Research on the Economic Scheduling Problem of Cogeneration Based on the Improved Artificial Hummingbird Algorithm

Abstract

With the increasing application of Combined Heat and Power (CHP) units, Combined Heat and Power Economic Dispatch (CHPED) has emerged as a significant issue in power system operations. To address the complex CHPED problem, this paper proposes an effective economic dispatch method based on the Improved Artificial Hummingbird Algorithm (IAHA). Given the complex constraints of the CHPED problem and the presence of valve point effects and prohibited operating zones, it requires the algorithm to have high traversal capability in the solution space and be resistant to becoming trapped in local optima. IAHA has introduced two key improvements based on the characteristics of the CHPED problem and the shortcomings of the standard Artificial Hummingbird Algorithm (AHA). Firstly, IAHA uses chaotic mapping to initialize the initial population, enhancing the algorithm’s traversal capability. Second, the guided foraging of the standard AHA has been modified to enhance the algorithm’s ability to escape from local optima. Simulation experiments were conducted on CHP systems at three different scales: 7 units, 24 units, and 48 units. Compared to other algorithms reported in the literature, the IAHA algorithm reduces the cost in the three testing systems by up to USD 18.04, 232.7894, and 870.7461. Compared to other swarm intelligence algorithms reported in the literature, the IAHA algorithm demonstrates significant advantages in terms of convergence accuracy and convergence speed. These results confirm that the IAHA algorithm is effective in solving the CHPED problem while overcoming the limitations of the standard AHA.

1. Introduction

Economic dispatch is one of the primary optimization problems in power system operations [1]. However, traditional power plants, which rely on coal-fired generation, suffer from low energy efficiency. During the energy conversion process, a significant amount of heat is lost and wasted. Combined Heat and Power (CHP) units improve energy conversion efficiency and avoid waste heat by generating electricity and heating simultaneously [2]. Over the years, CHP technology has garnered increasing attention from researchers due to its superior energy conversion efficiency [3,4]. According to the literature, CHP units can achieve 90% or higher energy efficiency [5] while reducing nearly 13% to 18% of pollutant gas emissions [6].

CHP systems typically consist of power-only units, CHP units, and heat-only units [7]. Due to the presence of Valve Point Effects and Prohibited Operating Zones (POZ) in traditional thermal power systems, the objective function of the Combined Heat and Power Economic Dispatch (CHPED) problem exhibits both nonlinear and non-convex characteristics [8]. CHPED aims to minimize the total production cost while satisfying various constraints, including VPE effects, power and heat generation capabilities, power-heat demand balance, and transmission losses [9].

Early studies proposed several classical mathematical methods, such as Lagrangian relaxation [10] and dual-quadratic programming [11], to address the CHPED problem. In response to these challenges, swarm intelligence optimization algorithms have gained significant attention for their ability to find optimal solutions under multiple constraints rapidly. Consequently, researchers began employing swarm intelligence optimization algorithms to tackle CHPED problems of varying scales. For example, Haghrah et al. [12] proposed an improved Real-Coded Genetic Algorithm based on Chaos Random Walk Mutation (RCGA-CRWM), introducing a simple method to determine the step sizes of random walks, which enhanced the algorithm’s convergence speed while avoiding premature convergence. Liu et al. [13] introduced a niching differential evolution algorithm (NDE) that balances the global and local search capabilities of the algorithm by using two strategies: the niching method and greedy selection, and verified its effectiveness on large-scale CHPED problems. An improved Marine Predator Optimization Algorithm (IMPOA) was proposed for the CHPED problem by Shaheen et al. [14], and its performance was tested using four testing systems to validate its efficiency and feasibility in solving CHPED problems of different scales. Ramachandran et al. [15] proposed the FRCSA-ABC algorithm by combining the Fuzzy adaptive Ranking-based Crow Search Algorithm (FRCSA) with an improved Artificial Bee Colony (ABC) algorithm. This algorithm incorporates three mechanisms that can achieve cost-effectiveness while ensuring global convergence. Sun et al. [16] introduced an Indicator & crowding Distance-Based Evolutionary Algorithm (IDBEA) specifically designed to address this non-convex and nonlinear problem. IDBEA was tested on three different types of CHPED systems. The simulation results demonstrate that the IDBEA algorithm has strong stability and superiority, and its solutions exhibit better convergence and diversity compared to other algorithms. Zhou et al. [17] proposed a novel Hybrid Neighbor Topology-based Multi-Agent Crossover algorithm (HNT-MACSO) to enhance the balance between exploration and exploitation, thereby improving the robustness of the CSO (Cat Swarm Optimization). Five CHP systems were tested, and the experimental results confirmed the effectiveness and superiority of HNT-MACSO in solving large-scale CHPED problems. Based on thermodynamics and heat transfer principles, Pattanaik et al. [18] proposed a heuristic optimization algorithm called Heat Transfer-based Search (HTS) to solve the CHPED problem, taking into account realistic constraints such as valve point effects, ramp rate limitations, and network losses. Ozkaya et al. [19] introduced a novel approach named Adaptive Fitness-Distance Balance-based Artificial Rabbits Optimization (AFDB-ARO) to address the CHPED problem. AFDB-ARO leverages the guidance mechanism of AFDB to enhance the exploration capabilities of ARO, thereby strengthening the balance between exploration and exploitation. Experimental results have confirmed the stability and superior performance of AFDB-ARO. Chen et al. [5] presented an improved Differential Evolution algorithm for large-scale CHPED problems, named SEDGCM, which enhanced performance by incorporating Gaussian-Cauchy mutation and parameter adaptation strategies, demonstrating the method’s excellent accuracy and stability in solving large-scale CHPED problems. For the non-convex and nonlinear CHPED problems in power systems, Nazari-Heris et al. [20] proposed a novel Multi-Person Harmony Search (MPHS) method. The performance of the proposed MPHS algorithm was validated on medium-scale and large-scale CHPED problems. Huang et al. [21] proposed the Heterogeneous Evolving Cuckoo Search (HECS) algorithm, which integrates learning strategies and applies heterogeneous evolutionary approaches, as well as a novel constraint handling mechanism, to tackle large-scale CHPED problems considering valve point effects. The effectiveness of HECS was proven with five CHP test systems. An improved artificial bee colony (IABC) algorithm was proposed to solve the CHPED problem by Rabiee et al. [22]. This algorithm has been applied to three testing systems, two of which are large-scale CHPED problems, demonstrating its superiority in handling non-convex and constrained CHPED problems. Spea et al. [23] employed a novel optimization algorithm, namely the Social Network Search (SNS) algorithm, to tackle the CHPED problem. SNS mimics the behavior of social network users to increase popularity, and experimental results have confirmed the effectiveness of SNS in various CHPED problems. By combining the Grey Wolf Optimizer (GWO) with Lévy Flight, Wang et al. [24] proposed the Lévy GWO algorithm, which overcomes the premature convergence problem of the GWO algorithm. They also introduced a constraint handling method, effectively solving scheduling problems that are difficult to handle due to numerous constraints.

In summary, swarm intelligence optimization algorithms can solve most optimization problems. However, it is important to note that no single intelligent optimization algorithm can efficiently solve all types of optimization problems and always find the best solution [25,26]. Therefore, it is necessary to combine real-world problems, improve existing swarm intelligence optimization algorithms, and explore more suitable solutions to problems.

The Artificial Hummingbird Algorithm (AHA) was introduced as a powerful algorithm by Weiguo Zhao, Liying Wang, and others in 2021 [27]. Their research indicates that AHA is a continuous algorithm that demonstrates good performance in the global optimization of complex functions. Due to its suitability for solving highly complex, non-convex problems that require high precision, it has been widely applied. As demonstrated in the literature [28], the AHA algorithm is applied to solve the complex problem of coordinated wind–solar–thermal generation scheduling problem in a multi-objective framework. The CHPED is a continuous problem, and in the actual situation, we often need to consider the fuel costs of three different types of units simultaneously, as well as the challenge of prohibited operating zones. Therefore, the CHPED problem is very complex and non-convex, requiring high traversal capability in the solution space and the ability of the algorithm to escape local optima. AHA, as a continuous algorithm, has a unique access table mechanism, flight patterns, and foraging behavior, which can help hummingbirds quickly and effectively find the optimal solution to highly complex and continuous problems. The characteristics of AHA are very consistent with those of the CHPED problem, so this paper employs AHA to solve the CHPED problem.

