Optimization of Cascade Reservoir Operation for Power Generation, Based on an Improved Lightning Search Algorithm

Abstract

Cascade reservoir operation can ensure the optimal use of water and hydro-energy resources and improve the overall efficiency of hydropower stations. A large number of studies have used meta-heuristic algorithms to optimize reservoir operation, but there are still problems such as the inability to find a global optimal solution and slow convergence speed. Lightning search algorithm (LSA) is a new meta-heuristic algorithm, which has the advantages such as high convergence speed and few parameters to be adjusted. However, there is no study on the application of LSA in reservoir operation. In this paper, LSA is used to solve the problem of reservoir operation optimization to verify its feasibility. We also propose an improved LSA algorithm, the frog-leaping–particle swarm optimization–LSA (FPLSA), which was improved by using multiple strategies, and we address the shortcomings of LSA such as low solution accuracy and the tendency to fall into local optima. After preliminary verification of ten test functions, the effect is significantly enhanced. Using the lower Jinsha River–Three Gorges cascade reservoirs as an example, the calculation is carried out and compared with other algorithms. The results show that the FPLSA performed better than the other algorithms in all of the indices measured which means it has stronger optimization ability. Under the premise of satisfying the constraints of cascade reservoirs, an approximate optimal solution could be found to provide an effective output strategy for cascade reservoir scheduling.

1. Introduction

Hydropower, a renewable energy source, has advantages including low operating costs, flexible startup and shutdown procedures, and high peak-shaving ability. Rational and scientific operation to improve water-resource utilization is an important means of improving the power-generation capacity of hydropower stations [1,2,3]. Reservoir operation can be divided into conventional and optimized operations. Using conventional operation, which uses an operation table, it is difficult to achieve a global optimum. To address this, there has been rapid development of optimized reservoir operation methods, which solve reservoir operation problems using optimization algorithms.

Reservoir operation optimization algorithms can be divided into traditional and intelligent optimization algorithms (Table 1). In the 1940s, Masse [4] first proposed a reservoir operation optimization model, setting a precedent for reservoir operation optimization. Traditional optimization algorithms, including linear programming [5], nonlinear programming [6], and dynamic programming [7], have since been applied to reservoir optimization and operation problems. Young [8] was the first to apply dynamic programming to reservoir operation. Dynamic programming successive approximation (DPSA) [9], incremental dynamic programming (IDP) [10], and mixed integer nonlinear programming (MINP) [11] and other algorithms have subsequently been studied and applied to reservoir operation.

Reservoir optimization and operation are high-dimensional optimization problems that are difficult to solve using traditional optimization algorithms [12]; such algorithms are often computationally intensive and tend to suffer from the “dimensional disaster” problem when solving high-dimensional problems [13]. Meta-heuristics algorithms provide new approaches to solving these problems and have improved efficiency, which have been widely studied in reservoir optimization operations. Meta-heuristics algorithms were partitioned into nine categories [14] including evolutionary-based, swarm-based, physics-based, human-based, bio-based, system-based, math-based, music-based, and probabilistic-based meta-heuristics algorithms.

The evolutionary-based algorithm is the initial development of the meta-heuristic algorithm, including genetic algorithm and differential evolution algorithm. Evolutionary-based algorithms update individuals through their own mutation, evolution, selection and replication, so as to obtain better solutions [15], which are characterized by strong applicability and widely used in reservoir optimal operation problems including irrigation, power generation, and water supply [16]. However, evolutionary-based algorithms have low computational efficiency and poor global search ability with high computational cost when dealing with complex high-dimensional problems. Therefore, some scholars improved the evolutionary-based algorithms and proposed sequential genetic algorithm [17] varying chromosome length GA [18], adaptive immune differential evolution algorithm [19], etc., to overcome the problem that the calculation time is long and the global optimal solution cannot be found in the reservoir optimal operation. The representative of swarm-based algorithm is particle swarm algorithm. Compared with evolutionary-based algorithms, its computational time is greatly reduced by increasing the randomness of the solution, and the convergence speed is also increased. A. Baltar [20] tested the performance of particle swarm optimization and genetic algorithm in reservoir optimization problems and the results show that the two algorithms have different characteristics. The genetic algorithm has stronger stability and particle swarm optimization has higher computational efficiency and convergence speed. The simulated annealing algorithm, as a typical physics-based algorithm, is widely used in reservoir optimization problems for its approach to global optimum, especially in irrigation problems [21]. It has been proven to minimize irrigation deficits in multiple reservoir cases [22].

There are many kinds of biological-based meta-heuristic algorithms, which can be subdivided into plant-based (invasive weed algorithm [23]), animal-based (whale algorithm [24], gray wolf algorithm [25]), insect-based (ant colony algorithm [26], bee colony algorithm [27]), microbial-based (Slime Mould algorithm [28]), and so on. These algorithms have high performance and adaptability in solving high-dimensional complex problems, so they have received much concern [29]. Mixing different algorithms could help overcome the shortcomings of a single algorithm and solve problems more efficiently [23]. The improved algorithms of these algorithms are hybrid whale optimization algorithm [30], hybrid algorithm of invasive weed optimization and cuckoo search algorithm [23], hybrid slime mold and arithmetic optimization algorithm [31], and so on. These are widely used in reservoir optimal operation problems, including short-term [32,33], medium- and long-term optimal operation [34,35,36]. In addition, teaching-learning algorithm (human-based) [37], water cycle algorithm (system-based) [38], sine cosine algorithm (math-based) [39], harmony search algorithm (music-based) [40], and cross-entropy algorithm (probabilistic based) [41] have been proven to be feasible in reservoir optimal operation. Nonetheless, there remains much room for improvement in intelligent optimization, to address problems such as sensitivity to initial values, slow convergence speed, and the tendency to fall into local optima.

Table 1. Category of algorithms used in reservoir operation optimization.

Category			Algorithms	Characteristics
Traditional optimization algorithms			Linear programming [4]	Huge computation and tend to suffer from the “dimensional disaster” problem
			Non-linear programming [5]
			Dynamic programming [6]
			Integer programming [10]
Meta-heuristics optimization algorithms	Evolutionary-based		Genetic algorithm [17]	High computational efficiency but difficult to achieve the global optimal solution
			Differential evolution algorithm [19]
	Swarm-based		Particle swarm algorithm [20]
	Physics-based		Simulated annealing algorithm [21]
	Human-based		Teaching-learning algorithm [37]
	Bio-based	Plant-based	Invasive weed algorithm [23]
		Animal-based	Whale algorithm [24]
		Insect-based	Ant colony algorithm [26]
		Microbial-based	Slime Mould algorithm [28]
	System-based		Water cycle algorithm [38]
	Math-based		Sine cosine algorithm [39]
	Music-based		Harmony search algorithm [40]
	Probabilistic-based		Cross-entropy algorithm [41]

Table 1. Category of algorithms used in reservoir operation optimization.

