An Improved Lion Swarm Optimization Algorithm With Chaotic Mutation Strategy and Boundary Mutation Strategy for Global Optimization

Lion swarm optimization (LSO) inspired by the natural division of labor among lion king, lionesses and lion cubs in a lion group, i.e., lion king guarding, lionesses hunting and lion cubs following, is a relatively novel swarm intelligent optimization technique. Due to its remarkable performance, the canonical LSO has been extensively researched. However, how to balance contradictions between the exploration and the exploitation and alleviate the premature convergence are two critical concerns that need to be dealt with in the LSO study. To address these two drawbacks, enhance the optimization performance, and broaden its application domain, an improved lion swarm optimization algorithm with chaotic mutation strategy and a boundary mutation strategy (CBLSO) is proposed in this paper. In the proposed algorithm, a chaotic mutation strategy based on chaotic cubic mapping is designed to enhance the exploration ability of the algorithm, while the boundary mutation strategy based on the concept of multilevel parallel is adopted to manage boundary constraint violations, which is beneficial for improving the exploitation ability of the algorithm. The proposed CBLSO is evaluated on 56 classic test functions and 30 CEC2014 benchmark functions, and is compared with quite a few state-of-the-art algorithms regarding often-used performance metrics. The experimental results demonstrate the superior performance of the embedded strategies on balancing the exploration and the exploitation. Furthermore, the proposed CBLSO is applied to the optimal dispatch problem of cascade hydropower stations based on a novel constraints handling method designed in this paper to validate its good practicability and performance. The experimental results of a case study on the optimal dispatch problem of China’s Wujiang cascade hydropower stations indicate that the proposed CBLSO can produce better and more reliable optimal results than the canonical LSO and other comparison algorithms with competitive speed. Thus, we can conclude that the proposed CBLSO is a competitive and effective alterative tool to solve complex numerical optimization problems and real-world optimization with complicated constraints.


I. INTRODUCTION
Global optimization problems (GOPs) that concern a large number of decision variables and complex constraints exist extensively in various fields of science [1], [2], engineering The associate editor coordinating the review of this manuscript and approving it for publication was Frederico Guimarães . design [3], [4], business and economics [5], which cannot be solved well within adequate time or specific accuracy by traditional deterministic methods. In order to address such optimization problems, a lot of computational intelligence methods, especially the meta-heuristic optimization algorithms that inspired natural organisms, social behaviors, biological behaviors and physical phenomena have emerged in the past few decades. Roughly, the main branch of meta-heuristic algorithms can be broadly classified into six main categories: stochastic-based algorithms, evolutionbased algorithms, swarm-based algorithms, physics-based algorithms, human-based algorithms, and ecosystem-based algorithms [6], [7], [8], [9], [10], [11], [12], [13], [14], which are shown in Fig. 1.
Stochastic-based algorithms use randomness to perform non-deterministic behaviors, usually different from the purely deterministic procedures. The most famous stochasticbased algorithms are Adaptive Random Search (ARS) [15], Random Search (RS) [16], Tabu Search (TS) [17], Hill Climbing (HC) [18], Local Search (LS) [19], and Guided Search (GLS) [19]. Evolution-based algorithms imitate the evolutionary behaviors, i.e., reproduction, mutation, recombination and selection of creatures found in nature. The search process in these algorithms usually starts with a randomly generated population, which further evolves over successive generations. For evolution-based algorithms, the main strength point is the best individuals are always combined together to form the new generation since it can promote the improvement of population over the course of iterations. Evolution Strategy (ES) [20], [21], Evolutionary Programming (EP) [22], Genetic Algorithm (GA) [23], Genetic Programming (GP) [24], Evolutionary Algorithm (EA) [25], and Differential Evolution (DE) [26], [27] can be considered as the most standard form of evolution-based algorithms. Swarmbased algorithms are swarm intelligence-based techniques, which simulate the intelligent social and individual behaviors of swarms, animals, herds, teams, or any group of creatures. Ant Colony Optimization (ACO) [28], Particle Swarm Optimization (PSO) [29], [30], Artificial Bee Colony (ABC) [31], Cuckoo Search (CS) [32], and Bat Algorithm (BA) [33] can be described as representative algorithms in Swarm-based algorithms. Some of the recent Swarm-based algorithms are Grey Wolf Optimizer (GWO) [34], Lion Swarm Optimization (LSO) [35], Squirrel Search Algorithm (SSA) [6], and Harris Hawks Optimization (HHO) [36]. Physics-based algorithms are motivated by the basic physical rules that exist in universe. The inspiring physical systems range from metallurgy, music, the interplay between culture and evolution, science (chemistry, physics, mathematics), and complex dynamic systems. Generally, a physics-based algorithm is a combination of local search techniques and global search techniques. Some of the prevailing algorithms of this category are Simulated Annealing (SA) [37], [38], Big Bang-Big Crunch (BB-BC) [39], Gravitational Search Algorithm (GSA) [40], and Atom Search Optimization (ASO) [41], [42]. Human-based algorithms are inspired by the human behaviors, while ecosystembased algorithms are mainly based on the natural ecosystem. The most state-of-the-art Human-based algorithms are Harmony Search (HS) [43], Imperialist Competitive Algorithm (ICA) [44], League Championship Algorithm (LCA) [45] and Teaching-Learning-Based Optimization (TLBO) [46], [47], while the most well-known ecosystem-based algorithms are Invasive Weed Optimization (IWO) [48], Biogeography-Based Optimization (BBO) [49], and Multi-species optimizer based on PSO (PS 2 O) [50].
Apart from the aforementioned algorithms, various improved versions based on the basic versions of existing meta-heuristic optimization algorithms are also proposed and have been successfully applied to almost all areas of operational research, data mining, neural networks, image and video processing, machine intelligence, and many other fields of knowledge. Thousands of research papers and dozens of books have been published [51], [52].
In spite of the rapid development of meta-heuristic optimization algorithms, to learn from the nature for developing a better algorithm to solve new and more complex GOPs is still in progress and new meta-heuristic optimization algorithms are still emerging, such as Marine Predators Algorithm (MPA) [53], Slime Mould Algorithm (SMA) [54], Golden Eagle Optimizer (GEO) [55], Artificial Hummingbird Algorithm (AHA) [56], and so on. In addition, according to the ''no free lunch'' (NFL) theory [57], there is no single metaheuristic optimization algorithms best suited for dealing with all GOPs. This means that a particular optimization method may show competitive outcomes on a set of optimization problems, but may not provide promising results on the other set of optimization problems. The NFL theory, certainly, forms the basis for the researchers to improve the existing algorithms or design new and powerful algorithms for better optimization now and then.
Lion swarm optimization (LSO) proposed by Liu et al. [35], is a new entrant in the domain of meta-heuristic algorithms based on the natural division of labor among lion king, lionesses and lion cubs in a lion group, i.e., lion king guarding, lionesses hunting and lion cubs following. Due to VOLUME 10, 2022 advantages, e.g., simple structure, few control parameter, ease of implementation, good robustness, and fast convergence speed, LSO has been widely used in solving various realworld optimization problems and many variants of LSO are also developed. Some excellent works have been reported: Yang and Wei [58] proposed an improved LSO algorithm for optimizing the long short-term memory recurrent neural network problem. In this algorithm, the chaos theory was introduced for accelerating the global convergence speed of the algorithm. Zhang and Jiang [59] presented a parallel discrete lion swarm optimization (PDSLO) algorithm for solving the traveling salesman problem (TSP). In the PDLSO, discrete coding and order crossover operators were firstly used to form a discrete lion swarm optimization (DLSO). And then, the complete 2-opt (C2-opt) algorithm was employed to enhance the local search ability of the PDLSO algorithm. In addition, the PDLSO has multiple populations and each sub-population independently ran the DLSO algorithm in parallel as well as transferred information among them by using the ring topology. Qiao et al. [60] proposed a novel hybrid lion swarm optimization algorithm to optimize the traditional least squares support vector machine model by combing lion swarm optimization algorithm and genetic algorithm. Guo and Jiang [61] presented an improved lion swarm optimization algorithm for solving the job-shop scheduling problem. In this algorithm, chaos search and gaussian perturbation strategy were added to the position of lions in the past dynasties, which could improve the optimization efficiency of the algorithm in the optimization process. Although these researches demonstrate that LSO has greater application potential, LSO stills has some noticeable deficiencies in solving some GOPs. Specially, when tackling the complex GOPs that have a high eccentric ellipse, a narrow curving valley, or GOPs characterized with multimodality, high-dimension and the existence of several local minima and many global minima, like most nature-inspired metaheuristic optimization algorithms, LSO fails to effectively find near-optimal solutions. This is mainly because of their stochastic nature, which may erect barriers for them to tackle some types of complex GOPs.
To address this weakness, chaotic mutation strategy based on chaotic cubic mapping and boundary mutation strategy based on the concept of multilevel parallel maybe two alternative technologies as they have been studied experimentally in multi-objective PSO for handling multi-objective optimization problems (MOPs) [62] and PSO for handling GOPs [63], respectively, and proven to be two effective and efficient technologies to improve the performance of PSO when tackling various categories of GOPs. Hence, it is reasonable to believe that the chaotic mutation strategy and the boundary mutation strategy can be introduced into LSO to address its premature convergence problem and improve the robustness as well as population diversity when tackling different types of GOPs.
In this paper, a novel LSO algorithm with a chaotic mutation strategy and a boundary mutation strategy (CBLSO) is proposed for GOPs and optimal dispatch problem of cascade hydropower stations. Compared with the standard LSO and other meta-heuristic optimization algorithms, the main features of our CBLSO algorithm can be stated as follows: (a) The CBLSO can achieve good performance with various updating strategies of the population which inherit from the standard LSO. Broadly, adaptability and choice of the fittest are two distinct features of nature that imitated by all meta-heuristic optimization algorithms. The population is recursively updated at each iteration according to the suitable updating mechanism. As previously stated, there are many meta-heuristic optimization algorithms inspired by animal social behaviors. Like PSO, ACO, ABC, BA, GWO, and SSA et al, imitating animal foraging behavior is also the solution updating strategy of the CBLSO. However, there are some evident differences between CBLSO and other algorithms. Similar to PSO and BA, a new direction for movement of lion individual is generated by considering the local best position (pbest) and the global best position (gbest) obtained by lion individual and lion group so far, while other algorithms (ACO, ABC, GWO, and SSA) do not adopt. Besides, PSO and BA update the population by using a single strategy, while CBLSO utilizes different strategies for different lion group. Thus, the position updating mechanism of CBLSO differs from PSO and BA. Although CBLSO, ABC, GWO, and SSA work on the effective division of labor, i.e., the whole population is divided into various regions. Different from ABC and GWO, CBLSO initially sorts the whole population in ascending order of fitness values and then divides it, which is controlled by the user. Although the dividing mode of CBLSO and SSA looks quite similar, they technically present some differences. In CBLSO, the highest hierarchy is the lion king and the number of lion king is only one, while the number of the highest hierarchy in SSA is the least ones. Besides, in CBLSO, for a certain hierarchy (lion cubs), the updating strategy is diversified, while the updating strategy used by SSA for a certain hierarchy is single. Hence, as discussed previously, we are confident that our CBLSO will be a good competitor for existing nature-inspired metaheuristic optimization algorithms.
(b) The introduced chaotic mutation strategy. In the standard LSO, both lionesses and lion cubs use gbest information possessed by the lion king to update the hunting behavior, directly or indirectly. Through this mechanism, the whole lion group may be easily trapped into a poor zone once the lion king is located around a local optimum. However, a chaotic mutation strategy based on chaotic cubic mapping is designed herein could improve the exploration ability of the CBLSO, since the regions where the lion king is located will be explored many times by lionesses and lion cubs. Hence, the whole lion group may be pulled toward the sparsest region. Its execution is determined by a pre-defined threshold, i.e., when the lion king's fitness value does not change or changes little in three consecutive generations, the chaotic mutation strategy is executed. The advantages of this lie in two aspects: (1) exploring more near-optimal solutions and (2) retaining the balance of the exploration ability and the exploitation ability of the algorithm. The efficiency of the chaotic mutation strategy is experimentally tested in Section 4.2.
(c) The introduced boundary mutation strategy. Different from the common boundary constraint approach used in the standard LSO, a boundary mutation strategy based on the concept of multilevel parallel is introduced in CBLSO for managing boundary constraint violations, which is beneficial for the maintaining of the population diversity and improving the global search ability. The advantage of the boundary mutation strategy is experimentally invested in Section 4.2.
Based on the above three main features, we are confident that the CBLSO will be effective and competitive. To evaluate the performance of the proposed algorithm, a comprehensive experiment study is conducted to compare the CBLSO with various state-of-the-art optimization algorithms on fifty-six classic test functions, encompassing Uni-modal, Multi-modal, Separable, Non-separable and Multi-dimension problems, and thirty 30-dimensional benchmark functions from the CEC 2014 test suite. The experiment results indicate that the CBLSO can provide better global optimal solutions with high quality. Besides, the results of non-parametric statistical tests demonstrate that the CBLSO performs significantly better than or at least comparable to the well-known algorithms and can be taken as a promising alternative optimization tool to solve GOPs. Finally, to validate the great potential of the CBLSO for real-world application, based on the novel constraints handling method designed in this paper, the CBLSO is applied to solve the optimal dispatch problem of cascade hydropower stations in Section 6. The obtained results show that the proposed algorithm can produce competitive results when compared with the standard LSO and other comparison algorithms.
The remainder of this paper is organized as follows. In Section 2, the mathematical model of the canonical LSO is presented. Details of the CBLSO are described in Section 3. In Section 4, the effects of the chaotic mutation strategy and the boundary mutation strategy are first well studied through a comparative experiment, and then a series of comparative experiments between the CBLSO and other stateof-the-art optimization algorithms, comparative statistical analysis of simulation results on 56 test problems are presented and discussed. Thereafter, In Section 5, the CBLSO is further compared with other well-known optimizers on 56 classic functions as well as other competitive optimization algorithms on the benchmark functions from the CEC 2014 test suite. Section 6 provides the application on real-world optimization problems. Finally, in Section 7, discussions, conclusions and some future works are given. All classic test functions are listed in the Appendix.

