doi: 10.3934/naco.2020057

A novel hybrid AGWO-PSO algorithm in mitigation of power network oscillations with STATCOM

Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, India, 826004

* Corresponding author: Ramesh Devarapalli

Received  July 2020 Revised  November 2020 Published  January 2021

The assimilation of flexible AC transmission (FACTS) controllers to the existing power network outweigh the numerous alternatives in enhancing the damping behavior for the inter-area /intra-area system oscillations of a power network. This paper provides a rigorous analysis in damping of oscillations in a power network. It utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics. A comprehensive system mathematical modelling has been developed for demonstrating the system behavior under different loading conditions. A novel hybrid augmented grey wolf optimization-particle swarm optimization (AGWO-PSO) is proposed for the coordinated design of controllers static synchronous compensator (STATCOM) and power system stabilizers (PSSs). A multi-objective function, comprising damping ratio improvement and drifting the real part to the left-hand side of S-plane of the system poles, has been developed to achieve the objective and the effectiveness of the proposed algorithms have been analyzed by monitoring the system performance under different loading conditions. Eigenvalue analysis and damping nature of the system states under perturbation have been presented for the proposed algorithms under different loading conditions, and the performance evaluation of the proposed algorithms have been done by means of time of execution and the convergence characteristics.

Citation: Ramesh Devarapalli, Biplab Bhattacharyya. A novel hybrid AGWO-PSO algorithm in mitigation of power network oscillations with STATCOM. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020057
References:
[1]

http://socr.ucla.edu/Applets.dir/F_Table.html. Google Scholar

[2]

V. S. AragónS. C. Esquivel and C. C. Coello, An immune algorithm with power redistribution for solving economic dispatch problems, Information Sciences, 295 (2015), 609-632.  doi: 10.1016/j.ins.2014.10.026.  Google Scholar

[3]

M. Basu, Squirrel search algorithm for multi-region combined heat and power economic dispatch incorporating renewable energy sources, Energy, 182 (2019), 296-305.   Google Scholar

[4]

A. CuevasM. Febrero and R. Fraiman, An anova test for functional data, Computational Statistics & Data Analysis, 47 (2004), 111-122.  doi: 10.1016/j.csda.2003.10.021.  Google Scholar

[5]

J. DerracS. GarcíaD. Molina and F. Herrera, A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms, Swarm and Evolutionary Computation, 1 (2011), 3-18.   Google Scholar

[6]

R. Devarapalli and B. Bhattacharyya, Application of modified harris hawks optimization in power system oscillations damping controller design, in 2019 8th International Conference on Power Systems (ICPS), IEEE, (2019), 1–6. Google Scholar

[7]

R. Devarapalli and B. Bhattacharyya, A framework for $h_{2} /h_\infty$ synthesis in damping power network oscillations with statcom, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 1–22. Google Scholar

[8]

R. Devarapalli and B. Bhattacharyya, Optimal parameter tuning of power oscillation damper by mhho algorithm, in 2019 20th International Conference on Intelligent System Application to Power Systems (ISAP), IEEE, 2019, 1–7. doi: 10.1007/978-981-13-1816-0.  Google Scholar

[9]

R. Devarapalli and B. Bhattacharyya, A hybrid modified grey wolf optimization-sine cosine algorithm-based power system stabilizer parameter tuning in a multimachine power system, Optimal Control Applications and Methods. Google Scholar

[10]

R. Devarapalli and B. Bhattacharyya, Optimal controller parameter tuning of pss using sine-cosine algorithm, in Metaheuristic and Evolutionary Computation: Algorithms and Applications, Springer, 2020,337–360. Google Scholar

[11]

R. Devarapalli, B. Bhattacharyya and J. K. Saw, Controller parameter tuning of a single machine infinite bus system with static synchronous compensator using antlion optimization algorithm for the power system stability improvement, Advanced Control for Applications: Engineering and Industrial Systems, 2 (2020), e45.  Google Scholar

[12]

R. DevarapalliB. Bhattacharyya and N. K. Sinha, An intelligent egwo-sca-cs algorithm for pss parameter tuning under system uncertainties, International Journal of Intelligent Systems, 35 (2020), 1520-1569.   Google Scholar

[13]

R. Devarapalli, B. Bhattacharyya, N. K. Sinha and B. Dey, Amended gwo approach based multi-machine power system stability enhancement, ISA Transactions. Google Scholar

[14]

M. Ebeed, S. Kamel and H. Youssef, Optimal setting of statcom based on voltage stability improvement and power loss minimization using moth-flame algorithm, in 2016 Eighteenth International Middle East Power Systems Conference (MEPCON), IEEE, 2016,815–820. Google Scholar

[15]

A. A. El-Fergany and H. M. Hasanien, Salp swarm optimizer to solve optimal power flow comprising voltage stability analysis, Neural Computing and Applications, 32 (2020), 5267-5283.   Google Scholar

[16]

S. Feng, P. Jiang and X. Wu, Suppression of power system forced oscillations based on pss with proportional-resonant controller, International Transactions on Electrical Energy Systems, 27 (2017), e2328. Google Scholar

[17]

B. Hekimoğlu, Sine-cosine algorithm-based optimization for automatic voltage regulator system, Transactions of the Institute of Measurement and Control, 41 (2019), 1761-1771.   Google Scholar

[18]

V. K. Kamboj, A novel hybrid pso–gwo approach for unit commitment problem, Neural Computing and Applications, 27 (2016), 1643-1655.   Google Scholar

[19]

R. KhosravanianV. MansouriD. A. Wood and M. R. Alipour, A comparative study of several metaheuristic algorithms for optimizing complex 3-d well-path designs, Journal of Petroleum Exploration and Production Technology, 8 (2018), 1487-1503.   Google Scholar

[20]

