A novel differential evolution algorithm for economic power dispatch problem

Pooja

doi:10.3934/naco.2021042

Article Contents

2023, Volume 13, Issue 1: 28-44. Doi: 10.3934/naco.2021042

This issue Previous Article General biconvex functions and bivariational inequalities Next Article Existence and well-posedness for excess demand equilibrium problems

A novel differential evolution algorithm for economic power dispatch problem

Pooja

Department of Electronics and Communication, University of Allahabad, Prayagraj, India

Received: March 2021

Revised: September 2021

Early access: October 2021

Published: March 2023

The paper is handled by Gerhard-Wilhelm Weber as guest editor

Abstract Full Text(HTML) Figure(1) / Table(9) Related Papers Cited by

Abstract

In power systems, Economic Power dispatch Problem (EPP) is an influential optimization problem which is a highly non-convex and non-linear optimization problem. In the current study, a novel version of Differential Evolution (NDE) is used to solve this particular problem. NDE algorithm enhances local and global search capability along with efficient utilization of time and space by making use of two elite features: selfadaptive control parameter and single population structure. The combined effect of these concepts improves the performance of Differential Evolution (DE) without compromising on quality of the solution and balances the exploitation and exploration capabilities of DE. The efficiency of NDE is validated by evaluating on three benchmark cases of the power system problem having constraints such as power balance and power generation along with nonsmooth cost function and is compared with other optimization algorithms. The Numerical outcomes uncovered that NDE performed well for all the benchmark cases and maintained a trade-off between convergence rate and efficiency.

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Mathematics Subject Classification: Primary: 68T20; Secondary: 90-08.

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Figure 1. Convergence graph between total cost and #FE for EPP: (a) 6-unit generating system, (b) 15-unit generating system, (c) 40-unit generating system

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Table 1. Survey on related work

Reference	Problem	Algorithms	Case Study
[17]	ELD with valve point effect	GA	13 generating units
[30]	ELD	EP	3, 13, and 40 generating units
[11]	ELD with generator constraints	PSO	6, 15, and 40 generating units
[15]	ELD with valve-point effect	HGA	13 and 40 generating units
[8]	ELD	SPSO	6, 15 and 40 generating units
[6]	ELD	CDE	6 and 13 generating units
[5]	ELD	BBO	6, 10, 20 and 40 generating units
[29]	ELD with generator constraints	IPSO	6 and 15 generating units
[24]	ELD	IHSWM	40 generating units
[28]	ELD	MPSO	3, 6, 15, and 40 generation units
[1]	ELD	HPSOTVAC	6, 15 and 38 generating units
[10]	Dynamic ELD	ADE	(1) 5-unit thermal system with Ploss for a 24-hours planning horizon; (2) 10-unit thermal system without Ploss for a 12-hours planning horizon; (3) 10-unit thermal system without Ploss for a 24-hours planning horizon
[2]	ELD	CBA	6, 13, 20, 40 and 160 generating units
[32]	Dynamic ELD	EA	(1) 5-unit thermal problems with and without Ploss (2) 10-unit thermal problems with and without Ploss; (3) 7-unit hydro-thermal problem without Ploss; (4) 19-unit solar-Cthermal system without Ploss; (5) 6-unit wind-Cthermal system with Ploss
[27]	Environmental/ Economic Dispatch	MOEA	(1) 6-generator 30-bus standard test system; (2) 13-generator 57-bus system; (3) 3-generator system; (4) 6-generator system; (5) 14-generator 118-bus system; (6) 40-generator system; (7) 10-generator system
[16]	Economic and Emission Dispatch	IDE	6 generating units
[25]	ELD	QPSO	6, 15 and 40 generating units
[20]	Renewable Energy Location	Robust Bi-Level Programming	Locating renewable energy sites
This research	EPP	NDE	6, 15 and 40 generating units
ELD: Economic Load Dispatch EPP:EconomicPower dispatch GA: Genetic Algorithm EP: Evolutionary Programming PSO: Particle Swarm Optimization HGA: Hybrid Genetic Algorithm approach based on Differential Evolution SPSO: Self-organizing Hierarchical Particle Swarm Optimization CDE: Chemotactic Differential Evolution Algorithm BBO: Biogeography-Based Optimization IPSO: Iteration Particle Swarm Optimization IHSWM: Improved Harmony Search with Wavelet Mutation MPSO: Particle Swarm Optimization by Evolutionary Technique HPSOTVAC: Hybrid Particle Swarm Optimization with Time-Varying Acceleration Coefficients ADE: Automated Differential Evolution CBA: Chaotic Bat Algorithm EA: Evolutionary Algorithms MOEA: Multi-Objective Evolutionary Algorithms IDE: Improved Differential Evolution Algorithm SG-QPSO: Novel Quantum-Behaved Particle Swarm Optimization Algorithm NDE: Novel Differential Evolution