II. CANONICAL LSO
Lion swarm optimization (LSO) algorithm is a Swarm-based optimization algorithm inspired by the hunting behavior of the lion group. In LSO, just as in real life, there are three kinds of lions: lion king, lionesses and lion cubs. Starting from an initial position of the search space to be optimized, the lion with the best fitness value is assumed to be the lion king. Then, a certain proportion of lions are selected as hunting lions (usually are lionesses) and they hunt together with other lionesses of the group. Once a prey found is better than the prey currently occupied by the lion king, the position of the prey will be possessed by the lion king. The lion cubs follow the lionesses to learn how to hunt or forage for food near the lion king. After reaching sexual maturity and they are not stronger than the lion king, the lion cubs will be driven out of the group and become nomadic lions. In order to survive, the nomadic lions will try to approach the best position in his or her memory. According to the natural division of labor and cooperate with each other, the lions of the group repeatedly search for the optimal of the objective function.

A. RANDOM INITIALIZATION
In LSO, each lion position represents a potential solution to the considered problem, and the quality of prey corresponds to the quality (fitness) of the associated solution. Specifically, for a D-dimensional GOP, an initial population P called lion group containing n solutions (lion positions) will be randomly generated in search space in the beginning. The position of the ith lion will be generated through the following equation: where x i,j represents the jth dimension of the ith lion; i = 1, 2, · · · , n; j = 1, 2, · · · , D and rand (0, 1) is a random number that distributed uniformly in the range [0,1]. x min,j and x max,j are the lower and upper bounds of the jth dimension, respectively. Up to now, the initialization of the population is completed. Thereafter, the number of adult lions, i.e., lion king and lioness, nLeader(2 ≤ nLeader ≤ n/2) will be defined through the following equation: where β is a random number in the range [0,1] that is acknowledged as the proportion factor of the adult lions (including lion king and lionesses). In LSO, the proportion of the adult lions in a group has an important effect on the final optimization result. The bigger the proportion of the adult lions, the fewer the number of the lion cubs. However, the updating pattern of the lion cubs' position is various, which may increase the otherness of the group and improve the exploration of the algorithm. To balance the exploitation and the exploration of LSO, the proportion factor of adult lions β is considered to be less than 0.5 for all cases in this paper [35]. Here, β is set to be 0.2 suggested by literature [64]. The number of lion cubs is n − nLeader.
After the population initialization, the fitness value of position for each lion is calculated by putting the values of decision variable (solution vector) into the user-defined fitness functions. The fitness value of each lion's position represents the quality of prey searched by it, i.e., optimal prey (possessed by lion king), normal prey (possessed by lioness) and little prey (possessed by lion cubs) and hence their probability of survival also.

