A Novel Hybrid Chaotic Jaya and Sequential Quadratic Programming Method for Robust Design of Power System Stabilizers and Static VAR Compensator
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)
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 and the maximum search length of CLS ; 2: Initialize: the iteration count l (l = 1), the initial conditions for the Lorenz system and for the Tent map; 3: Initialize the NP candidate solutions chaotically within their search boundaries using Equation (6); 4: Calculate the value of the objective functions of the initial solutions during iteration ; 5: Repeat 6: Identify the indexes assigned to the best candidate , and the worst candidate , and evaluate and ; 7: For To NP Do 8: Produce the new solution by updating the variables for the candidate solution using Equation (7); 9: Check the variables’ boundaries; {Compare the new solution with the old one }; 10: If Then , End If End For {Update the best candidate solution Xbest and its function value Fbest = F(Xbest) }; 11: For To NP Do 12: Initialize the iteration count of the CLS k (k = 1); 13: Repeat 14: Generate the chaotic variables for the Tent map using Equation (5); 15: Evaluate the components of the target vector Xt,i using Equation (8); 16: Check if the design variables are in the search space using Equation (9); {Compare the target vector with the candidate solution }; 17: If Then , , End If 18: ; {increment the iteration count of the CLS} 19: Until (End Repeat) {Update the best solution Xbest and the function value Fbest = F(Xbest) }; 20: If Then , End If End For 21: {increment the iteration count} 22: Until (End Repeat) 23: Let the best solution extracted by CJaya be the initial condition for SQP; 24: Use the SQP method for searching around 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
- The sensitivity factor 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
6. Numerical Experiments
6.1. Benchmark Functions and Parameter Settings
6.2. Results for Low-Dimensional Problems
6.2.1. Comparison of Solution Accuracy
6.2.2. Comparison of Convergence and Success Ratios
6.2.3. Statistical Tests
6.3. Results for High-Dimensional Problems
6.3.1. Comparison of Solution Accuracy
6.3.2. Comparison of Convergence and Success Ratios
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
7.4. Quantify the Enhancement of Proposed Approach
8. Conclusions and Perspectives
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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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 |
---|---|---|---|
Internal and the field voltages, respectively. | SVC susceptance. | ||
Rotor angle and speed, respectively. | Reference susceptance of SVC. | ||
Reference and terminal voltages, respectively. | Total number of buses. | ||
PSS and SVC stabilizing signals, respectively. | Total number of load buses. | ||
Gain of the PSS. | Real power injected at the slack bus s. | ||
Washout time constant of PSS. | Real power generation at the slack bus s. | ||
