International Journal of Renewable Energy Development Int. J. Renew. Energy Dev. 2023, 12 (3), 590-600 | 590 https://doi.org/10.14710/ijred.2023.52835 ISSN: 2252-4940/Β© 2023.The Author(s). Published by CBIORE Contents list available at IJRED website International Journal of Renewable Energy Development Journal homepage: https://ijred.undip.ac.id The application of equilibrium optimizer for solving modern economic load dispatch problem considering renewable energies and multiple-fuel thermal units Hung Duc Nguyena,b* , Khoa Hoang Truonga,b , Nhuan An Lea,b aDepartment of Power Delivery, Ho Chi Minh City University of Technology (HCMUT), 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam bVietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam Abstract. This study presents a modern version of the economic load dispatch (MELD) problem with the contribution of renewable energies and conventional energy, including wind, solar and thermal power plants. In the study, reduction of electricity generation cost is the first priority, while the use of multiple fuels in the thermal power plant is considered in addition to the consideration of all constraints of power plants. Two meta-heuristic algorithms, one conventional and one recently published, including Particle swarm optimization (PSO) and Equilibrium optimizer (EO), are applied to determine the optimal solutions for MELD. A power system with ten thermal power plants using multiple fossil fuels, one wind power plant, and three solar power plants is utilized to evaluate the performance of both PSO and EO. Unlike other previous studies, this paper considers the MELD problem with the change of load demands over one day with 24 periods as a real power system. In addition, the power generated by both wind and solar power plants varies at each period. The results obtained by applying the two algorithms indicate that EO is completely superior to PSO, and the solutions found by EO can satisfy all constraints. Particularly in Case 1 with different load demand values, EO achieves better total electricity production cost (TEGC) than PSO by 0.75%, 0.87%, 0.13%, and 0.45% for the loads of 2400 MW, 2500 MW, 2600 MW and 2700 MW. Moreover, EO also provides a faster response capability over PSO through the four subcases although EO and PSO are run by the same selection of control parameters. In Case 2, the high efficiency provided by EO is still maintained, though the scale of the considered problem has been substantially enlarged. Specifically, EO can save $51.2 compared to PSO for the minimum TEGC. The savings cost is equal to 0.33% for the whole schedule of 24 hours. With these results, EO is acknowledged as a favourable search method for dealing with the MELD problem. Besides, this study also points out the difference in performance between a modern meta-heuristic algorithm (EO) and the classical one (PSO). The modern metaheuristic algorithm with special structure is highly valuable for complicated problem as MELD. Keywords: Economic load dispatch; Particle swarm optimization; Equilibrium optimizer; multiple fuels; thermal generator; renewable energies. @ The author(s). Published by CBIORE. This is an open access article under the CC BY-SA license (http://creativecommons.org/licenses/by-sa/4.0/). Received: 25th Feb 2023; Revised: 16th April 2023; Accepted: 26th April 2023 ; Available online: 1st May 2023 1. Introduction Economic load dispatch (ELD) is considered one of the most crucial problems in power system operation. The determination of an optimal solution to ELD allows the operators to save more operation cost and reduce environmental damage (Xiang et al., 2021). Nowadays, the concept of the conventional economic load dispatch (CELD) is obsoleted and does not fit modern power systems anymore because of its static nature. In addition, CELD only considers thermal generators as the sole generating source. However, the modern economic load dispatch (MELD) was updated once multiple objective functions and renewable energy sources (RES) (Zhang et al., 2021; Shen et al., 2019; Li et al., 2020) were taken into account. Specifically, several objective functions can be listed, such as reducing the total electricity generating cost (TEGC), reducing the entire emissions (REE), etc. These objectives can be considered simultaneously or separately, depending on the different targets such as financial factor or/and technical factor. Besides, both CELD and MELD have a lot in common, such as the set of related constraints and * Corresponding author Email: hungnd@hcmut.edu.vn (H.D.Nguyen) the type of variable needed to be found. In terms of related constraints, several typical constraints can be named, such as the power balance constraint, the generator operational constraints, the multiple fuel constraint, etc. About the types of variables when dealing with both CELD and MELD problems, there are two of them, including the control and the dependent variables. These variables must be defined prior to any kind of computation. Particularly with these mentioned problems, control variables are the power generated by all the existent generators in the system except for the first generator, which is considered the dependent variable. In the whole process of solving such CELD and MELD problems, an optimal solution is acknowledged if only both control variables and dependent variables satisfy all the related constraints with the minimum value of the fitness function. The types of RES integrated with power systems are mainly solar and wind energies. Therefore, the thermal, wind and solar power plants supply enough power to loads in MELD, while only thermal power plants are in charge of the role in CELD (Hlalele et al., 2021; Kim and Kim, 2020). In Research Article https://doi.org/10.14710/ijred.2023.52835 https://doi.org/10.14710/ijred.2023.52835 mailto:hungnd@hcmut.edu.vn https://orcid.org/0000-0001-7831-3814 https://orcid.org/0000-0002-1237-0071 https://orcid.org/0000-0003-3880-471X http://crossmark.crossref.org/dialog/?doi=10.14710/ijred.2023.52835&domain=pdf H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |591 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE this study, MELD problem with the integration of RES into thermal power plants is solved. At the beginning of the foundation for the CELD problem, the solution for the problem is commonly given by the old fashion computing