B. SORTING AND DECLARATION
After calculating the fitness values of each lion's position, we sort the fitness values in ascending order. The lion with minimal fitness value is considered to be the lion king. The next best nLeader − 1 lions are declared as lionesses. The remaining lions are supposed to be lion cubs. Next, all the lions can conduct hunting behavior by adopting the position updating mechanism.

C. HUNTING BEHAVIORS OF LIONS
As mentioned previously, the LSO works on the effective division of the natural labor, i.e., the whole group is divided into various regions (including lion king, lioness and lion cubs). In each generation, it is assumed that different lions conduct different hunting behaviors to update their position. The dynamic hunting behavior can be mathematically modeled as follows: Lion king: To ensure the priority for prey than the other lions, the lion king may move in the range of the best food, i.e., the position that has the minimal fitness value. In this case, the new position of lion king can be obtained as follows: where γ is normally distributed random number in the range [0,1], t is the current iteration value, pbest i (t) is the historically best position of the ith lion at the current iteration, gbest (t) is the global best position of the lion group at the current iteration. Lionesses: Recognizing the position of prey, encircling them, and then attacking towards the prey is the usually hunting behavior of lioness. When lioness conducts the hunting behavior, they usually cooperate with another lioness. In this case, the new position of lionesses can be obtained as follows: where pbest c (t) is the historically best position of the cooperation lioness at the current iteration, α f is moving range disturbance factor of lionesses, defined as follows: where t max is the maximum iteration value, step denotes the maximal step value of the lioness' activity range computed as follows: wherex max andx min are maximal mean value and minimal mean value of each dimension, respectively. Lion cubs: As discussed above, three situations may occur during the dynamic hunting of the lion cubs. Note that the lioness followed by lion cubs is randomly selected from the lioness group, which means that the followed relationship between the lioness and the lion cubs is randomly established. In this case, the new position of lion cubs can be obtained as follows: where q is a random number in the range [0,1], pbest m (t) is the historically best position followed by lion cubs at the current iteration, α c is the moving range disturbance factor of lion cubs, gbest (t) is the position that ith lion cub is driven out of the scope of hunting. At the position that far from lion king, it is an elite opposition-based learning (OBL) strategy that has been verified to be an effective method for solving the optimization problems [65], [66]. The introduction of OBL can enlarge the search space of nomadic lion cubs and is helpful to improve the global exploration of the LSO. α c is defined as follows: where step denotes the maximal step value of lion cubs' activity range computed by Eq. (6): gbest (t) is defined as follows:

III. PROPOSED CBLSO ALGORITHM
In meta-heuristic optimization algorithms, generally, the optimization process may be divided into two conflicting milestones: the exploration and the exploitation, where enhancing one may result in degrading the other. In the early phase of the optimization process, the candidate solutions should be encouraged to explore the whole search space instead of clustering around the local optima, which is beneficial for improving the population diversity and causing a high exploration of the whole search space. In the later phase, the candidate solutions have to exploit the information gathered to converge towards the global optima, which is aimed at enhancing the quality of the solutions. Although there are various improvement methods to balance the two milestones and promote local optima avoidance, the works of literatures [62], [63] indicate that the chaotic mutation strategy and the boundary mutation strategy are better in tackling this issue. In the following subsections, the chaotic mutation strategy and the boundary mutation strategy are firstly described in detail, respectively. Then the CBLSO algorithm is proposed.

A. CHAOTIC MUTATION STRATEGY
Recently, following various realm of humans, a large number of chaotic maps developed by researchers, physicians and mathematicians are available in the optimization field. Out of all these available chaotic maps, the bulk of them has been applied widely in optimization algorithms for enhancing the local search ability of the algorithms and avoiding premature convergence effectively [67], [68], [69], [70]. To avoid the degeneration of the candidate solutions caused by an irregular random variation, a chaotic mutation strategy based on chaotic cubic mapping with exquisite internal structure is introduced to the LSO as the local exploitation, defined as follows [67], [68]: where x new i (t) and x i (t) are the state of the ith candidate solutions before and after mutation at the current iteration, respectively; η denotes the control factor of mutation, defined as follows: It should be noted that the value of η significantly affects the effectiveness of the chaotic mutation strategy. From Eq. (11), we can conclude that the value of η constructs an arithmetic progression with 1 as the first term, 0 as the last term, and −1/ (t max − 1) as the tolerance. η = 1 represents chaotic mutation plays the most vital role, while η = 0 represents chaotic mutation plays the least vital role. The chaotic sequences of z i are calculated as follows:

B. BOUNDARY MUTATION STRATEGY
Each GOP has various boundary constraints that limit the search space. When an individual moves out of the search space, an approach to managing boundary constraint violations will be employed. However, different boundary constraint approaches on the performance of the algorithm are different, which have been validated in works of literatures [71], [72]. When individuals move outside the predefined bounds of the multi-dimensional asymmetric search space, using the unified boundary constraint approaches to manage boundary constraint violations may lead to new boundary constraint violations or weak the search ability of individuals.
To solve this problem, based on the idea of a hierarchical GA, introduce the concept of the multilevel parallel, set the asymmetrical search spaces and the layer's parameters separately, and operate them in parallel style. If an individual's position violates the asymmetrical boundary for a specific dimension, it conducts mutation near the respond boundary and maintains the individual among the effective search space. Mathematically, the boundary mutation strategy is defined as: 2 , · · · , x max,D is the upper bounds of decision variables, while x min = x min,1 , x min,2 , · · · , x min,D is the lower bounds of decision variables, respectively. c is the parameter in the range [10 −2 , 10 −4 ].

C. FRAMEWORK OF CBLSO
The aforementioned subsections have described the main procedures of the LSO, the chaotic mutation strategy, and the boundary mutation strategy in detail, which compose the main components of the CBLSO. The main steps of the CBLSO are summarized as follows: In the initialization stage, a population with n lions are randomly generated by using Eq. (1). Then, set the numbers of the lion king, the lionesses, and the lion cubs. After evaluating the fitness value of each lion's position, the sorting, declaration, and random selection procedures are performed to sort the positions of lions in ascending order based on their fitness value and declare the lion king, the lionesses and the lion cubs. Then, the lionesses' pbest c and pbest m as the cooperation partner and the mother lioness followed by lion cub are selected, respectively. And the CBLSO turns to the main loop of the evolutionary process until the iteration value t or the number of function evaluations (denote as NFEs) reaches the predefined maximum iteration value t max or the maximum NFEs (denote as maxNFEs), defined as maxNFEs = n × t max .
During the evolutionary stage, the LSO search is first conducted. The values of the related parameters, i.e., α f , step, and α c are calculated according to Eq. (5), Eq. (6), and Eq. (8), respectively. The hunting behaviors of the lion king, the lionesses, and the lion cubs are updated by using Eq. (3), Eq. (4), Eq. (7), and the boundary mutation strategy illustrated in Section 3.2 is adopted at the same time. After the positional information for each lion is renewed, the fitness values of new lions are evaluated. Once the absolute value of difference for the lion king between tth iteration and t + 3th iteration is smaller than the predefined value (denote as u), usually is considered in the range [10 −6 , 10 −4 ]. The chaotic mutation strategy and the boundary mutation strategy introduced in Section 3.1 and Section 3.2 are called, respectively. Then, the fitness values of the mutant solutions are computed and the lionesses pbest c and pbest m , the values of related parameters, i.e., α f , step 1 , α c , and step 2 are updated again. Moreover, the lion king, the lionesses, and the lion cubs are sorted with 10 increments in the iteration. The above evolutionary phase will be repeated until the predefined termination condition is satisfied. At the end of the algorithm, the lion is reported as the final near-optimal solution. For the sake of clarity, the pseudo-code of the CBLSO is outlined in Fig. 2. and the complete flowchart of the CBLSO is described as Fig. 3.

