Multi-Objective Optimal Dispatch of Hydro-Wind-Solar Systems Using Hyper-Dominance Evolutionary Algorithm

Abstract

In response to the challenge of multi-objective optimal scheduling and efficient solution of hydropower stations under large-scale renewable energy integration, this study develops a multi-objective optimization model with the dual goals of maximizing total power generation and minimizing the variance of residual load. Four complementarity evaluation indicators are used to analyze the wind–solar complementarity characteristics. Building upon this foundation, Hyper-dominance Evolutionary Algorithm (HEA)—capable of efficiently solving high-dimensional problems—is introduced for the first time in the context of wind–solar–hydropower integrated scheduling. The case study results show that the HEA performs better than the benchmark algorithms, with the best mean Hypervolume and Inverted Generational Distance Plus across nine Walking Fish Group (WFG) series test functions. For the hydro-wind-solar scheduling problem, HEA obtains Pareto frontier solutions with both maximum power generation and minimal residual load variance, thus effectively solving the multi-objective scheduling problem of the hydropower system. This work provides a valuable reference for modeling and efficiently solving the multi-objective scheduling problem of hydropower in the context of emerging power systems. This work provides a valuable reference for the modeling and efficient solution of hydropower multi-objective scheduling problems in the context of emerging power systems.

1. Introduction

In response to the urgent need to address global climate change and achieve the ‘dual carbon’ goals, building a new power system dominated by renewable energy is the core strategy of China’s energy transition [1]. By the end of 2023, China’s total installed power generation capacity reached 292 million kilowatts, with wind and solar power accounting for over 36% of the total, marking the first time that clean energy installed capacity exceeded 50% [2]. However, the strong randomness, intermittency, and volatility of wind and solar power output pose significant challenges to grid peak regulation and stable operation [3]. The frequent occurrence of curtailed wind and solar power due to large-scale grid connection highlights the inadequacy of energy consumption and system regulation capabilities. In 2023, China’s curtailment rate for wind and solar power remained at 3.8% [4]. As a flexible and technically mature clean energy source, hydropower is transitioning from its traditional role as a ‘power provider’ to a dual role as a ‘power provider and regulator’ [4,5]. However, current dispatch practices face some challenges. Multi-objective conflicts are difficult to coordinate, and there are significant trade-offs between peak load regulation requirements, economic benefits, and ecological constraints [6]. Additionally, intelligent algorithms converge prematurely, and algorithms such as particle swarm optimization (PSO) are prone to getting stuck in local optima [7].

Renewable energy sources such as wind power and solar power often exhibit strong volatility, randomness, and intermittency. The uncertainty of wind and solar power output poses significant challenges to the safe and stable operation of power systems, thereby limiting their ability to integrate renewable energy [8]. Hydropower, as a flexible and rapidly responsive peak-shaving resource, can be integrated with wind and solar power to form a joint operation system, which is currently one of the most effective ways to solve the problem of large-scale intermittent energy consumption [9]. In response to the randomness and volatility of renewable energy output, some scholars have conducted research on multi-energy complementary scheduling with hydropower as the dominant resource, based on hydropower plant generation scheduling. For example, Ming et al. [10] established a hydro-solar complementary optimized dispatch model based on the criterion of renewable energy integration; Zhu et al. [11] considered the uncertainty of photovoltaic output and developed a short-term hydro-solar complementary dispatch model that balances power generation and output fluctuations; Zhang et al. [12] adopted a multi-objective optimized dispatch model with the objectives of maximizing the total power generation of the complementary system and minimizing the coefficient of variation in the output process; Me et al. [13] combined randomness and fuzziness based on cloud model concepts to construct a power cloud coupling model for wind-solar-hydro complementary power generation systems. These studies demonstrated the feasibility and effectiveness of hydroelectric power combined with wind and solar power stations from the perspective of power plant output characteristics.

Additionally, some scholars have focused on grid safety and stability, conducting a series of discussions on hydropower dispatch and operation issues under the context of renewable energy grid integration. Zhang et al. [14] established a random expected value peak-shaving model with the objective of minimizing residual load variance, achieving short-term joint dispatch of hydropower, wind power, and solar power; Jin et al. [15] in order to cope with the peaking pressure of stable grid operation under large-scale grid-connected power generation of new energy sources, constructed a hydro-wind-solar optimal scheduling model with long- and short-term coupling of regional grids and explored the impact of different wind-solar penetration rates on the peaking operation of hydropower.

Regarding solution algorithms, traditional optimization algorithms often involve significant computational complexity and are prone to the ‘curse of dimensionality’ when solving high-dimensional problems [16]. The hydro-wind-solar complementary optimization dispatch problem is precisely a high-dimensional optimization problem, making it difficult to solve using traditional optimization algorithms. Intelligent optimization algorithms offer a new approach to addressing this issue. Intelligent algorithms have rapidly developed over the past two decades due to their advantages of strong adaptability, robustness, and ease of parallel computation [17]. Compared to traditional optimization algorithms, they do not require an exact model of the problem itself, nor do they require the function to be continuous or differentiable, making them suitable for solving problems with highly complex objective functions or constraints [18,19]. However, they still have certain limitations. Genetic algorithms are simple in principle and easy to operate, but they are prone to getting stuck in local optima. Several commonly used particle swarm algorithms [20] can be processed in parallel and have the advantages of fast computation speed and good convergence performance when solving high-dimensional optimization problems, but they also have the drawbacks of premature convergence and getting stuck in local optima. Yazdi et al. [21] tackled the multi-objective design-parameter optimization problem for cascade reservoir systems by introducing a non-dominated sorting differential evolution algorithm, achieving effective Pareto front search under multiple objectives while reducing computational cost.