However, AHA suffers from drawbacks such as insufficient ergodicity and population diversity due to random initialization. Additionally, during the process of guiding foraging, if the fitness value of the new food source is poor, the guidance tends to be ineffective, resulting in the AHA easily falling into local optimality. To address these shortcomings, this paper first introduces chaotic mapping to initialize the population, improving the algorithm’s ergodicity and diversity. Secondly, the update mechanism of the AHA algorithm is modified to enhance its ability to escape the local optima. Finally, the constraint handling method is integrated with the proposed IAHA algorithm to solve the CHPED problem effectively.

To evaluate the performance of the proposed algorithm, three test systems composed of different units were selected for load distribution and scheduling optimization. Ultimately, a comprehensive comparative study was conducted between the IAHA algorithm and existing algorithms in the literature.

The remainder of this paper is organized as follows: Section 2 introduces the objective function and constraints of the CHPED problem. Section 3 details the proposed IAHA algorithm. Section 4 validates the performance of the IAHA algorithm using test functions. Section 5 analyzes the experimental results. Finally, Section 6 provides a conclusion to the paper.

2. Formulation of the CHPED Problem

Generally, a CHPED system is composed of three parts: power-only units, CHP units, and heat-only units. The objective of the CHPED problem is to determine the power output or the heat output of each unit to minimize the total cost of power generation while satisfying various equality and inequality constraints.

2.1. Objective Function

The objective function of a CHP system includes the generation costs of conventional power-only units, CHP units, and heat-only units, usually represented by Equation (1) [29]:

$$\min f_{\cos t}={\displaystyle \sum _{i=1}^{N_p}{C}_i}(P_i^p)+{\displaystyle \sum _{j=1}^{N_c}{C}_j(P_j^c,H_j^c)}+{\displaystyle \sum _{k=1}^{N_h}{C}_k(H_k^h)}$$

(1)

where $P_i^p$ and $H_k^h$ represent the power output of the $i$-th power-only unit and the heat output of the $k$-th heat-only unit, respectively; $P_j^c$ and $H_j^c$ represent the power and heat outputs of the $j$-th CHP unit, respectively; N_p, N_c, and N_h denote the number of power-only units, CHP units, and heat supply units, respectively; and C_i, C_j, and C_k are the operating costs of the power-only units, CHP units, and heat supply units, respectively and can be expressed is shown in Equations (2)–(4) [30]. Figure 1 is a schematic diagram of the threshold effect.

$$C_i(P_i^p)={\alpha }_i(P_i^p{)}^2+{\beta }_i{P}_i^p+{\gamma }_i+\left|e{e}_i\mathit{sin}(f{f}_i(P_i^{mi{n}_i^p}))\right|$$

(2)

$$C_j(P_j^c,H_j^c)=a_j(P_j^c{)}^2+b_j{P}_j^c+c_j+d_j(H_j^c{)}^2+e_j{H}_j^c+f_j{P}_j^c{H}_j^c$$

(3)

$$C_k(H_k^h)={\phi }_k(H_k^h{)}^2+{\eta }_k{H}_k^h+{\lambda }_k$$

(4)

2.2. Optimization Constraints

1. The load constraint condition is shown in Equations (5) and (6):

$${\displaystyle \sum _{i=1}^{N_p}{P}_i^P}+{\displaystyle \sum _{j=1}^{N_c}{P}_j^c}=P_D$$

(5)

$${\displaystyle \sum _{j=1}^{N_c}{H}_j^c}+{\displaystyle \sum _{k=1}^{N_h}{H}_k^h}=H_D$$

(6)

In Equations Equations (5) and (6), $P_D$ and $H_D$ represent the total electrical and thermal energy demands, respectively.

2. Power balance constraint

When considering grid losses, Equation (5) is modified to Equation (7):

$${\displaystyle \sum _{i=1}^{N_p}{P}_i^p+}{\displaystyle \sum _{j=1}^{N_c}{P}_j^c=P_D+P_L}$$

(7)

where $P_L$ is the grid loss. the mathematical description is shown in Equation (8) [31]:

$$P_L={\displaystyle \sum _{i=1}^{N_p}{\displaystyle \sum _{m=1}^{N_p}{P}_i^m{B}_{im}{P}_m^p+{\displaystyle \sum _{i=1}^{N_p}{\displaystyle \sum _{j=1}^{N_c}{P}_i^p{B}_{ij}{P}_j^c+}}}}{\displaystyle \sum _{j=1}^{N_c}{\displaystyle \sum _{n=1}^{N_c}{P}_j^c{B}_{jn}{P}_n^c}}$$

(8)

3. The unit output bounds constraint is shown in Equations (9)–(12):

$${P_i}^{p\min }\le {P_i}^p\le {P_i}^{p\max }$$

(9)

$${H_k}^{h\min }\le {H_k}^h\le {H_k}^{h\max }$$

(10)

$${P_j}^{c\min }({H_j}^c)\le {P_j}^c\le {P_j}^{c\max }({H_j}^c)$$

(11)

$${H_j}^{c\min }({P_j}^c)\le {H_j}^c\le {H_j}^{c\max }({P_j}^c)$$

(12)

In these equations, ${P_i}^{p\max }$ and ${P_i}^{p\min }$ are the maximum and minimum power outputs of the $i$-th power-only unit, ${H_k}^{h\max }$ and ${H_k}^{h\min }$ are the maximum and minimum heat outputs of the $k$-th heat supply unit, ${P_j}^{c\max }({H_j}^c)$ and ${P_j}^{c\min }({H_j}^c)$ are the maximum and minimum power outputs of the $j$-th CHP unit, and ${H_j}^{c\max }({P_j}^c)$ and ${H_j}^{c\min }({P_j}^c)$ are the maximum and minimum heat outputs of the $j$-th CHP unit.

2.3. Constraint Repair Technique

In the economic dispatch problem of CHP systems, there are numerous constraints. Therefore, dealing with these constraints effectively is crucial to avoid the local optima.

2.3.1. Equality Constraint Handling

In the CHPED problem, as the algorithm iterates and updates, some solutions may exceed their boundaries and become infeasible. For infeasible solutions, they can be discarded or repaired to become feasible solutions. Here, we use constraint repair to transform these individuals beyond the boundaries into feasible solutions (Figure 2).

1.

For the power-only units, $P_i^p$ is repaired as in Equation (13):

$$P_i^p=\left\{\begin{array}{ll}{P}_i^{p\min }\hfill & \mathrm{if} P_i^p\le P_i^{p\min } \\ P_i^{p\max }\hfill & \mathrm{if} P_i^p\ge P_i^{p\max }\\ P_i^p\hfill & otherwise\end{array}\right.$$

(13)

2.

For the heat-only units, $H_k^h$ is repaired as in Equation (14):

$$H_k^h=\left\{\begin{array}{ll}{h}_k^{h\min }\hfill & \mathrm{if} H_k^h\le h_k^{h\min }\\ h_k^{h\max }\hfill & \mathrm{if} H_k^h\ge h_k^{h\max }\\ H_k^h\hfill & otherwise\end{array}\right.$$

(14)

3.

For the CHP units, the power and heat output are interrelated and mutually restricted. $P_j^{c\min }(H_j^c)$ and $P_j^{c\max }(H_j^c)$ are calculated based on $H_j^c$. Then, $P_j^c$ is repaired as in Equation (15):

$$P_j^c=\left\{\begin{array}{ll}{P}_j^{c\min }(H_j^c)\hfill & \mathrm{if} P_j^c\le P_j^{c\min }(H_j^c)\\ P_j^{c\max }(H_j^c)\hfill & \mathrm{if} P_j^c\ge P_j^{c\max }(H_j^c)\\ P_j^c(H_j^c)\hfill & otherwise\end{array}\right.$$

(15)

Similarly, $H_j^{c\min }(P_j^c)$ and $H_j^{c\max }(P_j^c)$ are calculated based on $P_j^c$. Then, $H_j^c$ is repaired as in Equation (16):

$$H_j^c=\left\{\begin{array}{ll}{H}_j^{c\min }(P_j^c)\hfill & \mathrm{if} H_j^c\le H_j^{c\min }(P_j^c)\\ H_j^{c\max }(P_j^c)\hfill & \mathrm{if} H_j^c\ge H_j^{c\max }(P_j^c)\\ H_j^c\hfill & otherwise\end{array}\right.$$

(16)

2.3.2. Handling of Equality Constraints

1. Total load constraints of the heating and power networks

In the CHPED problem, given the differences in the scale of CHP units, grid losses are often neglected when handling CHPED problems of certain scales. Given that the grid loss values are relatively small for CHP units of specific scales considering grid losses, we assume that these values do not change significantly with fluctuations in unit power when correcting the unit power. Intelligent algorithms are characterized by individual random updates, and thus, when using intelligent algorithms to solve practical problems, it is common to encounter situations where some individuals do not satisfy the equality constraints. This paper adopts the following method to handle equality constraints:

