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doi: 10.3934/jimo.2021208
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Multi-objective optimization of multi-microgrid power dispatch under uncertainties using interval optimization

Shungen Luo 1, and  Xiuping Guo 1,2,,

1. 

School of Economics and Management, Southwest Jiaotong University, Chengdu, China
2. 

State key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications, Beijing, China

*Corresponding author: Xiuping Guo

Received  March 2021 Revised  August 2021 Early access December 2021

Fund Project: This work is partially supported by the National Natural Science Foundation of China [grant number 71471151] and by the Open Foundation of State key Laboratory of Networking and Switching Technology (Beijing University of Posts and Telecommunications) [SKLNST-2021-2-01]

The microgrid technology, which can dispatch power independently, is an effective way to increase the efficiency of energy utilization meanwhile develop and utilize the clean and renewable energy. However, the power generation of a single microgrid is unstable, because it is greatly affected by the external environment. Therefore, the development and application of the multi-microgrid system have gradually drawn various countries' attention. In order to minimize the operating cost and gaseous pollutant emission of the multi-microgrid system, which is composed of renewable energies and electric vehicles and so on, this paper builds a 24 hours day-ahead multi-objective complex constrained optimization model, using interval optimization to handle uncertainties of renewable energies. In view of the model characteristics, the metaheuristic strategies about initialization and repair of solution are designed. Furthermore, the fuzzy membership degree and Chebyshev function are used in parallel to decompose the multi-objective optimization problem, thus a multi-objective evolutionary algorithm based on hybrid decomposition (MOEA/HD) is constructed. Finally, the effectiveness of the metaheuristic strategies can be verified by analyzing the simulation results in this paper. Moreover, the results prove that the MOEA/HD is more efficient, which can get a higher-quality Pareto optimal solution set when compared to other algorithms.

Keywords: Power dispatch, interval optimization, multi-microgrid system, multi-objective optimization, uncertainty.
Mathematics Subject Classification: Primary: 49M37; Secondary: 68W50.
Citation: Shungen Luo, Xiuping Guo. Multi-objective optimization of multi-microgrid power dispatch under uncertainties using interval optimization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021208
References:
[1]

A. Akhil, The CERTS microgrid concept, White Paper for Transmission Reliability Program, U.S, 2002. Google Scholar

[2]

R. Baldick, B. H. Kim, C. Chase and Y. Luo, A fast distributed implementaion of optimal power flow, IEEE Transactions on Power Systems, 14 (1999), 858-864.   Google Scholar

[3]

F. Barbir and T. Gómez, Efficiency and economics of proton exchange membrane (PEM) fuel cells, International Journal of Hydrogen Energy, 21 (1996), 891-901.   Google Scholar

[4]

A. Chaouachi, R. M. Kamel, R. Andoulsi and K. Nagasaka, Multiobjective intelligent energy management for a microgrid, IEEE Transactions on Industrial Electronics, 60 (2013), 1688-1699.  doi: 10.1109/TIE.2012.2188873.  Google Scholar

[5]

K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, Goldberg. Wiley-Interscience Series in Systems and Optimization. John Wiley & Sons, Ltd., Chichester, 2001.  Google Scholar

[6]

K. Deb and H. Beyer, Self-adaptive genetic algorithms with simulated binary crossover, Evolutionary Computation, 9 (2001), 197-221.  doi: 10.1162/106365601750190406.  Google Scholar

[7]

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multi objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.   Google Scholar

[8]

F. Facchinei, A. Fischer and M. Herrich, An LP-Newton method: Nonsmooth equations, KKT systems, and nonisolated solutions, Math. Program., 146 (2014), 1-36.  doi: 10.1007/s10107-013-0676-6.  Google Scholar

[9]

H. Farzin, M. Fotuhi-Firuzabad and M. Moeini-Aghtaie, Developing a hierarchical scheme for outage management in multi-microgrids, IEEE Power Tech Conference, IEEE, 2015. doi: 10.1109/PTC.2015.7232575.  Google Scholar

[10]

N. Gil and J. Lopes, Hierarchical frequency control scheme for islanded multi-microgrids operation, 2007 IEEE Lausanne Power Tech, 2007. doi: 10.1109/PCT.2007.4538363.  Google Scholar

[11]

T. Gjengedal, S. Johansen and O. Hansen, A qualitative approach to economic-environment dispatch-treatment of multiple pollutants, IEEE Transactions on Energy Conversion, 7 (1992), 367-373.  doi: 10.1109/60.148554.  Google Scholar

[12]

C. Huang, D. Yue, S. Deng and J. Xie, Optimal scheduling of microgrid with multiple distributed resources using interval optimization, Energies, 10 (2017), 399-422.  doi: 10.3390/en10030339.  Google Scholar

[13]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, European J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[14]

P. Kou, D. Liang, L. Gao and F. Gao, Stochastic coordination of plug-in electric vehicles and wind turbines in microgrid: A model predictive control approach, IEEE Trans. Smart Grid, 7 (2016), 1537-1551.  doi: 10.1109/TSG.2015.2475316.  Google Scholar

[15]

X. Lu, K. Zhou and S. Yang, Multi-objective optimal dispatch of microgrid containing electric vehicles, Journal of Cleaner Production, 165 (2017), 1572-1581.  doi: 10.1016/j.jclepro.2017.07.221.  Google Scholar

[16]

T. Lv, Q. Ai and Y. Zhao, A bi-level multi-objective optimal operation of grid-connected microgrids, Electric Power Systems Research, 131 (2016), 60-70.  doi: 10.1016/j.epsr.2015.09.018.  Google Scholar

[17]

