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Article

Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations

1
School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
2
Hubei Key Laboratory of Digital River Basin Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
3
Institute of Water Resources and Hydropower, Huazhong University of Science and Technology, Wuhan 430074, China
4
China Yangtze Power Co., Ltd., Yichang 443000, China
5
Hubei Key Laboratory of Intelligent Yangtze and Hydroelectric Science, Yichang 443000, China
*
Author to whom correspondence should be addressed.
Submission received: 30 December 2024 / Revised: 19 January 2025 / Accepted: 24 January 2025 / Published: 26 January 2025
(This article belongs to the Section Energy Sustainability)

Abstract

:
Hydropower is a vital strategic component of China’s clean energy development. Its construction and optimized water resource allocation are crucial for addressing global energy challenges, promoting socio-economic development, and achieving sustainable development. However, the optimization scheduling of cascade hydropower stations is a large-scale, multi-constrained, and nonlinear problem. Traditional optimization methods suffer from low computational efficiency, while conventional intelligent algorithms still face issues like premature convergence and local optima, which severely hinder the full utilization of water resources. This study proposed an improved whale optimization algorithm, the Black-winged Differential-variant Whale Optimization Algorithm (BDWOA), which enhanced population diversity through a Logistic-Sine-Cosine combination chaotic map, improved algorithm flexibility with an adaptive adjustment strategy, and introduced the migration mechanism of the black-winged kite algorithm along with a differential mutation strategy to enhance the global search ability and convergence capacity. The BDWOA algorithm was tested using test functions with randomly generated simulated data, with its performance compared against five related optimization algorithms. Results indicate that the BDWOA achieved the optimal value with the fewest iterations, effectively overcoming the limitations of the original whale optimization algorithm. Further validation using actual runoff data for the cascade hydropower station optimization scheduling model showed that the BDWOA effectively enhanced power generation efficiency. In high-flow years, the average power generation increased by 8.3%, 6.5%, 6.8%, 4.1%, and 8.2% compared to the five algorithms while achieving the shortest computation time. Significant improvements in power generation were also observed in normal-flow and low-flow years. The scheduling solutions generated by the BDWOA can adapt to varying inflow conditions, offering an innovative approach to solving complex hydropower station optimization scheduling problems. This contributes to the sustainable utilization of water resources and supports the long-term development of renewable energy.

1. Introduction

Hydropower plays a crucial role in China’s clean energy development, and advancing hydropower construction has become a core task in the country’s power infrastructure development. China possesses abundant water resources, ranking among the top globally; however, its current utilization rate is low, and development efficiency remains limited. Improving resource utilization through scientific management is, therefore, essential. Consequently, the efficient allocation and use of water resources has become a key research focus in water conservancy engineering, with particular attention on the optimal scheduling of cascade hydropower stations. Cascade hydropower stations’ optimization scheduling involves the joint management of large-scale reservoir groups. It includes complex hydraulic and power connections. Optimal system scheduling is required under tasks such as flood control, power generation, and water supply. Due to its high dimensionality, nonlinearity, and multiple constraints, the problem faces the challenge of the “curse of dimensionality”. Effective scheduling methods are critical for improving the efficiency of water resource utilization and achieving multi-objective optimization, offering both theoretical and practical value.
In recent years, the optimal scheduling solution methods for cascade hydropower stations mainly include two categories: traditional optimization methods and intelligent optimization algorithms. In the 1940s and 1950s, international scholars carried out research on the optimal scheduling of cascade hydropower stations. Little [1] took the lead in applying the dynamic programming method to reservoir scheduling, creating a precedent for the application of mathematical planning theory to the field of reservoir scheduling. Since then, the linear programming method [2], the discrete differential dynamic programming method [3,4], the step-by-step optimization algorithms [5,6], and other traditional optimization methods have been successively proposed and applied to solve the optimal scheduling of reservoirs. The research on the optimal scheduling of reservoirs has made some progress.
However, traditional optimization methods often have a large computational volume. Especially when solving the ”dimensional disaster” problem encountered in the joint scheduling of cascade hydropower stations, the computational complexity will increase significantly, making the model solution extremely difficult. In recent years, the emerging intelligent optimization algorithms [7] have proposed a new solution to this problem, which is a kind of algorithm that simulates the biological, physical, or social behaviors and characteristics of nature to solve complex optimization problems. Such algorithms mainly include the following:
Evolutionary algorithms—such as genetic algorithms [8] and differential evolution algorithms [9,10]—which solve optimization problems by simulating biological processes like inheritance, crossover, and mutation. These algorithms have certain advantages in handling nonlinear, multi-objective, and complex constrained reservoir optimization scheduling problems. However, as the dimension of joint reservoir scheduling increases, their computational efficiency decreases, and they are prone to falling into local optima.
Physical algorithms—such as simulated annealing [11] and gravitational search algorithms [12]—typically rely on the simulation of real physical phenomena. These algorithms can effectively solve multi-objective, high-dimensional, and nonlinear problems. Gravitational search algorithms have been widely used in reservoir optimization scheduling research. In the ecological operation of cascade hydropower stations [13], they can effectively reduce unsuitable ecological flow rates. However, these algorithms have certain parameter limitations, leading to local convergence, and require further strategies to enhance performance.
Swarm intelligence optimization algorithms—including particle swarm optimization (PSO) and biological swarm algorithms. Particle swarm optimization is one of the most widely used algorithms in cascade reservoir optimization scheduling. Due to its strong global search ability, simple principles, and fewer parameters, it performs well in solving problems related to power generation and flood control [14,15], and is effective in addressing the “curse of dimensionality” in cascade reservoir scheduling. Biological swarm intelligence algorithms, known for their few parameters and ease of implementation, have also been frequently applied to reservoir optimization scheduling problems. Examples include gray wolf optimization [16], cuckoo search [17], whale optimization [18], artificial bee colony [19], and sea lion optimization [20]. These algorithms, when applied to reservoir optimization scheduling, can explore the global optimum in the solution space by simulating the collaborative behavior of biological groups in nature. They are particularly well-suited for solving nonlinear, multi-objective, high-dimensional, and complex constrained optimization problems. These algorithms can effectively address the multiple objectives in reservoir scheduling, such as flood control, power generation, and water supply, as well as complex constraints like water levels and flow rates.
Compared with traditional optimization methods, swarm intelligence algorithms have better adaptability and flexibility in dealing with complex problems with multiple phases and constraints. However, although these swarm intelligence optimization algorithms have shown better performance in solving reservoir scheduling problems, they still face some challenges. For example, these algorithms are prone to premature convergence and local optimal solutions, especially in complex steps of reservoir scheduling problems, where the optimization process may be limited as the dimensions increase and the constraints become more complex. Therefore, how to effectively improve the global search capability of these algorithms to overcome the local optimal and premature convergence problems remains the key to the current research. A large number of scholars have carried out research on this problem and made some progress.
The whale optimization algorithm (WOA) is a new type of intelligent optimization algorithm proposed by Seyedali Mirjalili [21] based on the predatory behavior of whales, which has the advantages of fewer parameters, simpler principles, and stronger search ability. Currently, scholars have conducted research around the advantages of WOA and applied it in various fields. For example, V. Lai et al. [22] used the whale optimization algorithm and Levy flight and distribution enhancement techniques to optimize the reservoir operation of the Klang Gate Dam, which was crucial for the development, operation, and management of multi-objective reservoirs for timely water supply; Kun Yang et al. [23] improved the original whale optimization algorithm based on binary coding and applied it to the short-term scheduling problem of the Three Gorges Hydropower Station; Wenchuan Wang et al. [24] improved the runoff prediction accuracy by optimizing the variational model decomposition through the whale optimization algorithm. Based on the above research background, although the whale optimization algorithm shows significant advantages in complex optimization problems, the limitations of the original algorithm are still significant when dealing with the scheduling problem of cascade hydropower stations, especially the tendency to fall into the local optimum and the slow convergence speed.
For this reason, this paper proposes an improved whale optimization algorithm to overcome the shortcomings of the existing WOA in the cascade hydropower station scheduling. The algorithm improves the original WOA by enhancing population diversity, incorporating adaptive strategies, and combining the newly proposed migration mechanism of the black-winged kite algorithm and differential mutation strategy. The resulting algorithm, the Black-winged Differential-variant Whale Optimization Algorithm (BDWOA), effectively improves optimization efficiency and development capability. It is applied to simulation calculations for the lower Jinsha River–Three Gorges cascade hydropower stations, verifying the algorithm’s ability and enhancing water resource utilization efficiency.
The remainder of this paper is organized as follows: Section 2 provides a detailed explanation of the construction of the cascade hydropower station optimization scheduling model, including the definition of the objective function and scheduling constraints. In Section 3, we introduce the fundamental principles of the whale optimization algorithm and explain the proposed improvement strategies. The performance of the improved algorithm is then validated using 10 benchmark test functions and compared with other algorithms. Section 4 presents a case study where the improved algorithm is applied to the power generation optimization scheduling of the downstream Jinsha River–Three Gorges cascade hydropower system, offering detailed methodologies, results, and discussions. Finally, Section 5 summarizes the main conclusions of this study and outlines potential directions for future research.

