1. Introduction
Smart energy networks need intelligent operation for optimally managing electrical and thermal energy requirements. The electrical and thermal energy demands can be managed through combined heat and power (CHP) plants. CHP plants are effectively contributing to the efficient operation of electrical and thermal energy networks as well as to the reduction in carbon emissions, facilitating an increasing penetration of intermittent renewable energy sources. CHPs are designed and operated to provide stable thermal and electrical energy supplies, and they are capable to overcome the load dynamics. Over the years, the most pressing challenges in electric power system operation have been the economic load dispatches from conventional power plants [
1,
2]. The energy efficiency of combined cycle plants is around 50–60%, since a majority of the energy is lost during the energy conversion process. The cogeneration system needs to be implemented in thermal power plants for utilizing the waste heat in a useful way to reduce the overall energy cost and increase the environmental benefits. In the utility market, cogeneration units are used as district power plants for supplying the electrical and thermal energy demands as well as to facilitate intermittent renewable energy sources. The use of a CHP system can increase fuel proficiency around 90%, reduce the operating charges by 10–40%, and decrease carbon emissions around 13–18% [
3,
4].
As a result, CHP economic dispatch (CHPED) can play an important role in increasing an overall system’s energy efficiency. The CHPED needs to be optimized considering different operational strategies (e.g., thermal or electrical prioritized operations, steam turbine operational limits, etc.). The CHEPD objective function becomes non-linear and non-convex due to the valve point loading (VPL) effect and prohibited operating zones (POZs) of a traditional thermal power system. Due to mutual dependency of heat and power in a cogeneration unit, the CHPED optimisation problem faces some technical challenges [
5,
6]. To better understanding, the feasible operating region of the cogeneration unit and transmission losses should be taken into consideration [
7].
Several optimisation strategies are proposed to resolve the CHPED problem, and they are classified in two types: mathematical and meta-heuristic. Mathematical methods consist of duel and quadratic programming [
8], Lagrangian relaxation [
9], two-layer Lagrangian relaxation algorithm [
10], branch and bound algorithms [
11], etc. These methods are not sufficiently suitable to address the non-convex, discontinues, and non-differentiable fuel cost functions. Therefore, there is a significant challenge to resolve CHPED problems. As a result, new metaheuristic strategies for CHPED problem formulation can be used to solve these shortcomings. These metaheuristic algorithms demonstrate the ability to locate optimal solutions through a global search. As a result, researchers applied several swarm intelligence optimisation techniques to resolve CHPED complications and find better solutions. Initially, researchers used a metaheuristic algorithm to solve the CHPED problem on a small scale. For example, in [
12], genetic algorithm with multiplier updating technique has been proposed with a penalty value to solve the CHPED issue. A stochastic model is introduced with improved particle swarm optimisation to solve CHPED in a real situation [
13]. In [
14], a self-adaptive real-coded GA (SARGA) based on mutation and crossover phenomena has been proposed for CHEPD, and, to address the equality and inequality constraints, a penalty mechanism has been implemented. In [
15], the authors have proposed a harmony search (HS) algorithm for minimising the fuel cost function by adjusting the pitch of basic search operators. In [
16], author(s) have employed the firefly algorithm (FA), influenced by the behaviour of fireflies, as they attract each other at different rates depending on the intensity of the light. Therefore, this intensity of the light is a good analogy to optimise any objective function value. In [
17], the authors have suggested a genetic algorithm based on an enhanced penalty function technique; to realize an equilibrium between exploration and probability, a distance-based constraint handling approach has been included to solve the CHPED problem. In [
18], the authors have proposed cuckoo search optimisation (CSO) algorithm with penalty factor to minimise the overall fuel cost function accurately. However, in [
19], the bender decomposition algorithm the CHPED problem has divided into master and sub-problem. The subproblem generates the bender cuts, which has been used to update the master problem. In [
20], a new variant of PSO has been proposed, where only the largest particle is chosen whose fitness is greater than the average fitness value. Whereas in [
21], the hybridization of genetic algorithm and harmony search has been introduced to solve the CHPED problem. In [
22], non-convexity has been replaced by convex formulations by using the two binary variables. However, in [
23], the authors have proposed simulated annealing with basic bio-geography-based optimisation, which increases the convergence capability and avoids finding local minima. The main disadvantage of mentioned algorithms is that they do not consider the valve point loading effects including losses. However, differential evolution (DE) and bee colony optimisation (BCO) overcome these drawbacks [
24,
25]. Similarly, in [
26], the authors have proposed novel self-adaptive learning method to solve the CHPED problem considering VPL, ramp rate limit, reserve constraints, and network losses.
