1. Introduction
In the process of converting fossil fuels into electrical power, conventional thermal power stations produce a large amount of thermal energy without adequate storage and use. This leads to huge losses of thermal energy. Therefore, the effectiveness of traditional power plants is reduced by 50% to 60% [
1,
2,
3]. Rising levels of carbon dioxide in the air and global warming have prompted the industry to integrate power and heat more efficiently within the energy networks. Hence, the effective planning of energy resources to meet the different loads has become more important. Therefore, it is required to have appropriate operational strategies for supplying the combined heat and power, considering the operational constraints of the CHP plants [
4]. Operation of CHP plants is very economical and can contribute effectively in overcoming the intermittence of renewable energy sources, as well as in primary/secondary control of the power system. CHP plants can increase the total efficiency by 90%, decrease the operational cost by about 10% to 40% with appropriate operational strategies, and reduce CO
2 releases by 13% to 18% [
5,
6]. The CHP plants need appropriate strategies for providing combined heat and power economic dispatching (CHEPD) for fulfilling the CHP demand, considering the technical and operational constraints. The motive behind CHPED is to find the profitable method for utilizing the existing generating plants to satisfy the electric power requirements and CHP requirements from CHP plants to fulfil the load demand and operating them within the constraints.
When optimising the CHPED, various operating techniques, such as constraints on the equality and inequality of power and heat units, must be considered [
7]. The CHEPD optimisation problem is no longer linear and convex because of the valve point loading (VPL) effect and prohibited operating zones (POZs) of a typical thermal power system. There are some technical difficulties with CHPED optimisation because of the joint reliance of heat-power in the CHP plant.
In the literature [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18], several optimising techniques have been recommended to fix the CHPED issues. These approaches can be divided as: (1) Mathematical and (2) Meta-heuristic. Mathematical techniques contain quadratic models [
8], the Lagrange method [
9], the bi-layer Lagrange approach [
10], branch-bound algorithm [
11], and so on. Such approaches are limited for handling the non-convexity of optimisation functions. As a result, resolving the CHPED issue is a significant challenge. These shortcomings can be addressed by developing some new methods for CHPED using metaheuristics. The metaheuristic techniques can be classified into single-objective and multi-objective frameworks to solve the CHPED optimisation process.
In single objective problem formulation, minimisation of the operating cost of the CHPED issue is the top priority. Thus, to resolve the CHPED issue, a penalty factor-based genetic algorithm (GA) is applied in [
12]. While in [
13], evolutionary programming (EP) is applied to formulate the CHPED issue. In [
14], a real-coded GA, based on self-adaptation is used to address the CHPED issue, based on mutation/crossover phenomena, and the constraints are handled by a penalty parameter. However, in [
15], researchers considered a harmony searching process to minimise the fuel price by the pitch factor adjustment of the search operators. A firefly algorithm is explained to formulate the CHPED issue in [
16]. An invasive weed technique is used to formulate the CHPED issue which is influenced by the environmental weed allocation and colonial procedures [
17]. While in [
18], the author proposed a hybrid biogeography-based method with simulated annealing (SABBO) to examine the constrained CHPED problem.
But in the research studies [
12,
13,
14,
15,
16,
17,
18], the VPL effect of a thermal power unit and transmission loss have not been looked at. To overcome these issues, an exchange market algorithm (EMA) is proposed to formulate the CHPED issue, considering the VPL effect and transmission loss [
19]. A cuckoo search algorithm (CSA) is also found suitable to resolve the constrained optimisation, due to fewer design variables and computational timings [
20]. The gravity search algorithm (GSA) was employed by Beigvand et al. [
21] to describe the CHPED difficulties. However, in [
22], to handle constraints, an improved differential evolution technique as well as a revised repair process were used. A modified group search optimisation (MGSO) is suggested in [
23] to solve the CHPED. In [
24], a self-regulating PSO is used to find the optimum scheduling of CHPED.