D. EQUATIONSTIME COMPLEXITY ANALYSIS
Any meta-heuristic optimization algorithms should have less computational complexity so that the real-world optimization problems can be solved in less computational efforts.  To investigate the effectiveness of the CBLSO on optimization problems, the time complexity analysis between the CBLSO and the LSO is carried out. The population size n is analyzed in the time complexity. The corresponding time complexity of each procedure in the CBLSO in terms of the worst-case of the computational time can be calculated as follows: (1) Initializing the parameters of the CBLSO takes the time complexity O(1).  (4) Sorting the population into a three-layered hierarchical population structure (i.e., the lion king, the lionesses and the lion cubs) costs O(nlogn).
(5) The pbest individual and the gbest individual are recorded. The time complexity of this operation at most is 2O(n).
(6) The cooperation partner and the mother lioness followed by lion cub are selected, respectively. This process at most requires 2O(n).  (12) Updating the pbest individual and the gbest individual as well as the cooperation partner and the mother lioness followed by lion cub, which needs 2O(n) and 2O(n).
(13) Sorting the population into a three-layered hierarchical population structure (i.e., lion king, lionesses and lion cubs) again. This process costs O(nlogn).
Consequently, the total time complexity of the CBLSO under the termination criterion is calculated as follows: To be simplified, the total time complexity can be regarded After calculating the time complexity of the LSO, we find that it possesses the same time complexity O (T · n log n) + O (t max · n). The results show that the time complexity of the LSO and the CBLSO is identical, indicating that the chaotic mutation strategy and the boundary mutation strategy not only improve the performance of the CBLSO but also does not decrease its computational efficiency.

IV. EXPERIMENTAL RESULTS AND ANALYSIS
In this section, a series of experimental studies are performed for rigorous assessing the performance of the CBLSO. Firstly, the related preparation works about the simulations are summarized, including the well-known classic test functions and the corresponding parameter settings. Secondly, to show the effect of the chaotic mutation strategy and the boundary mutation strategy introduced in CBLSO algorithm. The performance of the CBLSO is compared against three LSO variants: (1) the CLSO that only uses the chaotic mutation strategy and (2) the BLSO that replaces the unified boundary constraint approach with the boundary mutation strategy, while (3) the LSO uses no strategy at all, making it a standard LSO. Thirdly, the performance of the CBLSO is compared with some state-of-the-art optimization algorithms (GA [23], PSO [29], [30], ABC [31], CS [32], GSA [40], GWO [34], ASO [41], [42], and SMA [54]). Finally, in order to verify the optimization performance of the CBLSO further, more state-of-the-art algorithms are compared with CBLSO on CEC2014 test suite.

A. TEST FUNCTIONS AND EXPERIMENTAL SETUP
From the NFL theory [57], we can conclude that if we compare two searching methods or algorithms with all possible functions, the performance of any two algorithms will be, on average, the same. However, if the size of the functions used for a test is too small, it will be impossible to give a generalized and meaningful conclusion. Meanwhile, it also has a potential risk that the method is biased (optimized) towards the small size of test functions used for a test while such bias might be meaningless for other problems of interest [73]. In this study, a large enough set of test functions without any equality or inequality constraints, including 56 test functions collected from the related works of literature, are used for evaluating the performance of CBLSO. The detailed description of the set is given in the Appendix. These test functions can be classified into four categories on the basis of their functional features: Uni-modal (U), Multi-modal (M), Separable (S), and Non-separable (N The values of the common control parameters for the CBLSO and other algorithms used for comparison are population size, maxNFEs or t max . Different population size and maxNFEs or t max are set as for different experiments. Moreover, to balance the exploration and the exploitation of the CBLSO, c = 10 −2 and u = 10 −5 are adopted for all experiments as in [63] and [68]. In the following subsections, the effects of the chaotic mutation strategy and the boundary mutation strategy are shown first, and then, to validate the performance of the CBLSO in a convincing way, we compare the CBLSO with some other state-of-the-art algorithms. For each experiment, the best results are identified with boldfaced through the paper. All the experiments are implemented on the environment of Microsoft Windows 10 (64 bit) on Intel (R) Core (TM) i7-8700K CPU with 3.20 GHz 3.19 GHz and 64.0 GB (RAM). The programming software is MATLAB R2018b, and the software for Wilcoxon Signed Ranks Test (WSRT) is IBM SPSS Statistics 23.

