective values obtained by different algorithms over 30 runs on the 100-dimensional rotated problems: (

**a**) f

_{10}Rotated Schwefel 1.2; (

**b**) f

_{11}Rotated Schwefel 2.22; (

**c**) f

_{12}Rotated Zakharov; (

**d**) f

_{14}Rotated Rastrigin; (

**e**) f

_{15}Rotated Salomon; and (

**f**) f

_{16}Rotated Ackley.

**Figure 13.**Convergence characteristics of comparative algorithms on the 100-dimensional conventional functions: (

**a**) f

_{1}Sphere; (

**b**) f

_{4}Zakharov; (

**c**) f

_{5}Rosenbrock; (

**d**) f

_{7}Salomon; (

**e**) f

_{8}Ackley and (

**f**) f

_{9}Griewank.

**Figure 14.**Convergence characteristics of comparative algorithms on the 100-dimensional rotated functions: (

**a**) f

_{10}Rotated Schwefel 1.2; (

**b**) f

_{12}Rotated Zakharov; (

**c**) f

_{13}Rotated Rosenbrock; (

**d**) f

_{14}Rotated Rastrigin; (

**e**) f

_{16}Rotated Ackley; and (

**f**) f

_{17}Rotated Griewank.

**Figure 16.**Convergence characteristics of the objective function J with Jaya and CJaya-SQP algorithms.

**Figure 17.**Power system dynamic responses for scenario I: (

**a**) Speed deviation response of G1-G2; (

**b**) Speed deviation response of G3-G4; (

**c**) Speed deviation response of G1-G3; (

**d**) Speed deviation response of G1-G4; (

**e**) Rotor angle response of G1-G3; (

**f**) Rotor angle response of G2-G3; (

**g**) PSS output signal response at generator G1; (

**h**) PSS output signal response at generator G2; (

**i**) PSS output signal response at generator G4; and (

**j**) SVC susceptance response.

**Figure 18.**Power system dynamic responses for scenario II: (

**a**) Speed deviation response of G1-G2; (

**b**) Speed deviation response of G3-G4; (

**c**) Speed deviation response of G1-G3; (

**d**) Speed deviation response of G1-G4; (

**e**) Rotor angle response of G1-G3; (

**f**) Rotor angle response of G2-G3; (

**g**) PSS output signal response at generator G1; (

**h**) PSS output signal response at generator G2; (

**i**) PSS output signal response at generator G4; and (

**j**) SVC susceptance response.

**Figure 19.**Power systemdynamic responses for scenario III: (

**a**) Speed deviation response of G1-G2; (

**b**) Speed deviation response of G3-G4; (

**c**) Speed deviation response of G1-G3; (

**d**) Speed deviation response of G1-G4; (

**e**) Rotor angle response of G1-G3; (

**f**) Rotor angle response of G2-G3; (

**g**) PSS output signal response at generator G1; (

**h**) PSS output signal response at generator G2; (

**i**) PSS output signal response at generator G4; and (

**j**) SVC susceptance response.

**Figure 20.**Power systemdynamic responses for scenario IV: (

**a**) Speed deviation response of G1-G2; (

**b**) Speed deviation response of G3-G4; (

**c**) Speed deviation response of G1-G3; (

**d**) Speed deviation response of G1-G4; (

**e**) Rotor angle response of G1-G3; (

**f**) Rotor angle response of G2-G3; (

**g**) PSS output signal response at generator G1; (

**h**) PSS output signal response at generator G2; (

**i**) PSS output signal response at generator G4; and (

**j**) SVC susceptance response.

Characteristics | CJaya-SQP | Other Metaheuristic Optimizers |
---|---|---|

Algorithm formulation | Very simple formulation. | Sometimes very complicated formulations, especially for hybrid algorithms. |

Setting of control parameters | Not required. There are no algorithm-specific parameters. | Often required. The convergence behavior is very sensitive to tuning of algorithm-specific control parameters. |

Gradient information | The gradient-based SQP method is utilized for the adjustment of the best solution derived by CJaya. | Not utilized. |

Population diversity | The diversity of generated solutions is enhanced by embedding the chaotic maps to substitute random numbers in search equations and implementing chaotic local search strategy. | Not guaranteed. |

Solutions quality | There is a high probability of generating high quality solutions. Current best candidate can be improved in each design cycle. | Best candidate design of population is not necessarily improved during the update of decision variables. |

Elitism | Intrinsically elitist. There are no additional structural analyses. | Sometimes elitist. Additional structural analyses are usually required. |

Exploration/Exploitation | Naturally balanced with just one mathematical model. The exploitation and exploration capabilities are further enhanced by using chaos theory, a local search strategy and SQP technique. | Not necessarily balanced. |

Symbol | Variable Description | Symbol | Variable Description |
---|---|---|---|

${E}_{q}^{\prime},{E}_{fd}$ | Internal and the field voltages, respectively. | ${B}_{SVC}$ | SVC susceptance. |

$\delta ,\omega $ | Rotor angle and speed, respectively. | ${B}_{SVC}^{ref}$ | Reference susceptance of SVC. |

${V}_{ref},{V}_{t}$ | Reference and terminal voltages, respectively. | $N$ | Total number of buses. |

${u}_{PSS},{u}_{SVC}$ | PSS and SVC stabilizing signals, respectively. | ${N}_{L}$ | Total number of load buses. |

${K}_{PSS}$ | Gain of the PSS. | ${P}_{s}$ | Real power injected at the slack bus s. |

${T}_{wp}$ | Washout time constant of PSS. | ${P}_{gs}$ | Real power generation at the slack bus s. |

${T}_{1p}-{T}_{4p}$ | Lead-lag time constants of PSS. | ${P}_{ds}$ | Real power demand at the slack bus s. |

$\Delta {\omega}_{i}$ | Change in speed for machine i. | ${V}_{s}\angle {\alpha}_{s},{V}_{j}\angle {\alpha}_{j}$ | Voltages at the end buses s and j. |

${K}_{A},{T}_{A}$ | Gain and time constant of the excitation system, respectively. | ${Y}_{sj}\angle {\theta}_{sj}$ | $\mathrm{The}\left(s,j\right)\mathrm{th}$ element of the power system admittance matrix. |

${K}_{s},{T}_{s}$ | SVC gain and time constant, respectively. | ${P}_{D}$ | Total real power demand. |