Against this backdrop, this study introduces a multi-objective algorithm based on hyper-dominance, the Hyper-dominance Evolutionary Algorithm (HEA). which was proposed by Liu et al. in 2023, HEA dynamically controls the selection pressure through hyper-dominance, combines angular diversity strategy with population reselection, and efficiently solves the balance between convergence and diversity in high-dimensional objective optimization problems [22]. As the algorithm is relatively new and has not been cited by other scholars for the time being, this paper is the first attempt to apply the algorithm to the field of hydro-wind-solar multi-energy complementary optimization scheduling.

2. Research Objects and Methods

As of the end of 2024, hydropower accounts for 56.6% of Yunnan’s total installed capacity, while new energy sources account for 33.6%, making it a typical power market dominated by hydropower and clean energy. The total installed hydropower capacity in the province has exceeded 80 million kW. How to effectively leverage the flexibility of hydropower and facilitate its transformation has become a key focus of current research.

This study focuses on a hydropower station in Yunnan Province, aiming to provide a theoretical basis for the actual operation of the power station, effectively suppress the volatility and intermittency of wind power generation by giving full play to the flexible regulation capability of hydropower, so as to enhance the comprehensive power generation efficiency of the power station, ensure the safe and stable operation of the power grid, and ultimately provide a practical and operable basis and decision-making support for the construction of a new type of electric power system and the realization of the energy transition to green and low-carbon energy in Yunnan Province and even in the whole country. and decision-making support for building a new type of power system and realizing green and low-carbon energy transformation in Yunnan Province and the whole country.

2.1. Analysis of Wind and Solar Scenarios

2.1.1. Analysis of Output Change Trends Within the Year

China’s wind and solar power generation have continued to grow rapidly, with strong momentum in both installed capacity and grid integration. However, due to the inherent variability, randomness, and intermittency of wind and solar energy, significant challenges are posed to power system operations. The long-term output characteristics of wind power are influenced by factors such as seasonal variations and interannual fluctuations caused by climatic conditions [23]. For solar output, the most relevant factors are weather types and ambient temperature, which determine the seasonal behavior of solar output—specifically, its output characteristics vary as weather and temperature shift throughout the year [24].

The variation trend of wind power and solar power integration at the hydropower station in 2022 is shown in Figure 1 and Figure 2, respectively. It can be observed that wind energy exhibits significant random fluctuations while generally showing a seasonal trend of being stronger in winter and weaker in summer. As a region strongly influenced by monsoons, Yunnan experiences gradually intensifying winter winds that peak around January. In contrast, the summer monsoon develops in April and May as the Asian continent rapidly heats up, forming a thermal low-pressure system. Wind speeds reach their minimum around August. The distinction between dry and wet seasons is also pronounced: from November to April is Yunnan’s dry season, during which wind speeds are higher than in the rainy season, especially from February to April, when strong winds often occur on clear afternoons.

Solar power in Yunnan also shows high stochastic variability, with a general seasonal pattern of being stronger in spring and weaker in summer [25]. This trend is closely related to Yunnan’s climatic conditions. The rainy season spans from May to October, and the dry season from November to the following April. During the rainy season, overcast and rainy weather results in limited sunlight and reduced solar radiation, whereas the dry season is characterized by clear skies and ample sunshine, leading to higher solar potential. The fluctuation characteristics revealed by the wind and solar output curves in Yunnan are well aligned with the region’s climatic features.

2.1.2. Complementarity Assessment of Wind and Solar Power

This study adopts four complementary evaluation indicators to comprehensively analyze the complementarity and synergy between wind and solar power outputs: Pearson correlation coefficient, standard deviation-based complementarity rate (R-SD), Richard-Baker Flashiness complementarity rate (RBF), and first-order difference complementarity rate (R-FD). These indicators enable a multi-dimensional assessment of wind-solar characteristics from linear, global, local, and inter-temporal perspectives.

(1)

Pearson Correlation Coefficient

The Pearson correlation coefficient is a classical measure used to quantify the linear relationship between two random variables. It is defined as:

$$r={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^I\left(X_i-\overline{X}\right)\left(Y_i-\overline{Y}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^I{(X_i-\overline{X})}^2}\sqrt{{{\displaystyle \sum }}_{i=1}^I{(Y_i-\overline{Y})}^2}}}$$

(1)

where $X_i$ and $Y_i$ are the power outputs of variables X and Y at time $i$, and $\overline{X}$$\overline{Y}$ are their respective means.

The coefficient value ranges from −1 to 1. A value of 0 indicates no linear correlation; A positive value (0, 1] implies a strong positive linear relationship, where both variables increase together—while a negative value [−1, 0) indicates a strong inverse relationship, where one variable increases as the other decreases.

