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

: 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.


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 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 systembased 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.

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: where A is the hydropower station's output coefficient; Q t is its power-generating flow during time period t; H t is its head during time period t; ∆t is the length of each time period; and T is the total number of time periods.
(2) Constraints a. Water-balance constraint where V t and V t−1 are the reservoir capacities at the end of the time period on days t and t − 1, respectively; I t is the average inflow of the reservoir on day t; Q t is the average outflow of the reservoir on day t; and ∆t is the duration of a single time period.
where Z t and Z t−1 are the water levels at the end of the time period on days t and t − 1, respectively; ∆Z is the maximum possible variation in water level during the adjacent period.

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 where a and b are the upper and lower limits of the search space, p i and − p 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): where x T is a set of candidate solutions and a and 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, where x s is a set of candidate solutions and µ, 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): where e rand(µ i ) is the random exponential number, µ i is the distance between the stepped leader discharger p L and the spatial discharger p S i .If the energy E s i_new of the new spatial discharger p s i_new is greater than the energy E S i of the original spatial discharger p S i , then p S i is updated to position p s i_new .Otherwise, p S i 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 where µ is the shape parameter and σ is the scale parameter, reflecting the mining capacity at the current location.In the stepped leader discharger model, σ decreases exponentially as the discharger approaches the ground.Therefore, the direction of the stepped leader discharger p L in the next iteration is where normrand µ L , σ L is a random number from the generated normal distribution.If the energy E L new of the new stepped leader discharger p L new is greater than the energy E L of the original stepped leader discharger p L , p L is updated to take position p L new .Otherwise, p L remains unchanged until the next update.

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]: where α ∈ (0, 1).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, α = 0.49, and the meta-individuals are X = (x 1 , x 2 , . . . ,x m ), x i ∈ (0, 1), i = 1, 2, . . ., m, x i ∈ (0, 1), and i = 1, 2, . . ., m.
(2) The initial population is then obtained by mapping the meta-individuals to the decision-variable value space as shown in Equation ( 14): where a i and b i are the next and previous terms in the space, respectively, x i is the value of the meta-individual in ith dimensional space, and y 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 X 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 X l and least fit individual X w are selected.Finally, an update operation is performed.The worst individual in each subpopulation is updated as shown in Equation (15): where X w is the position of the updated individual and S is the individual jump.
Step length is calculated as shown in Equation ( 16): where rand is a (0, 1) uniformly distributed random number.
Step 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, X w is replaced by X new ; 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, X w is replaced by X new ; 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): where rand 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, X = (x 1 , x 2 , . . . ,x n ), n is the number of individuals, and the positions are updated as shown in Equation ( 19): 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, X n−i is replaced by X new ; otherwise, it is updated according to Equation (20), and individual fitness is again updated.If the new position is better than the original position, X n−i is replaced by X new ; 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): x where v i is the velocity of particle i, x i is the position of particle i, k is the number of iterations, w, c 1 , and c 2 are constants, r 1 and r 2 are (0, 1) uniformly distributed random numbers, p i is the best position in the path traveled by particle i, and p 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).

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 θ 1 , and particle discharger probability θ 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 powergeneration 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 E L of the stepped leader discharger p 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 E L new of the updated stepped leader discharger p L new exceeds that E L of the original stepped leader discharger p 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 E s i_new of new spatial discharger p s i_new exceeds that E S i of the original spatial discharger p S i , 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.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 θ , and particle discharger probability θ , 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

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 c 1 = 1.5, c 2 = 2, and w = 0.7.For DEPSO, the constants were CR = 0.5 and F = 0.8.The frog-leaping probability was θ 1 = 0.55 and particle discharger probability was θ 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 is a summary of the parameters of all algorithms and Table 3 shows the running environment, including hardware and software.The optimization results and convergence curves are shown in Figure 2 and Table 4, 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.Based on this comparison (Figure 2), for the 10 commonly used test functions, FPLSA achieved fast and accurate optimization, performing significantly be er 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.

Case Study
The Jinsha River is an important part of the upper reaches of the Yang e 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 bo om) 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 Yang e River Basin (Figure 3).Because WuDongDe and Bai-HeTan 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.

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.
charge management in the Yang e River Basin (Figure 3).Because Wu HeTan began full operation relatively late with insufficient data, the XiLu and Three Gorges reservoirs were selected for analysis in this study, con prehensiveness of the available data.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.

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.Tables 8-10 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.
ater 2023, 15, x FOR PEER REVIEW PSO, and DEPSO, respectively.For the dry year, the standard deviations 88.33%, and 66.13% lower than those of the LSA, PSO, and DEPSO, respecti Boxplots illustrate size relationships and the discreteness of the data, a reflect the advantages and disadvantages of various algorithms.The FPLS rower interquartile range of power generation than the other algorithms indicating its greater stability (Figure 5); it achieved higher average power ge the other algorithms, indicating that it achieves optimization more effecti other algorithms.Moreover, the FPLSA generated fewer discrete points t algorithms, indicating that it was less likely to fall into local optima.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.
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.Figures 6-8 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 be er 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 be er 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 opti- Figures 6-8 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.
year, the FPLSA converged to the almost optimal value after 20 iterations, and escaped local optima to a be er 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 be er 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 be er than the other algorithms in dry years.

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

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

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.
Figures 9-12 illustrate the power generation output processes for the XiLuoDu, Xi-angJiaBa, 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.Figures 13-15 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.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 pa ern 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 trend satisfies the hydropower operation rules.
(a) Inflow process (b) Outflow process

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 Tables 5-7 and Figures 4-8, 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 Figures 9-15, 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.

Figure 3 .
Figure 3.The Yang e River Basin, China, showing the locations of the hydropower stations on the River.

Figure 3 .
Figure 3.The Yang e River Basin, China, showing the locations of the hydropo Jinsha River.

Figure 3 .
Figure 3.The Yangtze River Basin, China, showing the locations of the hydropower stations on the Jinsha River.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 m 3 , 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 m 3 , 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 m 3 , and an average multi-year power-generation capacity of about 88 × 109 kW•h.Tables 5-7 describe these hydropower stations.

Figure 4 .
Figure 4. Standard deviation of power generation in different typical years.

Figure 4 .
Figure 4. Standard deviation of power generation in different typical years.

Figure 5 .
Figure 5. Boxplots of power generation in different typical years.

Figure 5 .
Figure 5. Boxplots of power generation in different typical years.

Figure 11 .
Figure 11.Power generation output process of Three Gorges Hydropower Station.

Figure 12 .
Figure 12.Total power generation output process.
Water 2023, 15, x FOR PEER REVIEW 20 of 24

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

Table 3 .
Running environment, including hardware and software.

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