Lead-lag time constants of PSS. | Real power demand at the slack bus s. | ||
Change in speed for machine i. | Voltages at the end buses s and j. | ||
Gain and time constant of the excitation system, respectively. | element of the power system admittance matrix. | ||
SVC gain and time constant, respectively. | Total real power demand. | ||
Gain of lead-lag circuits of SVC. | Real power loss. | ||
Washout time constant of SVC. | Real and reactive powers generated by machine i. | ||
Lead-lag time constants of SVC. |
Case | Lower Limit | Upper Limit | |
---|---|---|---|
PSS | KPSS | 1 | 100 |
Ti (i = 1,2,3,4) | 0.001 | 2 | |
SVC | Ksvc | 1 | 150 |
Ti (i = 1,2,3,4) | 0.001 | 2 |
Function | Formula | Range | T-Value |
---|---|---|---|
Category I: Conventional problems | |||
Sphere | [−100, 100] | 1 × 10−2 | |
Schwefel 1.2 | [−100, 100] | 1 × 10−5 | |
Schwefel 2.22 | [−10, 10] | 1 × 10−5 | |
Zakharov | [−5, 10] | 1 × 10−5 | |
Rosenbrock | [−2.048, 2.048] | 5 | |
Rastrigin | [−5.12, 5.12] | 1 × 10−5 | |
Salomon | [−100, 100] | 1 × 10−5 | |
Ackley | [−32.768, 32.768] | 1 × 10−5 | |
Griewank | [−600, 600] | 1 × 10−5 | |
Category II: Rotated problems | |||
Rotated Schwefel 1.2 | [−100, 100] | 1 × 10−5 | |
Rotated Schwefel 2.22 | [−10, 10] | 1 × 10−5 | |
Rotated Zakharov | [−5, 10] | 1 × 10−5 | |
Rotated Rosenbrock | [−2.048, 2.048] | 50 | |
Rotated Rastrigin | [−5.12, 5.12] | 1 × 10−5 | |
Rotated Salomon | [−100, 100] | 1 × 10−5 | |
Rotated Ackley | [−32.768, 32.768] | 1 × 10−5 | |
Rotated Griewank | [−600, 600] | 1 × 10−5 |
Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|
f1 | 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 | |
f2 | Mean | 5.54 × 10−1 | 4.54 × 10−3 | 7.89 × 103 | 4.39 × 10−4 | 8.73 × 10−33 | 2.57 × 104 | 0.00 |
SD | 5.53 × 10−1 | 4.90 × 10−3 | 2.03 × 103 | 2.31 × 10−4 | 3.13 × 10−33 | 6.56 × 103 | 0.00 | |
SEM | 1.01 × 10−1 | 8.95 × 10−4 | 3.71 × 102 | 4.23 × 10−5 | 5.72 × 10−34 | 1.19 × 103 | 0.00 | |
f3 | 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 | |
f4 | 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 | |
f5 | Mean | 2.49 × 101 | 1.95 × 101 | 1.36 × 101 | 2.66 × 101 | 1.12 × 101 | 1.90 × 101 | 3.12 × 10−1 |
SD | 1.79 × 100 | 2.24 × 100 | 7.26 × 100 | 1.40 × 100 | 1.16 × 100 | 1.91 × 101 | 1.32 | |
SEM | 3.27 × 10−1 | 4.09 × 10−1 | 1.33 × 100 | 2.55 × 10−1 | 2.13 × 10−1 | 3.48 × 100 | 2.41 × 10−1 | |
f6 | Mean | 2.74 × 101 | 4.91 × 101 | 3.31 × 10−15 | 6.30 × 101 | 4.98 × 101 | 2.05 × 102 | 0.00 |
SD | 1.81 × 101 | 1.70 × 101 | 1.71 × 10−14 | 1.82 × 101 | 1.56 × 101 | 1.70 × 101 | 0.00 | |
SEM | 3.31 × 100 | 3.11 × 100 | 3.13 × 10−15 | 3.32 × 100 | 2.85 × 100 | 3.11 × 100 | 0.00 | |
f7 | Mean | 2.07 × 10−1 | 4.37 × 10−1 | 1.15 × 100 | 1.75 × 101 | 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 × 100 | 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 | |
f8 | Mean | 3.10 × 10−2 | 5.95 × 10−1 | 4.96 × 10−13 | 1.76 × 101 | 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 × 100 | 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 | |