B. EFFECTS OF THE IMPROVED STRATEGY
In this section, the advantages of the chaotic mutation strategy and boundary mutation strategy introduced in the CBLSO algorithm are investigated by a simulation experiment. Three variants of the CBLSO called CLSO, BLSO, and LSO are adopted for comparison. Due to the space limitation, only 23 test functions shown in the Appendix is tested in this section, these functions include three US functions, i.e., F01, F02, and F04, four UN functions, i.e., F13, F14, F15, and F16, three MS functions, i.e., F18, F21, and F23, thirteen MN functions, i.e., F28, F30, F34, F35, F36, F37, F38, F41, F42, F43, F44, F45, and F46. For all algorithms, to ensure the convergence rate of CBLSO, CLSO, BLSO and LSO, the population size is respectively set to be 50 for all experiments. The maximum number of iterations is fixed to 4,000.
To make a comparison, all experiments are repeated for 30 independent runs and the results, including mean best solutions (Mean) and standard deviations (Std), are all summarized in Table 1. From Table 1, on the US functions, we can find that CBLSO provides the best results in terms of the Mean values and the Std values, followed by CLSO and BLSO, suggesting the positive effect of introducing the chaotic mutation strategy and the boundary mutation strategy in our CBLSO framework. In detail, all of the four algorithms have equal performance on functions F01 and F02. On F04, all algorithms cannot obtain a global optimum, but the Mean value provided by CBLSO outperforms the results of the other three algorithms, while the results obtained by the LSO and the BLSO are better than the results of the CLSO. The reason may be that F04 (Quartic function) is padded with noise and the random noise makes sure that the algorithm never gets the same value on the same point. For this function, the chaotic mutation may weaken the exploration ability of algorithm but combining the chaotic mutation and the boundary mutation can enhance the exploration ability of the algorithm. On the UN functions, we can see that on functions F13, F14, and F15, all algorithms have equal performance, whereas CBLSO cannot perform the best for F16. For this function, CLSO performs the best both in terms of the Mean and the Std. The reason may be that F16 (Rosenbrock function) is a function with a narrow curving valley. For this function, the boundary mutation is unfavorable for the algorithm to explore the space properly and keep up the direction changes, while the randomness of the chaotic system makes it more favorable for improving the exploration ability of the algorithm. However, it demonstrates that the chaotic mutation strategy and the boundary mutation strategy used in CBLSO indeed improves the performance of LSO. It also manifests that CLSO performs better than LSO and BLSO on F16 in terms of the Mean, which shows that the chaotic mutation strategy achieves better influences than the boundary mutation strategy. On the MS functions, all of the four algorithms show equal performance on F21, while CBLSO outperforms LSO, CLSO, and BLSO on all the remaining functions. On the MN functions, the results reported in Table 1 show that CBLSO performs the best for all MS problems. For all algorithms, there is no difference on functions, i.e., F30, F34, F36, F37, F38, F43, and F44 in terms of the Mean values. Besides, when we pay attention to Table 1, it is apparent that CBLSO has a good performance in comparison with the other algorithms on US functions but its performance decreases on the UN, MS, and MN functions. This is because US functions and UN functions are the easiest test problems that have no local optimum and there is only one global solution, and they are often used to examine the convergence rate of optimization algorithms. However, MS functions and MN functions usually have many or a few local optima and they have often employed the ability of the algorithms to escape from poor local optima and obtain the near-global optimum. Different algorithms have different advantages in dealing with these functions. Thus, the performance of the algorithm on solving these functions is different.
The maximum number of iterations is set to be 6,000. For every benchmark function, all algorithms are repeated for 30 independent runs. The experimental results of Mean, Std, and standard errors of means (SEM) are presented in Tables 2-5.
Firstly, we describe the results for US functions. It is apparent from Table 2 that the CBLSO is the most competitive algorithm that provides the best Mean values for all test functions, and then they are GWO and SMA that obtain the best Mean values on 3 of 4 test functions. GA, ABC, CS, GSA, and ASO perform similarly on function F01, while PSO gives poor results in terms of this indicator. It is also shown that, on F04, all algorithms cannot give a global optimum but the results provided by CBLSO outperform the results of the other six algorithms.
Next, we analyze the simulation results for UN functions ( We now pay attention to MS functions (Table 4). Regarding the best Mean values, SMA is the best algorithm providing the best Mean values on 7 out of 10 functions, followed by CS, GWO, and CBLSO, which obtains the best Mean values on 5 functions. PSO and ABC can be the third-best algorithm, which performs the best Mean values on 4 functions. And then GSA, ASO, and GA are the worst best algorithm which give the best Mean values only on 3, 3, and 2 functions, respectively. For all algorithms, it is clear from the results listed in Table 4 that there is no evident difference on F25. Only on 3 functions (F23, F26, and F27) that GA outperforms CBLSO but CBLSO outperforms GA on 6 functions (F18 to F22 and F24). On 3 functions (F18, F19, and F25) CBLSO and PSO show equal performance but on 5 of the remaining 7 functions CBLSO outperforms PSO while PSO outperforms CBLSO only on 2 functions (F20 and F27). The number of functions that CBLSO and CS show equal performance is the same as the number of CBLSO and CS show equal performance, but on 4 of the remaining 7 functions CBLSO outperforms CS while CS outperforms CBLSO only on 3 functions (F20, F26, and F27). The results between CBLSO and ABC as well as the results between CBLSO and GSA are identical to the results between CBLSO and CS as well as the results between CBLSO and PSO, respectively. It is evident from the results that CBLSO and GWO show equal performance on 5 functions but 4 of the remaining 5 functions CBLSO performs better than GWO while GWO performs better than CBLSO only on 1 function (F20). On functions (F18, F19, and F25     To intuitively analysis the performance of the CBLSO algorithm for solving Uni-modal problems and Multi-modal problems, the convergence progress of the average best-so-far solutions over 30 independent runs are plotted in Figs. 4, 6, 8, and 10, while the box-and-whisker diagrams of obtained optimal solutions over 30 independent runs are depicted in Figs. 5, 7, 9, and 11. Due to the space limitation, some representative curves of them are selected for illustration. We first take a look at the convergence process graphs of average best-so-far solutions (Figs. 4, 6, 8, and 10). For Uni-modal problems, which usually have no local optimums, the convergence rate of the algorithms is more meaningful than the final optimization results, because there are some other methods or algorithms that designed for optimizing Uni-modal problems, specifically. From Fig. 4 plotted on US problems (including F02, F03, and F04), we can see that CBLSO converges quickly thorough the whole stage of search process towards the best solution, while other eight algorithms converge slowly and finally traps into the local minimum in the late stage of search process, indicating that CBLSO possesses superior performance on such optimization problems. Different from the convergence characteristic shown in Fig. 4, according to Fig. 6 plotted on UN problems (including F06, F08, F10, F12, F13, and F16), it is observed that the convergence speed in the early stage of search process of CBLSO is not remarkable among all the nine algorithms, but it finally obtains the global optimal minimum in the late stage of search process, implying that CBLSO can effectively avoid a premature convergence and significantly improve the accuracy of solution. Thus, we can conclude that the chaotic mutation strategy and the boundary mutation strategy used in CBLSO indeed improve the algorithm's exploration and exploitation abilities. For multi-modal problems, which usually have many local optimums and almost are often difficult to be optimized, the final optimization results are usually more significant since the optimization results are the standard to measure the exploration and exploitation abilities of the algorithm in getting rid of local optimums and finding a better solution in a short time. From Fig. 8      we can see that the convergence speed of CBLSO is superior on F21, F22, F23, and F24. GA and PSO defeat CBLSO and other algorithms on F26 and F27 in the early stage of the search process but the solutions obtained by CBLSO are better than or similar to the solutions obtained by GA and PSO. And then according to Fig. 10 depicted on MN problems (including F29, F36, F42, F43, F45, F47, F49, F51, and F55), CBLSO defeats other eight algorithms in case of F36, F45, and F47 by providing very fast convergence speed and finally offers the best solutions. Although the convergence speed offered by CBLSO for other functions is not remarkable among all the nine algorithms, the solutions obtained by CBLSO are comparable.
We now analysis the box plots of optimal solutions obtained by each algorithm (Figs. 5, 7, 9, and 11). From Fig. 5 plotted on US problems (including F02, F03, and F04), it is observed that CBLSO maintains the fewer values and the shorter distribution of optimal solutions compared with the other peer algorithms on all problems, indicating its great superiority for solving US problems. Next, it can be seen from Fig. 7 plotted on UN problems (including F06, F08, F10, F12, F13, and F16) that the performance of CBLSO is satisfactory for all problems, especially on F16, CBLSO can provide a better optimal solution than other peer algorithms. Thus, based on the aforementioned studies, we can conclude that CBLSO is a more effective algorithm for optimizing Uni-modal problems. According to Fig. 9 on MS problems (including F21, F22, F23, F24, F26, and F27), we can find that CBLSO maintains the fewer values and the shorter distribution of optimal solutions compared with the other nine algorithms on F21, F22, F23, and F24. GA, CS, and ABC defeat CBLSO, PSO, GSA, GWO, ASO, and SMA in the case of F26 while GA defeats other eight algorithms in the case of F27. From Fig. 11 depicted on MN problems (including F29, VOLUME 10, 2022  F36, F42, F43, F45, F47, F49, F51, and F55), the complexity of the test problems is enhanced. We can find that CBLSO is superior on F29, F36, F42, F43, F45, F47, F49, and F51 whereas its performance is comparable to GA, CS, ABC, GSA, ASO, and SMA on F55, denoting a good and steady performance of CBLSO.

D. ROBUSTNESS ANALYSIS BETWEEN CBLSO AND OTHER NATURE-INSPIRED ALGORITHMS
The maxNFEs is set as 500,000 and other parameters are set to be the same as Section C. A trial is considered to be ''successful'' if the following inequality is satisfied [75]: where FOBJ ANAL indicates the known analytical minima, FOBJ BEST indicates the best function value provided by the method or algorithm. Control parameters of accuracy are considered to be ε rel = 10 −4 and ε abc = 10 −6 in all cases for the present work.
To compare the convergence speeds, we use the acceleration rate (AR) [76] which is defined as follows, based on the J. Liu, Y. Wu: Improved LSO Algorithm With CBLSO for Global Optimization where AR > 1 means CBLSO is faster. Note that, to minimize the effect of the stochastic nature of the algorithms on this metric, the reported NFEs for each problem is the mean value over 50 runs. The number of runs, for which the algorithm successfully reaches the accuracy of each test function is measured as the success rate (SR): SR = runs Success runs Total (19) where runs Success represents the number of runs that success, runs Total represents the total number of the independent runs. Also, the average acceleration rate (AR ave ) and the average success rate (SR ave ) over m test functions are calculated as follows: VOLUME 10, 2022 We compare the CBLSO algorithm with GA, PSO, CS, ABC, GSA, GWO, ASO, and SMA, in terms of convergence speed and robustness, and the results (i.e., NEFs, SR, AR, AR ave , and SR ave ) of 56 test problems are presented in Tables  6-7. Besides, to check whether CBLSO saves computation time, the results of the runtime (T) over 50 runs are also recorded in Table 7.
From the results in Table 6, it is clear that CS ranks the first in terms of average NFEs for all test problems, and then it is ABC. The CBLSO ranks the third. Although CS and ABC outperform our algorithm in terms of average NFEs for all test problems, the CBLSO ranks the first best results than the other six algorithms on NFEs for all test problems. As can be inferred from  21.5300, 239.9045, and 20.3052, respectively, which means CBLSO is faster than its competitors regarding to the overall average acceleration rates. Next, we pay attention to the SR and the execution time of 50 runs. As can be seen from Table 7 that GWO is the best algorithm in the case of average execution time and it also gets the first best results on execution time whereas our CBLSO ranks the sixth and the third, respectively. Moreover, when regarding the average SR and the best results on SR, CS performs the best (0.76 and 45), followed by our CBLSO algorithm (0.71 and 39). It also can be seen that there are 36 functions that CBLSO can solve 100% successfully (92.3% of the best results on SR), while the number for GA is 5 (71.4% of the best results on SR), for PSO is 23 (88.5% of the best results on SR), for CS is 40 (88.9% of the best results on SR), for ABC is 36 (94.7% of the best results on SR), for GSA is 27 (90.0% of the best results on SR), for GWO is 28 (96.5% of the best results on SR), for  ASO is 33 (82.5% of the best results on SR), and for SMA is 37 (92.5% of the best results on SR),. Thus, although GWO is better than our algorithm in terms of execution time, CBLSO achieves much better results than GWO in terms of Mean values, which can be seen from Tables 2-5. Besides, although CS has a superior performance in terms of SR and execution time, the percent of solving 100% successfully on problems of CBLSO is higher than CS. On the whole, the CBLSO is VOLUME 10, 2022 an effective algorithm with an acceptable convergence speed and stronger robustness.