2. Cascade Reservoir Optimization Scheduling Model

In order to improve the power generation efficiency of the gradient hydropower station, this paper sets the maximization of power generation as the objective function, takes the reservoir inflow process as the input variable, and improves the power generation efficiency by optimizing the operation of the reservoir under the constraints of reservoir safety and ecological environmental protection. The objective function and constraints of the optimal scheduling model of the terrace reservoir are as follows:
(1)
Objective function
E = m a x i = 1 I t = 1 T   A i Q i , t H i , t Δ t
where i is the serial number of the hydropower station; I is the number of gradient hydropower stations; t is the scheduling time period; T is the total time period of the transfer; A i is the output coefficient of the i th hydropower station; Q i , t and H i , t are the power generation flow rate and head of the i th hydropower station in the t th scheduling time period, with units of m3/s and m, respectively; Δ t is the length of each time period, and the length of time period of the present model is a ten-day period.
(2)
Constraints
① Water Balance Constraints
V i , t = V i , t 1 + I i , t R i , t × Δ t
where V i , t and V i , t 1 are the beginning and ending reservoir capacity of hydropower station i at time period t and time period t 1 , respectively, with units of m3; I i , t and R i , t are the incoming and outgoing flows of hydropower station i at time period t , respectively, with units of m3/s.
② Output Constraints
N i , t m i n N i , t N i , t m a x
where N i , t , N i , t m i n , and N i , t m a x are the actual output, minimum output, and maximum output of hydropower station i at time period t , respectively, with units of MW.
③ Flow Constraints
Q i , t m i n Q i , t Q i , t m a x
R i , t m i n R i , t R i , t m a x
where Q i , t m a x and Q i , t m i n are the maximum and minimum generation flow permitted by the scheduling regulations of hydropower station i at time period t ; R i , t m a x and R i , t m i n are the maximum and minimum release flow permitted by the scheduling regulations of hydropower station i at time period t , respectively, with units of m3/s.
④ Hydraulic Relationships
I i , t = R i 1 , t τ + q i , t
where I i , t is the inflow of hydropower station i at time period t ; R i 1 , t τ is the release flow of hydropower station i 1 at time period t τ ; q i , t is the interval inlet flow of hydropower station i at time period t , with units of m3/s; τ is the flow lag time, with units of seconds (s).
⑤ Water Level Constraints
Z i , t m i n Z i , t Z i , t m a x
Z i d o w n Z i , t + 1 Z i u p
where Z i , t m a x and Z i , t m i n are the upper and lower limits of the water level of hydropower station i at time period t ; Z i , t is the actual water level of hydropower station i at time period t ; Z i u p , Z i d o w n are the upper and lower limits of the variation in the water level of hydropower station i at time period t , respectively, with units of m.
⑥ Initial and Final Water Level Control Constraints
Z i , 0 = Z i , s t a r t
Z i , T = Z i , e n d
where Z i , s t a r t and Z i , e n d denote the beginning and end water levels of a hydropower station i in the scheduling period, respectively, with units of m.
⑦ Ecological Flow Constraints
R i , t m i n R e c o i , t m i n
where R e c o i , t m i n represents the minimum ecological flow of the hydropower station, with units of m3/s.

