An Enhanced Horned Lizard Optimization Algorithm for Flood Control Operation of Cascade Reservoirs

Abstract

The multi-reservoir flood control operation (MRFCO) problem is characterized by high dimensions and multiple constraints. These features pose significant challenges to algorithms aiming to solve the MRFCO problem, requiring them not only to handle high-dimensional variables effectively but also to manage constraints efficiently. The Horned Lizard Optimization Algorithm (HLOA) performs excellently in handling high-dimensional problems and effectively integrates with penalty functions to manage constraints. However, it still exhibits poor convergence when dealing with certain benchmark functions. Therefore, this paper proposes the Enhanced Horned Lizard Optimization Algorithm (EHLOA), which incorporates Circle initialization and two strategies for avoiding local optima, thereby enhancing HLOA’s convergence performance. Firstly, EHLOA was tested on benchmark functions, where it demonstrated strong robustness and scalability. Then, EHLOA was applied to the MRFCO problem at the upper section of Lanzhou of the Yellow River in China, showing excellent convergence capabilities and the ability to escape local optima. The reduction rates of flood peaks achieved by EHLOA for the two millennial floods and two decamillennial floods were 55.6%, 52.8%, 58.1%, and 56.4%, respectively. Additionally, the generated operation schemes showed that the reservoir volumes changes were reasonable, and the discharge processes were stable under EHLOA’s operation. Overall, EHLOA can be considered a reliable algorithm for addressing the MRFCO problem.

1. Introduction

As global warming intensifies, the enhancement of the hydrological cycle has led to an increased frequency of extreme precipitation events [1]. The amount of rainfall and the areas affected by these extreme precipitation events have become larger than before [2]. Moreover, some studies suggest that by the end of this century, the frequency of extreme precipitation events will increase globally by approximately 1.8 times, with a significant rise in the frequency of extreme precipitation also expected between 2041 and 2070 [3,4]. With the rising frequency and intensity of extreme precipitation, the frequency of flood events has also increased accordingly [1]. Over the past three years, several countries have experienced devastating floods. From December 2021 to early 2022, Malaysia faced one of the most severe floods in its history, with approximately 70,000 victims evacuated daily, 54 fatalities, and total losses amounting to approximately $1.46 billion [5]. In July 2021, catastrophic floods struck western Germany and Henan Province, China. In Germany, over 107,000 people were affected, 205 lives were lost, and economic losses reached $40 billion [6]. In Henan Province, the floods impacted 14.786 million people, resulting in 398 deaths and losses of $18.9 billion [6]. In 2022, unprecedented massive floods hit southern Pakistan, displacing 33 million people [7]. In summary, climate warming has led to an increase in the frequency of extreme precipitation, which directly results in a higher probability of extreme flood events. Moreover, as the frequency of extreme precipitation is expected to rise significantly in the coming decades, the losses caused by extreme flood events to humanity may become increasingly severe.

Reservoirs are crucial tools for flood control. By utilizing their flood storage capacity to store more floodwaters while ensuring their own safety, reservoirs can effectively reduce peak flood flows and mitigate downstream flood damages. When a reservoir operates independently, its flood detention capacity can be limited when dealing with large flood volumes, increasing the risk of downstream flooding [8]. Therefore, with the development of socio-economic conditions, there is a tendency to use multiple reservoirs for flood control, and a flood control system often includes several reservoirs [9]. Multi-reservoir systems are categorized into cascade reservoirs and parallel reservoirs. Both types can reduce maximum discharge, enhance flood control capacity, and lower flood risks in downstream areas through coordinated operating of multiple reservoirs [10,11]. However, this approach introduces several challenging issues during the operation process, such as how to rationally utilize the flood storage capacities of the various reservoirs and how to ensure the safety of each reservoir. Consequently, multi-reservoir flood control operation (MRFCO) is a high-dimensional, multi-constraint problem.

There are two main approaches to solving the problem of multi-reservoir flood control operation. The first approach involves the use of metaheuristic optimization algorithms [12,13,14], while the second approach uses mathematical algorithms that divide the flood control operation process into several stages for optimization [15,16]. The latter can be time-consuming and difficult to converge when dealing with long-duration flood events. In contrast, metaheuristic algorithms perform exceptionally well in high-dimensional optimization problems and are widely applied in commercial, engineering, and economic fields [17].

Solving the MRFCO problem using metaheuristic algorithms requires the establishment of an appropriate mathematical model, which typically comprises three components: the objective function, the variable bounds and constraints, and the handling of constraint violations. There are two primary methods for addressing constraint violations: employing metaheuristic algorithms with built-in constraint handling techniques [18] and adding a penalty function to the objective function [19]. The former method, though proven to work, tends to become trapped in local optima and struggles to converge, especially in high-dimensional optimization problems where the feasible region is small [20]. Additionally, it demonstrates inefficiency in high-dimensional contexts, making it less suitable for complex optimization tasks [21,22]. The latter method, the penalty function approach, is generally more efficient [23]. It assigns high penalty values to infeasible solutions, thereby forcing the algorithm to search within the feasible solution space. However, the performance of different metaheuristic algorithms when combined with penalty functions can vary [24]. Penalty functions can also cause the algorithm to get stuck in local optima in high-dimensional problems [25]. The ability to escape these local optima depends on the algorithm’s inherent capacity to overcome local optima challenges. Therefore, to effectively solve the MRFCO problem, it is crucial to find a metaheuristic algorithm that not only integrates well with the penalty function approach but also possesses a strong ability to escape local optima.

The Horned Lizard Optimization Algorithm (HLOA), proposed by Vázquez et al. [26] achieves global optimization by mathematically simulating various defensive behaviors of horned lizards. The performance of HLOA was tested on the optimization problems from IEEE CEC 2017 “Constrained Real-Parameter Optimization” [27], with variable dimensions of 10, 30, 50, and 100. The results indicated that HLOA performed best at dimensions of 50 and 100 and ranked second at dimensions of 30 and 10. It suggests that HLOA excels in handling high-dimensional constrained problems even more than low-dimensional ones. Additionally, HLOA was tested on five real-world constrained optimization problems using the penalty function method to handle constraints. The results demonstrated that HLOA delivered competitive results across all five problems and provided the best solutions for three of them. It indicates that the performance of HLOA combined with the penalty function approach is reliable.

However, HLOA still has areas for improvement. In their study on 63 commonly used unconstrained test functions, Vázquez et al. [26] found that HLOA performed poorly on some functions, becoming trapped in local optima, demonstrating that its global search capability and ability to escape local optima need enhancement.

Therefore, this paper combines three strategies to enhance HLOA’s performance, aiming to improve the algorithm’s ability to escape local optima, ensuring that it demonstrates strong robustness and convergence when handling both benchmark problems and the MRFCO problem. The novelties of this work are as follows:

• EHLOA demonstrates better robustness compared to widely used algorithms and highly cited algorithms from the past three years on both benchmark functions and the MRFCO problem.
• EHLOA exhibits superior convergence compared to widely used algorithms and highly cited algorithms from the past three years on both benchmark functions and the MRFCO problem.
• The flood control scheduling process provided by EHLOA is reasonable, making it a reliable algorithm for flood control operations.

The remainder of this paper is organized as follows: Section 2 introduces an overview of the study area and constructs the flood control operation model. Section 3 presents the basic framework of HLOA and explains the improvement strategies used in EHLOA. Section 4 tests the performance of EHLOA and its reliability in flood control operation through simulations on benchmark functions and the flood control operation model. Section 5 discusses the main findings of the study, and Section 6 presents the main conclusions of the paper. The flowchart of this study is presented in Figure 1.

2. Study Area and Flood Control Operation Model

2.1. Cascade Reservoirs and Flood Data

This section will introduce the study area of this paper, including the geographic location and composition of the cascade reservoirs, as well as the flood data used in this paper.

The Yellow River, the second longest river in China, traverses nine provinces and plays a crucial role in the agriculture, industry, and ecology of northern China. Floods in the upper reaches of the Yellow River generally take place between July and September. While the fluctuations of these floods are relatively mild, they tend to last for an extended period, approximately 40–45 days.