f9 | Mean | 4.35 × 10−3 | 1.44 × 10−2 | 1.08 × 10−3 | 2.70 × 102 | 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 × 101 | 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 × 100 | 8.89 × 10−4 | 2.54 × 10−2 | 0.00 | |
f10 | Mean | 4.67 × 10−2 | 1.83 × 10−3 | 5.54 × 103 | 4.55 × 10−4 | 5.68 × 10−33 | 1.30 × 104 | 0.00 |
SD | 4.06 × 10−2 | 1.86 × 10−3 | 1.59 × 103 | 1.78 × 10−4 | 1.71 × 10−33 | 5.21 × 103 | 0.00 | |
SEM | 7.41 × 10−3 | 3.40 × 10−4 | 2.90 × 102 | 3.25 × 10−5 | 3.12 × 10−34 | 9.52 × 102 | 0.00 | |
f11 | Mean | 6.36 × 10−12 | 1.30 × 10−1 | 7.58 × 10−2 | 2.48 × 10−2 | 9.37 × 10−18 | 2.15 × 101 | 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 × 101 | 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 × 100 | 0.00 | |
f12 | Mean | 1.76 × 10−11 | 4.11 × 10−21 | 5.01 × 100 | 6.51 × 10−1 | 2.48 × 10−34 | 4.53 × 101 | 0.00 |
SD | 5.10 × 10−11 | 1.68 × 10−20 | 5.90 × 100 | 2.82 × 10−1 | 4.93 × 10−35 | 2.46 × 102 | 0.00 | |
SEM | 9.32 × 10−12 | 3.07 × 10−21 | 1.07 × 100 | 5.15 × 10−2 | 9.01 × 10−36 | 4.49 × 101 | 0.00 | |
f13 | Mean | 2.39 × 101 | 1.93 × 101 | 2.39 × 101 | 2.64 × 101 | 1.22 × 101 | 2.89 × 101 | 2.87 × 101 |
SD | 1.19 × 100 | 1.49 × 100 | 2.95 × 100 | 1.60 × 100 | 1.88 × 100 | 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 | |
f14 | Mean | 1.53 × 102 | 3.92 × 101 | 1.30 × 102 | 5.47 × 101 | 5.86 × 101 | 2.25 × 102 | 0.00 |
SD | 4.63 × 101 | 1.12 × 101 | 1.35 × 101 | 1.46 × 101 | 1.90 × 101 | 1.82 × 101 | 0.00 | |
SEM | 8.45 × 100 | 2.04 × 100 | 2.47 × 100 | 2.68 × 100 | 3.48 × 100 | 3.32 × 100 | 0.00 | |
f15 | Mean | 2.09 × 10−1 | 4.60 × 10−1 | 1.16 × 100 | 1.75 × 101 | 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 × 100 | 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 | |
f16 | Mean | 3.85 × 10−2 | 1.83 × 100 | 7.09 × 100 | 1.88 × 101 | 1.80 × 10−14 | 7.75 × 10−2 | 8.88 × 10−16 |
SD | 2.11 × 10−1 | 6.11 × 10−1 | 7.21 × 100 | 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 × 100 | 4.95 × 10−2 | 1.24 × 10−15 | 5.48 × 10−2 | 0.00 | |
f17 | Mean | 4.99 × 10−3 | 1.03 × 10−2 | 2.31 × 10−5 | 2.73 × 102 | 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 × 101 | 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 × 100 | 3.42 × 10−4 | 3.29 × 10−2 | 0.00 |
Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|
f1 | MeanFEs | 15,963 | 9556 | 11,655 | 65,471 | 8593.3 | 37,517 | 3.33 |
SR (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |
f2 | MeanFEs | NaN | NaN | NaN | NaN | 19,125 | NaN | 8.06 |
SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |
f3 | MeanFEs | 32,199 | 25,373 | 35,040 | NaN | 25,160 | 76,022.67 | 10.43 |
SR (%) | 100 | 100 | 100 | 0 | 100 | 100 | 100 | |
f4 | MeanFEs | 25,470.67 | 16,504 | 20,672 | NaN | 14,408 | 56,173.33 | 1122.83 |
SR (%) | 100 | 100 | 100 | 0 | 100 | 100 | 100 | |
f5 | MeanFEs | NaN | NaN | 57,856 | NaN | NaN | 603,80 | 17,826.21 |
SR (%) | 0 | 0 | 16.66 | 0 | 0 | 13.33 | 96.66 | |
f6 | MeanFEs | NaN | NaN | 42,561.33 | NaN | NaN | NaN | 7.63 |
