Analysis of Heuristic Optimization Technique Solutions for Combined Heat-Power Economic Load Dispatch

: Thermal power plants use coal as a fuel to create electricity while wasting a signiﬁcant amount of energy as heat. If the heat and power plants are combined and used in cogeneration systems, it is possible to reuse the waste heat and hence enhance the overall efﬁciency of the power plant. In order to minimize production costs while taking system constraints into account, it is important to ﬁnd out the optimal operating point of power and heat for each unit. Combined heat and power production is now widely used to improve thermal efﬁciency, lower environmental emissions, and reduce power generation costs. In order to determine the best solutions to the combined heat and power economic dispatch problem, several traditional as well as innovative heuristic optimization approaches were employed. This study offers a thorough analysis of the use of heuristic optimization techniques for the solution of the combined heat and power economic dispatch problem. In this proposed work, the most well-known heuristic optimization methods are examined and used for the solution of various generating unit systems, such as 4, 7, 11, 24, 48, 84, and 96, taking into account various constraints. This study analyzes how various evolutionary approaches are performed for various test systems. The heuristic methodologies’ best outcomes for various case studies with restrictions are contrasted.


Introduction
Major worries about a number of causes, most notably climate change, the scarcity of oil and its consequent rise in price, population levels, and energy consumption, are rapidly dominating the world's energy supply and demand landscape.Finding a substitute for fossil fuels, especially petroleum fuels, is therefore crucial from an economic, environmental, and social standpoint [1].
Primary fossil fuels are converted somewhat inefficiently into electricity.The conventional generating plant achieves efficiencies of 50% to 60% only, because most of the heat energy is wasted during the conversion process and discharged into the environment [2].Cogeneration, also known as combined heat and power (CHP) generation, is an advanced and modern technology that outperforms conventional energy conversion systems and is also environmentally friendly [3].
CHP systems are systems that simultaneously provide consumers with electricity and meet heating demands [4].A tri-generation system (cooling, heating, and power generation) can be created by integrating thermally activated technologies into the CHP system to fulfill the consumer's cooling requirement [5].CHP systems can enhance the efficiencies of thermal power plants by over 90% and decrease their environmental effects [6].
CHP systems get more attention because they can enhance the economics and sustainability of electrical generating units [7].CHP economic dispatch can improve the efficiency of the energy conversion process in thermal power stations and reduce the cost of power generation [8].CHP units have the capacity to generate electricity from a range of fuels while simultaneously recovering and reusing the heat that would typically be lost during the creation of electricity [9].
Moreover, the use of CHP generation systems decreases pollutant emissions, such asCOx, SOx, and NOx [10][11][12][13][14][15].Because of these factors, researchers have focused increasingly on CHP units in recent years in an effort to fully explore their potential for meeting consumer demands for heat and electricity [16,17].The economic dispatch (ED) issue, which may be seen as the researchers' initial attempt to maximize the advantages of power systems, tries to determine the best scheduling of the generation units to reduce the fuel cost of power generation subject to operational and technological restrictions [18,19].The combined economic dispatch problem not only offers significant economic power generation advantages but also lessens the negative consequences of polluting gases [20][21][22].During studies in this area of research, it was found that many available articles showed the effectiveness of heuristic optimization for the solution of combined heat and power economic dispatch (CHPED), but no one had demonstrated a complete comparative study between all the proposed heuristic approaches in this research area.Due to this research gap, the authors of this article were motivated to conduct this study.
The objective of this research is to investigate the best heuristic optimization techniques used to address the nonconvex and non-smooth CHPED optimization issues.The bulk of the articles that used heuristic optimization techniques to find the optimum solution to the CHPED issue are discussed in this proposed article, to the best knowledge of the authors.To familiarize readers with the heuristic approaches used, a brief explanation of the utilized heuristic methods is given, and the most important contributions of each research work are introduced.Additionally, in order to create a helpful survey on the usage of heuristic approaches for the solution of the CHPED issue, the best solutions found in the articles under consideration are tallied.There are comparisons between the publications that have been examined in terms of objective function, restrictions, minimal operational cost, and computing time.This article will be of great use to scholars looking at the best generation planning for CHP systems.The rest of the document is structured as follows: Section 2 provides reviews of several heuristic optimization methods that have been applied to the CHPED issue in distinct case studies.The CHPED problem formulation is shown in Section 3. A thorough review of heuristic optimization techniques, handling various constraints and benefits of the heuristic techniques, are shown in Section 4. Comparative results in terms of cost generation and computation time taken in convergence by various optimization techniques are shown in Section 5, and the proposed work's conclusion is given in Section 6.