The cascaded reservoirs formed by Longyangxia (LYX) and Liujiaxia (LJX) bear the primary flood control tasks in the upper Yellow River due to their excellent flood mitigation capacity, playing a significant role in protecting the safety of Lanzhou, the second largest city in northwest China. This study is centered at the upper section of Lanzhou of Yellow River in China, where it conducts an in-depth analysis of flood control operation for the LYX and LJX cascade reservoirs across four different design floods.

The design floods used in this paper are based on typical floods from 1964 and 1967, provided by the Yellow River Conservancy Commission of China (http://www.yrcc.gov.cn, accessed on 20 May 2024). The 1964 flood is characterized by a sharp and narrow peak shape, while the 1967 flood features a flat-topped peak shape. Their characteristics include long durations (45 days) and large flood volumes. This study includes four different design floods: flood 1 (a millennial flood based on the typical 1964 flood), flood 2 (a millennial flood based on the typical 1967 flood), flood 3 (a decamillennial flood based on the typical 1964 flood), and flood 4 (a decamillennial flood based on the typical 1967 flood). Figure 2 illustrates the flood evolution process in the upper section of the Lanzhou section of the Yellow River.

Table 1. Parameter values used for flood control operation.

Reservoir	VL (Billion m3)	VD (Billion m3)	VC (Billion m3)	QOM,0.1% (m3/s)	QOM,0.01% (m3/s)
LYX	22.46	25.56	27.42	4000	6000
LJX	2.92	4.07	4.395	4510	7290

Note: V_L denotes the reservoir capacity at the dead water level; V_D denotes the reservoir capacity at the design flood level; V_C denotes the reservoir capacity at the check flood level; and QO_M,P (p = 0.1% and 0.01%) denotes the maximum allowable discharge flow of a reservoir corresponding to a millennial flood and a decamillennial flood.

details the characteristic parameters used in this study, including the reservoir capacity at the dead water level (V_L), the reservoir capacity at the design flood level (V_D) and the reservoir capacity at the check flood level (V_C), and the maximum allowable discharge flow (QO_M).

2.2. Flood Control Operation Model of Cascade Reservoirs

2.2.1. Objective Function of Flood Control Operation

This section will present the objective function for flood control operation and explain the rationale for its use. As shown in Figure 2, the primary flood protection target for the cascade reservoirs in the upstream of the Yellow River is Lanzhou. The last large reservoir upstream of Lanzhou is LJX, and its maximum discharge directly impacts the safety of Lanzhou. If the maximum discharge from LJX is relatively low, the operation is considered successful during the process of a large flood. Therefore, the decision variables (variables to be optimized) in this model are the outflow of LYX during the operation process (QO₁) and the outflow of LJX during the operation process (QO₂). The known variables input into the model include the inflow rate of LYX (QI₁) and the interval flows between the reservoirs during flood propagation (q₁, q₂, q₃). The decision variables and known variables will be input into the optimization algorithm along with the objective function and constraints. The optimization algorithm will output the minimum value of the objective function and the outflow processes of LYX and LJX during the flood operation period. The operation period (T) for this model is 45 days, and there are two decision variables (QO₁, QO₂), making the problem dimension for this model 90 (45 × 2). Simultaneously, the reservoirs must ensure their own safety, and the maximum storage capacity during the operation process should not exceed the flood control standards for the corresponding flood frequency. The objective function and the corresponding penalty function are as follows:

$$fun=\max \left({QO}_2\left(t\right)\right)+k\times penalty$$

(1)

$$\begin{array}{l}penalty={\displaystyle \sum _{t=1}^{T}f\left(t\right)+{\displaystyle \sum _{i=1}^{T}g\left(t\right)}}\\ \left\{\begin{array}{l}f\left(t\right)=0 if V_1\left(t\right)<V_{\max ,1}, \\ f\left(t\right)={{\displaystyle \left(V_1\left(t\right)-V_{\max ,1}\right)}}^2 else\end{array}\right.\\ \left\{\begin{array}{l}g\left(t\right)=0 if V_2\left(t\right)<V_{\max ,2}, \\ g\left(t\right)={{\displaystyle \left(V_2\left(t\right)-V_{\max ,2}\right)}}^2 else\end{array}\right.\end{array}$$

(2)

where QO₂(t) represents the discharge of LJX at the end of t period; V₁ and V₂ represent the reservoir volumes of LYX and LJX, respectively; and V_max,1 and V_max,2 represent the maximum storage capacities that LYX and LJX can reach during the flood control process, respectively.

2.2.2. Constraints of Flood Control Operation

(1)

Water balance constraint

$$V_i\left(t\right)=V_i\left(t-1\right)+\left[Q{I}_i\left(t\right)-{QO}_i\left(t\right)\right]\mathrm{\Delta }t$$

(3)

where V_i(t) and V_i(t − 1) are the reservoir volumes of reservoir i at the end of t period and t − 1 period; and QI_i(t) and QO_i(t) are the inflow and outflow of reservoir i at the end of period t.

(2)

Outflow constraint

To ensure the safety of downstream protection objects and the stability of the downstream river channel of a reservoir, two constraints are set as follows:

$${QO}_i\left(t\right)\le {QO}_{i,\max }$$

(4)

$$\left|{QO}_i\left(t\right)-{QO}_i\left(t-1\right)\right|\le \mathrm{\Delta }{QO}_{i,\max }$$

(5)

where QO_i,max is the maximum allowable outflow of reservoir i; and ΔQO_i,max is the maximum allowable fluctuation constraint for reservoir i.

(3)

Interval flow constraint

$$QO\left(i-1,t-\mathrm{\Delta }t\right)+q_{i-1,i}\left(\mathrm{\Delta }t\right)=QI\left(i,t\right)$$

(6)

where ∆t is equal to one day; and q_i−1,i(∆t) represents the inter flow between reservoir i − 1 and reservoir i.

(4)

Reservoir storage capacity constraint

$$V_i\left(t\right)\le V_{i,\max }$$

(7)

where V_i,max represents the maximum storage capacity that reservoir i can achieve during the operation process.

(5)

Initial condition constraint

$$V_i\left(1\right)=V_i\left(0\right)$$

(8)

where V_i(0) represents the reservoir volume corresponding to the flood control level of reservoir i.

(6)

Non-negative constraint

All variables in the flood control operation model of cascade reservoirs are non-negative.

3. HLOA Algorithm

3.1. Basic HLOA Algorithm

This section will introduce the basic HLOA algorithm, including its main framework and how it is inspired by the behavior of horned lizards. There is a type of horned lizard that inhabits the region from south central United States to northeastern Mexico. These lizards are adapted to extreme temperatures in arid or semi-arid areas. Their primary passive defense method is camouflage, which includes the ability to blend into their surroundings through changes in color, pattern, and shape. Another passive defense strategy is relocation or fleeing. Additionally, when threatened, these lizards can adopt aggressive strategies such as squirting blood to deter predators. The horned lizard’s skin can lighten or darken depending on whether it needs to reduce or increase solar gain. In high temperatures, they adopt a lighter color, while in low temperatures, their skin darkens. Dark skin does not reflect any color; instead, it absorbs all wavelengths of light and converts them into heat. The rapid change in the horned lizard’s skin color is due to the effect of temperature on its alpha-melanocyte-stimulating hormone (α-MSH). Inspired by these horned lizards, Vázquez et al. [26] proposed the HLOA algorithm, which incorporates a mathematical model based on the behaviors.