SR (%) | 0 | 0 | 100 | 0 | 0 | 0 | 100 | |
f7 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 20.46 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f8 | 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 | |
f9 | 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 | |
f10 | MeanFEs | NaN | NaN | NaN | NaN | 18,386.67 | NaN | 10.16 |
SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |
f11 | MeanFEs | 39,573.33 | 44,388 | NaN | NaN | 25,192 | NaN | 11.90 |
SR (%) | 100 | 66.66 | 0 | 0 | 100 | 0 | 100 | |
f12 | MeanFEs | 46,014.67 | 26,560 | NaN | NaN | 14,850.67 | NaN | 1278.66 |
SR (%) | 100 | 100 | 0 | 0 | 100 | 0 | 100 | |
f13 | MeanFEs | 5794.66 | 1226.66 | 7540 | 45,032 | 834.66 | 15,921.33 | 36.76 |
SR (%) | 100 | 100 | 100 | 100 | 100 | 100 | 100 | |
f14 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 9.33 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f15 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 23.56 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f16 | 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 | |
f17 | 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 |
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 |
Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|
f1 | Mean | 1.34 × 10−2 | 2.72 × 10−1 | 6.06 × 10−6 | 1.41 × 103 | 2.78 × 10−32 | 3.08 × 101 | 0.00 |
SD | 3.12 × 10−2 | 1.16 × 100 | 7.59 × 10−6 | 7.79 × 102 | 4.82 × 10−33 | 1.66 × 101 | 0.00 | |
SEM | 5.69 × 10−3 | 2.12 × 10−1 | 1.38 × 10−6 | 1.42 × 102 | 8.80 × 10−34 | 3.03 × 100 | 0.00 | |
f2 | Mean | 8.72 × 104 | 7.32 × 103 | 1.49 × 105 | 4.05 × 104 | 1.04 × 102 | 4.05 × 105 | 0.00 |
SD | 1.90 × 104 | 2.50 × 103 | 1.60 × 104 | 7.65 × 103 | 1.60 × 102 | 4.67 × 104 | 0.00 | |
SEM | 3.48 × 103 | 4.57 × 102 | 2.92 × 103 | 1.39 × 103 | 2.92 × 101 | 8.53 × 103 | 0.00 | |
f3 | Mean | 5.53 × 10−2 | 7.02 × 10−1 | 2.64 × 10−3 | 1.82 × 100 | 1.15 × 10−16 | 8.00 × 100 | 4.07 × 10−298 |
SD | 1.47 × 10−1 | 7.58 × 10−1 | 9.26 × 10−4 | 2.73 × 100 | 9.25 × 10−18 | 4.01 × 100 | 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 | |
f4 | Mean | 2.51 × 102 | 2.00 × 10−1 | 1.05 × 10−5 | 6.59 × 103 | 2.49 × 10−31 | 3.10 × 105 | 0.00 |
SD | 7.45 × 102 | 3.69 × 10−1 | 9.27 × 10−6 | 4.28 × 103 | 8.71 × 10−32 | 6.15 × 105 | 0.00 | |
SEM | 1.36 × 102 | 6.74 × 10−2 | 1.69 × 10−6 | 7.82 × 102 | 1.59 × 10−32 | 1.12 × 105 | 0.00 | |
f5 | Mean | 1.20 × 102 | 9.34 × 101 | 1.85 × 102 | 1.05 × 102 | 1.16 × 102 | 1.80 × 102 | 1.30 × 101 |
SD | 3.26 × 101 | 1.08 × 100 | 3.41 × 101 | 2.16 × 101 | 3.42 × 101 | 4.67 × 101 | 3.39 × 101 | |
SEM | 5.95 × 100 | 1.98 × 10−1 | 6.24 × 100 | 3.96 × 100 | 6.26 × 100 | 8.54 × 100 | 6.19 × 100 | |
f6 | Mean | 1.71 × 102 | 1.62 × 102 | 1.73 × 101 | 3.88 × 102 | 3.71 × 102 | 7.72 × 102 | 0.00 |
SD | 7.69 × 101 | 3.50 × 101 | 4.16 × 100 | 4.42 × 101 | 6.19 × 101 | 1.27 × 102 | 0.00 | |
SEM | 1.40 × 101 | 6.39 × 100 | 7.59 × 10−1 | 8.07 × 100 | 1.13 × 101 | 2.32 × 101 | 0.00 | |
f7 | Mean | 2.68 × 100 | 2.27 × 100 | 9.60 × 100 | 4.40 × 101 | 1.33 × 100 | 7.08 × 100 | 0.00 |