A. A. Moghaddam, A. Seifi, T. Niknam and M. R. Pahlavani, Multi-objective operation management of a renewable MG (micro-grid) with back-up micro-turbine/fuel cell/battery hybrid power source, Energy, 36 (2011), 6490-6507.  doi: 10.1016/j.energy.2011.09.017.  Google Scholar

[18]

F. A. Mohamed and H. N. Koivo, System modelling and online optimal management of microgrid using mesh adaptive direct search, International Journal of Electrical Power and Energy Systems, 32 (2010), 398-407.  doi: 10.1016/j.ijepes.2009.11.003.  Google Scholar

[19]

M. H. Moradi, M. Abedini and S. M. Hosseinian, Optimal operation of autonomous microgrid using HS–GA, International Journal of Electrical Power and Energy Systems, 77 (2016), 210-220.  doi: 10.1016/j.ijepes.2015.11.043.  Google Scholar

[20]

M. Muslu, Economic dispatch with environmental considerations: Trade off curves and emission reduction rates, Electric Power Systems Research, 71 (2004), 153-158.  doi: 10.1016/j.epsr.2004.01.009.  Google Scholar

[21]

Y. Qi, X. Ma, F. Liu, L. Jiao, J. Sun and J. Wu, MOEA/D with adaptive weight adjustment, Evolutionary Computation, 22 (2014), 231-264.  doi: 10.1162/EVCO_a_00109.  Google Scholar

[22]

J. Radosavljević, M. Jevtić and D. Klimenta, Energy and operation management of a microgrid using particle swarm optimization, Engineering Optimization, 48 (2016), 811-830.   Google Scholar

[23]

W. Saad, Z. Han and H. V. Poor, Coalitional game theory for cooperative micro-grid distribution networks, IEEE International Conf on Communications Workshops, IEEE, 2011. doi: 10.1109/iccw.2011.5963577.  Google Scholar

[24]

D. Saéz, F. A$\acute{\rm{v}}$ila, D. Olivares, C. Canizares and L. Mariń, Fuzzy prediction interval models for forecasting renewable resources and loads in microgrids, IEEE Transactions on Smart Grid, 6 (2015), 548-556.   Google Scholar

[25]

W. F. Tinney and C. E. Hart, Power flow solution by newton's method, IEEE Transactions on Power Apparatus and Systems, 86 (1967), 1449-1460.  doi: 10.1109/TPAS.1967.291823.  Google Scholar

[26]

J. Vasiljevska, J. P. Lopes and M. Matos, Multi-microgrid impact assessment using multi criteria decision aid methods, IEEE Power Tech Conference, IEEE, 2009. doi: 10.1109/PTC.2009.5282054.  Google Scholar

[27]

D. V. Veldhuizen and G. B. Lamont, On measuring multiobjective evolutionary algorithm performance, Proceedings of Evolutionary Computation, 2000. doi: 10.1109/CEC.2000.870296.  Google Scholar

[28]

J. L. Verdagay, Fuzzy Mathematical Programming, North-holland, Amsterdam, 1982. Google Scholar

[29]

S. Wang, X. Fan, L. Han and L. Ge, Improved interval optimization method based on differential evolution for microgrid economic dispatch, Electric Machines and Power Systems, 43 (2015), 1882-1890.  doi: 10.1080/15325008.2015.1057783.  Google Scholar

[30]

Y. Wang, Q. Xia and C. Kang, Unit commitment with volatile node injections by using interval optimization, IEEE Transactions on Power Systems, 26 (2011), 1705-1713.  doi: 10.1109/TPWRS.2010.2100050.  Google Scholar

[31]

C. Wei, Z. M. Fadlullah, N. Kato and A. Takeuchi, GT-CFS: A game theoretic coalition formulation strategy for reducing power loss in micro grids, IEEE Trans on Parallel and Distributed Systems, 25 (2014), 2307-2317.  doi: 10.1109/TPDS.2013.178.  Google Scholar

[32]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, John Wiley and Sons, New York, 1996. Google Scholar

[33]

H. Wu, X. Liu and M. Ding, Dynamic economic dispatch of a microgrid: Mathematical models and solution algorithm, International Journal of Electrical Power and Energy Systems, 63 (2014), 336-346.  doi: 10.1016/j.ijepes.2014.06.002.  Google Scholar

[34]

Q. Xiao, X. Guo and D. Li, Partial disassembly line balancing under uncertainty: Robust optimization models and an improved migrating birds optimization algorithm, International Journal of Production Research, 59 (2021), 2977-2995.  doi: 10.1080/00207543.2020.1744765.  Google Scholar

[35]

N. Yu, J. S. Kang, C. C. Chang, T. Y. Lee and D. Y. Lee, Robust economic optimization and environmental policy analysis for microgrid planning: An application to Taichung Industrial Park, Taiwan, Energy, 113 (2016), 671-682.  doi: 10.1016/j.energy.2016.07.066.  Google Scholar

[36]

Q. Zhang and H. Li, MOEA/D: A multi objective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731.   Google Scholar

[37]

Q. Zhang, H. Li, D. Maringer and E. Tsang, MOEA/D with NBI-style chebyshev approach for portfolio management, IEEE Congress on Evolutionary Computation, 2010. Google Scholar

[38]

E. Zitzler and L. Thiele, Multi-objective evolutionary algorithms: A comparative case study and the strength pareto approach, IEEE Transactions on Evolutionary Computation, 3 (1999), 257-271.   Google Scholar

[39]

E. V. Zyl and A. P. Engelbrecht, A subspace-based method for PSO initialization, IEEE Symposium Series on Computational Intelligence. IEEE, 2016. Google Scholar

show all references

References:
[1]

A. Akhil, The CERTS microgrid concept, White Paper for Transmission Reliability Program, U.S, 2002. Google Scholar

[2]

