An Improved Artificial Ecosystem Algorithm for Economic Dispatch with Combined Heat and Power Units

Abstract

The most effective use of numerous Combined Heat and Power Units (CHPUs) is a challenging issue that requires strong approaches to handle the Economic Dispatch (ED) with CHPUs. It aims at minimizing the fuel costs by managing the Power-Only Units (POUs), CHPUs, and Heat-Only Units (HOUs). The transmission losses are also integrated, which increases the non-convexity of the ED problem. This paper proposes a Modified Artificial Ecosystem Algorithm (MAEA) motivated by three energy transfer processes in an ecosystem: production, consumption, and decomposition. The MAEA incorporates a Fitness Distance Balance Model (FDBM) with the basic AEA to improve the quality of the solution in non-linear and multivariate optimization environments. The FDBM is a selection approach meant to find individuals which will provide the most to the searching pathways within a population as part of a reliable and productive approach. Consequently, the diversity and intensification processes are carried out in a balanced manner. The basic AEA and the proposed MAEA are performed, in a comparative manner considering the 7-unit and 48-unit test systems. According to numerical data, the proposed MAEA shows a robustness improvement of 97.31% and 96.63% for the 7-unit system and 46.03% and 60.57% for the 48-unit system, with and without the power losses, respectively. On the side of convergence, based on the average statistics, the proposed MAEA shows a considerable improvement of 47% and 43% of the total number of iterations for the 7-unit system and 13% and 20% of the total number of iterations for the 48-unit system, with and without the power losses, respectively. Thus, the suggested MAEA provides significant improvements in the robustness and convergence properties. The proposed MAEA also provides superior performance compared with different reported results, which indicates a promising solution methodology based on the proposed MAEA.

1. Introduction

1.1. Motivation

In numerous academic fields, meta-heuristics have steadily gained popularity for handling difficult optimization problems [1]. Traditional optimization procedures are undervalued due to concerns related to local optimal stagnation [2]. To overcome such issues, meta-heuristics optimization procedures are followed which involve the most effective and influential strategies for identifying optimal solutions. Because of the clear growth in manufacturing and residential needs, the world’s usage of electrical and thermal energy has recently become materialistic. In order to diminish the drawbacks of conventional facilities, energy planners have been instructed to include heat and power sources in addition to renewable energy sources. The reduction of pollution emissions that contribute to global warming has also received international attention [3]. One of the national energy policy initiatives by China is to focus on the development of efficient, secure, and sustainable energy sources with an ideal management system [4]. Economic load dispatch is a critical optimization problem in power systems that necessitates good generator coordination, control, and management [5]. Because of the imposed identical and uneven constraints, it exhibits non-linear performance. In response, it has been recognized as a difficult multi-modal optimizing problem to address [6]. Therefore, it is critical that a search is undertaken for efficient, robust, and highly convergent optimization solutions to the non-linear and complex ED optimization problem, which considers CHPUs.

1.2. Literature Survey

For this reason, a systematic learned PSO has been blended with a sequential quadratic programming technique and employed for ED optimization of the power system [6]. However, the primary objective task that was taken into consideration was the reduction of gasoline expenses. In [7], a multi-objective pigeon-inspired algorithm was used to solve the ED problem involving emissions minimization; however, in three scenarios explored, only 6-unit and 14-unit systems were considered in detail. The ED problem was solved using a dispersed fixed step-size optimizer in [8] while taking into account the cost function of the distributed energy resources. However, the traditional quadratic model was applied while excluding the actual effects of the valve point loadings.

In standard thermal power plants, a substantial amount of thermal energy is wasted and released into the environment via cooling towers, flue gas, or other methods. The efficiency of converting carbon fuels into electrical energy is therefore just 50% to 60%, despite the most efficient contemporary combined cycle plants. By gathering and using waste heat, CHPUs raise the energy conversion efficiency of these typical units from 50–60% to the order of 90% [9]. An essential issue for managing the operation of these units is the ED model combined with CHPUs [10]. Traditionally, an ED incorporating CHPUs manages the Power-Only Unit, CHP, and HOUs to save fuel expenditures. However, the production and use of energy are closely related to environmental concerns.

The integrated hybrid energy systems can meet a variety of energy demands with increasing productivity and efficiency. This lays the groundwork for creating a low-carbon, sustainable method of economic and social advancement. Additionally, during the past few decades, combined heat and power systems have been associated with energy savings and reduced environmental impact. Such systems have alerted the scientific community to further research and developments of renewable-based combined heat and power configurations in the domestic and commercial sectors, which served the objective [11,12]. By managing Power-Only Units, CHPs, and HOUs, the CHPEED challenge seeks to reduce fuel costs and emissions [13]. Moreover, in order to maintain the performance of the power, heat, and CHPUs, certain inequality limits must be satisfied. Additionally, the mutual dependence of the CHPUs must be maintained because it may have an impact on how the CHPEED problem is solved [14]. The challenging ED problem with CHPUs has been addressed using a wide variety of MAs. According to the primary objectives, the published research on the ED problem using CHPUs that has used metaheuristic methodologies to solve this problem can be split into two types. To achieve the lowest operating costs, the first category involves creating efficient optimization methods for systems incorporating thermal plants, CHPUs, and boilers. The investigation of all practically pertinent restrictions, such as transmission loss, valve-point impacts, and environmental difficulties with the heat and power supply of the ED issue with CHPUs, falls under the second category. Some of the most intriguing works in the first category include the following: as shown in [15], the ED problem with CHPUs was solved using the GSA by examining network system losses and the valve-point effect of POUs. To address the production cost reduction of the ED problem with the CHPU problem, specifically examining the valve-point effect of POUs, the CSA was employed in [16]. Both investigations looked at network losses and valve-point consequences. The environmental concerns, however, were not considered.