methods such as the gradient method, the quadratic method, and the conventional iterative method. The real efficiency of these methods seems to be good and acceptable as the theorem study. However, at the time, MELD is widely studied and the scale of the problem is enlarged so much over CELD. Besides, many complicate constraints are also taken into account. As a result, the whole complicated degree of MELD is increased substantially. The application of old fashion computing methods begins showing more drawbacks such as a low response, a poor accuracy. Luckily, in two past decades, computing methods have witnessed a huge leap forward in meta-heuristic methods to cope with high- degree complicated problems. There have been a huge number of meta-heuristic applied to MELD problem, such as Genetic algorithm (GA) (Chen and Chang, 1995), Particle swarm optimization (PSO) (Park et al., 2005), Artificial algae algorithm (AAA) (Kumar and Dhillon, 2018), Modified differential evolution (MDE) (Nguyen et al., 2018), Grey wolf optimal (Pradhan et al., 2016), Tunicate Swarm Optimizer (TSO) in (Hien et al., 2021), differential evolution (DE) in (Parouha and Das, 2018), improved firework algorithm (IFA) (Zare et al., 2021), quantum PSO (QPSO) (Xin-gang et al., 2020), improved Manta ray optimization (IMRO) (Hassan et al., 2021), simplex search-based PSO (SSM-PSO) (Chopra et al., 2021), improved bird swarm optimization (IBSO) (Fu et al., 2020), Distributed roust optimization (DRO) (Chang et al., 2021), hybrid grey wolf optimizer (Al-Betar et al., 2020), Pattern search and Sequential quadratic programing-based genetic algorithm (PS-SQP-GA) (Alsumait et al., 2010), Double weight-based PSO (DW-PSO) (Kheshti et al., 2018), improved bacterial foraging algorithm (IBFA) (Pandit et al., 2012), acceleration coefficients-based PSO (AC-PSO) (Ghasemi et al., 2019), Nondominated sorting-based genetic algorithm (NSGA) (Basu, 2008), Chaotic differential evolution (CDE) (Coelho and Mariani, 2006), biogeography optimization (BO) (Xiong and Shi, 2018), and Ameliorated dragonfly algorithm (ADA) (Suresh et al., 2019). These algorithms have shown a good performance as compared to deterministic algorithms based on Lagrange relaxation, newton, gradient search (Vaisakh and Reddy, 2013; Nguyen et al., 2019); however, these algorithms have not been applied to deal with the complex fuel cost functions of thermal units with three fuel options (Chen et al., 2020; Pham et al., 2022). Multiple fuel options can be used for generating units in thermal power plants and the different options can bring more benefits to thermal power plants. But the units with several fuel options can bring more challenges to optimization tools since the generation of unit is not continuous within allowable ranges (Dieu et al., 2013; Jeyakumar et al., 2006). Almost applied algorithms for the multiple fuel units are strong and not much dependent on Lagrange function, excluding augmented Lagrange Hopfield network (ALHN) (Dieu et al., 2013). In fact, these applied methods are comprised of Modified PSO (Jeyakumar et al., 2006), Differential evolution (Noman and Iba, 2008), Self- Adaptive Differential Evolution (SDE) (Balamurugan and Subramanian, 2007), adaptive real coded genetic algorithm (ARC-GA) (Amjady and Nasiri-Rad, 2010), and Improved evolutionary programming (IEP) (Park et al., 1998). These algorithms have reported good results with minimum fuel cost and high stable search ability; however, the search speed was almost neglected once the comparison of setting parameters was not implemented. In this study, the original version of PSO (Kennedy and Eberhart, 1995) and one modern meta-heuristic algorithm called Equilibrium optimizer (EO) (Faramarzi et al., 2020) are applied to determine the optimal solution for MELD with multiple fuel options and renewable energies. The multiple fuel options are examined in the operation cost function belonging to thermal generators. The renewable generating sources, including one wind farm and three solar power plants, and the variation of power output values from these sources within 24 hours are included. The first priority of this study is to reduce the total operation cost as much as possible. The application of both EO and PSO is considered to be a good example for evaluating the performance of a modern meta-heuristic algorithm and a classical algorithm. Briefly, the novelties of this study can be seen on different aspects, including proposing the MELD problem where varied load demands are employed with a day and the 24-single periods, the variation of power generated from both wind and solar power plants is examined throughout the 24 periods, and the difference in raw performance between the modern meta- heuristic algorithm and the classical one is clarified and pointed out based on the results. The key contributions of the study are stated in the four claims. Firstly, the proposed MELD is successfully solved considering the presence of both solar and wind power plants. Secondly, the optimal solution for MELD is determined under the consideration of the multiple fuel constraints. Thirdly, EO is proved to be the best applied method for the proposed problem through the comparisons of the results reached by PSO and other previous studies. Lastly, the superiority of a modern meta-heuristic algorithm (EO) over the classical one (PSO) is proved and demonstrated by results and figures. In this study, two power systems with different complicated levels are considered for reaching the objective function of reducing the total electric generation cost over optimal schedule horizon, one hour for the first system and 24 hours for the second system. The two systems use the same ten thermal generating units in which each thermal generating unit can use two or three fuel types for electric production. The total cost values from the two systems are comparison criteria to evaluating the performance of EO and other algorithms. 2. Problem descriptions. 2.1 Main objective function As mentioned earlier, there are three types of generating sources in this study, including thermal power plants, wind power plants, and solar power plants. However, only the operation process of thermal generators consumes fossil fuels. So, the major target of the problem is to reduce the total electricity generation cost (TEGC) from thermal power plants. The target can be formulated as follows. 