R. Baldick, B. H. Kim, C. Chase and Y. Luo, A fast distributed implementaion of optimal power flow, IEEE Transactions on Power Systems, 14 (1999), 858-864.   Google Scholar

[3]

F. Barbir and T. Gómez, Efficiency and economics of proton exchange membrane (PEM) fuel cells, International Journal of Hydrogen Energy, 21 (1996), 891-901.   Google Scholar

[4]

A. Chaouachi, R. M. Kamel, R. Andoulsi and K. Nagasaka, Multiobjective intelligent energy management for a microgrid, IEEE Transactions on Industrial Electronics, 60 (2013), 1688-1699.  doi: 10.1109/TIE.2012.2188873.  Google Scholar

[5]

K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, Goldberg. Wiley-Interscience Series in Systems and Optimization. John Wiley & Sons, Ltd., Chichester, 2001.  Google Scholar

[6]

K. Deb and H. Beyer, Self-adaptive genetic algorithms with simulated binary crossover, Evolutionary Computation, 9 (2001), 197-221.  doi: 10.1162/106365601750190406.  Google Scholar

[7]

K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, A fast and elitist multi objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation, 6 (2002), 182-197.   Google Scholar

[8]

F. Facchinei, A. Fischer and M. Herrich, An LP-Newton method: Nonsmooth equations, KKT systems, and nonisolated solutions, Math. Program., 146 (2014), 1-36.  doi: 10.1007/s10107-013-0676-6.  Google Scholar

[9]

H. Farzin, M. Fotuhi-Firuzabad and M. Moeini-Aghtaie, Developing a hierarchical scheme for outage management in multi-microgrids, IEEE Power Tech Conference, IEEE, 2015. doi: 10.1109/PTC.2015.7232575.  Google Scholar

[10]

N. Gil and J. Lopes, Hierarchical frequency control scheme for islanded multi-microgrids operation, 2007 IEEE Lausanne Power Tech, 2007. doi: 10.1109/PCT.2007.4538363.  Google Scholar

[11]

T. Gjengedal, S. Johansen and O. Hansen, A qualitative approach to economic-environment dispatch-treatment of multiple pollutants, IEEE Transactions on Energy Conversion, 7 (1992), 367-373.  doi: 10.1109/60.148554.  Google Scholar

[12]

C. Huang, D. Yue, S. Deng and J. Xie, Optimal scheduling of microgrid with multiple distributed resources using interval optimization, Energies, 10 (2017), 399-422.  doi: 10.3390/en10030339.  Google Scholar

[13]

C. Jiang, X. Han, G. R. Liu and G. P. Liu, A nonlinear interval number programming method for uncertain optimization problems, European J. Oper. Res., 188 (2008), 1-13.  doi: 10.1016/j.ejor.2007.03.031.  Google Scholar

[14]

P. Kou, D. Liang, L. Gao and F. Gao, Stochastic coordination of plug-in electric vehicles and wind turbines in microgrid: A model predictive control approach, IEEE Trans. Smart Grid, 7 (2016), 1537-1551.  doi: 10.1109/TSG.2015.2475316.  Google Scholar

[15]

X. Lu, K. Zhou and S. Yang, Multi-objective optimal dispatch of microgrid containing electric vehicles, Journal of Cleaner Production, 165 (2017), 1572-1581.  doi: 10.1016/j.jclepro.2017.07.221.  Google Scholar

[16]

T. Lv, Q. Ai and Y. Zhao, A bi-level multi-objective optimal operation of grid-connected microgrids, Electric Power Systems Research, 131 (2016), 60-70.  doi: 10.1016/j.epsr.2015.09.018.  Google Scholar

[17]

A. A. Moghaddam, A. Seifi, T. Niknam and M. R. Pahlavani, Multi-objective operation management of a renewable MG (micro-grid) with back-up micro-turbine/fuel cell/battery hybrid power source, Energy, 36 (2011), 6490-6507.  doi: 10.1016/j.energy.2011.09.017.  Google Scholar

[18]

F. A. Mohamed and H. N. Koivo, System modelling and online optimal management of microgrid using mesh adaptive direct search, International Journal of Electrical Power and Energy Systems, 32 (2010), 398-407.  doi: 10.1016/j.ijepes.2009.11.003.  Google Scholar

[19]

M. H. Moradi, M. Abedini and S. M. Hosseinian, Optimal operation of autonomous microgrid using HS–GA, International Journal of Electrical Power and Energy Systems, 77 (2016), 210-220.  doi: 10.1016/j.ijepes.2015.11.043.  Google Scholar

[20]

M. Muslu, Economic dispatch with environmental considerations: Trade off curves and emission reduction rates, Electric Power Systems Research, 71 (2004), 153-158.  doi: 10.1016/j.epsr.2004.01.009.  Google Scholar

[21]

Y. Qi, X. Ma, F. Liu, L. Jiao, J. Sun and J. Wu, MOEA/D with adaptive weight adjustment, Evolutionary Computation, 22 (2014), 231-264.  doi: 10.1162/EVCO_a_00109.  Google Scholar

[22]

J. Radosavljević, M. Jevtić and D. Klimenta, Energy and operation management of a microgrid using particle swarm optimization, Engineering Optimization, 48 (2016), 811-830.   Google Scholar

[23]

W. Saad, Z. Han and H. V. Poor, Coalitional game theory for cooperative micro-grid distribution networks, IEEE International Conf on Communications Workshops, IEEE, 2011. doi: 10.1109/iccw.2011.5963577.  Google Scholar

[24]

D. Saéz, F. A$\acute{\rm{v}}$ila, D. Olivares, C. Canizares and L. Mariń, Fuzzy prediction interval models for forecasting renewable resources and loads in microgrids, IEEE Transactions on Smart Grid, 6 (2015), 548-556.   Google Scholar