To solve the ED issue under varying CHPU operating conditions with minimal computational effort, [17] implements a DRL approach. Artificial neural networks have also been used to try to fix the ED issue using CHPUs [18]. Practical limitations including valve-point influence, transmission power loss, and environmental factors were not considered in [17,18]. Considering the transmission loss and valve-point impact, a heap optimizer was used in [19] on large-scale 84-unit and 96-unit systems. In addition, the optimal ED problem with the CHPU problem has been solved using a composite firefly and self-regulating PSO technique [20]. In addition, a differential evolution with migrating variables was performed to address the ED problem with the CHPU problem in [21], and the cuckoo optimization approach was combined with a penalty function to address the ED problem with the CHPU issue in [22]. The probability of the MPA [23] failing when prey is lost has been reduced, due to the partitioning of the iterations into three separate and continuous sections. In [24], the authors offer a MPHS for an ED problem involving CHPU optimization with 84 units, considering the effects of valve-point loading on thermal power plants. Together, the heap optimizer and the jellyfish optimizer were used to solve a 96-unit ED problem in the CHPU system while also considering the potential for unit outages (as studied in [25]). Most MAs, despite their impressive results, are highly sensitive to changes in user-defined parameters. Another drawback is that the MAs may not reliably converge to the global optimum. These worries have seized the interest of researchers, and they have begun developing hybrid versions as one of the valid metrics, as hybridization is a crucial part of high-performing algorithms.

Modifying and applying the AEA [26] in engineering contexts is straightforward and requires only minor adjustments. All ecosystems involve three forms of energy transfer (production, consumption, and breakdown), and AEA considers all three to obtain the optimal fitness score. Meanwhile, the consumption approach can help to strengthen the discovery, exploration, and exploitation of space. The production operation enables AEA to generate a new member at irregular intervals. Due to its robustness and powerful global searching capabilities, the AEA approach has been applied to a variety of real-world optimization engineering problems, such as distributed generation and capacitor allocation in power delivery networks [27], the minimization of regression test suites [28], the optimization of filter parameters [29], the optimization of demand-side management for hybridized energy sources [30], the representation of PV cells [31], and the identification of fuel-cell parameters [32].

1.3. Paper Contribution

This study introduces a novel improved MAEA, or Modified Artificial Ecosystem Algorithm, for solving the ED problem with CHPUs, both with and without power losses. One novel method for enhancing the solution quality in non-linear and multivariate optimization settings is to combine the AEA with the FDBM. A successful application of AEA combined with the FDBM technique for the power flow constrained by the transient stability level in power networks has been reported [33]. Therefore, the processes of diversification and intensification are carried out in harmony. The 7-unit and 48-unit test systems are used to conduct a comparative analysis of the standard AEA and the proposed MAEA. The results demonstrated that the proposed method was superior to the standard AEA in locating the global optimal solution. In early tests, the planned MAEA demonstrated impressive problem-solving abilities. This criterion suggests that the changes made to the design of this AEA throughout the decomposition stage were successful in producing results that were closer to the real-world behaviour of the algorithm being simulated. A few of the most important things that this research has added are:

• An FDBM is developed in collaboration with an AEA to create a unique MAEA with improved performance.
• The basic AEA and the proposed MAEA have been assessed in solving the ED problem including CHPUs with and without power losses.
• The proposed MAEA shows greater performance compared with several other reported algorithms in the literature.
• Furthermore, the suggested MAEA is stated to be more resilient and stable than the basic AEA.

1.4. Key Segments of the Paper

This paper is divided into five key segments. The first segment is the introduction section which describes the problem context, literature review, and the hypothesis based on the gap analysis of the previously published research. The second segment describes the modelling of the ED combining CHPUs in terms of the main objective function and the practical regarding constraints. The third segment describes the main structure of the basic AEA and the processes of the developed MAEA. The fourth segment compares the MAEA’s simulated outcomes considering two practical systems of the 7-unit and 48-unit test systems. The last segment provides a concluding note to this work.

2. ED Problem with CHPUs

The key players in the ED in efforts to supply the electricity and heat loads in the facilities and buildings are depicted in Figure 1. The core purpose of the ED combining CHPUs would be to identify the economic potential rate for heat generated by HOUs, power generated by POUs, and both power and heat generated by CHPUs, such that fuel costs are maintained to a minimum while heat and power needs and restrictions are met [34]. Thus, the generation cost objective (F) may be stated as:

$$F={\displaystyle \sum _{m=1}^{{}_{N_{GU}}}{C}_m(P{g}_m)}+{\displaystyle \sum _{n=1}^{{}_{N_{HU}}}{C}_n(H{g}_n)}+{\displaystyle \sum _{k=1}^{N_{CHPU}}{C}_k}(P{g}_k,H{g}_k)$$

(1)

where

$$C_m(P{g}_m)=\alpha 1_m{(P{g}_m)}^2+\alpha 2_{m}P{g}_m+\alpha 3_m+\left|\alpha 4_m\sin (\alpha 5_m(P{g}_{m,\mathit{min}}-P{g}_m))\right|$$

(2)

$$C_n(H{g}_n)=\phi 1_n{(H{g}_n)}^2+\phi 2_{n}H{g}_n+\phi 3_n$$

(3)

$$C_k(P{g}_k,H{g}_k)=\beta 1_k{(P{g}_k)}^2+\beta 2_{k}P{g}_k{}_i+\beta 3_k+\beta 4_k{(H{g}_k)}^2+\beta 5_{k}H{g}_k+\beta 6_{k}H{g}_{k}P{g}_k$$

(4)

Figure 1. Key players in the ED problem with CHPUs [35].