(2)

Standard Deviation-Based Complementarity Rate

As a key metric of time series fluctuation, standard deviation measures the dispersion of data around its mean. A larger standard deviation indicates stronger fluctuations. For a time series $X\left(x_1,x_2,\dots ,x_T\right)$, the standard deviation is defined as:

$$\sigma (X)=\sqrt{{\displaystyle \frac{1}{T}}\sum _{t=1}^T{\left(x_t-\overline{x}\right)}^2}$$

(2)

where $\sigma (X)$ denotes the standard deviation of sequence x, T represents the number of segments defined within the time window, $\overline{x}$ is the arithmetic mean of the sequence.

This paper proposes a fluctuation-based method to quantify wind-solar complementarity. Let the wind and solar output sequences be $N_w(t)$ and $N_s(t)$, respectively, and the combined output sequence be $N_{ws}(t)=N_w(t)+N_s(t)$ The criterion for evaluating complementarity is as follows:

If the fluctuation of $N_{ws}(t)$ equals the sum of the fluctuations of $N_w(t)$ and $N_s(t)$, the two outputs vary in the same direction, indicating no complementarity. Conversely, if the fluctuation of $N_{ws}(t)$ is less than the sum of the individual fluctuations, this implies anti-phase variation and the presence of complementary potential. Based on this, the standard deviation complementarity rate is defined as:

$$R-SD={\displaystyle \frac{\left(\sigma \left(N^{\mathrm{w}}\right)+\sigma \left(N^s\right)\right))-\sigma (N^{sp})}{\left(\sigma \left(N^{\mathrm{w}}\right)+\sigma \left(N^s\right)\right)}}\times 100\%$$

(3)

where $R-SD$ denotes the standard deviation complementarity rate of wind and solar outputs; $\sigma \left(N^{\mathrm{w}}\right)$, $\sigma \left(N^s\right)$, and $\sigma \left(N^{ws}\right)$ represent the standard deviations of the wind, solar, and combined wind-solar output sequences, respectively.

The metric $R-SD$ reflects the fluctuation intensity of individual and combined outputs. Its value ranges from 0 to 1. The closer $R-SD$ is to 1, the better the wind-solar complementarity. An ideal complementary state occurs when $R-SD$ = 1, indicating that fluctuations fully offset each other. When $R-SD$ = 0, there is no evidence of complementarity during the observed time period.

(3)

Richard-Baker Flashiness Complementarity Rate

The RBF index is a single-pass fluctuation analysis method used to quantify the transient fluctuation characteristics of power output sequences. Given a time series $X\left(x_1,x_2,\dots ,x_T\right)$, the RBF index is calculated as:

$$RBF\left(X\right)=\sum _{t=2}^{T-1}0.5\times \left(\left|x_t-x_{t-1}\right|+\left|x_{t+1}-x_t\right|\right)/\sum _{t=1}^T{x}_t$$

(4)

where $RBF\left(X\right)$ denotes the RBF index value of time series X

The complementarity criterion based on the RBF index is as follows: When the wind and solar output sequences $N_w(t)$ and$N_s(t)$ are completely synchronized over a given time period, the RBF index of the combined output sequence $N_{ws}(t)$ equals the sum of the individual RBF indices. In this case, wind and solar outputs exhibit no complementarity. Conversely, if the RBF index of $N_{ws}(t)$ is lower than the sum of the individual indices, it indicates the presence of anti-phase fluctuations and complementary potential. The formula is as follows:

$$R-RBF={\displaystyle \frac{\left(RBF\left(N^{\mathrm{w}}\right)+RBF\left(N^s\right)\right)-RBF\left(N^{\mathrm{ws}}\right)}{\left(RBF\left(N^{\mathrm{w}}\right)+RBF\left(N^s\right)\right)}}\times 100\%$$

(5)

where $R-RBF$ denotes the complementarity rate based on the RBF index; $RBF\left(N^{\mathrm{w}}\right)$, $RBF\left(N^s\right)$, and $RBF\left(N^{\mathrm{ws}}\right)$ represent the RBF indices of wind power, solar power, and their combined output sequence, respectively.

The value of $R-RBF$ ranges from 0 to 1. A value approaching 1 indicates stronger complementarity. When $R-RBF$= 1, the output fluctuations of wind and solar are perfectly offset. In contrast, $R-RBF$= 0 implies no complementarity under the given time condition.

(4)

First-Order Difference Complementarity Rate

As a fundamental tool for analyzing the dynamic characteristics of time series, the first-order difference ($\mathsf{\Delta }X$) can directly reflect the magnitude of output changes between adjacent time steps. A larger absolute value of $\mathsf{\Delta }X$ implies stronger short-term fluctuations in power output. For a time series $X\left(x_1,x_2,\dots ,x_T\right)$, the average absolute first-order difference is given by:

$$\overline{\left|\mathsf{\Delta }X\right|}={\displaystyle \frac{1}{T-1}}\sum _{t=1}^{T-1}\left|x_{t+1}-x_t\right|$$

(6)

where $\overline{\left|\mathsf{\Delta }X\right|}$ denotes the average absolute first-order difference of time series X.