[25]

W. F. Tinney and C. E. Hart, Power flow solution by newton's method, IEEE Transactions on Power Apparatus and Systems, 86 (1967), 1449-1460.  doi: 10.1109/TPAS.1967.291823.  Google Scholar

[26]

J. Vasiljevska, J. P. Lopes and M. Matos, Multi-microgrid impact assessment using multi criteria decision aid methods, IEEE Power Tech Conference, IEEE, 2009. doi: 10.1109/PTC.2009.5282054.  Google Scholar

[27]

D. V. Veldhuizen and G. B. Lamont, On measuring multiobjective evolutionary algorithm performance, Proceedings of Evolutionary Computation, 2000. doi: 10.1109/CEC.2000.870296.  Google Scholar

[28]

J. L. Verdagay, Fuzzy Mathematical Programming, North-holland, Amsterdam, 1982. Google Scholar

[29]

S. Wang, X. Fan, L. Han and L. Ge, Improved interval optimization method based on differential evolution for microgrid economic dispatch, Electric Machines and Power Systems, 43 (2015), 1882-1890.  doi: 10.1080/15325008.2015.1057783.  Google Scholar

[30]

Y. Wang, Q. Xia and C. Kang, Unit commitment with volatile node injections by using interval optimization, IEEE Transactions on Power Systems, 26 (2011), 1705-1713.  doi: 10.1109/TPWRS.2010.2100050.  Google Scholar

[31]

C. Wei, Z. M. Fadlullah, N. Kato and A. Takeuchi, GT-CFS: A game theoretic coalition formulation strategy for reducing power loss in micro grids, IEEE Trans on Parallel and Distributed Systems, 25 (2014), 2307-2317.  doi: 10.1109/TPDS.2013.178.  Google Scholar

[32]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, John Wiley and Sons, New York, 1996. Google Scholar

[33]

H. Wu, X. Liu and M. Ding, Dynamic economic dispatch of a microgrid: Mathematical models and solution algorithm, International Journal of Electrical Power and Energy Systems, 63 (2014), 336-346.  doi: 10.1016/j.ijepes.2014.06.002.  Google Scholar

[34]

Q. Xiao, X. Guo and D. Li, Partial disassembly line balancing under uncertainty: Robust optimization models and an improved migrating birds optimization algorithm, International Journal of Production Research, 59 (2021), 2977-2995.  doi: 10.1080/00207543.2020.1744765.  Google Scholar

[35]

N. Yu, J. S. Kang, C. C. Chang, T. Y. Lee and D. Y. Lee, Robust economic optimization and environmental policy analysis for microgrid planning: An application to Taichung Industrial Park, Taiwan, Energy, 113 (2016), 671-682.  doi: 10.1016/j.energy.2016.07.066.  Google Scholar

[36]

Q. Zhang and H. Li, MOEA/D: A multi objective evolutionary algorithm based on decomposition, IEEE Transactions on Evolutionary Computation, 11 (2007), 712-731.   Google Scholar

[37]

Q. Zhang, H. Li, D. Maringer and E. Tsang, MOEA/D with NBI-style chebyshev approach for portfolio management, IEEE Congress on Evolutionary Computation, 2010. Google Scholar

[38]

E. Zitzler and L. Thiele, Multi-objective evolutionary algorithms: A comparative case study and the strength pareto approach, IEEE Transactions on Evolutionary Computation, 3 (1999), 257-271.   Google Scholar

[39]

E. V. Zyl and A. P. Engelbrecht, A subspace-based method for PSO initialization, IEEE Symposium Series on Computational Intelligence. IEEE, 2016. Google Scholar

Figure 1.  The framework of the multi-microgrid system
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Figure 2.  Three possible relations between an interval and a real number
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Figure 3.  The flow chart about the repair of ESU's power dispatch scheme
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Figure 4.  The flow chart about the repair of EV's power dispatch scheme
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Figure 5.  Power prediction intervals of PVs and WTs in 24 h
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Figure 6.  The objective values of the Pareto optimal solutions
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Figure 7.  The boxplot group of the $ CS $ metric values
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Figure 8.  A group of three-dimensional Surface of the mean of $ CS $ metric values
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Figure 9.  Run time of each algorithm
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Figure 10.  The scatter diagram of non-dominated set
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Figure 11.  Cost and emission interval in each period
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Figure 12.  Active power of DG, ESU and EV in each period
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Figure 13.  Energy exchange
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Table 1.  The difference between this paper and existing relevant research
Research field Differences
Interval Existing research   Existing studies on IO, such as literature [29], mostly don't consider the risk preference of decision makers.
Optimization This paper   Considering the risk preference of decision makers, this paper creates a multi-objective IO model with the possibility degree.
Handling Existing research   In existing studies, such as literature [16], the method of handling complex constraints is lack of detailed description
Constraints This paper   In this paper, the metaheuristic strategies about initialization and repair of solution are designed to deal with the complex constraints.
MOEA/D Existing research   The existing studies, such as literature [36], mostly use Chebyshev function as the decomposition strategy, and do not apply hybrid decomposition strategy to update the solution.
This paper   In this paper, Chebyshev function and fuzzy membership are used in parallel to improve the conventional MOEA/D to MOEA/HD to solve the complex constrained model.
Table Options