Furthermore, the inequality requirements of this problem should always be fulfilled in regard to the capacity of the POUs, HOUs, and CHPUs, as shown in Equations (5)–(8):

$$P{g}_m^{\mathit{min}}\le P{g}_m\le P{g}_m^{\mathit{max}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}m=1:N_{GU}$$

(5)

$$H{g}_n^{\mathit{min}}\le H{g}_n\le H{g}_n^{\mathit{max}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}n=1:N_{HU}$$

(6)

$$P{g}_k^{\mathit{min}}\le P{g}_k\le P{g}_k^{\mathit{max}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1:N_{CHPU}$$

(7)

$$H{g}_k^{\mathit{min}}\le H{g}_k\le H{g}_k^{\mathit{max}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k=1:N_{CHPU}$$

(8)

Furthermore, the equality requirements of this problem should always be satisfied in the perspective of heat and power balance, as addressed in Equations (9) and (10), and as described in the following:

$${\displaystyle \sum _{i=1}^{N_{CHPU}}H{g}_i}+{\displaystyle \sum _{j=1}^{N_{HU}}H{g}_j}=Hea{t}_D$$

(9)

$${\displaystyle \sum _{i=1}^{N_{CHPU}}P{g}_i}+{\displaystyle \sum _{k=1}^{N_{GU}}P{g}_k}=Powe{r}_D$$

(10)

The power loss represents an important phenomenon that occurs in power system networks due to the flow of the output power from the generation units to the customers. It is of great importance since it is usually modelled in a highly non-linear form that represents further complexity to the ED model incorporating CHPUs. Thus, the integration of transmission losses might result in additional non-convexity for the issue, which is described in Equation (11) as a proportion of the output power of the POUs, HOUs, and CHPUs:

$$P_{Loss}={\displaystyle \sum _{j=1}^{N_{GU}}{\displaystyle \sum _{i=1}^{N_{GU}}{B}_{ji}P{g}_j}}P{g}_i+{\displaystyle \sum _{j=1}^{N_{GU}}{\displaystyle \sum _{i=1}^{N_{HU}}{B}_{ji}P{g}_j}}H{g}_i+{\displaystyle \sum _{j=1}^{N_{HU}}{\displaystyle \sum _{i=1}^{N_{CHPU}}{B}_{ji}H{g}_j}}H{g}_j$$

(11)

where P_Loss is the total losses; B_ji is the coefficient element in the B-matrix that describes line losses correlating the units.

Accordingly, Equation (10) can be reformulated as follows:

$${\displaystyle \sum _{i=1}^{N_{CHPU}}P{g}_i}+{\displaystyle \sum _{k=1}^{N_{GU}}P{g}_k}=Powe{r}_D+P_{Loss}$$

(12)

3. Proposed MAEA for Solving the ED Problem with CHPUs

3.1. Artificial Ecosystem Algorithm

The AEA is influenced by three energy transfer processes inside an ecosystem: production, consumption, and decomposition. The production operation enables the AEA to construct a new solution represented at random, that may replace the prior member amongst the global optimum (YA_Best) and a randomized individual (YA_R) created at irregular intervals in the solution space. The following is how the production operation may be quantified [26]:

$$\begin{gathered} Y{A}_1(it+1)=Y{A}_{Best}(it)\times (1-q_1\times (1-\frac{it}{T_{\mathit{max}}}))+q_1\hspace{0.17em}\times Y{A}_R\times (1-\frac{it}{T_{\mathit{max}}}) \\ Y{A}_1(it+1)=Y{A}_{Best}(it)\times (1-q_1\times (1-\frac{it}{T_{\mathit{max}}}))+q_1\hspace{0.17em}\times Y{A}_R\times (1-\frac{it}{T_{\mathit{max}}}) \end{gathered}$$

(13)

$$Y{A}_R=LB+q\times (UB-LB)$$

(14)

wherever it corresponds to the present iteration; T_max and P_M represent the maximum number of repetitions and the size of the population, accordingly, whereas UB and LB represent the upper and lower limits, respectively. In addition, q₁ and q represent a randomized value and a randomized vector inside the domain [0, 1].