The complementarity criterion based on first-order difference is defined as follows: For wind and solar output sequences $N_w(t)$ and $N_s(t)$, if the changes in both series are synchronized, the first-order difference of their combined sequence $N_{ws}(t)$ equals the sum of the individual first-order differences, indicating no complementarity. Conversely, if the first-order difference of $N_{ws}(t)$ is lower than the sum of the individual differences, it indicates anti-phase fluctuations and potential complementarity. The first-order difference complementarity rate is defined as:

$$R-FD={\displaystyle \frac{\left(\left|\mathsf{\Delta }{N}^{\mathrm{w}}\right|+\left|\mathsf{\Delta }{N}^{\mathrm{p}}\right|\right)-\left|\mathsf{\Delta }{N}^{\mathrm{wp}}\right|}{\left(\left|\mathsf{\Delta }{N}^{\mathrm{w}}\right|+\left|\mathsf{\Delta }{N}^{\mathrm{p}}\right|\right)}}\times 100\%$$

(7)

where $R-FD$ denotes the first-order difference complementarity rate of wind and solar power outputs; $\left|\mathsf{\Delta }{N}^{\mathrm{w}}\right|$, $\left|\mathsf{\Delta }{N}^{\mathrm{p}}\right|$, and $\left|\mathsf{\Delta }{N}^{\mathrm{ws}}\right|$ represent the first-order difference indicators of wind power, solar power, and their combined output sequence, respectively. $R-FD$ takes values in the range [0, 1]. A value closer to 1 indicates better wind-solar complementarity. An ideal state of complementarity is achieved when $R-FD$ = 1, implying that fluctuations are completely offset. In contrast, $R-FD$ = 0 suggests no complementarity exists within the analyzed time window.

From 1 January to 31 December 2022, a daily complementarity evaluation was conducted on wind and solar outputs to assess their intraday complementarity characteristics. The results, illustrated in the corresponding Figure 3, Figure 4, Figure 5 and Figure 6, reveal significant variability in wind-solar complementarity under different operational conditions. Specifically, the ranges of variation for the Pearson correlation coefficient, standard deviation complementarity rate, RBF complementarity rate, and first-order difference complementarity rate are −0.977 to 0.885, 0 to 0.847, 0.559 to 0.901, and 0 to 0.555, respectively. These findings indicate that certain days exhibit high synergy (positive correlation), while others reflect strong complementarity (anti-phase fluctuations). The presence of frequent peaks and troughs in the result curves highlights the strong intermittency and volatility of wind and solar generation, as well as the impact of extreme or abnormal weather events on wind-solar complementarity.

2.2. Optimisation Dispatch Model for Hydro-Wind-Solar Hybrid Power Generation

2.2.1. Objective Function

The coordinated operation of wind, solar, and hydropower can effectively leverage the flexibility of hydropower to mitigate the volatility and intermittency of wind and solar generation. However, such an approach may, to some extent, affect the long-term operational benefits of hydropower systems. To balance generation-side economic efficiency with grid-side peaking requirements, this section proposes a mid- to long-term multi-objective optimization scheduling model. The model takes the maximization of total system generation and the minimization of the mean square deviation of residual load as dual objectives. Using a daily time step and a monthly scheduling horizon, the model comprehensively considers the output characteristics of generation resources and grid load profiles and formulates two objective functions: maximizing total power generation and minimizing the variance of residual load.

(1)

Generation-side: Maximizing total system generation

$$\left\{\begin{array}{l}\max F_1=\max \left\{{\displaystyle \sum _{t=1}^T{N}_t·\Delta t}\right\}\\ N_t=N_t^h+N_t^w+N_t^s\end{array}\right.$$

(8)

(2)

Grid-side: Minimizing the mean square error of the residual load

$$\left\{\begin{array}{l}\min F_2=\min \left\{\sqrt{{\displaystyle \frac{1}{\mathrm{T}}}{\displaystyle \sum _{t=1}^T}{\left(N_t^{\mathrm{r}}-\overline{N_t^r}\right)}^2}\right\}\\ N_t^r=P_t-N_t\\ \overline{N_t^r}={\displaystyle \frac{1}{T}}{\displaystyle \sum _{t=1}^T{N}_t^r}\end{array}\right.$$

(9)

where $F_1$, $F_2$ represent total system generation and total system mean square error of the residual load, respectively; $N_t$ denote the system’s total generation at time $t$, $N_t^h$, $N_t^w$, $N_t^s$ refer to hydropower, wind, and solar generation at time $t$, in MW; $N_t^r$ and $\overline{N_t^r}$ represent the system residual load and the average value of the system residual load, respectively. $P_t$ is the system load at time $t$, in MW; $T$ denotes the total number of time intervals within the scheduling period, and $\Delta t$ is the duration of each scheduling interval, measured in days.

2.2.2. Constraints

(1)

Water Balance Constraints

$$V_{t+1}=V_t+(R_t-Q_t)\Delta t$$

(10)

where $V_{t+1}$ and $V_t$ are the initial and final reservoir storage volumes of the hydropower station during time period $t$, respectively; $R_t$ and $Q_t$ correspond to the inflow and outflow of the hydropower station during time period $t$; $\Delta t$ is the duration of time period $t$.