Download as excel

Research field Differences
Interval Existing research   Existing studies on IO, such as literature [29], mostly don't consider the risk preference of decision makers.
Optimization This paper   Considering the risk preference of decision makers, this paper creates a multi-objective IO model with the possibility degree.
Handling Existing research   In existing studies, such as literature [16], the method of handling complex constraints is lack of detailed description
Constraints This paper   In this paper, the metaheuristic strategies about initialization and repair of solution are designed to deal with the complex constraints.
MOEA/D Existing research   The existing studies, such as literature [36], mostly use Chebyshev function as the decomposition strategy, and do not apply hybrid decomposition strategy to update the solution.
This paper   In this paper, Chebyshev function and fuzzy membership are used in parallel to improve the conventional MOEA/D to MOEA/HD to solve the complex constrained model.
Table 2.  The real-time market prices
Periods $ {\boldsymbol{S_{\rm{Mac}}}} $(CNY/kwh)
00:00-08:00 0.3
08:00-12:00 0.95
12:00-17:00 0.56
17:00-21:00 0.95
21:00-24:00 0.56
Table Options

Download as excel

Periods $ {\boldsymbol{S_{\rm{Mac}}}} $(CNY/kwh)
00:00-08:00 0.3
08:00-12:00 0.95
12:00-17:00 0.56
17:00-21:00 0.95
21:00-24:00 0.56
Table 3.  The parameters of PVs and WTs
Bus No. Name $ {\boldsymbol{P_{rate}}} $(kW) $ {\boldsymbol{Q_{min}}} $(kvar) $ {\boldsymbol{Q_{max}}} $(kvar)
3 PV1 30 -15 15
5 WT1 35 -10 10
7 WT2 35 -10 10
8 PV2 30 -15 15
10 WT3 45 -10 10
11 PV3 30 -15 15
12 PV4 40 -20 20
13 WT4 45 -10 10
14 PV5 40 -20 20
Table Options

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Bus No. Name $ {\boldsymbol{P_{rate}}} $(kW) $ {\boldsymbol{Q_{min}}} $(kvar) $ {\boldsymbol{Q_{max}}} $(kvar)
3 PV1 30 -15 15
5 WT1 35 -10 10
7 WT2 35 -10 10
8 PV2 30 -15 15
10 WT3 45 -10 10
11 PV3 30 -15 15
12 PV4 40 -20 20
13 WT4 45 -10 10
14 PV5 40 -20 20
Table 4.  The parameters of MOEA/HD
Control parameter Combination Number
1 2 3 4 5 6 7 8 9
CN 100 100 100 200 200 200 300 300 300
ps 0.6 0.8 0.9 0.6 0.8 0.9 0.6 0.8 0.9
pm 0.05 0.08 0.1 0.08 0.1 0.05 0.1 0.05 0.08
Z 20 50 100 100 20 50 50 100 20
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Control parameter Combination Number
1 2 3 4 5 6 7 8 9
CN 100 100 100 200 200 200 300 300 300
ps 0.6 0.8 0.9 0.6 0.8 0.9 0.6 0.8 0.9
pm 0.05 0.08 0.1 0.08 0.1 0.05 0.1 0.05 0.08
Z 20 50 100 100 20 50 50 100 20
Table 5.  $ Sa $ metric value
Possibility degree $ \xi $=0 $ {\boldsymbol{Sa }}$ metric value
2MGs 3MGs 4MGs 5MGs 6MGs 7MGs 8MGs 9MGs 10MGs
MOEA/CD 0.341 0.442 0.556 0.384 0.553 0.607 0.490 0.418 0.535
Best MOEA/FD 0.429 0.486 0.545 0.545 0.475 0.476 0.498 0.623 0.454
value NSGAII 0.549 0.562 0.684 0.559 0.442 0.745 0.554 0.608 0.526
MOEA/HD 0.365 0.382 0.460 0.425 0.441 0.502 0.559 0.420 0.413
MOEA/CD 0.598 0.731 0.662 0.758 0.808 0.795 0.547 0.540 0.599
Worst MOEA/FD 0.462 0.622 0.583 0.682 0.587 0.592 0.601 0.640 0.626
value NSGAII 0.648 0.737 1.110 0.606 1.058 0.875 0.742 1.148 0.590
MOEA/HD 0.485 0.487 0.555 0.518 0.499 0.675 0.580 0.711 0.574
MOEA/CD 0.428 0.588 0.606 0.582 0.713 0.679 0.513 0.492 0.576
Mean MOEA/FD 0.443 0.560 0.567 0.620 0.540 0.516 0.552 0.633 0.551
value NSGAII 0.597 0.665 0.830 0.581 0.795 0.791 0.652 0.903 0.551
MOEA/HD 0.433 0.434 0.504 0.463 0.467 0.562 0.570 0.613 0.509
MOEA/CD 0.120 0.118 0.043 0.154 0.113 0.083 0.024 0.053 0.029
Standard MOEA/FD 0.014 0.056 0.016 0.057 0.048 0.054 0.042 0.007 0.072
deviation NSGAII 0.040 0.075 0.199 0.019 0.259 0.059 0.077 0.223 0.028
MOEA/HD 0.050 0.043 0.039 0.040 0.024 0.080 0.008 0.137 0.069
${\boldsymbol{ \xi}} $=0.5
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Best MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.456 0.561 0.600 0.796 0.543 0.685 0.726 0.784 0.572
Worst MOEA/FD 0.538 0.570 0.910 0.555 0.687 0.590 0.789 0.824 0.907
value NSGAII 0.611 0.790 0.750 1.170 0.780 0.686 1.062 0.704 0.829
MOEA/HD 0.699 0.494 0.576 0.731 0.578 0.516 0.647 0.649 0.757
MOEA/CD 0.447 0.456 0.527 0.690 0.525 0.633 0.641 0.632 0.552
Mean MOEA/FD 0.491 0.547 0.764 0.490 0.537 0.569 0.635 0.632 0.666
value NSGAII 0.574 0.658 0.715 0.951 0.637 0.634 0.839 0.691 0.708
MOEA/HD 0.570 0.485 0.500 0.597 0.468 0.459 0.559 0.550 0.616
MOEA/CD 0.009 0.101 0.058 0.080 0.017 0.056 0.075 0.117 0.015
Standard MOEA/FD 0.058 0.016 0.153 0.056 0.122 0.024 0.109 0.150 0.178
deviation NSGAII 0.033 0.177 0.026 0.167 0.125 0.051 0.166 0.010 0.144
MOEA/HD 0.105 0.012 0.057 0.118 0.079 0.052 0.066 0.082 0.130
$ {\boldsymbol{\xi }}$=1
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Best MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.456 0.561 0.600 0.796 0.543 0.685 0.726 0.784 0.572
Worst MOEA/FD 0.538 0.570 0.910 0.555 0.687 0.590 0.789 0.824 0.907
value NSGAII 0.611 0.790 0.750 1.170 0.780 0.686 1.062 0.704 0.829
MOEA/HD 0.699 0.494 0.576 0.731 0.578 0.516 0.647 0.649 0.757
MOEA/CD 0.447 0.456 0.527 0.690 0.525 0.633 0.641 0.632 0.552
Mean MOEA/FD 0.491 0.547 0.764 0.490 0.537 0.569 0.635 0.632 0.666
value NSGAII 0.574 0.658 0.715 0.951 0.637 0.634 0.839 0.691 0.708
MOEA/HD 0.570 0.485 0.500 0.597 0.468 0.459 0.559 0.550 0.616
MOEA/CD 0.009 0.101 0.058 0.080 0.017 0.056 0.075 0.117 0.015
Standard MOEA/FD 0.058 0.016 0.153 0.056 0.122 0.024 0.109 0.150 0.178
deviation NSGAII 0.033 0.177 0.026 0.167 0.125 0.051 0.166 0.010 0.144
MOEA/HD 0.105 0.012 0.057 0.118 0.079 0.052 0.066 0.082 0.130
Table Options