P. Kundur, N. J. Balu and M. G. Lauby, Power System Stability and Control, Vol. 7, McGraw-hill New York, 1994. Google Scholar

[21]

Z. LiD. Zou and Z. Kong, A harmony search variant and a useful constraint handling method for the dynamic economic emission dispatch problems considering transmission loss, Engineering Applications of Artificial Intelligence, 84 (2019), 18-40.   Google Scholar

[22]

S. Mirjalili, Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowledge-Based Systems, 89 (2015), 228-249.   Google Scholar

[23]

S. MirjaliliS. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.   Google Scholar

[24]

M. Molga and C. Smutnicki, Test functions for optimization needs, Test Functions for Optimization Needs, 101 (2005), 48. Google Scholar

[25]

K. Padiyar et al., Power System Dynamics: Stability and Control, John Wiley New York, 1996. Google Scholar

[26]

B. V. PatilL. P. M. I. SampathA. Krishnan and F. Y. Eddy, Decentralized nonlinear model predictive control of a multimachine power system, International Journal of Electrical Power & Energy Systems, 106 (2019), 358-372.   Google Scholar

[27]

M. H. QaisH. M. Hasanien and S. Alghuwainem, Augmented grey wolf optimizer for grid-connected pmsg-based wind energy conversion systems, Applied Soft Computing, 69 (2018), 504-515.   Google Scholar

[28]

M. Rahmatian and S. Seyedtabaii, Multi-machine optimal power system stabilizers design based on system stability and nonlinearity indices using hyper-spherical search method, International Journal of Electrical Power & Energy Systems, 105 (2019), 729-740.   Google Scholar

[29]

S. Raj and B. Bhattacharyya, Optimal placement of tcsc and svc for reactive power planning using whale optimization algorithm, Swarm and Evolutionary Computation, 40 (2018), 131-143.   Google Scholar

[30]

A. Salgotra and S. Pan, Model based pi power system stabilizer design for damping low frequency oscillations in power systems, ISA Transactions, 76 (2018), 110-121.   Google Scholar

[31]

M. SarailooN. E. Wu and J. S. Bay, Transient stability assessment of large lossy power systems, IET Generation, Transmission & Distribution, 12 (2017), 1822-1830.   Google Scholar

[32]

S. Saurav, V. K. Gupta and S. K. Mishra, Moth-flame optimization based algorithm for facts devices allocation in a power system, in 2017 International Conference on Innovations in Information, Embedded and Communication Systems (ICIIECS), IEEE, 2017, 1–7. Google Scholar

[33]

I. Stojanović, I. Brajević, P. S. Stanimirović, L. A. Kazakovtsev and Z. Zdravev, Application of heuristic and metaheuristic algorithms in solving constrained weber problem with feasible region bounded by arcs, Mathematical Problems in Engineering, 2017. doi: 10.1155/2017/8306732.  Google Scholar

[34]

R. K. Varma, Introduction to facts controllers, in 2009 IEEE/PES Power Systems Conference and Exposition, IEEE, 2009, 1–6. Google Scholar

[35]

Z. WangZ. LiuJ. WangX. JiangS. LiuY. LiuG. Sheng and T. Liu, The application of analytical mechanics in a multimachine power system, Turkish Journal of Electrical Engineering & Computer Sciences, 26 (2018), 1530-1540.   Google Scholar

[36]

D. H. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, 1 (1997), 67-82.   Google Scholar

[37]

G. Yuan and W. Yang, Study on optimization of economic dispatching of electric power system based on hybrid intelligent algorithms (pso and afsa), Energy, 183 (2019), 926-935.   Google Scholar

[38]

P. ZhaoW. YaoS. WangJ. Wen and S. Cheng, Decentralized nonlinear synergetic power system stabilizers design for power system stability enhancement, International Transactions on Electrical Energy Systems, 24 (2014), 1356-1368.   Google Scholar

show all references

References:
[1]

http://socr.ucla.edu/Applets.dir/F_Table.html. Google Scholar

[2]

V. S. AragónS. C. Esquivel and C. C. Coello, An immune algorithm with power redistribution for solving economic dispatch problems, Information Sciences, 295 (2015), 609-632.  doi: 10.1016/j.ins.2014.10.026.  Google Scholar

[3]

M. Basu, Squirrel search algorithm for multi-region combined heat and power economic dispatch incorporating renewable energy sources, Energy, 182 (2019), 296-305.   Google Scholar

[4]

A. CuevasM. Febrero and R. Fraiman, An anova test for functional data, Computational Statistics & Data Analysis, 47 (2004), 111-122.  doi: 10.1016/j.csda.2003.10.021.  Google Scholar

[5]

J. DerracS. GarcíaD. Molina and F. Herrera, A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms, Swarm and Evolutionary Computation, 1 (2011), 3-18.   Google Scholar

[6]

R. Devarapalli and B. Bhattacharyya, Application of modified harris hawks optimization in power system oscillations damping controller design, in 2019 8th International Conference on Power Systems (ICPS), IEEE, (2019), 1–6. Google Scholar

[7]

R. Devarapalli and B. Bhattacharyya, A framework for $h_{2} /h_\infty$ synthesis in damping power network oscillations with statcom, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, 1–22. Google Scholar

[8]

R. Devarapalli and B. Bhattacharyya, Optimal parameter tuning of power oscillation damper by mhho algorithm, in 2019 20th International Conference on Intelligent System Application to Power Systems (ISAP), IEEE, 2019, 1–7. doi: 10.1007/978-981-13-1816-0.  Google Scholar

[9]

R. Devarapalli and B. Bhattacharyya, A hybrid modified grey wolf optimization-sine cosine algorithm-based power system stabilizer parameter tuning in a multimachine power system, Optimal Control Applications and Methods. Google Scholar

[10]

R. Devarapalli and B. Bhattacharyya, Optimal controller parameter tuning of pss using sine-cosine algorithm, in Metaheuristic and Evolutionary Computation: Algorithms and Applications, Springer, 2020,337–360. Google Scholar