(2)

Water Level Constraints

$$Z_t^{\min }\le Z_t\le Z_t^{\max }$$

(11)

where $Z_t^{\min }$ and $Z_t^{\max }$ are the minimum and maximum allowable water levels of the hydropower station’s reservoir during time period $t$, respectively.

(3)

Reservoir Characteristic Constraints

$$\left\{\begin{array}{l}{V}_t=f^{zv}(Z_t)\\ Z_t^d=f^{zq}(Q_t)\end{array}\right.$$

(12)

where $Z_t$ represents the average upstream water level of reservoir during period $t$, and $Z_t^d$ denotes the average downstream water level. $f^{zv}(\cdot )$ and $f^{zq}(\cdot )$ represent the water level–storage and water level–discharge characteristic curves of reservoir, respectively.

(4)

Output Characteristic Constraints

$$N_t^{\mathrm{h}}=f^N(H_t,Q_g)$$

(13)

where $f^N(\cdot )$ is the power output calculation function for reservoir. $H_t$ is the net head of reservoir during period $t$ and $Q_g$ is the generating discharge of reservoir at time $t$.

(5)

Output Constraints

$$N_{t,min}^{\mathrm{h}}⩽N_t^{\mathrm{h}}⩽N_{t,max}^{\mathrm{h}}$$

(14)

where $N_t^{\mathrm{h}}$ is the output of the hydropower station during time period $t$, MW; $N_{t,min}^{\mathrm{h}}$ and $N_{t,max}^{\mathrm{h}}$ are the minimum and maximum allowable outputs of the hydropower station during time period $t$, MW; $N_t^w$ is the output of the wind farm during time period $t$, MW; $N_{t,min}^w$ and $N_{t,max}^w$ are the minimum and maximum allowable outputs of the wind farm during time period $t$, MW; $N_t^s$ is the output of the solar farm during time period $t$; $N_{t,min}^s$ and $N_{t,max}^s$ are the minimum and maximum allowable outputs of the solar farm during time period $t$, MW.

(6)

Outflow Constraints

$$Q_t^{\min }\le Q_t\le Q_t^{\max }$$

(15)

where $Q_t^{\min }$ and $Q_t^{\max }$ are the minimum and maximum allowable outflow of the hydropower station during time period $t$, respectively.

(7)

Water Level Fluctuation Constraints

$$Z_t^{down}\le \Delta Z_t\le Z_t^{\mathrm{up}}$$

(16)

where $Z_t^{down}$ and $Z_t^{\mathrm{up}}$ are the minimum and maximum allowable water level drop and rise limits of the hydropower station’s reservoir during time period $t$, respectively.

(8)

Boundary Constraints

$$\left\{\begin{array}{l}{Z}_0=Z_{start}\\ Z_T=Z_{end}\end{array}\right.$$

(17)

where $Z_{start}$ and $Z_{end}$ are the initial and final water levels of the reservoir during the scheduling period, respectively.

3. Model Solution Based on the HEA

3.1. HEA

Hydropower optimal scheduling is inherently a complex and challenging problem characterized by uncertainty [26], high dimensionality, nonlinearity, and non-convexity. The integration of wind and solar power further increases the complexity of the problem. HEA (Hyper-dominance Evolutionary Algorithm) is a multi-objective optimization algorithm based on hyper-dominance. By quantifying the convergence of solutions, dynamically adjusting selection pressure, improving diversity maintenance strategies, and refining population reselection mechanisms, HEA effectively balances convergence and diversity. It has demonstrated strong performance in solving hydropower optimization problems. The algorithm workflow is as follows:

STEP 1: Initialization

• Randomly generate a population P of size N in the decision space.
• Create a set of uniformly distributed reference vectors V in the M-dimensional objective space.
• Evaluate all individuals in P on each objective and record the ideal point $z^{*}$ and nadir point $z^{nad}$.
• Initialize an empty elite archive $S=\varnothing$ and set hyper-dominance tolerance T = 0.

STEP 2: Offspring Creation

• Apply genetic operators (e.g., SBX crossover, polynomial mutation) to P to produce an offspring population $P_{off}$ of size N.
• Merge offspring with the current archive: $P^{\prime }=P_{off}\cup S$.

STEP 3: Normalization & Hyper-Dominance Filtering

• Update $z^{*}$ and $z^{nad}$ based on P’ and linearly normalize each objective value to [0, 1].
• For each solution in P’, compute its hyper-dominance score $h\left(x\right)$ (the count of solutions it strictly dominates).
• Remove all solutions whose $h\left(x\right)<T$, thereby discarding weakly converged individuals.

STEP 4: Elite Archive Update

• Associate each remaining solution with the closest reference vector in V by smallest angle.
• From each vector’s associated cluster, select the solution with the highest $h\left(x\right)$ to form a provisional archive $S\prime$.
• If $\left|S^{\prime }\right|<N$, fill up to $N$ by adding the next-best solutions by hyper-dominance from the residual pool.
• $Set S\leftarrow S\prime$.

STEP 5: Parent Re-Selection and Tolerance Adjustment

• Perform tournament or rank-based selection on $S$ (favoring higher $h\left(x\right)$) to form the next parent population P of size N.
• Increase T according to a linear schedule (e.g., $T\leftarrow T+\Delta T$) up to a maximum $Tmax$, shifting emphasis gradually from diversity toward convergence.

