Reservoir Scheduling Using a Multi-Objective Cuckoo Search Algorithm under Climate Change in Jinsha River, China

: Changes in rainfall and streamﬂow due to climate change have an adverse impact on hydropower generation reliability and scheduling of cascade hydropower stations. To estimate the impact of climate change on hydropower, a combination of climate, hydrological, and hydropower scheduling models is needed. Here, we take the Jinsha River as an example to estimate the impact of climate change on total power generation of the cascade hydropower stations and residual load variance of the power grid. These two goals are solved by applying an improved multi-objective cuckoo search algorithm, and a variety of strategies for the optimal dispatch of hydropower stations are adopted to improve the efﬁciency of the algorithm. Using streamﬂow prediction results of CMIP5 climate data, in conjunction with the Xinanjiang model, the estimated results for the next 30 years were obtained. The results indicated that the negative correlation between total power generation and residual load variance under the RCP 2.6 scenario was weaker than that under the RCP 8.5. Moreover, the average power generation and the average residual load variance in RCP 2.6 was signiﬁcantly larger than that in RCP 8.5. Thus, reducing carbon emissions is not only beneﬁcial to ecological sustainability, but also has a positive impact on hydropower generation. Our approaches are also applicable for cascade reservoirs in other river catchments worldwide to estimate impact of climate change on hydropower development.


Introduction
Hydropower is one of the most important renewable, and environmentally sustainable energy resources. It can store energy at low costs, maintains lower operating and maintenance costs, and can operate with great stability [1,2]; thus, when integrated with intermittent energy sources, such as solar and wind, can make a significant contribution to the consistency of the energy grid [3,4]. As shown in the 2019 BP Statistical Review of World Energy, global primary energy consumption has grown rapidly in recent years, with hydroelectric generation increasing by 3.1%; however, carbon emissions grew by 2.0% in 2018, the fastest over the previous seven years. The emissions of greenhouse gases from anthropogenic activity are the primary mechanism driving the changing global climate, imposing severe stress on the global ecosystem with potentially immeasurable consequences. Climate change is having a direct effect on meteorological factors such as temperature, rainfall, and evaporation, and indirectly affects other factors such as soil moisture content and runoff, resulting in the spatiotemporal redistribution of water resources [5]. It is projected that annual mean streamflow will increase in the high-latitudes and wet tropical regions, models, and ultimately be used to assess the impact of climate change on water resources. The conceptual hydrological model is a physical approach to describing the relationship between hydrological factors by generalizing complex phenomena [25,26]. The distributed hydrological model, such as TOP model, SWAT, Mike SHE, and VIC, divides the watershed into multiple sub-simulation units, and performs separate hydrological simulations to better reflect the characteristics of medium-and large-scale watersheds [27,28].
After completing a rainfall runoff simulation, a power system model can be applied to estimate the impacts on energy production. For example, Queiroz et al. introduced spatiotemporal information of future climate change into the operation plan of Brazil's hydropower system to evaluate its impact on the revenue of hydropower stations [29]. By applying GCM projections, the trend of hydropower demand over the next 30 years can be obtained [30]. Other objectives can also be estimated through this method. For example, Zhai et al. predicted the trend of inflow of the Jinsha river through two climate models and carried out a qualitative analysis of the possible strategies of hydropower stations for flood control purposes in the future [31]. Zhang et al. studied the derivation and the adaption of operating rules for the irrigation reservoir under climate change [32].
To best estimate the impact of climate change on hydropower, a combination of climate, hydrological, and power system models is needed. The accuracy of the hydrological model, and the efficiency of the power system model, will determine the accuracy of the climate change impact prediction. Uncertainty of the results mainly comes from three aspects: Observational Uncertainty, Model Structural Uncertainty, and Parametric Uncertainty. In order to evaluate the uncertainty, probabilistic analysis methods, such as Expectancy Method, Utility Function Method, and Model Analysis, are mainly used. Given the uncertain factors a random change within a certain range, analyze and determine the probability distribution of this change, so as to calculate its expected value and standard deviation. The flowchart of the method used in this paper is shown in Figure 1. Details will be explained in the following sections.