In [
27], the authors have been proposed cuckoo search algorithm (CSA) to resolve the CHPED issue. CSA is relatively easy to implement and provides better alternatives for CHPED problems due to its fewer control parameters and computational timings. In [
28], the authors have used the gravity search algorithm (GSA) to explain the CHPED issue. The gravitation law of moving particles inspired GSA, which includes various physical quantities such as the gravitational constant, mass, Euclidean distance, gravitation, acceleration, and velocity. GSA gives a better solution for the CHPED problem, having minimal fuel prices by dynamic adjustment of control parameters. In [
29], cuckoo optimisation has been applied as a powerful and robust solution technique for large scale units. An improved differential evolution algorithm has been proposed, which enhances the performance of basic DE with a modified repair process to handle equality constraints [
30]. In [
31], the authors have suggested a new optimisation method for analysing heat and power demand uncertainties using information gap decision theory and interval analysis to explain the CHPED optimisation process. In [
32], a Lagrangian-relaxation-based alternative approach has used to divide the non-convex area of the CHP unit into several convex regions using the Big M method to solve the CHPED issue efficiently.
To improve system performance and accuracy, several other studies have been projected to explain the CHPED optimisation task. For example, in [
33], the acceleration constants have dynamically altered to increase the optimum search and avoid premature convergence during CHPED optimisation. However, in [
34], the authors have suggested an oppositional teaching learning-based (OTLBO) algorithm to improve CHP plant solution accuracy and speed of convergence. In [
35], to solve CHPED effectively, the authors have suggested a crisscross optimisation algorithm in which two exploration operator crossings exist, flat and perpendicular. Whereas in [
36], a real coded genetic algorithm (RCGA) through progressive transformation evolution has been applied on six benchmark systems of the CHP plant, considering VPL and network losses.
Thermal power plant’s prohibited operating zones (POZs) make the system more challenging due to the discontinuous nature of fuel supply. It increases the complexity to solve the CHPED system. To solve CHPED with POZs, only a few optimisation algorithms have been implemented. In [
37,
38], the author proposed group search optimisation to reduce the operating cost as well as enhance the solution accuracy of the CHP economic dispatch problem. To handle the non-convex CHPED problem, an oppositional-based group search optimisation based on guess and opposite guess has been developed [
39]. The authors of [
40] have suggested a heat transfer search technique that included three stages: conduction, convection, and radiation. In contrast, the authors of [
41] have reported enhanced GA, employing unique crossover and mutation techniques. To recover mutation offspring and assist them in reaching a feasible range, an effective constraint handling technique has been used to resolve the CHPED issue. To address the CHPED problem with different constraints, a bio-geography-based learning particle swarm optimisation is developed, which uses migration operator to attain the best position [
42]. In [
43], the authors have proposed a modified PSO, where Gaussian random variables are taken to improve the global search of particles. However, in [
44], a self-regulating PSO has given the higher speed of convergence, when considering VPL and POZs. Differential evolution with a dual population has been planned to solve the CHPED problem for multifuel selections of generating units [
45]. In [
46], differential evolution with migrating variables has been used to increase the direction of search and replace those variables that were targeted earlier. A mesh adaptive direct search algorithm has been used in CHP dispatch in [
47], but it has not sufficiently covered impacts of POZs. Similarly, presented algorithms in [
48,
49] for CHP dispatching, POZs and VPLs have not been adequately considered for finding operations’ of CHPs.