3. Black-Winged Differential-Variant Whale Optimization Algorithm

3.1. Whale Optimization Algorithm

The WOA is a mathematical model constructed to simulate the behavior of whale predation. The algorithm uses a random or optimal search to simulate the hunting behavior of chasing prey, and a spiral search to simulate the bubble net attack mechanism of humpback whales, which forage through encircling predation, spiral bubble attack, and random hunting mechanisms. In the WOA algorithm, the positions of all whales are feasible solutions to the problem, and the computation starts with a set of random solutions, and in each iteration, the position of a new individual is updated according to the randomly selected individual or the best individual obtained so far, and the position of the optimal individual is the optimal solution of the model. The WOA consists of three phases: encircling predation, spiral bubble attack, and random search predation. The main processes and principles of the algorithm are as follows:
(1)
Individual Initialization
A population with individual dimension D and population size N is randomly generated as in Equation (12):
X m = x 1 x 2 . . . x N = x 1 , 1 x 1 , 2 . . . x 1 , D x 2 , 1 x 2 , 2 . . . x 2 , D . . . . . . x i , j . . . x N , 1 x N , 2 . . . x N D
where X m denotes the vector of population individuals under the current m iterations; x i , j denotes the j th dimension value of the i th individual, x i , j a j ,   b j , where a j and b j are the upper and lower bounds of the algorithm’s decision variables, respectively; and x i , j is initialized according to Equation (13) within a j ,   b j .
x i , j = r a n d × b j a j + a j     i = 1 ,   2 ,   ,   N ,   j = 1 ,   2 ,   ,   D
where rand represents a random number randomly distributed over the interval.
(2)
Encircling Predation
In the WOA, the whales are able to determine the position of the prey and try to encircle it. At the initial stage of the algorithm, it is assumed that the current position of the best candidate solution is the position of the target prey, and once the position of the target prey is determined, the rest of the whales gradually approach the target to complete the encirclement by adjusting their own positions. The behavior of encircling the prey is expressed mathematically as follows:
X m + 1 = X * ( m ) A × D m
D m =   C × X * ( m ) X ( m )
A = 2 × a × r a
C = 2 × r
where m is the current number of iterations; A and C are the coefficients; X is the current solution location; X * is the current optimal solution location; r is a random vector in [0, 1]; and a is the convergence factor, which varies linearly from 2 to reduce to 0 during iterations. This is shown in Equation (18):
a = 2 2 m M a x i t e r
where M a x i t e r is the maximum number of iterations.
(3)
Spiral Bubble Attack
In the spiral bubble attack phase, whales approach prey progressively by means of a spiraling upward motion while simultaneously closing the envelope to complete the capture. The spiral bubble attack predation strategy consists of two main steps: shrinking encirclement and spiral updating.
Shrinking encirclement: This behavior is based on the linear convergence factor a defined in Equation (18). During the iterative process, as the parameter a decreases, the individual whale adjusts its position and approaches the prey until the encirclement process is completed.
Spiral update: In the process of narrowing the encirclement, the whale gradually approaches the prey along a spiral trajectory. Its movement path can be described by the following mathematical expression:
X m + 1 = X * ( m ) + D * × e b l × c o s 2 π l
D * = | X * ( m ) X ( m ) |
where D * is the distance between the i th individual and the optimal individual; b is a constant used to define the logarithmic spiral shape; and l is a random number in [−1,1].
In order to simulate both the contraction–enclosure mechanism and the spiral renewal mechanism of whales, it is assumed that there is a 50% probability of choosing between the contraction–enclosure mechanism or the spiral renewal, and the mathematical expression is represented as follows:
X m + 1 = X * ( m ) A × D m X * ( m ) + D * × e b l × cos 2 π l p < 0.5 p 0.5
where p is a random number in the range of [0, 1].
(4)
Random Search
In addition to the spiral bubble attack predation strategy, whales balance the exploration and development phases of the algorithm by randomly searching for prey. This process is accomplished by adjusting the value of the parameter A . When A 1 , the algorithm no longer selects the optimal individual for position updating but uses a randomly selected whale to update the position, as shown in the following mathematical expression:
X m + 1 = X r a n d A × D r a n d
D r a n d = C × X r a n d X
where X r a n d is a randomly selected individual whale.

3.2. Whale Optimization Algorithm Improvement Strategy

(1)
Combinatorial Chaos Mapping
Chaotic mappings are characterized by a mixture of randomness, sensitivity, and determinism, which enhances the global search performance of the algorithm. However, a single chaotic mapping—such as Logistic mapping, Sine mapping, Fuch mapping, Tent mapping, or Cosine mapping—although capable of generating chaotic sequences, exhibits relatively limited chaotic properties and easily falls into periodic trajectories. Therefore, this paper proposes to use Logistic-Sine-Cosine combined chaotic mapping to initialize the population, which integrates the complex chaotic dynamics of the Logistic map with the oscillatory characteristics of Sine and Cosine maps. This enables the algorithm to be both globally explored and locally exploited, improving the diversity of the initial population, reducing the likelihood of premature convergence. Figure 1 illustrates the population distribution of the combined chaotic mapping and single chaotic mapping. It can be observed from the figure that the population distribution of the Logistic-Sine-Cosine chaotic mapping is more uniform.
The Logistic-Sine-Cosine chaotic mapping first generates chaotic sequences according to the following Equation (24):
x i + 1 = cos ( π 4 r x i 1 x i + 1 r sin ( π x i ) 0.5 ) ,   r [ 0,1 ]
where r is the chaos parameter, which is a random number in the range of (0, 1); and x i is the current chaotic sequence state value. Then, the chaotic sequence is mapped into the value space of the decision variable; the specific steps are as follows:
(1) N population individuals are generated according to the Logistic-Sine-Cosine chaotic mapping method, each with dimension D. The two-dimensional individual expression mapped using Equation (24) is as follows:
L S C = L x 11 L x 12 . . . L x 1 D L x 21 L x 22 . . . L x 2 D . . . . . . L x i j . . . L x N 1 L x N 2 . . . L x N D
where i = 1 ,   2   . . . .   N ;   j = 1 ,   2   . . . .   D , L x i j represents the chaotic mapping value of the j th dimension for the i th individual.
(2) In accordance with Equation (26)
y i j = a i j + L x i j × ( b i j a i j )
The generated chaotic sequence values are mapped to the value space of decision variables to obtain the initial population, where, a i j and b i j are the lower and upper bounds of the value space, respectively; L x i j represents the j th dimensional chaotic value of the i th chaotic individual; y i j is the value of the j th dimensional space of the i th initial solution individual.
(2)
Adaptive Weights
The original whale optimization algorithm was developed locally at a later stage without corresponding weights to the update formula. As the search process deepens, it may lead to individual whales staying near the theoretical position or even falling into local extremes. Meanwhile, in order to synchronize the encircling and spiral marching processes, the original algorithm sets the probability thresholds both at 0.5, and the hunting strategy is selected by comparing the randomly generated p -values with the probability thresholds. As the number of iterations increases, this equal probability of hunting leads to problems such as the algorithm falling into local optimization. Based on the above problems, the present invention uses adaptive weight values to adjust the update formula, and proposes an adaptive critical value to balance the global and local search abilities.
The corresponding adaptive weight ω expression is as follows:
ω = ω e n d + ( ω s t a r t ω e n d ) 1 1 + e a b × m M a x i t e r
where m is the current iteration number and M a x i t e r represents the number of iterations; ω s t a r t and ω e n d represent the weights at the beginning and end of the iteration. To ensure the algorithm maintains strong global search capability in the early stages of optimization, these parameters are set based on experience as ω s t a r t = 0.9 , ω e n d = 0.4 , a = 3.6 , and b = 0.1 . At the beginning of the algorithm, larger weight coefficients confer a strong global search capability. As the number of iterations increases, the weight coefficients gradually decrease, at which time the algorithm performs a fine search in the neighborhood of the optimal solution with smaller weight coefficients by spiral marching, thus avoiding falling into the local optimum. We added adaptive weights to improve the shortcomings of the random search and spiral predation phase in the later stage of local search as it is easy to fall into the local extremes. The equations displaying the improvement to Equations (19) and (22) are as follows:
X m + 1 = X * ( m ) + ω × D * × e b l × c o s ( 2 π l )
X m + 1 = X r a n d ω × A × D r a n d
In addition, the exponential adaptive value p 1 is expressed as follows:
p 1 = e α × m M a x i t e r
where α is the decay coefficient in the exponential decay formula, directly affecting the rate of decay. Setting α = 0.8 results in a slower decay, allowing the optimization process to maintain its global search capability for a longer duration. At the beginning of the iteration, the adaptive threshold is larger, at which time the algorithm performs a global search with higher probability and faster speed when p p 1 . As the iteration advances, the probability threshold slowly decreases to close to 0. Since p > p 1 at this time, the algorithm will more likely choose the spiral traveling method to update the leader’s position. By constantly updating the adaptive probability threshold, the local and global search ability of the algorithm is adjusted so that the whale group gradually moves closer to the optimal solution, thus improving the convergence accuracy of the algorithm.
(3)
Improvements Based on the Migration Behavior of Black-Winged Kites
Black-winged kite algorithm [25] (BKA) is an optimization algorithm derived from simulating the migration and predation behavior of black-winged kites. Each black-winged kite represents a feasible solution, and the main update mechanisms include the following: attack behavior and migration behavior. The attack behavior mechanism is used for global search, and the migration strategy can dynamically select the excellent optimal solution to ensure the success of migration. The present invention improves the original whale optimization algorithm based on the migration principle of the black-winged kite, and uses the migration behavior mechanism as a step in the whale update mechanism to strengthen the local search capability of the original whale optimization algorithm. The equation for the black-winged kite migration behavior update is as follows:
X m + 1 = X r a n d ω × A × D r a n d r a n d > θ
X m + 1 = X m + C 0 ,   1 × X m X b e s t m r a n d θ
where r a n d represents the random number in the range of (0, 1); θ is the migration probability. Referring to the update probability in the whale optimization algorithm, θ = 0.5 is chosen here. At this point, the algorithm has the probability to carry out the black-winged kite migration update, otherwise, it proceeds to continue with the whale algorithm update.
(4)
Improvements Based on Differential Mutation
A differential evolutionary algorithm is an evolutionary computing technique that contains three main operations: mutation, crossover, and selection. Mutation is to utilize the difference vectors between different individuals to perturb the individuals, which can effectively improve the search ability of the algorithm. There are various mutation strategies. The present invention improves the contraction encirclement phase of the original whale by combining the mutation strategy of DE/current-to-best/1. The DE/current-to-best/1 expression is as follows:
V i , G = X i , G + F X b e s t , G X i , G + F X r 1 , G X r 2 , G
where F is the variation operator, which controls the degree of perturbation; X i , G is the current individual; X b e s t , G X i , G is the difference variable generated by the optimal individual and the current individual; and X r , G X r , G is the difference vector generated by the random individual.
In order to improve the original whale algorithm’s shortcomings of poor searching ability in the process of contraction encirclement and tendency to fall into the local optimum, Equation (14) is improved by adding the difference variable and variation operator for perturbation, which can improve the ability of population diversity and global search, and the improved formula is as follows:
λ = f × X * ( m ) X m + f × X r 1 , G m X r 2 , G m X ( m + 1 ) = X * ( m ) A × D m + λ
where f takes a random number in the range of [0, 2]. The improved equation adds the variational difference vector λ for perturbation on the basis of the original equation, which effectively improves the algorithm’s optimization seeking ability.