Step 1: Take an individual X_i as a solution and divide X_i into two parts: $X_i^P$ and $X_i^H$. Here, $X_i^P$ = [P₁…P_Np,$P_1^C$…$P_{Nc}^C$], $X_i^H$ = [$H_1^C$… $H_{Nc}^C$, H₁…H_Nh]. $X_i^H$ includes the generation from power-only units and CHP units in the system. $X_i^H$ includes the heating units from heating-only units and the heating components from CHP units in the system. Then, calculate the constraint violation values $f_{power}$ and $f_{heat}$ in Equations (17) and (18):

$$f_{power}=P_D+P_L-({\displaystyle \sum _{i=1}^{N_p}{P}_i^P}+{\displaystyle \sum _{j=1}^{N_c}{P}_j^c})$$

(17)

$$f_{heat}=H_D-({\displaystyle \sum _{j=1}^{Nc}{H}_j^c}+{\displaystyle \sum _{k=1}^{N_h}{H}_k^h})$$

(18)

Step 2: Repair the power demand constraint. If $f_{power}$> 0, select the first variable $X_{idx}^P$ in $X_i^P$, and repair $X_{idx}^P$ shown in Equation (19):

$$X_{idx}^P=\min (X_{idx}^P+f_{power},P_{idx}^{\max })$$

(19)

Step 3: If the value of $X_{idx}^P$ is equal to $P_{idx}^{\max }$, it indicates that $X_{idx}^P+f_{power}$ exceeds its boundary. In this case, select the next variable $X_{jdx}^P$, assign the portion of $X_{idx}^P+f_{power}$ that exceeds the boundary to $X_{jdx}^P$, and use Equation (19) to determine whether $X_{jdx}^P$. exceeds its boundary. Repeat this step until $f_{power}$ = 0. If, after all variables in $X_i^P$ have been adjusted, the value of $f_{power}$ is still not equal to 0, then discard that individual.

Step 4: When $f_{power}$ < 0, select the first variable $X_{idx}^P$ in $X_i^P$, and repair $X_{idx}^P$ shown in Equation (20):

$$X_{idx}^P=\max (X_{idx}^P-f_{power},P_{idx}^{\min })$$

(20)

Step 5: If the value of $X_{idx}^P$ is equal to $P_{idx}^{\min }$, it indicates that $X_{idx}^P-f_{power}$ exceeds its boundary. In this case, select the next variable $X_{jdx}^P$, assign the portion of $X_{idx}^P-f_{power}$ that exceeds the boundary to $X_{jdx}^P$, and use Equation (20) to determine whether $X_{jdx}^P$ exceeds its boundary. Repeat this step until $f_{power}$ = 0. If, after all variables in $X_i^P$ have been adjusted, the value of $f_{power}$ is still not equal to 0, then discard that individual.

Step 6: Repair the heating demand constraint. If $f_{heat}$ > 0, select the last variable $X_{idx}^H$ in $X_i^H$ and perform the following repair $X_{idx}^H$ shown in Equation (21):

$$X_{idx}^H=\min (X_{idx}^H+f_{heat},H_{idx}^{\max })$$

(21)

Step 7: If the value of $X_{idx}^H$ is $H_{idx}^{\max }$, it indicates that the value of $X_{idx}^H+f_{heat}$ exceeds its boundary. Choose the previous variable $X_{jdx}^H$ and assign the portion of $X_{idx}^H+f_{heat}$ that exceeds the boundary to $X_{jdx}^H$. Use Equation (21) to check if $X_{jdx}^H$ exceeds its boundary. Repeat this step until $f_{heat}$ = 0. If the value of $f_{heat}$ still cannot be zero after all variables in $X_i^H$ have been repaired, discard this individual.

Step 8: When $f_{heat}$ < 0, select the last variable $X_{idx}^H$ in $X_i^H$ and perform the following repair $X_{idx}^H$ shown in Equation (22):

$$X_{idx}^H=\max (X_{idx}^H-f_{heat},H_{idx}^{\min })$$

(22)

Step 9: If the value of $X_{idx}^H$ is $H_{idx}^{\min }$, it indicates that the value of $X_{idx}^H-f_{heat}$ exceeds its boundary. Choose the previous variable $X_{jdx}^H$ and assign the portion of $X_{idx}^H-f_{heat}$ that exceeds the boundary to $X_{jdx}^H$. Use Equation (22) to check if $X_{jdx}^H$ exceeds its boundary. Repeat this step until $f_{heat}$ = 0. If the value of $f_{heat}$ still cannot be zero after all variables in $X_i^H$ have been repaired, discard this individual.

3. Improved Artificial Hummingbird Algorithm

3.1. Introduction of the Artificial Hummingbird Algorithm

The Artificial Hummingbird Algorithm is a novel metaheuristic algorithm proposed by Weiguo Zhao, Liying Wang, and others in 2021 [27]. The algorithm is inspired by the foraging process of hummingbirds for nectar. The basic principle of the algorithm is to use mathematical models to simulate the flight patterns, foraging methods, and memory abilities of hummingbirds during foraging activities.

3.1.1. Flight Patterns

In the AHA, hummingbirds have three types of flight patterns: axial flight, diagonal flight, and omnidirectional flight. These flight patterns are reflected in the algorithm as control over different dimensional directions of the solution space, represented by vector $D$.

(1) Axial Flight: The hummingbird flies along a specific coordinate axis direction in space. This flight pattern allows the hummingbird to explore in only one dimensional direction. Its mathematical description is shown in Equation (23):

$$D^{(i)}=\left\{\begin{array}{l}1 if i = randi([1,d])\hfill \\ 0 \mathrm{else}\hfill \end{array}\right.i=1,\mathrm{\dots },d$$

(23)

In this context,$randi \left(\right[1,d\left]\right)$ is a random integer between 1 and $d$. The 1 in vector $D$ indicates that the hummingbird can fly in the direction of that dimension, meaning the corresponding position in the solution space for that dimension is allowed to be updated.

(2) Diagonal Flight: The hummingbird flies in a diagonal direction formed by randomly chosen coordinate axes. This flight pattern preserves at least one dimension that is not updated. The mathematical description is shown in Equation (24):

$$\begin{array}{}\begin{array}{l}{D}^{(i)}=\left\{\begin{array}{ll}1\hspace{1em}if\hspace{1em}i=P(j),j\in [1,k]\hfill & \\ & i=1,\dots ,d\\ 0\hspace{1em}else\hfill & \end{array}\right.\hfill \end{array}\\ \begin{array}{l}s.t. P=randperm(k)\hfill \\ \hspace{1em}\hspace{1em}k=[2,⌈r_1(d-2)⌉+1]\hfill \end{array}\end{array}$$

(24)

$Randperm\left(k\right)$ represents a random permutation of integers from 1 to $k$, and $r_1$ is a random number within $(0,1]$. When the value of vector $D$ is 1, it indicates that the hummingbird can update the corresponding position in the solution space for that dimension.

(3) Omnidirectional Flight: The hummingbird flies in directions formed by combinations of all coordinate axes, meaning all dimensions in the solution space are allowed to be updated. The mathematical description is shown in Equation (25):

$$D^{\left(i\right)}=1i=1,...,d$$

(25)

3.1.2. Foraging Methods

Hummingbirds have three primary foraging strategies: guided foraging, territorial foraging, and migratory foraging. In the algorithm, these strategies correspond to global search, local search, and mutation search, respectively.

(1) Guided Foraging: In guided foraging, hummingbirds always target the location with the highest unvisited rank as their food source. If there are multiple targets with the same unvisited rank, the one with the highest nectar replenishment rate is chosen. The hummingbird then explores between the target and its current position to find a better food source. Each hummingbird retains a memory of previously visited locations, recording the visit ranks in a visitation table. After each foraging, the unvisited rank of locations that have not been visited increases, while the rank of visited locations drops to the lowest level. The mathematical description of guided foraging is as shown in Equation (26):

$$v_i(t+1)=x_{i,tar}\left(t\right)+a·D·\left(x_i\right(t)-x_{i,tar}(t\left)\right)$$

(26)

where $t$ represents number of iterations, $a$ is the guided step size, with a being a value of $rand$, $x_{i,tar}\left(t\right)$ is the position of the target food source, and $x_i\left(t\right)$ is the current position of hummingbird $i$. The position of the $i-$th food source is updated as shown in Equation (27):

$$x_i(t+1)=\left\{\begin{array}{ll}{x}_i(t)\hfill & f(x_i(t))\le f(v_i(t+1))\hfill \\ v_i(t+1)\hfill & f(x_i(t))>f(v_i(t+1))\hfill \end{array}\right.$$

(27)

When the fitness value $f\left(x_i\right(t\left)\right)$ of the target food source is better than that of the hummingbird’s current food source (as per Equation (28)), the hummingbird will abandon the current source and move to the new food source.