It is required to investigate a more effective metaheuristic algorithm, which can solve a large and complex non-linear and non-convex CHPED problem. Metaheuristic algorithm can only provide the global optimum solutions. They cannot give accurate solutions. Most of the techniques explained by the different authors are complicated due to consideration of a large number of parameters. This paper focuses on the use of a simple algorithm to solve the CHPED problem, and it is based on [
50]. In the considered algorithm, a random communication occurs between all the candidates throughout the iteration process. The simplicity of the considered algorithm is that there are only two design parameters such as population size and number of iterations. To address the CHPED problem more accurately, the feasible operating region of the cogeneration unit and network losses are taken into account with the objective of minimizing fuel cost. To handle equality and inequality constraints, a suitable penalty factor-based constraint handling strategy has introduced. The VPL and POZs are used for considering the feasible operating region of a CHP plant.
The CHEPD formulations have been presented in [
51,
52,
53,
54,
55] and they have used in their presented soft computing techniques for CHP dispatch. The VPLs description have been provided in [
56], but not covered sufficiently role of POZs. Also, some computing techniques have been described in [
57,
58,
59,
60] with CHP operations’ constraints for maintaining the heat and power balance operation within the considered energy network.
In this work, to validate the proposed algorithm, three typical test case systems, including 4-units, 7-units, and 24-units, are considered. The results of these cases are compared with the well-established algorithm to show the superiority among them. The following sections of the article are structured as: (i)
Section 2 presents the mathematical modelling of the CHP economic dispatch optimisation problem; (ii)
Section 3 represents the proposed algorithm; (iii)
Section 4 provides the analysis of considered three cases; And (iv) finally, the conclusion and future scope of the article are given in
Section 5.
3. Optimisation Algorithm
The issues (discussed in
Section 2) need to be addressed more effectively. For the viable operational area of the CHP, power system network losses or an optimal allocation of power output and heat generation is required. In this work, a metaheuristic optimisation algorithm is proposed to solve the CHPED to minimise the fuel supply, thus satisfying the constraints. To handle equality and inequality constraints, an external penalty factor-based constraint handling technique is used. This paper focuses on use of a simple algorithm to solve the CHPED problem and it is based on [
50]. The programming technique is outlined in the steps, given below. The algorithm flow chart is shown in the
Figure 1.
Step 1. Design the fitness function: Formulate the mathematical equation as a fitness function f(x) to be minimised.
Step 2. Initialization of system input parameters: Input the CHPED designing variable for the power and heat units, electrical power demand and heat demand, population size, and termination criteria. Consider the number of design variables, and population size and candidate solution are set to be m and n, respectively.
Step 3. Select the required solutions: Find the minimum value of fuel cost F(X) of CHP economic dispatch for the best particle as F(X)best and the maximum value of F(X) as F(X)worst throughout the algorithm step. Identify the corresponding power and heat generation of the CHP plant, according to the minimum and maximum values of fuel cost. Where, the value of the design variable during the iteration for the particle.
Step 4. Modify the solutions: Modify the solution based on the minimum and maximum fuel cost of the CHP system and the random interaction between the generated values of power and heat. Modifying the power and heat generation values, marked by the following Equation (15), which is based on [
50]:
where
is the best particle value for variable j during the
iteration,
is the worst particle value for variable j during the
iteration,
is the updated value of
, and
and
are two random numbers for the
variable during the
iteration between (0,1).
shows any particle result k is related with any randomly chosen particle solution i, and information is replaced based on their fitness value. If the objective function value of the particle is better than the objective function value of the solution, then is replaced with . However, if the objective function value of the solution is better than the objective function value of the solution, then is replaced with . Similarly, if the objective function value of the solution is better than the objective function value of the solution, then becomes . On the other hand, if the objective function value of the solution is better than the objective function value of the solution, then “” becomes .
Step 5. Report the required solutions: As a check, if the new value of fuel cost F(X)new is corresponding to the modified value of power and heat generation , then take the modified value in place of the previous value, otherwise, keep the previous value. Repeat the same process until the termination criteria has been satisfied and the optimum solution has been found out.