𝑅𝑒𝑑𝑒𝑐𝑖𝑛𝑔 𝑇𝐸𝐺𝐢 = βˆ‘ 𝐸𝐺𝐢𝑔 𝐺 𝑔=1 (1) Where EGCg ($/h) is the fuel cost of the gth thermal generator and formulated by (Park et al., 2005): 𝐸𝐺𝐢𝑔 = 𝑔 + 𝛿𝑔𝑃𝐺𝑔 + 𝛾𝑔𝑃𝐺𝑔 2 π‘€π‘–π‘‘β„Ž 𝑔 = 1,…,𝐺 (2) TEGC is the total fuel cost of all G generators working in one hour meanwhile 𝐸𝐺𝐢𝑔 is the fuel cost of one generator only. So, the unit of TEGC is $ but that is $/h for EGCg. The cost function of the thermal generator, 𝐸𝐺𝐢𝑔described in Equation (2) is depicted at Fig.1a. H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |592 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE Fig. 1 Models of fuel cost function for thermal units: a) One fuel option; b) multiple fuel options In addition, the multiple fuels aspect is also considered in this study, as shown in Fig. 1b. According to (Dieu et al., 2013), The mathematical expression of 𝐸𝐺𝐢𝑔 is represented by: 𝐸𝐺𝐢𝑔 = { 𝑔1 + 𝛿𝑔1𝑃𝐺𝑔 + 𝛾𝑔1𝑃𝐺𝑔 2 , 𝑓𝑒𝑒𝑙 1,𝑃𝐺𝑔1,𝑙𝑠 ≀ 𝑃𝐺𝑔 ≀ 𝑃𝐺𝑔1,β„Žπ‘  𝑔2 + 𝛿𝑔2𝑃𝐺𝑔 + 𝛾𝑔2𝑃𝐺𝑔 2 , 𝑓𝑒𝑒𝑙 2,𝑃𝐺𝑔2,𝑙𝑠 ≀ 𝑃𝐺𝑔 ≀ 𝑃𝐺𝑔2,β„Žπ‘  … 𝑔𝑀 + 𝛿𝑔𝑀𝑃𝐺𝑔 + 𝛾𝑔𝑀𝑃𝐺𝑔 2, 𝑓𝑒𝑒𝑙 𝑀,𝑃𝐺𝑔𝑀,𝑙𝑠 ≀ 𝑃𝐺𝑔 ≀ 𝑃𝐺𝑔𝑀,β„Žπ‘  (3) Where 𝑔1, 𝛿𝑔1, and 𝛾𝑔1 are fuel consumption factors while using fuel 1 of generator g; 𝑃𝐺𝑔1,𝑙𝑠 and 𝑃𝐺𝑔1,β„Žπ‘  are the lowest and highest power generated by generator g whilst using fuel 1. Similarly, 𝑔2, 𝛿𝑔2 and 𝛾𝑔2 are fuel consumption factors of generator g whilst using fuel 2. 𝑃𝐺𝑔2,𝑙𝑠 and 𝑃𝐺𝑔2,β„Žπ‘  are the lowest and highest power generated by generator g whilst using the fuel 2. Finally, 𝑔𝑀, 𝛿𝑔𝑀, and 𝛾𝑔𝑀 are fuel consumption factors while using the fuel 𝑀 of generator g; and 𝑃𝐺𝑔𝑀,𝑙𝑠 and 𝑃𝐺𝑔𝑀,β„Žπ‘  are the lowest and highest power generated by generator g whilst using the fuel 𝑀. 2.2 Constraints There are important constraints that must be respected while solving both CELD and MELD. They are described one by one as follows: β€’ Power balance constraint: This constraint is mainly about the relationship between the generating side and the demand side. According to (Duong et al., 2021), the mathematical expression of the constraint is presented by Eq 4: 𝑃𝐷 + 𝑃𝐿 = βˆ‘ 𝑃𝐺𝑔 𝐺 𝑔=1 + π‘ƒπ‘Š + 𝑃𝑆 (4) In Equation (4), PL is calculated by using the Equation (5) below (Dieu et al., 2013): 𝑃𝐿 = βˆ‘ βˆ‘ π‘ƒπΊπ‘”π΅π‘”β„Žπ‘ƒπΊβ„Ž 𝐺 β„Ž=1 𝐺 𝑔=1 + βˆ‘ 𝐡0𝑔𝑃𝐺𝑔 𝐺 𝑔=1 + 𝐡00 (5) β€’ Generator operational constraints: This constraint is about the working limit belonging to thermal generators. That means power output must allocate in the allowed range as depicted in Equation (6). Besides, the working limitation of a typical thermal generator is illustrated in Fig. 2. According to (Dieu et al., 2013), this constraint is formulated as follow: 𝑃𝐺𝑔,π‘šπ‘–π‘› ≀ 𝑃𝐺𝑔 ≀ 𝑃𝐺𝑔,π‘šπ‘Žπ‘₯ (6) β€’ The electricity producing constraint of solar power plant: according to (Phan et al., 2021) the power generated by solar power plant is limited as follows: βˆ‘π‘ƒπ‘†πΊ,π‘ž 𝑁𝑆 π‘ž ≀ 80% Γ— 𝑃𝐷 (7) 𝑃𝑆𝐺,π‘ž π‘šπ‘–π‘› ≀ |𝑃𝑆𝐺,π‘ž| ≀ 𝑃𝑆𝐺,π‘ž π‘šπ‘Žπ‘₯ (8) 3. The applied methods To determine the optimal solution for MELD in this study, two meta-heuristic methods, Particle swarm optimization (PSO) (Kennedy and Eberhart, 1995) and Equilibrium optimizer (EO) (Faramarzi et al., 2020) are applied to solve the given problem. PSO is inspired by the foraging practice of animal swarms in real life such as fish, bird, ox, etc., while EO is inspired by physical law named equilibrium state of mass. The key difference between PSO and EO is their update process for new solutions, which will be described in the following subsections: Fig. 2 The working limitation of a thermal generator H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |593 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE 3.1 The original version of PSO The update process for new solutions belonging to PSO includes two main steps: the velocity update and the new position update. These steps are clarified by mathematical Equations (9) and (10) as follows: 𝑉𝑖 𝑛𝑒𝑀 = 𝑉𝑖 + π‘π‘ž1 Γ— 𝛾1 Γ— (𝑃𝐡𝑒𝑠𝑑,𝑖 βˆ’ 𝑃𝑖) + π‘π‘ž2 Γ— 𝛾2 Γ— (𝑃𝐺𝑏𝑒𝑠𝑑,𝑖 βˆ’ 𝑃𝑖) (9) 𝑃𝑖 𝑛𝑒𝑀 = 𝑃𝑖 + 𝑉𝑖 𝑛𝑒𝑀 (10) 3.2 The Equilibrium optimizer (EO) The update process of EO is conducted based on the references around the four best solutions at each iteration. The key elements of the whole process are described as follows: 𝑆𝑖 𝑛𝑒𝑀 = 𝑆𝑠 + (𝑆𝑖 βˆ’ 𝑆𝑠)𝐸π‘₯ + 𝐺𝑒 π‘£π‘Ÿ (1 βˆ’ 𝐸π‘₯) π‘€π‘–π‘‘β„Ž 𝑖 = 1, . . . ,𝑁𝑃 (11) In Equation (11), the exponential term (Ex) and the generation rate (Ge) are calculated by using the Equations below: 𝐸π‘₯ = πœƒ1𝑠𝑖𝑔𝑛(πœ” βˆ’ 0.5)(𝑒 βˆ’π‘‘π‘“.π‘£π‘Ÿ βˆ’ 1) (12) Where πœƒ1 is set by 2, and the integer number (πœ”) is a random value between 0 and 1. Besides, the time length factor (𝑑𝑓) is a value that changes with each iteration, and its value depends entirely on the maximum iteration number (πΌπ‘‘π‘€π‘Žπ‘₯) and the current iteration number (𝐼𝑑). 𝑑𝑓 can be determined by using the Equation (13) 𝑑𝑓 = (1 βˆ’ 𝐻 π»π‘€π‘Žπ‘₯ ) πœƒ2( 𝐻 π»π‘€π‘Žπ‘₯ ) (13) where 𝛽2 is set by 1 (Faramarzi et al., 2020), 𝐻 is the current iteration, and π»π‘€π‘Žπ‘₯ is the maximum quantity of iteration. About the generation rate (Ge), this term can be found by using the Equation (14) below: 𝐺𝑒 = 𝐸π‘₯.𝛿.(𝑆𝑠 βˆ’ π‘£π‘Ÿ.𝑆𝑖) (14) Where, 𝛿 = { π‘Ÿπ‘‘3 2 𝑖𝑓 π‘Ÿπ‘‘4 β‰₯ 𝑐𝑓 0 𝑒𝑙𝑠𝑒 (15) Where 𝒄𝒇 is set by 0.5 (Faramarzi et al., 2020). 4. Results and discussions The study is conducted on a personal computer with a Central processing unit (CPU) of 2.0 GHz and 8 GB of Random-Access Memory (RAM). The coding is implemented in MATLAB software with version 2018b. The two study cases are implemented as follows: - Study Case 1: The system with ten thermal units using multiple fuels for four load demand cases, including 2400, 2500, 2600, and 2700 MW. Data of the system is taken from (Park et al., 1998). - Study Case 2: One wind and three solar photovoltaic power plants are integrated into the above ten-unit system. The power output supplied by wind farms and solar power plants over one day are taken from the studies (Zhang et al., 2017) and (Augusteen et al., 2016), respectively. Fig. 3 The results obtained by PSO and EO after 50 independent runs corresponding to different load demand values: a) Load of 2400MW, b) Load of 2500MW, c) Load of 2600MW, and d) Load of 2700MW H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |594 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE 4.1 The determination of population and maximum iteration Initially, the determination of optimal settings for the initial control parameters, including