Category			Algorithms	Characteristics
Traditional optimization algorithms			Linear programming [4]	Huge computation and tend to suffer from the “dimensional disaster” problem
			Non-linear programming [5]
			Dynamic programming [6]
			Integer programming [10]
Meta-heuristics optimization algorithms	Evolutionary-based		Genetic algorithm [17]	High computational efficiency but difficult to achieve the global optimal solution
			Differential evolution algorithm [19]
	Swarm-based		Particle swarm algorithm [20]
	Physics-based		Simulated annealing algorithm [21]
	Human-based		Teaching-learning algorithm [37]
	Bio-based	Plant-based	Invasive weed algorithm [23]
		Animal-based	Whale algorithm [24]
		Insect-based	Ant colony algorithm [26]
		Microbial-based	Slime Mould algorithm [28]
	System-based		Water cycle algorithm [38]
	Math-based		Sine cosine algorithm [39]
	Music-based		Harmony search algorithm [40]
	Probabilistic-based		Cross-entropy algorithm [41]

The lightning search algorithm (LSA) of Shareef et al. [42], a new heuristic optimization algorithm, uses the mechanism of lightning formation (the process whereby lightning strikes the ground from the clouds) to simulate the optimization process; in the LSA, the optimal solution is simulated assuming stepped leader propagation when simulating the first lightning strike on the ground from the clouds. The LSA has the advantages of strong search capability, high accuracy, and fewer parameters requiring adjustment [43], although it also has shortcomings such as poor stability and a tendency to fall into local optima. Methods to improve the LSA include the heuristic LSA (QLSA), based on the theory of quantum theory [43], and the dynamic adjustment coefficient-based LSA [44]. LSA and its improvements have been applied in various fields, including the design of fuzzy logic controllers [43]; stochastic operation of a power grid, including electric vehicles and renewable energy [44]; models of composite exchange membrane fuel cells [45]; and the layout of wind power plants [46]. LSA is a system-based algorithm in the classification of meta-heuristic algorithms which also includes the water cycle algorithm that has been proven to be applicable to the field of reservoir operation [38]. It shows that the system-based algorithm has certain feasibility in the field of reservoir optimal operation, but there is no research on the application of LSA in reservoir operation.

Based on the above research status, the goal of this study is to solve the following problems:

(1) The meta-heuristic algorithm still has problems such as falling into local optimum, low accuracy, and slow convergence speed when solving reservoir scheduling problems, and there is still much room for improvement.

(2) There is no research on the application of LSA in the field of reservoir operation, and its effectiveness needs to be verified.

(3) There are still some defects in the LSA algorithm, which can be further improved.

In view of the above problems, this paper applied chaotic initialization (mapping), the frog-leaping algorithm (FLA), and the PSO to improve the LSA, which is applied to the reservoir operation. In this paper, Section 2 introduces the reservoir optimization operation model, the improvement strategy of LSA, and the preliminary verification of the test functions. Section 3 introduces a case study of cascade reservoirs on the Jinsha River, China. Section 4 compares the results of FPLSA and other algorithms in the case study and Section 5 discusses the results. In the end, Section 6 summarizes the conclusions of this paper and analyzes the shortcomings of the study and the future research direction.

2. Models and Methods

2.1. Models

The objectives in optimizing cascade reservoir operation are typically to maximize the power-generation capacity or output. The timescale can be hourly, daily, monthly, or longer. We consider maximum power generation as the objective function. Operation optimization was performed as follows: based on the known initial and final water levels, inflow process, and the flow intervals of each reservoir, the water-level operating process of each reservoir is optimized to maximize total power generation given the reservoirs’ constraints in terms of water-level restriction, water-level variation, discharge flows, and discharge capacity. The objective function and constraints of the reservoir optimization model are as follows in Equations (1)–(6) [47,48,49]:

(1) Objectives:

$$\mathrm{F}=\mathrm{m}\mathrm{a}\mathrm{x}\sum _{\mathrm{t}}^{\mathrm{T}}\mathrm{A}{\mathrm{Q}}_{\mathrm{t}}{\mathrm{H}}_{\mathrm{t}}∆\mathrm{t}$$

(1)

where $\mathrm{A}$ is the hydropower station’s output coefficient; ${\mathrm{Q}}_{\mathrm{t}}$ is its power-generating flow during time period t; ${\mathrm{H}}_{\mathrm{t}}$ is its head during time period $\mathrm{t}$; $∆\mathrm{t}$ is the length of each time period; and $\mathrm{T}$ is the total number of time periods.

(2) Constraints

a. Water-balance constraint

$${\mathrm{V}}_{\mathrm{t}}={\mathrm{V}}_{\mathrm{t}-1}+\left({\mathrm{I}}_{\mathrm{t}}-{\mathrm{Q}}_{\mathrm{t}}\right)·∆\mathrm{t}$$

(2)

where ${\mathrm{V}}_{\mathrm{t}}$ and ${\mathrm{V}}_{\mathrm{t}-1}$ are the reservoir capacities at the end of the time period on days $\mathrm{t}$ and $\mathrm{t}$ − 1, respectively; ${\mathrm{I}}_{\mathrm{t}}$ is the average inflow of the reservoir on day $\mathrm{t}$; ${\mathrm{Q}}_{\mathrm{t}}$ is the average outflow of the reservoir on day $\mathrm{t}$; and $∆\mathrm{t}$ is the duration of a single time period.

b. Water-level constraint

$${\mathrm{Z}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le {\mathrm{Z}}_{\mathrm{t}}\le {\mathrm{Z}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(3)

where ${\mathrm{Z}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{Z}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are, respectively, the minimum and maximum water-level limits of the reservoir on day $\mathrm{t}$.

c. Outflow constraint

$${\mathrm{Q}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le {\mathrm{Q}}_{\mathrm{t}}\le {\mathrm{Q}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(4)

where ${\mathrm{Q}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{Q}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are, respectively, the minimum and maximum outflow limits of the reservoir on day $\mathrm{t}$.

d. Power-output constraint

$${\mathrm{N}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le {\mathrm{N}}_{\mathrm{t}}\le {\mathrm{N}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

(5)

where ${\mathrm{N}}_{\mathrm{t}}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{N}}_{\mathrm{t}}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are, respectively, the minimum and maximum output limits of the power station on day $\mathrm{t}$.

e. Water-level variation constraint

$$\left|{\mathrm{Z}}_{\mathrm{t}}-{\mathrm{Z}}_{\mathrm{t}-1}\right|\le ∆\mathrm{Z}$$

(6)

where ${\mathrm{Z}}_{\mathrm{t}}$ and ${\mathrm{Z}}_{\mathrm{t}-1}$ are the water levels at the end of the time period on days $\mathrm{t}$ and $\mathrm{t}$ − 1, respectively; $∆\mathrm{Z}$ is the maximum possible variation in water level during the adjacent period.