[11]

R. Devarapalli, B. Bhattacharyya and J. K. Saw, Controller parameter tuning of a single machine infinite bus system with static synchronous compensator using antlion optimization algorithm for the power system stability improvement, Advanced Control for Applications: Engineering and Industrial Systems, 2 (2020), e45.  Google Scholar

[12]

R. DevarapalliB. Bhattacharyya and N. K. Sinha, An intelligent egwo-sca-cs algorithm for pss parameter tuning under system uncertainties, International Journal of Intelligent Systems, 35 (2020), 1520-1569.   Google Scholar

[13]

R. Devarapalli, B. Bhattacharyya, N. K. Sinha and B. Dey, Amended gwo approach based multi-machine power system stability enhancement, ISA Transactions. Google Scholar

[14]

M. Ebeed, S. Kamel and H. Youssef, Optimal setting of statcom based on voltage stability improvement and power loss minimization using moth-flame algorithm, in 2016 Eighteenth International Middle East Power Systems Conference (MEPCON), IEEE, 2016,815–820. Google Scholar

[15]

A. A. El-Fergany and H. M. Hasanien, Salp swarm optimizer to solve optimal power flow comprising voltage stability analysis, Neural Computing and Applications, 32 (2020), 5267-5283.   Google Scholar

[16]

S. Feng, P. Jiang and X. Wu, Suppression of power system forced oscillations based on pss with proportional-resonant controller, International Transactions on Electrical Energy Systems, 27 (2017), e2328. Google Scholar

[17]

B. Hekimoğlu, Sine-cosine algorithm-based optimization for automatic voltage regulator system, Transactions of the Institute of Measurement and Control, 41 (2019), 1761-1771.   Google Scholar

[18]

V. K. Kamboj, A novel hybrid pso–gwo approach for unit commitment problem, Neural Computing and Applications, 27 (2016), 1643-1655.   Google Scholar

[19]

R. KhosravanianV. MansouriD. A. Wood and M. R. Alipour, A comparative study of several metaheuristic algorithms for optimizing complex 3-d well-path designs, Journal of Petroleum Exploration and Production Technology, 8 (2018), 1487-1503.   Google Scholar

[20]

P. Kundur, N. J. Balu and M. G. Lauby, Power System Stability and Control, Vol. 7, McGraw-hill New York, 1994. Google Scholar

[21]

Z. LiD. Zou and Z. Kong, A harmony search variant and a useful constraint handling method for the dynamic economic emission dispatch problems considering transmission loss, Engineering Applications of Artificial Intelligence, 84 (2019), 18-40.   Google Scholar

[22]

S. Mirjalili, Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowledge-Based Systems, 89 (2015), 228-249.   Google Scholar

[23]

S. MirjaliliS. M. Mirjalili and A. Lewis, Grey wolf optimizer, Advances in Engineering Software, 69 (2014), 46-61.   Google Scholar

[24]

M. Molga and C. Smutnicki, Test functions for optimization needs, Test Functions for Optimization Needs, 101 (2005), 48. Google Scholar

[25]

K. Padiyar et al., Power System Dynamics: Stability and Control, John Wiley New York, 1996. Google Scholar

[26]

B. V. PatilL. P. M. I. SampathA. Krishnan and F. Y. Eddy, Decentralized nonlinear model predictive control of a multimachine power system, International Journal of Electrical Power & Energy Systems, 106 (2019), 358-372.   Google Scholar

[27]

M. H. QaisH. M. Hasanien and S. Alghuwainem, Augmented grey wolf optimizer for grid-connected pmsg-based wind energy conversion systems, Applied Soft Computing, 69 (2018), 504-515.   Google Scholar

[28]

M. Rahmatian and S. Seyedtabaii, Multi-machine optimal power system stabilizers design based on system stability and nonlinearity indices using hyper-spherical search method, International Journal of Electrical Power & Energy Systems, 105 (2019), 729-740.   Google Scholar

[29]

S. Raj and B. Bhattacharyya, Optimal placement of tcsc and svc for reactive power planning using whale optimization algorithm, Swarm and Evolutionary Computation, 40 (2018), 131-143.   Google Scholar

[30]

A. Salgotra and S. Pan, Model based pi power system stabilizer design for damping low frequency oscillations in power systems, ISA Transactions, 76 (2018), 110-121.   Google Scholar

[31]

M. SarailooN. E. Wu and J. S. Bay, Transient stability assessment of large lossy power systems, IET Generation, Transmission & Distribution, 12 (2017), 1822-1830.   Google Scholar

[32]

S. Saurav, V. K. Gupta and S. K. Mishra, Moth-flame optimization based algorithm for facts devices allocation in a power system, in 2017 International Conference on Innovations in Information, Embedded and Communication Systems (ICIIECS), IEEE, 2017, 1–7. Google Scholar

[33]

I. Stojanović, I. Brajević, P. S. Stanimirović, L. A. Kazakovtsev and Z. Zdravev, Application of heuristic and metaheuristic algorithms in solving constrained weber problem with feasible region bounded by arcs, Mathematical Problems in Engineering, 2017. doi: 10.1155/2017/8306732.  Google Scholar

[34]

R. K. Varma, Introduction to facts controllers, in 2009 IEEE/PES Power Systems Conference and Exposition, IEEE, 2009, 1–6. Google Scholar

[35]

Z. WangZ. LiuJ. WangX. JiangS. LiuY. LiuG. Sheng and T. Liu, The application of analytical mechanics in a multimachine power system, Turkish Journal of Electrical Engineering & Computer Sciences, 26 (2018), 1530-1540.   Google Scholar

[36]

D. H. Wolpert and W. G. Macready, No free lunch theorems for optimization, IEEE Transactions on Evolutionary Computation, 1 (1997), 67-82.   Google Scholar