(2) Territorial Foraging: In territorial foraging, the hummingbird searches for a better food source in the vicinity outside its territory. The mathematical description of territorial foraging is shown in Equation (28):

$$v_i(t+1)=x_i+b·D\cdot x_i\left(t\right)$$

(28)

where $b$ is the flight step size, with $b$ being a value of $rand$.

(3) Migratory Foraging: After guided or territorial foraging, the hummingbird at the worst food source position needs to migrate to a distant location and quickly find the next food source. The mathematical description of migratory foraging is shown in Equation (29):

$$x_{wor}(t+1)=Low+r·(Up-Low)$$

(29)

where $x_{wor}$ is the food source with the lowest nectar replenishment rate in the population, and $Up$ and $Low$ are the upper and lower bounds of the solution space, respectively.

3.2. Improving the Artificial Hummingbird Algorithm

Since the CHPED problem considers three types of units simultaneously and involves highly complex constraints, as well as valve point effects and prohibited operating zones, the algorithm used to solve the CHPED problem must have high traversal capability in the solution space and avoid becoming trapped in the local optima. To address the drawbacks of the AHA and solve the CHPED problem more effectively, an IAHA is proposed instead.

3.2.1. Initial Solution

The standard AHA initializes the population using a random initialization method. However, this approach suffers from insufficient traversal and low population diversity, which can negatively impact the convergence speed and optimization accuracy of the AHA. In order to solve these problems, this paper proposes to use chaotic mapping to initialize the population of the algorithm.

Chaos is a complex and widespread phenomenon in nonlinear deterministic systems, reflecting the interaction between order and disorder and between determinism and randomness [32]. Chaotic behavior is confined to a specific region, and its trajectory is complex, unique, and never repeats. Although chaos appears random, it is actually a deterministic system [33].

Chaotic mapping is characterized by randomness, traversal, and regularity. It can traverse the search space non-repetitively within a certain range, according to its own rules. The initial population generated in this way outperforms the traditional random initialization method in terms of solution accuracy and convergence speed, enhancing the population diversity and providing strong adaptability and explorability within the solution space. Therefore, chaotic mapping effectively compensates for the shortcomings of the standard AHA. The mathematical description of chaotic mapping initialization is shown in Equations (30) and (31):

$$\left\{\begin{array}{l}\beta (1)=rand()\hfill \\ \beta (i+1)=\sin (\pi \cdot \beta (i))\hfill \end{array}\right.$$

(30)

$$X_i=\beta ·(Up-Low)+Low$$

(31)

where the strategy is chosen as the chaotic map, and β is the chaotic mapping coefficient. X_i represents the population generated through chaotic mapping.

3.2.2. Modify Update Mechanism

In the standard AHA, whenever the position of a food source is updated, the priority level of the new food sources that have not been visited is usually set to the highest. However, during experiments, it was observed that, if the new food source’s nectar replenishment rate (fitness value) is poor, the hummingbird might fail to utilize the new food source effectively during guided foraging, potentially leading the entire AHA algorithm to become trapped in a local optimum (Figure 3).

To overcome this issue, the original update rule has been modified. Specifically, the priority of the new food source is elevated to the highest level only if its nectar replenishment rate exceeds the average nectar replenishment rate of all the current food sources. Otherwise, the priority levels of the existing food sources remain unchanged. This adjustment not only enhances the algorithm’s convergence speed but also maintains its optimization capability.

The mathematical description of the new update rule is as expressed as Equations (32)–(34):

$$X_i\left(newFoodIndex\right)=x_{i,new}+rand·D·(x_i-x_{i,new})$$

(32)

$$X_i\left(oldFoodIndex\right)=x_{i,old}+rand·D·(x_i-x_{i,new})$$

(33)

$$new{X}_i=\left\{\begin{array}{ll}{X}_i(newFoodIndex)\hfill & if newFit<{\displaystyle \sum _{i=1}^{N}Fit}/N\hfill \\ X_i(oldFoodIndex)\hfill & otherwise\hfill \end{array}\right.$$

(34)

where $X_i(newFoodIndex)$ represents the position of the new food source, $X_i(oldFoodIndex)$ represents the position of the old food source, $x_{i,new}$ represents the position of the target food source, $x_{i,old}$ represents the position of the old food source, and xi is the position of the current food source. $new{X}_i$ is the new individual generated using the updated rule.

4. Performance Verification of the IAHA

To validate the effectiveness and rationality of the IAHA, it was compared with four other metaheuristic algorithms, which acronyms and parameters are listed in

Table 1. List of algorithm abbreviations.

Table of Abbreviations	Abbreviations
1. Artificial Hummingbird Algorithm	AHA [27]
2. Ant Lion Optimizer	ALO [35]
3. Grey Wolf Optimizer	GWO [34]
4. Whale Optimization Algorithm	WOA [36]
5. Improved Artificial Hummingbird Algorithm	IAHA

. The experiments were conducted using six well-known benchmark functions: F1 (Sphere), F2 (Schwefel 2.22), F3 (Schwefel 1.2), F4 (Rosenbrock), F5 (Schwefel), and F6 (Ackley), selected from the 23 benchmark test functions mentioned in the literature [34]. These experiments demonstrated the effectiveness and superiority of the IAHA. As shown in Table 1, the selected benchmark functions include both unimodal and multimodal test functions. Unimodal functions have only one global optimum, making it easy to determine, whereas multimodal functions have multiple local optima, making it difficult to find the global optimum. Therefore, the chosen benchmark functions can effectively evaluate the algorithm’s performance in escaping the local optima and convergence speed. To verify the robustness of IAHA across different dimensions of the same test function, the algorithm’s ability was tested using dimensions of 10, 30, and 50 for both unimodal and multimodal test functions, and each dimension was tested separately 30 times. The maximum number of iterations was uniformly set to 1000. When the evaluation times of the function reach the maximum number of iterations, the optimization process of the algorithm will stop. All experiments in this study were conducted on a PC equipped with a 12th Gen Intel(R) Core (TM) i7-12700H 2.30 GHz processor and 16 GB of RAM. Table 1 shows the names and abbreviations of five algorithms.

Table 2. Algorithm parameters table.

Algorithms	Parameter Settings	Iteration
Artificial Hummingbird Algorithm	NP = 30	1000
Ant Lion Optimizer	NP = 30	1000
Grey Wolf Optimizer	NP = 30	1000
Whale Optimization Algorithm	NP = 30, b = 1	1000
Improved Artificial Hummingbird Algorithm	NP = 30	1000

shows the parameter settings and iteration of five algorithms

Table 3. Basic test functions.

Function Name	Dim	Range	Fmin
Sphere	10/30/50	[−100, 100]	0
Schwefel 2.22	10/30/50	[−10, 10]	0
Schwefel 1.2	10/30/50	[−100, 100]	0
Rosenbrock	10/30/50	[−30, 30]	0
Schwefel	10/30/50	[−500, 500]	−12,569.5
Ackley	10/30/50	[−32, 32]	0

shows the names and parameters of six test functions. The experimental results are shown in

Table 4. Comparison of the IAHA with other algorithms (10 dimensions).