2.2. Methods

2.2.1. LSA

The basic LSA is based on the natural phenomenon of lightning. When lightning is formed, it forms a “discharger” body of particles in the air, which passes rapidly through the atmosphere, creating an initial particle channel and forming a stepped leader through the collisions between particles. Owing to the probabilistic and tortuous nature of lightning propagation, lightning is predictable and random in terms of where it reaches the ground. In this algorithm, it is assumed that each discharging body produces a particle channel via collisions, forming a stepped leader. The concept of “dischargers” is similar to that concept of individuals in the population in a differential evolution algorithm, and each discharger can be considered a set of candidate solutions in the optimization problem. By simulating three types of dischargers (transitional, spatial, and stepped leader dischargers), a mathematical model of the stochastic distribution function was established for solving optimization problems [42,50].

In the early stages of lightning formation, the discharge moves rapidly through air and loses energy when it collides with other molecules and atoms in the air. When dischargers follow a long path, they can no longer ionize or explore a larger space, but can only ionize particles within a small surrounding space. In the LSA, the discharger’s energy is used to control the global and local searches.

As lightning reaches the ground, the discharger bifurcates in two ways. The first type of bifurcation creates two symmetric channels, expressed as

$${\overline{\mathrm{p}}}_{\mathrm{i}}=\mathrm{a}+\mathrm{b}-{\mathrm{p}}_{\mathrm{i}}$$

(7)

where $\mathrm{a}$ and $\mathrm{b}$ are the upper and lower limits of the search space, ${\mathrm{p}}_{\mathrm{i}}$ and ${\overline{\mathrm{p}}}_{\mathrm{i}}$ are the symmetric and original channels formed via bifurcation. To maintain a constant population size, the energy of the two channels is compared, and one channel is retained.

The second bifurcation type eliminates the worst channel. The algorithm iteratively increases the channel time. The worst channel is eliminated. To maintain a constant population size, the optimal channel value is assigned to the worst channel and the channel time is reset.

The LSA is calculated by simulating the discharge process of the transitional, spatial, and stepped leader dischargers, as follows:

a. Transitional discharger

The transitional discharger creates the initial population, which propagates randomly downward from the thundercloud. Thus, it obeys a uniform distribution, and its probability density function can be expressed as Equation (8):

$$\mathrm{f}\left({\mathrm{x}}^{\mathrm{T}}\right)=\left\{\begin{array}{c}\frac{1}{\mathrm{b}-\mathrm{a}} a\le {\mathrm{x}}^{\mathrm{T}}\le b\\ 0 {\mathrm{x}}^{\mathrm{T}}<a \mathrm{or} {\mathrm{x}}^{\mathrm{T}}>b\end{array}\right.$$

(8)

where ${\mathrm{x}}^{\mathrm{T}}$ is a set of candidate solutions and $\mathrm{a}$ and $\mathrm{b}$ are the upper and lower limits of the solution space, respectively.

b. Spatial discharger

The spatial discharger attempts to reach the optimal position of the stepped leader. Its position can be modeled using an exponential distribution,

$$\mathrm{f}\left({\mathrm{x}}^{\mathrm{T}}\right)=\left\{\begin{array}{c} \frac{{\mathrm{e}}^{-{\mathrm{x}}^2/\mathsf{\mu }}}{\mathsf{\mu }} {\mathrm{x}}^{\mathrm{s}}\ge 0\\ 0 {\mathrm{x}}^{\mathrm{s}}\le 0\end{array}\right.$$

(9)

where ${\mathrm{x}}^{\mathrm{s}}$ is a set of candidate solutions and $\mathsf{\mu }$, the shape parameter, serves to control the direction of the next iteration. Thus, the spatial discharge in the next iteration can be expressed as Equation (10):

$${\mathrm{p}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}={\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}\pm {\mathrm{e}}^{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left({\mathsf{\mu }}_{\mathrm{i}}\right)}$$

(10)

where ${\mathrm{e}}^{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left({\mathsf{\mu }}_{\mathrm{i}}\right)}$ is the random exponential number, ${\mathsf{\mu }}_{\mathrm{i}}$ is the distance between the stepped leader discharger ${\mathrm{p}}^{\mathrm{L}}$ and the spatial discharger ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}$. If the energy ${\mathrm{E}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}$ of the new spatial discharger ${\mathrm{p}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}$ is greater than the energy ${\mathrm{E}}_{\mathrm{i}}^{\mathrm{S}}$ of the original spatial discharger ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}$, then ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}$ is updated to position ${\mathrm{p}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}$. Otherwise, ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}$ remains unchanged until the next update is performed.

c. Stepped leader discharger

The stepped leader discharger can be modeled using a normal distribution, with the probability density function

$$\mathrm{f}\left({\mathrm{x}}^{\mathrm{L}}\right)=\frac{1}{\mathsf{\sigma }\sqrt{2\mathsf{\pi }}}{\mathrm{e}}^{-\frac{{(\mathrm{x}\mathrm{L}-\mathsf{\mu })}^2}{2{\mathsf{\sigma }}^2}}$$

(11)

where $\mathsf{\mu }$ is the shape parameter and $\mathsf{\sigma }$ is the scale parameter, reflecting the mining capacity at the current location. In the stepped leader discharger model, $\mathsf{\sigma }$ decreases exponentially as the discharger approaches the ground. Therefore, the direction of the stepped leader discharger ${\mathrm{p}}^{\mathrm{L}}$ in the next iteration is

$${\mathrm{p}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}={\mathrm{p}}^{\mathrm{L}}+\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}({\mathsf{\mu }}^{\mathrm{L}},{\mathsf{\sigma }}^{\mathrm{L}})$$

(12)

where $\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}({\mathsf{\mu }}^{\mathrm{L}},{\mathsf{\sigma }}^{\mathrm{L}})$ is a random number from the generated normal distribution. If the energy ${\mathrm{E}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}$ of the new stepped leader discharger ${\mathrm{p}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}$ is greater than the energy ${\mathrm{E}}^{\mathrm{L}}$ of the original stepped leader discharger ${\mathrm{p}}^{\mathrm{L}}$, ${\mathrm{p}}^{\mathrm{L}}$ is updated to take position ${\mathrm{p}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}$. Otherwise, ${\mathrm{p}}^{\mathrm{L}}$ remains unchanged until the next update.