NGSA-II
In general, real-world engineering problems must consider multiple goals, and tradeoffs are required for goal optimization; therefore, one must consider the relationship between conflicting targets to obtain the most reasonable multi-objective solution that maximizes total benefits. The concept of "dominance" is commonly used in multi-objective optimization algorithms. Pareto optimal solutions are feasible, and not dominated by other arbitrary solutions in the set. The multidimensional space formed by the target values of all Pareto optimal solutions is the Pareto optimal front; at present, the main methods for obtaining the frontal space are mathematical programming and intelligent algorithms.
Elitist nondominated sorting genetic algorithm v. II (NSGA-II) is a popular genetic algorithm-based multi-objective optimization method. It uses an elite strategy and maintains population diversity through the operator of crowding comparison. The basic process of NSGA-II is: (1) Randomly generate an initial population of size N. Then, sort the initial population using the non-dominated method. (2) Generate new subpopulations through selection, crossover, and mutation operations of the evolutionary algorithm. (3) Merge the parent and child populations and select the most fit individuals to form the next-generation population based on a dominance relationship and crowding degree. (4) Repeat the above processes until the termination condition is met.

Improved Cuckoo Search
The cuckoo search (CS) algorithm is a meta-heuristic method developed by Yang and Deb (2009). It is inspired by the obligate brood parasitism of some cuckoo species [33].  Elitist nondominated sorting genetic algorithm v. II (NSGA-II) is a popula algorithm-based multi-objective optimization method. It uses an elite strategy a tains population diversity through the operator of crowding comparison. The b cess of NSGA-II is: (1) Randomly generate an initial population of size N. Then, sort the initial po using the non-dominated method. (2) Generate new subpopulations through selection, crossover, and mutation op of the evolutionary algorithm. The most important feature of the CS algorithm is the use of Lévy flight characteristics, where a number of studies have shown that the behavior of many insects and larger animals exhibited the typical characteristics of a Lévy flight, shown by a power-law behavior. Inspired by this action, a new solution was generated as follows: Water 2021, 13, 1803

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Research has shown that the strength of the local convergence ability of basic CS insufficiently and several improvements have been proposed to improve its performance: (1) Dynamic parameter adjustment strategy In the basic CS algorithm, the abandon probability (p a ) is a fixed value that affects the convergence speed of the algorithm. The improved solution turns p a into a dynamic adjustment using Equation (2): where p s a and p e a are the start and end of p a , respectively, and CE and NE are the current and maximum evaluation times, respectively.
(2) Differential strategy for Lévy flight Originally, Lévy flights were described according to Equation (3), although the details of its implementation are not provided. Different interpretations of this formula have been suggested by researchers, and in the present study, we implemented the equation using the differential strategy as follows: where x i and x j are randomly chosen nests, sl is the step size, and Levy(u,c) is the sample value generated by the Lévy distribution. The probability density function of Levy(u,c) is calculated according to Equation (4): (

3) Revised solution
The basic CS generates a new solution and compares it with randomly chosen ones from the nests. If the randomly chosen solution is inferior, it is replaced; however, the two solutions are irrelevant, and the conversion is inadequate, and the source solution of the new solution should be replaced accordingly. The pseudo-code of the improved cuckoo search (ICS) Algorithm 1 is as follows:

Algorithm 1. Improved cuckoo search
Objective function f (x), x = (x 1 , . . . , x d ) T Initialize default parameters Generate initial population of n host nests x i (i = 1, 2, . . . , n) While (t < MaxEvaluation) or (stop criterion) Select two solution x i , x j from host nests randomly For d=1, . . . , D do Init the worst nest x worst End if End while

Multi-Objective Cuckoo Search
The multi-objective cuckoo search (MoCS) in the present study was based on the NSGA-II algorithm and the improved cuckoo search algorithm described above. The MoCS procedure is explained as follows ( Figure 2): (1) Generate a random initial population, and classify the individuals using the nondominated sorting method. A new population can then be generated using the ICS algorithm. (2) Merge the parent and child populations, employ a fast, non-dominated sorting and crowding degree calculation in the mixed population, and select the most fit individuals to form the next generation.  The multi-objective cuckoo search (MoCS) in the present study was b NSGA-II algorithm and the improved cuckoo search algorithm described MoCS procedure is explained as follows ( Figure 2): (1) Generate a random initial population, and classify the individuals usi dominated sorting method. A new population can then be generated us algorithm. (2) Merge the parent and child populations, employ a fast, non-dominated crowding degree calculation in the mixed population, and select the mos uals to form the next generation. (3) Repeat the processes above until the termination conditions are met.

Gradient Multi-Objective Cuckoo Search for Reservoir Scheduling
A reasonable scheduling plan can achieve efficient trade-offs for various conflicting objectives [34]. The objectives and constraints of the multi-objective long-term hydropower generation (MLTHG) are described in this section, as is the application of the MoCS for MLTHG.