NO.	Statistics	ALO	GWO	WOA	AHA	IAHA
F1	Mean	2.67 × 10−9	4.07 × 10−117	1.02 × 10−157	2.12 × 10−277	0.00
	Sta	1.16 × 10−9	1.99 × 10−116	4.69 × 10−157	0.00	0.00
	Best	1.09 × 10−9	1.08 × 10−123	8.64 × 10−174	1.02 × 10−310	0.00
	Time (s)	160.24	160.28	160.33	155.18	155.24
	Winner	1	1	1	1
F2	Mean	2.69 × 10−1	2.98 × 10−67	1.49 × 10−106	4.24 × 10−144	3.25 × 10−226
	Sta	7.77 × 10−1	4.99 × 10−67	6.20 × 10−106	1.56 × 10−143	0.00
	Best	6.57 × 10−6	1.78 × 10−69	9.19 × 10−106	1.13 × 10−154	1.74 × 10−242
	Time (s)	158.32	158.37	158.42	153.29	153.35
	Winner	1	1	1	1
F3	Mean	9.38 × 10−6	5.46 × 10−52	2.35 × 101	1.73 × 10−244	0.00
	Sta	1.05 × 10−5	2.03 × 10−51	8.07 × 101	0.00	0.00
	Best	1.09 × 10−7	1.10 × 10−64	3.56 × 10−10	1.70 × 10−278	0.00
	Time (s)	163.46	163.56	163.66	158.39	158.49
	Winner	1	1	1	1
F4	Mean	5.21 × 10−1	0.65 × 101	0.62 × 101	0.45 × 101	1.87 × 10−4
	Sta	1.04 × 102	0.09 × 101	0.04 × 101	0.04 × 101	6.58 × 10−4
	Best	5.15 × 10−4	0.47 × 101	0.55 × 101	0.34 × 101	4.4 × 10−7
	Time (s)	159.53	159.59	159.66	154.44	154.50
	Winner	1	1	1	1
F5	Mean	−2.47 × 103	−2.71 × 103	−3.37 × 103	−4.19 × 103	−4.19 × 103
	Sta	6.42 × 10−2	2.96 × 10−2	6.06 × 10−2	2.80 × 10−7	3.13 × 10−3
	Best	−4.19 × 103	−3.23 × 103	−4.19 × 103	−4.19 × 103	−4.19 × 103
	Time (s)	157.85	157.91	157.99	152.83	152.89
	Winner	1	1	1	−1
F6	Mean	1.60 × 10−1	4.67 × 10−15	4.09 × 10−15	8.88 × 10−16	8.88 × 10−16
	Sta	5.03 × 10−1	9.01 × 10−16	2.16 × 10−15	0.00	0.00
	Best	1.50 × 10−5	4.44 × 10−15	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Time (s)	344.87	344.92	344.98	339.82	339.87
	Winner	1	1	1	−1

,

Table 5. Comparison of the IAHA with other algorithms (30 dimensions).

NO.	Statistics	ALO	GWO	WOA	AHA	IAHA
F1	Mean	1.02 × 10−5	6.40 × 10−59	2.16 × 10−149	3.05 × 10−284	0.00
	Sta	7.63 × 10−6	1.37 × 10−58	1.04 × 10−148	0.00	0.00
	Best	1.16 × 10−6	2.20 × 10−62	7.01 × 10−169	1.54 × 10−315	0.00
	Time (s)	663.6472	663.7312	663.7962	647.4758	649.5391
	Winner	1	1	1	1
F2	Mean	4.34 × 101	7.84 × 10−35	8.90 × 10−104	5.01 × 10−148	1.05 × 10−224
	Sta	5.26 × 101	4.84 × 10−35	4.51 × 10−103	1.71 × 10−147	0.00
	Best	1.99 × 10−2	4.96 × 10−36	1.06 × 10−112	1.06 × 10−162	3.71 × 10−245
	Time (s)	520.2035	520.2883	520.3485	506.1642	506.2296
	Winner	1	1	1	1
F3	Mean	1.09 × 103	5.24 × 10−15	2.07 × 104	2.00 × 10−264	0.00
	Sta	6.43 × 102	1.47 × 10−14	9.64 × 103	0.00	0.00
	Best	3.89 × 102	1.01 × 10−20	3.30 × 103	5.68 × 10−301	0.00
	Time (s)	456.5424	456.7693	456.9743	442.0417	442.2414
	Winner	1	1	1	1
F4	Mean	2.84 × 102	2.70 × 101	2.71 × 101	2.57 × 101	1.20 × 10−3
	Sta	4.02 × 102	5.49 × 10−1	4.71 × 10−1	3.81 × 10−1	3.90 × 10−3
	Best	2.12 × 101	2.52 × 101	2.66 × 101	2.51 × 101	1.71 × 10−7
	Time (s)	439.1047	439.2087	439.2835	424.8770	424.9433
	Winner	1	1	1	1
F5	Mean	−5.57 × 103	−5.79 × 103	−1.14 × 104	−1.21 × 104	−1.26 × 104
	Sta	4.79 × 102	8.10 × 102	1.51 × 103	3.94 × 102	1.16 × 10−2
	Best	−8.07 × 103	−6.94 × 103	−1.26 × 104	−1.26 × 104	−1.26 × 104
	Time (s)	438.1611	438.2655	438.3335	423.9423	424.0128
	Winner	1	1	1	0
F6	Mean	0.21 × 101	1.55 × 10−14	4.44 × 10−15	8.88 × 10−16	8.88 × 10−16
	Sta	7.53 × 10−1	1.71 × 10−15	1.87 × 10−15	0.00	0.00
	Best	9.13 × 10−4	1.15 × 10−14	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Time (s)	436.6612	436.7514	436.8130	422.4533	422.5068
	Winner	1	1	1	−1

and

Table 6. Comparison of the IAHA with other algorithms (50 dimensions).

NO.	Statistics	ALO	GWO	WOA	AHA	IAHA
F1	Mean	7.41 × 10−4	2.16 × 10−43	9.72 × 10−151	1.58 × 10−287	0.00
	Sta	2.94 × 10−4	6.42 × 10−43	5.27 × 10−150	0.00	0.00
	Best	3.09 × 10−4	4.78 × 10−45	6.77 × 10−168	0.00	0.00
	Time (s)	1144.81	1144.93	1144.99	1121.32	1121.38
	Winner	1	1	1	1
F2	Mean	1.07 × 102	4.55 × 10−26	1.61 × 10−102	7.94 × 10−146	7.25 × 10−222
	Sta	8.67 × 101	2.46 × 10−26	7.53 × 10−102	4.35 × 10−145	0.00
	Best	0.45 × 101	9.83 × 10−27	5.53 × 10−115	6.94 × 10−167	3.95 × 10−250
	Time (s)	727.8588	727.9880	728.0518	704.1420	704.2151
	Winner	1	1	1	1
F3	Mean	9.17 × 103	5.97 × 10−6	1.23 × 105	5.91 × 10−261	0.00
	Sta	2.77 × 103	2.03 × 10−5	3.57 × 104	0.00	0.00
	Best	3.54 × 103	2.03 × 10−11	5.96 × 104	5.55 × 10−302	0.00
	Time (s)	1277.53	1277.91	1278.24	1252.25	1252.57
	Winner	1	1	1	1
F4	Mean	3.37e × 102	4.71 × 101	4.75 × 101	4.62 × 101	2.92 × 10−4
	Sta	3.59e × 102	8.62 × 10−1	5.83 × 10−1	4.76 × 10−1	6.99 × 10−4
	Best	4.26 × 101	4.54 × 101	4.67 × 101	4.55 × 101	8.76 × 10−7
	Time (s)	1430.55	1430.71	1430.80	1405.94	1406.01
	Winner	1	1	1	1
F5	Mean	−9.57 × 103	−8.86 × 103	−1.75 × 104	−1.92 × 104	−2.09 × 104
	Sta	1.58 × 103	1.55 × 103	3.17 × 103	5.59 × 102	1.85 × 10−2
	Best	−1.49 × 104	−1.11 × 104	−2.09 × 104	−2.02 × 104	−2.09 × 104
	Time (s)	1390.45	1390.64	1390.77	1360.35	1360.68
	Winner	1	1	1	1
F6	Mean	0.45 × 101	3.39 × 10−14	4.09 × 10−15	8.88 × 10−16	8.88 × 10−16
	Sta	0.26 × 101	4.39 × 10−15	2.69 × 10−15	0.00	0.00
	Best	0.24 × 101	2.93 × 10−14	8.88 × 10−16	8.88 × 10−16	8.88 × 10−16
	Time (s)	1126.10	1126.46	1126.64	1057.07	1057.31
	Winner	1	1	1	−1

. In Figure 4, Figure 5 and Figure 6, subgraphs a-f show the iterative curves of different algorithms on the test function F1–6

5. Experimental Results

In this section, the IAHA algorithm is applied to solve the CHPED problem, and three systems of different sizes are adopted. To verify its effectiveness, we compared the IAHA with various other algorithms across different test systems.