${K}_{SVC}$ | Gain of lead-lag circuits of SVC. | ${P}_{Loss}$ | Real power loss. |

${T}_{ws}$ | Washout time constant of SVC. | ${P}_{gi},{Q}_{gi}$ | Real and reactive powers generated by machine i. |

${T}_{1s}-{T}_{4s}$ | Lead-lag time constants of SVC. |

Case | Lower Limit | Upper Limit | |
---|---|---|---|

PSS | K_{PSS} | 1 | 100 |

T_{i} (i = 1,2,3,4) | 0.001 | 2 | |

SVC | K_{svc} | 1 | 150 |

T_{i} (i = 1,2,3,4) | 0.001 | 2 |

Function | Formula | Range | T-Value |
---|---|---|---|

Category I: Conventional problems | |||

Sphere | ${f}_{1}(x)={\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}$ | [−100, 100] | 1 × 10^{−2} |

Schwefel 1.2 | ${f}_{2}(x)={{\displaystyle {\sum}_{i=1}^{D}\left({\displaystyle {\sum}_{j=1}^{i}{x}_{j}}\right)}}^{2}$ | [−100, 100] | 1 × 10^{−5} |

Schwefel 2.22 | ${f}_{3}(x)={\displaystyle {\sum}_{i=1}^{D}|{x}_{i}|}+{\displaystyle {\prod}_{i=1}^{D}|{x}_{i}|}$ | [−10, 10] | 1 × 10^{−5} |

Zakharov | ${f}_{4}(x)={{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}+\left({\displaystyle {\sum}_{i=1}^{D}0.5i{x}_{i}}\right)}}^{2}+{\left({\displaystyle {\sum}_{i=1}^{D}0.5i{x}_{i}}\right)}^{4}$ | [−5, 10] | 1 × 10^{−5} |

Rosenbrock | ${f}_{5}(x)={\displaystyle {\sum}_{i=1}^{D-1}\left[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}\right]}$ | [−2.048, 2.048] | 5 |

Rastrigin | ${f}_{6}(x)={\displaystyle {\sum}_{i=1}^{D}[{x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10]}$ | [−5.12, 5.12] | 1 × 10^{−5} |

Salomon | ${f}_{7}(x)=1-\mathrm{cos}\left(2\pi \sqrt{{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}}\right)+0.1\sqrt{{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}}$ | [−100, 100] | 1 × 10^{−5} |

Ackley | ${f}_{8}(x)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}}\right)-\mathrm{exp}\left(\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)}\right)+20+e$ | [−32.768, 32.768] | 1 × 10^{−5} |

Griewank | ${f}_{9}(x)={\displaystyle {\sum}_{i=1}^{D}\frac{{x}_{i}^{2}}{4000}}-{\displaystyle {\prod}_{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)}+1$ | [−600, 600] | 1 × 10^{−5} |

Category II: Rotated problems | |||

Rotated Schwefel 1.2 | ${f}_{10}(x)=\begin{array}{cc}{{\displaystyle {\sum}_{i=1}^{D}\left({\displaystyle {\sum}_{j=1}^{i}{y}_{j}}\right)}}^{2}& y=M\times x\end{array}$ | [−100, 100] | 1 × 10^{−5} |

Rotated Schwefel 2.22 | ${f}_{11}(x)=\begin{array}{cc}{\displaystyle {\sum}_{i=1}^{D}|{y}_{i}|}+{\displaystyle {\prod}_{i=1}^{D}|{y}_{i}|}& y=M\times x\end{array}$ | [−10, 10] | 1 × 10^{−5} |

Rotated Zakharov | ${f}_{12}(x)=\begin{array}{cc}{{\displaystyle {\sum}_{i=1}^{D}{y}_{i}^{2}+\left({\displaystyle {\sum}_{i=1}^{D}0.5i{y}_{i}}\right)}}^{2}+{\left({\displaystyle {\sum}_{i=1}^{D}0.5i{y}_{i}}\right)}^{4}& y=M\times x\end{array}$ | [−5, 10] | 1 × 10^{−5} |

Rotated Rosenbrock | ${f}_{13}(x)=\begin{array}{cc}{\displaystyle {\sum}_{i=1}^{D-1}\left[100{({y}_{i+1}-{y}_{i}^{2})}^{2}+{({y}_{i}-1)}^{2}\right]}& y=M\times x\end{array}$ | [−2.048, 2.048] | 50 |

Rotated Rastrigin | ${f}_{14}(x)=\begin{array}{cc}{\displaystyle {\sum}_{i=1}^{D}[{y}_{i}^{2}-10\mathrm{cos}(2\pi {y}_{i})+10]}& y=M\times x\end{array}$ | [−5.12, 5.12] | 1 × 10^{−5} |

Rotated Salomon | ${f}_{15}(x)=\begin{array}{cc}1-\mathrm{cos}\left(2\pi \sqrt{{\displaystyle {\sum}_{i=1}^{D}{y}_{i}^{2}}}\right)+0.1\sqrt{{\displaystyle {\sum}_{i=1}^{D}{y}_{i}^{2}}}& y=M\times x\end{array}$ | [−100, 100] | 1 × 10^{−5} |

Rotated Ackley | ${f}_{16}(x)=\begin{array}{cc}-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}{y}_{i}^{2}}}\right)-\mathrm{exp}\left(\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}\mathrm{cos}\left(2\pi {y}_{i}\right)}\right)+20+e& y=M\times x\end{array}$ | [−32.768, 32.768] | 1 × 10^{−5} |

Rotated Griewank | ${f}_{17}(x)=\begin{array}{cc}{\displaystyle {\sum}_{i=1}^{D}\frac{{y}_{i}^{2}}{4000}}-{\displaystyle {\prod}_{i=1}^{D}\mathrm{cos}\left(\frac{{y}_{i}}{\sqrt{i}}\right)}+1& y=M\times x\end{array}$ | [−600, 600] | 1 × 10^{−5} |

**Table 5.**Statistical results obtained by comparative algorithms for 30-dimensional functions over 30 runs with 80,000 function evaluations.

Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|

f_{1} | Mean | 1.78 × 10^{−29} | 8.32 × 10^{−39} | 5.67 × 10^{−16} | 3.29 × 10^{−5} | 4.72 × 10^{−34} | 2.45 × 10^{−10} | 0.00 |

SD | 1.87 × 10^{−29} | 1.55 × 10^{−38} | 1.01 × 10^{−16} | 4.95 × 10^{−6} | 5.85 × 10^{−35} | 1.75 × 10^{−10} | 0.00 | |

SEM | 3.42 × 10^{−30} | 2.83 × 10^{−39} | 1.85 × 10^{−17} | 9.04 × 10^{−7} | 1.06 × 10^{−35} | 3.20 × 10^{−11} | 0.00 | |

f_{2} | Mean | 5.54 × 10^{−1} | 4.54 × 10^{−3} | 7.89 × 10^{3} | 4.39 × 10^{−4} | 8.73 × 10^{−33} | 2.57 × 10^{4} | 0.00 |

SD | 5.53 × 10^{−1} | 4.90 × 10^{−3} | 2.03 × 10^{3} | 2.31 × 10^{−4} | 3.13 × 10^{−33} | 6.56 × 10^{3} | 0.00 | |

SEM | 1.01 × 10^{−1} | 8.95 × 10^{−4} | 3.71 × 10^{2} | 4.23 × 10^{−5} | 5.72 × 10^{−34} | 1.19 × 10^{3} | 0.00 | |

f_{3} | Mean | 1.21 × 10^{−16} | 4.93 × 10^{−16} | 6.26 × 10^{−15} | 2.55 × 10^{−2} | 9.17 × 10^{−18} | 4.75 × 10^{−6} | 6.80 × 10^{−297} |

SD | 9.13 × 10^{−17} | 2.21 × 10^{−15} | 2.83 × 10^{−15} | 2.27 × 10^{−3} | 7.23 × 10^{−19} | 2.38 × 10^{−6} | 0.00 | |

SEM | 1.67 × 10^{−17} | 4.03 × 10^{−16} | 5.17 × 10^{−16} | 4.15 × 10^{−4} | 1.32 × 10^{−19} | 4.35 × 10^{−7} | 0.00 | |

f_{4} | Mean | 8.00 × 10^{−28} | 1.38 × 10^{−38} | 5.39 × 10^{−16} | 4.99 × 10^{−1} | 1.65 × 10^{−34} | 5.68 × 10^{−10} | 0.00 |

SD | 1.82 × 10^{−27} | 2.65 × 10^{−38} | 9.65 × 10^{−17} | 2.95 × 10^{−1} | 2.87 × 10^{−35} | 3.76 × 10^{−10} | 0.00 | |

SEM | 3.33 × 10^{−28} | 4.83 × 10^{−39} | 1.76 × 10^{−17} | 5.40 × 10^{−2} | 5.25 × 10^{−36} | 6.86 × 10^{−11} | 0.00 | |

f_{5} | Mean | 2.49 × 10^{1} | 1.95 × 10^{1} | 1.36 × 10^{1} | 2.66 × 10^{1} | 1.12 × 10^{1} | 1.90 × 10^{1} | 3.12 × 10^{−1} |

SD | 1.79 × 10^{0} | 2.24 × 10^{0} | 7.26 × 10^{0} | 1.40 × 10^{0} | 1.16 × 10^{0} | 1.91 × 10^{1} | 1.32 | |

SEM | 3.27 × 10^{−1} | 4.09 × 10^{−1} | 1.33 × 10^{0} | 2.55 × 10^{−1} | 2.13 × 10^{−1} | 3.48 × 10^{0} | 2.41 × 10^{−1} | |

f_{6} | Mean | 2.74 × 10^{1} | 4.91 × 10^{1} | 3.31 × 10^{−15} | 6.30 × 10^{1} | 4.98 × 10^{1} | 2.05 × 10^{2} | 0.00 |

SD | 1.81 × 10^{1} | 1.70 × 10^{1} | 1.71 × 10^{−14} | 1.82 × 10^{1} | 1.56 × 10^{1} | 1.70 × 10^{1} | 0.00 | |

SEM | 3.31 × 10^{0} | 3.11 × 10^{0} | 3.13 × 10^{−15} | 3.32 × 10^{0} | 2.85 × 10^{0} | 3.11 × 10^{0} | 0.00 | |

f_{7} | Mean | 2.07 × 10^{−1} | 4.37 × 10^{−1} | 1.15 × 10^{0} | 1.75 × 10^{1} | 1.96 × 10^{−1} | 4.04 × 10^{−1} | 0.00 |

SD | 2.54 × 10^{−2} | 9.64 × 10^{−2} | 1.73 × 10^{−1} | 1.47 × 10^{0} | 1.82 × 10^{−2} | 7.07 × 10^{−2} | 0.00 | |

SEM | 4.63 × 10^{−3} | 1.76 × 10^{−2} | 3.16 × 10^{−2} | 2.69 × 10^{−1} | 3.33 × 10^{−3} | 1.29 × 10^{−2} | 0.00 | |

f_{8} | Mean | 3.10 × 10^{−2} | 5.95 × 10^{−1} | 4.96 × 10^{−13} | 1.76 × 10^{1} | 1.85 × 10^{−14} | 4.46 × 10^{−2} | 8.88 × 10^{−16} |

SD | 1.70 × 10^{−1} | 6.83 × 10^{−1} | 1.87 × 10^{−13} | 4.80 × 10^{0} | 5.15 × 10^{−15} | 2.44 × 10^{−1} | 0.00 | |

SEM | 3.10 × 10^{−2} | 1.25 × 10^{−1} | 3.42 × 10^{−14} | 8.77 × 10^{−1} | 9.40 × 10^{−16} | 4.46 × 10^{−2} | 0.00 | |

f_{9} | Mean | 4.35 × 10^{−3} | 1.44 × 10^{−2} | 1.08 × 10^{−3} | 2.70 × 10^{2} | 3.53 × 10^{−3} | 9.17 × 10^{−2} | 0.00 |

SD | 7.30 × 10^{−3} | 1.51 × 10^{−2} | 2.84 × 10^{−3} | 3.62 × 10^{1} | 4.87 × 10^{−3} | 1.39 × 10^{−1} | 0.00 | |

SEM | 1.33 × 10^{−3} | 2.75 × 10^{−3} | 5.18 × 10^{−4} | 6.61 × 10^{0} | 8.89 × 10^{−4} | 2.54 × 10^{−2} | 0.00 | |

f_{10} | Mean | 4.67 × 10^{−2} | 1.83 × 10^{−3} | 5.54 × 10^{3} | 4.55 × 10^{−4} | 5.68 × 10^{−33} | 1.30 × 10^{4} | 0.00 |