In the consumption framework, Levy flying is incorporated, which can conveniently traverse the search area. It simulates the food quest of various species such as lions and cuckoos as a numerical operation. Levy flight is a randomized walk which may cover the search region successfully since the length of some stages is much longer in the long run, implying that it can achieve the global optimum. As a result, Levy flying is commonly used to increase the optimizing efficiency of metaheuristic algorithms [26]. However, there appear to be two drawbacks to such movement: intricacy and the necessity to adjust multiple settings. As a result, given the properties of Levy flight, a parameter-free randomized walk, termed consumption parameter (CP), is derived as shown in Equation (15).

$$CP=0.5\times \frac{v_1}{\left|v_2\right|},v_1\approx N(0,1) \& v_2\approx N(0,1)$$

(15)

wherein N(0,1) denotes a normal distribution with a mean of zero and a standard deviation of one. As a result, this consumption parameter could aid various sorts of consumers in implementing three consuming tactics. The first method is Herbivore, in which the consumer could only consume what the producer produces. This behaviour can be represented mathematically as described in the following:

$$Y{A}_k(it+1)=Y{A}_k(it)+CP\times (\hspace{0.17em}Y{A}_k(it)-Y{A}_1(it)),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\in [2:P_M]$$

(16)

The second technique is called Carnivore, in which the consumer could only devour a consumer having the highest degree of energy at irregular intervals. This behaviour could be represented mathematically as follows:

$$Y{A}_k(it+1)=Y{A}_k(it)+CP\times (\hspace{0.17em}Y{A}_k(it)-Y{A}_1(it)),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\in [3:P_M]$$

(17)

The third technique is called Omnivore, in which the consumer could devour both the consumer and the producer at random. This is how this behaviour may be demonstrated:

$$Y{A}_k(it+1)=Y{A}_k(it)+CP\times (q_2·Y{A}_k(it)-Y{A}_1(it))+(1-q_2)(Y{A}_k(it)-Y{A}_j(it)),\hspace{0.17em}k=3:P_M,j=q(\left[\begin{array}{cc}2& k-1\end{array}\right])$$

(18)

The individual position in a population could be upgraded in the decomposition, as shown in Equation (19). Therefore, to some extent, this approach exemplifies exploitation since it allows the subsequent place of every individual in the solution to be distributed around the global optimum of the best solution, which is stated as the decomposer. The following describes the decomposition behaviour:

$$Y{A}_k(it+1)=Y{A}_{Best}(it)+3\times N(0,1)\times ((q_3·q(\left[\begin{array}{cc}1& 2\end{array}\right])-1)·Y{A}_{Best}(it)-(2·q_3-1)·Y{A}_i(it)),\hspace{0.17em}\hspace{0.17em}k=1:P_M$$

(19)

Based on the above illustrations, the main steps of the basic AEA can be depicted as in Figure 2.

Figure 2. Main steps of AEA.

3.2. Proposed MAEA with FDBM

The purpose of developing the FDBM selecting approach is to discover participants who will make the greatest contribution to the search operations in a systematic and efficient way. Therefore, it is possible to ensure that the varying and strengthening actions are carried out in a balanced manner. The Euclidean distance metric could be employed to calculate the distance between the solutions and the preferred opportunity (YA_Best). Consequently, the distance (D_k) between each individual and the optimal choice is calculated as described in the following:

$$D_k={\left({{\displaystyle \sum _{d=1}^{Dim}(Y{A}_{k,d}-Y{A}_{Best,d})}}^2 \right)}^{0.5} k=1:P_M$$

(20)

After that, the rating grade of each design choice is established. The rating grade is computed using the normalized fitness (NF) and normalized distance (ND). They may be assessed for every member (k) as follows:

$$N{D}_k=\frac{D_k-D_{k,\mathit{min}}}{D_{k,\mathit{max}}-D_{k,\mathit{min}}} i=1:P_M$$

(21)

$$N{F}_k=\frac{F_k-F_{k,\mathit{min}}}{F_{k,\mathit{max}}-F_{k,\mathit{min}}} k=1:P_M$$

(22)

Relying on this, the inclusion of normalized numerical quantities is designed to prevent these features from overpowering the target computation. As a result, each individual’s (k) grade (GR) may be calculated as follows:

$$G{R}_k=N{D}_k+N{F}_k k=1:P_M$$

(23)

Following the determination of all individuals’ grades, a roulette wheel selection process [36] is used to choose an alternative by including a high probability of getting a high grade (YA_FDBM). As a result, the decomposition step stated in Equation (19) is improved by combining the FDBM:

$$Y{A}_k(it+1)=Y{A}_{FDBM}(it)+3\times N(0,1)·((q_3·q(\left[\begin{array}{cc}1& 2\end{array}\right])-1)·Y{A}_{Best}(it)-(2·q_3-1)·Y{A}_k(it)),\hspace{0.17em}\hspace{0.17em}k=1:P_M$$

(24)

The MAEA process is depicted in Figure 3. It starts by randomly forming a population. The first seeking individuals adjust their locations according to Equation (13) with each repeat, but the subsequent participants have the same opportunity to alter their placements by selecting Herbivore according to Equation (16), Carnivore according to Equation (17), or Omnivore according to Equation (18). Adjustment may be allowable when a participant obtains a greater fitness trait. The FDBM is then triggered to select an alternative by including a high probability of receiving a good grade. To accomplish this, each member’s distance from the optimum choice is calculated as shown in Equation (20). The normalized objective functions and distance scores of the prospects are therefore assessed utilizing Equations (21) and (22), whereas the rating grades of the solutions are computed in the second step of the FDBM approach as shown in Equation (23). Equation (24) would then be used to change the placement of each component. Participants might be created at random in the seeking space whenever there is gap far from the upper or lower borders throughout the upgrade sequence. All changes are made continually until the AEA procedure is completed, by the inclusion of a termination criterion. Eventually, the best candidate is chosen.

Figure 3. Main steps of the proposed MAEA.