3.3. Test Functions to Verify Performance

Ten test functions (six single-peak and four multi-peak functions) were used for testing to verify the performance of the algorithm. The BDWOA algorithm is compared and analyzed with the standard whale optimization algorithm (WOA), the whale optimization algorithm (BWOA) under the black-winged kite strategy, the whale optimization algorithm under the differential variational strategy (DWOA), standard particle swarm (PSO), and particle swarm under the Levy flight strategy (LPSO), and it is tested whether the optimality-seeking, convergence, and robustness performance of the BDWOA algorithm are excellent. In order to ensure the reasonableness of the experiment, the number of iterations of each algorithm is uniformly set at M a x _ i t e r = 500 , the population size is N = 30 , and the dimension of the population is D = 20 , where the algorithm parameters are set as follows in the WOA algorithm: b = 1 , the migration probability θ = 0.5 , and the adaptive weighting factor ω s t a r t = 0.9 , ω e n d = 0.4 . The test function is shown in the following Table 1, and the optimization results and convergence curves are shown in Table 2 and Figure 2 below.
Through the comparison and analysis of the above results, it can be seen that among the ten test functions, the BDWOA proposed in this paper has the best effect in the process of iterative optimization compared with the other five algorithms, with high optimization accuracy and faster convergence speed at the same time. It shows that the improvement strategy proposed in this paper is effective and feasible, and has good searching and optimization ability in the optimization of mathematical problems.
Although the BDWOA algorithm performs well in mathematical problem testing, its effect in practical engineering applications is still unclear. Therefore, further exploration and verification of the applicability and performance of the BDWOA algorithm in the cascade reservoir optimization scheduling problem are needed.

3.4. Model Solving Process

According to the above improvement strategy, the flow of the BDWOA algorithm for solving the optimal scheduling problem of power generation from a terrace reservoir is as follows:
Step 1: Set the parameters of the BDWOA algorithm—the population size is N, the individual dimension is D, the number of algorithm iterations is M a x _ i t e r , the migration probability is θ , the weights at the start and end of the iteration are ω s t a r t and ω e n d , and the adaptive weight control factor is k. Read the basic data of the hydropower station, including the reservoir storage-capacity curve, outflow discharge curve, inflow discharge of each station, and the inflow and outflow of each station during the scheduling period. The decision variable is the reservoir water level process for each time period, which corresponds to the population individuals in the algorithm. The size of the scheduling time period T is also the individual dimension D.
Step 2: Initialize and generate N whale individuals with D dimensions using the Logistic-Sine-Cosine chaotic mapping and Equation (26), forming the initial population of N individuals. Each individual represents the water level process of a cascade reservoir. Compute the fitness F of each individual and evaluate the fitness of the initial population, recording the individual best values and the global best value. Check whether the generated water level process meets the constraint conditions. If it does, proceed to the next step; if not, adjust the water levels.
Step 3: Calculate the whale convergence factor a, coefficient vectors A and C, adaptive threshold p 1 , adaptive weight ω , and random probability p . Check if p p 1 holds. If it does, proceed to the next step; if not, update the individual position using the spiral predation update mechanism, combined with the adaptive weight ω , and update the position according to Equation (28).
Step 4: Check if the condition A < 1 is satisfied. If it is, enter the encircling predation phase, calculate the differential mutation operator λ —which effectively increases the diversity of the original population—compute the distance vector D m using Equation (15), set f as a random number between 0 and 2, and update the individual position according to Equation (34). If the condition is not satisfied, proceed to the next step.
Step 5: Generate a random number r a n d . During the random search phase, the individuals generated in this phase cannot be guaranteed to be optimal. Here, the black-winged kite migration mechanism is used to guide the randomly generated individuals toward better individuals, enhancing the algorithm’s optimization performance. Additionally, the migration probability θ is introduced to balance the algorithm’s exploration and exploitation capabilities. Check if r a n d θ θ = 0.5 holds. If it does, update the individual position using the black-winged kite migration mechanism, according to Equation (32). If it does not, update the individual position based on the whale’s random search mechanism, using the calculated adaptive weight ω and Equation (31).
Step 6: After updating the individual positions based on Step 3, Step 4, and Step 5, the new generation of individuals is obtained. Calculate the fitness values of the new individuals, and update the individual best positions and global best position.
Step 7: Check if the maximum number of iterations M a x _ i t e r has been reached. If not, increment the iteration count and return to Step 3 to continue the process. If the maximum iteration count M a x _ i t e r is reached, determine the global best position, and output the global optimal solution as the cascade reservoir optimization scheduling scheme.
In summary, the flowchart of the BDWOA algorithm for solving the optimal scheduling problem for power generation in a terrace reservoir is shown in the following Figure 3.