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Possibility degree $ \xi $=0 $ {\boldsymbol{Sa }}$ metric value
2MGs 3MGs 4MGs 5MGs 6MGs 7MGs 8MGs 9MGs 10MGs
MOEA/CD 0.341 0.442 0.556 0.384 0.553 0.607 0.490 0.418 0.535
Best MOEA/FD 0.429 0.486 0.545 0.545 0.475 0.476 0.498 0.623 0.454
value NSGAII 0.549 0.562 0.684 0.559 0.442 0.745 0.554 0.608 0.526
MOEA/HD 0.365 0.382 0.460 0.425 0.441 0.502 0.559 0.420 0.413
MOEA/CD 0.598 0.731 0.662 0.758 0.808 0.795 0.547 0.540 0.599
Worst MOEA/FD 0.462 0.622 0.583 0.682 0.587 0.592 0.601 0.640 0.626
value NSGAII 0.648 0.737 1.110 0.606 1.058 0.875 0.742 1.148 0.590
MOEA/HD 0.485 0.487 0.555 0.518 0.499 0.675 0.580 0.711 0.574
MOEA/CD 0.428 0.588 0.606 0.582 0.713 0.679 0.513 0.492 0.576
Mean MOEA/FD 0.443 0.560 0.567 0.620 0.540 0.516 0.552 0.633 0.551
value NSGAII 0.597 0.665 0.830 0.581 0.795 0.791 0.652 0.903 0.551
MOEA/HD 0.433 0.434 0.504 0.463 0.467 0.562 0.570 0.613 0.509
MOEA/CD 0.120 0.118 0.043 0.154 0.113 0.083 0.024 0.053 0.029
Standard MOEA/FD 0.014 0.056 0.016 0.057 0.048 0.054 0.042 0.007 0.072
deviation NSGAII 0.040 0.075 0.199 0.019 0.259 0.059 0.077 0.223 0.028
MOEA/HD 0.050 0.043 0.039 0.040 0.024 0.080 0.008 0.137 0.069
${\boldsymbol{ \xi}} $=0.5
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Best MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.456 0.561 0.600 0.796 0.543 0.685 0.726 0.784 0.572
Worst MOEA/FD 0.538 0.570 0.910 0.555 0.687 0.590 0.789 0.824 0.907
value NSGAII 0.611 0.790 0.750 1.170 0.780 0.686 1.062 0.704 0.829
MOEA/HD 0.699 0.494 0.576 0.731 0.578 0.516 0.647 0.649 0.757
MOEA/CD 0.447 0.456 0.527 0.690 0.525 0.633 0.641 0.632 0.552
Mean MOEA/FD 0.491 0.547 0.764 0.490 0.537 0.569 0.635 0.632 0.666
value NSGAII 0.574 0.658 0.715 0.951 0.637 0.634 0.839 0.691 0.708
MOEA/HD 0.570 0.485 0.500 0.597 0.468 0.459 0.559 0.550 0.616
MOEA/CD 0.009 0.101 0.058 0.080 0.017 0.056 0.075 0.117 0.015
Standard MOEA/FD 0.058 0.016 0.153 0.056 0.122 0.024 0.109 0.150 0.178
deviation NSGAII 0.033 0.177 0.026 0.167 0.125 0.051 0.166 0.010 0.144
MOEA/HD 0.105 0.012 0.057 0.118 0.079 0.052 0.066 0.082 0.130
$ {\boldsymbol{\xi }}$=1
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Best MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.456 0.561 0.600 0.796 0.543 0.685 0.726 0.784 0.572
Worst MOEA/FD 0.538 0.570 0.910 0.555 0.687 0.590 0.789 0.824 0.907
value NSGAII 0.611 0.790 0.750 1.170 0.780 0.686 1.062 0.704 0.829
MOEA/HD 0.699 0.494 0.576 0.731 0.578 0.516 0.647 0.649 0.757
MOEA/CD 0.447 0.456 0.527 0.690 0.525 0.633 0.641 0.632 0.552
Mean MOEA/FD 0.491 0.547 0.764 0.490 0.537 0.569 0.635 0.632 0.666
value NSGAII 0.574 0.658 0.715 0.951 0.637 0.634 0.839 0.691 0.708
MOEA/HD 0.570 0.485 0.500 0.597 0.468 0.459 0.559 0.550 0.616
MOEA/CD 0.009 0.101 0.058 0.080 0.017 0.056 0.075 0.117 0.015
Standard MOEA/FD 0.058 0.016 0.153 0.056 0.122 0.024 0.109 0.150 0.178
deviation NSGAII 0.033 0.177 0.026 0.167 0.125 0.051 0.166 0.010 0.144
MOEA/HD 0.105 0.012 0.057 0.118 0.079 0.052 0.066 0.082 0.130
Table 6.  MS metric value