2.2.2. LSA Improvement Strategy

To address the shortcomings mentioned earlier, we propose improving the LSA by incorporating chaotic initialization, the FLA, and PSO.

a. Chaotic initialization-based improvement

In mathematics, “chaotic” describes irregularly distributed systems. Chaotic searching is characterized by nonlinearity, ergodicity, randomness, and initial-value sensitivity, features that improve an algorithm’s search performance [51]. Chaotic mapping methods include logistic mapping, Lozi mapping, Chebyshev mapping, tent mapping, and cubic mapping. References [52,53] have shown that using chaotic tent mapping to initialize the population can effectively improve the diversity of the population. Therefore, we use chaotic tent mapping to initialize the population in FPLSA, such that the initial solutions were distributed as uniformly as possible in the solution space. First, tent mapping is used to generate chaotic sequences, as shown in Equation (13) [53,54]:

$${\mathrm{x}}_{\mathrm{n}+1}=\mathrm{f}\left({\mathrm{x}}_{\mathrm{n}}\right)=\left\{\begin{array}{c}\frac{{\mathrm{x}}_{\mathrm{n}}}{\mathsf{\alpha }} , {\mathrm{x}}_{\mathrm{n}} ϵ [0,\alpha ] \\ \frac{1-{\mathrm{x}}_{\mathrm{n}}}{1-\mathsf{\alpha }}, {\mathrm{x}}_{\mathrm{n}} ϵ [\alpha ,1]\end{array}\right.$$

(13)

where $\mathsf{\alpha }\in \left(\mathrm{0,1}\right)$. Subsequently, the chaotic sequence is mapped to the decision-variable space of the problem being optimized, as follows:

(1) According to Equation (13), n m-dimensional individuals are generated, $\mathsf{\alpha }$ = 0.49, and the meta-individuals are $\mathrm{X}=({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{m}})$, ${\mathrm{x}}_{\mathrm{i}}\in \left(\mathrm{0,1}\right)$, $\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{m}$, ${\mathrm{x}}_{\mathrm{i}}\in \left(\mathrm{0,1}\right)$, and $\mathrm{i}=\mathrm{1,2},\dots ,\mathrm{m}$.

(2) The initial population is then obtained by mapping the meta-individuals to the decision-variable value space as shown in Equation (14):

$${\mathrm{y}}_{\mathrm{i}}={\mathrm{a}}_{\mathrm{i}}+{\mathrm{x}}_{\mathrm{i}}\times ({\mathrm{b}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{i}})$$

(14)

where ${\mathrm{a}}_{\mathrm{i}}$ and ${\mathrm{b}}_{\mathrm{i}}$ are the next and previous terms in the space, respectively, ${\mathrm{x}}_{\mathrm{i}}$ is the value of the meta-individual in ith dimensional space, and ${\mathrm{y}}_{\mathrm{i}}$ is the value of the initial solution individual in the ith dimensional space.

b. FLA-based improvement

The FLA was developed to simulate the behavior of a group of frogs searching for food in a field [55]. Each frog represents a set of feasible solutions to the problem to be optimized. The algorithm flow is divided into three steps. First, a global search is conducted. The initialized frogs are first ranked according to their fitness, and globally fittest individual ${\mathrm{X}}_{\mathrm{g}}$ is selected. Second, a local search is performed. Grouping is applied to divide the population into subpopulations, and for each subpopulation, the locally fittest individual ${\mathrm{X}}_{\mathrm{l}}$ and least fit individual ${\mathrm{X}}_{\mathrm{w}}$ are selected. Finally, an update operation is performed. The worst individual in each subpopulation is updated as shown in Equation (15):

$${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathrm{X}}_{\mathrm{w}}+\mathrm{S}$$

(15)

where ${\mathrm{X}}_{\mathrm{w}}$ is the position of the updated individual and $\mathrm{S}$ is the individual jump. Step length is calculated as shown in Equation (16):

$$\mathrm{S}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\times ({\mathrm{X}}_{\mathrm{l}}-{\mathrm{X}}_{\mathrm{w}})$$

(16)

$$\mathrm{S}=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\times ({\mathrm{X}}_{\mathrm{g}}-{\mathrm{X}}_{\mathrm{w}})$$

(17)

where $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}$ is a (0, 1) uniformly distributed random number. Step $\mathrm{S}$ is first updated according to Equation (16), and the updated fitness of the individual is calculated. If the new position is better than the original position, ${\mathrm{X}}_{\mathrm{w}}$ is replaced by ${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}$; otherwise, the position is updated according to Equation (17), and the individual’s fitness is updated. Again, if the position is better than the original position, ${\mathrm{X}}_{\mathrm{w}}$ is replaced by ${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}$; otherwise, a new position is randomly generated within the feasible solution range. Following the update, all individuals in all subpopulations are mixed again, and the iterations continue until the algorithm’s stop-condition is satisfied.

The FLA combines global and local searches. In each subgroup, the individual with the lowest fitness moves in the direction of the best individual, improving the algorithm’s optimization efficiency and convergence speed. Our improved LSA method has some limitations using this FLA. First, the individuals with the lowest fitness will move only toward the optimal individual, causing the convergence of all of the updated individuals. Second, owing to the limitations imposed by grouping, only a few poorly adapted individuals are updated in each iteration of the FLA, hindering the overall optimization efficiency.

To solve the convergence problem, the position is updated probabilistically when the FLA and LSA are combined. To do this, Equation (15) is modified as shown in Equation (18):

$${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}=\left\{\begin{array}{c}{\mathrm{X}}_{\mathrm{w}} , \mathrm{rand}<\theta \\ {\mathrm{X}}_{\mathrm{w}}+\mathrm{S} , \mathrm{rand}\ge \theta \end{array}\right.$$

(18)

where $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}$ is a (0, 1) uniformly distributed random number. Based on Equation (18), each FLA-based update has a specific probability. If an FLA-based update does not occur, an LSA-based update occurs.