4. Case Study

4.1. Overview of the Study Area

This study selects the Xiluodu–Xiangjiaba–Three Gorges cascade hydropower station system as the research area. The lower reaches of the Jinsha River have abundant water resources, with concentrated riverbed drops and abundant water flow, providing favorable conditions for hydropower development. Currently, four large hydropower stations—Wudongde, Baihetan, Xiluodu, and Xiangjiaba—are built sequentially along the river, forming the world’s largest cascade reservoir group along with the Three Gorges Hydropower Station. The total installed capacity is 67.3 million kW, making it the core project for water resources scheduling and management in the Yangtze River Basin. It also effectively improves economic power generation and water demand in the Central China region. Due to the later commissioning of Wudongde and Baihetan, the actual operational data for these stations is relatively short. To ensure data completeness and the general applicability of the experiment, this study selects the Xiluodu, Xiangjiaba, and Three Gorges reservoirs for cascade scheduling analysis. The topological diagram of the cascade hydropower stations is shown in Figure 4.
The Xiluodu Hydropower Station has a normal reservoir water level of 600 m, a dead storage level of 540 m, a total reservoir capacity of 12.67 billion m3, an adjustable reservoir capacity of 6.46 billion m3, and a flood control capacity of 4.65 billion m3. Its average annual power generation is approximately 64 billion kW·h, with incomplete annual regulation capability.
The Xiangjiaba Hydropower Station is located on the downstream section of the Jinsha River, at the border of Yibin County in Sichuan Province and Shuihu County in Yunnan Province. It is 157 km upstream from the Xiluodu Hydropower Station and 33 km downstream from Yibin City. Its long-term average flow is 4570 m3/s, with a normal water level of 380 m, an installed capacity of 6.4 million kW, and a long-term average power generation of approximately 31 billion kW·h. The total storage capacity is 5.163 billion m3, offering partial seasonal regulation capability.
The Three Gorges Hydropower Station, located in Yichang City, Hubei Province, is the largest hydropower station in the world with a total installed capacity of 22.5 million kW. The normal water level is 175 m, the dead water level is 145 m, the regulating storage capacity is 39.3 billion m3, and the long-term average power generation is approximately 85 billion kW·h. It offers incomplete annual regulation. The specific parameters of the three hydropower stations are shown in Table 3 below.

4.2. Experimental Results

To ensure that the experimental results are more representative and scientifically robust, three typical runoff scenarios—a high-flow year, normal-flow year, and low-flow year—are selected for the calculations. Using a ten-day period as the time scale and one year as the scheduling period, a total of six algorithms—BDWOA, WOA, BWOA, DWOA, PSO, and LPSO—are used to solve the optimal scheduling model for the cascade hydropower stations.
In terms of algorithm parameters, the maximum number of iterations for each algorithm is set to M a x _ i t e r = 300 , the population size N = 30 , and the dimension D = 36 . Other algorithm parameters are set as specified in Section 3.4. To verify the robustness of the experimental results, each algorithm is independently run 20 times. The average value, maximum value, minimum value, standard deviation, and computational time are recorded. The detailed results are shown in Table 4.
From Table 4, it can be observed that the BDWOA algorithm outperforms the other five algorithms in cascade reservoir optimization scheduling across the three typical year scenarios.
Firstly, taking the average power generation results from 20 independent runs as an example, compared with the WOA, BWOA, DWOA, PSO, and LPSO algorithms, the BDWOA algorithm increased the mean power generation in high-flow years by 8.3%, 6.5%, 6.8%, 4.1%, and 8.2%, respectively; in normal-flow years by 7.1%, 5.9%, 6.5%, 4.4%, and 7.8%, respectively; and in low-flow years by 7.0%, 5.8%, 6.7%, 5.1%, and 8.0%, respectively. This indicates that the BDWOA algorithm demonstrates stronger optimization capabilities, achieving more significant improvements in power generation across different typical year scenarios.
Secondly, the standard deviation of power generation for the BDWOA algorithm is lower than that of the other five algorithms in all typical year scenarios, indicating that its results show less fluctuation across multiple runs, thus exhibiting higher stability. Moreover, when comparing across the years, the standard deviation in the low-flow year is generally higher than in the normal-flow and high-flow years. This may be due to the more cautious scheduling strategy required in low-flow years, where water resources are scarcer and the flexibility in scheduling is limited.
Finally, in terms of computational time, the BDWOA algorithm consistently outperforms the other five algorithms, requiring less time to find solutions across all typical year scenarios. This indicates that the BDWOA algorithm not only maintains high optimization performance but also offers higher computational efficiency. Therefore, the BDWOA algorithm can provide excellent scheduling solutions in a shorter time, effectively supporting enhanced water resource utilization efficiency.
A variance analysis of the results from 20 independent runs compares the performance of the algorithms. A one-way analysis of variance (ANOVA) tests whether there are significant differences between the algorithms, followed by post hoc analysis using Tukey’s HSD test based on the results. The results are shown in Table 5 and Figure 5, Figure 6 and Figure 7. From the data in Table 5, the p-values for high-flow years, normal-flow years, and low-flow years are all much smaller than 0.05, indicating significant differences between the groups of algorithms. Further analysis of Tukey’s HSD test results in Figure 5, Figure 6 and Figure 7 reveals that in all three typical years, the BDWOA algorithm shows the largest mean differences compared to other algorithms. This further demonstrates that the BDWOA outperforms the five algorithms—WOA, BWOA, DWOA, LPSO, and PSO—in improving power generation.
In high-flow years, the BDWOA exhibits the most significant differences, followed by LPSO. In normal-flow years, the BDWOA shows the most pronounced mean differences when compared to WOA, DWOA, BWOA, LPSO, and PSO. There is no significant difference between BWOA and DWOA, nor between PSO and WOA. In low-flow years, the differences between PSO and WOA are not significant, while LPSO shows significant differences compared to DWOA, BWOA, PSO, and WOA. This suggests that LPSO outperforms all methods except the BDWOA in addressing power generation problems.
Figure 8 presents the boxplots of annual power generation obtained from 20 independent runs for the six algorithms under three typical year scenarios: a high-flow year, normal-flow year, and low-flow year. From Figure 8, it can be observed that the BDWOA algorithm outperforms the other five algorithms under different hydrological scenarios, particularly demonstrating significant advantages in power generation stability.
In the high-flow year scenario, the BDWOA algorithm shows shorter box and whisker lengths with no outliers, indicating higher stability. In contrast, the WOA, BWOA, and LPSO algorithms exhibit similar box lengths but have lower average power generation compared to the BDWOA algorithm, suggesting greater result fluctuations and less favorable overall performance. In the normal-flow year scenario, the BDWOA algorithm again maintains the shortest box and whisker lengths, indicating higher stability in power generation. In comparison, the BWOA algorithm has the longest box length, showing the largest power generation fluctuations and the lowest stability. In the low-flow year scenario, the BDWOA algorithm continues to perform the best, with the shortest box and whisker lengths and the highest average power generation. However, the WOA algorithm performs the worst, exhibiting the longest box length and the largest power generation fluctuations, while the PSO algorithm has the lowest mean power generation. These results demonstrate that the BDWOA algorithm achieves smaller power generation fluctuations and the highest power generation across all typical year scenarios compared to the other five algorithms.
Based on the results from the three typical year scenarios, the BDWOA algorithm exhibits the smallest power generation fluctuation range and consistently achieves the highest average power generation in the high-flow year, normal-flow year, and low-flow year scenarios, demonstrating its robustness under different hydrological conditions. A horizontal comparison of the BDWOA algorithm’s performance in the high-flow year, normal-flow year, and low-flow year reveals that the box and whisker lengths in the low-flow year and high-flow year scenarios are relatively longer, with fluctuations significantly higher than those in the normal-flow year. This indicates that in the low-flow year scenario, where water resources are more limited, the BDWOA algorithm’s solution process is more influenced by hydrological conditions, and the stability of the scheduling scheme decreases; however, it still outperforms the other algorithms.
Based on the results in Figure 8 and Table 4, it can be further concluded that the proposed BDWOA algorithm not only maintains high power generation across all typical year scenarios but also exhibits lower result fluctuations, demonstrating stronger optimization capabilities and stability. This makes it a more optimal solution for cascade reservoir scheduling.
Table 6 and Figure 9 describe the power generation iteration processes of the BDWOA, WOA, BWOA, DWOA, PSO, and LPSO algorithms under each typical year scenario. From the figure, it can be seen that the BDWOA algorithm converges to a clearly higher optimal value compared to the other five algorithms and has a faster convergence rate. In the high-flow year, the algorithm converges to a value close to the optimal solution within 74 iterations; in the low-flow year, it converges to the optimal solution within 100 iterations; and in the normal-flow year, it converges to a value close to the optimal solution within 120 iterations. In comparison, in the high-flow year and low-flow year, the WOA, BWOA, LPSO, and PSO algorithms generally converge around 150 iterations, and in the normal-flow year, even after 300 iterations, they do not fully converge. The DWOA algorithm converges earlier but its optimization ability is weaker.
In conclusion, the improved algorithm proposed in this paper integrates several optimization techniques, including Logistic-Sine-Cosine chaotic mapping, adaptive strategies, the black-winged kite migration mechanism, and differential mutation strategies, which significantly enhance its optimization capability and convergence speed. The combined effect of these four improvements strengthens the algorithm’s ability to make local adjustments, reducing the likelihood of becoming trapped in local optima and effectively balancing global and local search capabilities. These advantages allow the improved BDWOA algorithm to exhibit excellent performance in solving the cascade reservoir optimization scheduling problem. It not only offers practical value in improving power generation efficiency but also demonstrates higher computational efficiency and result accuracy. The findings indicate that the improved algorithm is well-suited to meet the complex optimization scheduling needs of cascade hydropower stations, providing strong support for the efficient management and rational utilization of water resources.