Literature Review
Researchers have been interested in the CHP economic dispatch problem recently, and the answer has been found in earlier literature utilizing a variety of conventional and modern heuristic methods.In the early 1990s, research started in the field of CHP problems and suggested a quadratic programming method for the solution of this problem for 15 generating unit systems [1].The Lagrangian relaxation technique was proposed for the solution to the CPH problem.They considered one case study for four generating units with load balance and power generation limits [2].
A classical method called Benders decomposition is used for the solution in cases of four and five generating units with two and three co-generation systems with inequality constraints [3].To obtain optimum results, the four generating units (two co-generation and one heat unit) were optimized by improved PSO (SPSO) for the load demand of 200 MW and the heat demand of 115 MWth [4].A novel bee colony optimization algorithm [5] and AI (artificial immune) systems [6] were suggested for the study of a 4-generating unit system of CHPED for the load demand of 600 MW and 150 MWth.
Similarly, the firefly algorithm was used for the optimization of CHED for the four generating units, where units 2 and 3 have co-generation and unit 4 has heat.The proposed optimization is used to obtain the global results of power and heat demand of 200 MW and 115 MWth, respectively [7].For the test data of single heat area and power area systems with loads and heat demands of 200 MW and 115 MWth, respectively, MADS-PSO and, for various power and heat demands, MADS-DACE and MADS-DACE were used [8].
For the solution of nonlinear CHPED, they suggested and demonstrated the effectiveness of TVAC-PSO.The proposed technique was tested on two case study data sets.In the first test case, they considered a four-unit system for the power and heat demand of 200 MW and 115 MWth, respectively.In the second case, they considered a five-unit system for three different load conditions [9].
For the optimization of a large unit data set of CHPED along with constraints, crisscross optimization was used, and it was found that the proposed techniques gave a global solution for such a large data set.They solved six different cases, and in all cases, the proposed techniques gave the best-optimized solutions compared to the other algorithms [10].Five different cases of CHPED were considered and global solutions were obtained using the exchange market algorithm.This algorithm was tested for different loads in different data sets (small and large data), and it was found that it gave the global solution in all considered test data [11].
Based on the behavior of humpback whales, the WOA optimization used for the solution of CHPED considered test cases of 24, 84, and 96 data points.The WOA performs well for non-convex nonlinear optimization problems [12].A crossover and mutation-based improved GA was given for the solution to the CHPED problems [13].
For the non-linear combination of heat and power dispatch systems, the AMPSO algorithm was proposed.To improve the efficiency of the proposed technique, the Taguchi approach was used [14].A hybrid algorithm is suggested by article [15] for solving the CHP economic emission dispatch problem in such a way as to reduce the cost of generation and emission.Similarly, a deep study discusses the different optimization techniques recommended by the researchers for the solution of the CHP economic emission dispatch problem in article [16].
The CHP economic dispatch problem was solved using the HBOA optimization algorithm, which is based on the interaction of coworkers and employees [17].A combination of HBA and JSA, commonly known as HBJSA, was used for the solution of the CHP economic load dispatch.The proposed methods overcame the problems associated with HBA and JSA, easily handled the constraints, and solved the nonlinear CHP problem [18].For the solution to nonlinear CHP, the economic dispatch group search method was suggested, which is based on opposition [19].The foundation of GSA is the gravitational law, which helps particles move in the search space used to solve CHPED [20].
To solve the CHPED problem with various constraints, biogeography-based particle swarm optimization was suggested.In this PSO, particles update their position by using the migration operator [21].Based on the cuckoo bird's reproduction behavior, CSA techniques were proposed for the CHPED with a valve-point loading problem solution [22].The CPH problem was solved using a hybrid approach that combined PPS and CSO for local and global search, respectively [23].Article [24] demonstrates the use of the group search optimizer for CHP dispatch problems.However, a hybrid (TVAC-GSA-PSO) method was used to solve the large-scale, complex CHPED problems [25].One more hybrid method (bat algorithm + artificial bee colony) with a chaotic-based self-adaptive search strategy known as CSA-BA-ABC was suggested in article [26] to solve the large-scale, non-differential CHP economic dispatch.