3.1.1. Crypsis Behavior

The horned lizard’s skin color often changes with its external environment, this behavior can help it better hide from predators. To simulate the horned lizard’s constantly changing colors, the variability of sine and cosine functions is utilized. This variability increases the algorithm’s randomness, enhancing its ability to perform global searches. The mathematical formula for this is as follows [26]:

$$\overrightarrow{x_i}(t+1)=\overrightarrow{x_{best}}(t)+\left(2-{\displaystyle \frac{2\cdot t}{\mathrm{Max}\_\mathrm{iter}}}\right)\left[c_1\left(\sin (\overrightarrow{x_{r_1}}(t))-\cos (\overrightarrow{x_{r_2}}(t))\right)-{(-1)}^{\sigma }{c}_2\left(\cos (\overrightarrow{x_{r3}}(t))-\sin (\overrightarrow{x_{r4}}(t))\right)\right]$$

(9)

where $\overrightarrow{x_i}(t+1)$ is the new search agent position in the solution search space for the generation t + 1; $\overrightarrow{x_{best}}(t)$ is the best search agent for the generation t; Max_iter represent the maximum number of iterations; and r₁, r₂, r₃ and r₄ are random numbers generated within the range of 1 to the population size. $\overrightarrow{x_{r_1}}(t)$, $\overrightarrow{x_{r_2}}(t)$, $\overrightarrow{x_{r3}}(t)$ and $\overrightarrow{x_{r4}}(t)$ are the r₁, r₂, r₃, r₄-th search agent selected; σ is a binary value, which can be either 0 or 1; and, c₁, c₂, with c₁ ≠ c₂, are random numbers.

3.1.2. Skin Darkening or Lightening

Horned lizards can lighten or darken their skin depending on whether they need to reduce or increase their solar heat gain. The color changes in horned lizard skin are described by Equations (10) and (11), where Equation (10) represents the skin-lightening strategy, and Equation (11) represents the skin-darkening strategy [26].

$${\overrightarrow{x}}_{worst}(t)={\overrightarrow{x}}_{best}(t)+{\displaystyle \frac{1}{2}}Ligh{t}_1\sin \left({\overrightarrow{x}}_{r_1}(t)-{\overrightarrow{x}}_{r_2}(t)\right)-{(-1)}^{\sigma }{\displaystyle \frac{1}{2}}Ligh{t}_2\sin \left({\overrightarrow{x}}_{r_3}(t)-{\overrightarrow{x}}_{r_4}(t)\right)$$

(10)

$${\overrightarrow{x}}_{worst}(t)={\overrightarrow{x}}_{best}(t)+{\displaystyle \frac{1}{2}}Dar{k}_1\sin \left({\overrightarrow{x}}_{r_1}(t)-{\overrightarrow{x}}_{r_2}(t)\right)-{(-1)}^{\sigma }{\displaystyle \frac{1}{2}}Dar{k}_2\sin \left({\overrightarrow{x}}_{r_3}(t)-{\overrightarrow{x}}_{r_4}(t)\right)$$

(11)

where ${\overrightarrow{x}}_{worst}(t)$ is the worst search agent for the generation t; and Light₁, Light₂, Dark₁, Dark₂ are random numbers.

3.1.3. Blood-Squirting

Horned lizards defend themselves by squirting blood from their eyes to deter predators. This defensive behavior is defined as a projectile motion. By utilizing projectile motion, the current solution is combined with the best solution, allowing the algorithm to move closer to the optimal solution while maintaining population diversity. The mathematical formula as follows [26]:

$${\overrightarrow{x}}_i(t+1)=\left[\cos \left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{t}{\mathrm{Max}\_\mathrm{iter}}}\right)+{\epsilon }_1\right]{\overrightarrow{x}}_{best}(t)+\left[\sin \left({\displaystyle \frac{\pi }{2}}-{\displaystyle \frac{\pi }{2}}{\displaystyle \frac{t}{\mathrm{Max}\_\mathrm{iter}}}\right)-\mathrm{g}+{\epsilon }_1\right]\overrightarrow{x_i}(t)$$

(12)

where ε₁ is set to 1 × 10⁻⁶ g is set to 0.009807 to represent the gravity of the earth; and $\overrightarrow{x_i}(t)$ is the current search agent.

3.1.4. Move-to-Escape

Horned lizards can move quickly and randomly in nature to evade predators. A function that includes both local and global movements is introduced to simulate this behavior. This combination of local and global movements reduces the likelihood of the algorithm becoming trapped in local optima [26]:

$${\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_{best}(t)+walk({\displaystyle \frac{1}{2}}-{\epsilon }_2){\overrightarrow{x}}_i(t)$$

(13)

where walk is a random number generated between −1 and 1; ε₂ is a random number generated from a standard Cauchy distribution with the mean and standard deviation set to 0 and 1.

3.1.5. Alpha Melanophore Stimulating Hormone (α-MSH) Rate

The color change in horned lizard skin can be attributed to the effect of temperature on melanophore. Inspired by this behavior, the rate of melanophore is introduced to assess the quality of the current solution, determining whether the solution enhancement procedure needs to be executed. The rate of melanophore activity in horned lizards is defined as follows [26]:

$$melanophore(i)={\displaystyle \frac{Fitnes{s}_{\max }-Fitness(i)}{Fitnes{s}_{\max }-Fitnes{s}_{\min }}}$$

(14)

where Fitness_min and Fitness_max are the best and the worst fitness values in the current t generation, respectively; and Fitness(i) is the current fitness value of the i-th search agent.

The melanophore activity rate is a value within the interval [0, 1]. If its value is less than 0.3, the search agent will be replaced.

$${\overrightarrow{x}}_i(t)={\overrightarrow{x}}_{best}(t)+{\displaystyle \frac{1}{2}}[{\overrightarrow{x}}_{r_1}(t)-{(-1)}^{\sigma }{\overrightarrow{x}}_{r_2}(t)]$$

(15)

The flowchart of HLOA is shown in Figure 3.

3.2. Enhanced HLOA Algorithm

To enhance HLOA’s convergence capability, chaotic initialization and two solution enhancement strategies were adopted. Chaotic initialization can generate higher-quality initial solutions, helping the algorithm converge to smaller values within a limited number of iterations. The two solution enhancement strategies can increase the algorithm’s ability to escape local optima.

3.2.1. Circle Map Initialization

Circle map initialization is a type of chaotic initialization that effectively helps the algorithm generate higher-quality solutions during the initialization phase. Due to the ergodicity and randomness of chaotic search, it can easily escape local optima. Therefore, chaotic initialization is often incorporated into metaheuristic optimization algorithms [28]. Chaos can enhance the algorithm’s performance on benchmark functions [29,30] and improve its efficiency in solving engineering problems [31]. It also makes the algorithm perform exceptionally well in handling high-dimensional optimization problems [32]. The Circle map has been proven to be one of the better chaotic mapping methods [33]. Therefore, in EHLOA, the Circle map is used for initialization instead of random initialization [33].

$$\begin{array}{l}{x}_{i+1}=x_i+\mathrm{b}-{\displaystyle \frac{\mathrm{a}}{2\pi }}\sin \left(2\pi x_i\right)\mathrm{mod}\left(1\right)\\ {{\displaystyle x}}_n^j=lb+x_{i+1}\times \left(ub-lb\right), n=1,2,\dots ,Np \mathrm{and} j=1,2,\dots , \mathit{dim}\end{array}$$

(16)

where a and b are set to 0.5 and 0.2, respectively; x_i+1 is a number obtained through the circle map; x_i is a number obtained through random initialization. The mod (1) ensures that the value of x_i+1 stays within the range of 0 to 1; ${{\displaystyle x}}_n^j$ represents the initial position of the n th search agent’s j th search variable; lb and ub represent the lower and upper bounds of the problem to be solved, respectively; N_P represents the population size of the algorithm; and dim represents the number of decision variables.