4. Simulation Results

The acquired findings for the ED incorporating CHPUs were contrasted with the basic AEA to illustrate the effectiveness of the proposed MAEA. Both techniques are tested on two standard test systems. The two selected tested networks have different configurations and scalability called 7- and 48-unit systems. The first test system consists of two CHPUs, four POUs, and one HOU. As mentioned in [37], system data is stated as loss coefficients, fuel prices, and CHPU restrictions. For this system, the loading level of power and heat are 600 MW and 150 MWth. The second test system comprises 48-unit systems as mentioned in [38] which illustrates that 4700 MW and 2500 MWth are the load demand and heat demand, respectively, and it has 10 HOUs, 26 POUs, and 12 CHPUs. For the basic AEA and the proposed MAEA, the number of individuals is taken as 100 and the numbers of iterations are 300 and 3000, respectively, for the first and second system. MATLAB 2017b is used to execute the simulated implementations.

Based on the consideration of power losses, four cases are included in the study, which can be summarized as follows:

• Case 1: Minimization of the fuel costs without loss consideration for the 7-unit system.
• Case 2: Minimization of the fuel costs considering the power losses for the 7-unit system.
• Case 3: Minimization of the fuel costs without loss consideration for the 48-unit system.
• Case 4: Minimization of the fuel costs considering the power losses for the 48-unit system.

4.1. Implementation for Case 1

The suggested MAEA and basic AEA are employed to solve the ED with CHPUs in order to minimize fuel expenditures without accounting for losses. Table 1 depicts the optimal operating parameters of the POUs, CHPUs, and HOUs depending upon the suggested MAEA and the essential AEA in this instance. According to this data, the suggested MAEA achieves remarkable results by having the lowest fuel costs of 10,092.18 USD/h. To obtain these circumstances, the proposed MAEA sets the operational settings to 44.76, 98.56, 112.68, 209.82, 94.19, and 40 MW for the power outputs and 27.18, 74.99, and 47.82 MWth for the heat units. The basic AEA, on the other hand, attains fuel costs of 10,092.41 USD/h.

Table 1. Optimal operational settings and related costs of the proposed MAEA and the basic AEA of Case 1.

	Outputs	AEA	Proposed MAEA
Power-only units	Pg 1	44.70016	44.75768
	Pg 2	98.56597	98.56182
	Pg 3	112.681	112.6768
	Pg 4	209.8095	209.8153
CHP 1	Pg 5	94.24102	94.18733
	Hg 5	26.88296	27.18475
CHP 2	Pg 6	40.00238	40.00106
	Hg 6	74.95064	74.99904
Heat-only unit	Hg 7	48.1664	47.81621
Costs (USD/h)		10,092.41375	10,092.18153

In addition, Figure 4 displays the obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 1. As shown, the superior performance of the proposed MAEA is declared over the basic AEA in all simulated runs. The improvement percentage ranges from the very small value of 0.0023% to 0.89%.

Figure 4. Obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 1.

Based on that outcome in Figure 4, Table 2 records the robustness metrics of the proposed MAEA and the basic AEA of Case 1 in terms of the minimum, mean, maximum, and standard deviation. As shown, superior resilience performance related to the proposed MAEA is declared over the basic AEA. The proposed MAEA acquires the lowest minimum, mean, maximum, and standard deviation of 10,092.18, 10,093.32, 10,095.17, and 0.734646 USD/h, with improvements of 0.0023, 0.145, 0.892, and 97.31%, respectively.

Table 2. Robustness metrics of the proposed MAEA and the basic AEA of Case 1.

Costs (USD/h)	AEO	Proposed MAEA	Improvement %
Minimum	10,092.41	10,092.18	0.002301
Mean	10,108.01	10,093.32	0.145364
Maximum	10,186.05	10,095.17	0.892249
Standard Deviation	27.37743	0.734646	97.3166

In addition, Figure 5 displays the convergence rates of the proposed MAEA and the basic AEA related to the best run, worst run, and the average of all simulated runs. As demonstrated, the suggested MAEA has superior convergence features in its evolution in terms of lowering fuel expenditures throughout the duration of iterations. Despite achieving lower fitness values in the first 130 iterations, the AEA remained in a local optimal zone, particularly for its best run. The difference between the best run, worst run, and the average of all runs of the MAEA and AEA of Case 1 is shown in Figure 6, confirming the considerable improvement of the proposed MAEA after about 47%, 7%, and 20% of the total number of iterations for the average, best, and worst situations.

Figure 5. Convergence rates of the proposed MAEA and the basic AEA of Case 1.

Figure 6. Percentage difference for the best and worst run and the average of all runs of the MAEA and AEA of Case 1.

4.2. Implementation for Case 2

The suggested MAEA and basic AEA are employed to solve the ED with CHPUs in order to minimize fuel expenditures, taking into consideration the power losses. Table 3 illustrates the optimal operating parameters of the POUs, CHPUs, and HOUs depending upon the suggested MAEA and the essential AEA in this instance. The suggested MAEA achieves remarkable results by having the lowest fuel costs of 10,095.02 USD/h. To obtain these circumstances, the proposed MAEA sets the operational settings to 45.17, 98.54, 112.69, 209.82, 94.6, and 40 MW for the power outputs and 24.73, 75, and 50.27 MWth for the heat units. The basic AEA, on the other hand, attains fuel costs of 10,092.18 USD/h.

Table 3. Optimal operational settings and related costs of the proposed MAEA and the basic AEA of Case 2.