Power Generation Objective
One of the optimization goals of the MLTHG is to maximize total power generation of cascade hydropower stations, using the following formula Equation (5): where f 1 (kW·h) is the total energy production of the cascade hydropower system, T is the period count, N is the number of reservoirs, k i is the output coefficient of reservoir i, Q i,t (m 3 /s) is the generation of outflow through hydropower units of reservoir i at time t; H i,t (m) is the net water head of hydropower reservoir i at time t, and ∆t is the length of the time interval.

Residual Load Variance Objective
An important function of a hydropower station is to optimally respond to the power grid, thus minimizing the residual load; however, this objective is not easily resolved, and researchers have carried out a variety of targeted improvements towards this end. In the present study, the objective was to minimize the residual load variance of the power grid, according to the following formula Equation (6): where f 2 (kW) is the residual load variance of the power grid, P t (kW) is the total power output of the cascade hydropower stations over period t, and P (kW) is the average power output of all periods. Notably, it is necessary to maintain a low water level during the flood season due to flood control requirements; thus, this season was not considered in the formula.

Constraints
(1) Hydraulic connection (Equation (7)): where I i,t (m 3 /s) is the inflow of reservoir i at time t; O i−1,t (m 3 /s) is the outflow of reservoir i − 1 at time t, R i,t (m 3 /s) is the interval inflow between reservoir i − 1 and i; S i−1,t (m 3 /s) is the spillage of the upstream reservoir i − 1, and Q i−1,t (m 3 /s) is the generation flow of the upstream reservoir i − 1.
(2) Water-balance constraint (Equation (8)): where V i,t (m 3 ) is the storage of reservoir i in time t. (3) Water-level constraints (Equations (9) and (10)): Water 2021, 13, 1803 where Z min i,t (m) and Z max i,t (m) are the lower and upper bounds of the water level, respectively; and Z step i (m) is the limitation of the water-level variation. (4) Outflow constraint (Equation (11)): where O min i,t (m 3 /s) and O max i,t (m 3 /s) are the minimum and maximum outflows of reservoir i, respectively. (5) Output constraint (Equation (12)): where N min i,t (kW·h) and N max i,t (kW·h) are the minimum and maximum power output of reservoir i, respectively. (6) Boundary condition (Equation (13)): where Z i,start (m) and Z i,end (m) are the initial and terminal water levels of reservoir i, respectively.

Solution Encoding and Initialization
When applying the MoCS to address multi-objective power generation scheduling, the water levels or outflows of the hydropower stations are often chosen as the decision variable. In the present study, the water level was employed, and encoded as (Equation (14)): where z d i (m) is the water level at hydropower station i and time d. The initial value of z d i is randomly generated as: where RAND(0-1) is a randomly generated value between 0 and 1; and z max i (m) and z min i (m) are the upper and lower boundaries of reservoir i, respectively. When a new individual is randomly generated, there is a high probability that it will not meet the constraints, and it will be replaced with new individuals until the new individuals meet the constraints.
In the evolution stage, the individual does not meet the constraints and is corrected by adjusting the water level to the feasible range as follows: where z min i and z max i are determined by the water level, outflow, and power output constraints, respectively according to Equations (17)- (19): Water 2021, 13, 1803 where Q min t (m 3 /s) and Q max t (m 3 /s) are the minimum and maximum outflows of the reservoir; Q Pmin t (m 3 /s) is the minimum outflow for a guaranteed output; ∆z (m) is the limitation of the water-level change, Z(V) and V(Z) are the relationship between the water level and the storage.
A combination of forward and reverse corrections was used when applying the correction to an unfeasible range. First, the water level was adjusted from time period 1 to T. If the correction failed midway, attempts were made to correct the water level from period T to the breakpoint. If the correction was unsuccessful, the fitness of the individual was marked as 0.

Gradient Search Strategy
Following the gradient search strategy of the MoCS algorithm, the solution was adjusted by a small gradient, as shown in Figure 3. If the water level of a reservoir changed, both its power generation capacity and its downstream reservoirs changed.
where (m³/s) and (m³/s) are the minimum and maximu ervoir; (m³/s) is the minimum outflow for a guaranteed out itation of the water-level change, and are the relations level and the storage.
A combination of forward and reverse corrections was used w rection to an unfeasible range. First, the water level was adjusted fr If the correction failed midway, attempts were made to correct the w T to the breakpoint. If the correction was unsuccessful, the fitness marked as 0.

Gradient Search Strategy
Following the gradient search strategy of the MoCS algorithm justed by a small gradient, as shown in Figure 3. If the water level o both its power generation capacity and its downstream reservoirs c .
Accordingly, the partial derivative of power generation E (kW• dient Δl is (Equation (21)): where if 0, the water level is adjusted by Δl; otherwise, the a Detail derivation process can be found in [35]. A flowchart of the strategy for the cascade system is shown in Figure 4.
Accordingly, the partial derivative of power generation E (kW·h), with respect to gradient ∆l is (Equation (21)): where if ∂E ∂l > 0, the water level is adjusted by ∆l; otherwise, the adjustment is rejected. Detail derivation process can be found in [35]. A flowchart of the gradient-based search strategy for the cascade system is shown in Figure 4.