E. COMPREHENSIVE SIGNIFICANCE ANALYSIS
The comparison of algorithms does not guarantee the efficacy and superiority of the proposed algorithm. The possibility of getting good results, by chance, cannot be ignored. To provide more accurate conclusions, a non-parametric statistical test named Wilcoxon Signed-Rank Test (WSRT) [77], [78] is used to better compare the overall performance of the algorithms. WSRT is a pair-wise test that aims at detecting the statistically significant difference between the behaviors of two algorithms. The null hypothesis H 0 for a two-sided test is: ''there is no difference between the median of the solutions provided by algorithm A and the median of the solutions provided by algorithm B for the same test function'' [77], [78]. To determine whether algorithm A achieves a statistically better solution than algorithm B, or if not, whether the alternative hypothesis is valid, the sum of the ranks by produced by WSRT is examined. When using WSRT, the R + and R − regarding the comparisons between the two algorithms are computed firstly. Once the R + and R − have been achieved, the p-values can be computed. Usually, there are two types of WSRT: single-problem statistical analysis and multi-problem statistical analysis. In this part, WSRT at a significant level of α = 0.05 is used for multi-problem statistical analysis.
For single-problem statistical analysis, the Mean values of 30 runs for each test problem presented in Tables 2-5 are used for the sample data. The single-problem-based statistical comparisons between the CBLSO algorithm and one of the referred algorithms by WSRT as well as the corresponding statistical results for each test problem in 30 runs are shown in Tables 2-5, respectively. In these tables, '+' means the case in which the null hypothesis is rejected and CBLSO shows a better performance in the single-problem-based statistical comparisons tests at 95% significance level (α = 0.05); '-' means the case in which the null hypothesis is rejected and CBLSO shows a worse performance; and '=' means a case in which no statistically significant difference between LSO and the other algorithms exists. From the results in Tables 2-5, we can see that the CBLSO significantly performs better than GA, PSO, CS, ABC, GSA, GWO, ASO, and SMA on the US problems. For the UN problems, the CBLSO significantly outperforms other peer algorithms except for SMA. For the MS problems, CBLSO significantly improves other peer algorithms and achieves comparable results compared with SMA. For the MN problems, the performance of LSO is not inferior to that of GA, PSO, GSA, and GWO, and it seems that CBLSO is as competitive as other peer algorithms.
For multi-problem-based statistical analysis, the Mean values of the best solutions of 30 runs for each test problem are used for the pair-wise comparisons. In this part, fiftysix Mean values of all benchmark functions in Tables 2-5 are used for the sample data of WSRT. The multi-problembased statistical comparisons between the CBLSO algorithm and one of the referred algorithms by WSRT as well as the corresponding statistical results for each test problem in 30 runs are shown in Table 8. According to Table 8, we can find that for US and MS problems, there is no statistically significant difference between CBLSO and its competitors. For UN problems, the CBLSO is statistically significantly better than GA and GWO, and performs as competitive as PSO, CS, ABC, GSA, ASO, and SMA. For MN problems, CBLSO performs significantly better than GA, PSO, and GSA, and shows no statistically significant difference between CBLSO and ABC, GWO, ASO, and SMA, although the performance of CBLSO is inferior to that of the CS. As for all problems, the CBLSO shows an improvement over GA, PSO, GSA, and GWO, and achieves comparable results compared with CS, ABC, ASO, and SMA. Thus, we can conclude that CBLSO is a superior and competitive algorithm for numerous functions.

F. FURTHER COMPARISON ON CEC2014 TEST SUITE
In this section, the performance of CBLSO is further compared with some renowned algorithms on 30 benchmark functions from the CEC2014 competition with D = 30 [79]. These functions consist of Uni-modal problems (FC01-FC03), Multi-modal problems (FC04-FC16), Hybrid and Composite type with varying difficulty levels (FC17-FC22 and FC23-FC30). Specifically, the range of the search space is in [−100, 100] d . The algorithms used here are listed in Table 9. In the case of the problems with D = 30, the population size is set to 50 and the maxNFEs is set to D * 5000. The CBLSO algorithm is run independently 30 times for each benchmark function, and the values of Mean, Std, and best solution (Best) of different algorithms are shown in Table 10.   Note that the fitness function is the error between the real optimal solution and the solution obtained by the algorithm. Assume that the real global optimal solution is x * , and the best solution provided by the optimization algorithm is y * . Then, |f (y * )−f (x * )| is selected to be the fitness function [80]. In Table 10, the data for these compared algorithms are provided by Chen et al. [80] and Nenavath et al. [81]. The best results are shown in bold. Table 10 shows that CBLSO performs best for functions FC23, FC24, FC25, FC27, FC28, FC29, and FC30 according to the value of Mean. jDE performs best for functions FC03, FC04, FC08, FC10, FC20, FC21, FC25, and FC26 according to the value of Mean. SaDE performs best in terms of the value of Mean for functions FC02, FC06, FC07, FC18, FC19, and FC26. The performance in terms of Mean with LBSA is better than the others for functions FC09, FC12, FC14, FC16, and FC22, while SCA-PSO is better than the others for functions FC01, FC11, and FC17. For function FC13, the best value of Mean is provided by BSA. For function FECE15, FDR-PSO has the smallest mean value among all algorithms. The three TLBOs, SCA-PSO and CBLSO provide the same mean value for function FC24, which is the smallest of the twelve algorithms. Considering   the best solutions, CBLSO performs best for functions FC23,  FC24, FC25, FC26, FC27, FC28, FC29, and FC30. jDE  performs best for 9 functions (FC02, FC03, FC05, FC08,  FC10, FC18, FC20, FC25, and FC26), while SaDE performs  best for 6 functions (FC02, FC06, FC07, FC21 and FC26). For functions FC05, the jDE, BSA, LBSA, and SCA-PSO have the same best solutions, which is the smallest of the twelve algorithms. For functions FC24, the best solutions obtained by TLBO, ETLBO, DGSTLBO, and CBLSO can produce are the smallest. For function FC25, jDE, TLBO, ETLBO, DGSTLBO and CBLSO provide the smallest best solutions among all algorithms, while for function FC26, all the algorithms can obtain the best solutions. FDAR-PSO is better than other algorithms in terms of the best solutions for functions FC15, FC16, and FC19. ETLBO outperforms other algorithms in terms of the best solutions for function FC14, BSA outperforms other algorithms in terms of the best solutions for function FC12 and FC13, LBSA outperforms other algorithms in terms of the best solutions for function FC04 and FC09, and SCA-PSO outperforms other algorithms in terms of the best solutions for function FC01, FC11, and FC22. When we pay attention to the value of Mean of the twelve algorithms for 30 functions, we can find that CBLSO, jDE, and SaDE, rank in first place for seven functions, BSA and SCA-PSO for six functions, ETLBO for three functions, BSA for two functions, and FDR-PSO, TLBO, and DGSTLBO for one function. The statistical analysis results conducted by the multi-problem-based WSRT at a significant level of α = 0.05 in Table 11 show that CBLSO significantly outperforms FDR-PSO, FIPS, CLPSO, jDE, SaDE, ETLBO, DGSTLBO, BSA, and LBSA in terms of p-value, although the statistical results between CBLSO and TLBO, CBLSO, and SCA-PSO are not significant, suggesting that CBLSO is very effective. Thus, it can be concluded that CBLSO can show strong and competitive performance compared with other algorithms.