population and maximum iteration, is one of the first hurdles that must be overcome. To deal with the first hurdle, different simulations have been conducted before the certain values for the population and the maximum number of iterations are established. Due to the high complexity posted by the set of constraints, the large scale of search space explicitly seen in Case 2, and the non-convex characteristic of both CELD and MELD, EO and PSO are executed with different settings of population and maximum iteration. For Study Case 1, the population is set to 30, 40, 50, and 60, respectively. The maximum iteration is also varied from 80, 90, 100, and 150, respectively. The results obtained by all experiments revealed that the value of the population strongly influences the quality of the solutions and the execution time of each iteration, while the iteration number highly involves the quality of the solutions and the execution time of an independent run. As a result, when the population is set to a high value, the quality of the solutions will be improved; however, each iteration will require more time to complete. If maximum iteration is fixed at a high value, the quality of the solution at the end of each independent run could be enhanced, but more time would be consumed to finish a run. More importantly, if both population and maximum iteration are set to large values, the applied methods will result in the same optimal solutions after a long execution time. In that circumstance, the performance of the two applied methods cannot be evaluated precisely and reliably. By analysing the results obtained by the mentioned experiments, 50 and 100 are considered the optimal setting for population and maximum iteration. These settings also perfectly serve the initial purpose of evaluating the performance of the EO and PSO. Study Case 2 is more complicated than Study Case 1 by considering 24 periods instead of one period. The setting of Case 1 is applied for each hour of Case 2. So, Case 2 takes more simulation time than Case 1 as a result. 4.2 Results of Case 1 Fig. 3 presents the results obtained by PSO and EO for 50 trial runs. The four subfigures have the same characteristic that EO can reach approximately the same solution for over 50 runs but those from PSO highly fluctuate. In addition, PSO cannot reach the same best solution as EO. Fig. 4 shows the best run of PSO and EO for the four cases. Specifically, for the first load demand case with 2400 MW, EO only needs over 50 iterations to reach the best value of the considered fitness function, while PSO cannot perform the same even if the last iteration is used on this best run. For the last three cases of load demand, EO still maintains its fast-response capability over PSO by reaching the fitness value with fewer iterations. Particularly, with load demands of 2500, 2600, and 2700 (MW), EO also requires approximately 60 iterations to obtain the best value, while PSO cannot achieve any similar value on the three comparisons. Clearly, EO is much faster than PSO in finding the most optimal generation solutions. Fig. 5 shows the best cost for the four cases obtained by PSO, EO and other methods. In the figure, PSO is the worst method while EO can reach the same or slightly smaller cost than others. It should be emphasized here that EO only search solution by using 50 and 100 for population and iterations, whereas others must use higher than 100 for iterations excluding ALHN (Dieu et al., 2013), which is a deterministic algorithm. Because of this evidence, the superiority of EO over other previous methods is not deniable, so this study focuses on analyzing the efficiency of EO and PSO for each subfigure corresponding to each level of load demand. Particularly, in the first case of load demand in Fig. 5 with 2400 MW, the TEGC achieved by EO is only $481.723, while the similar value obtained by PSO is up to $485.317. The savings cost saved by EO over PSO in this case is approximately $3.6, corresponding to 0.75%. On the three remaining load demand levels, the TGEC values achieved by EO are, respectively, $526.239, $574.381, and $623.809. The similar values resulted by PSO are, Fig. 4 The best convergences obtained by PSO and EO among 50 independent runs corresponding to different load demand values: a) Load of 2400MW, b) Load of 2500MW, c) Load of 2600MW, and d) Load of 2700MW H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |595 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE respectively, $530.826, $575.128, and $626.259. The savings costs saved by EO over PSO in these cases are $4.587, $0.747, and $2.784, corresponding to 0.87%, 0.13%, and 0.45%, respectively. As stated in the study (Nguyen et al., 2021), a method with better minimum, mean and maximum objective function is more effective than others. So, EO is more suitable than PSO for the system. 4.3 The results of Case 2 In the study case, the integrated system supplies electricity to loads over 24 hours as shown in Fig. 6. In addition, the generation of the wind and solar photovoltaic power plants are also given in Fig. 6. Both PSO and EO are applied to determine the optimal generation of the ten thermal generators and the summary of 50 runs is given in Fig. 7. In the figure, three comparison criteria are employed to analyse the efficiency of the two applied methods while dealing with the larger scale of the MELD problem, including the Minimum TEGC within 24 hours (Minimum), Average TEGC within 24 hours (Average), and Maximum TEGC within 24 hours (Maximum). The green bars stand for the results obtained by PSO, while the blue ones represent the similar values achieved by EO. For the first criterion, EO achieves $15384.5 of the TEGC, while that of PSO is $15435.7. It is easy to figure out that EO has saved $51.2, or 0.33%, over PSO on this criterion in an operation day. Next, in Fig. 5 The comparison between two applied methods and other methods on different values of load demand: : a) Load of 2400MW, b) Load of 2500MW, c) Load of 2600MW, and d) Load of 2700MW Fig. 6 Load demand and generation of wind and solar plants over 24 hours 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 4 0 0 2 4 0 0 2 4 0 0 3 3 0 0 3 3 0 0 3 3 0 0 3 3 0 0 3 3 0 0 3 4 0 0 3 4 0 0 3 3 0 0 3 3 0 0 3 0 0 0 3 0 0 0 3 0 0 0 3 0 0 0 3 0 0 0 3 0 0 0 2 0 0 0 2 0 0 0 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 2000 2500 3000 3500 