Abstract
This article presents an innovative method to address the economic load dispatch (ELD) problem in power systems incorporating renewable energy sources within thermal units. Employing the 2m point estimation technique for determining renewable energy output power, the proposed approach effectively addresses challenges associated with renewablebased ELD by utilizing the artificial electric field method. Inspired by the electrostatic force principle among charged particles, this approach guides particles toward optimal solutions within the search space. Validation on power systems featuring 3, 5, 6, 15, and 40 units demonstrates superior performance compared to established algorithms, confirmed by the Wilcoxon signedrank test. The research contributes to the advancement of sustainable and efficient power systems.
Similar content being viewed by others
Availability of data and materials
Data available within the article or its supplementary materials.
Abbreviations
 \(\textrm{a}_{\textrm{i}}\),\(\textrm{b}_{\textrm{i}}, \textrm{c}_{\textrm{i}}\),\(\textrm{e}_{\textrm{i}}\), \(\textrm{f}_{\textrm{i}}\) :

Cost variable of the \(\textrm{i}_{\textrm{th}}\) thermal unit
 \(\textrm{N}\) :

Count of connected generators
 \(\textrm{Ti}\) :

Generated power of \(\textrm{i\;th}\) thermal generator in MW
 \(\textrm{W}_{\textrm{p}}, \textrm{S}_{\textrm{p}}\) :

Generated power of windsolar in MW
 \(\textrm{C}_{\textrm{w}}\) :

Cost variable of wind in $/h
 \(\textrm{N}_{\textrm{W}}, \textrm{N}_{\textrm{s}}\) :

Count of wind and solar unit
 Bidl:

Bid price of \(\textrm{l\;th}\) solar unit
 TL:

Transmission loss
 TD:

Power requirement
 \(\textrm{B}_{\textrm{ij}}, \textrm{B}_{0\textrm{i}}, \textrm{B}_{00}\) :

Loss matrix
 \(\textrm{T}_{\textrm{imin}}, \textrm{T}_{\textrm{imax}}\) :

Boundary limit Minimummaximum power \(\textrm{i\;th}\) unit
 \(\textrm{S}_{\textrm{hp}}, \textrm{S}_{\textrm{cp}}\) :

Weibull variables
 \(\textrm{v, v}_{\textrm{r}}\) :

Instant and rated haste of wind unit
 \(\textrm{v}_{\textrm{in}}, \textrm{v}_{\textrm{out}}\) :

Cut in–cut out the haste of wind unit
 \(\textrm{W}_{\textrm{p}}, \textrm{W}_{\textrm{pt}}\) :

Instant and rated power of wind unit
 \(\upomega , \uppsi\) :

Beta variables
 \(\Gamma\) :

Gamma objectives
 \(\textrm{S}_{\mathrm{rad(t)}}\) :

Cellular solar radiation at time t
 \(\textrm{S}_{\textrm{rad,stc}}\) :

Solar radiation in normal circumstances
 \(\textrm{S}_{\textrm{P,stc}}\) :

Solar power in normal circumstances
 \(\upgamma\) :

Temperature variable in %/°\(\hbox {C}\)
 Tcell:

The degree of heat in a solar cell
 \(\textrm{T}_{\textrm{cell,stc}}\) :

The solar cell’s temperature under the usual test conditions
 NOT:

The cell’s typical operating temperature
 \(\textrm{N}_{\textrm{sc}}, \textrm{N}_{\textrm{pc}}\) :

Number of solar cells in series and parallel
 \(\upmu , \upsigma\) :

The average and standard deviation
 \(\textrm{I}_{\textrm{k}}\) :

Input constant
 \(\textrm{S}_{\textrm{e}}\) :

Total electricity production, including solar and wind
 \(\textrm{z}_{\textrm{l}}\) :

Uncertainty in the input variable
 \(\mathrm{Qi(t)}, \mathrm{Qj(t)}\) :

Charges of \(\textrm{i\;th}\) and \(\textrm{j\;th}\) fleck
 K(t):

Coulomb’s variable
 E:

Modestly positive constant
 \(\textrm{R}_{\textrm{ij(t)}}\) :

The distance in Euclid between two particles
 \(\upalpha\), K_{0} :

Parameter and starting point
 \(\textrm{F}_{\textrm{i}}, \textrm{M}_{\textrm{i}}\) :

Force and mass of the \(\textrm{i\;th}\) particle
 \(\textrm{V}_{\textrm{i}}, \textrm{X}_{\textrm{i}}\) :