The suggested FS technique in Article [27] was used to solve the CHP economic dispatch in such a way that the cost should be low and fulfill the constraints.Optimization techniques based on the Kho-Kho game (a game played between two teams) were proposed for the CHPED problem [28].The CHPED problem was solved for a large data system (48 units) with a minimum total operation cost [29].To minimize the overall fuel cost of cogeneration units, an improved marine predator optimization algorithm was used [30].
For the solution to the CHP dispatch problem with valve-point loading effects and prohibited operating zones, a wavelet-mutated slime mold technique was used [31].For the purpose of calculating the system-wide additional costs associated with optimum dispatch using the search optimization approach, an explicit formula was created [32].For the operation of CHP, a demand response algorithm was used [33].The heat transfer search technique, which follows the laws of thermodynamics and heat transfer, was used to find the solution to complicated CHP economic dispatch problems [34].
To address the CHP dynamic economic dispatch, a new differential evolution method that has an attractive component and gives mutant vectors more possibilities to find prospective locations utilizing migrating variables was proposed [35].The TVAC-PSO was suggested to address the multi-objective CHPEED and dynamic economic emission dispatch challenges in the context of operational limitations [36].
It was suggested to combine particle swarm optimization algorithms with enthusiasmaided teaching and learning-based optimization algorithms to simultaneously reduce overall generation costs while taking constraints into account [37].For handling a very large CHPED (140 bus system), an article proposed a multiobjective technique that was based on a chaotic opposition-based strategy [38].The usefulness of the group search method in solving the CHPED issue was reported in an article [39].
For the CHPED problem, a genetic algorithm method was suggested [40].An efficient tool called the search algorithm was proposed to solve the CHPED with ramp rate constraints [41].The authors of [42] suggested a cuckoo optimization algorithm to solve the CHP in such a way that energy production costs are minimized.In order to solve the CHPED problem, a paper proposed an IDE approach that makes use of mutation operators, dynamical crossover, and population randomization [43].
To solve CHP dispatch problems with bounded and feasible operating regions, researchers used a TLBO approach.In this method, an opposition-based learning approach was incorporated so that convergence speed was enhanced and the simulation results were improved [44].
Grey wolf optimization techniques were used for the solution of CHPD.The effectiveness of the proposed algorithm was tested on the data of 4, 7, 11, and 24 units [45].To reduce the generation cost and environmental emissions, a multiobjective fuzzy-operated system was proposed for the CHPED problem [46].For the optimization of the power economic dispatch problem along with valve loading and multiple fuel constraints, an improved genetic algorithm approach was proposed.The proposed algorithm was a combination of an improved genetic algorithm and multiplier updating [47].
Using penalty and binary concepts, researchers discussed a cuckoo algorithm for the optimized CHPED problem [48].A Mühlenbein mutation-based coded genetic algorithm was presented for the solution of the CHP economic dispatch problem.Such mutations enhance the convergence process and improve the results [49].A multi-player harmony search technique was recommended for the resolution of a non-linear large-scale CHPED issue.The proposed methods were evaluated using data from CHPED 24-and 84-unit systems [50].An MPSO with a Gaussian random variable was suggested for the optimization of the CHPED problem.The proposed technique had good convergence speed and gave a global solution to the problem [51].Other authors created a new cuckoo search with elitist CSA to address the issue of CHP economic load dispatch [52].The CHPED problem was suggested to be solved using improved particle swarm optimization in Article [53].