Possibility degree $ {\boldsymbol{\xi }}$=0 ${\boldsymbol{ MS }}$ metric value
2MGs 3MGs 4MGs 5MGs 6MGs 7MGs 8MGs 9MGs 10MGs
MOEA/CD 0.948 0.857 0.940 0.948 0.929 0.895 0.820 0.885 0.906
Best MOEA/FD 0.958 0.953 0.881 1 0.976 1 0.899 0.927 0.910
result NSGAII 0.654 0.614 0.606 0.683 0.700 0.627 0.522 0.647 0.686
MOEA/HD 1 1 1 0.985 1 1 1 1 0.991
MOEA/CD 0.880 0.831 0.907 0.822 0.896 0.801 0.795 0.865 0.888
Worst MOEA/FD 0.935 0.868 0.832 0.944 0.914 0.942 0.782 0.899 0.857
result NSGAII 0.599 0.550 0.515 0.621 0.677 0.490 0.473 0.617 0.612
MOEA/HD 0.913 0.965 0.925 0.944 0.961 0.991 0.959 0.896 0.967
MOEA/CD 0.916 0.844 0.923 0.887 0.914 0.844 0.805 0.874 0.898
Mean MOEA/FD 0.946 0.923 0.854 0.974 0.942 0.967 0.823 0.909 0.889
value NSGAII 0.633 0.590 0.556 0.657 0.689 0.576 0.504 0.627 0.642
MOEA/HD 0.959 0.985 0.956 0.971 0.987 0.997 0.983 0.96 0.978
MOEA/CD 0.028 0.011 0.014 0.052 0.013 0.039 0.010 0.008 0.008
Standard MOEA/FD 0.009 0.038 0.020 0.023 0.025 0.024 0.054 0.013 0.023
deviation NSGAII 0.024 0.029 0.038 0.026 0.009 0.061 0.022 0.014 0.032
MOEA/HD 0.036 0.015 0.032 0.019 0.018 0.004 0.018 0.046 0.01
${\boldsymbol{ \xi }}$=0.5
MOEA/CD 0.884 0.901 0.882 0.901 0.945 0.838 0.921 0.912 0.913
Best MOEA/FD 0.870 1 1 1 0.919 0.975 0.956 1 0.987
value NSGAII 0.629 0.576 0.748 0.774 0.587 0.66 0.672 0.575 0.617
MOEA/HD 1 1 1 0.957 1 1 1 0.966 1
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Worst MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.864 0.832 0.869 0.876 0.872 0.822 0.871 0.895 0.905
Mean MOEA/FD 0.862 0.994 0.966 0.973 0.888 0.952 0.921 0.99 0.949
value NSGAII 0.610 0.543 0.726 0.720 0.554 0.647 0.604 0.560 0.601
MOEA/HD 0.987 0.974 0.942 0.937 0.987 0.971 0.968 0.960 0.975
MOEA/CD 0.017 0.051 0.011 0.018 0.052 0.012 0.039 0.020 0.051
Standard MOEA/FD 0.006 0.009 0.029 0.019 0.022 0.025 0.040 0.010 0.030
deviation NSGAII 0.017 0.025 0.017 0.038 0.031 0.013 0.051 0.011 0.012
MOEA/HD 0.018 0.028 0.049 0.014 0.018 0.041 0.034 0.004 0.035
${\boldsymbol{ \xi }}$=1
MOEA/CD 0.886 0.972 0.8794 0.882 0.963 0.780 0.889 0.950 0.784
Best MOEA/FD 0.909 0.921 0.954 1 0.990 0.890 0.987 1 1
value NSGAII 0.540 0.802 0.571 0.631 0.615 0.558 0.913 0.754 0.772
MOEA/HD 1 0.970 1 1 0.968 1 1 1 1
MOEA/CD 0.843 0.779 0.855 0.861 0.826 0.808 0.827 0.867 0.900
Worst MOEA/FD 0.858 0.982 0.930 0.957 0.869 0.917 0.865 0.976 0.912
value NSGAII 0.587 0.515 0.708 0.692 0.512 0.629 0.549 0.550 0.586
MOEA/HD 0.962 0.936 0.881 0.924 0.962 0.912 0.921 0.956 0.926
MOEA/CD 0.826 0.938 0.847 0.814 0.957 0.747 0.872 0.868 0.727
Mean MOEA/FD 0.903 0.902 0.899 0.929 0.966 0.871 0.964 0.998 0.902
value NSGAII 0.520 0.773 0.532 0.589 0.611 0.515 0.870 0.720 0.669
MOEA/HD 0.987 0.961 0.958 0.979 0.957 0.936 0.995 0.941 0.988
MOEA/CD 0.045 0.025 0.038 0.049 0.005 0.028 0.018 0.058 0.052
Standard MOEA/FD 0.005 0.025 0.041 0.060 0.0310 0.022 0.026 0.002 0.069
deviation NSGAII 0.017 0.022 0.034 0.030 0.004 0.031 0.030 0.0280 0.0930
MOEA/HD 0.018 0.007 0.036 0.021 0.014 0.056 0.005 0.044 0.011
Table Options