The POZs of thermal power units make the CHPED problem more difficult to address due to the fuel supply. Only a few optimisation procedures are realised including POZs. For handling the non-convexity of the CHPED issue, a group search optimisation, based on opposition is applied, which is based on the opposition guess of the search particles [
25]. While in [
26], the best outcome of CHPED is attained by the enhanced particle swarm optimisation (PSO), wherever the Gaussian distribution factor is used to increase the global searching phenomena. In [
27], the group search optimisation for lowering the operational price and improve the accuracy of the CHPED issue is explained. However, in [
28], a hybrid bat-arbitrary bee colony method is established to obtain the benefits over the basic bat and bee colony methods. A heat transfer search technique has been proposed, considering the conduction, convection, and radiation to effectively solve the CHPED problem [
29]. Moreover, Zou et al. [
30] have described the improved GA, with distinct crossover and mutation steps. However, in [
31], a self-regulating PSO gives a better convergence speed when VPL and POZs are taken. To report the CHPED issue with various limitations, a bio-geography-based PSO is established, which uses migration operators to reach the best location [
32].
However, in [
33], generation fuel cost is minimised by a hybrid metaheuristic algorithm considering different operational constraints. Where as in [
34], a high level CHPED framework, including 24-units and 84-units has been formulated using a multi-player harmony search method. An optimal model of CHPED is introduced in [
35], where electric boilers are used to reduce cost and wind curtailment. While in [
36], a hybrid heap-based jellyfish searching method is applied to explain a non-convex CHPED issue. Where the exploration/exploitation is used to allocate optimum results of thermal and electrical energy generation.
Several research studies have been carried out for a multi-objective framework, where the objective function is taken as the minimisation of cost, as well as emissions. As in [
37], the problem is solved via deterministic and stochastic methods, including various complications. While in [
38], the multi-objective multi versus optimisation (MOMVO) algorithm is proposed to resolve the environmental CHPED issue. A hybrid enhanced GA and PSO is proposed to obtain the best outcomes of a combined economic/environmental dispatch (CEED) issue [
39]. However, in [
40], a kho-kho optimisation (KKO) technique is applied to resolve the CHPED and CEED issues.
However, apart from the above research studies, some of the studies have been carried out, where metaheuristic and mathematical methods were used together. As in [
41], a time varying accelerative coefficient-based PSO (TVAC-PSO) was created to examine the CHPED issue. In [
42], Basu proposed a non-dominated sorting GA-II for analysing the CHPED issue. This is employed in conjunction with a real-coded genetic algorithm because binary values create complexities in search spaces with high precision. While in [
43], a real coded GA (RC-GA) through progressive transformation evolution is proposed for the CHP unit, considering VPL and network losses. In [
44], Jena et al. suggested a Gaussian genetic change in the basic DE for enhancing the search functionality. While in [
45], TVAC-PSO is adopted to explain the economic emission dispatch issue considering losses. The Monte-Carlo technique is used to implement a stochastic model to handle the real-world scenario. In [
46], Beigvand et al. have suggested a hybrid gravitational search TVAC-PSO method to explain the large-scale CHPED issue. In [
47], a Lagrangian-relaxation-based alternative method is implemented where the non-convexity of the CHP component is separated in numerous convex areas utilizing the Big-M technique for solving the CHPED issue successfully. In [
48], the exchanged market algorithm EMA is joined with the non-dominated TVAC-PSO, to resolve the dispatch problem
The state-of-the-art literature review has found that the mathematical modelling of CHP units with a thermal power unit is quite complex. Because of the VPL effect and the restricted working region of thermal power units, the entire solution for CHPED becomes non-convex, non-linear, and non-differentiable. Furthermore, the viability of the CHP unit is reliant on both power generation and heat production. To solve this complex system, several mathematical and metaheuristic-based methodologies have been proposed. However, most of them have struggled to find an optimal solution due to various generating unit constraints. Many algorithms have their own algorithm-specific parameters. The tuning of such variables makes the CHPED issue more challenging.
For solving the above difficulties, in the current research study, three simple metaheuristic algorithms, such as (i) Rao 3 algorithm [
49], (ii) Jaya algorithm [
50], and (iii) hybrid CHPED algorithm (i.e., developed by the authors, based on (i) and (ii)) are applied for solving the complex CHPED issues. The author’s contribution in the present study is explained by the following points.