	Outputs	AEA	Proposed MAEA
Power-only units	Pg 1	45.17078	45.17078
	Pg 2	98.53982	98.53982
	Pg 3	112.6899	112.6899
	Pg 4	209.8158	209.8158
CHP 1	Pg 5	94.59907	94.59907
	Hg 5	24.72766	24.72766
CHP 2	Pg 6	40	40
	Hg 6	75.00086	75.00086
Heat-only unit	Hg 7	50.27148	50.27148
Costs (USD/h)		10,095.12	10,095.02

In addition, Figure 7 displays the obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 2. As shown, the superior performance of the proposed MAEA is declared over the basic AEA in all simulated runs. The improvement percentage ranges from the very small value of 0.0009% to 0.73%.

Figure 7. Obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 2.

Based on the outcome in Figure 7, Table 4 records the robustness metrics of the proposed MAEA and the basic AEA of Case 2. As shown, superior resilience performance related to the proposed MAEA is declared over the basic AEA. The proposed MAEA acquires the lowest minimum, mean, maximum, and standard deviation of 10,092.18, 10,093.32, 10,095.17, and 0.734646 USD/h with improvements of 0.0009, 0.115, 0.73, and 96.63%, respectively.

Table 4. Robustness metrics of the proposed MAEA and the basic AEA of Case 2.

Costs (USD/h)	AEO	Proposed MAEA	Improvement %
Minimum	10,095.11736	10,095.02453	0.000919468
Mean	10,107.45372	10,095.84203	0.114882388
Maximum	10,172.61916	10,097.86343	0.734872038
Standard Deviation	23.09259359	0.777264037	96.63414144

In addition, comparisons are made with the results of powerful optimization algorithms used in solving ED problems in the literature. For this purpose, Table 5 displays the comparative assessment of the AEA and the proposed MAEA with reported algorithms of TVAC-PSO [38], IGA [39], ECSA [40], PSO [41], TVAC-PSO [41], LCA [42], CPSO [43], WVO [44], WVO-PSO [44], RCGA [45], BCO [45], and DE [43,46]. As shown, the suggested MAEA provides better-performing features compared with the others.

Table 5. Comparative Results for Case 2 for the 7-Unit System.

Optimizer	Costs (USD/h)
Proposed MAEA	10,095.02453
AEO	10,095.11736
TVAC-PSO [38]	10,100.3000
IGA [39]	10,107.9071
ECSA [40]	10,121.9466
PSO [41]	10,178.4311
TVAC-PSO [41]	10,244.0200
LCA [42]	12,451.4000
CPSO [43]	10,325.3000
WVO-PSO [44]	10,372.0000
WVO [44]	10,317.0000
RCGA [45]	10,667.0000
BCO [45]	10,317.0000
DE [46]	10,317.0000
DE [43]	10,317.0000

In addition, Figure 8 displays the convergence rates of the proposed MAEA and the basic AEA related to the best run, worst run, and the average of all simulated runs. As demonstrated, the suggested MAEA has superior convergence features in its evolution in terms of lowering fuel expenditures throughout the duration of iterations. Despite achieving lower fitness values in the first 140 iterations, the AEA remained in a local optimal zone, particularly for its best run. Focusing on the average and worst performance of the AEA and MAEA, the difference between the obtained convergence of the MAEA and AEA of Case 2 is shown in Figure 9, confirming the considerable improvement of the proposed MAEA after about 43% and 28% of the total number of iterations for the average and worst situations.

Figure 8. Convergence rates of the proposed MAEA and the basic AEA of Case 2.

Figure 9. Percentage difference for the worst run and the average of all runs of the MAEA and AEA of Case 2.

4.3. Implementation for Case 3

In this case, the 48-unit system is considered where the load demand and heat demand are 4700 MW and 2500 MWth, respectively. The suggested MAEA and basic AEA are employed to solve the ED with CHPUs to minimize the fuel cost without considering the losses. Table 6 depicts the optimal settings of the POUs, CHPUs, and HOUs. According to this data, the suggested MAEA achieves remarkable results by having the lowest fuel costs of 116,897.9 USD/h. The basic AEA, on the other hand, attains fuel costs of 118,881.4 USD/h.

Table 6. Optimal operational settings and related costs of the proposed MAEA and the basic AEA of Case 3.