Particle’s location and speed
References
Dhillon J, Parti S, Kothari D (1993) Stochastic economic emission load dispatch. Electr Power Syst Res 26(3):179–186
Bhattacharjee K, Shah K, Soni J (2022) Solving economic dispatch using artificial eco systembased optimization. Electr Power Compon Syst 49(11–12):1034–1051
Soni J, Bhattacharjee K (2022) Sooty tern optimization algorithm for solving the multiobjective dynamic economic emission dispatch problem. Int J Swarm Intell Res (IJSIR) 13(1):1–15
Patel N, Bhattacharjee K (2020) A comparative study of economic load dispatch using sine cosine algorithm. Sci Iran 27(3):1467–1480
Bhattacharjee K, Bhattacharya A, nee Dey S.H (2014) Oppositional real coded chemical reaction based optimization to solve shortterm hydrothermal scheduling problems. Int J Electr Power Energy Syst 63:145–157
Kempton W, Letendre S (1997) Electric vehicle as a new source of power for electric vehicles. Transp Res 2:157–175
Soni J, Bhattacharjee K (2024) A multiobjective economic emission dispatch problem in microgrid with high penetration of renewable energy sources using equilibrium optimizer. Electr Eng 342:103780
Verma D, Soni J, Kalathia D, Bhattacharjee K (2022) Sine cosine algorithm for solving economic load dispatch problem with penetration of renewables. Int J Swarm Intell Res (IJSIR) 13(1):1–21
Galus MD, Andersson G (2008) Demand management of grid connected plugin hybrid electric vehicles (phev). In: 2008 IEEE energy 2030 conference. IEEE, pp 1–8
Soni JM, Pandya MH (2018) Power quality enhancement for PV rooftop and Bess in islanded mode. In: 2018 4th international conference on electrical energy systems (ICEES). IEEE, pp 242–247
Soni J, Bhattacharjee K (2024) Equilibrium optimizer for multiobjective dynamic economic emission dispatch integration with plugin electric vehicles and renewable sources. Multiscale Multidiscip Model Exp Des 1–17
Bhattacharjee K, Bhattacharya A, Shah K, Patel N (2022) Backtracking search optimization applied to solve shortterm electrical real power generation of hydrothermal plant. Eng Optim 54(9):1525–1543
Ma H, Yang Z, You P, Fei M (2017) Multiobjective biogeographybased optimization for dynamic economic emission load dispatch considering plugin electric vehicles charging. Energy 135:101–111
Qu B, Qiao B, Zhu Y, Jiao Y, Xiao J, Wang X (2017) Using multiobjective evolutionary algorithm to solve dynamic environment and economic dispatch with EVS. In: International conference on swarm intelligence. Springer, pp 31–39
Zou D, Li S, Xuan K, Ouyang H (2022) A NSGAII variant for the dynamic economic emission dispatch considering plugin electric vehicles. Comput Ind Eng 173:108717
Bhattacharjee K, Patel N (2020) An experimental study regarding economic load dispatch using search group optimization. Sci Iran 27(6):3175–3189
Bhattacharjee K, Bhattacharya A, nee Dey SH (2015) Backtracking search optimization based economic environmental power dispatch problems. Int J Electr Power Energy Syst 73:830–842
Chen F, Zhou J, Wang C, Li C, Lu P (2017) A modified gravitational search algorithm based on a nondominated sorting genetic approach for hydrothermalwind economic emission dispatching. Energy 121:276–291
Liu G, Zhu YL, Jiang W (2018) Windthermal dynamic economic emission dispatch with a hybrid multiobjective algorithm based on wind speed statistical analysis. IET Gener Transm Distrib 12(17):3972–3984
Basu M (2019) Multiarea dynamic economic emission dispatch of hydrowindthermal power system. Renew Energy Focus 28:11–35
Zhu Z, Wang J, Baloch MH (2016) Dynamic economic emission dispatch using modified NSGAII. Int Trans Electr Energy Syst 26(12):2684–2698
Kheshti M, Ding L, Ma S, Zhao B (2018) Double weighted particle swarm optimization to nonconvex wind penetrated emission/economic dispatch and multiple fuel option systems. Renew Energy 125:1021–1037
Zhao J, Wen F, Dong ZY, Xue Y, Wong KP (2012) Optimal dispatch of electric vehicles and wind power using enhanced particle swarm optimization. IEEE Trans Ind Inform 8(4):889–899
Jin J, Zhou D, Zhou P, Miao Z (2014) Environmental/economic power dispatch with wind power. Renew Energy 71:234–242