3.2.2. Trap Escaping Operator (TEO)

Trap Escaping Operator (TEO) is a strategy designed to help the algorithm avoid becoming trapped in the local optima. In HLOA, Strategies 4 and 5 can be considered as enhancing the quality of the population to increase the likelihood of escaping local optima. However, regarding Strategy 5 in HLOA and its Equation (15), although it includes random numbers, it may not be sufficient to allow the population to escape local optima. Moreover, it completely ignores the role of the current solution, which may reduce the diversity of the population in the early stages of iteration. Therefore, TEO is used to replace Equation (15). The essence of TEO is to generate enhanced quality solutions by combining the best position, the current velocity position, and the average position of each dimension. The parameters μ₁ and μ₂ in TEO are highly random, making it easier for the population to escape local optima and contributing to increased diversity within the population. The parameter δ is designed to maintain a balance between exploration and exploitation within the population. The TEO formula is inspired by the TAO strategy in the Newton–Raphson-based optimizer [34]. The mathematical expression of the TEO is as follows:

$$\begin{array}{l}\delta ={\left(1-\left({\displaystyle \frac{2\times t}{\mathrm{Max}\_\mathrm{iter}}}\right)\right)}^5\\ {\mu }_1=\beta \times 3\times rand+(1-\beta )\\ {\mu }_2=\beta \times rand+(1-\beta )\\ {\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_{best}(t)+\left(2\cdot rand-1\right)\cdot \left({\mu }_1\cdot {\overrightarrow{x}}_{best}(t)-\mu 2\cdot {\overrightarrow{x}}_i(t)\right)+\left(rand-0.5\right)\cdot \delta \cdot \left({\mu }_1\cdot {\overrightarrow{x}}_{avg}(t)-{\mu }_2\cdot {\overrightarrow{x}}_i(t)\right)\end{array}$$

(17)

where t represents the current iteration number; μ₁ and μ₂ are random numbers based on β; β denotes a binary number, either 1 or 0; rand denotes the random number; and ${\overrightarrow{x}}_{avg}(t)$ is a vector obtained by combining the average positions of all search agents in each dimension.

3.2.3. Sudden Attack Model

Horned lizards possess the ability to crypsis, which not only confuses predators but also serves as an excellent tool for hunting prey. Their color closely matches their surroundings, making them difficult to detect. When they spot their prey, they often lie in wait, seizing the moment when the prey is unaware, and then use their tongues to capture and devour the prey. Inspired by this unique hunting behavior, a sudden attack mathematical model was established to mimic this behavior. The sudden attack can enable the algorithm to systematically change the search step length and direction, helping to increase population diversity and escape local optima. The mathematical model for sudden attack is as follows:

$$\begin{array}{l}While iter=25,50,75 \dots \\ {\overrightarrow{x}}_i(t+1)={\overrightarrow{x}}_i(t)\left(1+\gamma \left(0.5-rand\right)\right)\end{array}$$

(18)

where γ is set to 6.

The pseudocode of EHLOA is presented in

Table 2. Pseudo code of the EHLOA algorithm.

Pseudo code of enhanced Horned Lizard Optimization Algorithm

1: Initialize random variables such as p. Specify the number of search agents, Population Size, and Maximum number of iterations.

2: Generate the initial population by Equation (16)

3: While (t < max number of iterations)

4:   for each search agent

5:      if t/25 = 1,2…, n? then

6:       compute sudden attack, update the positions by Equation (18)

7:      else

8:        if p > 0.5? then

9:         compute Crypsis in HLOA, update the positions by Equation (9)

10:       else

11:          if t is an even number? then

12:           compute Blood-squirting in HLOA, update the positions by Equation (12)

13:          else

14:           compute Move-to-escape in HLOA, update the positions by Equation (13)

15:          end if

16:       end if

17:     end if

18:     Replace the worst search agent by randomly select Equation (10) or Equation (11)

19:     if Alpha melanophore rate lower than 0.3? then

20:        compute trap escaping operator in EHLOA, update the positions by Equation (17)

21:     end if

22:     Calculate xnew, the fitness value of the new search agents

23:     t = t + 1

24:     end for

25:     end while

26:     find the lowest fitness value, Display it and its position xbest

.

4. Experiments and Results

The experimental simulations in this section were conducted on a laptop equipped with the macOS Sonoma operating system, featuring 8 GB of RAM and an Apple M1 chip. The simulations were carried out using MATLAB R2023a. To evaluate the performance of EHLOA, it will be compared with HLOA and three classical algorithms widely used in engineering problems: Grey Wolf Optimizer (GWO) [35], Whale optimization algorithm (WOA) [36], Particle swarm optimization (PSO) [37], as well as three highly cited algorithms proposed in the last three years: Honey badger algorithm (HBA) [38], Golden jackal optimization algorithm (GJO) [39], and Dwarf mongoose optimization algorithm (DMO) [40].

4.1. Experiments 1: In Benchmark Functions

In this section, the tests will be conducted using the F1–F13 functions from the CEC2005 benchmark set [41], where F1–F7 are unimodal test functions and F8–F13 are multimodal test functions. To assess the robustness of the algorithms, these test functions will have dimensions of 30, 100. All algorithms will independently run 30 times on these test functions. The maximum number of iterations for each algorithm will be set to 1000, and the population size will be set to 30. The detailed expressions and optimal values of the test functions are provided in

Table A1. Details of the benchmark function.

Fun	Dimension	Range	Theoretical Optimization Value
$f_1\left(x\right)={\displaystyle \sum _{i=1}^n{x}_i^2}$	30/100	[−100, 100]	0
$f_2\left(x\right)={\displaystyle \sum _{i=1}^n\left|x_i\right|+{\displaystyle \prod _{i=1}^n\left|x_i\right|}}$	30/100	[−10, 10]	0
$f_3\left(x\right)={\displaystyle \sum _{i=1}^n{\left({\displaystyle \sum _{i=1}^i{x}_i^2}\right)}^2}$	30/100	[−100, 100]	0
$f_4\left(x\right)=\max \left\{\left|x_i\right|,1\le i\le n\right\}$	30/100	[−100, 100]	0
$f_5\left(x\right)={\displaystyle \sum _{i=1}^n\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]}$	30/100	[−30, 30]	0
$f_6\left(x\right)={\displaystyle \sum _{i=1}^n{\left(⌊x_i+0.5⌋\right)}^2}$	30/100	[−100, 100]	0
$f_7\left(x\right)={\displaystyle \sum _{i=1}^{n}i{x}_i^4}+random[0,1)$	30/100	[−1.28, 1.28]	0
$f_8\left(x\right)={\displaystyle \sum _{i=1}^n-x_i}\sin \left(\sqrt{\left|x_i\right|}\right)$	30/100	[−500, 500]	418.9829 × Dimension
$f_9\left(x\right)={\displaystyle \sum _{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]}$	30/100	[−5.12, 5.12]	0
$f_{10}\left(x\right)=-20\exp \left(-0.2\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i^2}}\right)-\exp \left(-\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\cos \left(2\pi x_i\right)}}\right)+20+e$	30/100	[−32, 32]	0
$f_{11}\left(x\right)=1+{\displaystyle \frac{1}{400}}{\displaystyle \sum _{i=1}^n{x_i}^2-{\displaystyle \prod _{i=1}^n\cos \left(\frac{x_i}{\sqrt{i+1}}\right)}}$	30/100	[−600, 600]	0
$\begin{array}{l}{f}_{12}(x)={\displaystyle \frac{\pi }{n}}\left\{10{\sin }^2\left(\pi y_i\right)+{\displaystyle \sum _{i=1}^{n-1}}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]\right.\\ \left.+{\left(y_n-1\right)}^2\right\}+{\displaystyle \sum _{i=1}^n}u\left(x_i,10,100,4\right)\\ y_i=1+{\displaystyle \frac{1}{4}}\left(x_i+1\right)\\ u\left(x_i,a,k,m\right)=\left\{\begin{array}{cc}k{\left(x_i-a\right)}^m,& x_i>a,\\ 0,& -a\le x_i\le a,\\ k{\left(-x_i-a\right)}^m,& x_i<-a.\end{array}\right.\end{array}$	30/100	[−50, 50]	0
$\begin{array}{l}{f}_{13}\left(x\right)=0.1\left\{{\sin }^2\left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^n{\left(x_i-1\right)}^2\left[1+{\sin }^2\left(3\pi x_i+1\right)\right]}\right.\\ \left.+\left(x_n-1\right)\left[1+{\sin }^2\left(2\pi x_n\right)\right]\right\}+{\displaystyle \sum _{i=1}^{n}u\left(x_i,5,100,4\right)}\end{array}$	30/100	[−50, 50]	0

in the Appendix A, and the important parameter settings for the algorithms are consistent with the settings in their references—please refer to [35,36,37,38,39,40].