4000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 P o w e r fr o m s o la r a n d w in d p o w e r p la n ts ( M W ) P o w e r d e m a n d ( M W ) Hour Power demand (MW) Wind power (MW) Solar power (MW) H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |596 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE the average and maximum criteria, the savings costs saved by EO over PSO are even larger. Specifically, the savings costs in these criteria are $325.7 and $1132.1, corresponding to 2.12% and 7.31%, respectively. Clearly, EO completely outperforms PSO while dealing with the large-scale MELD. Fig. 8 shows the fuel cost at each hour for the best, mean, and worst solutions of over 50 solutions obtained by PSO and EO, corresponding with the subfigures a, b, and c. In terms of the best solution presented in Fig. 8a, EO always offers a better value of TEGC per hour. Particularly, regardless of the variation in load demand within 24 hours, the TEGC values found by EO at each hour are lower than the ones obtained by PSO at the same time. However, the differences between the TEGC values resulting from EO compared to the similar ones belonging to Fig. 7 Summary of 50 runs found by PSO and EO 15435.7 15720.3 16624.9 15384.5 15394.6 15492.8 14500 15000 15500 16000 16500 17000 Minimum Average Maximum T E G C ( $ ) PSO EO Fig. 8 The TEGC of each hour obtained by PSO and EO: a) TEGC of the best run, b) TEGC of the mean run, c) TEGC of the worst run H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |597 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE PSO are not much in Fig. 8a. Hence, the outstanding characteristics of EO over PSO are not clearly shown. In Fig. 8b, the efficiency of a modern meta-heuristic algorithm such as EO starts to reveal itself. In the figure, the contrast in TEGC values obtained by EO and PSO starts to depart from each other in all hours and is easy to observe. EO always results in lower values of TEGC over PSO throughout the 24 hours. Noticeably, the TEGCs found by EO are always better than PSO even in the high load demand hours. Finally, the observations on Fig. 8c indicate that the high efficiency of EO is enhanced when compared to PSO. Particularly, the better degree of TEGC values resulted by EO can be seen vividly for 24 hours, regardless of high demand hours or low demand ones. In addition to that, the TEGC values from 8 to 17 hours found by EO compared PSO are substantially better than the same period as mentioned in Fig. 8b. The most optimal generations for the ten thermal units corresponding to the best solution are reported in Fig. 9. In general, the generation rate of the units is dependent on the load level at each hour and units with higher generation is more effective than units with lower generation. 4.3 Discussion on performance of EO and on renewable energies EO is a metaheuristic algorithm mainly based on randomization, exploitation, and exploration. Randomization is a general characteristic for approximately all algorithms belonging to the metaheuristic family. So, EO as well as PSO are sensitive to settings of population, iterations and run number, leading to different results for the number of trials. Because of the unexpected characteristic, EO has been implemented for 50 trials for each study case to summarize the best, mean and worst results for comparisons. Fig. 4 above indicates the very stable ability of EO in reaching the best solution for four study cases. EO found the same solution quality for the 50 runs, which can be seen via the line of fuel cost, meanwhile PSO must suffer very high fluctuation among the 50 runs, especially the deviation between the best and the worst runs. The two algorithms were tested by using the same population of 50 and the same iteration of 100. About the structure, EO only use one main Equation (11) to update solutions, here they are generation of thermal units. PSO has updated velocity and location in which velocity is equivalent to an increased interval and location is solution. The algorithms have the same characteristic of using randomization, but EO almost do not have sensitivity to the randomization. Here, the setting of population and iteration are 50 and 100, not having enough impact on the change of EO but they highly influence the fluctuations of PSO. As compared to other previous algorithms shown in previous studies, the best performance of EO cannot be shown in terms of reaching less fuel costs than these algorithms. EO only reached smaller cost than several algorithms such as SDE, IEP, and ARDGA. Other algorithms did not show all simulated results as the study, so the full comparison is impossible to carry out. On the other hand, presentation of settings of iterations and population was not done in the studies too. However, the use of 50 and 100 for population and iteration is not high setting for the algorithm. And the comparison with PSO is the evidence for this statement. PSO could not reach the best solution although it was run 50 trials. So, EO is a very effective algorithm for this research. In the second case of testing EO performance, the supplied power to load over 24 hours with an additional supply from solar and wind power plants are both employed. The second case is much more complicated than the first case in terms of multiple hours and the presence of renewable energies-based generating units. The uncertainty of wind and solar have not been considered, and this is the major shortcoming of the study. However, the power sources have a high contribution to form a load curve for one day within 24 hours and reduce the fuel cost from thermal units. The fixed power of the renewable energies- based units can guarantee energy security, i.e., load demand can be satisfied all hours. However, there are other cases that the real power of renewable power plants is smaller than predetermined values. This case is serious for the power system and frequency can be reduced to a smaller value than rated frequency. For this case, thermal units or battery energy storage system (BESS) as considered by (Kheiter et al., 2022) can supply more power to compensate for the lack of the renewable energies-based units. To reach the purpose, BESS must save enough energy for discharge meanwhile the thermal units copes with the challenge of increasing power or starting up shut down units. This unexpected issue should be solved