SD | 4.06 × 10^{−2} | 1.86 × 10^{−3} | 1.59 × 10^{3} | 1.78 × 10^{−4} | 1.71 × 10^{−33} | 5.21 × 10^{3} | 0.00 | |

SEM | 7.41 × 10^{−3} | 3.40 × 10^{−4} | 2.90 × 10^{2} | 3.25 × 10^{−5} | 3.12 × 10^{−34} | 9.52 × 10^{2} | 0.00 | |

f_{11} | Mean | 6.36 × 10^{−12} | 1.30 × 10^{−1} | 7.58 × 10^{−2} | 2.48 × 10^{−2} | 9.37 × 10^{−18} | 2.15 × 10^{1} | 1.91 × 10^{−302} |

SD | 3.41 × 10^{−11} | 5.63 × 10^{−1} | 8.39 × 10^{−2} | 2.41 × 10^{−3} | 7.10 × 10^{−19} | 3.46 × 10^{1} | 0.00 | |

SEM | 6.23 × 10^{−12} | 1.03 × 10^{−1} | 1.53 × 10^{−2} | 4.41 × 10^{−4} | 1.29 × 10^{−19} | 6.31 × 10^{0} | 0.00 | |

f_{12} | Mean | 1.76 × 10^{−11} | 4.11 × 10^{−21} | 5.01 × 10^{0} | 6.51 × 10^{−1} | 2.48 × 10^{−34} | 4.53 × 10^{1} | 0.00 |

SD | 5.10 × 10^{−11} | 1.68 × 10^{−20} | 5.90 × 10^{0} | 2.82 × 10^{−1} | 4.93 × 10^{−35} | 2.46 × 10^{2} | 0.00 | |

SEM | 9.32 × 10^{−12} | 3.07 × 10^{−21} | 1.07 × 10^{0} | 5.15 × 10^{−2} | 9.01 × 10^{−36} | 4.49 × 10^{1} | 0.00 | |

f_{13} | Mean | 2.39 × 10^{1} | 1.93 × 10^{1} | 2.39 × 10^{1} | 2.64 × 10^{1} | 1.22 × 10^{1} | 2.89 × 10^{1} | 2.87 × 10^{1} |

SD | 1.19 × 10^{0} | 1.49 × 10^{0} | 2.95 × 10^{0} | 1.60 × 10^{0} | 1.88 × 10^{0} | 7.64 × 10^{−1} | 2.74 × 10^{−2} | |

SEM | 2.17 × 10^{−1} | 2.73 × 10^{−1} | 5.39 × 10^{−1} | 2.93 × 10^{−1} | 3.44 × 10^{−1} | 1.39 × 10^{−1} | 5.00 × 10^{−3} | |

f_{14} | Mean | 1.53 × 10^{2} | 3.92 × 10^{1} | 1.30 × 10^{2} | 5.47 × 10^{1} | 5.86 × 10^{1} | 2.25 × 10^{2} | 0.00 |

SD | 4.63 × 10^{1} | 1.12 × 10^{1} | 1.35 × 10^{1} | 1.46 × 10^{1} | 1.90 × 10^{1} | 1.82 × 10^{1} | 0.00 | |

SEM | 8.45 × 10^{0} | 2.04 × 10^{0} | 2.47 × 10^{0} | 2.68 × 10^{0} | 3.48 × 10^{0} | 3.32 × 10^{0} | 0.00 | |

f_{15} | Mean | 2.09 × 10^{−1} | 4.60 × 10^{−1} | 1.16 × 10^{0} | 1.75 × 10^{1} | 1.93 × 10^{−1} | 3.97 × 10^{−1} | 0.00 |

SD | 4.04 × 10^{−2} | 9.68 × 10^{−2} | 1.17 × 10^{−1} | 1.26 × 10^{0} | 2.53 × 10^{−2} | 6.09 × 10^{−2} | 0.00 | |

SEM | 7.38 × 10^{−3} | 1.77 × 10^{−2} | 2.14 × 10^{−2} | 2.30 × 10^{−1} | 4.63 × 10^{−3} | 1.11 × 10^{−2} | 0.00 | |

f_{16} | Mean | 3.85 × 10^{−2} | 1.83 × 10^{0} | 7.09 × 10^{0} | 1.88 × 10^{1} | 1.80 × 10^{−14} | 7.75 × 10^{−2} | 8.88 × 10^{−16} |

SD | 2.11 × 10^{−1} | 6.11 × 10^{−1} | 7.21 × 10^{0} | 2.71 × 10^{−1} | 6.79 × 10^{−15} | 3.00 × 10^{−1} | 0.00 | |

SEM | 3.85 × 10^{−2} | 1.11 × 10^{−1} | 1.31 × 10^{0} | 4.95 × 10^{−2} | 1.24 × 10^{−15} | 5.48 × 10^{−2} | 0.00 | |

f_{17} | Mean | 4.99 × 10^{−3} | 1.03 × 10^{−2} | 2.31 × 10^{−5} | 2.73 × 10^{2} | 4.93 × 10^{−4} | 1.33 × 10^{−1} | 0.00 |

SD | 1.06 × 10^{−2} | 1.58 × 10^{−2} | 6.46 × 10^{−5} | 3.20 × 10^{1} | 1.87 × 10^{−3} | 1.80 × 10^{−1} | 0.00 | |

SEM | 1.93 × 10^{−3} | 2.89 × 10^{−3} | 1.18 × 10^{−5} | 5.85 × 10^{0} | 3.42 × 10^{−4} | 3.29 × 10^{−2} | 0.00 |

**Table 6.**Mean function evaluations required to reach the threshold value and success rate by various algorithms for 30-dimensional functions over 30 runs.

Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|

f_{1} | MeanFEs | 15,963 | 9556 | 11,655 | 65,471 | 8593.3 | 37,517 | 3.33 |

SR (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

f_{2} | MeanFEs | NaN | NaN | NaN | NaN | 19,125 | NaN | 8.06 |

SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |

f_{3} | MeanFEs | 32,199 | 25,373 | 35,040 | NaN | 25,160 | 76,022.67 | 10.43 |

SR (%) | 100 | 100 | 100 | 0 | 100 | 100 | 100 | |

f_{4} | MeanFEs | 25,470.67 | 16,504 | 20,672 | NaN | 14,408 | 56,173.33 | 1122.83 |

SR (%) | 100 | 100 | 100 | 0 | 100 | 100 | 100 | |

f_{5} | MeanFEs | NaN | NaN | 57,856 | NaN | NaN | 603,80 | 17,826.21 |

SR (%) | 0 | 0 | 16.66 | 0 | 0 | 13.33 | 96.66 | |

f_{6} | MeanFEs | NaN | NaN | 42,561.33 | NaN | NaN | NaN | 7.63 |

SR (%) | 0 | 0 | 100 | 0 | 0 | 0 | 100 | |

f_{7} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 20.46 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{8} | MeanFEs | 32,580.69 | 23,925 | 41,117.33 | NaN | 23,965.33 | 77,400 | 27.5 |

SR (%) | 96.66 | 53.33 | 100 | 0 | 100 | 86.66 | 100 | |

f_{9} | MeanFEs | 24,464.76 | 16,008.89 | 31,547.69 | NaN | 16,073.68 | 61,088 | 23.5 |

SR (%) | 70 | 30 | 86.66 | 0 | 63.33 | 16.66 | 100 | |

f_{10} | MeanFEs | NaN | NaN | NaN | NaN | 18,386.67 | NaN | 10.16 |

SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |

f_{11} | MeanFEs | 39,573.33 | 44,388 | NaN | NaN | 25,192 | NaN | 11.90 |

SR (%) | 100 | 66.66 | 0 | 0 | 100 | 0 | 100 | |

f_{12} | MeanFEs | 46,014.67 | 26,560 | NaN | NaN | 14,850.67 | NaN | 1278.66 |

SR (%) | 100 | 100 | 0 | 0 | 100 | 0 | 100 | |

f_{13} | MeanFEs | 5794.66 | 1226.66 | 7540 | 45,032 | 834.66 | 15,921.33 | 36.76 |

SR (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |

f_{14} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 9.33 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{15} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 23.56 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{16} | MeanFEs | 33,384.83 | 22,480 | NaN | NaN | 23,966.67 | 78,511.11 | 29.83 |

SR (%) | 96.66 | 3.33 | 0 | 0 | 100 | 30 | 100 | |

f_{17} | MeanFEs | 26,200 | 17,637.33 | 47,376.67 | NaN | 16,048.57 | 64,885 | 25.43 |

SR (%) | 66.66 | 50 | 80 | 0 | 93.33 | 26.66 | 100 |

**Table 7.**Friedman rank test for the ‘Mean’ and ‘SEM’ solutions obtained for 30-dimensional functions over 30 runs.

Test for Mean Solutions | Test for SEM Solutions | ||||||
---|---|---|---|---|---|---|---|

Algorithms | Friedman Value | Normalized Value | Rank | Algorithms | Friedman value | Normalized Value | Rank |

DE | 3.94 | 3.05 | 3 | DE | 4.05 | 3.85 | 3 |

PSO | 4.05 | 3.13 | 4 | PSO | 4.23 | 4.02 | 4 |

ABC | 4.52 | 3.50 | 5 | ABC | 4.82 | 4.59 | 5 |

IWO | 5.82 | 4.51 | 6 | IWO | 5.52 | 5.25 | 6 |

FA | 2.47 | 1.91 | 2 | FA | 2.70 | 2.57 | 2 |

Jaya | 5.88 | 4.55 | 7 | Jaya | 5.58 | 5.31 | 7 |

CJaya-SQP | 1.29 | 1.00 | 1 | CJaya-SQP | 1.05 | 1.00 | 1 |

**Table 8.**Statistical results obtained by comparative algorithms for 100-dimensional functions over 30 runs with 80,000 function evaluations.

Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|

f_{1} | Mean | 1.34 × 10^{−2} | 2.72 × 10^{−1} | 6.06 × 10^{−6} | 1.41 × 10^{3} | 2.78 × 10^{−32} | 3.08 × 10^{1} | 0.00 |

SD | 3.12 × 10^{−2} | 1.16 × 10^{0} | 7.59 × 10^{−6} | 7.79 × 10^{2} | 4.82 × 10^{−33} | 1.66 × 10^{1} | 0.00 | |

SEM | 5.69 × 10^{−3} | 2.12 × 10^{−1} | 1.38 × 10^{−6} | 1.42 × 10^{2} | 8.80 × 10^{−34} | 3.03 × 10^{0} | 0.00 | |

f_{2} | Mean | 8.72 × 10^{4} | 7.32 × 10^{3} | 1.49 × 10^{5} | 4.05 × 10^{4} | 1.04 × 10^{2} | 4.05 × 10^{5} | 0.00 |

SD | 1.90 × 10^{4} | 2.50 × 10^{3} | 1.60 × 10^{4} | 7.65 × 10^{3} | 1.60 × 10^{2} | 4.67 × 10^{4} | 0.00 | |

SEM | 3.48 × 10^{3} | 4.57 × 10^{2} | 2.92 × 10^{3} | 1.39 × 10^{3} | 2.92 × 10^{1} | 8.53 × 10^{3} | 0.00 | |

f_{3} | Mean | 5.53 × 10^{−2} | 7.02 × 10^{−1} | 2.64 × 10^{−3} | 1.82 × 10^{0} | 1.15 × 10^{−16} | 8.00 × 10^{0} | 4.07 × 10^{−298} |

SD | 1.47 × 10^{−1} | 7.58 × 10^{−1} | 9.26 × 10^{−4} | 2.73 × 10^{0} | 9.25 × 10^{−18} | 4.01 × 10^{0} | 0.00 | |

SEM | 2.69 × 10^{−2} | 1.38 × 10^{−1} | 1.69 × 10^{−4} | 5.00 × 10^{−1} | 1.69 × 10^{−18} | 7.33 × 10^{−1} | 0.00 | |

f_{4} | Mean | 2.51 × 10^{2} | 2.00 × 10^{−1} | 1.05 × 10^{−5} | 6.59 × 10^{3} | 2.49 × 10^{−31} | 3.10 × 10^{5} | 0.00 |