STEP 6: Termination and Output

• If the maximum number of evaluations or generations is reached, terminate and return $S$ as the final Pareto-approximate set; otherwise, go back to STEP 2.

In summary, The following Algorithm 1 presents the pseudocode of the HEA for optimal multi-objective hydro-wind-solar scheduling. And the flowchart of the HEA is shown in Figure 7.

Algorithm 1: HEA

Input: Inflow runoff sequence for the scheduling period, wind-solar output, boundary conditions for scheduling operation, and various operational constraints. Output: Pareto frontier solutions and corresponding water level, outflow, and output processes. 1: Initialize population P and reference vectors V; 2: Evaluate fitness of P; $3: \mathrm{Calculate} z^{*}$$\mathrm{and} z^{nad}$; 4: S = ∅; 5: T = 0; 6: iteration = 0; 7: while iteration < MaxFEs/N do 8:   Update P by GA optimizer; $9: P\leftarrow S\cup P$; $10: \mathrm{normalize} \mathrm{P} \mathrm{by} z^{*}$$\mathrm{and} z^{nad}$; $11: (P,hd)\leftarrow Hyper Dominance(P,T)$; $12: \mathrm{Calculate} z^{*}$ $\mathrm{and} z^{nad}$; $13: (S,hd)\leftarrow Diversity Preservation(P,V,N,hd)$; 14:     Update T; $15: iteration\to iteration+1$; 16: end

3.2. Performance Evaluation Indicators for Multi-Objective Algorithms

To evaluate the diversity and uniformity of the Pareto non-dominated solution set, this study adopts two widely used performance indicators: Hyper-volume (HV) and Inverted Generational Distance Plus (IGDp).

HV measures the volume of the objective space dominated by the obtained solution set relative to a given reference point. It simultaneously reflects the convergence, spread, and density of solutions. A larger HV value indicates better overall algorithm performance. IGDp calculates the distance from each reference Pareto front point to the nearest point in the obtained solution set. It takes into account both convergence and distribution. Compared to the original IGD metric, IGDp improves sensitivity to the quality of solution distribution and fairness, avoiding excessive penalization of already dominated regions and enhancing Pareto compliance. A smaller IGDp value indicates better algorithmic performance. The formulas for computing Hyper-volume and Inverted Generational Distance Plus are shown in Equations (18) and (19), respectively.

$$HV\left(ND,r\right)=Vol\left(\underset{f\in ND}{\cup }[f_1,r_1]\times \cdots [f_m,r_m]\right)$$

(18)

where $ND$ is the current set of non-dominated solutions, $f\in ND$ is a solution in the set, with objective values $f=\left(f_1,f_2,\dots ,f_m\right)$, $r=\left(r_1,r_2,\dots ,r_m\right)$ is the reference point, typically set to the worst objective values, $Vol(\cdot )$ is the Lebesgue measure in $m$-dimensional space, representing the volume.$\left[f_i,r_i\right]$ is the interval from the $i-th$ objective value $f_i$ to the corresponding coordinate $r_i$ of the reference point, $m$ is the number of objectives.

$$IGDp(Z,A)={\displaystyle \frac{{\displaystyle \sum _{j=1}^{\left|z\right|}}min\sqrt{{\displaystyle \sum _{k=1}^M}{\left(max\{a_k-z_k,0\}\right)}^2}}{Z}}$$

(19)

where $Z$ refers to the Pareto frontier, which consists of a set of optimal solutions.$A$ is the solution set obtained by the algorithm. $\left|Z\right|$ is the number of reference points in $Z$, $z_j=\left(z_{j1},z_{j2},\dots ,z_{jM}\right)$ is the $j-th$ point in the reference set. $a=\left(a_1,a_2,\dots ,a_M\right)\in A$ is a point in the obtained solution set. $M$ is the number of objectives. $\max \left\{\left.a_k-z_k,0\right\}\right.$ considers only the part where the solution is worse than the reference, thus avoiding negative bias (the key distinction between IGD and IGDp).

3.3. Function Testing and Analysis

To validate the superiority of the HEA, this study employs the Walking Fish Group (WFG) series of multi-objective benchmark functions (WFG1–9) [27]. These functions are specifically designed to represent a wide range of problem characteristics, including separable or highly coupled variable structures (e.g., WFG2/3/6/8/9), biased search spaces (e.g., WFG1/7/9), deceptive objective features (e.g., WFG5/9), and multimodal landscapes (e.g., WFG4/9), as well as various Pareto front shapes and dependency forms. These features allow the WFG test suite to comprehensively evaluate the convergence, diversity, and robustness of multi-objective optimization algorithms from multiple dimensions. Compared to traditional DTLZ/ZDT benchmarks with fixed structures, the WFG suite offers a more targeted diagnosis of algorithmic weaknesses and facilitates more precise performance assessments. The characteristics of WFG series test functions are shown in

Table 1. WFG series test functions characteristics.