[37]

G. Yuan and W. Yang, Study on optimization of economic dispatching of electric power system based on hybrid intelligent algorithms (pso and afsa), Energy, 183 (2019), 926-935.   Google Scholar

[38]

P. ZhaoW. YaoS. WangJ. Wen and S. Cheng, Decentralized nonlinear synergetic power system stabilizers design for power system stability enhancement, International Transactions on Electrical Energy Systems, 24 (2014), 1356-1368.   Google Scholar

Figure 1.  Schematic of the mathematically modelled System to coordinate STATCOM and PSS controllers
Figure 2.  Power System Stabilizer block diagram representation
Figure 3.  Desired location of eigenvalues [9]
Figure 4.  Governance hierarchy of GWs
Figure 5.  Regulation vectors adjustment for catching the prey
Figure 6.  Attacking and searching strategy of GWs
Figure 7.  (a) 2D representation of F1-F23, (b) Search space, (c) Average fitness obtained over the iterations, (d) box and whisker plot, (e) Convergence curves
Figure 8.  Convergence characteristics of the proposed optimization algorithms under (a) Light Load, (b) Nominal Load and (c) Heavy Load condition
Figure 9.  Execution time (in sec.) of proposed heuristic methods under different loading conditions
Figure 10.  Damping behavior of system states for 10% perturbation under light load with different algorithms
Figure 11.  Damping behavior of system states for 10% perturbation under light load with different algorithms
Figure 12.  Damping behavior of system states for 10% perturbation under heavy load with different algorithms
Table 1.  Application of different meta-heuristic techniques for research problems in the electric power system
Suggested Algorithm Performance comparison with algorithm Problem solution Considered Test system Ref.
SCA ZN, DE, ABC, BBO controller parameters of AVR system Sample test system [17]
SSA MSA, DE, MDE, PSO, TSA OPF problem with voltage stability objective IEEE 57- and 118-bus [15]
Variant of HSA Other variants of HSA Economic Emission problem 5, 6, 10, 14- test systems [21]
SSOA GWO, PSO, DE, EP MRCHPED problem three region test system [3]
SCA ALO, DA Power system oscillation damping 2 machine with STATCOM [10]
Hybrid PSO-AFSA PSO, AFSA ELD problem 5, 10 Units system [37]
ALO PSO PSS parameter tuning SMIB with STATCOM [11]
IA Algorithms from literature Economic Dispatch problem 8 test systems [2]
HHO modified GWO, WOA, ALO Stabilizers coordination SMIB with STATCOM [6,8]
Suggested Algorithm Performance comparison with algorithm Problem solution Considered Test system Ref.
SCA ZN, DE, ABC, BBO controller parameters of AVR system Sample test system [17]
SSA MSA, DE, MDE, PSO, TSA OPF problem with voltage stability objective IEEE 57- and 118-bus [15]
Variant of HSA Other variants of HSA Economic Emission problem 5, 6, 10, 14- test systems [21]
SSOA GWO, PSO, DE, EP MRCHPED problem three region test system [3]
SCA ALO, DA Power system oscillation damping 2 machine with STATCOM [10]
Hybrid PSO-AFSA PSO, AFSA ELD problem 5, 10 Units system [37]
ALO PSO PSS parameter tuning SMIB with STATCOM [11]
IA Algorithms from literature Economic Dispatch problem 8 test systems [2]
HHO modified GWO, WOA, ALO Stabilizers coordination SMIB with STATCOM [6,8]
Table 2.  Mean, Standard deviation (SD), Best and worst values obtained with different algorithms
$\textbf{Functions}$ $\textbf{PSO}$ $\textbf{MFO}$ $\textbf{GWO}$ $\textbf{AGWO}$ $\textbf{AGWOPSO}$
$\textbf{F1: Sphere}$ Mean 1.21E-27 1.26E-25 1.81E-05 2.74E-25 9.59E-06
SD 1.40E-27 2.17E-25 8.12E-06 5.90E-25 4.67E-06