5.1. Population Coding

In the CHPED problem, each individual is composed of the power and heat outputs from power-only units, CHP units, and heat-only units. When the AHA is employed to solve the CHPED problem, the solution matrix of the algorithm consists of two parts: the number of individuals and the number of units. Here, the number of solutions is equivalent to the number of hummingbirds in the AHA, and the number of units corresponds to the dimensions in the AHA. To better represent these two sub-problems, this paper encodes individuals in the following manner as Equation (35):

$$X=\left[\begin{array}{llll}{P_{11}}^p\cdots P_{1N_p}^p,\hfill & P_{11}^c\cdots P_{1N_c}^c,\hfill & H_{11}^c\cdots H_{1N_c}^c,\hfill & {H_{11}}^h\cdots H_{1N_h}^h\hfill \\ {P_{21}}^p\cdots P_{2N_p}^p,\hfill & P_{21}^c\cdots P_{2N_c}^c,\hfill & H_{21}^c\cdots H_{2N_c}^c,\hfill & {H_{21}}^h\cdots H_{2N_h}^h\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ {P_{N1}}^p\cdots P_{N{N}_p}^p,\hfill & P_{N1}^c\cdots P_{N{N}_c}^c,\hfill & H_{N1}^c\cdots H_{N{N}_c}^c,\hfill & {H_{N1}}^h\cdots H_{N{N}_h}^h\hfill \end{array}\right]$$

(35)

where X is the population matrix, and N is the number of individuals. ${P_{11}}^p$ represents the power output of the first power-only unit in the first individual, while $P_{1N_p}^p$ represents the power output of the last power-only unit in the first individual. $P_{11}^c$ denotes the power output of the first CHP unit in the first individual, and $P_{1N_c}^c$ represents the power output of the last CHP unit in the first individual. $H_{11}^c$ stands for the heat output of the first CHP unit in the first individual, and $H_{1N_c}^c$ represents the heat output of the last CHP unit in the first individual. ${H_{11}}^h$ is the heat output of the first heat-only unit in the first individual, and $H_{1N_h}^h$ represents the heat output of the last heat-only unit in the first individual.

Similarly, ${P_{N1}}^p$ is the power output of the first power-only unit in the last individual, and $P_{N{N}_p}^p$ is the power output of the last power-only unit in the last individual. $P_{N1}^c$ is the power output of the first CHP unit in the last individual, and $P_{N{N}_c}^c$ is the power output of the last CHP unit in the last individual. $H_{N1}^c$ is the heat output of the first CHP unit in the last individual, and $H_{N{N}_c}^c$ is the heat output of the last CHP unit in the last individual. ${H_{N1}}^h$ is the heat output of the first heat-only unit in the last individual, and $H_{N{N}_h}^h$ is the heat output of the last heat-only unit in the last individual (Figure 7).

5.2. Test System 1: 7-Unit CHPED Problem

The first test system is a seven-unit CHPED problem, taken from reference [37]. It is a classic small-scale CHPED problem consisting of four power-only units, two CHP units, and one heat-only unit. In Test System 1, grid losses, valve point effects, and operational regions were considered. The maximum number of iterations was set to MaxIt = 1000, and the population size NP = 100. Accordingly,

Table 7. Power-only units cost function parameters of Test System 1.

Power-Only Units	αi	βi	γi	eei	ffi	Pimin	Pimax
1	0.008	2	25	100	0.042	10	75
2	0.003	1.8	60	140	0.04	20	125
3	0.0012	2.1	100	160	0.038	30	175
4	0.001	2	120	180	0.037	40	250

,

Table 8. CHP units cost function parameters of Test System 1.

CHP Units	aj	bj	cj	dj	ej	fj	Feasible Region Coordinates
5	0.0345	14.5	2650	0.03	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
6	0.0435	36	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]

and

Table 9. Heat-only units cost function parameters of Test System 1.

Heat-Only Units	φk	ηk	λk	Hmin	Hmax
7	0.038	2.0109	950	0	2695.20

list the relevant parameters of the operational cost functions for each unit. The data were sourced from [38].

The grid loss coefficient is provided by Equation (36) [24]:

$$B=\left[\left.\begin{array}{llllll}49\hfill & 14\hfill & 15\hfill & 15\hfill & 20\hfill & 25\hfill \\ 14\hfill & 45\hfill & 16\hfill & 20\hfill & 18\hfill & 19\hfill \\ 15\hfill & 16\hfill & 39\hfill & 10\hfill & 12\hfill & 15\hfill \\ 15\hfill & 20\hfill & 10\hfill & 40\hfill & 14\hfill & 11\hfill \\ 20\hfill & 18\hfill & 12\hfill & 14\hfill & 35\hfill & 17\hfill \\ 25\hfill & 19\hfill & 15\hfill & 11\hfill & 17\hfill & 39\hfill \end{array}\right]\right.\times {10}^{-7}$$

(36)

According to Reference [37], the power load of Test System 1 is 600 MW, and the thermal load (Hp) is 150 MWth.

Table 10. Test system 1 results statistics.

Optimizer	OBC (USD)	Average	Worst	maxFES	Time (s)
IAHA	10,093.75	10,093.76	10,093.78	1000	2.67
AHA	10,095.25	10,097.22	10,098.25	1000	2.23
GWO	10,117.52	10,123.10	10,126.94	1000	18.37
WOA	10,326.40	10,341.20	10,345.72	1000	18.41
HTS [18]	10,104.2707	10,104.4054	10,104.7031	100	2.4405
BLPSO [39]	10,101.3079	10,101.5626	10,102.1864	20,000	6.18
ISNS [40]	10,094.4196	NA	NA	NA	NA
Levy-GWO [24]	10,111.79	NA	NA	NA	NA

lists the optimal, average, and worst costs, as well as the running time, of various algorithms in four experiments: IAHA, AHA, GWO, and WOA. Additionally, Table 10 also compares the results from HTS [18], BLPSO [39], ISNS [40], and Levy-GWO [24] in the literature.

From Table 10 and

Table 11. Test system 1 unit output power of different algorithms.

Items	GWO	WOA	AHA	IAHA
P1 (MW)	42.59	75	45.50	45.48
P2 (MW)	98.55	125	98.53	98.54
P3 (MW)	112.47	139.65	112.69	112.67
P4 (MW)	209.77	125.30	209.85	209.82
P5 (MW)	97.25	95.67	94.01	94.07
P6 (MW)	40.01	40.00	40.03	40.00
H5 (MWth)	9.12	18.41	28.25	27.84
H6 (MWth)	74.96	74.99	74.69	74.99
H7 (MWth)	65.92	56.59	47.06	47.16
Best Cost (USD)	10,117.52	10,326.40	10,095.25	10,093.75
Average cost (USD)	10,123.10	10,341.20	10,097.22	10,093.76
Worst cost (USD)	10,126.94	10,345.72	10,098.25	10,093.78
Time (s)	18.37	18.41	2.23	2.67

, it can be seen that, compared to all algorithms in experiments and the literature, IAHA has achieved the best results in terms of optimal, average, and worst costs. The optimal cost obtained by IAHA is only USD 10,093.75, which is lower than the optimal values obtained by other algorithms in Table 10. Compared to the WOA in the experiment, IAHA achieved the best cost savings of up to USD 232.65. Compared to the Levy-GWO in the literature [24], IAHA achieves an optimal cost savings of up to USD 18.04. Considering the average cost, IAHA also obtained a smaller value (i.e., USD 10,093.76), indicating that IAHA also has good stability.

From Table 11, it can be seen that the power output of each unit obtained by IAHA is more stable, with few cases where the power output is at the power boundary. This is mainly due to AHA’s unique flying, foraging, and memory abilities, which can quickly and accurately find the optimal solution. Meanwhile, IAHA is slightly inferior to AHA in terms of algorithm runtime. This is because, compared to AHA, IAHA’s search space is more extensive, but at the same time, it enhances the population diversity of the algorithm and the ability to jump out of the local optimum.

The convergence processes of the four algorithms are shown in Figure 8. From Figure 8, it can be seen that the convergence speed and accuracy of the IAHA are higher than those of the other algorithms shown. The electrical and thermal energy values obtained by the IAHA algorithm are very close to the required power loads.

5.3. Test System 2: 24-Unit CHPED Problem

Test System 2 is a 24-unit CHPED problem, taken from Reference [41]. It is a classic medium-scale CHPED problem consisting of 13 power-only units, 6 CHP units, and 5 heat-only units. In Test System 2, valve point effects and operational regions were considered. The maximum number of iterations was set to MaxIt = 4000, and the population size NP = 150. Accordingly,

Table 12. Power-only units cost function parameters of Test System 2.

Power-Only Units	αi	βi	γi	eei	ffi	Pimin	Pimax
1	0.00028	8.10	550	300	0.035	0	680
2	0.00056	8.10	309	200	0.042	0	360
3	0.00056	8.10	309	200	0.042	0	360
4	0.00324	7.74	240	150	0.063	60	180
5	0.00324	7.74	240	150	0.063	60	180
6	0.00324	7.74	240	150	0.063	60	180
7	0.00324	7.74	240	150	0.063	60	180
8	0.00324	7.74	240	150	0.063	60	180
9	0.00324	7.74	240	150	0.063	60	180
10	0.00284	8.60	126	100	0.084	40	120
11	0.00284	8.60	126	100	0.084	40	120
12	0.00284	8.60	126	100	0.084	55	120
13	0.00284	8.60	126	100	0.084	55	120

,

Table 13. CHP units cost function parameters of Test System 2.