Single Entry External Archive
Elite individuals were inserted into the concept of an external archive that was introduced in the MoCS algorithm. When the number of individuals in the archive exceeded the limit, the non-optimal solution was removed based on the non-dominated sorting and crowding degree calculation. Water 2021, 13, x FOR PEER REVIEW 10 of 29

Single Entry External Archive
Elite individuals were inserted into the concept of an external archive that was introduced in the MoCS algorithm. When the number of individuals in the archive exceeded the limit, the non-optimal solution was removed based on the non-dominated sorting and crowding degree calculation.
Notably, if all of the elite individuals were inserted into the external archive at the same time, some may be concentrated. The crowding degrees of these individuals will be small, and there is a high probability that they will be removed during selection, which is not conducive to the generation of the Pareto front. Therefore, it was necessary to insert only one elite individual at a time, with the subsequent individual inserted only after the selection was completed. The Pareto optimal front will be more uniform following this strategy shown in Figure 5. Notably, if all of the elite individuals were inserted into the external archive at the same time, some may be concentrated. The crowding degrees of these individuals will be small, and there is a high probability that they will be removed during selection, which is not conducive to the generation of the Pareto front. Therefore, it was necessary to insert only one elite individual at a time, with the subsequent individual inserted only after the selection was completed. The Pareto optimal front will be more uniform following this strategy shown in Figure 5.

Self-Tuning Divergent Operator Strategy
A divergent operator strategy was introduced to increase population diversity. During new individual generation, a certain number from the population were randomly selected and underwent single-objective evolution separately. Simultaneously, the remaining individuals executed the original evolution selection strategy.
The number of individuals performing single-target evolution (n) was determined by the number of targets ( Figure 6). If the objective maximum changed, the amount of the corresponding single-target individuals increased, and if no change was observed, this number decreased. The purpose of this strategy was to expand the algorithm's ability to search for edges.
Water 2021, 13, x FOR PEER REVIEW 1 Figure 5. Single entry external archive.

Self-Tuning Divergent Operator Strategy
A divergent operator strategy was introduced to increase population diversity ing new individual generation, a certain number from the population were random lected and underwent single-objective evolution separately. Simultaneously, the re ing individuals executed the original evolution selection strategy.
The number of individuals performing single-target evolution (n) was determ by the number of targets ( Figure 6). If the objective maximum changed, the amount corresponding single-target individuals increased, and if no change was observed number decreased. The purpose of this strategy was to expand the algorithmʹs abi search for edges.

Self-Tuning Divergent Operator Strategy
A divergent operator strategy was introduced to increase population diversity ing new individual generation, a certain number from the population were random lected and underwent single-objective evolution separately. Simultaneously, the re ing individuals executed the original evolution selection strategy.
The number of individuals performing single-target evolution (n) was deter by the number of targets ( Figure 6). If the objective maximum changed, the amount corresponding single-target individuals increased, and if no change was observe number decreased. The purpose of this strategy was to expand the algorithmʹs abi search for edges.   Figure 8.
The future climate change scenarios are obtained from the Inter-Sectoral Impact Model Intercomparison Project (ISI-MIP) [36]. Continuous daily precipitation and temperature data on a horizontal grid with 0.5 • × 0.5 • resolution for the period 1960-2099 were offered by five GCMs, including GFDL-ESM2M, HADGEM2-ES, IPSL-CM5A-LR, MIROC-ESM-CHEM, and NORESM1-M. Output of the climate model was spatially interpolated using a bilinear interpolation method, and the data has already been bias-corrected using a trendpreserving bias correction method [37]. The bias correction method modifies the monthly mean and daily variability of the simulated data to match the observations. Change of average daily precipitation and temperature (maximum and minimum) between each GCM and observations are shown in Figures 9-11. The main hydrological model used for streamflow forecasting is the Xinanjiang (XAJ) model. The daily precipitation of each sub-catchment is evaluated following: where P is the average precipitation of the sub-catchment, P i is the precipitation of the grid box i, and S i is the area of grid box i. N is the number of grid boxes in the sub-catchment. The daily potential evapotranspiration of each grid is calculated using Hargreaves method [38]: where H A and H E are the parameters with standard values of 0.0023 and 0.5, respectively. R e is the extraterrestrial radiation. T is the mean temperature (T = (T max + T min )/2) and ∆T is the air temperature range (∆T = T max -T min ). The daily evapotranspiration of each sub-catchment is calculated in the same way with precipitation.
The parameters of XAJ model are calibrated using the observed hydrological data and the calibrate method is based on ICS. The outflow of each sub-catchment is routed using the Muskingum method. The precipitation and temperature data from 1961 to 2050 given by five GCMs in three different scenarios (RCP2.6, RCP4.5, RCP8.5) was input to the parameterized XAJ model, and the daily streamflow in the context of climate change can be simulated. Detailed prediction methods can be found in [39].  (http://data.cma.cn/ accessed on 25 May 2021), respectively. T tional meteorological stations over the Chinese mainland. It c observed data from 1961 to the present. We use data from 208 the Jinsha River, shown in Figure 8. The future climate change scenarios are obtained from Model Intercomparison Project (ISI-MIP) [36]. Continuous da perature data on a horizontal grid with 0.5° × 0.5° resolution for offered by five GCMs, including GFDL-ESM2M, HADGEM2 ROC-ESM-CHEM, and NORESM1-M. Output of the climate m lated using a bilinear interpolation method, and the data has a using a trend-preserving bias correction method [37]. The bias c the monthly mean and daily variability of the simulated data Change of average daily precipitation and temperature (max tween each GCM and observations are shown in Figures 9-11.      The main hydrological model used for streamflow forecasting is the Xinanjiang (XA model. The daily precipitation of each sub-catchment is evaluated following: where P is the average precipitation of the sub-catchment, Pi is the precipitation of grid box i, and Si is the area of grid box i. N is the number of grid boxes in the sub-cat ment.
The daily potential evapotranspiration of each grid is calculated using Hargreav method [38]:

Modeling of MLTHG in Jinsha River
The Jinsha River is in the upper reaches of the Yangtze River. It is born at the junction of Qinghai and Sichuan Province, and ends at the mainstream of the Yangtze River in Yibin. There is strong hydrological potential along the river, particularly in the lower reaches where elevation decreases dramatically. Four large cascade hydropower station-Wudongde, Baihetan, Xiluodu, and Xiangjiaba-are located along the lower reaches of the river (Figure 12). The main primary station parameters are listed in Table 1, and the topology structure is shown in Figure 13. the river (Figure 12). The main primary station parameters topology structure is shown in Figure 13.  There are two hydrological stations in the basin-H streamflow of the four hydropower stations was projected The inflow of Wudongde was generalized using the strea and the interval inflow between Baihetan and Xiluodu was flow of the Pingshan station according to Equations (22) an  the river (Figure 12). The main primary station parameters are listed in Table 1, and the topology structure is shown in Figure 13.  There are two hydrological stations in the basin-Huatan and Pinshan, and the streamflow of the four hydropower stations was projected using the Xinanjiang model. The inflow of Wudongde was generalized using the streamflow of the Huatan station, and the interval inflow between Baihetan and Xiluodu was generalized using the streamflow of the Pingshan station according to Equations (22) and (23): The power generation at time t can be obtained by Equation (24): There are two hydrological stations in the basin-Huatan and Pinshan, and the streamflow of the four hydropower stations was projected using the Xinanjiang model. The inflow of Wudongde was generalized using the streamflow of the Huatan station, and the interval inflow between Baihetan and Xiluodu was generalized using the streamflow of the Pingshan station according to Equations (22) and (23): The power generation at time t can be obtained by Equation (24): where E i,t (kW·h) is the hydropower generation of reservoir i at time t; Q i,t (m 3 /s) is the generation flow through the hydropower units of reservoir i at time t; k i is the comprehensive benefit coefficient of reservoir i, which reflects the hydro-generating unit efficiency; ∆t is the length of time interval, and H i,t (m) is the net water head of reservoir i at time t, defined by Equation (25): where Z i,t−1 (m) and Z i,t (m) are the beginning and end water levels of reservoir i at time t, respectively; Zd i,t (m) is the water level under the dam; and ∆H i (m) is the water head loss of reservoir i. Overall, the objective of the MLTHG scheduling model was to find an optimal set of water releases or storage volumes to maximize the MLTHG of cascade hydropower stations, which can be described mathematically according to Equation (26): where E (kW·h) is the total hydropower generation of the cascade reservoirs, T is the number of time periods, and N is the number of reservoirs.