in the future work and the power system will become more effective in the future. Besides, the uncertainties and the mathematical models of renewable energy sources, including solar and wind power plants, as considered in (Kaur et al., 2021; Khamharnphol et al., Fig. 9 The most optimal generations of each thermal unit over 24 hours 0 500 1000 1500 2000 2500 3000 3500 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 P o w e r o u tp u t (M W ) Hour Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 Unit 9 Unit 10 H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |598 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE 2022), should be employed while solving the MELD problem as another complex constraint for assessing the efficiency of a modern meta-heuristic algorithm such as EO. This implementation is also a great method to improve the overall quality of the study and make it closer to practice, where the penetration rate of these sources is growing. 5. Conclusion In this study, one conventional ten-thermal unit power system and one integrated ten-thermal unit, three-solar photovoltaic plant and one wind-plant power systems are successfully solved by conventional meta-heuristic algorithm, named Particle swarm optimization and a modern one, named Equilibrium Optimizer. The evaluation of results about the minimum and maximum total electricity generating cost values indicated that EO is completely superior to PSO. Specifically, EO provides a quick response capability, lower fluctuations of fitness values for fifty independent runs, and fast convergence to optimal results. Therefore, EO has been acknowledged as a highly effective method for solving MELD problems. In the future, the MELD should be expanded with the consideration of higher-degree complicated constraints such as prohibited operation zone, valve point effects and uncertainties of renewable. In addition to highly valuable contributions above, this study also copes with the following shortcomings: ➒ Other constraints of thermal generating units are still not evaluated on this study such as valve point effects, ramp-rate limits, prohibited zones, etc. ➒ The study only considers the case that both wind and solar power plants generate enough power to meet the demand as predicted. For other cases with a mismatch between the power generated by wind and solar power plants and the load demand, the evaluation in these cases is not taken place. As a result, no actions are proposed to deal with these scenarios while power from wind and solar power plants is lower or higher than load demand. ➒ The uncertainties of the wind and solar power plants is not clearly discussed and evaluated strictly. By fully acknowledging the shortcomings, there are many improvements that must be conducted in the future. ➒ More high-complex constraints related to MELD must be taken into account. ➒ The uncertainties of power generated by wind and solar power plants must be fully evaluated while solving MELD. ➒ More scenarios of power system operation must be employed and analyzed. Especially, as there is a mismatch between the forecast and the real power production from wind and solar power plants. Consequences as well as solutions for the cases have to be calculated and proposed to reach the least impact. Nomenclature G The number of thermal generators 𝑔, 𝛿𝑔, and 𝛾𝑔 The fuel coefficients PGg The power output of thermal generator g (MW) 𝑃𝐷 The power demand required by the load (MW) 𝑃𝐿 The total power loss of the transmission process (MW) βˆ‘ 𝑃𝑔 𝐺 𝑔=1 The total power generated by all thermal generators (MW) π‘ƒπ‘Š and 𝑃𝑆 The power supplied by wind and solar generators (MW) Bgh, B0g, and B00 The loss factors PGg and PGh The power injected by the generators g and h (MW) 𝑃𝐺𝑔,π‘šπ‘–π‘› and 𝑃𝐺𝑔,π‘šπ‘Žπ‘₯ The working limitations of generator g (MW) βˆ‘ 𝑃𝑆𝐺,π‘ž 𝑁𝑆 π‘ž The total power generated by all solar power plants (MW) 𝑃𝑆𝐺,π‘ž(MW) Active power generated by solar power plant q (MW) 𝑃𝑆𝐺,π‘ž π‘šπ‘–π‘› and 𝑃𝑆𝐺,π‘ž π‘šπ‘Žπ‘₯ The minimum and maximum power generated by solar generator q (MW) NP The population number 𝑉𝑖 𝑛𝑒𝑀 and 𝑃𝑖 𝑛𝑒𝑀 The new velocity and new position of the individual i. 𝑉𝑖 and 𝑃𝑖 The current velocity and current position of individual i qc1 and cq2 The accelerating factors 𝛾1 and 𝛾2 The random numbers in the interval between 0 and 1 𝑃𝐡𝑒𝑠𝑑,𝑖 and 𝑃𝐺𝑏𝑒𝑠𝑑,𝑖 The best position at the time considered and the best position at all times of the individual i. vr A random value in the range of (0, 1) π‘Ÿπ‘‘3 and π‘Ÿπ‘‘4 The random values in the interval between 0 and 1 𝑐𝑓 The comparative factor Acknowledgments We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study. Author Contributions: H.D.N.: Methodology, formal analysis, writingβ€”original draft, K.H.T.; supervision, resources, project administration, N.A.L.; Conceptualization, writingβ€”review and editing, project administration, validation. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Ho Chi Minh City University of Technology (HCMUT), VNU-HCM. Conflicts of Interest: Authors declare no conflict of interest. References Al-Betar, M.A., Awadallah, M.A. & Krishan, M.M. (2020). A non-convex economic load dispatch problem with valve loading effect using a hybrid grey wolf optimizer. Neural Computing and Applications, 32 (16), 12127–12154. https://doi.org/10.1007/s00521-019-04284-9. Alsumait, J.S., Sykulski, J.K. & Al-Othman, A.K. (2010). A hybrid GA– PS–SQP method to solve power system valve-point economic dispatch problems. Applied Energy, 87 (5), 1773–1781. https://doi.org/10.1016/j.apenergy.2009.10.007. Amjady, N. & Nasiri-Rad, H. (2010). Solution of nonconvex and nonsmooth economic dispatch by a new Adaptive Real Coded Genetic Algorithm. Expert Systems with Applications, 37 (7), 5239– 5245. https://doi.org/10.1016/j.eswa.2009.12.084. Augusteen, W.A., Geetha, S. and Rengaraj, R. (2016). Economic dispatch incorporation solar energy using particle swarm optimization. Paper presented at 2016 3rd International Conference on Electrical Energy Systems (ICEES). 