Function	Decomposability	Modality (Unimodal/Multimodal)	Parameter Dependence	Pareto Front Shape
WFG 1	Decomposable	Unimodal	None	Regular
WFG 2	Decomposable	Unimodal	None	Disconnected
WFG 3	Non-decomposable	Unimodal	None	Regular
WFG 4	Non-decomposable	Multimodel	None	Regular
WFG 5	Non-decomposable	Multimodel	Yes	Regular
WFG 6	Decomposable	Unimodal	Yes	Convex with bias
WFG 7	Decomposable	Unimodal	None	Mixed linear-convex
WFG 8	Decomposable	Unimodal	None	Mixed concave-convex
WFG 9	Non-decomposable	Multimodel	Yes	Complex (multiply connected)

.

To provide a comparative analysis, four well-established algorithms—Multi-Objective Particle Swarm Optimization (MOPSO), Multi-Objective Besiege and Conquer Algorithm (MOBCA), Non-dominated Sorting Genetic Algorithm III (NSGA-III), and large-scale multi-objective optimization framework (LSMOF)—are selected as baselines. For fairness, all algorithms are configured with the same population size of 50 and a maximum number of 500 iterations. Other parameters are set according to the recommended values in the respective original literature.

To reduce the influence of randomness and enhance the statistical significance of the experimental results, each algorithm is independently executed 20 times on each test function. Both the average, median, and variance results of the 20 runs are recorded and used for comparative analysis. The results are presented in

Table 2. WFG Series Test Function HV Indicator.

Function	HV	HEA	MOPSO	MOBCA	NSGA III	LSMOF
WFG 1	mean	9.31 × 10−1	3.22 × 10−1	3.50 × 10−1	9.29 × 10−1	7.07 × 10−1
	median	9.31 × 10−1	3.26 × 10−1	3.54 × 10−1	9.31 × 10−1	7.12 × 10−1
	variance	2.45 × 10−1	4.25 × 10−2	2.77 × 10−2	3.73 × 10−3	5.52 × 10−2
WFG 2	mean	9.21 × 10−1	7.27 × 10−1	8.34 × 10−1	9.13 × 10−1	8.92 × 10−1
	median	9.21 × 10−1	8.02 × 10−1	8.36 × 10−1	9.13 × 10−1	8.94 × 10−1
	variance	2.55 × 10−3	1.59 × 10−1	1.31 × 10−2	1.72 × 10−3	6.86 × 10−3
WFG 3	mean	3.90 × 10−1	7.58 × 10−2	1.55 × 10−1	3.69 × 10−1	3.79 × 10−1
	median	3.90 × 10−1	5.77 × 10−2	1.58 × 10−1	3.70 × 10−1	3.79 × 10−1
	variance	3.82 × 10−3	7.44 × 10−2	2.23 × 10−2	8.44 × 10−3	4.43 × 10−3
WFG 4	mean	5.36 × 10−1	1.10 × 10−1	3.35 × 10−1	5.35 × 10−1	4.92 × 10−1
	median	5.36 × 10−1	1.08 × 10−1	3.34 × 10−1	5.35 × 10−1	4.95 × 10−1
	variance	5.86 × 10−4	9.78 × 10−3	1.28 × 10−2	1.15 × 10−3	1.47 × 10−2
WFG 5	mean	4.97 × 10−1	7.36 × 10−2	3.99 × 10−1	4.97 × 10−1	4.62 × 10−1
	median	4.97 × 10−1	6.83 × 10−2	4.01 × 10−1	4.97 × 10−1	4.61 × 10−1
	variance	3.02 × 10−5	2.21 × 10−2	1.02 × 10−2	1.95 × 10−4	6.49 × 10−3
WFG 6	mean	4.80 × 10−1	1.78 × 10−1	3.24 × 10−1	4.79 × 10−1	4.73 × 10−1
	median	4.83 × 10−1	1.63 × 10−1	3.27 × 10−1	4.80 × 10−1	4.79 × 10−1
	variance	1.69 × 10−2	4.58 × 10−2	1.72 × 10−2	1.40 × 10−2	2.35 × 10−2
WFG 7	mean	5.36 × 10−1	2.09 × 10−1	3.06 × 10−1	5.35 × 10−1	4.80 × 10−1
	median	5.36 × 10−1	2.21 × 10−1	3.05 × 10−1	5.35 × 10−1	4.90 × 10−1
	variance	3.94 × 10−4	5.87 × 10−2	2.63 × 10−2	6.13 × 10−4	3.08 × 10−2
WFG 8	mean	4.52 × 10−1	9.81 × 10−2	2.36 × 10−1	4.45 × 10−1	4.12 × 10−1
	median	4.52 × 10−1	1.01 × 10−1	2.39 × 10−1	4.46 × 10−1	4.13 × 10−1
	variance	2.73 × 10−3	6.54 × 10−2	1.49 × 10−2	3.30 × 10−3	9.39 × 10−3
WFG 9	mean	4.84 × 10−1	2.40 × 10−1	3.41 × 10−1	4.91 × 10−1	4.58 × 10−1
	median	5.03 × 10−1	2.49 × 10−1	3.38 × 10−1	5.01 × 10−1	4.60 × 10−1
	variance	4.37 × 10−2	5.17 × 10−2	1.56 × 10−2	3.65 × 10−2	1.32 × 10−2

and

Table 3. WFG series test function IGDp indicator.