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Possibility degree $ {\boldsymbol{\xi }}$=0 ${\boldsymbol{ MS }}$ metric value
2MGs 3MGs 4MGs 5MGs 6MGs 7MGs 8MGs 9MGs 10MGs
MOEA/CD 0.948 0.857 0.940 0.948 0.929 0.895 0.820 0.885 0.906
Best MOEA/FD 0.958 0.953 0.881 1 0.976 1 0.899 0.927 0.910
result NSGAII 0.654 0.614 0.606 0.683 0.700 0.627 0.522 0.647 0.686
MOEA/HD 1 1 1 0.985 1 1 1 1 0.991
MOEA/CD 0.880 0.831 0.907 0.822 0.896 0.801 0.795 0.865 0.888
Worst MOEA/FD 0.935 0.868 0.832 0.944 0.914 0.942 0.782 0.899 0.857
result NSGAII 0.599 0.550 0.515 0.621 0.677 0.490 0.473 0.617 0.612
MOEA/HD 0.913 0.965 0.925 0.944 0.961 0.991 0.959 0.896 0.967
MOEA/CD 0.916 0.844 0.923 0.887 0.914 0.844 0.805 0.874 0.898
Mean MOEA/FD 0.946 0.923 0.854 0.974 0.942 0.967 0.823 0.909 0.889
value NSGAII 0.633 0.590 0.556 0.657 0.689 0.576 0.504 0.627 0.642
MOEA/HD 0.959 0.985 0.956 0.971 0.987 0.997 0.983 0.96 0.978
MOEA/CD 0.028 0.011 0.014 0.052 0.013 0.039 0.010 0.008 0.008
Standard MOEA/FD 0.009 0.038 0.020 0.023 0.025 0.024 0.054 0.013 0.023
deviation NSGAII 0.024 0.029 0.038 0.026 0.009 0.061 0.022 0.014 0.032
MOEA/HD 0.036 0.015 0.032 0.019 0.018 0.004 0.018 0.046 0.01
${\boldsymbol{ \xi }}$=0.5
MOEA/CD 0.884 0.901 0.882 0.901 0.945 0.838 0.921 0.912 0.913
Best MOEA/FD 0.870 1 1 1 0.919 0.975 0.956 1 0.987
value NSGAII 0.629 0.576 0.748 0.774 0.587 0.66 0.672 0.575 0.617
MOEA/HD 1 1 1 0.957 1 1 1 0.966 1
MOEA/CD 0.435 0.320 0.454 0.602 0.502 0.556 0.543 0.498 0.535
Worst MOEA/FD 0.409 0.535 0.553 0.419 0.387 0.535 0.546 0.458 0.483
value NSGAII 0.530 0.408 0.686 0.766 0.475 0.566 0.664 0.680 0.506
MOEA/HD 0.442 0.469 0.439 0.444 0.396 0.390 0.488 0.449 0.444
MOEA/CD 0.864 0.832 0.869 0.876 0.872 0.822 0.871 0.895 0.905
Mean MOEA/FD 0.862 0.994 0.966 0.973 0.888 0.952 0.921 0.99 0.949
value NSGAII 0.610 0.543 0.726 0.720 0.554 0.647 0.604 0.560 0.601
MOEA/HD 0.987 0.974 0.942 0.937 0.987 0.971 0.968 0.960 0.975
MOEA/CD 0.017 0.051 0.011 0.018 0.052 0.012 0.039 0.020 0.051
Standard MOEA/FD 0.006 0.009 0.029 0.019 0.022 0.025 0.040 0.010 0.030
deviation NSGAII 0.017 0.025 0.017 0.038 0.031 0.013 0.051 0.011 0.012
MOEA/HD 0.018 0.028 0.049 0.014 0.018 0.041 0.034 0.004 0.035
${\boldsymbol{ \xi }}$=1
MOEA/CD 0.886 0.972 0.8794 0.882 0.963 0.780 0.889 0.950 0.784
Best MOEA/FD 0.909 0.921 0.954 1 0.990 0.890 0.987 1 1
value NSGAII 0.540 0.802 0.571 0.631 0.615 0.558 0.913 0.754 0.772
MOEA/HD 1 0.970 1 1 0.968 1 1 1 1
MOEA/CD 0.843 0.779 0.855 0.861 0.826 0.808 0.827 0.867 0.900
Worst MOEA/FD 0.858 0.982 0.930 0.957 0.869 0.917 0.865 0.976 0.912
value NSGAII 0.587 0.515 0.708 0.692 0.512 0.629 0.549 0.550 0.586
MOEA/HD 0.962 0.936 0.881 0.924 0.962 0.912 0.921 0.956 0.926
MOEA/CD 0.826 0.938 0.847 0.814 0.957 0.747 0.872 0.868 0.727
Mean MOEA/FD 0.903 0.902 0.899 0.929 0.966 0.871 0.964 0.998 0.902
value NSGAII 0.520 0.773 0.532 0.589 0.611 0.515 0.870 0.720 0.669
MOEA/HD 0.987 0.961 0.958 0.979 0.957 0.936 0.995 0.941 0.988
MOEA/CD 0.045 0.025 0.038 0.049 0.005 0.028 0.018 0.058 0.052
Standard MOEA/FD 0.005 0.025 0.041 0.060 0.0310 0.022 0.026 0.002 0.069
deviation NSGAII 0.017 0.022 0.034 0.030 0.004 0.031 0.030 0.0280 0.0930
MOEA/HD 0.018 0.007 0.036 0.021 0.014 0.056 0.005 0.044 0.011
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