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Table 2. Cost coefficient and bound values for 6-unit system

Unit no.	$ a_i $ (MW)	$ b_i $ (MW)	$ c_i $ (MW)	$ pwr_i^{min} $ (MW)	$ pwr_i^{max} $ (MW)
1	0.007	7	240	100	500
2	0.0095	10	200	50	200
3	0.009	8.5	220	80	300
4	0.009	11	200	50	150
5	0.008	10.5	220	50	200
6	0.0075	12	190	50	120

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Table 3. Ramp rate and prohibited zones limits for 6-unit system

Unit no.	$ UR_i $	$ DR_i $	$ pwr_i(0) $	Zone-1	Zone-2
1	80	120	440	210-240	350-380
2	50	90	170	90-110	140-160
3	65	100	200	150-170	210-240
4	50	90	150	80-90	110-120
5	50	90	190	90-110	140-150
6	50	90	110	75-85	100-105

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Table 4. Experimental results of NDE, DE and other comparative algorithms for 6-unit system

Unit no.	NDE	DE	GA	PSO
1	441.8657	446.7157	474.8066	447.497
2	169.6242	172.7829	178.6363	173.3221
3	259.2367	259.1119	262.2089	263.4745
4	139.5649	142.234	134.2826	139.0594
5	160.22	165.8878	151.9039	165.4761
6	105.0001	88.735	74.1812	87.128
$ pwr_L $	13.0289	12.4673	13.0217	12.9584
Total output power	1, 276.03	1, 275.47	1, 276.03	1, 276.01
Min cost (＄/hr)	15, 444.09	15, 449.48	15, 459	15, 450
Mean cost (＄/hr)	15, 448.48	15, 452.28	15, 469	15, 454

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Table 5. Cost coefficient and bound values for 15-unit system

Unit no.	$ a_i $ (MW)	$ b_i $ (MW)	$ c_i $ (MW)	$ pwr_i^{min} $ (MW)	$ pwr_i^{max} $ (MW)
1	0.000299	10.1	671	455	150
2	0.000183	10.2	574	455	150
3	0. 001126	8.8	374	130	20
4	0. 001126	8.8	374	130	20
5	0.000205	10.4	461	470	150
6	0.000301	10.1	630	460	135
7	0.000364	9.8	548	465	135
8	0.000338	11.2	227	300	60
9	0.000807	11.2	173	162	25
10	0. 001203	10.7	175	160	25
11	0. 003586	10.2	186	80	20
12	0. 005513	9.9	230	80	20
13	0.000371	13.1	225	85	25
14	0.001929	12.1	309	55	15
15	0.004447	12.4	323	55	15

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Table 6. Ramp rate and prohibited zones limits for 15-unit system