Faramarzi A, Heidarinejad M, Stephens B, Mirjalili S (2020) Equilibrium optimizer: a novel optimization algorithm. KnowlBased Syst 191:105190
Varzaneh ZA, Hossein S, Mood SE, Javidi MM (2022) A new hybrid feature selection based on improved equilibrium optimization. Chemom Intell Lab Syst 228:104618
Basu M (2019) Multiarea dynamic economic emission dispatch of hydrowindthermal power system. Renew Energy Focus 28:11–35
Chen MR, Zeng GQ, Lu KD (2019) Constrained multiobjective population extremal optimization based economicemission dispatch incorporating renewable energy resources. Renew Energy 143:277–294
Ding Y, Cano ZP, Yu A, Lu J, Chen Z (2019) Automotive Liion batteries: current status and future perspectives. Electrochem Energy Rev 2(1):1–28
Gupta S, Abderazek H, Yıldız BS, Yildiz AR, Mirjalili S, Sait SM (2021) Comparison of metaheuristic optimization algorithms for solving constrained mechanical design optimization problems. Expert Syst Appl 183:115351
Zhang Y, Le J, Liao X, Zheng F, Liu K, An X (2018) Multiobjective hydrothermalwind coordination scheduling integrated with largescale electric vehicles using IMOPSO. Renew Energy 128:91–107
Shao C, Wang X, Wang X, Du C, Dang C, Liu S (2014) Cooperative dispatch of wind generation and electric vehicles with battery storage capacity constraints in SCUC. IEEE Trans Smart Grid 5(5):2219–2226
Soni J, Bhattacharjee K (2024) Integrating renewable energy sources and electric vehicles in dynamic economic emission dispatch: an oppositionalbased equilibrium optimizer approach. Eng Optim 1–35
Shao C, Wang X, Wang X, Du C, Dang C, Liu S (2014) Cooperative dispatch of wind generation and electric vehicles with battery storage capacity constraints in SCUC. IEEE Trans Smart Grid 5(5):2219–2226
Qu B, Qiao B, Zhu Y, Liang J, Wang L (2017) Dynamic power dispatch considering electric vehicles and wind power using decomposition based multiobjective evolutionary algorithm. Energies 10(12):1991
Qiao B, Liu J (2020) Multiobjective dynamic economic emission dispatch based on electric vehicles and wind power integrated system using differential evolution algorithm. Renew Energy 154:316–336
Zhang Q, Li H (2007) MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans Evol Comput 11(6):712–731
Soni J, Bhattacharjee K (2023) Equilibrium optimiser for the economic load dispatch problem with multiple fuel option and renewable sources. Int J Ambient Energy 44(1):2386–2397
Yadav A, Kumar N et al (2020) Artificial electric field algorithm for engineering optimization problems. Expert Syst Appl 149:113308
Yadav A et al (2019) AEFA: artificial electric field algorithm for global optimization. Swarm Evol Comput 48:93–108
Soni J, Bhattacharjee K (2024) Multiobjective dynamic economic emission dispatch integration with renewable energy sources and plugin electrical vehicle using equilibrium optimizer. Environ Dev Sustain 26(4):8555–8586
Basu M (2016) Multiobjective optimal reactive power dispatch using multiobjective differential evolution. Int J Electr Power Energy Syst 82:213–224
Basu M (2014) Fuel constrained economic emission dispatch using nondominated sorting genetic algorithmII. Energy 78:649–664
Ghasemi M, Akbari E, Zand M, Hadipour M, Ghavidel S, Li L (2019) An efficient modified HPSOTVACbased dynamic economic dispatch of generating units. Electr Power Compon Syst 47(19–20):1826–1840
Author information
Authors and Affiliations
Contributions
Diwakar Verma: Methodology, software, writing original Draft, data curation, conceptualization, investigation, validation, resources, project administration, format analysis, visualization, supervision. Jatin Soni: Writing original Draft, data curation, conceptualization, investigation, validation, format analysis, visualization, supervision. Kuntal Bhattacharjee: Investigation, validation, format analysis, visualization, supervision.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Verma, D., Soni, J. & Bhattacharjee, K. A novel artificial electric field strategy for economic load dispatch problem with renewable penetration. Evol. Intel. (2024). https://doi.org/10.1007/s12065024009463
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12065024009463