SD | 7.45 × 10^{2} | 3.69 × 10^{−1} | 9.27 × 10^{−6} | 4.28 × 10^{3} | 8.71 × 10^{−32} | 6.15 × 10^{5} | 0.00 | |

SEM | 1.36 × 10^{2} | 6.74 × 10^{−2} | 1.69 × 10^{−6} | 7.82 × 10^{2} | 1.59 × 10^{−32} | 1.12 × 10^{5} | 0.00 | |

f_{5} | Mean | 1.20 × 10^{2} | 9.34 × 10^{1} | 1.85 × 10^{2} | 1.05 × 10^{2} | 1.16 × 10^{2} | 1.80 × 10^{2} | 1.30 × 10^{1} |

SD | 3.26 × 10^{1} | 1.08 × 10^{0} | 3.41 × 10^{1} | 2.16 × 10^{1} | 3.42 × 10^{1} | 4.67 × 10^{1} | 3.39 × 10^{1} | |

SEM | 5.95 × 10^{0} | 1.98 × 10^{−1} | 6.24 × 10^{0} | 3.96 × 10^{0} | 6.26 × 10^{0} | 8.54 × 10^{0} | 6.19 × 10^{0} | |

f_{6} | Mean | 1.71 × 10^{2} | 1.62 × 10^{2} | 1.73 × 10^{1} | 3.88 × 10^{2} | 3.71 × 10^{2} | 7.72 × 10^{2} | 0.00 |

SD | 7.69 × 10^{1} | 3.50 × 10^{1} | 4.16 × 10^{0} | 4.42 × 10^{1} | 6.19 × 10^{1} | 1.27 × 10^{2} | 0.00 | |

SEM | 1.40 × 10^{1} | 6.39 × 10^{0} | 7.59 × 10^{−1} | 8.07 × 10^{0} | 1.13 × 10^{1} | 2.32 × 10^{1} | 0.00 | |

f_{7} | Mean | 2.68 × 10^{0} | 2.27 × 10^{0} | 9.60 × 10^{0} | 4.40 × 10^{1} | 1.33 × 10^{0} | 7.08 × 10^{0} | 0.00 |

SD | 7.94 × 10^{−1} | 5.72 × 10^{−1} | 7.20 × 10^{−1} | 1.40 × 10^{0} | 1.63 × 10^{−1} | 8.45 × 10^{−1} | 0.00 | |

SEM | 1.45 × 10^{−1} | 1.04 × 10^{−1} | 1.31 × 10^{−1} | 2.56 × 10^{−1} | 2.97 × 10^{−2} | 1.54 × 10^{−1} | 0.00 | |

f_{8} | Mean | 2.84 × 10^{0} | 3.89 × 10^{0} | 4.14 × 10^{−2} | 1.92 × 10^{1} | 1.36 × 10^{0} | 5.51 × 10^{0} | 8.88 × 10^{−16} |

SD | 1.10 × 10^{0} | 9.08 × 10^{−1} | 2.94 × 10^{−2} | 8.75 × 10^{−2} | 7.94 × 10^{−1} | 1.16 × 10^{0} | 0.00 | |

SEM | 2.01 × 10^{−1} | 1.65 × 10^{−1} | 5.38 × 10^{−3} | 1.59 × 10^{−2} | 1.45 × 10^{−1} | 2.13 × 10^{−1} | 0.00 | |

f_{9} | Mean | 5.07 × 10^{−2} | 2.30 × 10^{−1} | 4.15 × 10^{−3} | 1.71 × 10^{3} | 3.28 × 10^{−3} | 1.25 × 10^{0} | 0.00 |

SD | 6.99 × 10^{−2} | 3.87 × 10^{−1} | 9.14 × 10^{−3} | 8.95 × 10^{1} | 5.93 × 10^{−3} | 1.57 × 10^{−1} | 0.00 | |

SEM | 1.27 × 10^{−2} | 7.07 × 10^{−2} | 1.67 × 10^{−3} | 1.63 × 10^{1} | 1.08 × 10^{−3} | 2.86 × 10^{−2} | 0.00 | |

f_{10} | Mean | 4.11 × 10^{4} | 4.75 × 10^{3} | 1.16 × 10^{5} | 4.24 × 10^{4} | 3.82 × 10^{1} | 3.37 × 10^{5} | 0.00 |

SD | 1.13 × 10^{4} | 1.66 × 10^{3} | 1.72 × 10^{4} | 6.87 × 10^{3} | 3.28 × 10^{1} | 4.32 × 10^{4} | 0.00 | |

SEM | 2.07 × 10^{3} | 3.04 × 10^{2} | 3.14 × 10^{3} | 1.25 × 10^{3} | 5.99 × 10^{0} | 7.89 × 10^{3} | 0.00 | |

f_{11} | Mean | 6.76 × 10^{0} | 5.26 × 10^{0} | 1.36 × 10^{1} | 3.17 × 10^{0} | 8.83 × 10^{0} | 3.23 × 10^{12} | 1.54 × 10^{−296} |

SD | 3.64 × 10^{0} | 3.51 × 10^{0} | 5.57 × 10^{0} | 4.48 × 10^{0} | 7.97 × 10^{0} | 1.77 × 10^{13} | 0.00 | |

SEM | 6.65 × 10^{−1} | 6.41 × 10^{−1} | 1.01 × 10^{0} | 8.19 × 10^{−1} | 1.45 × 10^{0} | 3.23 × 10^{12} | 0.00 | |

f_{12} | Mean | 1.56 × 10^{2} | 1.12 × 10^{1} | 4.11 × 10^{2} | 6.75 × 10^{3} | 1.38 × 10^{−18} | 2.81 × 10^{6} | 0.00 |

SD | 2.53 × 10^{2} | 2.40 × 10^{1} | 3.49 × 10^{2} | 6.01 × 10^{3} | 2.52 × 10^{−18} | 6.79 × 10^{6} | 0.00 | |

SEM | 4.63 × 10^{1} | 4.39 × 10^{0} | 6.37 × 10^{1} | 1.09 × 10^{3} | 4.60 × 10^{−19} | 1.24 × 10^{6} | 0.00 | |

f_{13} | Mean | 9.81 × 10^{1} | 9.44 × 10^{1} | 9.65 × 10^{1} | 1.01 × 10^{2} | 9.45 × 10^{1} | 2.53 × 10^{2} | 9.89 × 10^{1} |