The hybrid CHPED optimisation algorithm is a newly developed algorithm by the authors to solve the constrained optimisation problem;
The hybrid CHPED algorithm is developed by the combination of the basic Jaya algorithm and Rao 3 algorithm;
The hybrid CHPED algorithm is used to solve the constrained and unconstrained optimisation problems of CHP operations;
To handle all of the constraints, the exterior penalty factor method is used to obtain the desired solutions.
All three methods are used by selecting the best and worst candidates and they have only two designing variables: size of the population and iterations. There is no need for any other algorithm specific variables. The Rao 3 method is based on random interactions between the candidate throughout the iteration, but it is not required in the developed hybrid CHPED algorithm. To analyse the CHPED problem accurately, the VPL effect and POZs of the power plants are taken. For a better understanding, the viable working areas of the CHP unit are taken into account for minimising the total operating price. In this study, two test case systems, a 5-unit and a 24-unit, are used to evaluate the proposed technique (details are presented in
Section 4). The results of the developed hybrid CHPED algorithm is compared with the basic Jaya algorithm and Rao 3 algorithm. It is found that the hybrid CHPED optimisation algorithm performs better. The results of these three methods are also compared to that of a well-known research method to indicate their superiority.
The article is organised as follows:
Section 2 provides the computational models of the CHP economic load dispatch optimisation.
Section 3 contains the proposed technique.
Section 4 provides an evaluation of the two cases studied and last,
Section 5 provides the paper’s conclusion and the future scope.
3. Implementation of the Optimisation Algorithms in the CHPED Formulation
For solving the CHPED issue effectively, three distinct optimisation algorithms are employed: (1) the Rao 3 algorithm [
49], (2) Jaya algorithm [
50], and (3) hybrid CHPED algorithm. The hybrid CHPED algorithm is developed by the authors, which is a combination of the basic Jaya algorithm and Rao 3 algorithm. All three algorithms are based on the best and worst candidate selection. In the Rao 3 algorithm, there is a random communication among the candidates, but it is not required in the hybrid CHPED algorithm, which makes it simpler than the Rao 3 algorithm. Furthermore, the hybrid combination performs better, as compared to the individual algorithm, in relation to accuracy and reliability. The mathematical equations for Jaya, Rao 3, and hybrid CHPED algorithm are expressed by Equation (14) to Equation (16), respectively. In this study, the programming technique is outlined in the following steps. The flow charts of three algorithms are demonstrated in
Figure 2.
Step 1. Design the fitness function: Formulate the mathematical equation for the total operational price of the CHPED problem as fitness function F (Z) to be minimised;
Step 2. Allocate the input variables: Insert the input variables for power, CHP, and heat units. Assign the required thermal and electrical power demands. Furthermore, fix the population size (m) and termination criteria (n);
Step 3. Select the feasible solutions: Obtain the minimum and maximum value of F(Z), as F(Z)BEST and F(Z)WORST during the iteration. Select the respective power and heat output of F(X). Where, is the value of designing parameter for particle throughout iteration;
Step 4. Update the solutions: Update the results on the basis of lower and higher values of F(Z) for the Jaya algorithm, Rao 3 algorithm, and hybrid CHPED algorithm, based on Equations (14)–(16). Modify the power and heat output results given by the following equation:
where,
and
are the best and worst outcomes of parameter j for
iteration.
is the improved result of
.
and
are random values between 0 and 1.
In Equations (14) and (16), the terms “ ”and “” are close to the best and worst outcomes, respectively.
In Equation (15), represent a random interaction between the candidate ‘k’ and ‘l’, based on the fitness values and exchange the information. Set , if the fitness value of candidate k is better than candidate l, otherwise it will be . Similarly for the term it is applicable, if candidate k is better than candidate l, then replace the term with otherwise it will be .
Step 5. Locate the final outcome: If is better than , then choose the updated value instead of the earlier value, or else keep the earlier value. Furthermore, report the optimum values of power and heat production for the CHPED problem. Continue this process until the termination requirements are met.