Best 7.00E-30 2.04E-27 6.75E-06 7.01E-27 3.26E-06
Worst 5.18E-27 1.11E-24 3.63E-05 3.01E-24 2.08E-05
$\textbf{F2: Schwefel 2.22}$ Mean 1.18E-16 1.45E-15 1.63E-03 1.76E-15 1.09E-03
SD 7.79E-17 1.51E-15 5.46E-04 1.15E-15 2.13E-04
Best 1.10E-17 2.02E-16 9.45E-04 4.12E-16 7.33E-04
Worst 3.57E-16 6.04E-15 3.83E-03 6.38E-15 1.52E-03
$\textbf{F3: Schwefel 1.2}$ Mean 1.30E-05 2.32E-04 4.84E-01 1.49E-05 1.42E-01
SD 3.68E-05 5.55E-04 5.71E-01 2.35E-05 1.12E-01
Best 2.15E-09 3.43E-07 2.36E-02 7.99E-09 8.57E-03
Worst 1.79E-04 2.46E-03 3.25E+00 9.74E-05 4.45E-01
$\textbf{F4: Schwefel 2.21}$ Mean 1.15E-06 2.19E-06 3.30E-01 1.93E-06 1.90E-01
SD 1.09E-06 2.82E-06 1.09E-01 1.18E-06 4.73E-02
Best 1.29E-07 3.03E-07 1.62E-01 1.66E-07 1.11E-01
Worst 4.80E-06 1.54E-05 5.45E-01 4.72E-06 2.78E-01
$\textbf{F5: Rosenbrock}$ Mean 2.95E+01 2.95E+01 3.11E+01 2.97E+01 3.85E+01
SD 7.79E-01 8.06E-01 6.25E-01 9.26E-01 2.14E+01
Best 2.84E+01 2.81E+01 2.97E+01 2.78E+01 2.89E+01
Worst 3.14E+01 3.14E+01 3.18E+01 3.14E+01 1.03E+02
$\textbf{F6: Step}$ Mean 8.33E-01 8.07E-01 4.48E+00 1.15E+00 5.20E+00
SD 4.07E-01 4.69E-01 5.37E-01 5.07E-01 6.25E-01
Best 6.19E-05 2.74E-07 3.58E+00 2.55E-01 3.86E+00
Worst 1.64E+00 1.82E+00 5.23E+00 2.46E+00 6.31E+00
$\textbf{F7: Quartic}$ Mean 1.67E-03 2.11E-03 3.53E-02 2.60E-03 2.93E-02
SD 9.55E-04 1.27E-03 9.30E-03 8.54E-04 8.34E-03
Best 3.43E-04 8.07E-04 2.13E-02 1.11E-03 1.39E-02
Worst 4.54E-03 6.70E-03 6.09E-02 4.59E-03 5.91E-02
$\textbf{F8: Schwefel}$ Mean -6.58E+03 -6.33E+03 -5.72E+03 -1.24E+04 -1.25E+04
SD 1.15E+03 1.23E+03 1.09E+03 8.64E+02 1.05E+03
Best -8.55E+03 -8.26E+03 -7.71E+03 -1.37E+04 -1.35E+04
Worst -3.34E+03 -3.64E+03 -2.28E+03 -9.89E+03 -7.61E+03
$\textbf{F9: Rastrigin}$ Mean 2.20E+00 3.31E+00 5.81E+01 5.17E+00 4.07E+01
SD 3.09E+00 4.75E+00 4.16E+01 5.68E+00 1.12E+01
Best 6.20E-14 6.20E-14 3.21E+01 6.20E-14 2.11E+01
Worst 1.24E+01 1.63E+01 2.69E+02 1.78E+01 7.27E+01
$\textbf{F10: Ackley}$ Mean 1.15E-13 2.24E-13 3.78E+00 2.62E-13 3.73E+00
SD 1.95E-14 6.46E-14 1.01E-01 6.88E-14 8.85E-02
Best 8.24E-14 1.25E-13 3.35E+00 1.52E-13 3.50E+00
Worst 1.56E-13 3.77E-13 3.90E+00 4.23E-13 3.85E+00
$\textbf{F11: Griewank}$ Mean 4.34E-03 3.58E-03 9.42E-03 6.42E-03 6.74E-03
SD 8.26E-03 7.71E-03 1.17E-02 1.10E-02 1.05E-02
Best 0.00E+00 0.00E+00 1.25E-05 0.00E+00 5.45E-06
Worst 2.52E-02 2.78E-02 2.65E-02 4.36E-02 2.68E-02
$\textbf{F12: Penalized}$ Mean 4.72E-02 3.70E-02 4.97E+00 6.52E-02 5.02E+00
SD 2.32E-02 1.44E-02 1.38E+00 3.38E-02 1.45E+00
Best 7.19E-03 1.39E-02 2.08E+00 1.11E-05 2.43E+00
Worst 1.29E-01 7.27E-02 8.08E+00 1.24E-01 8.91E+00
$\textbf{F13: Penalize 2}$ Mean 7.08E-01 6.16E-01 3.52E+00 7.88E-01 2.28E+00
SD 2.93E-01 2.06E-01 5.98E-01 2.66E-01 7.11E-01
Best 2.17E-01 1.02E-01 2.35E+00 3.38E-01 9.85E-01
Worst 1.37E+00 9.03E-01 4.91E+00 1.43E+00 3.37E+00
$\textbf{F14: Foxholes}$ Mean 6.63E+00 7.34E+00 1.11E+01 6.73E+00 1.37E+01
SD 5.22E+00 5.43E+00 4.50E+00 5.56E+00 6.05E+00
Best 1.09E+00 1.09E+00 2.17E+00 1.09E+00 1.09E+00
Worst 1.38E+01 1.38E+01 2.00E+01 1.38E+01 2.50E+01
$\textbf{F15: Kowalik}$ Mean 6.26E-03 5.54E-03 2.32E-03 4.38E-04 1.53E-03
SD 9.79E-03 9.36E-03 5.66E-03 3.21E-04 6.23E-03
Best 3.36E-04 3.36E-04 3.35E-04 3.36E-04 3.35E-04
Worst 2.22E-02 2.22E-02 2.29E-02 1.74E-03 3.45E-02
$\textbf{F16: Six-hump }\; \textbf{Camel-Back}$ Mean -1.13E+00 -1.13E+00 -1.12E+00 -1.13E+00 -1.12E+00
SD 3.15E-08 9.19E-12 8.76E-03 4.23E-08 8.76E-03
Best -1.13E+00 -1.13E+00 -1.13E+00 -1.13E+00 -1.13E+00
Worst -1.13E+00 -1.13E+00 -1.09E+00 -1.13E+00 -1.09E+00
$\textbf{F17: Branin}$ Mean 4.34E-01 4.34E-01 5.22E-01 4.34E-01 4.34E-01