Table A2. Optimization results of EHLOA and comparison algorithms on benchmark functions (dimensions = 30).

Fun.	Index	EHLOA	HLOA	GWO	WOA	PSO	HBA	GJO	DMO
F1	Min	0	0	2.85 × 10−61	7.49 × 10−167	3.25 × 10−6	5.06 × 10−288	4.08 × 10−117	1.02 × 10−4
	Mean	0	0	7.68 × 10−59	1.48 × 10−148	4.50 × 10−5	1.33 × 10−276	1.25 × 10−110	4.35 × 10−4
	Std	0	0	2.25 × 10−58	7.39 × 10−148	6.73 × 10−5	0	6.81 × 10−110	3.35 × 10−4
F2	Min	0	8.99 × 10−264	1.58 × 10−35	9.65 × 10−115	8.69 × 10−5	8.21 × 10−152	2.99 × 10−68	3.21 × 10−4
	Mean	0	1.06 × 10−243	1.16 × 10−34	2.79 × 10−100	0.33	1.06 × 10−146	2.20 × 10−66	7.69 × 10−3
	Std	0	0	9.46 × 10−35	1.53 × 10−99	1.83	2.68 × 10−146	3.49 × 10−66	1.28 × 10−2
F3	Min	0	0	3.92 × 10−20	2.29 × 103	1.76 × 102	4.71 × 10−218	1.34 × 10−49	1.81 × 104
	Mean	0	0	7.19 × 10−14	2.04 × 104	8.11 × 102	4.01 × 10−200	5.9 × 10−38	5.14 × 104
	Std	0	0	3.73 × 10−13	1.06 × 104	1.29 × 103	0	2.47 × 10−37	1.45 × 104
F4	Min	0	3.62 × 10−255	1.05 × 10−15	7.7 × 10−1	2.47	6.05 × 10−124	1.66 × 10−36	3.54 × 102
	Mean	0	2.71 × 10−239	2.03 × 10−14	3.50 × 101	4.55	5.23 × 10−117	5.88 × 10−33	4.54 × 102
	Std	0	0	3.68 × 10−14	2.93 × 101	9.6 × 101	2.12 × 10−116	2.08 × 10−32	4.82
F5	Min	2.95 × 10−12	4.13 × 10−4	2.62 × 101	2.69 × 101	2.02 × 101	2.11 × 101	2.62 × 101	3.57 × 102
	Mean	4.25 × 10−2	2.2 × 101	2.71 × 101	2.73 × 101	1.83 × 102	2.18 × 101	2.75 × 101	2.43 × 103
	Std	2 × 10−2	1.23 × 101	7.62 × 10−1	6.28 × 10−1	5.42 × 102	4.56 × 10−1	7.43 × 10−1	2.24 × 103
F6	Min	3.13 × 10−11	5.89 × 10−6	1.64 × 10−5	1.04 × 10−2	4.07 × 10−6	2.62 × 10−9	1.5	9.28 × 10−5
	Mean	1.95 × 10−5	1.72 × 10−4	5.24 × 10−1	1.09 × 10−1	6.39 × 10−5	1.97 × 10−8	2.75	4.93 × 10−4
	Std	2.55 × 10−5	1.91 × 10−4	2.89 × 10−1	1.18 × 10−1	9.14 × 10−5	1.06 × 10−7	4.71 × 10−1	2.99 × 10−4
F7	Min	1.54 × 10−6	3.44 × 10−6	2.05 × 10−4	4.28 × 10−5	9.03 × 10−3	3.71 × 10−5	4.99 × 10−5	5.1 × 10−2
	Mean	4.70 × 10−5	1.53 × 10−4	9.20 × 10−4	1.36 × 10−3	2.41 × 10−2	1.65 × 10−3	1.89 × 10−5	8.25 × 10−2
	Std	3.94 × 10−5	1.48 × 10−4	4.43 × 10−4	1.52 × 10−3	7.63 × 10−3	1.02 × 10−4	1.31 × 10−4	1.74 × 10−2
F8	Min	−12,569.487	−10,417.75	−7318.73	−12,569.486	−8897.80	−10,291.76	−6410.95	infinity
	Mean	−11,525.40	−7412.19	−6039.21	−11,330.87	−7793.66	−8865.33	−3928.43	infinity
	Std	1548.62	995.25	718.53	1633.55	539.36	952.32	1053.01	infinity
F9	Min	0	0	0	0	2.36 × 101	0	0	1.25 × 102
	Mean	0	0	8.1 × 10−1	0	4.38 × 101	0	0	2.02 × 102
	Std	0	0	2.13	0	1.67 × 101	0	0	2.56 × 101
F10	Min	4.4 × 10−16	4.4 × 10−16	7.55 × 10−15	4.4 × 10−16	3.1 × 10−4	4.4 × 10−16	3.99 × 10−15	5.91 × 10−3
	Mean	4.4 × 10−16	4.4 × 10−16	1.59 × 10−14	4.1 × 10−15	8.1 × 10−3	4.4 × 10−16	4.47 × 10−15	1.53 × 10−2
	Std	0	0	3.02 × 10−15	2.37 × 10−15	1.86 × 10−2	0	1.23 × 10−15	1.63 × 10−2
F11	Min	0	0	0	0	8.05 × 10−6	0	0	1.49 × 10−3
	Mean	0	0	1.38 × 10−3	0	1.85 × 10−2	0	0	5.98 × 10−2
	Std	0	0	4.55 × 10−3	0	8.08 × 10−3	0	0	1.1 × 10−1
F12	Min	1.98 × 10−9	9.66 × 10−8	9.14 × 10−3	6.33 × 10−4	7.28 × 10−8	4.55 × 10−10	7.84 × 10−2	6.21
	Mean	1.69 × 10−6	6.92 × 10−3	4.24 × 10−2	6.08 × 10−3	6.22 × 10−2	1.66 × 10−8	2.63 × 10−1	1.53 × 101
	Std	2.16 × 10−6	2.6 × 10−2	2.56 × 10−2	5.29 × 10−3	1.03 × 10−1	3.97 × 10−8	1.79 × 10−1	5.03
F13	Min	4.49 × 10−8	6.04 × 10−7	1.00 × 10−1	3.83 × 10−2	2.63 × 10−6	1.12 × 10−7	1.20	9.91
	Mean	1.73 × 10−2	1.18 × 10−1	5.34 × 10−1	2.44 × 10−1	3.49 × 10−3	1.51 × 10−1	1.61	2.86 × 101
	Std	6.55 × 10−2	3.81 × 10−1	2.10 × 10−1	1.50 × 10−1	5.06 × 10−3	1.85 × 10−1	2.41 × 10−1	3.37 × 101

and

Table A3. Optimization results of EHLOA and comparison algorithms on benchmark functions (dimensions = 100).