SD | 7.94 × 10−1 | 5.72 × 10−1 | 7.20 × 10−1 | 1.40 × 100 | 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 | |
f8 | Mean | 2.84 × 100 | 3.89 × 100 | 4.14 × 10−2 | 1.92 × 101 | 1.36 × 100 | 5.51 × 100 | 8.88 × 10−16 |
SD | 1.10 × 100 | 9.08 × 10−1 | 2.94 × 10−2 | 8.75 × 10−2 | 7.94 × 10−1 | 1.16 × 100 | 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 | |
f9 | Mean | 5.07 × 10−2 | 2.30 × 10−1 | 4.15 × 10−3 | 1.71 × 103 | 3.28 × 10−3 | 1.25 × 100 | 0.00 |
SD | 6.99 × 10−2 | 3.87 × 10−1 | 9.14 × 10−3 | 8.95 × 101 | 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 × 101 | 1.08 × 10−3 | 2.86 × 10−2 | 0.00 | |
f10 | Mean | 4.11 × 104 | 4.75 × 103 | 1.16 × 105 | 4.24 × 104 | 3.82 × 101 | 3.37 × 105 | 0.00 |
SD | 1.13 × 104 | 1.66 × 103 | 1.72 × 104 | 6.87 × 103 | 3.28 × 101 | 4.32 × 104 | 0.00 | |
SEM | 2.07 × 103 | 3.04 × 102 | 3.14 × 103 | 1.25 × 103 | 5.99 × 100 | 7.89 × 103 | 0.00 | |
f11 | Mean | 6.76 × 100 | 5.26 × 100 | 1.36 × 101 | 3.17 × 100 | 8.83 × 100 | 3.23 × 1012 | 1.54 × 10−296 |
SD | 3.64 × 100 | 3.51 × 100 | 5.57 × 100 | 4.48 × 100 | 7.97 × 100 | 1.77 × 1013 | 0.00 | |
SEM | 6.65 × 10−1 | 6.41 × 10−1 | 1.01 × 100 | 8.19 × 10−1 | 1.45 × 100 | 3.23 × 1012 | 0.00 | |
f12 | Mean | 1.56 × 102 | 1.12 × 101 | 4.11 × 102 | 6.75 × 103 | 1.38 × 10−18 | 2.81 × 106 | 0.00 |
SD | 2.53 × 102 | 2.40 × 101 | 3.49 × 102 | 6.01 × 103 | 2.52 × 10−18 | 6.79 × 106 | 0.00 | |
SEM | 4.63 × 101 | 4.39 × 100 | 6.37 × 101 | 1.09 × 103 | 4.60 × 10−19 | 1.24 × 106 | 0.00 | |
f13 | Mean | 9.81 × 101 | 9.44 × 101 | 9.65 × 101 | 1.01 × 102 | 9.45 × 101 | 2.53 × 102 | 9.89 × 101 |
SD | 9.85 × 100 | 2.00 × 100 | 1.50 × 100 | 1.57 × 101 | 1.80 × 101 | 1.82 × 102 | 2.12 × 10−2 | |
SEM | 1.79 × 100 | 3.65 × 10−1 | 2.74 × 10−1 | 2.87 × 100 | 3.29 × 100 | 3.33 × 101 | 3.87 × 10−3 | |
f14 | Mean | 8.33 × 102 | 1.38 × 102 | 5.72 × 102 | 3.68 × 102 | 3.60 × 102 | 9.88 × 102 | 0.00 |
SD | 1.29 × 102 | 2.69 × 101 | 3.98 × 101 | 5.52 × 101 | 6.10 × 101 | 4.36 × 101 | 0.00 | |
SEM | 2.36 × 101 | 4.92 × 100 | 7.27 × 100 | 1.00 × 101 | 1.11 × 101 | 7.97 × 100 | 0.00 | |
f15 | Mean | 2.38 × 100 | 2.40 × 100 | 9.63 × 100 | 4.43 × 101 | 1.25 × 100 | 7.29 × 100 | 0.00 |
SD | 7.40 × 10−1 | 4.94 × 10−1 | 7.86 × 10−1 | 1.71 × 100 | 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 | |
f16 | Mean | 2.51 × 100 | 4.87 × 100 | 1.89 × 101 | 1.93 × 101 | 1.95 × 100 | 5.41 × 100 | 8.88 × 10−16 |
SD | 6.20 × 10−1 | 8.23 × 10−1 | 2.68 × 100 | 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 | |
f17 | Mean | 1.28 × 10−2 | 9.16 × 10−2 | 1.10 × 10−3 | 1.72e × 103 | 4.76 × 10−3 | 1.42 × 100 | 0.00 |
SD | 1.27 × 10−2 | 1.36 × 10−1 | 1.75 × 10−3 | 8.97 × 101 | 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 × 101 | 1.16 × 10−3 | 1.08 × 10−1 | 0.00 |
Function | DE | PSO | ABC | IWO | FA | Jaya | CJaya-SQP | |
---|---|---|---|---|---|---|---|---|
f1 | MeanFEs | 74,322 | 66,373 | 43,039 | NaN | 12704 | NaN | 9.86 |
SR (%) | 66.66 | 60 | 100 | 0 | 100 | 0 | 100 | |