SD 4.68E-06 3.05E-04 4.82E-01 1.62E-06 3.12E-07
Best 4.34E-01 4.34E-01 4.34E-01 4.34E-01 4.34E-01
Worst 4.34E-01 4.35E-01 3.07E+00 4.34E-01 4.34E-01
$\textbf{F18: Goldstein-Price}$ Mean 3.27E+00 3.27E+00 4.26E+00 3.27E+00 1.41E+01
SD 7.76E-05 5.59E-05 5.38E+00 6.06E-05 1.44E+01
Best 3.27E+00 3.27E+00 3.27E+00 3.27E+00 3.27E+00
Worst 3.27E+00 3.27E+00 3.27E+01 3.27E+00 3.27E+01
$\textbf{F19: Hartman 3}$ Mean -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
SD 2.19E-03 2.30E-03 6.26E-04 4.28E-05 7.99E-06
Best -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
Worst -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
$\textbf{F20: Hartman 6}$ Mean -3.53E+00 -3.57E+00 -3.59E+00 -3.62E+00 -3.62E+00
SD 9.68E-02 6.93E-02 6.05E-02 2.38E-02 3.29E-02
Best -3.62E+00 -3.62E+00 -3.62E+00 -3.62E+00 -3.62E+00
Worst -3.37E+00 -3.42E+00 -3.49E+00 -3.49E+00 -3.49E+00
$\textbf{F21: Shekel5}$ Mean -1.02E+01 -1.03E+01 -7.45E+00 -9.52E+00 -5.84E+00
SD 2.10E+00 1.91E+00 4.00E+00 2.67E+00 3.43E+00
Best -1.11E+01 -1.11E+01 -1.11E+01 -1.11E+01 -1.11E+01
Worst -5.52E+00 -5.52E+00 -2.87E+00 -2.93E+00 -1.48E+00
$\textbf{F22: Shekel7}$ Mean -1.13E+01 -1.14E+01 -7.49E+00 -1.00E+01 -5.89E+00
SD 9.28E-04 2.21E-07 4.20E+00 2.48E+00 3.30E+00
Best -1.14E+01 -1.14E+01 -1.14E+01 -1.14E+01 -1.14E+01
Worst -1.13E+01 -1.14E+01 -3.00E+00 -5.55E+00 -1.53E+00
$\textbf{F23: Shekel10}$ Mean -1.11E+01 -1.13E+01 -8.14E+00 -1.03E+01 -7.10E+00
SD 1.50E+00 1.07E+00 4.20E+00 2.51E+00 4.02E+00
Best -1.15E+01 -1.15E+01 -1.15E+01 -1.15E+01 -1.15E+01
Worst -5.60E+00 -5.65E+00 -2.64E+00 -4.09E+00 -1.86E+00
$\textbf{Functions}$ $\textbf{PSO}$ $\textbf{MFO}$ $\textbf{GWO}$ $\textbf{AGWO}$ $\textbf{AGWOPSO}$
$\textbf{F1: Sphere}$ Mean 1.21E-27 1.26E-25 1.81E-05 2.74E-25 9.59E-06
SD 1.40E-27 2.17E-25 8.12E-06 5.90E-25 4.67E-06
Best 7.00E-30 2.04E-27 6.75E-06 7.01E-27 3.26E-06
Worst 5.18E-27 1.11E-24 3.63E-05 3.01E-24 2.08E-05
$\textbf{F2: Schwefel 2.22}$ Mean 1.18E-16 1.45E-15 1.63E-03 1.76E-15 1.09E-03
SD 7.79E-17 1.51E-15 5.46E-04 1.15E-15 2.13E-04
Best 1.10E-17 2.02E-16 9.45E-04 4.12E-16 7.33E-04
Worst 3.57E-16 6.04E-15 3.83E-03 6.38E-15 1.52E-03
$\textbf{F3: Schwefel 1.2}$ Mean 1.30E-05 2.32E-04 4.84E-01 1.49E-05 1.42E-01
SD 3.68E-05 5.55E-04 5.71E-01 2.35E-05 1.12E-01
Best 2.15E-09 3.43E-07 2.36E-02 7.99E-09 8.57E-03
Worst 1.79E-04 2.46E-03 3.25E+00 9.74E-05 4.45E-01
$\textbf{F4: Schwefel 2.21}$ Mean 1.15E-06 2.19E-06 3.30E-01 1.93E-06 1.90E-01
SD 1.09E-06 2.82E-06 1.09E-01 1.18E-06 4.73E-02
Best 1.29E-07 3.03E-07 1.62E-01 1.66E-07 1.11E-01
Worst 4.80E-06 1.54E-05 5.45E-01 4.72E-06 2.78E-01
$\textbf{F5: Rosenbrock}$ Mean 2.95E+01 2.95E+01 3.11E+01 2.97E+01 3.85E+01
SD 7.79E-01 8.06E-01 6.25E-01 9.26E-01 2.14E+01
Best 2.84E+01 2.81E+01 2.97E+01 2.78E+01 2.89E+01
Worst 3.14E+01 3.14E+01 3.18E+01 3.14E+01 1.03E+02
$\textbf{F6: Step}$ Mean 8.33E-01 8.07E-01 4.48E+00 1.15E+00 5.20E+00
SD 4.07E-01 4.69E-01 5.37E-01 5.07E-01 6.25E-01
Best 6.19E-05 2.74E-07 3.58E+00 2.55E-01 3.86E+00
Worst 1.64E+00 1.82E+00 5.23E+00 2.46E+00 6.31E+00
$\textbf{F7: Quartic}$ Mean 1.67E-03 2.11E-03 3.53E-02 2.60E-03 2.93E-02
SD 9.55E-04 1.27E-03 9.30E-03 8.54E-04 8.34E-03
Best 3.43E-04 8.07E-04 2.13E-02 1.11E-03 1.39E-02
Worst 4.54E-03 6.70E-03 6.09E-02 4.59E-03 5.91E-02
$\textbf{F8: Schwefel}$ Mean -6.58E+03 -6.33E+03 -5.72E+03 -1.24E+04 -1.25E+04
SD 1.15E+03 1.23E+03 1.09E+03 8.64E+02 1.05E+03
Best -8.55E+03 -8.26E+03 -7.71E+03 -1.37E+04 -1.35E+04
Worst -3.34E+03 -3.64E+03 -2.28E+03 -9.89E+03 -7.61E+03