4.3. Scheduling Results Analysis

The above research results indicate that the BDWOA algorithm demonstrates higher solution efficiency in cascade reservoir optimization scheduling. From a longitudinal comparison, the convergence speed of this algorithm is significantly faster in the high-flow year compared to the normal-flow and low-flow years. This may be due to the abundant water resources in the high-flow year, where the flood season generally allows for full generation, thereby limiting the search space for water levels and output. As a result, the algorithm is more likely to quickly converge to the optimal solution. In contrast, during the normal-flow and low-flow years, due to insufficient inflow, the full-generation periods are fewer, and the algorithm has to explore the optimal strategy in a larger output search space, leading to slower convergence.
For the Xiluodu, Xiangjiaba, and Three Gorges cascade hydropower stations, six different algorithms were used to solve the optimization problem, and the output processes for the high-flow year, normal-flow year, and low-flow year scenarios were obtained as shown in Figure 10. The results show that the BDWOA algorithm consistently yields better average output in all three scenarios compared to the other five algorithms. Furthermore, in the high-flow year, the output processes of all algorithms show relatively consistent trends throughout most periods. This is primarily because, during the high-flow year, when the goal is to maximize power generation, the ample water resources during the flood season allow for full generation in most periods, causing the output strategies of all algorithms to converge, with the search space for the population being relatively limited. In contrast, during the normal-flow and low-flow years, due to limited inflows and fewer full-generation periods, the populations of all algorithms explore the optimal strategy in a larger output search space, leading to significant differences in the output processes of different algorithms. This variation reflects the adaptability of each algorithm to cascade reservoir scheduling strategies under different runoff scenarios and also indicates that, in practical scheduling, appropriate strategies can be chosen based on different hydrological conditions.
The inflow, outflow, and water level-output processes of the Xiluodu, Xiangjiaba, and Three Gorges cascade hydropower stations were solved using the BDWOA algorithm, as shown in Figure 11, Figure 12 and Figure 13. Analyzing the inflow and outflow processes under different typical years, it can be observed that the inflow processes of Xiluodu and Xiangjiaba are similar, with their inflow curves almost overlapping. Compared to the Three Gorges Reservoir, the inflows at Xiluodu and Xiangjiaba differ in absolute values but the overall inflow trends of the three reservoirs remain consistent.
Examining the outflow process of the Three Gorges Reservoir reveals that, in a high-flow year, the outflow is significantly higher than in normal-flow and low-flow years, with notable peak flow fluctuations. This indicates the reservoir’s strong regulation capacity and the temporal–spatial uneven distribution of flow during the flood season. Meanwhile, the outflow at Xiluodu and Xiangjiaba also increases in a high-flow year but with relatively smaller fluctuations, reflecting their comparatively stable regulation roles across different hydrological conditions.
This flow process analysis demonstrates that the BDWOA algorithm can adapt to varying hydrological conditions by flexibly selecting appropriate power generation strategies. During high-flow years, utilize water resources effectively. Regulate the outflows of the Three Gorges, Xiluodu, and Xiangjiaba reservoirs to improve hydropower generation efficiency. In periods of water shortage, optimize outflow strategies to ensure downstream ecological flow requirements are met and achieve rational water resource allocation. The analysis of this scheduling scheme provides essential data support for multi-objective water resource management within the basin, contributing to power generation benefits and ecological protection.
Figure 14, Figure 15 and Figure 16 present the power generation–water level processes of the Xiluodu, Xiangjiaba, and Three Gorges reservoirs under different typical year scenarios, as calculated by the BDWOA algorithm. Analyzing the results in these figures reveals that, under the scheduling scheme derived by the BDWOA algorithm, the Xiluodu, Xiangjiaba, and Three Gorges hydropower stations maintain levels close to their maximum output during most flood season stages in high-flow years, with relatively significant water level variations. This indicates that the algorithm effectively utilizes reservoir regulation capacity and optimizes power generation efficiency when water resources are abundant. In normal-flow and low-flow years, due to relatively limited inflows, the scheduling scheme generated by the BDWOA algorithm reasonably reduces power generation during non-flood periods. At the Xiluodu hydropower station, the output remains at lower levels during most periods except for essential stages in the flood season, a trend also observed at Xiangjiaba and Three Gorges. The amplitude of water level variations is significantly reduced. This characteristic demonstrates that the BDWOA algorithm can adapt its scheduling strategy under different hydrological conditions, achieving efficient power generation in high-flow years while ensuring the safe operation of hydropower stations in normal-flow and low-flow years, balancing other water resource demands.
Overall, the BDWOA algorithm exhibits strong adaptability and excellent scheduling performance across different typical year scenarios. It flexibly adjusts the power generation and water level scheduling of each reservoir based on inflow conditions, ensuring the optimal utilization of water resources.