Problem Formulation of CHPED
Thermal generating units, cogeneration units, and heat-only units were taken into consideration for the problem formulation of the CHPED.The heat-power viable operation zone of a combined cycle cogeneration unit is depicted in Figure 1.The KLMNOP boundary curve encloses the viable operation zone.
with elitist CSA to address the issue of CHP economic load dispatch [52].The CHPED problem was suggested to be solved using improved particle swarm optimization in Article [53].

Problem Formulation of CHPED
Thermal generating units, cogeneration units, and heat-only units were taken into consideration for the problem formulation of the CHPED.The heat-power viable operation zone of a combined cycle cogeneration unit is depicted in Figure 1.The KLMNOP boundary curve encloses the viable operation zone.
The heat capacity rises along the boundary curve LM as the generation of electricity falls, whereas it falls along the curve MN.It is obvious that the unit's maximum output power is reached along the KL curve.On the other hand, the MN curve is where the unit produces the most heat.The main goal of the issue is to estimate the heat and power generation rates for each unit in order to minimize the cost of heat and power generation while meeting the heat and power demands.The CHP load dispatch problem is represented mathematically as follows: where   (   ) is total generation cost,   (   ,    ) is generation cost with CHP units, and   (  ℎ ) is generation cost using heat-only units.Np, Nc, and Ck denote the number of power only units, CHP units, and heat only units, respectively.Similarly, i, j and k show the number of power-only units, CHP units, and heat-only units.
The quadratic cost function of power-only units is given as where   (   ) denotes fuel cost of the i th generating units, and ai, bi, and ci are the cost coefficients of power-only units.A combined heat-power cogeneration system is given as And a heat-only unit is defined as The heat capacity rises along the boundary curve LM as the generation of electricity falls, whereas it falls along the curve MN.It is obvious that the unit's maximum output power is reached along the KL curve.On the other hand, the MN curve is where the unit produces the most heat.
The main goal of the issue is to estimate the heat and power generation rates for each unit in order to minimize the cost of heat and power generation while meeting the heat and power demands.The CHP load dispatch problem is represented mathematically as follows: where C i (P P i ) is total generation cost, C j (P c j , H c j ) is generation cost with CHP units, and C k (H h k ) is generation cost using heat-only units.Np, Nc, and C k denote the number of power only units, CHP units, and heat only units, respectively.Similarly, i, j and k show the number of power-only units, CHP units, and heat-only units.
The quadratic cost function of power-only units is given as where C i (P p i ) denotes fuel cost of the i th generating units, and a i , b i , and c i are the cost coefficients of power-only units.
A combined heat-power cogeneration system is given as And a heat-only unit is defined as

Problem Formulation with Valve-Point Effects
In traditional thermal power plants, a large number of steam valves are utilized to increase turbine speed when the load is high.The plant's cost function is altered as a result of opening the valve in this way.A sinusoidal element is added to the quadratic cost function of traditional thermal units to simulate valve-point consequences.The valve-point effect is taken into account, which creates a non-convex optimization issue.The cost function for power-producing units under the influence of valve loading is expressed as where a i , b i , and c i are the fuel coefficients, and d i and e i are the valve-loading coefficients.

Constraints
When power is generated at thermal power plants, it faces many limitations called constraints.The following constraints are considered when the CHP problem is formulated: 3.2.1.Power Balance Generated power must be equal to the load demand plus the loss of power in the transmission line.It is defined as follows: where P p i is the power generated by the i th generating units, and P d and P Loss are the demand of power and power loss in the transmission line, respectively.Power loss in the transmission line is given as where B il , B ij , and B jm are the line loss coefficients.