$\textbf{F9: Rastrigin}$ Mean 2.20E+00 3.31E+00 5.81E+01 5.17E+00 4.07E+01
SD 3.09E+00 4.75E+00 4.16E+01 5.68E+00 1.12E+01
Best 6.20E-14 6.20E-14 3.21E+01 6.20E-14 2.11E+01
Worst 1.24E+01 1.63E+01 2.69E+02 1.78E+01 7.27E+01
$\textbf{F10: Ackley}$ Mean 1.15E-13 2.24E-13 3.78E+00 2.62E-13 3.73E+00
SD 1.95E-14 6.46E-14 1.01E-01 6.88E-14 8.85E-02
Best 8.24E-14 1.25E-13 3.35E+00 1.52E-13 3.50E+00
Worst 1.56E-13 3.77E-13 3.90E+00 4.23E-13 3.85E+00
$\textbf{F11: Griewank}$ Mean 4.34E-03 3.58E-03 9.42E-03 6.42E-03 6.74E-03
SD 8.26E-03 7.71E-03 1.17E-02 1.10E-02 1.05E-02
Best 0.00E+00 0.00E+00 1.25E-05 0.00E+00 5.45E-06
Worst 2.52E-02 2.78E-02 2.65E-02 4.36E-02 2.68E-02
$\textbf{F12: Penalized}$ Mean 4.72E-02 3.70E-02 4.97E+00 6.52E-02 5.02E+00
SD 2.32E-02 1.44E-02 1.38E+00 3.38E-02 1.45E+00
Best 7.19E-03 1.39E-02 2.08E+00 1.11E-05 2.43E+00
Worst 1.29E-01 7.27E-02 8.08E+00 1.24E-01 8.91E+00
$\textbf{F13: Penalize 2}$ Mean 7.08E-01 6.16E-01 3.52E+00 7.88E-01 2.28E+00
SD 2.93E-01 2.06E-01 5.98E-01 2.66E-01 7.11E-01
Best 2.17E-01 1.02E-01 2.35E+00 3.38E-01 9.85E-01
Worst 1.37E+00 9.03E-01 4.91E+00 1.43E+00 3.37E+00
$\textbf{F14: Foxholes}$ Mean 6.63E+00 7.34E+00 1.11E+01 6.73E+00 1.37E+01
SD 5.22E+00 5.43E+00 4.50E+00 5.56E+00 6.05E+00
Best 1.09E+00 1.09E+00 2.17E+00 1.09E+00 1.09E+00
Worst 1.38E+01 1.38E+01 2.00E+01 1.38E+01 2.50E+01
$\textbf{F15: Kowalik}$ Mean 6.26E-03 5.54E-03 2.32E-03 4.38E-04 1.53E-03
SD 9.79E-03 9.36E-03 5.66E-03 3.21E-04 6.23E-03
Best 3.36E-04 3.36E-04 3.35E-04 3.36E-04 3.35E-04
Worst 2.22E-02 2.22E-02 2.29E-02 1.74E-03 3.45E-02
$\textbf{F16: Six-hump }\; \textbf{Camel-Back}$ Mean -1.13E+00 -1.13E+00 -1.12E+00 -1.13E+00 -1.12E+00
SD 3.15E-08 9.19E-12 8.76E-03 4.23E-08 8.76E-03
Best -1.13E+00 -1.13E+00 -1.13E+00 -1.13E+00 -1.13E+00
Worst -1.13E+00 -1.13E+00 -1.09E+00 -1.13E+00 -1.09E+00
$\textbf{F17: Branin}$ Mean 4.34E-01 4.34E-01 5.22E-01 4.34E-01 4.34E-01
SD 4.68E-06 3.05E-04 4.82E-01 1.62E-06 3.12E-07
Best 4.34E-01 4.34E-01 4.34E-01 4.34E-01 4.34E-01
Worst 4.34E-01 4.35E-01 3.07E+00 4.34E-01 4.34E-01
$\textbf{F18: Goldstein-Price}$ Mean 3.27E+00 3.27E+00 4.26E+00 3.27E+00 1.41E+01
SD 7.76E-05 5.59E-05 5.38E+00 6.06E-05 1.44E+01
Best 3.27E+00 3.27E+00 3.27E+00 3.27E+00 3.27E+00
Worst 3.27E+00 3.27E+00 3.27E+01 3.27E+00 3.27E+01
$\textbf{F19: Hartman 3}$ Mean -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
SD 2.19E-03 2.30E-03 6.26E-04 4.28E-05 7.99E-06
Best -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
Worst -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00 -4.21E+00
$\textbf{F20: Hartman 6}$ Mean -3.53E+00 -3.57E+00 -3.59E+00 -3.62E+00 -3.62E+00
SD 9.68E-02 6.93E-02 6.05E-02 2.38E-02 3.29E-02
Best -3.62E+00 -3.62E+00 -3.62E+00 -3.62E+00 -3.62E+00
Worst -3.37E+00 -3.42E+00 -3.49E+00 -3.49E+00 -3.49E+00
$\textbf{F21: Shekel5}$ Mean -1.02E+01 -1.03E+01 -7.45E+00 -9.52E+00 -5.84E+00
SD 2.10E+00 1.91E+00 4.00E+00 2.67E+00 3.43E+00
Best -1.11E+01 -1.11E+01 -1.11E+01 -1.11E+01 -1.11E+01
Worst -5.52E+00 -5.52E+00 -2.87E+00 -2.93E+00 -1.48E+00
$\textbf{F22: Shekel7}$ Mean -1.13E+01 -1.14E+01 -7.49E+00 -1.00E+01 -5.89E+00
SD 9.28E-04 2.21E-07 4.20E+00 2.48E+00 3.30E+00
Best -1.14E+01 -1.14E+01 -1.14E+01 -1.14E+01 -1.14E+01
Worst -1.13E+01 -1.14E+01 -3.00E+00 -5.55E+00 -1.53E+00
$\textbf{F23: Shekel10}$ Mean -1.11E+01 -1.13E+01 -8.14E+00 -1.03E+01 -7.10E+00
SD 1.50E+00 1.07E+00 4.20E+00 2.51E+00 4.02E+00
Best -1.15E+01 -1.15E+01 -1.15E+01 -1.15E+01 -1.15E+01
Worst -5.60E+00 -5.65E+00 -2.64E+00 -4.09E+00 -1.86E+00
Table 3.  Wilcoxon paired signed ranks test for the test system