SD | 9.85 × 10^{0} | 2.00 × 10^{0} | 1.50 × 10^{0} | 1.57 × 10^{1} | 1.80 × 10^{1} | 1.82 × 10^{2} | 2.12 × 10^{−2} | |

SEM | 1.79 × 10^{0} | 3.65 × 10^{−1} | 2.74 × 10^{−1} | 2.87 × 10^{0} | 3.29 × 10^{0} | 3.33 × 10^{1} | 3.87 × 10^{−3} | |

f_{14} | Mean | 8.33 × 10^{2} | 1.38 × 10^{2} | 5.72 × 10^{2} | 3.68 × 10^{2} | 3.60 × 10^{2} | 9.88 × 10^{2} | 0.00 |

SD | 1.29 × 10^{2} | 2.69 × 10^{1} | 3.98 × 10^{1} | 5.52 × 10^{1} | 6.10 × 10^{1} | 4.36 × 10^{1} | 0.00 | |

SEM | 2.36 × 10^{1} | 4.92 × 10^{0} | 7.27 × 10^{0} | 1.00 × 10^{1} | 1.11 × 10^{1} | 7.97 × 10^{0} | 0.00 | |

f_{15} | Mean | 2.38 × 10^{0} | 2.40 × 10^{0} | 9.63 × 10^{0} | 4.43 × 10^{1} | 1.25 × 10^{0} | 7.29 × 10^{0} | 0.00 |

SD | 7.40 × 10^{−1} | 4.94 × 10^{−1} | 7.86 × 10^{−1} | 1.71 × 10^{0} | 1.52 × 10^{−1} | 8.45 × 10^{−1} | 0.00 | |

SEM | 1.35 × 10^{−1} | 9.02 × 10^{−2} | 1.43 × 10^{−1} | 3.13 × 10^{−1} | 2.78 × 10^{−2} | 1.54 × 10^{−1} | 0.00 | |

f_{16} | Mean | 2.51 × 10^{0} | 4.87 × 10^{0} | 1.89 × 10^{1} | 1.93 × 10^{1} | 1.95 × 10^{0} | 5.41 × 10^{0} | 8.88 × 10^{−16} |

SD | 6.20 × 10^{−1} | 8.23 × 10^{−1} | 2.68 × 10^{0} | 7.75 × 10^{−2} | 3.91 × 10^{−1} | 9.34 × 10^{−1} | 0.00 | |

SEM | 1.13 × 10^{−1} | 1.50 × 10^{−1} | 4.90 × 10^{−1} | 1.41 × 10^{−2} | 7.14 × 10^{−2} | 1.70 × 10^{−1} | 0.00 | |

f_{17} | Mean | 1.28 × 10^{−2} | 9.16 × 10^{−2} | 1.10 × 10^{−3} | 1.72e × 10^{3} | 4.76 × 10^{−3} | 1.42 × 10^{0} | 0.00 |

SD | 1.27 × 10^{−2} | 1.36 × 10^{−1} | 1.75 × 10^{−3} | 8.97 × 10^{1} | 6.39 × 10^{−3} | 5.93 × 10^{−1} | 0.00 | |

SEM | 2.31 × 10^{−3} | 2.49 × 10^{−2} | 3.20 × 10^{−4} | 1.63 × 10^{1} | 1.16 × 10^{−3} | 1.08 × 10^{−1} | 0.00 |

**Table 9.**Mean function evaluations required to reach the threshold value and success rate by various algorithms for 100-dimensional functions over 30 runs.

Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|

f_{1} | MeanFEs | 74,322 | 66,373 | 43,039 | NaN | 12704 | NaN | 9.86 |

SR (%) | 66.66 | 60 | 100 | 0 | 100 | 0 | 100 | |

f_{2} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 12.5 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{3} | MeanFEs | NaN | NaN | NaN | NaN | 30140 | NaN | 14.3 |

SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |

f_{4} | MeanFEs | NaN | NaN | 76,290.91 | NaN | 21,966.67 | NaN | 1498.66 |

SR (%) | 0 | 0 | 73.33 | 0 | 100 | 0 | 100 | |

f_{5} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 55,755.38 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 86.66 | |

f_{6} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 12.03 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{7} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 23.10 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{8} | MeanFEs | NaN | NaN | NaN | NaN | 26,793.33 | NaN | 31.16 |

SR (%) | 0 | 0 | 0 | 0 | 20 | 0 | 100 | |

f_{9} | MeanFEs | NaN | NaN | 79,240 | NaN | 19,405.71 | NaN | 24.53 |

SR (%) | 0 | 0 | 3.33 | 0 | 70 | 0 | 100 | |

f_{10} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 16.43 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{11} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 22.19 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{12} | MeanFEs | NaN | NaN | NaN | NaN | 32,870.67 | NaN | 1564 |

SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |

f_{13} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | NaN |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

f_{14} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 20.76 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{15} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 26.86 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{16} | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 33.73 |

SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

f_{17} | MeanFEs | NaN | NaN | NaN | NaN | 19,456.47 | NaN | 29.36 |

SR (%) | 0 | 0 | 0 | 0 | 56.66 | 0 | 100 |

**Table 10.**Friedman rank test for the ‘Mean’ and ‘SEM’ solutions obtained for 100-dimensional functions over 30 runs.

Test for Mean Solutions | Test for SEM Solutions | ||||||
---|---|---|---|---|---|---|---|

Algorithms | Friedman Value | Normalized Value | Rank | Algorithms | Friedman Value | Normalized Value | Rank |

DE | 4.17 | 3.39 | 4 | DE | 4.58 | 3.91 | 5 |

PSO | 3.52 | 2.86 | 3 | PSO | 3.58 | 3.05 | 3 |

ABC | 4.35 | 3.53 | 5 | ABC | 3.82 | 3.26 | 4 |

IWO | 5.70 | 4.63 | 6 | IWO | 5.05 | 4.31 | 6 |

FA | 2.64 | 2.14 | 2 | FA | 3.35 | 2.86 | 2 |

Jaya | 6.35 | 5.16 | 7 | Jaya | 6.41 | 5.47 | 7 |

CJaya-SQP | 1.23 | 1.00 | 1 | CJaya-SQP | 1.17 | 1.00 | 1 |

Eigenvalues | Frequency | Modes | Damping Ratio | Generators | State Variables | $\mathit{P}\mathit{F}$ |
---|---|---|---|---|---|---|

