Multi-objective optimization of multi-microgrid power dispatch under uncertainties using interval optimization

Shungen Luo; Xiuping Guo

doi:10.3934/jimo.2021208

Article Contents

2023, Volume 19, Issue 2: 823-851. 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
^*Corresponding author: Xiuping Guo

Received: February 28, 2021

Revised: July 31, 2021

Early access: December 2021

Published: February 2023

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]

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Abstract

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:

Mathematics Subject Classification: Primary: 49M37; Secondary: 68W50.

Citation:

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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.

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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

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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

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Table 4. The parameters of MOEA/HD

Control parameter	Combination Number
Control parameter	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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Table 5. $ Sa $ metric value

Possibility degree $ \xi $=0		$ {\boldsymbol{Sa }}$ metric value
Possibility degree $ \xi $=0		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

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Table 6. MS metric value

Possibility degree $ {\boldsymbol{\xi }}$=0		${\boldsymbol{ MS }}$ metric value
Possibility degree $ {\boldsymbol{\xi }}$=0		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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References

[1]	A. Akhil, The CERTS microgrid concept, White Paper for Transmission Reliability Program, U.S, 2002.
[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.
[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.
[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.
[5]	K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, Goldberg. Wiley-Interscience Series in Systems and Optimization. John Wiley & Sons, Ltd., Chichester, 2001.
[6]	K. Deb and H. Beyer, Self-adaptive genetic algorithms with simulated binary crossover, Evolutionary Computation, 9 (2001), 197-221. doi: 10.1162/106365601750190406.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	J. L. Verdagay, Fuzzy Mathematical Programming, North-holland, Amsterdam, 1982.
[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.
[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.
[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.
[32]	A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, John Wiley and Sons, New York, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[39]	E. V. Zyl and A. P. Engelbrecht, A subspace-based method for PSO initialization, IEEE Symposium Series on Computational Intelligence. IEEE, 2016.