Performance of GMoCS
To evaluate the performance of the GMoCS, the optimal results of the cascade hydropower stations were compared with those obtained using MoCS. The parameters of GMoCS were: ns = 40, p s a = 0.3, p e a = 0.1, sl = 0.01, u = 0, c = 1.5, and the maximum evaluation time was set to 12000. The initial and end water levels of the reservoirs were set to their normal levels (Table 1), and each month was divided into three periods of~10 days each, resulting in 36 time intervals per year. Each method was run 100 times, independently.
Three typical streamflows of Pingshan station (Figure 14) chosen by the annual runoff frequency curve -dry (frequency 75%), normal (50%), and wet (25%) years-are used as the input of MLTHG. Water levels of the four reservoirs at different time intervals (excluding the 36th) were chosen as the optimization parameters, making the problem dimension = 140. Equations (5) and (6) are the fitness functions of MLTHG, and the greater the power generation and the smaller the residual load variance, the better the solution. The simulation results are shown in Figure 15.
The Pareto optimal frontier obtained by GMoCS was superior to the results of MoCS (Figure 15). MoCS fell into a local optimum when solving the MLTHG; whereas the results obtained by GMoCS indicated that the strategies proposed in the present study effectively improved the algorithm's search ability. It can be seen that total power generation of the cascade hydropower stations and residual load variance of the power grid were inversely correlated. These two objectives were conflicting and mutual. The scheduling processes of the 2nd and 49th solutions of wet year are shown in Figure 16 as an example. The difference between the two solutions is mainly in the period before the flood season (June, July). Due to the large inflow during the flood season, the power generation will increase significantly. The 2nd solution increases the power generation capacity for a period of time before the flood season by lowering the water level in advance. This makes the growth of power generation before the flood season relatively smooth, which is conducive to reducing the residual load variance of the power grid. The 49th solution maintains high water level for the period before flood season, resulting in an increase in the total power generation. While the residual load variance of the power grid is larger than that in 2nd solution.
In this section, we focus on the trade-off between the two objectives of power generation and residual load variance. Research done by Geressu et al. [34] shows that when the financial benefit of the reservoirs is increased, the downstream release will be adversely affected. This is similar to our research. Only a compromise plan can be chosen if we consider two conflicting goals. Moreover, as explained by Zhai et al. [31], during the flood season the cascade hydropower stations are operated below flood-limiting water level in order to reserve adequate reservoir storage for flood prevention. We also considered the same restriction, the change of the water level is mainly in the non-flood season as shown in Figure 16.

Climate Change on Multi-Objective Scheduling of Cascade Hydropower Stations
The impact of climate change on the total power generation of the cascade hydropower stations and residual load variance of the power grid over the next 30 years was estimated. First, streamflow prediction data of Pingshan station for the years 2021-2050 were prepared (Figure 17), and prediction results were obtained by applying the CMIP5 climate data and the Xinanjiang model (detailed prediction methods can be found in [39]). Then, the multi-objective scheduling results under three climate change scenarios-RCP2.6, RCP4.5, and RCP8.5-using five climate models-GFDL-ESM2M, HADGEM2-ES, IPSL-CM5A-LR, MIROC-ESM-CHEM, and NORESM1-M-were obtained with 30 years of daily streamflow data as the input. Accordingly, 450 solution sets were obtained. Owing to the inaccuracy of the predictions of various climate models, the Pareto frontiers also showed large levels of uncertainty.   The Pareto optimal frontier obtained by GMoCS was superior to the results of MoCS (Figure 15). MoCS fell into a local optimum when solving the MLTHG; whereas the results obtained by GMoCS indicated that the strategies proposed in the present study effectively improved the algorithm's search ability. It can be seen that total power generation of the climate data and the Xinanjiang model (detailed prediction methods can be found in [39]). Then, the multi-objective scheduling results under three climate change scenarios-RCP2.6, RCP4.5, and RCP8.5-using five climate models-GFDL-ESM2M, HADGEM2-ES, IPSL-CM5A-LR, MIROC-ESM-CHEM, and NORESM1-M-were obtained with 30 years of daily streamflow data as the input. Accordingly, 450 solution sets were obtained.
Owing to the inaccuracy of the predictions of various climate models, the Pareto frontiers also showed large levels of uncertainty.