Heat Balance
The production of heat is always equal to the demand for heat, called the heat balance, and it is formulated as follows: where H j is the heat generated due to the co-generation system, H k is the heat generated due to the heat-only unit, and H d is the head demand.

Heuristic Optimization Techniques Analysis
The numerous demands make it difficult to include cogeneration units in the economic dispatch of the power system.The cogeneration units' mutual dependence on each other's heat-power capacity makes it difficult to economically dispatch cogeneration units into the power grid due to the numerous demands (for both heat and electricity).Researchers have put forth a variety of heuristic methods for optimizing the CHPED problem.The many heuristic methods proposed for CHPED optimization are shown in Table 1.This approach prevents premature convergence and increases the precision of the solution.
[22] Cuckoo search algorithm (CSA) Valve point, power losses, and power balance 4 and 5 units Controls parameters in such a way that they evaluate the high-quality solution and take less computational time.
[23] CPSO Prohibited operating zones, valve point, and transmission losses 4, 7, and 24 units Enhances the quality of the answer while requiring fewer function evaluations.
[25] Hybrid TVAC-GSA-PSO Power balance and generation limits 24 units, 48 units, This technology is robust in evaluating the minimum generation cost with less expensive solutions.
[26] CSA-BA-ABC Power and heat balance and prohibited operation zones

5-and 7-unit test system
Delivering a high-quality solution with more economic benefits and no convergence issues [27] SFS Power balance and generation limits

5-and 7-unit test system
It is possible to avoid local minima and require less computing time.

5-and 7-unit test system
This method imitates the special technique the chasing squad used to touch the runners team.Toimprove the efficiency and simulation solution, Gaussian random variables were used.

Test case 1
The first case considered the test data of a four-unit system with one available poweronly unit, two CHP units, and one available heat-only unit.The test data for this case is taken from the articles [7- 9,11,13,27,32,37].All the optimization techniques were tested for the power and heat demands of 200 MW and 115 MWth, respectively.Table 2 shows the comparative results of FA [7], MADS-DACE [8], TVAC-PSO [9], CSO [10], EMA [11], IGA-NCM [13], SFS [27], ETLBOIPSO [37], and GWO [45] for the load demand of 200 MW and heat demand of 115 MWth. Figure 2 shows the total costs obtained by different optimization techniques for the load and heat demand of 200 MW and 115 MWth, respectively; out of all the techniques, it is shown that ETLBOIPSO [37] gave the best results (total cost = $9178.9934),whereas the other techniques gave almost the same results.Figure 2 shows the total costs obtained by different optimization techniques for the load and heat demand of 200 MW and 115 MWth, respectively; out of all the techniques, it is shown that ETLBOIPSO [37] gave the best results (total cost = $9178.9934),whereas the other techniques gave almost the same results.
Figure 3 shows the total costs obtained by different optimization techniques for a load demand of 600 MW and a heat demand of 150 MWth.Out of all the techniques, CSO [10] gave the best results, whereas AIS [6] gave the worst results for this case study.
Figure 3 shows the total costs obtained by different optimization techniques for a load demand of 600 MW and a heat demand of 150 MWth.Out of all the techniques, CSO [10] gave the best results, whereas AIS [6] gave the worst results for this case study.
Figure 4 shows the total costs obtained by different optimization techniques for a load and heat demand of 2350 MW and 1250 MWth, respectively.In this case, ETLBOI-PSO [37] gave the best results, whereas HBOA [17] gave the worst results.

Test Case 4
This case study took data from a large system with non-convex fuel costs [10,11,18,[29][30][31]36,37].These large test system had 48 units (26 power-only units, twelve CHP units, and ten heat-only units).The comparative results of the new heuristic optimization techniques for the power and heat demands of 4700 MW and 2500 MWth, respectively, are shown in Table 5.Compared to all the other techniques, KKO [28] gavea minimum cost of $115,422, whereas the OQNLP [29] technique reported a generation cost of $116,993.2,which was the maximum, compared to the other methods.
Table 5. Costs obtained by different heuristic optimization techniques for the 48-unit system (power and heat demands of 4700 MW and 2500 MWth).
Figure 5 shows the total costs obtained by different optimization techniques for a load and heat demand of 4700 MW and 2500 MWth, respectively.In this case, ETLBOI-PSO [37] gave the best results, whereas OQNLP [29] gave the worst results.