f2 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 12.5 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f3 | MeanFEs | NaN | NaN | NaN | NaN | 30140 | NaN | 14.3 |
SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |
f4 | MeanFEs | NaN | NaN | 76,290.91 | NaN | 21,966.67 | NaN | 1498.66 |
SR (%) | 0 | 0 | 73.33 | 0 | 100 | 0 | 100 | |
f5 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 55,755.38 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 86.66 | |
f6 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 12.03 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f7 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 23.10 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f8 | MeanFEs | NaN | NaN | NaN | NaN | 26,793.33 | NaN | 31.16 |
SR (%) | 0 | 0 | 0 | 0 | 20 | 0 | 100 | |
f9 | MeanFEs | NaN | NaN | 79,240 | NaN | 19,405.71 | NaN | 24.53 |
SR (%) | 0 | 0 | 3.33 | 0 | 70 | 0 | 100 | |
f10 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 16.43 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f11 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 22.19 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f12 | MeanFEs | NaN | NaN | NaN | NaN | 32,870.67 | NaN | 1564 |
SR (%) | 0 | 0 | 0 | 0 | 100 | 0 | 100 | |
f13 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | NaN |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
f14 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 20.76 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f15 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 26.86 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f16 | MeanFEs | NaN | NaN | NaN | NaN | NaN | NaN | 33.73 |
SR (%) | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |
f17 | MeanFEs | NaN | NaN | NaN | NaN | 19,456.47 | NaN | 29.36 |
SR (%) | 0 | 0 | 0 | 0 | 56.66 | 0 | 100 |
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 | |
---|---|---|---|---|---|---|
0.0720 ± j3.7192 | 0.5919 | Interarea | −0.0194 | 1.0000, 1.0000 | ||
0.6012, 0.6012 | ||||||
0.7181, 0.7181 | ||||||
0.7445, 0.7445 | ||||||
−0.2166 ± j5.9569 | 0.9481 | Local | 0.0363 | 0.8053, 0.8053 | ||
1.0000, 1.0000 | ||||||
0.0316, 0.0316 | ||||||
0.0119, 0.0119 | ||||||
−0.2161 ± j6.1273 | 0.9752 | Local | 0.0352 | 0.0145, 0.0145 | ||
0.0229, 0.0229 | ||||||
0.9941, 0.9941 | ||||||
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 |
T1 | 1.1324 | 1.1831 | 1.2220 | 0.1464 | 0.9337 | 0.7207 | 1.8364 | 1.3573 |
T2 | 0.2664 | 0.6197 | 0.5310 | 0.8140 | 0.6893 | 0.5714 | 1.3917 | 0.9393 |
T3 | 1.4371 | 1.1725 | 1.4376 | 1.6985 | 0.8732 | 1.5084 | 1.3293 | 1.0909 |
T4 | 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 | ITAE1 (tsim = 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 | ITAE2 (tsim = 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 |
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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
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 StyleWelhazi, 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