Fun.	Index	EHLOA	HLOA	GWO	WOA	PSO	HBA	GJO	DMO
F1	Min	0	0	1.58 × 10−30	1.17 × 10−166	1.60 × 102	5.26 × 10−262	4.55 × 10−62	1.97 × 105
	Mean	0	0	2.63 × 10−29	1.05 × 10−148	6.46 × 102	7.56 × 10−248	5.52 × 10−60	2.51 × 105
	Std	0	0	4.42 × 10−29	5.75 × 10−148	1.83 × 103	0	1.19 × 10−59	2.27 × 105
F2	Min	0	8.25 × 10−259	2.19 × 10−18	8.62 × 10−113	6.06	8.56 × 10−138	5.51 × 10−38	1.62 × 1016
	Mean	0	2.36 × 10−239	6.54 × 10−18	3.83 × 10−103	2.08 × 101	6.82 × 10−132	4.71 × 10−37	7.98 × 1034
	Std	0	0	3.04 × 10−18	1.99 × 10−102	1.20 × 101	3.41 × 10−131	1.65 × 10−36	2.96 × 1035
F3	Min	0	0	2.00 × 10−2	5.44 × 106	4.34 × 104	1.57 × 10−186	3.85 × 10−19	7.15 × 105
	Mean	0	0	9.95	8.87 × 106	6.45 × 104	4.02 × 10−165	2.00 × 10−4	1.06 × 106
	Std	0	0	1.92 × 101	1.58 × 106	1.30 × 104	0	6.00 × 10−3	2.23 × 105
F4	Min	0	3.40 × 10−256	3.87 × 10−5	1.79 × 101	2.60 × 101	9.63 × 10−89	6.92 × 10−13	9.27 × 101
	Mean	0	5.82 × 10−241	1.00 × 10−2	7.67 × 101	3.34 × 101	7.27 × 10−83	1.53	9.53 × 101
	Std	0	0	4.00 × 10−2	2.26 × 101	3.09	2.18 × 10−82	5.20	1.04
F5	Min	9.09 × 10−11	4.13 × 10−4	9.59 × 101	9.67 × 101	1.61 × 104	9.24 × 101	9.62 × 102	9.64 × 108
	Mean	8.59 × 10−4	2.20 × 101	9.77 × 101	9.77 × 101	5.59 × 104	9.57 × 101	9.82 × 101	1.12 × 109
	Std	2.58 × 10−3	1.23 × 101	8.00 × 10−1	4.40 × 10−1	3.23 × 104	1.99	6.02 × 10−1	8.29 × 107
F6	Min	2.91 × 10−10	5.89 × 10−6	6.74	9.10 × 10−1	1.23 × 102	2.24	1.55 × 101	1.99 × 105
	Mean	9.21 × 10−5	1.72 × 10−4	9.11	2.13	6.74 × 102	4.16	1.68 × 101	2.43 × 106
	Std	1.15 × 10−5	1.91 × 10−4	1.02	9.51 × 10−1	1.83 × 103	8.41 × 10−1	8.36 × 10−1	2.11 × 105
F7	Min	1.34 × 10−6	3.44 × 10−6	1.14 × 10−3	6.55 × 10−5	5.33 × 10−1	2.44 × 10−5	5.31 × 10−5	1.06 × 103
	Mean	9.21 × 10−5	1.53 × 10−4	2.43 × 10−3	1.31 × 10−3	8.10 × 10−1	2.68 × 10−4	5.98 × 10−4	1.77 × 103
	Std	1.15 × 10−5	1.48 × 10−4	1.21 × 10−3	1.53 × 10−3	1.70 × 10−1	2.92 × 10−4	4.33 × 10−4	2.08 × 102
F8	Min	−41,898.29	−25,513.15	−20,216.25	−41,897.14	−25,484.57	−28,539.74	−16,502.35	Inf
	Mean	−37,887.25	−22,227.35	−15,699.06	−37,799.01	−21,852.99	−24,870.91	−9351.13	Inf
	Std	5381.91	1793.71	3511.17	5440.29	1453.26	3297.17	3940.74	Inf
F9	Min	0	0	2.27 × 10−13	0	2.09 × 102	0.	0	1.45 × 103
	Mean	0	0	3.71 × 10−1	3.78 × 10−15	2.94 × 102	0	0	1.57 × 103
	Std	0	0	1.12	2.07 × 10−14	3.97 × 101	0	0	5.36 × 101
F10	Min	4.44 × 10−16	4.44 × 10−16	9.99 × 10−14	4.44 × 10−16	3.58	4.44 × 10−16	7.55 × 10−15	2.05 × 102
	Mean	4.44 × 10−16	4.44 × 10−16	1.13 × 10−13	3.28 × 10−15	4.81	2.66	9.44 × 10−15	2.08 × 102
	Std	0	0	1.10 × 10−14	2.36 × 10−15	1.28	6.89	2.91 × 10−15	9.45 × 10−2
F11	Min	0	0	0	0	2.47	0	0	1.91 × 103
	Mean	0	0	2.8 × 10−3	0	6.47	0	0	2.24 × 103
	Std	0	0	8.6 × 10−3	0	1.65	0	0	1.53 × 102
F12	Min	1.33 × 10−13	9.66 × 10−8	1.38 × 10−1	7.23 × 10−3	1.96 × 101	2.59 × 10−2	4.71 × 10−1	1.76 × 109
	Mean	5.43 × 10−7	1.04 × 10−3	2.55 × 10−1	1.68 × 10−2	9.66 × 101	4.52 × 10−2	6.19 × 10−1	2.71 × 109
	Std	6.12 × 10−7	5.68 × 10−3	7.55 × 10−2	6.3 × 10−3	2.25 × 102	1.45 × 10−2	7.49 × 10−2	3.07 × 108
F13	Min	2.49 × 10−9	1.28 × 10−5	5.58	7.62 × 10−1	6.23 × 102	5.77	7.88	3.89 × 109
	Mean	7.60 × 10−3	9.41 × 10−3	6.29	1.97	7.32 × 103	7.41	8.46	4.93 × 109
	Std	4.06 × 10−2	2.61 × 10−2	3.38 × 10−1	7.41 × 10−1	7.50 × 103	7.95 × 10−1	2.95 × 10−1	3.96 × 109

in the Appendix A present the minimum values, mean values, and standard deviations obtained by eight algorithms after 30 independent runs on benchmark functions with 30 and 100 dimensions, respectively. To enhance readability, the best results for each of the three statistical indicators on each benchmark function are highlighted.

As shown in Table A2 in the Appendix A, EHLOA achieved the best minimum values for all benchmark functions except F12. It also obtained the best mean values for all benchmark functions except F6 and F12, and the best standard deviations for all benchmark functions except F6, F8, and F12. From Table A3 in the Appendix A, it can be seen that EHLOA achieved the best minimum values and mean values in all cases under the 100-dimensional benchmark functions. Additionally, EHLOA obtained the best standard deviations for all benchmark functions except F6 and F13.

To comprehensively evaluate the performance of EHLOA, Wilcoxon signed-rank tests [42] were conducted to compare EHLOA’s results on each benchmark function with those of the comparison algorithms. This test determines whether there are statistically significant differences between EHLOA and the comparison algorithms on the benchmark functions. The tests were performed at the 5% significance level. When the p-value is less than 0.05, the result is denoted as “+” or “−”. “+” indicates that EHLOA performs better on the given benchmark function, while “−” indicates that EHLOA performs worse. When the p-value is greater than 0.05, the result is denoted as “=“, indicating that EHLOA and the comparison algorithm have similar performance on the given benchmark function.

Table A4. Wilcoxon signed-rank test simulation results: EHLOA and comparative algorithms on benchmark functions (dimensions = 30).

Fun.	Index	HLOA	GWO	WOA	PSO	HBA	GJO	DMO
F1	p-value	1	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	+	+	+
F2	p-value	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F3	p-value	1	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	+	+	+
F4	p-value	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F5	p-value	1.74 × 10−4	4.86 × 10−5	4.86 × 10−5	2.35 × 10−6	5.31 × 10−5	2.35 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F6	p-value	2.16 × 10−5	1.73 × 10−6	1.73 × 10−6	1.04 × 10−3	1.92 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	−	+	+
F7	p-value	5.31 × 10−5	1.73 × 10−6	2.13 × 10−6	1.73 × 10−6	3.18 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F8	p-value	1.73 × 10−6	1.73 × 10−6	1	1.73 × 10−6	6.64 × 10−4	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	=	+	+	+	+
F9	p-value	1	3.13 × 10−2	1	1.73 × 10−6	1	1	1.73 × 10−6
	+/=/−	=	+	=	+	=	=	+
F10	p-value	1	1.73 × 10−6	7.12 × 10−6	1.73 × 10−6	1	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	=	+	+
F11	p-value	1	0.25	1	1.73 × 10−6	1	1	1.73 × 10−6
	+/=/−	=	=	=	+	=	=	+
F12	p-value	8.31 × 10−4	1.73 × 10−6	1.73 × 10−6	9.32 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	−	+	+
F13	p-value	1.25 × 10−2	1.73 × 10−6	6.98 × 10−6	0.32	2.22 × 10−4	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	=	+	+	+
Total	(+/=/−)	8/5/0	12/1/0	10/3/0	12/1/0	8/3/2	10/3/0	13/0/0

and

Table A5. Wilcoxon signed-rank test simulation results: EHLOA and comparative algorithms on benchmark functions (dimensions = 100).