Test Case 5
In this case, a large test system of 84 units was taken into consideration.This test case had 40 generating, 24 cogeneration, and 20 heat-only units [12,30,31].The test results of various optimization techniques for the 84-unit system (5000 MWth and 12,700 MW of heat and power demands, respectively) are shown in Table 6.The MPHS [50] techniques reported a minimum generation cost of $288,157.4297,which was a minimum compared to all the other techniques, and it took 76.65 s to compute, which was also the least amount of computation time, compared to all the other listed techniques in Table 6.

Test Case 5
In this case, a large test system of 84 units was taken into consideration.This test case had 40 generating, 24 cogeneration, and 20 heat-only units [12,30,31].The test results of various optimization techniques for the 84-unit system (5000 MWth and 12,700 MW of heat and power demands, respectively) are shown in Table 6.The MPHS [50] techniques reported a minimum generation cost of $288,157.4297,which was a minimum compared to all the other techniques, and it took 76.65 s to compute, which was also the least amount of computation time, compared to all the other listed techniques in Table 6. Figure 6 shows the total costs obtained by the different optimization techniques for a load and heat demand of 12,700 MW and 5000 MWth, respectively.In this case, CLWSMA [31] gave the best results, whereas CODED GENETIC ALGORITHM [49] gave the worst results.

Test Case 6
In this case, again, a larger test data set of a 96-unit system was available, with 52 traditional power units, 24 cogeneration units, and 20 heat-only units [12,17,30,31].All the algorithms' comparative results are listed in Table 7 for the load demand of 9400 MW and the heat demand of 5000 MWth.The CLWSMA [31] method reported a minimum generation cost of $235,083.367,which was the least, compared to the other listed techniques in Table 7.

Test Case 6
In this case, again, a larger test data set of a 96-unit system was available, with 52 traditional power units, 24 cogeneration units, and 20 heat-only units [12,17,30,31].All the algorithms' comparative results are listed in Table 7 for the load demand of 9400 MW and the heat demand of 5000 MWth.The CLWSMA [31] method reported a minimum generation cost of $235,083.367,which was the least, compared to the other listed techniques in Table 7. Figure 7 shows the total costs obtained by different optimization techniques for a load and heat demand of 9400 MW and 5000 MWth, respectively.In this case, HBOA [17] gave the best results, whereas RCGA-IMM [49] gave the worst results.

Conclusions
This article presents a deep analysis of various heuristic optimization techniques used for the optimum solution of CHPED.The CHPED problem is formulated along with various constraints shown in Table 1, which increase the complexity of the system and make classical optimization methods ineffective at finding an optimal solution.Numerous population-based heuristic optimization approaches have now been used to solve the CHPED issue in order to address the deficiencies of traditional optimization techniques.In this article, we consider many of the heuristic optimization techniques shown in Table 1, which are used to solve the CHPED problems with different load and heat demand conditions.Some methods are used to solve small generating units, such as 4, 7, and 24 units, while others are used for large generating units, such as 48-, 84-, and 96-unit systems.In this article, we try to show the effectiveness of optimization techniques for small generating units in a large available unit system.This study covered six cases for different unit systems.It is observed that almost all techniques are able to solve the CHPED problem in a very short amount of computation time.As in case 1, almost all the methods give the same results; only the computation time is different.The WOA [12], the heap-based optimization algorithm (HBOA) [17], the hybrid heap-based and jellyfish search algorithms [18], the modified group search optimizer [24], the comprehensive learning wavelet-mutated slime mold algorithm [31], the differential evolution [35], and the MPHS [50] techniques are found effective for small as well as large generating unit systems.For all the methods, the results are similar to each other, but in some cases, the results from two to three techniques are better in terms of minimum generation cost, which is already explained in the results analysis section.