To ensure that more poor-fitness individuals are updated, we modified the grouping and leaping methods. The fitness of all the individuals in the population is calculated and ranked. In the resulting population, $\mathrm{X}=({\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}})$, $\mathrm{n}$ is the number of individuals, and the positions are updated as shown in Equation (19):

$${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathrm{X}}_{\mathrm{n}-\mathrm{i}}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\times \left({\mathrm{X}}_{\mathrm{i}}-{\mathrm{X}}_{\mathrm{n}-\mathrm{i}}\right),\mathrm{i}=\mathrm{0,1},\dots ,\mathrm{n}/2$$

(19)

$${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathrm{X}}_{\mathrm{n}-\mathrm{i}}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\times \left({\mathrm{X}}_{\mathrm{g}}-{\mathrm{X}}_{\mathrm{n}-\mathrm{i}}\right),\mathrm{i}=\mathrm{0,1},\dots ,\mathrm{n}/2$$

(20)

The positions of individuals are first updated according to Equation (19), and individual fitness is updated. If the revised position is better than the original position, ${\mathrm{X}}_{\mathrm{n}-\mathrm{i}}$ is replaced by ${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}$; otherwise, it is updated according to Equation (20), and individual fitness is again updated. If the new position is better than the original position, ${\mathrm{X}}_{\mathrm{n}-\mathrm{i}}$ is replaced by ${\mathrm{X}}_{\mathrm{n}\mathrm{e}\mathrm{w}}$; otherwise, a new position is generated randomly within a feasible solution range to replace the original position. After completing the update operation, all individuals in all subpopulations are mixed again for subsequent updates.

c. PSO-based improvement

When updating positions via the LSA, discharger updates can only occur if the new position provides greater fitness; this tends to lead the algorithm into local optima. Incorporating PSO into the search addresses this problem. PSO records the location of the optimal solution in the population and in its own path process, and continuously searches for updates accordingly [56], thus making it easier for the LSA to escape local optima.

PSO generates updates by sharing information within the population based on the search paths of memorized individuals and their adaptation to the environment, so that each individual moves closer to the optimal solution. In PSO, individuals are referred to as particles, and the particle’s position comprises a set of feasible solutions. Each particle’s velocity and optimal position in its own path, as well as the global optimal position in the population, determine its update direction. A particle’s velocity and position update are expressed as shown in Equations (21) and (22):

$$\begin{gathered} {\mathrm{v}}_{\mathrm{i}}\left(\mathrm{k}+1\right)=\mathrm{w}{\mathrm{v}}_{\mathrm{i}}\left(\mathrm{k}\right)+{\mathrm{c}}_1{\mathrm{r}}_1\left({\mathrm{p}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{k}\right)\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\mathrm{c}}_2{\mathrm{r}}_2\left({\mathrm{p}}_{\mathrm{g}}-{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{k}\right)\right) \end{gathered}$$

(21)

$${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{k}+1\right)={\mathrm{x}}_{\mathrm{i}}\left(\mathrm{k}\right)+{\mathrm{v}}_{\mathrm{i}}\left(\mathrm{k}+1\right)$$

(22)

where ${\mathrm{v}}_{\mathrm{i}}$ is the velocity of particle i, ${\mathrm{x}}_{\mathrm{i}}$ is the position of particle i, k is the number of iterations, $\mathrm{w}$, ${\mathrm{c}}_1$, and ${\mathrm{c}}_2$ are constants, ${\mathrm{r}}_1$ and ${\mathrm{r}}_2$ are (0, 1) uniformly distributed random numbers, ${\mathrm{p}}_{\mathrm{i}}$ is the best position in the path traveled by particle i, and ${\mathrm{p}}_{\mathrm{g}}$ is the optimal position in the population.

PSO, which introduces the concept of particle dischargers, updates particle dischargers probabilistically. In the improved LSA method, specific probabilities are associated with spatial discharger updates via Equations (21) and (22). Otherwise, the spatial dischargers are updated via Equations (9) and (10).

2.3. Optimization Procedure

The improved FPLSA procedure (as described in Section 2.2.2) for optimizing cascade reservoir power generation has the following eight steps (Figure 1):

Step 1: The FPLSA parameters, including the population size N, maximum number of iterations M, maximum channel time T, individual dimension D, frog-leaping probability ${\mathsf{\theta }}_1$, and particle discharger probability ${\mathsf{\theta }}_1$, are set. The cascade-reservoir data, including the normal storage level, water-level storage-capacity curve, discharge capacity curve, and daily inflow and outflow for the discharge period, are obtained. The objective function is set as the maximum power-generation capacity. The objective function returns the total power-generation capacity of the cascade reservoirs; this value reflects the extent of adaptation. The decision variable is the water level at each timepoint. The constraints of the power generation and operation models are entered.

Step 2: N transitional dischargers are generated via tent mapping initialization, producing an initial population containing N individuals, each of which represents the water-level process of a cascade reservoir. Discharger energy is evaluated by calculating individual fitness. The next step is to determine whether or not the water-level process satisfies the constraints (as described in Section 2.1). If it does, step 3 is performed; otherwise, the water-level process is corrected according to the constraints.

Step 3: This step involves determining whether to perform frog-leaping updates. If the frog-leaping update condition is satisfied, the individuals are sorted according to their fitness, and frog-leaping updates are performed using the improved frog-leaping strategy. If the frog-leaping update condition is not satisfied, step 4 is performed.

Step 4: The energy ${\mathrm{E}}^{\mathrm{L}}$ of the stepped leader discharger ${\mathrm{p}}^{\mathrm{L}}$ is determined and the pilots with the best and worst positions are identified. The next step is to determine whether the maximum channel time has been reached. If it has, the worst channel pilot is eliminated, and the channel time is reset; if not, the algorithm proceeds to step 5.

Step 5: The direction of the stepped leader discharger is then updated according to its update rules; the stepped leader discharger and its energy are then updated. If the energy ${\mathrm{E}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}$ of the updated stepped leader discharger ${\mathrm{p}}_{\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{L}}$ exceeds that ${\mathrm{E}}^{\mathrm{L}}$ of the original stepped leader discharger ${\mathrm{p}}^{\mathrm{L}}$, its position is updated. Otherwise, its position remains unchanged.

Step 6: The next step is to determine whether to generate a particle discharger. If the condition for generating a particle discharger is satisfied, the particle discharger is updated. If not, the spatial discharger direction is updated according to its update rules. The spatial discharger and its energy are then updated. If the energy ${\mathrm{E}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}$ of new spatial discharger ${\mathrm{p}}_{\mathrm{i}\_\mathrm{n}\mathrm{e}\mathrm{w}}^{\mathrm{s}}$ exceeds that ${\mathrm{E}}_{\mathrm{i}}^{\mathrm{S}}$ of the original spatial discharger ${\mathrm{p}}_{\mathrm{i}}^{\mathrm{S}}$, the spatial discharger position is updated. Otherwise, it retains its position.