2016. pp. 67–73. https://doi.org/10.1109/ICEES.2016.7510618. Balamurugan, R. & Subramanian, S. (2007). Self-adaptive differential evolution-based power economic dispatch of generators with valve- point effects and multiple fuel options. International Journal of Electrical and Computer Engineering, 1 (3), 543–550. https://doi.org/10.1007/s00521-019-04284-9 https://doi.org/10.1016/j.apenergy.2009.10.007 https://doi.org/10.1016/j.eswa.2009.12.084 https://doi.org/10.1109/ICEES.2016.7510618 H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |599 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE Basu, M. (2008). Dynamic economic emission dispatch using nondominated sorting genetic algorithm-II. International Journal of Electrical Power & Energy Systems, 30(2), 140–149. https://doi.org/10.1016/j.ijepes.2007.06.009. Chang, X., Xu, Y., Sun, H., Khan, I. (2021). A distributed robust optimization approach for the economic dispatch of flexible resources. International Journal of Electrical Power & Energy Systems, 124, 106360. https://doi.org/10.1016/j.ijepes.2020.106360. Chen, C., Zou, D. & Li, C. (2020). Improved Jaya Algorithm for Economic Dispatch Considering Valve-Point Effect and Multi-Fuel Options. IEEE Access, 8: 84981–84995. https://doi.org/10.1109/ACCESS.2020.2992616 . Chen, P.-H. & Chang, H.-C. (1995) Large-scale economic dispatch by genetic algorithm. IEEE Transactions on Power Systems, 10 (4), 1919– 1926. https://doi.org/10.1109/59.476058. Chopra, N., Brar, Y.S. & Dhillon, J.S. (2021). An improved particle swarm optimization using simplex-based deterministic approach for economic-emission power dispatch problem. Electrical Engineering, 103(3), 1347–1365. https://doi.org/10.1007/s00202-020-01164-7 . Coelho, Ld.S. & Mariani, V.C. (2006) Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect. IEEE Transactions on Power Systems, 21(2), 989–996. https://doi.org/10.1109/TPWRS.2006.873410. Dieu, V.N., Ongsakul, W. & Polprasert, J. (2013) The augmented Lagrange Hopfield network for economic dispatch with multiple fuel options. Mathematical and Computer Modelling, 57 (1), 30–39. https://doi.org/10.1016/j.mcm.2011.03.041. Duong, M.Q., Nguyen, T.T. & Nguyen, T.T. (2021) Optimal Placement of Wind Power Plants in Transmission Power Networks by Applying an Effectively Proposed Metaheuristic Algorithm Li, Y. (ed.). Mathematical Problems in Engineering, 2021, 1015367. https://doi.org/10.1155/2021/1015367. Faramarzi, A., Heidarinejad, M., Stephens, B., & Mirjalili, S. (2020). Equilibrium optimizer: A novel optimization algorithm. Knowledge- Based Systems, 191, 105190. https://doi.org/10.1016/j.knosys.2019.105190. Fu, C., Zhang, S. & Chao, K.-H. (2020). Energy Management of a Power System for Economic Load Dispatch Using the Artificial Intelligent Algorithm. Electronics, 9(1). https://doi.org/10.3390/electronics9010108. Ghasemi, M., Akbari, E., Zand, M., Hadipour, M., Ghavidel, S., & Li, L. (2019) An Efficient Modified HPSO-TVAC-Based Dynamic Economic Dispatch of Generating Units. Electric Power Components and Systems, 47 (19–20), 1826–1840. https://doi.org/10.1080/15325008.2020.1731876. Hassan, M.H., Houssein, E. H., Mahdy, M.A., & Kamel, S. (2021). An improved Manta ray foraging optimizer for cost-effective emission dispatch problems. Engineering Applications of Artificial Intelligence, 100, 104155. https://doi.org/10.1016/j.engappai.2021.104155. Hien, C.T., Ha, P.T., Phan-Van, T. H., & Pham, T. M. (2021). Multi- Period Economic Load Dispatch with Wind Power Using a Novel Metaheuristic. GMSARN International Journal, 16 (2022), 165–173. Hlalele, T.G., Zhang, J., Naidoo, R. M., & Bansal, R. C. (2021). Multi- objective economic dispatch with residential demand response programme under renewable obligation. Energy, 218, 119473. https://doi.org/10.1016/j.energy.2020.119473. Jeyakumar, D.N., Jayabarathi, T. & Raghunathan, T. (2006). Particle swarm optimization for various types of economic dispatch problems. International Journal of Electrical Power & Energy Systems, 28 (1), 36–42. https://doi.org/10.1016/j.ijepes.2005.09.004. Kennedy, J. & Eberhart, R. (1995) β€œParticle swarm optimization.” In Proceedings of ICNN’95 - International Conference on Neural Networks. 1995. pp. 1942–1948 vol.4. https://doi.org/10.1109/ICNN.1995.488968. Kheshti, M., Ding, L., Ma, S., & Zhao, B. (2018). Double weighted particle swarm optimization to non-convex wind penetrated emission/economic dispatch and multiple fuel option systems. Renewable Energy, 125, 1021–1037. https://doi.org/10.1016/j.renene.2018.03.024. Kim, J. & Kim, K.-K.K. (2020). Dynamic programming for scalable just- in-time economic dispatch with non-convex constraints and anytime participation. International Journal of Electrical Power & Energy Systems, 123, 106217. https://doi.org/10.1016/j.ijepes.2020.106217. Kumar, M. and Dhillon, J.S. (2018). Hybrid artificial algae algorithm for economic load dispatch. Applied Soft Computing, 71: 89–109. https://doi.org/10.1016/j.asoc.2018.06.035. Kheiter, A., Souag, S., Chaouch, A., Boukortt, A., Bekkouche, B., & Guezgouz, M. (2022). Energy Management Strategy based on marine predators algorithm for grid-connected microgrid. International Journal of Renewable Energy Development, 11(3), 751– 765. https://doi.org/10.14710/ijred.2022.42797. Kaur, S., Brar, Y.S. and Dhillon, J.S. (2021). Short-term hydro-thermal- wind-solar power scheduling: A case study of kanyakumari region of India. International Journal of Renewable Energy Development, 10(3), 635–651. https://doi.org/10.14710/ijred.2021.35558. Khamharnphol, R. Kamdar, I., Waewsak, J., Chaichan, W., Khunpetch, S., Chiwamongkhonkarn, S., & Gagnon, Y. (2022). Microgrid Hybrid Solar/Wind/diesel and Battery Energy Storage Power Generation System: Application to Koh Samui, Southern Thailand. International Journal of Renewable Energy Development, 12(2), 216–226. https://doi.org/10.14710/ijred.2023.47761. Li, X., Wang, W., Wang, H., Wu, J., Fan, X., & Xu, Q. (2020). Dynamic environmental economic dispatch of hybrid renewable energy systems based on tradable green certificates. Energy, 193, 116699. https://doi.org/10.1016/j.energy.2019.116699. Nguyen, T. T., Vu, Q. N., Duong, M. Q., & Le, V. D. (2018). Modified differential evolution algorithm: A novel approach to optimize the operation of hydrothermal power systems while considering the different constraints and valve point loading effects. Energies, 11(3), 540. https://doi.org/10.3390/en11030540. Nguyen, T. T., Vo, D. N., Tran, H. V., & Le, V. D. (2019). Optimal dispatch of reactive power using modified stochastic fractal search algorithm. Complexity, 2019, Article ID 4670820, 28 pages. https://doi.org/10.1155/2019/4670820 . Nguyen, T. T., Nguyen, T. T., Duong, L. T., & Truong, V. A. (2021). An effective method