Fun.	Index	HLOA	GWO	WOA	PSO	HBA	GJO	DMO
F1	p-value	1	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	+	+	+
F2	p-value	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F3	p-value	1	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	+	+	+
F4	p-value	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F5	p-value	2.35 × 10−6	1.73 × 10−6	4.86 × 10−5	2.35 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F6	p-value	4.45 × 10−5	1.73 × 10−6	1.73 × 10−6	1.04 × 10−3	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F7	p-value	2.11 × 10−3	1.73 × 10−6	2.13 × 10−6	1.73 × 10−6	4.45 × 10−5	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F8	p-value	1.73 × 10−6	1.73 × 10−6	1	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	=	+	+	+	+
F9	p-value	1	1.48 × 10−6	1	1.73 × 10−6	1	1	1.73 × 10−6
	+/=/−	=	+	=	+	=	=	+
F10	p-value	1	1.61 × 10−6	2.93 × 10−5	1.73 × 10−6	0.12	1.73 × 10−6	1.73 × 10−6
	+/=/−	=	+	+	+	=	+	+
F11	p-value	1	0.25	1	1.73 × 10−6	1	1	1.73 × 10−6
	+/=/−	=	=	=	+	=	=	+
F12	p-value	2.35 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
F13	p-value	1.39 × 10−2	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6	1.73 × 10−6
	+/=/−	+	+	+	+	+	+	+
Total	(+/=/−)	8/5/0	12/1/0	10/3/0	13/0/0	10/3/0	10/2/0	13/0/0

in Appendix A present the p-values and Wilcoxon signed-rank test results for the 30-dimensional and 100-dimensional benchmark functions, respectively.

It can be found from Table A4 in Appendix A that EHLOA performs worse than the comparison algorithm HBA only on F6 and F12 for the 30-dimensional benchmark functions. The remaining results indicate that EHLOA’s performance is not inferior to the comparison algorithms, and it outperforms them in most cases. Specifically, EHLOA outperforms HLOA and HBA 8 times, GWO and PSO 12 times, WOA and GJO 10 times, and it outperforms DMO on all benchmark functions. In Table A5 in the Appendix A, EHLOA does not perform worse than any of the comparison algorithms on any benchmark function for the 100-dimensional benchmark functions. Specifically, EHLOA outperforms HLOA 8 times, GWO 12 times, WOA, HBA, and GJO 10 times, and it outperforms PSO and DMO on all benchmark functions.

4.2. Experiment 2: In the Flood Control Operation Model

In this section, the performance of 8 algorithms will be analyzed after independently running the flood control operation model 30 times under the conditions of a population size of 100 and a maximum of 1000 iterations.

Table 3. Optimization results of EHLOA and comparison algorithms on benchmark flood control operation model.

Flood Type	Index	EHLOA	HLOA	GWO	WOA	PSO	HBA	GJO	DMO
Flood1	Min	4160.89	4201.44	4276.81	4319.78	N/A	4261.15	4421.58	N/A
	Mean	4194.12	4295.04	4418.68	4393.19	N/A	4375.08	4504.88	N/A
Flood 2	Min	4408.79	4412.10	4452.01	4467.01	N/A	4430.41	4484.03	N/A
	Mean	4421.13	4449.07	4487.36	4489.90	N/A	4468.28	4507.46	N/A
Flood 3	Min	4521.82	4588.62	4562	4756.93	4943.81	4619.97	4646.38	5253.01
	Mean	4592.95	4680.02	4598.69	4923.78	4986.57	4713.98	4767.07	5332.82
Flood 4	Min	4701.87	4753.19	4749.66	4964.86	4950.83	4796.51	4821.83	5302.54
	Mean	4754.32	4845.17	4780.11	5037.51	4971.29	4865.93	4897.72	5379.37

presents the minimum and mean values of the objective function obtained by EHLOA and the comparison algorithms after 30 independent runs on four typical flood datasets. In Table 3, the best value in each row is highlighted in bold, and N/A indicates that the algorithm could not find a feasible solution. As shown in Table 3, EHLOA achieved the best minimum and mean values across all four floods. Except for Flood 2, EHLOA’s minimum values after 30 runs were more than 40 lower than the second-best algorithm’s minimum values in the other three floods. However, in Flood 2, EHLOA’s minimum value was only slightly lower than that of HLOA. In Flood 1, EHLOA had the largest lead in mean value, outperforming the second-best HLOA by more than 100. In Flood 3, EHLOA’s lead in the mean value was the smallest, just slightly better than GWO. In both Flood 2 and Flood 4, EHLOA’s mean values were more than 20 lower than the second-best mean values.

Figure 4a,b shows the iterative processes of EHLOA, HLOA, HBA, GWO, WOA, and GJO when operating flood 1 and flood 2, respectively. These figures present the iterative process for the minimum fitness values. Please note that due to the large penalty function coefficients, fitness values exceeding 10,000 are uniformly replaced with a fitness value of 10,000. Additionally, the value 4510 is marked in the figure, representing the discharge limit of LJX under a millennial flood. Values below 4510 indicate that the algorithm has found a feasible region. From Figure 4a, it can be seen that the fitness values for all six algorithms exceed 10,000 at the start of the iterations. However, EHLOA quickly converges to below 4510 and achieves the best fitness value at the end of the iterations. Observing the iterative curves of GWO, GJO, and WOA, it is evident that they remain at the value of 4510 for an extended period, Additionally, GJO and GWO converge to 4510 significantly slower than the other four algorithms. As observed in Figure 4b, EHLOA starts with a fitness value around 4510, while the other five algorithms have initial fitness values exceeding 10,000. Moreover, the fitness values for both GWO and GJO do not drop below 10,000 until after 800 iterations. By the end of the iterations, the fitness values of EHLOA and HLOA are nearly the same and are better than those of the other four algorithms.

Figure 5a,b shows the iterative processes of EHLOA, HLOA, HBA, GWO, WOA, GJO, PSO, and DMO when operating flood 3 and flood 4, respectively. These figures present the iterative process for the minimum fitness values too. As shown in Figure 5a,b, EHLOA starts with a fitness value that is more than 300 lower than the other seven algorithms at the beginning of the operation process. Except for DMO, EHLOA has the fastest convergence rate. Although DMO converges quickly, its final fitness values in Figure 5a,b are much higher than those of EHLOA. Additionally, EHLOA achieves the best fitness values for both flood 3 and flood 4, maintaining a noticeable lead over the second-best algorithm.

4.3. Analysis of the Flood Control Operation Process Obtained by EHLOA

In this section, the scheme with the smallest fitness value obtained by EHLOA when operating four floods will be analyzed.

Table 4. Best peak reduction rates achieved by EHLOA for four floods.

Flood Type	Peak Values of the Design Floods (m3/s)	Peak Values after Operation (m3/s)	Peak Reduction Rates (%)
flood 1	9368	4161	55.6
flood 2	9343	4409	52.8
flood 3	10,800	4522	58.1
flood 4	10,791	4702	56.4

Note: flood 1 and flood 2 denote a millennial flood based on the typical 1964 and 1967 flood, respectively; and flood 3 and flood 4 denote a decamillennial flood based on the typical 1964 and 1967 flood, respectively.

presents the best peak reduction rates achieved by EHLOA for four floods. As shown in Table 4, under EHLOA operation, the peak reduction rates of the cascade reservoirs are all above 50%, with higher peak reduction rates for the peak-shaped floods compared to the flat-topped floods.