V. REAL APPLICATION ON OPTIMAL DISPATCH OF CASCADE HYDROPOWER STATIONS
The problem considered in this study is called the optimal dispatch of cascade hydropower stations. Traditional optimization methods or algorithms may face trouble with problems that having many equality or inequality constraints as well as interdependent relationships between decision variables and may require more NFEs [91]. Many water resources and hydrological optimal dispatch problems often involve many equality or inequality constraints as well as interdependent relationships among decision variables. Moreover, the size of the decision variables is often large. This paper utilizes these aspects to investigate the performance of CBLSO in tackling these problems.
In this section, the optimal dispatch problem of Wujiang cascade hydropower stations in Guizhou province, southwest of China [91], is employed to evaluate the great potential of CBLSO for real application. In this problem, there is often interdependence among one or more decision variables and    has a series of equality and inequality constraints, which is very representative to validate the feasibility and effectiveness of CBLSO for solving real problems. Finally, the results obtained by CBLSO are validated and compared statistically with other well-known optimization algorithms, i.e., GA, variant of PSO, improved CS and standard LSO.

A. WUJIANG CASCADE HYDROPOWER STATIONS
The Wujiang River lies to the right bank of the upper Yangtze River and is the largest river in Guizhou province, southwest of China. The mainstream length of the Wujiang River is 1,037 km and the drainage area is 87,920 km 2 and 66,849 km 2 is in the district of Guizhou province, which accounts for 76% of the drainage area. The index map of the Wujiang River basin is presented in Fig.12. The rainfall of the Wujiang River is quite abundant and the annual average rainfall varies between 900 and 1,400 mm. Because of the representative continental monsoon climate, the rainfall presents a typical season characteristic that about 80% of the annual rainfall centralizes from April to August [92].
The Wujiang cascade hydropower station is one of thirteen large power generation bases in China. In this paper, 4 cascade reservoirs: Hong jiadu (HJD) reservoir, Dong feng (DF) reservoir, Suo fengying (SFY) reservoir, and Wu jiangdu (WJD) reservoir are selected as the object of the case study shown in Fig.12. Since the SFY is a daily regulation reservoir and the regulation storage is too small, the optimal dispatch model used in this paper is only applicable to the optimal dispatch of the HJD, DF, and WID reservoir are shown in Fig.13.

B. MODEL FORMULATION
The objective function of cascade hydropower stations adopted here is to maximize the annual energy production of the cascade hydropower stations, subject to some constraints (including technical and physical). Because the water level has an important impact on the storage capacity of the reservoir, therefore, the change of the outflow is indicated by the variation of the water level. Moreover, the output of hydropower station goes hand in hand with turbine release and hydraulic head determined by the water level of the period, dynamically. Based on the above discussion, once we can determine the upstream water level, we can calculate the outflow through Eq. (27). Furthermore, the downstream water level goes hand in hand with the outflow of the reservoir, if we enhance the outflow, the downstream water level will be higher, thus will lead the hydraulic head to be lower [91]. The objective function is expressed as follows: where E represents the annual energy production for the cascade hydropower stations (kW·h); S num is the number for the hydropower stations; T is the size for the dispatch period; t denotes the current period; i denotes the current station; A i is the power coefficient for the station i; H i,t is the average  hydraulic head for the station i in period t(m); Q i,t is the outflow for the station i in period t(m 3 /s); t is the dispatch period (h); N i,t is the output of the station i in period t(kW).

C. CONSTRAINTS
In the cascade hydropower stations, the entire cascade hydropower stations should be subject to the constraints introduced by the interaction of hydropower stations or reservoirs, while each hydropower station should be subject to their constraints [93]. Specifically, the constraints can be expressed as follows: (1) Water level constraint where Z i,t denotes the water level for the reservoir i in period t (m); Z i,t andZ i,t are lower and upper bounds of water level for the reservoir i in period t (m), respectively.
(2) Power output constraint For each hydropower station where N i,t andN i,t are lower and upper bounds of output for the station i in period t (kW), respectively; N t andN t are lower and upper bounds of output for the entire cascade hydropower station in period t (kW), respectively.
(3) Outflow constraint where Q i,t andQ i,t are lower and upper bounds of outflow for the reservoir i in period t (m 3 /s), respectively. (4) Water balance equation where V i,t is the volume for the reservoir i storage at the beginning of period t (m 3 );V i,t+1 is the volume for the reservoir i storage at the end of period t (m 3 ); I i,t is the inflow for the reservoir i in period t (m 3 /s); Q i,t is the outflow for the reservoir i in period t(m 3 /s). (5) Hydraulic connection equation where α f is the inflow for the reservoir t +1 in period t(m 3 /s); Q i,t is the outflow for the reservoir t + 1 in period t (m 3 /s); q i,t is the interval inflow into the reservoir t + 1 in period t (m 3 /s). (6) Boundary constraint where Z i,b is the initial water level for the reservoir i(m); Z i,e is the final water level for the reservoir i(m). Water levels, power outputs, reservoir inflows, and reservoir outflows are all variables that will be calculated, Historical monthly inflow sequences of reservoirs, curve of reservoir water level-reservoir storage capacity, curve of reservoir outflow-reservoir downstream water level and values of control parameters that used for reservoirs or hydropower stations are all known. Furthermore, we also assume that the generality loss is ignored and the evaporation loss of water can be canceled out by rainfall.

D. CONSTRAINTS HANDLING METHOD
The critical step in developing a successful application for efficiently solving the optimal dispatch of cascade hydropower stations is appropriately handling equality constraints and inequality constraints. In this section, to effectively handle the complex constraints of the optimal dispatch of cascade hydropower stations without degrading the optimization method's computation efficiency, a novel constraint handling method according to the characteristics of the different constraints and the different power generation mechanisms of hydropower stations or reservoirs is designed.
Water level constraints shown in Eq. (23) are inequality constraints. For these constraints, the handling method considering the feasible boundaries is carried out as follows: Power output constraints shown in Eqs. (24) and (25) and outflow constraints shown in Eq. (26) are inequality constraints. For power output constraints Eq. (24) and outflow VOLUME 10, 2022 131290 VOLUME 10, 2022 Through the designed constraint handling method, it is possible to guarantee that the feasible solutions always have priorities than the infeasible solutions. What's more, when the solutions are out of the feasible regions, Eq. (32) can effectively evaluate the distance between the infeasible solutions and the bounds of the feasible regions, allowing the search to move quickly to the feasible regions.

E. SIMULATION SCENARIOS
Suppose the four reservoirs studied here are considered as a whole. The guaranteed power output for Wujiang cascade hydropower station should be higher than the minimum of the power output of the entire cascade hydropower station (680 MW). Because the interval inflow is quite abundant, the ecological runoff will be negligible. The basic characteristics of the Wujiang cascade hydropower station are listed in Table 12.
The simulation about the optimal dispatch of Wujiang cascade hydropower station for three scenarios is done on a monthly basis. The dispatch periods are 12 months, which range from May of the current year to April of the next year. Based on the hydrological frequency analysis of historical monthly inflow sequence data, we choose three typical years: 75% guaranteed rates of water supply (from 1951. 5 to 1952.4), name as the wet year (Scenario 1), 50% guaranteed rates of water supply (from 1985.5 to 1986.4), name as the normal year (Scenario 2), 25% guaranteed rates of water supply (from 1963.5 to 1963.4), name as the dry year (Scenario 3).

F. PARAMETER SETTINGS
The CBLSO along with the comparison algorithms include GA [23], PSO [29], [30], ABC [31], CS [32], GSA [40], GWO [34], LSO [35], ASO [41], [42], and SMA [54] are applied to solve the optimal dispatch problem of cascade hydropower stations. For all algorithms, the population size n is set to be 100, and the maximum number of iterations t max is set to be 60,000. The results composed of best (Best), average (Average), worst (Worst), and standard deviation (Std) values for 10 independent runs on three scenarios (wet year, normal year, and dry year) are presented in Table 13, respectively. In addition, the details of default parameter settings for GA, PSO, ABC, CS, GSA, and GWO, ASO, and SMA are kept the same as in Subsection IV. The details of default parameter settings for LSO, and CBLSO are recommended by [64].