Outputs	AEA	Proposed MAEA	Outputs	AEA	Proposed MAEA
Pg 1	448.8807	538.5761	Pg 32	40.15046	53.42016
Pg 2	153.4517	224.6881	Pg 33	81.22367	105.7268
Pg 3	297.4129	150.6271	Pg 34	54.23016	40.71791
Pg 4	159.7331	109.8798	Pg 35	159.8071	145.7548
Pg 5	109.8657	159.6088	Pg 36	40.35118	58.07748
Pg 6	109.8665	109.6811	Pg 37	18.34234	11.87948
Pg 7	159.7313	109.9305	Pg 38	58.66311	35.5726
Pg 8	159.5867	111.388	Hg 27	157.8226	111.8012
Pg 9	109.8644	109.9677	Hg 28	77.27785	82.64235
Pg 10	113.2198	77.44919	Hg 29	106.0673	115.5393
Pg 11	84.37659	114.8267	Hg 30	96.1183	75.15452
Pg 12	69.79648	92.7831	Hg 31	40.07545	40.54104
Pg 13	108.2384	55.06172	Hg 32	22.3367	28.37245
Pg 14	269.1298	359.1073	Hg 33	104.9259	118.6701
Pg 15	18.09279	300.7246	Hg 34	87.28232	75.61917
Pg 16	299.1923	299.6896	Hg 35	149.024	141.13
Pg 17	134.9289	109.9425	Hg 36	75.30099	90.60269
Pg 18	159.7199	110.3213	Hg 37	43.57571	40.79744
Pg 19	133.4154	159.7346	Hg 38	30.7564	20.25903
Pg 20	159.7371	109.9029	Hg 39	418.0359	419.3306
Pg 21	109.4822	109.8995	Hg 40	60	59.99817
Pg 22	109.8535	110.3726	Hg 41	59.99961	59.02255
Pg 23	77.06126	77.56081	Hg 42	119.9991	119.9959
Pg 24	114.9288	77.73939	Hg 43	119.896	119.9999
Pg 25	92.40386	72.898	Hg 44	371.5652	420.5329
Pg 26	109.2187	92.54764	Hg 45	59.99897	59.99902
Pg 27	175.4811	93.49093	Hg 46	59.99318	59.99546
Pg 28	42.63888	48.85531	Hg 47	119.9613	119.9966
Pg 29	83.2744	100.1503	Hg 48	119.9873	119.9998
Pg 30	64.4735	40.18019	Costs (USD/h)	118,881.4	116,897.9
Pg 31	10.17506	11.26554

In addition, Figure 10 displays the obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 3. As shown, the superior performance of the proposed MAEA is declared over the basic AEA in all simulated runs. The improvement percentage ranges from the very small value of 1.67% to 3.99%.

Figure 10. Obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 3.

Added to that, Table 7 records the corresponding robustness metrics of the proposed MAEA and the basic AEA of Case 3. The proposed MAEA greatly outperforms the basic AEA. The proposed MAEA acquires the lowest minimum, mean, maximum, and standard deviation of 116,897.89, 118,004.35, 119,424.03, and 597.05 USD/h with improvements of 1.69, 1.7, 4, and 46.03%, respectively.

Table 7. Robustness metrics of the proposed MAEA and the basic AEA of Case 3.

Costs (USD/h)	AEO	Proposed MAEA	Improvement %
Minimum	118,881.4473	116,897.8879	1.668518838
Mean	120,045.6955	118,004.3493	1.70047432
Maximum	124,396.4722	119,424.0332	3.997250827
Standard Deviation	1106.34051	597.0478043	46.03399236

In addition, comparisons are made with the results of powerful optimization algorithms used in solving ED problems in the literature. For this purpose, Table 8 displays the comparative assessment of the AEA and proposed MAEA with reported algorithms of CPSO [41], GSA [15], MRFO [47], TVAC-PSO [41], MVO [47], and SSA [47]. As shown, the suggested MAEA provides better-performing features compared with the others.

Table 8. Comparative Results for Case 3 for the 48-Unit System.

Optimizer	Best Costs (USD/h)	Mean Costs (USD/h)	Worst Costs (USD/h)
Proposed MAEA	116,897.8879	118,004.3493	119,424.0332
AEO	118,881.4473	120,045.6955	124,396.4722
GSA [15]	119,775.9	-	-
MRFO [47]	117,336.9	117,875.4	118,217.5
CPSO [41]	120,918.9	-	-
TVAC-PSO [41]	118,962.5	-	-
MVO [47]	117,657.9	118,724	119,249.3
SSA [47]	120,174.1	121,110.2	122,636.8

Figure 11 displays the convergence rates of the proposed MAEA and the basic AEA related to the best run, worst run, and the average of all simulated runs. The suggested MAEA has superior convergence features in its evolution in terms of lowering fuel expenditures throughout the duration of iterations. Despite achieving lower fitness values in the first 200 iterations, the AEA remained in a local optimal zone, particularly for its best run. The difference between the best run, worst run, and the average of all runs of the MAEA and AEA of Case 3 is shown in Figure 12, confirming the considerable improvement of the proposed MAEA after about 13%, 6.2%, and 4.5% of the total number of iterations for the average, best, and worst situations.

Figure 11. Convergence rates of the proposed MAEA and the basic AEA of Case 3.

Figure 12. Percentage difference for the best, worst run, and the average of all runs of the MAEA and AEA of Case 3.

4.4. Implementation for Case 4

The suggested MAEA and basic AEA are employed to solve the ED with CHPUs to minimise fuel expenditures, taking into consideration the power losses. Table 9 illustrates the optimal settings of the POUs, CHPUs, and HOUs. According to this data, the suggested MAEA achieves remarkable results by having the lowest fuel costs of 118,134.96 USD/h. The basic AEA, on the other hand, attains fuel costs of 118,793.85 USD/h.

Table 9. Optimal operational settings and related costs of the proposed MAEA and the basic AEA of Case 4.