Power Generation Objective
As explained in Section 2.3.7, the boundary of the Pareto frontier is close to that obtained through single-objective optimization. First, the optimal solution for power generation was obtained with the operator. Figure 18 displays average total power generation results per 10 years for the next 30 years under the three climate change scenarios, and Figure 18a shows the average generation results for scenario RCP2.6. The average generation results of the five GCMs were 2266, 2238, and 2188 (10 8 kWh), respectively, while the uncertainty ranges were similar to each other: (−4.4%, 4.0%), (−5.2%, 3.9%), and (−4.3%, 5.4%). Figure 18b,c show the results under scenarios RCP4.5 and RCP8.5, which both displayed much larger uncertainty ratios than RCP2.6. Moreover, the average power generation in RCP2.6 was significantly larger than that in RCP8.5, and when compared with the average power generation in RCP4.5, RCP2.6 was larger for the first 20 years, but smaller following decade.
tained through single-objective optimization. First, the optimal solution for power generation was obtained with the operator. Figure 18 displays average total power generation results per 10 years for the next 30 years under the three climate change scenarios, and Figure 18a shows the average generation results for scenario RCP2.6. The average generation results of the five GCMs were 2266, 2238, and 2188 (10 8 kWh), respectively, while the uncertainty ranges were similar to each other: (−4.4%, 4.0%), (−5.2%, 3.9%), and (−4.3%, 5.4%). Figure 18b,c show the results under scenarios RCP4.5 and RCP8.5, which both displayed much larger uncertainty ratios than RCP2.6. Moreover, the average power generation in RCP2.6 was significantly larger than that in RCP8.5, and when compared with the average power generation in RCP4.5, RCP2.6 was larger for the first 20 years, but smaller following decade.

Residual Load Variance Objective
The optimal solution for the residual load variance was obtained by using the respective operator. Figure 19 displays the predicted results for the next 30 years under each scenario. Figure 19a shows the results under scenario RCP2.6. The average residual load variance results for the five GCMs were 910, 916, and 867 (10 8 kWh), for 2021-2030, 2031-2040, 2041-2050, respectively; while the uncertainty ranges were (13.6%, 6.0%), (6.1, 6.5%) and (−9.7%, 17.5%). Figure 19b,c show the results under scenarios RCP4.5 and RCP8.5, respectively, with relatively larger uncertainty ranges. Similar to the characteristics of power generation in the preceding section, the average residual load variance in RCP2.6 was larger than RCP8.5 at each interval, and larger than RCP4.5 for the first 20 years, but smaller following decade.

Residual Load Variance Objective
The optimal solution for the residual load variance was obtained by using the respective operator. Figure 19 displays the predicted results for the next 30 years under each scenario. Figure 19a shows the results under scenario RCP2.6. The average residual load variance results for the five GCMs were 910, 916, and 867 (10 8 kWh), for 2021-2030, 2031-2040, 2041-2050, respectively; while the uncertainty ranges were (13.6%, 6.0%), (6.1, 6.5%) and (−9.7%, 17.5%). Figure 19b,c show the results under scenarios RCP4.5 and RCP8.5, respectively, with relatively larger uncertainty ranges. Similar to the characteristics of power generation in the preceding section, the average residual load variance in RCP2.6 was larger than RCP8.5 at each interval, and larger than RCP4.5 for the first 20 years, but smaller following decade.

Combination of the Two Objectives
The impact of climate change on the combination of the two objectives was evaluated using the contradiction between the two goals (Equation (29)

Combination of the Two Objectives
The impact of climate change on the combination of the two objectives was evaluated using the contradiction between the two goals (Equation (29)): where C is the variation ratio; f max is the maximum (minimum) value of the solution i. When C was larger, the reduced power generation due to the increase in the residual load variance was larger, implying a clear contradiction between the two goals; however, when C was smaller, the reduced power generation due to the increase in the residual load variance was decreased, implying a more subtle contradiction between the two objectives. Figure 20 shows the predicted variation ratios of the total power generation and residual load variance over the next 30 years, at decadal intervals. The results indicated that: under scenario RCP2.6 scenario indicated the average ratio of change between 2021-2030 and 2041-2050 was similar; under scenario RCP4.5, the average variation ratio in 2021-2030 was significantly higher than that in 2031-2040 or 2041-2050; however, the variation ratios in 2031-2040 and 2041-2050 were similar to those under RCP2.6; and the average variation ratio under the RCP8.5 scenario was even higher than that of RCP2.6.
In this section, we focus on the impact of climate change on two objectives: total power generation and residual load variance. The key factor associated with climate change and hydropower generation is streamflow, which is the same with the research done by Zhang [32] and Zhai [31]. However, due to the different climate scenarios selected, there is a difference in the predicted results. Zhang used a climate scenario with reduced rainfall and increased evaporation and concluded that agricultural profits are declining. Zhai adopted a climate scenario with increased discharge and concluded that extreme hydrological events are increasing. Trend of the objectives that are greatly affected by streamflow is generally depended on the trend of streamflow. This is also reflected in our article.
As can be seen in this paper, the results of RCP2.6 were more positive than those of RCP8.5. This is the same with the research done by Nam [20]. Nam carried out the assessment of vulnerable seasons for paddy irrigation under two climate change scenarios: RCP4.5 and RCP8.5. Results show that the total duration of the vulnerable irrigation seasons is longer in RCP8.5 than that in RCP4.5. Therefore, it is necessary to reduce carbon emissions to mitigate the adverse effects of climate change.