to solve the problem of electric distribution network reconfiguration considering distributed generations for energy loss reduction. Neural Computing and Applications, 33, 1625- 1641. https://doi.org/10.1007/s00521-020-05092-2. Noman, N. & Iba, H. (2008). Differential evolution for economic load dispatch problems. Electric Power Systems Research, 78 (8), 1322– 1331. https://doi.org/10.1016/j.epsr.2007.11.007. Pandit, N., Tripathi, A., Tapaswi, S., & Pandit, M. (2012). An improved bacterial foraging algorithm for combined static/dynamic environmental economic dispatch. Applied Soft Computing, 12 (11), 3500–3513. https://doi.org/10.1016/j.asoc.2012.06.011. Park, J.B., Lee, K. S., Shin, J. R., & Lee, K. Y. (2005). A particle swarm optimization for economic dispatch with nonsmooth cost functions. IEEE Transactions on Power Systems, 20 (1), 34–42. https://doi.org/10.1109/TPWRS.2004.831275. Park, Y.M., Won, J.R. & Park, J.B. (1998). A new approach to economic load dispatch based on improved evolutionary programming. Engineering Intelligent Systems for Electrical Engineering and Communications, 6 (2), 103–110. Parouha, R.P. & Das, K.N. (2018). Economic load dispatch using memory based differential evolution. International Journal of Bio- Inspired Computation, 11 (3), 159–170. https://doi.org/10.1504/IJBIC.2018.091700. Pham, L.H., Dinh, B.H. & Nguyen, T.T. (2022). Optimal power flow for an integrated wind-solar-hydro-thermal power system considering uncertainty of wind speed and solar radiation. Neural Computing and Applications, 34 (13), 10655–10689. https://doi.org/10.1007/s00521-022-07000-2 . Phan, V.-D., Duong, M.Q., Doan, M. M., & Nguyen, T. T. (2021). Optimal Distributed Photovoltaic Units Placement in Radial Distribution System Considering Harmonic Distortion Limitation. International Journal on Electrical Engineering & Informatics, 13 (2). https://doi.org/10.15676/ijeei.2020.13.2.7 Pradhan, M., Roy, P.K. & Pal, T. (2016). Grey wolf optimization applied to economic load dispatch problems. International Journal of Electrical Power & Energy Systems, 83, 325–334. https://doi.org/10.1016/j.ijepes.2016.04.034. Shen, X., Zou, D., Duan, N., & Zhang, Q. (2019). An efficient fitness- based differential evolution algorithm and a constraint handling technique for dynamic economic emission dispatch. Energy, 186, 115801. https://doi.org/10.1016/j.energy.2019.07.131. Suresh, V., Sreejith, S., Sudabattula, S.K., & Kamboj, V. K. (2019). Demand response-integrated economic dispatch incorporating https://doi.org/10.1016/j.ijepes.2007.06.009 https://doi.org/10.1016/j.ijepes.2020.106360 https://doi.org/10.1109/ACCESS.2020.2992616 https://doi.org/10.1109/59.476058 https://doi.org/10.1007/s00202-020-01164-7 https://doi.org/10.1109/TPWRS.2006.873410 https://doi.org/10.1016/j.mcm.2011.03.041 https://doi.org/10.1155/2021/1015367 https://doi.org/10.1016/j.knosys.2019.105190 https://doi.org/10.3390/electronics9010108 https://doi.org/10.1080/15325008.2020.1731876 https://doi.org/10.1016/j.engappai.2021.104155 https://doi.org/10.1016/j.energy.2020.119473 https://doi.org/10.1016/j.ijepes.2005.09.004 https://doi.org/10.1109/ICNN.1995.488968 https://doi.org/10.1016/j.renene.2018.03.024 https://doi.org/10.1016/j.ijepes.2020.106217 https://doi.org/10.1016/j.asoc.2018.06.035 https://doi.org/10.14710/ijred.2022.42797 https://doi.org/10.1016/j.energy.2019.116699 https://doi.org/10.3390/en11030540 https://doi.org/10.1155/2019/4670820 https://doi.org/10.1007/s00521-020-05092-2 https://doi.org/10.1016/j.epsr.2007.11.007 https://doi.org/10.1016/j.asoc.2012.06.011 https://doi.org/10.1109/TPWRS.2004.831275 https://doi.org/10.1504/IJBIC.2018.091700 https://doi.org/10.1007/s00521-022-07000-2 https://doi.org/10.15676/ijeei.2020.13.2.7 https://doi.org/10.1016/j.ijepes.2016.04.034 https://doi.org/10.1016/j.energy.2019.07.131 H.D.Nguyen et al Int. J. Renew. Energy Dev 2023, 12(3), 590-600 |600 ISSN: 2252-4940/Β© 2023. The Author(s). Published by CBIORE renewable energy sources using ameliorated dragonfly algorithm. Electrical Engineering, 101 (2), 421–442. https://doi.org/10.1007/s00202-019-00792-y. Vaisakh, K. & Reddy, A.S. (2013). MSFLA/GHS/SFLA-GHS/SDE algorithms for economic dispatch problem considering multiple fuels and valve point loadings. Applied Soft Computing, 13 (11), 4281–4291. https://doi.org/10.1016/j.asoc.2013.07.001. Xiang, Y., Wu, G., Shen, X., Ma, Y., Gou, J., Xu, W., & Liu, J. (2021). Low-carbon economic dispatch of electricity-gas systems. Energy, 226, 120267. https://doi.org/10.1016/j.energy.2021.120267 . Xin-gang, Z., Ze-qi, Z., Yi-min, X., & Jin, M. (2020). Economic- environmental dispatch of microgrid based on improved quantum particle swarm optimization. Energy, 195, 117014. https://doi.org/10.1016/j.energy.2020.117014. Xiong, G. & Shi, D. (2018). Hybrid biogeography-based optimization with brain storm optimization for non-convex dynamic economic dispatch with valve-point effects. Energy, 157, 424–435. https://doi.org/10.1016/j.energy.2018.05.180 . Zare, M., Narimani, M. R., Malekpour, M., Azizipanah-Abarghooee, R., & Terzija, V. (2021). Reserve constrained dynamic economic dispatch in multi-area power systems: An improved fireworks algorithm. International Journal of Electrical Power & Energy Systems, 126, 106579. https://doi.org/10.1016/j.ijepes.2020.106579. Zhang, H., Liang, S., Ou, M., & Wei, M. (2021). An asynchronous distributed gradient algorithm for economic dispatch over stochastic networks. International Journal of Electrical Power & Energy Systems, 124, 106240. https://doi.org/10.1016/j.ijepes.2020.106240 . Zhang, H., Yue, D., Xie, X., Dou, C., & Sun, F. (2017). Gradient decent based multi-objective cultural differential evolution for short-term hydrothermal optimal scheduling of economic emission with integrating wind power and photovoltaic power. Energy, 122, 748– 766. https://doi.org/10.1016/j.energy.2017.01.083. Β© 2023. The Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-ShareAlike 4.0 (CC BY-SA) International License (http://creativecommons.org/licenses/by-sa/4.0/) https://doi.org/10.1007/s00202-019-00792-y https://doi.org/10.1016/j.asoc.2013.07.001 https://doi.org/10.1016/j.energy.2021.120267 https://doi.org/10.1016/j.energy.2020.117014 https://doi.org/10.1016/j.energy.2018.05.180 https://doi.org/10.1016/j.ijepes.2020.106579 https://doi.org/10.1016/j.ijepes.2020.106240 https://doi.org/10.1016/j.energy.2017.01.083