Outputs	AEA	Proposed MAEA	Outputs	AEA	Proposed MAEA
Pg 1	538.5587406	628.6477795	Pg 32	38.76118435	63.12739336
Pg 2	224.500532	299.3039097	Pg 33	88.68870343	146.019954
Pg 3	224.4082699	224.4413162	Pg 34	42.77532917	50.56025758
Pg 4	159.7326419	110.1098241	Pg 35	139.9313947	111.1809533
Pg 5	109.8653469	109.9778609	Pg 36	64.33155077	41.38648745
Pg 6	110.0410418	109.9393484	Pg 37	17.10618711	21.37570912
Pg 7	159.7343364	109.9133488	Pg 38	51.53918299	42.68203447
Pg 8	109.6188942	110.043928	Hg 27	124.6343646	116.4793542
Pg 9	109.8371821	109.938183	Hg 28	104.1938155	76.07240708
Pg 10	77.4032053	48.92271876	Hg 29	104.8875567	106.7188473
Pg 11	40.00026194	77.44715085	Hg 30	100.112622	84.40365244
Pg 12	92.61659623	94.12721908	Hg 31	44.08211567	41.31323057
Pg 13	69.13694475	92.38136273	Hg 32	21.70901773	32.76518388
Pg 14	538.5591303	448.8213154	Hg 33	109.1153478	141.283024
Pg 15	305.3626551	150.3625883	Hg 34	77.39491135	84.08846428
Pg 16	75.71780424	224.5198314	Hg 35	137.8725067	121.7181569
Pg 17	109.8666626	109.8560048	Hg 36	96.0050104	76.19104695
Pg 18	110.3999059	110.5899462	Hg 37	43.04592942	44.86858389
Pg 19	160.1264504	110.0674115	Hg 38	27.51826351	23.48428876
Pg 20	109.8878115	159.7957779	Hg 39	380.963592	415.0460685
Pg 21	109.8694694	109.9903509	Hg 40	59.96320817	59.86965016
Pg 22	109.8484983	160.5171237	Hg 41	59.99673171	59.97150217
Pg 23	97.5173401	77.50518542	Hg 42	119.9998655	119.9989975
Pg 24	77.40055945	77.50615461	Hg 43	119.9938832	119.7248381
Pg 25	92.63089118	92.55833553	Hg 44	408.8797	416.2126319
Pg 26	92.41551521	92.7814134	Hg 45	59.99971606	59.98718434
Pg 27	116.3421498	101.8906582	Hg 46	59.9999997	59.85924821
Pg 28	73.81758739	41.24267045	Hg 47	119.6365482	119.9649134
Pg 29	81.15521833	84.45117414	Hg 48	119.9952941	119.9787256
Pg 30	69.0898591	50.89534319	Costs (USD/h)	118,793.8535	118,134.9569
Pg 31	19.52394366	13.06729569

In addition, Figure 13 displays the obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 4. As shown, the superior performance of the proposed MAEA is declared over the basic AEA in all simulated runs. The improvement percentage ranges from the very small value of 0.55% to 3.87%.

Figure 13. Obtained costs for all simulated runs of the proposed MAEA and the basic AEA of Case 4.

In addition, Table 10 records the robustness metrics of the proposed MAEA and the basic AEA of Case 4. As shown, superior resilience performance related to the proposed MAEA is declared over the basic AEA. The proposed MAEA acquires the lowest minimum, mean, maximum, and standard deviation of 118,134.96, 118,925.83, 120,226.61, and 489.6 USD/h with improvements of 0.55, 1.44, 3.87, and 60.57%, respectively.

Table 10. Robustness metrics of the proposed MAEA and the basic AEA of Case 4.

Costs (USD/h)	AEO	Proposed MAEA	Improvement %
Minimum	118,793.8535	118,134.9569	0.554655419
Mean	120,660.8568	118,925.8259	1.437940105
Maximum	125,071.3754	120,226.6133	3.87359788
Standard Deviation	1241.686276	489.6017384	60.56961023

In addition, the convergence rates of the proposed MAEA and the basic AEA related to the best run, worst run, and the average of all simulated runs are displayed in Figure 14. The suggested MAEA has superior convergence features in its evolution in terms of lowering fuel expenditures throughout the duration of iterations. Despite achieving lower fitness values in the first 500 iterations, the AEA remained in a local optimal zone, particularly for its best run. The difference between the best run, worst run, and the average of all runs of the MAEA and AEA of Case 4 is shown in Figure 15, confirming the considerable improvement of the proposed MAEA after about 20%, 27%, and 10% of the total number of iterations for the average, best, and worst situations.

Figure 14. Convergence rates of the proposed MAEA and the basic AEA of Case 4.

Figure 15. Percentage difference for the worst run and the average of all runs of the MAEA and AEA of Case 4.

5. Conclusions

In this paper, a promising solution methodology based on a novel Modified Artificial Ecosystem Algorithm (MAEA) with superior performance and significant convergence has been proposed for solving the Economic Dispatch (ED) with Combined Heat and Power Units (CHPUs). The proposed MAEA combines the original AEA with a Fitness Distance Balance Model (FDBM) to increase solution quality in non-linear and multivariate optimization contexts. The FDBM was used as a method of selecting individuals which will contribute the most to the seeking paths within a community in a dependable and productive manner. As a result, the processes of diversification and intensification were carried out in a balanced manner. Both algorithms have been carried out in comparison using the 7-unit and 48-unit test systems. The suggested MAEA significantly outperforms the basic AEA with and without loss considerations. The suggested MAEA indicates superior resilience over the basic AEA by acquiring the lowest minimum, mean, maximum, and standard deviation. In addition, the suggested MAEA has superior convergence features in its evolution in terms of lowering fuel expenditures throughout the duration of iterations. As a further future study, applied methodology via the suggested MAEA is recommended for the optimal ED of cogeneration units considering the variability of electricity prices on the market which is a significant issue. Even though many CHPUs benefit from a tariff model with a fixed offtake price, it is recommended that the model is upgraded with external market signals in determining the optimal dispatch scenario.