Unit no.	$ UR_i $	$ DR_i $	$ pwr_i(0) $	Zone-1	Zone-2	Zone-3
1	180	120	400	150-150	150-150	150-150
2	180	120	300	185-255	305-335	430-450
3	130	130	105	20-20	20-20	20-20
4	130	130	100	20-20	20-20	20-20
5	80	120	90	180-200	305-335	390-430
6	80	120	400	230-255	335-395	430-455
7	80	120	350	135-135	135-135	135-135
8	65	100	95	60-60	60-60	60-60
9	60	100	105	60-60	25-25	25-25
10	60	100	110	25-25	25-25	25-25
11	80	80	60	20-20	20-20	20-20
12	80	80	40	30-40	55-65	20-20
13	80	80	30	25-25	25-25	25-25
14	55	55	20	15-15	15-15	15-15
15	55	55	20	15-15	15-15	15-15

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Table 7. For 15-unit system (Experimental outcomes and comparison)

Unit no.	NDE	DE	MPSO	GA	PSO	IPSO
1	446.3316	454.998	455	415.31	455	455
2	366.7521	379.996	455	359.72	380	380
3	127.9874	129.9991	130	104.43	130	129.97
4	129.1781	129.9899	130	74.99	130	130
5	165.6126	169.9968	286.4128	380.28	170	169.93
6	423.1885	429.9944	460	426.79	460	459.88
7	415.0202	429.9944	465	341.32	430	429.25
8	132.9585	120.1228	60	124.79	60	60.43
9	124.8283	47.6016	25	133.14	71.05	74.78
10	82.68406	146.0069	37.5603	89.26	159.85	158.02
11	68.83981	79.99735	20	60.06	80	80
12	71.96576	79.9997	80	50	80	78.57
13	37.01169	25.0118	25	38.77	25	25
14	25.78839	16.8516	15	41.94	15	15
15	24.5418	20.6135	15	22.64	15	15
$ Pow_L $	29.8923	31.1792	28.9734	38.278	30.908	30.858
Total Output Power	2, 659.89	2, 661.20	2, 658.97	2, 668.40	2, 660.90	2, 660.80
Mean Cost (＄/Hr)	32, 561.89	32, 747.16	32, 569.95	33, 113	32, 708	32, 709

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Table 8. Cost coefficient and bound values for 40-unit system

Unit no.	$ a_i $ (MW)	$ b_i $ (MW)	$ c_i $ (MW)	$ pwr_i^{min} $ (MW)	$ pwr_i^{max} $ (MW)
1	0.00708	9.15	1728.3	114	36
2	0.00313	7.97	647.85	114	36
3	0.00313	7.95	649.69	120	60
4	0.00313	7.97	647.83	190	80
5	0.00313	7.97	647.81	97	47
6	0.00298	6.63	785.96	140	68
7	0.00298	6.63	785.96	300	110
8	0.00284	6.66	794.53	300	135
9	0.00284	6.66	794.53	300	135
10	0.00277	7.1	801.32	300	130
11	0.00277	7.1	801.32	375	94
12	0.52124	3.33	1055.1	375	94
13	0.52124	3.33	1055.1	500	125
14	0.52124	3.33	1055.1	500	125
15	0.0114	5.35	148.89	500	125
16	0.0016	6.43	222.92	500	125
17	0.0016	6.43	222.92	500	220
18	0.0016	6.43	222.92	500	220
19	0.0001	8.95	107.87	550	242
20	0.0001	8.62	116.58	550	242
21	0.0001	8.62	116.58	550	254
22	0.0161	5.88	307.45	550	254
23	0.0161	5.88	307.45	550	254
24	0.0161	5.88	307.45	550	254
25	0.00313	7.97	647.83	550	254
26	0.00708	9.15	1728.3	550	254
27	0.00313	7.97	647.85	150	10
28	0.00313	7.95	649.69	150	10
29	0.00313	7.97	647.83	150	10
30	0.00313	7.97	647.81	97	47
31	0.00298	6.63	785.96	190	60
32	0.00298	6.63	785.96	190	60
33	0.00284	6.66	794.53	190	60
34	0.00284	6.66	794.53	200	90
35	0.00277	7.1	801.32	200	90
36	0.00277	7.1	801.32	200	90
37	0.52124	3.33	1055.1	110	25
38	0.52124	3.33	1055.1	110	25
39	0.52124	3.33	1055.1	110	25
40	0.0114	5.35	148.89	550	242