Step 7: The next step is to determine whether the particle or spatial discharger energy exceeds that of the stepped leader discharger. If not, the discharger’s position remains unchanged. If it does, the evidence for bifurcation is examined. If bifurcation has not occurred, the stepped leader discharger and energy are updated. If bifurcation has occurred, the bifurcation point is treated as a symmetric channel, the channel with the lower energy is eliminated, and the stepped leader discharger and its energy are updated.

Step 8: Finally, the algorithm examines whether or not the maximum number of iterations has been reached. If not, the number of iterations and channel time are increased, and the algorithm returns to Step 3 for iterative calculation. If the channel time has been reached, the preferred guide is determined, and the optimal solution is output as the cascade reservoir operation scheme.

2.4. Test Functions

To verify the superiority of the FPLSA, we used 10 commonly used test functions, including six single-peaked and four multi-peaked functions, to compare the standard LSA, standard PSO, and improved PSO (DEPSO) algorithms.

The total number of iterations (M) was set to 500, population size N to 50, and variable dimensions (D) to 30. For the LSA, a maximum channel time of five was used. The PSO constants (as described in Section 2.2.2) were ${\mathrm{c}}_1=1.5$, ${\mathrm{c}}_2=2$, and $\mathrm{w}=0.7$. For DEPSO, the constants were $\mathrm{C}\mathrm{R}=0.5$ and $\mathrm{F}=0.8$. The frog-leaping probability was ${\mathsf{\theta }}_1=0.55$ and particle discharger probability was ${\mathsf{\theta }}_2=0.45$ in FPLSA. We set the other parameters in the FPLSA with reference to those used for the LSA and PSO. References [57,58,59,60,61,62] explain the meaning of the parameters of the algorithms for comparison and how the values are selected.

Table 2. Summary of the parameters of all algorithms, including FPLSA, LSA, PSO, DEPSO.

Algorithm	Parameter	Value	Explanation
FPLSA */LSA	T	5	maximum channel time
FPLSA *	θ1	0.55	frog-leaping probability
	θ2	0.45	particle discharger probability
DEPSO */PSO	c1	1.5	cognitive factor, reflects the trends of the approach to the local optimum
	c2	2	social factor, reflects the trends of the approach to the global optimum
	w	0.7	coefficient of inertia weight, reflects the tendency of particles to maintain their current state
DEPSO *	CR	0.5	predefined crossover probability
	F	0.8	scaling factor

* Frog-leaping–particle swarm optimization (PSO)–lightning search algorithm (LSA); DEPSO, improved PSO.

is a summary of the parameters of all algorithms and

Table 3. Running environment, including hardware and software.

Item		Parameter
Hardware environment	CPU	1.2 GHz, 4-core, Intel Core i7
	RAM	16 G
	ROM	500 G
Software environment	Operating system	macOS 10.15.7
	Programming language	Java
	JDK version	1.8.0_91

shows the running environment, including hardware and software. The optimization results and convergence curves are shown in Figure 2 and

Table 4. Calculations using ten groups of test functions for different algorithms.

No.	Type	Test Function	Range	Optimized Value
				FPLSA	LSA	PSO	DEPSO
f1	Single peak	${\mathrm{f}}_1\left(x\right)={\sum }_{i=1}^n{x}_i^2$	[−100, 100]	0.0136	0.8799	0.2593	0.0382
f2	Single peak	${\mathrm{f}}_2\left(x\right)={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	[−10, 10]	0.0125	0.1363	0.9986	0.0486
f3	Single peak	${\mathrm{f}}_3\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	[−100, 100]	284.3669	11,930.8066	535.2708	12,406.0228
f4	Single peak	${\mathrm{f}}_4\left(x\right)={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	[−100, 100]	11.0366	26.1959	13.2687	20.2326
f5	Single peak	${\mathrm{f}}_5\left(x\right)={\sum }_{i=1}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	[−30, 30]	27.9801	365.2011	43.4538	164.0295
f6	Single peak	${\mathrm{f}}_6\left(x\right)={\sum }_{i=1}^n{\left|x_i+0.5\right|}^2$	[−100, 100]	0.0353	0.6714	0.6951	0.0351
f7	Multi-peak	${\mathrm{f}}_7\left(x\right)={\sum }_{i=1}^n\left[x_i^2-10\cos \left(2\pi x_i\right)+10\right]$	[−5.12, 5.12]	17.4902	154.1762	124.4625	35.6399
f8	Multi-peak	$$\begin{gathered} {\mathrm{f}}_8\left(x\right)=-20\exp \left(-0.2\sqrt{\frac{1}{n}{\sum }_{i=1}^n{x}_i^2}\right)- \\ \exp \left(\frac{1}{n}{\sum }_{i=1}^n\cos \left(2\pi x_i\right)\right)+20+\mathrm{e} \end{gathered}$$	[−32, 32]	1.5035	3.2920	4.6902	3.1585
f9	Multi-peak	${\mathrm{f}}_9\left(x\right)=\frac{1}{4000}{\sum }_{i=1}^{\pi }{x}_i^2-{\prod }_{i=1}^n\cos \frac{x_i}{\sqrt{i}}+1$	[−600, 600]	1.0053	1.0044	1.1890	20.2088
f10	Multi-peak	$$\begin{gathered} {\mathrm{f}}_{10}\left(x\right)=0.1\left\{\sin \left(3\pi x_1\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\sum }_{i=1}^n{\left(x_i-1\right)}^2\left[1 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\sin }^2\left(3\pi x_i+1\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.{\left(x_n-1\right)}^2\left[1 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\sin \left(2\pi x_n\right)\right]\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\sum }_{i=1}^{n}u\left(x_i,5100,4\right)\right. \end{gathered}$$	[−50, 50]	0.0372	4.6783	8.7650	1.4780

, respectively.

Based on this comparison (Figure 2), for the 10 commonly used test functions, FPLSA achieved fast and accurate optimization, performing significantly better than the original unimproved LSA and the other algorithms alone. The improved LSA is therefore effective. To further verify the feasibility and efficiency of the improved algorithm in cascade reservoir operation optimization, we applied it to a real-world engineering problem, using a case study of cascade reservoirs on the Jinsha River, China.

3. Case Study

The Jinsha River is an important part of the upper reaches of the Yangtze River. The main stream flows through four provinces of China, the Qinghai, Tibet, Sichuan, and Yunnan provinces. The Jinsha River, which is rich in water resources, is 3500 km long and has a natural drop of ca. 5100 m; its large water volume and concentrated drop make it suitable for hydropower energy development. There are currently four hydropower stations on this river, namely (from top to bottom) WuDongDe, BaiHeTan, XiLuoDu, and XiangJiaBa; all four were scheduled to operate by December 2022. Together with the Three Gorges Hydropower Station, they form the largest group of cascade reservoirs globally, with a total installed capacity of 67.3 × 106 kW, providing the backbone of water-resource discharge management in the Yangtze River Basin (Figure 3). Because WuDongDe and BaiHeTan began full operation relatively late with insufficient data, the XiLuoDu, XiangJiaBa, and Three Gorges reservoirs were selected for analysis in this study, considering the comprehensiveness of the available data.