G. RESULTS ANALYSIS AND DISCUSSION
From Table 13, we can find that, compared with other peer algorithms, the Average and Std values provided by the CBLSO algorithm are quite competitive, indicating its effective and steady performance. In detail, CBLSO presents the Average values of generation on three scenarios are 122.5168, 103.8529, and 99.7140 (10 8 kW·h), respectively. By observing Table 13, we can find that the results on three scenarios obtained by CBLSO are also obviously better than the results provided by LSO. In addition, the worst solutions obtained by CBLSO on three scenarios are also relatively the least, indicating that the premature convergence's influence on CBLSO is limited owing to its effective structure and updating strategies. What's more, it is worth mentioning that the inflow of Scenario 3 is limited, making it more challenging to allocate the water resources rationally than Scenarios 1 and 2. However, the results yielded by CBLSO are comparable to the results yielded by other comparison algorithms, suggesting the superior search ability of CBLSO. To arrive at more precise conclusions, the WSRT at a significant level of α = 0.05 is used for statistical analysis is conducted, and the statistical results are shown in Table 14. It can be seen from this table that although CBLSO cannot outperform GA, CS, ABC, GSA, GWO, and ASO on Scenario 1, GA, PSO, CS, ABC, GSA, GWO, and ASO on Scenario 2, and GA, GSA, GWO, and ASO on Scenarios 3, CBLSO shows a statistical significance over SMA on the three scenarios, CS and ABC on Scenario 3, and, achieves comparable results with PSO on Scenario 3 and Scenario 3, with LSO on all three scenarios. implying that CBLSO is capable of performing comparably well to other comparison algorithms.
To intuitively show each optimization algorithm's convergence characteristic and check whether the constraints of the problem are met or not, the average best-so-far power generations for three scenarios (wet year, normal year, and dry year) of all algorithms, the monthly mean water level processes of 10 schemes for the HJD reservoir, the DF reservoir, and the WJD reservoir obtained from three scenarios (wet year, normal year, and dry year) of CBLSO are plotted in Fig. 14 and Fig. 15, respectively. Meanwhile, to present the rationality and the correctness of the operation results, the monthly water level processes, the monthly power output processes, the monthly power generation processes, the discharge and power output processes, the monthly reservoir storages of each reservoir, and the stack of colors of monthly power out for each reservoir on 3 representative schedule results (i.e., scheme 1, scheme 5, and scheme 10) obtained by DMSDL-HHO are shown in Figs. 16-21, respectively. Firstly, we describe the results obtained in the convergence graphs of the average power generation (Fig. 14). From   Fig. 14, we can find that CS, ABC, GWO, and ASO converge slowly in the early stage of the search process for all three scenarios, while CBLSO presents a different convergence characteristic, i.e., it consistently converges in the early stage of the search process and finally achieves the best solution in the early half stage of the search process for all three scenarios, indicating that it is more effective. In particular, the convergence speed of CBLSO is faster than almost all other peer algorithms in the early stage of the search process. Thus, we can conclude that CBLSO can effectively alleviate the premature convergence and significantly enhance its optimization performance during its execution compared with other peer algorithms.
Next, we compare the algorithms in terms of the monthly water level processes obtained by CBLSO and comparison algorithms (Figs. 15 and 16). From Figs. 15 and 16, we can find that the water levels of each reservoir all locate in the rational ranges, indicating that the constraint handling method designed in this study can meet the practical requirement of the multi-reservoir system. It also can be seen from Figs. 15 and 16 that the water level of the HJD reservoir decreases suddenly and then gradually rises in the initial stage of the dispatch periods. This is because the HJD reservoir's inflow in the initial stage of the dispatch periods is quite VOLUME 10, 2022 limited (only 88 m 3 /s). To meet the guarantee power output requirement of the whole cascade system (680 MW), the HJD reservoir must increase the outflow to compensate the downstream reservoirs, which results in a lower water level of itself. However, in the following dispatch periods, the inflows of the three reservoirs are abundant enough. In the precondition of meeting the guarantee power output requirement of the whole system, the HJD reservoir's outflow can be decreased to increase the storage volume of its own. Thus, the reservoir can use its storage volume to generate more energy in the following dispatch periods. In addition, it also can use the water stored during the wet seasons to compensate for the downstream reservoirs during the dry seasons. Meanwhile, from Fig. 16, we can see that the water levels of the three reservoirs are nearly identical for the wet year and are slightly different for the normal year and the dry year. The reason may be that the water supply of the wet year is abundant, which is helpful for optimal dispatch, but the water supply of the normal year and the dry year is relatively short, which make it more difficult to allocate the water resources than the wet year.
Thirdly, we pay attention to the results obtained in the power output processes and the power generation processes (Figs. 17 and 18). The optimal results indicate that the power output and the power generation on April, July, and August for the wet year, on April, June, and July for the normal year, and on June, July, and August for the dry year are near zero. The reason may be that the HJD reservoir's inflows in these dispatch periods are abundant, the reservoir needs to decrease the outflows of itself to rapidly increase its storage volume, which not only can utilize the higher hydraulic head to generate more output and more power generation during the following dispatch period but also can utilize the stored water resources to compensate for the downstream reservoirs in the dry periods. This phenomenon can also be explained by the variations in the water levels shown in Figs. 15 and 16. We now analyze the results obtained in the discharge and power output processes (Fig. 19). From Fig. 19, we can find that the water discharge rate and the power output are all within the boundaries of constraints, suggesting that the proposed constraint handling method is reasonable and effective. In addition, the changes of the water discharge rate processes and the power output processes are all following the variations of the water levels, the power out, and the power generation, which further indicating the rationality and effectiveness of the proposed constraint handling method.
Finally, we compare the results obtained in the monthly reservoir storages and the Stack of colors of monthly power output for each reservoir (Figs. 20 and 21). It is shown by Fig. 20 that the monthly reservoir storages of the 3 schemes for Wet year, Normal year, and Dry year are all within the boundaries of the constraints. What's more, the results displayed in Fig. 21 show that the monthly power outputs all meet the whole system's guarantee power output requirement. That is to say, the dispatch results provided by CBLSO are reasonable, and the constraint handling method proposed in this study is effective.

VI. CONCLUSION
In this paper, a novel LSO with a chaotic mutation strategy and boundary mutation strategy (CBLSO) is presented to enhance the performance of LSO from the viewpoint of mutation strategy. Since some shortcomings of the standard LSO, such as premature convergence and limited global search ability, CBLSO is designed to address these issues. In this proposed algorithm, a chaotic mutation strategy based on chaotic cubic mapping is used to effectively improve the exploration ability for avoiding premature convergence and a boundary mutation strategy based on the concept of multilevel parallel is designed to manage boundary constraint violations for keeping and maintaining the population diversity. A comparative experiment is implemented to explore the effectiveness of the chaotic mutation strategy and the boundary mutation strategy. When compared with three variants of LSO algorithms (standard LSO, CLSO, and BLSO), the experimental results demonstrate that the effect of the chaotic mutation strategy is more evident than the effect of the boundary mutation strategy but introduces the two strategies in our CBLSO can achieve better positive effects.
The CBLSO is evaluated on 56 test functions and compared with 8 state-of-the-art algorithms: GA, PSO, CS, ABC, GSA, GWO, ASO, and SMA. The experimental results in terms of some often-used performance indicators and statistical analysis illustrate that our CBLSO is the most salient algorithm since it performs best on most of the test problems. The experiment results of the robustness analysis, the comprehensive significance analysis, and the study of high-dimension functions demonstrate that CBLSO can provide comparable results with remarkable convergence behavior compared to the other reported algorithms. Sequentially, the time complexity analysis of CBLSO shows that it is the same computational efficiency in contrast to standard LSO. Moreover, the results obtained by CBLSO on CEC 2014 are further compared with those provided by some well-known algorithms. It is shown by the experiment results that the proposed CBLSO performs better than or at least comparable to other algorithms and can be taken as a promising alternative optimization tool to solve GOPs. Finally, CBLSO is applied to the optimal dispatch problem of cascade hydropower stations to investigate its great potential for real-world applications. The obtained results show that the algorithm can produce competitive results when compared with standard LSO and other outstanding algorithms.
Summarizing, according to the test problems, the robustness analysis, the comprehensive significance analysis, the comparative experiments and the real-world application, the CBLSO is the most promising algorithm in our study. In our future work, the following future research can be focused on: (1) LSO can be further improved according to employ other strategies or operators used by other algorithms. (2) Other real-world engineering and practical problems can be used to determine whether the CBLSO can present good results with the parameter settings provided in this study. (3) Other heuristic algorithms should be reinforced by the chaotic mutation strategy and boundary mutation strategy used in this paper. (4) The CBLSO can be developed for solving optimization problems with nonlinear constraints. (5) Extending the proposed CBLSO for handling multi-objective optimization problems and many-objective optimization problems is also meaningful work. The coefficients used in Hartman 3,4 function and Hartman 6,4 function can be seen in [64].