Figure 6 and Figure 7 illustrate the variations in reservoir storage capacity, inflow, and discharge flow for floods 1 to 2, respectively. To better demonstrate the flood control effectiveness of the reservoir, the inflow of LJX uses the design flood, rather than the outflow from LYX during the operating process. Due to the peak shape and operating process of the decamillennial floods (flood 3 and flood 4) being essentially the same as those of the millennial floods, their detailed operating processes are included in Appendix B without further analysis.

Figure 6a shows that the pre-release period for LJX lasts up to 11 days, with its storage capacity being pre-released to 2.38 million m³ on the 11th day. LYX and LJX approach their flood control storage capacities (V_L), on the 27th and 24th days, respectively. After this, the storage capacities gradually decrease, but LYX’s storage capacity rises again on the 40th day, while LJX’s storage capacity rises again on the 41st and 42nd days. Figure 5b indicates that LYX’s maximum inflow occurs on the 22nd day, after which the inflow gradually decreases. However, LYX’s inflow rate significantly increases again between the 38th and 41st days. The discharge process for LYX exhibits noticeable fluctuations but does not exceed the fluctuation constraint of 1000 m³/s. Figure 6c illustrates that LJX’s maximum inflow occurs on the 17th day, after which the inflow gradually decreases. However, LJX’s inflow rate significantly increases again on the 41st and 42nd days, rising to 5164 m³/s on the 42nd day. Overall, the discharge process for LJX is stable, with no significant fluctuations except on the 37th day.

Figure 7a shows that the pre-release period for LJX lasts 9 days, with its storage capacity being pre-released to 2.46 billion m³ on the 9th day. LYX and LJX approach their flood control storage capacities (V_L), on the 29th day, and there is no significant rebound in their storage capacities afterward. Figure 7b indicates that LYX’s maximum inflow occurs on the 19th day. Although the inflow decreases between the 20th and 24th days, there are noticeable increases on the 22nd and 24th days. After the 24th day, LYX’s inflow declines rapidly, with a slight increase on the 43rd day, but it does not exceed 3000 m³/s. LYX’s discharge exhibits considerable fluctuations from the 1st to the 27th day, but these do not exceed the fluctuation constraint. After the 27th day, LYX’s discharge becomes stable with no significant fluctuations. Figure 7c illustrates that LJX’s maximum inflow occurs on the 20st day, followed by a gradual decrease. However, there is a noticeable increase on the 23rd day. Although there are rebounds after the 35th day, the inflow does not exceed 3600 m³/s.

5. Discussion

5.1. Robustness and Scalability of EHLOA

EHLOA demonstrates a good scalability. Comparing the results from Table A2, Table A3, Table A4 and Table A5 in the Appendix A, it can be seen that EHLOA’s performance on the 100-dimensional benchmark functions is even better than on the 30-dimensional ones. For the 30-dimensional benchmark functions, HBA was able to achieve the best minimum values and means on F6 and F12. However, for the 100-dimensional benchmark functions, EHLOA achieved the best minimum values and means across all benchmark functions. Additionally, according to the Wilcoxon test results in Table A4 and Table A5 in Appendix A, EHLOA outperformed the comparison algorithms three more times on the 100-dimensional problems than on the 30-dimensional ones. Interestingly, according to the results from Table A2 and Table A3 in Appendix A, EHLOA’s average and minimum values on benchmark functions F5 and F13 are lower for the 100-dimensional problems than for the 30-dimensional ones. This demonstrates that EHLOA’s performance does not degrade with increasing problem dimensionality and may even excel in handling high-dimensional problems.

EHLOA has demonstrated a good robustness. As shown in Table A2 and Table A3 in Appendix A, the comparison algorithms performed poorly on the unimodal function F5 and the multimodal function F8, with the minimum and mean values after 30 runs being significantly distant from the theoretical optimal values. On F5, the mean value of EHLOA is below 0, while the mean values of all comparison algorithms are above 20. On F8, the minimum values of all comparison algorithms, except for WOA, are more than 10,000 away from the theoretical optimal value after 30 runs, whereas the minimum value of EHLOA equals the theoretical optimal value. F5 and F8 present a substantial challenge due to the presence of numerous local optima, causing many algorithms to become trapped in these local optima, resulting in poor performance [43,44]. EHLOA’s excellent performance on the most challenging unimodal and multimodal functions in the CEC2005 test set demonstrates its ability to handle different types of problems, verifying its robustness and ability to escape local optima.

5.2. Reliability of EHLOA in a Multi-Reservoir Flood Control Operation

From Table 3, it can be seen that whether it is a millennial flood (flood 1, flood 2) or a decamillennial flood (flood 3, flood 4), the mean objective function value for operating peak-shaped floods (flood 1, flood 3) are lower than those for operation flat-topped floods (flood 2, flood 4) for all algorithms. This indicates that flat-topped floods are more difficult to operate than peak-shaped floods. EHLOA achieved the lowest mean fitness values and lowest minimum values in all four floods, demonstrating its good robustness in the MRFCO problem. It performs best regardless of whether it is operation peak-shaped or flat-topped floods. The fitness curves corresponding to the best fitness values of each algorithm for the four floods show that EHLOA consistently obtained the lowest best fitness values, further illustrating its excellent robustness in the MRFCO problem.

The significantly lower initial fitness values of EHLOA compared to the other algorithms when operating floods 2 to 4 suggest that the Circle map initialization strategy likely enables EHLOA to find feasible solutions in less time for the MRFCO problem. EHLOA exhibited the fastest convergence speed and the highest convergence accuracy (lowest fitness value) across all four floods, indicating strong convergence performance in the MRFCO problem.

Additionally, when operating flood 1, the fitness values of GWO, WOA, and GJO remained at 4510 for an extended period. This is because 4510 is the discharge limit of LJX for a millennial flood; a fitness value of 4510 implies that the algorithm releases a large flow to prevent LYX and LJX from exceeding their flood control capacity, resulting in a penalty function value of 0. However, the final operation results show that fitness values can be below 4510, indicating that 4510 is a local optimum. The iterative curve of EHLOA did not stay at 4510, illustrating that EHLOA has a strong ability to escape local optima in the MRFCO problem and can effectively integrate with penalty functions to handle constraints.

From the flood operation results and the operation process, the schemes provided by EHLOA are reliable. Firstly, under the operation of EHLOA, the peak reduction rates of the cascade reservoirs for all four floods are above 50%, proving that EHLOA has strong peak reduction capabilities for different types of floods. Secondly, the discharge processes of LJX are very stable, which greatly protects the stability of downstream slopes of the cascade reservoirs.

Finally, the changes in storage capacity of LYX and LJX during the operating of flood 1 and flood 2 show significant differences. The pre-release period for flood 1 is two days longer than that for flood 2, due to the much larger total flood volume in the first 10 days of the 1967 design flood compared to the 1964 design flood. During the operating of flood 1, both LYX and LJX exhibit a noticeable rebound in storage capacity towards the end of the operating period, whereas this phenomenon does not occur during the operating of flood 2. This is because the total flood volume in the last 10 days of the 1964 design flood is much larger than that of the 1967 design flood. Therefore, the 1964 design flood is characterized by smaller early and peak flows but larger later flows, while the 1967 design flood has larger early and peak flows but smaller later flows. EHLOA can flexibly adjust the pre-release duration and provide different operating schemes based on the varying flood patterns of these two types of floods.

6. Conclusions

The main conclusions of this paper are summarized as follows:

(1) EHLOA has excellent robustness and scalability.

(2) EHLOA can be considered a reliable algorithm for solving multi-reservoir flood control operation problems.

The methods used in this paper to enhance EHLOA’s performance are improving the quality of initial solutions and strengthening the ability to escape local optima. However, balancing exploration and exploitation is also crucial for metaheuristic algorithms. Therefore, future work will analyze the exploration-to-exploitation ratio of HLOA to determine if improvements are needed. Additionally, EHLOA will be tested on more benchmark functions and applied to parallel reservoir flood control operation to further validate its performance.