Algorithm p-value $ +/-/\backsim $
Light Load
(Pe1=Pe2=0.3& Qe1=Qe2=0.1)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 1.7344e-06 +
Nominal Load
(Pe1=Pe2=0.8& Qe1=Qe2=0.6)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 1.7344e-06 +
Heavy Load
(Pe1=Pe2=1.3& Qe1=Qe2=1.0)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 7.7122e-04 +
Algorithm p-value $ +/-/\backsim $
Light Load
(Pe1=Pe2=0.3& Qe1=Qe2=0.1)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 1.7344e-06 +
Nominal Load
(Pe1=Pe2=0.8& Qe1=Qe2=0.6)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 1.7344e-06 +
Heavy Load
(Pe1=Pe2=1.3& Qe1=Qe2=1.0)
AGWO-PSO Versus MFO 1.7344e-06 +
AGWO-PSO Versus GWO 1.7344e-06 +
AGWO-PSO Versus PSO 1.7344e-06 +
AGWO-PSO Versus AGWO 7.7122e-04 +
Table 4.  Tuned parameters of controller parameters with the proposed metaheuristic optimization algorithms
T11 T21 K1 T12 T22 K2 me de TOE Jmin
Light Load MFO 2 0.353621 17.1851 0.896599 2 50 0.877263 0 21.1012 20634.638
Pe1=Pe2=0.3 & GWO 1.99226 0.631084 20.0048 2 0.408557 7.68251 1 0.553048 19.8576 19744.427
PSO 1.99616 0.531169 17.1061 1.99706 0.376092 7.67036 0.93685 0.471697 22.568 19790.2423
AGWO 2 0.814356 20.0335 0.223326 0.349201 0.145617 1 0.606624 20.511 19749.174
Qe1=Qe2=0.1 AGWO-PSO 1.91829 0.315199 16.476 2 0.367394 7.72613 0.759916 0 21.5609 19657.1023
Nominal Load MFO 2 0.34587 6.5268 2 0.23173 4.0409 0.71636 1 25.2134 20069.6872
Pe1=Pe2=0.8 & GWO 2 0.20802 4.8299 2 0.18249 4.344 1 0.00080834 21.9609 19897.141
PSO 2 0.34586 6.5267 2 0.23173 4.0409 0.71631 0.99996 23.2164 19918.5198
AGWO 2 0.30786 5.9268 2 0.22043 2.9841 0.71539 1 20.9591 19457.3614
Qe1=Qe2=0.6 AGWO-PSO 2 0.21304 5.271 2 0.20313 5.4271 1 0 20.5927 19425.0108
Heavy Load MFO 2 0.12956 3.698 2 0.1436 5.036 1 0 21.3626 20880.619
Pe1=Pe2=1.3 & GWO 2 0.12949 3.7016 2 0.14394 5.0545 1 0.00074374 20.4907 20388.983
PSO 0.709342 0.168539 19.6737 0.69435 0.174195 19.1828 0.235767 0 22.1254 20396.0086
AGWO 2 0.13122 3.7291 2 0.13306 4.9783 1 0 20.5043 20387.3276
Qe1=Qe2=1.0 AGWO-PSO 2 0.12329 3.5156 2 0.14564 5.9558 1 0 22.1768 20387.1437
T11 T21 K1 T12 T22 K2 me de TOE Jmin
Light Load MFO 2 0.353621 17.1851 0.896599 2 50 0.877263 0 21.1012 20634.638
Pe1=Pe2=0.3 & GWO 1.99226 0.631084 20.0048 2 0.408557 7.68251 1 0.553048 19.8576 19744.427
PSO 1.99616 0.531169 17.1061 1.99706 0.376092 7.67036 0.93685 0.471697 22.568 19790.2423
AGWO 2 0.814356 20.0335 0.223326 0.349201 0.145617 1 0.606624 20.511 19749.174
Qe1=Qe2=0.1 AGWO-PSO 1.91829 0.315199 16.476 2 0.367394 7.72613 0.759916 0 21.5609 19657.1023
Nominal Load MFO 2 0.34587 6.5268 2 0.23173 4.0409 0.71636 1 25.2134 20069.6872
Pe1=Pe2=0.8 & GWO 2 0.20802 4.8299 2 0.18249 4.344 1 0.00080834 21.9609 19897.141
PSO 2 0.34586 6.5267 2 0.23173 4.0409 0.71631 0.99996 23.2164 19918.5198
AGWO 2 0.30786 5.9268 2 0.22043 2.9841 0.71539 1 20.9591 19457.3614
Qe1=Qe2=0.6 AGWO-PSO 2 0.21304 5.271 2 0.20313 5.4271 1 0 20.5927 19425.0108
Heavy Load MFO 2 0.12956 3.698 2 0.1436 5.036 1 0 21.3626 20880.619
Pe1=Pe2=1.3 & GWO 2 0.12949 3.7016 2 0.14394 5.0545 1 0.00074374 20.4907 20388.983
PSO 0.709342 0.168539 19.6737 0.69435 0.174195 19.1828 0.235767 0 22.1254 20396.0086
AGWO 2 0.13122 3.7291 2 0.13306 4.9783 1 0 20.5043 20387.3276
Qe1=Qe2=1.0 AGWO-PSO 2 0.12329 3.5156 2 0.14564 5.9558 1 0 22.1768 20387.1437
Table 5.  ANOVA test for the test system under different loading conditions
Table 6.  System Eigenvalues & Damping ratios under various loading conditions with the proposed optimization algorithms
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