CHP Units	aj	bj	cj	dj	ej	fj	Feasible Region Coordinates
14	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
15	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
16	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
17	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
18	0.1035	34.5	2650	0.025	2.203	0.051	[20, 0], [10, 40], [45, 55], [60, 0]
19	0.0720	20.0	1565	0.020	2.340	0.040	[35, 0]. [35, 20], [90, 45], [90, 25], [105, 0]

and

Table 14. Heat-only units cost function parameters of Test System 2.

Heat-Only Units	φk	ηk	λk	Hmin	Hmax
20	0.038	2.0109	950	0	2695.20
21	0.038	2.0109	950	0	60
22	0.038	2.0109	950	0	60
23	0.052	3.0651	480	0	120
24	0.052	3.0651	480	0	120

list the relevant parameters of the operational cost functions for each unit. The data are sourced from [38].

According to Reference [41], the power load of Test System 2 is 2350 MW, and the thermal load (Hp) is 1250 MWth.

Table 15. Test system 2 results statistics.

Optimizer	OBC ($)	Average	Worst	maxFES	Time (s)
IAHA	57,876.5508	57,894.9375	57,915.0069	4000	26.4515
AHA	57,996.9548	57,998.8079	58,012.4878	4000	24.8515
GWO	59,521.2456	59,670.0777	59,781.7602	4000	565.0260
WOA	61,883.1815	62,242.9196	62,345.7564	4000	570.1440
Proposed CHPED Algorithm [6]	57,895.6050	57,896.7356	57,897.8129	1000	11.2710
HTS [18]	57,959.4100	57,959.9200	57,960.7300	200	6.6877
ISNS [40]	58,082.5371	58,302.4427	58,556.2406	NA	NA
SNS [40]	58,109.3402	58,362.8765	58,710.1045	NA	NA

lists the optimal, average, and worst costs, as well as the running time, of various algorithms in four experiments: IAHA, AHA, GWO, and WOA. In addition, Table 15 also compares the results from the proposed CHPED algorithm [6], HTS [18], ISNS [40], and SNS [40] in the literature.

From Table 15 and

Table 16. Test system 2 unit output power of different algorithms.

Items	GWO	WOA	AHA	IAHA
P1 (MW)	538.4775	451.0720	538.5587	628.3185
P2 (MW)	299.3808	184.2383	149.5996	299.2009
P3 (MW)	55.4767	305.4657	299.1799	149.6151
P4 (MW)	110.9189	136.9736	159.7323	109.8666
P5 (MW)	98.0100	130.1374	109.8661	109.8666
P6 (MW)	159.6166	112.6991	109.8663	159.7331
P7 (MW)	112.2540	98.0116	109.8666	109.8666
P8 (MW)	79.3887	138.1284	109.8663	109.8666
P9 (MW)	159.9315	112.0139	109.8665	109.8666
P10 (MW)	77.2408	52.7221	76.8571	40.0000
P11 (MW)	42.0301	42.7646	76.3756	77.3999
P12 (MW)	91.9634	101.0753	92.0504	92.3999
P13 (MW)	92.0569	79.1348	92.1309	55.0000
P14 (MW)	141.3902	83.8794	94.0647	86.0327
P15 (MW)	43.5152	83.3590	40.0000	40.0012
P16 (MW)	141.4363	83.0794	95.9973	87.9656
P17 (MW)	45.2182	68.2452	40.0108	40.0000
P18 (MW)	23.8396	35.6075	11.1110	10.0000
P19 (MW)	37.8548	51.3427	35.0000	35.0003
H14 (MWth)	137.9082	87.8725	112.1328	107.6252
H15 (MWth)	78.0325	104.2315	74.9981	74.9992
H16 (MWth)	138.7176	105.9643	113.2173	108.7099
H17 (MWth)	79.5025	98.7734	75.0074	74.9981
H18 (MWth)	45.9133	35.6416	40.4767	40.0006
H19 (MWth)	20.9020	16.8535	19.9984	19.9985
H20 (MWth)	389.8931	449.8231	454.1693	463.6685
H21 (MWth)	59.7296	50.8744	60.0000	60.0000
H22 (MWth)	59.8629	59.9934	60.0000	60.0000
H23 (MWth)	119.9360	120.0000	120.0000	120.0000
H24 (MWth)	119.6025	119.9723	120.0000	120.0000
Best cost ($)	59,521.2456	61,883.1815	57,996.9548	57,876.5508
Average cost ($)	59,670.0777	62,242.9196	57,998.8079	57,894.9375
Worst cost ($)	59,781.7602	62,345.7564	58,012.4878	57,915.0069
Time (s)	565.0260	570.1440	24.8515	26.4515

, it can be seen that, compared to all algorithms in the experiment and literature, IAHA has obtained the best results in terms of the optimal, average, and worst costs. The optimal cost obtained by IAHA is only USD 57,876.5508, which is lower than the optimal values obtained by other algorithms in Table 15. Compared to the WOA in the experiment, IAHA achieved the best cost savings of up to USD 4006.6307. Compared to the SNS in the literature [40], IAHA achieves the best cost savings of up to USD 232.7894. Considering the average cost, IAHA also obtained a smaller value (i.e., 57,894.9375), indicating that IAHA also has good stability.

From Table 16, it can be seen that the power output of each unit obtained by IAHA is more stable, with few cases where the power output is at the power boundary. In terms of algorithm run time, IAHA shows a significant advantage over other algorithms, primarily due to the unique memory capability of AHA. Each hummingbird retains a memory of positions with higher nectar replenishment rates during iterations, which significantly reduces the algorithm’s run time.

As shown in Figure 9, compared to the other three algorithms, both the convergence speed and accuracy of IAHA are improved.

Test Case 3 is a 48-unit CHPED problem taken from Reference [41]. It is a classic large-scale CHPED problem consisting of 26 power-only units, 12 CHP units, and 10 heat-only units. In Test System 3, valve point effects and operational regions were considered. The maximum number of iterations is set to MaxIt = 20,000, and the population size NP = 200. Accordingly,

Table 17. Power-only units cost function parameters of Test System 3.

Power-Only Units	αi	βi	γi	eei	ffi	Pimin	Pimax
1	0.00028	8.10	550	300	0.035	0	680
2	0.00056	8.10	309	200	0.042	0	360
3	0.00056	8.10	309	200	0.042	0	360
4	0.00324	7.74	240	150	0.063	60	180
5	0.00324	7.74	240	150	0.063	60	180
6	0.00324	7.74	240	150	0.063	60	180
7	0.00324	7.74	240	150	0.063	60	180
8	0.00324	7.74	240	150	0.063	60	180
9	0.00324	7.74	240	150	0.063	60	180
10	0.00284	8.60	126	100	0.084	40	120
11	0.00284	8.60	126	100	0.084	40	120
12	0.00284	8.60	126	100	0.084	55	120
13	0.00284	8.60	126	100	0.084	55	120
14	0.00028	8.10	550	300	0.035	0	680
15	0.00056	8.10	309	200	0.042	0	360
16	0.00056	8.10	309	200	0.042	0	360
17	0.00324	7.74	240	150	0.063	60	180
18	0.00324	7.74	240	150	0.063	60	180
19	0.00324	7.74	240	150	0.063	60	180
20	0.00324	7.74	240	150	0.063	60	180
21	0.00324	7.74	240	150	0.063	60	180
22	0.00324	7.74	240	150	0.063	60	180
23	0.00284	8.60	126	100	0.084	40	120
24	0.00284	8.60	126	100	0.084	40	120
25	0.00284	8.60	126	100	0.084	55	120
26	0.00284	8.60	126	100	0.084	55	120

,

Table 18. CHP units cost function parameters of Test System 3.

CHP Units	aj	bj	cj	dj	ej	fj	Feasible Region Coordinates
27	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
28	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
29	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
30	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
31	0.1035	34.5	2650	0.025	2.203	0.051	[20, 0], [10, 40], [45, 55], [60, 0]
32	0.0720	20.0	1565	0.020	2.340	0.040	[35, 0]. [35, 20], [90. 45], [90, 25], [105, 0]
33	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
34	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
35	0.0345	14.5	2650	0.030	4.2	0.031	[98.8, 0], [81, 104.8], [215, 180], [247, 0]
36	0.0435	36.0	1250	0.027	0.6	0.011	[44, 0], [44, 15.9], [40, 75], [110.2, 135.6], [125.8, 32.4], [125.8, 0]
37	0.1035	34.5	2650	0.025	2.203	0.051	[20, 0], [10, 40], [45, 55], [60, 0]
38	0.0720	20.0	1565	0.020	2.340	0.040	[35, 0]. [35, 20], [90, 45], [90, 25], [105, 0]

and

Table 19. Heat-only units cost function parameters of Test System 3.