5. Conclusions and Future Work

This study first defined an optimization scheduling model for cascade hydropower stations, aiming to maximize power generation while incorporating corresponding scheduling constraints. Based on the original whale optimization algorithm (WOA), an improved algorithm—Black-winged Differential-variant Whale Optimization Algorithm (BDWOA)—was proposed. The performance of the BDWOA was evaluated using ten benchmark test functions and compared with PSO, LPSO, WOA, DWOA, and BWOA. The results indicated that the BDWOA outperformed the other five algorithms in terms of convergence speed and optimization capability. Finally, the BDWOA was applied to solve the power generation optimization scheduling model for the cascade hydropower stations from the lower Jinsha River to the Three Gorges. This study selected three typical years—high-flow, normal-flow, and dry-flow years—to compare and analyze the performance of the BDWOA and the other five algorithms in solving scheduling schemes. The results showed that the BDWOA achieved the highest power generation in all three years. After detailed analysis and discussion, the following conclusions were drawn:
(1)
The BDWOA improved optimization and search efficiency by introducing Logistic-Sine-Cosine chaotic mapping, adaptive weights, the black-winged kite migration mechanism, and the differential mutation strategy. These enhancements help it effectively avoid becoming trapped in local optima. Compared to the other five algorithms, the BDWOA demonstrated superior performance across ten test functions, specifically by requiring the fewest iterations for convergence and achieving the smallest objective function values in each set of test functions;
(2)
In the application of cascade hydropower station optimization scheduling, the BDWOA demonstrated excellent solution efficiency and effectively increased the overall power generation of the cascade hydropower stations. In the optimization model with the goal of maximizing power generation, the BDWOA significantly outperformed the comparison algorithms. Although the BDWOA algorithm’s results varied slightly across different typical years, it consistently achieved the shortest solving time, the highest power generation, and the smallest standard deviation. For example, in wet years, the solving time was 52 s, the power generation was 2127.86 × 109 kW·h, and the standard deviation was 21.77, outperforming the other five algorithms. This indicated that BDWOA’s convergence process was less prone to becoming trapped in local optima, while also demonstrating a faster convergence speed and higher stability;
(3)
The proposed BDWOA algorithm enhances adaptability to complex water resource scheduling problems and demonstrates strong application potential for sustainable energy development. Under the constraints of water level, flow, and power output, the BDWOA was able to develop appropriate scheduling strategies based on varying inflow conditions, making more efficient use of water resources and maximizing the power generation benefits of cascade hydropower stations, thereby providing strong support for the sustainable development of renewable energy.
The BDWOA algorithm effectively improves upon the original method and increases power generation in cascade hydropower stations. However, its general applicability remains limited. Real-world reservoir scheduling involves uncertainties, such as extreme weather from climate change and the spatiotemporal variability of watershed rainfall, which have yet to be fully integrated into the constraints. Future research should further investigate the application of the BDWOA in managing uncertainty risks in cascade hydropower scheduling, as well as its potential for multi-objective optimization scheduling problems. Such efforts would enhance the algorithm’s applicability and practical value in addressing complex water resource scheduling challenges.

Author Contributions

Conceptualization, M.Z. and Y.Y.; methodology, M.Z.; software, M.Z. and Z.L.; validation, Z.L., S.Z. and Y.Y.; formal analysis, M.Z.; data curation, M.Z. and R.B.; writing—original draft preparation, M.Z.; writing—review and editing, supervision, Y.Y.; funding acquisition, L.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Natural Science Foundation of China (No. 52379011) and the National Key R&D Program “Multi-service Convergence Knowledge Platform Construction Technology and Intelligent Management Decision-making Method” (2023YFC3209104).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author due to their storage by the hydrological station, which involves sensitive environmental information and potential privacy concerns. Additionally, access to the data must comply with relevant legal regulations and ethical requirements. To ensure lawful use and protect the rights of the data provider, access to the research data requires a formal application and approval process.

Conflicts of Interest

Yuqi Yang was employed by the China Yangtze Power Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Abbreviations

BDWOABlack-winged Differential-variant Whale Optimization Algorithm
WOAWhale optimization algorithm
BWOAWhale optimization algorithm under the black-winged kite strategy
DWOAWhale optimization algorithm under the differential variational strategy
PSOStandard particle swarm
LPSOParticle swarm under the Levy flight strategy