The XiLuoDu Hydropower Station, at the junction of the Sichuan and Yunnan provinces, has a normal storage level of 600 m, a dead water level of 540 m, a regulating reservoir capacity of 6.46 × 109 m³, installed capacity of 12.6 × 106 kW, and a multi-year average power-generation capacity of ca. 64 × 109 kW·h. The XiangJiaBa Hydropower Station, at the exit of the canyon at the junction of Yibin County (Sichuan), and Shuifu County (Yunnan) and the final hydropower station on the lower reaches of the Jinsha River, has a normal storage level of 380 m, a dead water level of 370 m, a regulating reservoir capacity of 0.903 × 109 m³, an installed capacity of 6.4 × 106 kW, and an average multi-year power-generation capacity of ca. 31 × 109 kW·h. The Three Gorges Hydropower Station (Yichang City, Hubei), the largest hydropower station globally, has a total installed capacity of 22.5 × 106 kW, a normal storage level of 175 m, a dead water level of 145 m, a regulating reservoir capacity of 39.3 × 109 m³, and an average multi-year power-generation capacity of about 88 × 109 kW·h.

Table 5. XiLuoDu Hydropower Station.

Type	Total Storage Capacity (108 m3)	Normal Water Storage Level (m)	Dead Water Level (m)	Dead Storage Capacity (108 m3)
Large (I)	126.7	600	540	51.1
Beneficial Reservoir Capacity (108 m3)	Flood Control storage Capacity (108 m3)	Adjustment Performance	Installed Capacity (MW)	Average Annual Power Generation (108 kW·h)
64.6	46.5	Incomplete annual reconciliation	12,600	640

,

Table 6. XiangJiaBa Hydropower Station.

Type	Total Storage Capacity (108 m3)	Normal Water Storage Level (m)	Dead Water Level (m)	Dead Storage Capacity (108 m3)
Large (I)	51.63	380	370	42.6
Beneficial Reservoir Capacity (108 m3)	Flood Control Storage Capacity (108 m3)	Adjustment Performance	Installed Capacity (MW)	Average Annual Power Generation (108 kW·h)
9.03	9.03	Incomplete seasonal adjustment	6000	307.47

and

Table 7. Three Gorges Hydropower Station.

Type	Total Storage Capacity (108 m3)	Normal Water Storage Level (m)	Dead Water Level (m)	Dead Storage Capacity (108 m3)
Large (I)	393	175	145	171.5
Beneficial Reservoir Capacity (108 m3)	Flood Control Storage Capacity (108 m3)	Adjustment Performance	Installed Capacity (MW)	Average Annual Power Generation (108 kW·h)
165	221.5	Incomplete annual reconciliation	22,500	884

describe these hydropower stations.

To ensure that our experimental results were representative, we selected three typical years, 2020, 2015, and 2016, as wet, normal, and dry years, respectively. For model optimization, we used 1–30 September each year as the operation period, with a daily time-scale, and applied the FPLSA, LSA, PSO, and DEPSO algorithms, using the parameter settings and running environment described in Section 2.3.

4. Results

To obtain more convincing results, 200 solutions were obtained using each of the four algorithms. The optimal value of each result was recorded, and the maximum, minimum, mean, and standard deviation of the 200 solution results were calculated.

Table 8. Electricity generation capacity in 2020 (wet year).

Indicator	FPLSA	LSA	PSO	DEPSO
Average (108 kW·h)	376.8393	376.7282	376.6328	376.6824
Maximum (108 kW·h)	377.0155	377.0131	377.0133	376.9387
Minimum (108 kW·h)	376.6913	376.1986	376.6130	376.5425
Standard deviation	0.0476	0.0564	0.1210	0.0583

,

Table 9. Electricity generation capacity in 2015 (normal year).

Indicator	FPLSA	LSA	PSO	DEPSO
Average (108 kW·h)	375.2046	374.3585	374.7516	374.7003
Maximum (108 kW·h)	375.4058	374.5739	374.8157	374.8609
Minimum (108 kW·h)	375.1348	374.0311	374.1027	374.5952
Standard deviation	0.0578	0.1118	0.1258	0.0603

and

Table 10. Electricity generation capacity in 2016 (dry year).

Indicator	FPLSA	LSA	PSO	DEPSO
Average (108 kW·h)	268.3505	265.9566	266.5051	266.2722
Maximum (108 kW·h)	268.3926	267.3617	267.3772	266.9771
Minimum (108 kW·h)	268.1579	265.4089	264.5911	265.5615
Standard deviation	0.0831	0.5322	0.6773	0.2333

present the power generation capacity during September (30 days) in 2020 (wet year), 2015 (normal year), and 2016 (dry year), respectively, obtained using the four algorithms.

In the wet year (2020), rainfall and inflow were high; as a result, generation capacity was considerably higher than in the drier years. In the dry year (2016), rainfall and runoff were low, hence power generation was lower, while power generation in the normal year (2015) was intermediate. The FPLSA achieved the highest mean, maximum, and minimum optimal solution values than that of the other three algorithms in the wet year (2020); its mean power generation was 0.02%, 0.05%, and 0.04% higher than that of the LSA, PSO, and DEPSO algorithms, respectively. In the normal year (2015), the superiority of the FPLSA algorithm was more remarkable, with mean power generation that was 0.23%, 0.12%, and 0.13% higher than that of the LSA, PSO, and DEPSO algorithms, respectively. The FPLSA algorithm achieved the most substantial effects for the dry year (2016), with mean power-generation values 0.90%, 0.69%, and 0.13% higher than those of the LSA, PSO, and DEPSO algorithms, respectively.

Standard deviation can reflect the stability of an algorithm. For the wet year, the FPLSA achieved standard deviations that were 15.69%, 60.67%, and 18.35% lower than those of the LSA, PSO, and DEPSO algorithms, respectively (Figure 4). For the normal year, its standard deviations were 48.33%, 54.07%, and 4.20% lower than those of the LSA, PSO, and DEPSO, respectively. For the dry year, the standard deviations were 85.15%, 88.33%, and 66.13% lower than those of the LSA, PSO, and DEPSO, respectively.