Heat-Only Units	φk	ηk	λk	Hmin	Hmax
39	0.038	2.0109	950	0	2695.20
40	0.038	2.0109	950	0	60
41	0.038	2.0109	950	0	60
42	0.052	3.0651	480	0	120
43	0.052	3.0651	480	0	120
44	0.038	2.0109	950	0	2695.20
45	0.038	2.0109	950	0	60
46	0.038	2.0109	950	0	60
47	0.052	3.0651	480	0	120
48	0.052	3.0651	480	0	120

list the relevant parameters of the operational cost functions for each unit. The data are sourced from [41].

According to Reference [5], the power load of Test System 3 is 4700 MW, and the thermal load (Hp) is 2500 MWth.

Table 20. Test system 3 results statistics.

Optimizer	OBC ($)	Average	Worst	maxFES	Time (s)
IAHA	116,048.1539	116,111.1857	116,149.3838	20,000	272.8630
AHA	116,125.5048	116,181.8785	116,241.6110	20,000	233.1710
GWO	125,338.4898	125,443.0591	125,665.7251	20,000	8866.4285
WOA	130,045.9012	130,747.9537	131,933.1849	20,000	9142.3150
HTS [18]	116,918.90	116,924.75	116,935.38	300	6.9056
ISNS [40]	116,582.0833	117,059.3330	117,512.2505	NA	NA
SNS [40]	116,710.7282	117,150.5584	117,658.2387	NA	NA
HBJSA [42]	1161,40.34	NA	NA	900	NA

lists the optimal, average, and worst costs, as well as the running time, of various algorithms in four experiments: IAHA, AHA, GWO, and WOA. In addition, Table 20 also compares the results from HTS [18], ISNS [40], SNS [40], and HBJSA [42] in the literature.

From Table 20 and

Table 21. Test system 3 unit output power of different algorithms.

Items	GWO	WOA	AHA	IAHA
P1 (MW)	621.8124	537.9541	628.4013	628.3187
P2 (MW)	331.0854	235.9879	299.1948	299.1993
P3 (MW)	322.0548	240.8566	299.1995	299.1993
P4 (MW)	114.9221	81.7905	159.7334	159.7331
P5 (MW)	123.8439	169.6766	159.7148	159.7331
P6 (MW)	116.7211	143.8568	159.7354	159.7332
P7 (MW)	178.0947	99.9090	159.7024	109.8666
P8 (MW)	140.5066	113.1946	159.7163	109.8666
P9 (MW)	132.9864	113.8379	109.8675	109.8666
P10 (MW)	54.7267	79.4349	77.3979	77.3999
P11 (MW)	91.9046	96.2005	77.3641	77.3999
P12 (MW)	91.3963	79.1883	92.0451	92.3999
P13 (MW)	96.1018	62.2822	92.3741	120.0000
P14 (MW)	152.3511	346.2562	179.5187	179.5196
P15 (MW)	27.3960	229.3590	149.5985	159.6051
P16 (MW)	225.0296	185.5076	149.5965	299.1993
P17 (MW)	149.4663	126.4464	109.8680	109.8666
P18 (MW)	98.2292	85.2319	159.7187	109.8666
P19 (MW)	122.1164	123.8847	109.8712	109.8666
P20 (MW)	115.9117	96.0316	159.7310	109.8665
P21 (MW)	83.0334	87.2576	109.8668	109.8666
P22 (MW)	70.2352	71.6450	109.8670	159.7331
P23 (MW)	57.6080	90.1981	77.3999	77.3999
P24 (MW)	60.6206	63.5091	77.4017	77.4000
P25 (MW)	77.9824	84.0430	92.4006	92.3999
P26 (MW)	85.3007	71.0718	92.3995	92.3999
P27 (MW)	119.4496	136.6224	117.6223	103.1602
P28 (MW)	41.6005	82.3386	40.0007	40.0001
P29 (MW)	152.1515	83.1478	85.3225	87.0991
P30 (MW)	68.5788	46.8083	40.0026	40.0000
P31 (MW)	27.4758	37.3482	10.0159	10.0000
P32 (MW)	49.1511	78.1141	35.0136	35.0000
P33 (MW)	136.6217	162.7575	101.7523	87.8648
P34 (MW)	50.5656	50.4175	40.0009	40.0000
P35 (MW)	178.9121	172.8019	93.5791	92.1699
P36 (MW)	44.4178	59.0721	40.0035	40.0000
P37 (MW)	43.4296	17.4293	10.0010	10.0000
P38 (MW)	36.2045	57.5305	35.0008	35.0000
H27 (MWth)	126.3787	105.1637	125.3533	117.2372
H28 (MWth)	76.3796	110.8909	74.9982	74.9982
H29 (MWth)	143.4418	92.1800	107.2264	108.2237
H30 (MWth)	58.6803	80.8750	74.9999	74.9981
H31 (MWth)	43.1512	50.1953	40.0074	40.0006
H32 (MWth)	16.7663	17.1735	20.0042	19.9984
H33 (MWth)	134.5339	125.9003	116.4470	108.6534
H34 (MWth)	84.1183	65.7802	74.9982	74.9981
H35 (MWth)	158.9286	139.0957	111.8603	111.0694
H36 (MWth)	78.8115	80.3278	75.0004	74.9981
H37 (MWth)	40.4266	10.2830	40.0000	40.0006
H38 (MWth)	20.5459	16.9296	19.9987	19.9984
H39 (MWth)	422.2681	406.9643	444.2832	457.2425
H40 (MWth)	45.8771	48.1419	59.9983	60.0000
H41 (MWth)	40.0031	43.5554	59.9986	60.0000
H42 (MWth)	101.5608	80.9111	119.9982	120.0000
H43 (MWth)	115.0466	69.9596	119.9999	120.0000
H44 (MWth)	433.0814	595.6730	454.8273	457.5834
H45 (MWth)	60.0000	60.0000	60.0000	60.0000
H46 (MWth)	60.0000	60.0000	60.0000	60.0000
H47 (MWth)	120.0000	120.0000	120.0000	120.0000
H48 (MWth)	120.0000	120.0000	120.0000	120.0000
Best cost ($)	125,338.4898	130,045.9012	116,125.5048	116,048.1539
Average cost ($)	125,443.0591	130,747.9537	116,181.8785	116,111.1857
Worst cost ($)	125,665.7251	131,933.1849	116,241.6110	116,149.3838
Time (s)	9142.3150	8866.4285	233.1710	272.8630

, it can be seen that, compared to all algorithms in the experiments and literature, IAHA has achieved the best results in terms of the optimal, average, and worst costs. The optimal cost obtained by IAHA is only USD 116,048.1539, which is lower than the optimal values obtained by other algorithms in Table 20. Compared to the WOA in the experiment, IAHA achieved the best cost savings of up to USD 13,997.7473. Compared to the SNS in the literature [40], IAHA achieves the best cost savings of up to USD 870.7461. Considering the average cost, IAHA also obtained a smaller value (i.e., 116,111.1857), indicating that IAHA also has good stability.

From Table 21, it can be seen that the power output of each unit obtained by IAHA is more stable, with few cases where the power output is at the power boundary. Additionally, due to AHA’s unique access table mechanism and memory capability, both IAHA and AHA have significant advantages in processing large-scale CHPED problems in terms of run time.

As shown in Figure 10, both the convergence speed and accuracy of IAHA are superior to the other three algorithms.

6. Summary

This paper introduces an Improved Artificial Hummingbird Algorithm (abbreviated as IAHA) designed to address complex CHPED problems more effectively. The algorithm employs chaotic mapping for population initialization, enhances the iterative mechanism of the original IAHA, boosts the algorithm’s traversal capability and population diversity, overcomes its shortcomings, and mitigates the risk of premature convergence. We evaluated the performance of IAHA using benchmark test functions and compared it with several other commonly used algorithms. The results demonstrate that IAHA can discover high-quality solutions within a relatively short period. Furthermore, simulation experiments were conducted on three CHPED using IAHA, with the test units including 7 units, 24 units, and 48 units, respectively. The results indicate that IAHA is capable of generating optimal solutions, confirming its effectiveness in solving CHPED problems.

However, there are also some limitations in this study. Firstly, due to the enhanced traversal capability of IAHA based on the original AHA, the exploration space for hummingbirds expands, resulting in a slight decrease in computational speed. Additionally, more factors have not been taken into account when solving the CHPED problem. In the future, we plan to incorporate factors such as dynamic economic dispatch and environmental pollution into the CHPED framework, which will help in addressing real-world challenges.