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Figure 1. Population distribution plots for combined chaos and single chaos.
Figure 1. Population distribution plots for combined chaos and single chaos.
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Figure 2. Convergence process of different algorithms with 10 sets of test functions.
Figure 2. Convergence process of different algorithms with 10 sets of test functions.
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Figure 3. Flowchart of BDWOA algorithm.
Figure 3. Flowchart of BDWOA algorithm.
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Figure 4. Topology diagram of the Xiluodu–Xiangjiaba–Three Gorges cascade reservoir system.
Figure 4. Topology diagram of the Xiluodu–Xiangjiaba–Three Gorges cascade reservoir system.
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Figure 5. Tukey’s HSD test in high-flow year.
Figure 5. Tukey’s HSD test in high-flow year.
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Figure 6. Tukey’s HSD test in normal-flow year.
Figure 6. Tukey’s HSD test in normal-flow year.
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Figure 7. Tukey’s HSD test in low-flow year.
Figure 7. Tukey’s HSD test in low-flow year.
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Figure 8. Boxplots of power generation under typical year scenarios.
Figure 8. Boxplots of power generation under typical year scenarios.
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Figure 9. Power generation convergence process under each typical year scenario.
Figure 9. Power generation convergence process under each typical year scenario.
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Figure 10. Output processes of each algorithm under different typical scenarios.
Figure 10. Output processes of each algorithm under different typical scenarios.
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Figure 11. Inflow and outflow processes solved by the BDWOA algorithm under the high-flow year scenario.
Figure 11. Inflow and outflow processes solved by the BDWOA algorithm under the high-flow year scenario.
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Figure 12. Inflow and outflow processes solved by the BDWOA algorithm under the normal-flow year scenario.
Figure 12. Inflow and outflow processes solved by the BDWOA algorithm under the normal-flow year scenario.
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Figure 13. Inflow and outflow processes solved by the BDWOA algorithm under the low-flow year scenario.
Figure 13. Inflow and outflow processes solved by the BDWOA algorithm under the low-flow year scenario.
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Figure 14. Power generation–water level processes of cascade hydropower stations in the high-flow year scenario.
Figure 14. Power generation–water level processes of cascade hydropower stations in the high-flow year scenario.
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Figure 15. Power generation–water level processes of cascade hydropower stations in the normal-flow year scenario.
Figure 15. Power generation–water level processes of cascade hydropower stations in the normal-flow year scenario.
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Figure 16. Power generation–water level processes of cascade hydropower stations in the low-flow year scenario.
Figure 16. Power generation–water level processes of cascade hydropower stations in the low-flow year scenario.
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Table 1. Characterization table for 10 groups of test functions.
Table 1. Characterization table for 10 groups of test functions.
Functions NameTest Function ExpressionsFeasibility
Domain
Optimum Value
F1 f ( x ) = i = 1 n   x i 2 [−100,100]0
F2 f ( x ) = i = 1 n   j 1 i   x j 2 [−100,100]0
F3 f ( x ) = m a x i   | x i | [−100,100]0
F4 f ( x ) = i = 1 n 1   [ 100 ( x i + 1 x i 2 ) 2 + ( x i 1 ) 2 ] [−30,30]0
F5 f ( x ) = i = 1 n   i x i 4 + random   [ 0,1 ] [−1.28,1.28]0
F6 f ( x ) = i = 1 n   x i + 0.5 2 [−100,100]0
F7 f ( x ) = i = 1 n   x i 2 10 c o s ( 2 π x i ) + 10 [−5.12,5.12]0
F8 f ( x ) = 1 + 1 4000 i = 1 n   x i 2 i = 1 n   cos x i i [−600,600]0
F9 f x = s i n 2 ( π w 1 ) + i = 1 n 1   ( w i 1 ) 2 1 + 10 sin 2 π w i + 1 + ( w n 1 ) 2 1 + s i n 2 ( 2 π w n )
Among w i = 1 + x i 1 4
[−50,50]0
F10 f ( x ) = 10 d + i = 1 d   x i 2 10 cos ( 2 π x i ) [−5.12,5.12]0
Table 2. Statistics of optimal value results of different algorithms under 10 sets of test functions.
Table 2. Statistics of optimal value results of different algorithms under 10 sets of test functions.
Functions NameOptimum Value
PSOLPSOWOADWOABWOABDWOA
F113,056.5916,615.953.38 × 10−295.12 × 10−192.52 × 10−321.18 × 10−64
F234,457.6839,553.327.79 × 10−61.041.29 × 10−55.27 × 10−30
F34.422.441.23 × 10−61.45 × 10−44.28 × 10−56.61 × 10−23
F45.04 × 1081.34 × 1091816.8316.2216.17
F510,536.439,530.140.970.251.150
F61.79 × 1082,332,424.2000.250
F713,281.019869.78010.4900
F83.830.0700.0300
F91.760.670.540.540.30.27
F1006503.18002.84 × 10−140
Table 3. Characteristics of the Xiluodu, Xiangjiaba, and Three Gorges cascade hydropower stations.
Table 3. Characteristics of the Xiluodu, Xiangjiaba, and Three Gorges cascade hydropower stations.
Power Station Characteristic ParametersXiluoduXiangjiabaThree Gorges
Regulation PerformanceInadequate Annual RegulationInadequate Seasonal RegulationInadequate Annual Regulation
Total reservoir capacity (109 m3)126.751.63393
Flood control reservoir capacity (109 m3)46.59.03221.5
Normal water level (m)600380175
Dead water level (m)540370145
Guaranteed output (MW)379520094990
Installed capacity (MW)12,600640022,500
Average annual power generation (109 kWh)640310884
Minimum discharge (m3/s)170017006000
Table 4. Statistical results of operation under each typical year scenario.
Table 4. Statistical results of operation under each typical year scenario.
Typical Year ScenarioObjective Value (×109 kW·h)Computation Time (s)
MethodsAverage ValueBest ValueWorst ValueStandard Deviation
High-flow yearBDWOA2127.862176.342092.1121.7752.03
WOA1965.382002.711907.1726.3164.91
BWOA2008.862045.061971.6923.4771.72
DWOA1991.952045.221950.9425.3466.54
LPSO2045.012083.921771.5923.7966.23
PSO1967.442026.091771.5928.6569.61
Normal-flow yearBDWOA1951.631984.571922.9214.2455.81
WOA1821.721867.021779.2923.8970.21
BWOA1843.411905.411791.9830.8966.67
DWOA1833.051870.011786.1122.7666.51
LPSO1869.251902.131833.0317.2967.24
PSO1809.951859.941771.5921.3070.98
Low-flow yearBDWOA1916.711947.451883.1715.1752.52
WOA1791.861845.441756.4327.0466.79
BWOA1811.411875.881771.3925.5368.45
DWOA1796.051845.331756.3524.3867.21
LPSO1823.611854.091796.4315.5168.78
PSO1774.791813.721741.4216.6264.17
Table 5. ANOVA test results.
Table 5. ANOVA test results.
Runoff ScenarioF-Valuep-Value
High-flow year114.90867.7646 × 10−43
Normal-flow year101.03393.1967 × 10−40
Low-flow year107.42281.8468 × 10−41
Table 6. Optimal values of different algorithms under each typical year scenario (×109 kW·h).
Table 6. Optimal values of different algorithms under each typical year scenario (×109 kW·h).
AlgorithmTypical Year Scenario
High-Flow YearNormal-Flow YearLow-Flow Year
BDWOA2132.641950.181918.49
WOA1969.551814.121782.23
BWOA2010.721841.711819.16
DWOA1988.191825.591791.61
PSO2042.181865.991824.07
LPSO1972.841809.061768.64
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Zhang, M.; Liu, Z.; Bao, R.; Zhu, S.; Mo, L.; Yang, Y. Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability 2025, 17, 1018. https://doi.org/10.3390/su17031018

AMA Style

Zhang M, Liu Z, Bao R, Zhu S, Mo L, Yang Y. Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability. 2025; 17(3):1018. https://doi.org/10.3390/su17031018

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Zhang, Mi, Zixuan Liu, Rungang Bao, Shuli Zhu, Li Mo, and Yuqi Yang. 2025. "Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations" Sustainability 17, no. 3: 1018. https://doi.org/10.3390/su17031018

APA Style

Zhang, M., Liu, Z., Bao, R., Zhu, S., Mo, L., & Yang, Y. (2025). Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability, 17(3), 1018. https://doi.org/10.3390/su17031018

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