0.0720 ± j3.7192 | 0.5919 | Interarea | −0.0194 | $\mathrm{G}1$ | $\delta ,\omega $ | 1.0000, 1.0000 |

$\mathrm{G}2$ | $\delta ,\omega $ | 0.6012, 0.6012 | ||||

$\mathrm{G}3$ | $\delta ,\omega $ | 0.7181, 0.7181 | ||||

$\mathrm{G}4$ | $\delta ,\omega $ | 0.7445, 0.7445 | ||||

−0.2166 ± j5.9569 | 0.9481 | Local | 0.0363 | $\mathrm{G}1$ | $\delta ,\omega $ | 0.8053, 0.8053 |

$\mathrm{G}2$ | $\delta ,\omega $ | 1.0000, 1.0000 | ||||

$\mathrm{G}3$ | $\delta ,\omega $ | 0.0316, 0.0316 | ||||

$\mathrm{G}4$ | $\delta ,\omega $ | 0.0119, 0.0119 | ||||

−0.2161 ± j6.1273 | 0.9752 | Local | 0.0352 | $\mathrm{G}1$ | $\delta ,\omega $ | 0.0145, 0.0145 |

$\mathrm{G}2$ | $\delta ,\omega $ | 0.0229, 0.0229 | ||||

$\mathrm{G}3$ | $\delta ,\omega $ | 0.9941, 0.9941 | ||||

$\mathrm{G}4$ | $\delta ,\omega $ | 1.0000, 1.0000 |

Bus No. | Base Case | % Load Increase | ||||
---|---|---|---|---|---|---|

10 | 15 | 20 | 25 | 30 | ||

Bus 7 | 0.0038 | 0.0055 | 0.0064 | 0.0073 | 0.0081 | 0.0087 |

Bus 8 | 0.0279 | 0.0390 | 0.0448 | 0.0509 | 0.0574 | 0.0646 |

Uncoordinated Design | Coordinated Design | |||||||
---|---|---|---|---|---|---|---|---|

PSS-G1 | PSS-G2 | PSS-G4 | SVC | PSS-G1 | PSS-G2 | PSS-G4 | SVC | |

K | 17.5059 | 28.0717 | 11.5571 | 33.3058 | 32.2919 | 30.2838 | 9.2705 | 11.6793 |

T_{1} | 1.1324 | 1.1831 | 1.2220 | 0.1464 | 0.9337 | 0.7207 | 1.8364 | 1.3573 |

T_{2} | 0.2664 | 0.6197 | 0.5310 | 0.8140 | 0.6893 | 0.5714 | 1.3917 | 0.9393 |

T_{3} | 1.4371 | 1.1725 | 1.4376 | 1.6985 | 0.8732 | 1.5084 | 1.3293 | 1.0909 |

T_{4} | 1.0159 | 1.3452 | 1.1442 | 1.2474 | 0.6431 | 0.6918 | 0.5001 | 1.3336 |

Case | Eigenvalues | Damping Ratios |
---|---|---|

CJaya-SQP-PSSs | −2.7659 ± j7.7039 | 0.3379 |

−2.1961 ± j1.8797 | 0.7597 | |

−1.2187 ± j2.0866 | 0.5043 | |

CJaya-SQP-SVC | 0.0731 ± j3.7180 | −0.0197 |

−0.2164 ± j5.9569 | 0.0363 | |

−0.2167 ± j6.1283 | 0.0353 | |

CJaya-SQP-PSSs&SVC | −2.7881 ± j7.6715 | 0.3416 |

−2.6014 ± j1.9392 | 0.8018 | |

−2.0017 ± j3.2075 | 0.5294 |

Controller type | ITAE_{1} (t_{sim} = 10 s) | ||||
---|---|---|---|---|---|

Scenario I | Scenario II | Scenario III | Scenario IV | Total | |

CJaya-SQP-PSSs | 0.0141 | 0.0192 | 9.0501 | 3.9026 | 12.9860 |

CJaya-SQP-SVC | 0.3100 | 0.6942 | 29.7768 | 19.3631 | 50.1441 |

CJaya-SQP-PSSs&SVC | 0.0049 | 0.0122 | 0.0778 | 0.0988 | 0.1937 |

Controller type | ITAE_{2} (t_{sim} = 10 s) | ||||
---|---|---|---|---|---|

Scenario I | Scenario II | Scenario III | Scenario IV | Total | |

CJaya-SQP-PSSs | 0.0064 | 0.0095 | 0.0698 | 0.4432 | 0.5289 |

CJaya-SQP-SVC | 0.0286 | 0.0468 | 0.0941 | 0.9933 | 1.1628 |

CJaya-SQP-PSSs&SVC | 0.0016 | 0.0040 | 0.0075 | 0.0097 | 0.0228 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Welhazi, Y.; Guesmi, T.; Alshammari, B.M.; Alqunun, K.; Alateeq, A.; Almalaq, Y.; Alsabhan, R.; Abdallah, H.H.
A Novel Hybrid Chaotic Jaya and Sequential Quadratic Programming Method for Robust Design of Power System Stabilizers and Static VAR Compensator. *Energies* **2022**, *15*, 860.
https://doi.org/10.3390/en15030860

**AMA Style**

Welhazi Y, Guesmi T, Alshammari BM, Alqunun K, Alateeq A, Almalaq Y, Alsabhan R, Abdallah HH.
A Novel Hybrid Chaotic Jaya and Sequential Quadratic Programming Method for Robust Design of Power System Stabilizers and Static VAR Compensator. *Energies*. 2022; 15(3):860.
https://doi.org/10.3390/en15030860

**Chicago/Turabian Style**

Welhazi, Yosra, Tawfik Guesmi, Badr M. Alshammari, Khalid Alqunun, Ayoob Alateeq, Yasser Almalaq, Robaya Alsabhan, and Hsan Hadj Abdallah.
2022. "A Novel Hybrid Chaotic Jaya and Sequential Quadratic Programming Method for Robust Design of Power System Stabilizers and Static VAR Compensator" *Energies* 15, no. 3: 860.
https://doi.org/10.3390/en15030860