Combination of the Two Objectives
The impact of climate change on the combination of the two objectives was evaluated using the contradiction between the two goals (Equation (29) where C is the variation ratio; and are the maximum and minimum values of the power generation objective in the solution set, respectively; and are the maximum and minimum values of the residual load variance objective in the solution set, respectively; , is the maximum (minimum) value of the solution i. When C was larger, the reduced power generation due to the increase in the residual load variance was larger, implying a clear contradiction between the two goals; however, when C was smaller, the reduced power generation due to the increase in the residual load variance was decreased, implying a more subtle contradiction between the two objectives. Figure 20 shows the predicted variation ratios of the total power generation and residual load variance over the next 30 years, at decadal intervals. The results indicated that: under scenario RCP2.6 scenario indicated the average ratio of change between 2021-2030 and 2041-2050 was similar; under scenario RCP4.5, the average variation ratio in 2021-2030 was significantly higher than that in 2031-2040 or 2041-2050; however, the variation ratios in 2031-2040 and 2041-2050 were similar to those under RCP2.6; and the average variation ratio under the RCP8.5 scenario was even higher than that of RCP2.6. In this section, we focus on the impact of climate change on two objectives: total power generation and residual load variance. The key factor associated with climate change and hydropower generation is streamflow, which is the same with the research done by Zhang [32] and Zhai [31]. However, due to the different climate scenarios selected, there is a difference in the predicted results. Zhang used a climate scenario with reduced rainfall and increased evaporation and concluded that agricultural profits are declining. Zhai adopted a climate scenario with increased discharge and concluded that extreme hydrological events are increasing. Trend of the objectives that are greatly affected by streamflow is generally depended on the trend of streamflow. This is also reflected in our article.
As can be seen in this paper, the results of RCP2.6 were more positive than those of RCP8.5. This is the same with the research done by Nam [20]. Nam carried out the assessment of vulnerable seasons for paddy irrigation under two climate change scenarios: RCP4.5 and RCP8.5. Results show that the total duration of the vulnerable irrigation seasons is longer in RCP8.5 than that in RCP4.5. Therefore, it is necessary to reduce carbon emissions to mitigate the adverse effects of climate change.

Conclusions
In the present research, a method to estimate the impact of climate change on hydropower was discussed, and a comprehensive impact of climate change assessment was made on the power generation of cascade hydropower stations in the lower reaches of the Jinsha River and residual load variance of the power grid. Climate change is affecting the amount of available water for power generation in a variety of ways, thereby altering the cascade of hydropower stations. The most common method to estimate the impacts of climate change on hydropower generation is to run a hydrological model under climate

Conclusions
In the present research, a method to estimate the impact of climate change on hydropower was discussed, and a comprehensive impact of climate change assessment was made on the power generation of cascade hydropower stations in the lower reaches of the Jinsha River and residual load variance of the power grid. Climate change is affecting the amount of available water for power generation in a variety of ways, thereby altering the cascade of hydropower stations. The most common method to estimate the impacts of climate change on hydropower generation is to run a hydrological model under climate change conditions, obtain the changes in streamflow, and then assess the correlated impact on hydropower. An improved multi-objective cuckoo search algorithm has been proposed, and a variety of strategies for the optimal dispatch of hydropower stations, such as the gradient search strategy, single-entry external archive, and self-turning divergent operator strategy, were adopted to improve the efficiency of the algorithm.
In the case studies, the improved and unimproved algorithms were compared. The optimal results of the three typical years obtained by MoCS and GMoCS showed that the Pareto optimal frontier obtained by GMoCS was vastly superior to the results obtained by MoCS. The results also showed that the trends of total power generation of cascade hydropower stations and residual load variance of power grid were inversely correlated. After simulating the average variation ratios of five climate models under the three climate change scenarios, it was revealed that the strength of this relationship under the RCP 2.6 scenario was weaker than that under the RCP 8.5.
Overall, the impact of climate change on hydropower is reflected in the total power generation, as well as other indicators, including residual load variance. Although the results obtained through different climate models did not produce a unified conclusion, the results of RCP2.6 were more positive than those of RCP8.5, indicating that reducing carbon emissions is not only beneficial to ecological sustainability, but also has a positive impact on hydropower generation. Our approaches are also applicable for cascade reservoirs in other river catchments worldwide to estimate impact of climate change on hydropower development.