Boxplots illustrate size relationships and the discreteness of the data, and intuitively reflect the advantages and disadvantages of various algorithms. The FPLSA had a narrower interquartile range of power generation than the other algorithms for each year, indicating its greater stability (Figure 5); it achieved higher average power generation than the other algorithms, indicating that it achieves optimization more effectively than the other algorithms. Moreover, the FPLSA generated fewer discrete points than the other algorithms, indicating that it was less likely to fall into local optima.

Figure 6, Figure 7 and Figure 8 illustrate the optimization search processes of the algorithms. For the wet year, the FPLSA converged to the almost optimal value after 20 iterations, and escaped local optima to a better solution after ca. 170 iterations, while the LSA, PSO, and DEPSO converged to almost optimal values after 80, 40, and 50 iterations, respectively. This indicates that, in a wet year, the FPLSA convergences faster and is less likely than the other algorithms to fall into a local optimal solution. For the normal year, the FPLSA converged to the optimal value after ca. 35 iterations, while the LSA and DEPSO converged after ca. 80 and 45 iterations, respectively. Although PSO converged at 30 iterations, it continued to search for the optimal value in subsequent iterations, indicating that it had not yet converged on the optimal solution. Based on these findings, for a normal year, the FPLSA achieves better optimization than the other algorithms. For the dry year, while PSO converged to the optimal value in ca. 15 iterations, its convergence was again unsatisfactory. In contrast, the FPLSA, LSA, and DEPSO did not converge to the optimal value even in 300 iterations. However, after 70 iterations, the FPLSA achieved substantially higher optimal values than the other algorithms. Therefore, the FPLSA also optimizes better than the other algorithms in dry years.

5. Discussion

Based on the above results, the wet year required the fewest iterations for each algorithm to converge to the optimal value; the normal year required more, and the dry year the most. In wet years, with high inflow, hydropower stations operate close to full power generation. Therefore, for a wet year, if maximum power generation is the objective function, the range of available output process strategies is small, and the algorithm can easily locate the optimal solution, thus converging faster. In normal and dry years, with insufficient inflow to reach full generation, the solution space was larger, making it more difficult for the algorithms to locate the optimal solution, and thus leading to slower convergence.

Figure 9, Figure 10, Figure 11 and Figure 12 illustrate the power generation output processes for the XiLuoDu, XiangJiaBa, and Three Gorges hydropower stations, respectively. For the wet year, the algorithms generated similar power output curves. This is because the objective function, maximizing power generation, requires high power-generating flow. The wet year had the highest runoff volume, making it possible to achieve maximum power generation while meeting the constraints. For the wet year, the modeled daily output of each power station is essentially close to the maximum possible output, thereby maximizing the objective function. For the wet year, the FPLSA achieved an average optimal power output <0.1% higher than that of the other algorithms. In normal and dry years, with less rainfall and inflow, the inflow of each station was insufficient to achieve full-generation capacity, and there is more scope to choose a generation strategy. The variance in power output was lowest for the wet year, intermediate for the normal year, and highest for the dry year. FPLSA achieved higher average power generation than the other algorithms in the normal and dry years, achieving the largest difference (up to 0.9%) in the dry year. This indicates that the FPLSA optimizes power output more effectively than the other algorithms when the solution space for generation strategies is larger.

Figure 13, Figure 14 and Figure 15 illustrate the inflow and outflow processes of XiLuoDu, XiangJiaBa and Three Gorges hydropower stations in different typical years. The outflow of Three Gorges is basically close to the maximum outflow due to the large amount of inflow in the wet year, while the outflow process of Three Gorges Power Station has the same trend as the inflow process in other years. It shows that the algorithm chooses the strategy of generating more power during the period of high water inflow to fully utilize the water resources and ensure that the hydropower station operates at high capacity. The outflow and inflow processes of XiLuoDu and XiangJiaBa have the same pattern as that of the Three Gorges. The runoff processes of XiLuoDu and XiangJiaBa are similar in each typical year, but differ greatly from that of the Three Gorges. This is due to regional differences. XiLuoDu and XiangJiaBa are closer together and have similar hydrometeorological conditions. XiangJiaBa and the Three Gorges are farther away from each other, and the difference in hydrometeorological conditions makes the interval runoff more variable. The analysis shows that the flow and output process change under different conditions in different typical years. However, the power generation strategies are generally consistent, and the overall trend satisfies the hydropower operation rules.

6. Conclusions and Future Research

In this paper, we proposed a new particle frog jump lightning search algorithm by combining it with chaotic initialization (mapping), the FLA, and PSO. Ten test functions are applied for preliminary validation. The optimal operation model of the maximum power generation capacity is constructed with the boundary conditions of cascade hydropower reservoirs. The results of FPLSA, LSA, PSO, and DE algorithms were compared in three typical years of different runoff scenarios. After analysis and discussion, the following conclusions are drawn.

(1) The FPLSA, improved by using multiple strategies, addresses the shortcomings of the standard LSA, including low solution accuracy and the tendency to fall into local optima. From the test results of ten test functions, FPLSA has better performance in both single-peak and multi-peak functions. Table 1 shows that FPLSA has better global search and local search capabilities in optimization and can find the optimal solution more accurately. Figure 2 shows that FPLSA converges faster than other algorithms.

(2) In the optimal operation model of cascade hydropower reservoirs, FPLSA has excellent performance. From the results of Table 5, Table 6 and Table 7 and Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, in the maximum power generation model constructed by taking XiLuoDu, XiangJiaBa, and Three Gorges cascade hydropower stations as examples, FPLSA achieved the best power output generation with higher solution accuracy, faster convergence speed, better search capability and greater stability among the results of multiple trials.

(3) FPLSA can reasonably optimize the allocation of water resources in typical years under different runoff conditions. From the output process and runoff process in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15, FPLSA makes optimal use of water resources to generate power to ensure the high-capacity operation of hydropower stations under the condition of meeting various constraints, including water-level constraint, water-level variation constraint, outflow constraint, power-output constraint, and water-balance constraint, which can provide an efficient power generation strategy for hydropower station operation decision makers.

Although the results of FPLSA in actual cascade hydropower reservoirs simulations can meet the operational needs of hydropower stations, the research in this paper has certain limitations. Future research directions can include the following aspects: (1) Taking the maximum power generation as the objective function is only considered by the power supply side which focuses on the power generation benefit. In practical engineering applications, there are many other factors involved, such as ecological goals and grid peak shaving needs. The operation effect of other objective function models using FPLSA in reservoir optimal operation needs to be verified in the future. (2) The improved FPLSA algorithm in this paper is currently only applicable to the single objective function model. However, the optimal operation of reservoirs often requires multiple objectives to be considered together, and studying the multi-objective model of FPSLA is conducive to better integration with engineering practice.