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Table 9. For 40-unit system (Experimental outcomes and comparison)

Unit no.	NDE	DE	IHSWM	HPSOTVAC	SPSO	BBO
1	90.59008	113.035	113.9088	113.9907	113.97	110.8158
2	102.8105	110.0641	110.9064	113.2932	114	111.0896
3	95.41785	96.508	97.402	120	109.19	97.40261
4	160.0214	180.7317	179.7332	175.0364	179.77	179.7549
5	81.69155	87.3028	88.7117	91	97	88.20832
6	107.7849	110.2516	139.9991	140	91.01	139.9886
7	278.066	259.0112	259.6372	260.3635	259.87	259.5935
8	268.3649	284.6521	284.6106	288.1256	286.99	284.6174
9	266.8508	286.1955	284.6024	286.9435	284.09	284.6479
10	228.3208	131.3615	130	130	204.05	130.0298
11	286.0884	243.8422	168.7992	170	168.4	94.01459
12	159.1006	168.7969	168.806	170	94	94.26367
13	309.307	302.4883	214.7593	210.0287	212.3	304.5153
14	346.2984	394.1255	394.2774	390.0677	393.76	394.264
15	382.4365	306.0514	304.5207	307.6247	303.62	304.5057
16	354.3069	394.3787	394.2762	300.0056	392.05	394.2472
17	442.9098	489.715	489.2787	487.0486	489.49	489.3273
18	440.5635	402.2923	489.2875	485.0793	489.35	489.3047
19	503.2849	516.0751	511.2827	510.541	512.39	511.3087
20	481.891	512.4736	511.2768	511.3472	511.21	511.2495
21	506.4032	524.042	523.2884	524.9522	522.61	523.3217
22	496.2304	524.0563	523.2794	526	523.65	523.3144
23	512.3193	524.3457	523.2772	523.9211	523.06	523.3629
24	511.7555	524.2132	523.2928	525.612	520.72	523.2883
25	507.4037	525.7952	523.3047	521.02	524.86	523.2989
26	493.267	522.6361	523.2872	520.1457	525.22	523.2802
27	10.30096	10.1786	10	10	10	10.02817
28	19.53625	12.3112	10.0022	10	10	10.00321
29	10.68301	10.8716	10.0018	10	10	10.0288
30	87.11869	90.3572	88.362	89.7002	87.64	88.14595
31	184.1883	187.5783	190	190	190	189.9913
32	174.4745	167.4291	190	190	190	189.9888
33	169.4674	177.4801	189.9935	190	190	189.9998
34	171.0925	166.2373	164.7992	167.0209	200	164.8452
35	176.4929	185.5927	164.8923	200	167.18	192.9876
36	172.1041	173.4381	164.864	200	172.12	199.9876
37	94.5799	89.3767	110	110	110	109.9941
38	85.26457	91.0112	110	110	110	109.9992
39	93.85836	91.6815	109.9965	110	95.58	109.9833
40	493.3801	512.0169	511.2828	511.1323	510.85	511.2794
$ Pow_{loss} $	0	0.001	0	0	0	0.28
Total Output Power	10, 500.00	10, 500.00	10, 500.00	10, 500.00	10, 500.00	10, 500.28
Min Cost (＄/Hr)	1, 21, 721.62	1, 21, 974.50	1, 21, 416.26	1, 21, 070.64	1, 22, 049.66	1, 21, 479.50
Mean Cost (＄/Hr)	1, 21, 992.20	1, 22, 580.30	1, 21, 553.42	1, 21, 075.74	1, 22, 327.36	1, 21, 512.05

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