Conclusions
This article presents a deep analysis of various heuristic optimization techniques used for the optimum solution of CHPED.The CHPED problem is formulated along with various constraints shown in Table 1, which increase the complexity of the system and make classical optimization methods ineffective at finding an optimal solution.Numerous populationbased heuristic optimization approaches have now been used to solve the CHPED issue in order to address the deficiencies of traditional optimization techniques.In this article, we consider many of the heuristic optimization techniques shown in Table 1, which are used to solve the CHPED problems with different load and heat demand conditions.Some methods are used to solve small generating units, such as 4, 7, and 24 units, while others are used for large generating units, such as 48-, 84-, and 96-unit systems.In this article, we try to show the effectiveness of optimization techniques for small generating units in a large available unit system.This study covered six cases for different unit systems.It is observed that almost all techniques are able to solve the CHPED problem in a very short amount of computation time.As in case 1, almost all the methods give the same results; only the computation time is different.The WOA [12], the heap-based optimization algorithm (HBOA) [17], the hybrid heap-based and jellyfish search algorithms [18], the modified group search optimizer [24], the comprehensive learning wavelet-mutated slime mold algorithm [31], the differential evolution [35], and the MPHS [50] techniques are found effective for small as well as large generating unit systems.For all the methods, the results are similar to each other, but in some cases, the results from two to three techniques are better in terms of minimum generation cost, which is already explained in the results analysis section.

Figure 1 .
Figure 1.Feasible operating region for a co-generation system.

Figure 1 .
Figure 1.Feasible operating region for a co-generation system.

Figure 2 .Figure 2 .
Figure 2.Total costs for load demand of 200 MW and heat demand of 115 MWth.

Figure 3 .Table 3 .Figure 3 .
Figure 3.Total costs for load demand of 600 MW and heat demand of 150 MWth.Table 3.Comparative performances of the various optimization techniques for the 7-unit system for the load demand of 600 MW and heat demand of 150 MWth.Optimum Results of TVAC-CSO EMA IGA-NCM HTS GSO GWO RCGA-I Appl.Sci.2023, 13, 10380 13 of 19

Figure 4 .
Figure 4. Total costs for power and heat demands of 2350 MW and 1250 MWth, respectively.

Figure 4 .
Figure 4. Total costs for power and heat demands of 2350 MW and 1250 MWth, respectively.

Figure 5 .
Figure 5.Total costs for power and heat demands of 4700 MW and 2500 MWth, respectively.

Figure 5 .
Figure 5.Total costs for power and heat demands of 4700 MW and 2500 MWth, respectively.

19 Figure 6 .
Figure 6.Total costs for power and heat demands of 12,700 MW and 5000 MWth, respectively.

Figure 6 .
Figure 6.Total costs for power and heat demands of 12,700 MW and 5000 MWth, respectively.

19 Figure 7 .
Figure 7.Total costs for load and heat demands of 9400 MW and 5000 MWth.

Figure 7 .
Figure 7.Total costs for load and heat demands of 9400 MW and 5000 MWth.

Table 1 .
Different optimization approaches used for the solution of the CHPED problem.

Table 2 .
Results obtained for the four generating units with two co-generation units and one heat unit for the load demand of 200 MW and 115 MWth.

Table 3 .
Comparative performances of the various optimization techniques for the 7-unit system for the load demand of 600 MW and heat demand of 150 MWth.

Table 4 .
Competitive results for a 24-unit system for the power and heat demands of 2350 MW and 1250 MWth.

Table 6 .
Test results for the test data of 84-unit system.

Table 6 .
Test results for the test data of 84-unit system.

Table 7 .
Cost of 96-unit system for the load demand of 9400MW and heat demand of 5000 MWth.

Table 7 .
Cost of 96-unit system for the load demand of 9400 MW and heat demand of 5000 MWth.