Function	IGDp	HEA	MOPSO	MOBCA	NSGAIII	LSMOF
WFG 1	mean	9.96 × 10−2	1.63	1.31	1.03 × 10−1	5.23 × 10−1
	median	1.00 × 10−1	1.62	1.31	9.81 × 10−2	5.09 × 10−1
	variance	4.93 × 10−3	2.56 × 10−1	8.72 × 10−2	5.43 × 10−3	1.06 × 10−1
WFG 2	mean	8.28 × 10−2	6.91 × 10−1	2.47 × 10−1	9.55 × 10−2	1.87 × 10−1
	median	9.54 × 10−2	2.90 × 10−1	2.49 × 10−1	8.28 × 10−2	1.90 × 10−1
	variance	5.49 × 10−3	8.28 × 10−1	3.09 × 10−2	3.30 × 10−3	2.83 × 10−2
WFG 3	mean	7.21 × 10−2	7.56 × 10−1	5.56 × 10−1	1.13 × 10−1	1.13 × 10−1
	median	1.15 × 10−1	7.74 × 10−1	5.55 × 10−1	7.01 × 10−2	1.14 × 10−1
	variance	8.29 × 10−3	2.48 × 10−1	4.99 × 10−2	2.21 × 10−2	1.80 × 10−2
WFG 4	mean	1.13 × 10−1	1.74	4.72 × 10−1	1.14 × 10−1	1.73 × 10−1
	median	1.14 × 10−1	1.65	4.55 × 10−1	1.12 × 10−1	1.68 × 10−1
	variance	7.88 × 10−4	2.45 × 10−1	7.23 × 10−2	1.49 × 10−3	2.17 × 10−2
WFG 5	mean	1.65 × 10−1	1.73	3.43 × 10−1	1.65 × 10−1	2.07 × 10−1
	median	1.65 × 10−1	1.74	3.33 × 10−1	1.65 × 10−1	2.06 × 10−1
	variance	4.83 × 10−3	3.33 × 10−1	3.65 × 10−1	2.31 × 10−4	1.00 × 10−2
WFG 6	mean	1.90 × 10−1	1.20	4.80 × 10−1	1.91 × 10−1	2.00 × 10−1
	median	1.89 × 10−1	1.25	4.73 × 10−1	1.85 × 10−1	1.87 × 10−1
	variance	2.47 × 10−2	2.78 × 10−1	4.74 × 10−2	2.05 × 10−2	3.61 × 10−2
WFG 7	mean	1.13 × 10−1	1.26	5.20 × 10−1	1.14 × 10−1	1.91 × 10−1
	median	1.14 × 10−1	1.12	5.15 × 10−1	1.13 × 10−1	1.75 × 10−1
	variance	6.18 × 10−4	4.18 × 10−1	7.45 × 10−2	8.75 × 10−4	4.52 × 10−2
WFG 8	mean	2.41 × 10−1	1.53	7.12 × 10−1	2.49 × 10−1	2.88 × 10−1
	median	2.48 × 10−1	1.56	6.96 × 10−1	2.40 × 10−1	2.87 × 10−1
	variance	3.95 × 10−3	2.91 × 10−1	6.48 × 10−2	5.46 × 10−3	1.38 × 10−2
WFG 9	mean	1.83 × 10−1	1.00	4.50 × 10−1	1.70 × 10−1	2.13 × 10−1
	median	1.52 × 10−1	9.69 × 10−1	4.50 × 10−1	1.53 × 10−1	2.10 × 10−1
	variance	6.63 × 10−2	3.25 × 10−2	4.38 × 10−2	5.68 × 10−2	1.86 × 10−2

, where the best-performing algorithm on each test function is highlighted in dark gray. In addition, the results are visualized in the form of radar charts to provide a more intuitive comparison of the performance distributions across all algorithms.

As shown in Table 2 and Table 3, the HEA demonstrates strong overall performance across the WFG benchmark suite. When executed independently 20 times, HEA yields average, median, and variance values for both HV and IGDp that are superior to those of all other algorithms on more than 90% of the test functions, with only a few isolated cases where its performance is slightly inferior to that of the mainstream genetic algorithm NSGA-III.

Figure 8 and Figure 9 further illustrate this advantage: the distributions of HV average and median values for HEA consistently appear on the outermost ring, while the corresponding IGDp distributions are located on the innermost ring. This indicates that HEA not only converges efficiently to the true Pareto front across a range of complex multi-objective problems but also maintains strong solution diversity and robustness. These results comprehensively validate the overall superiority of the HEA.

In addition, the HEA and the NSGA-III algorithm are discussed in focus below. On the WFG test suite, HEA and NSGA-III demonstrate nearly identical average performance in terms of both the HV and IGDp metrics. However, HEA exhibits two major advantages: lower result variability and reduced time complexity. First, the average standard deviation of HEA over 20 independent runs on the nine WFG problems is lower than that of NSGA-III, indicating that HEA produces more stable and reproducible solution sets. Second, regarding algorithmic complexity, the core non-dominated sorting and reference point assignment in NSGA-III lead to an overall time complexity of approximately $O(M{N}^2)$. In contrast, HEA achieves a complexity of $O(N\log N+MN)$ through hyper-dominance degree filtering and reference vector clustering. Experimental results demonstrate that under identical population sizes and iteration counts, HEA requires shorter average computation time than NSGA-III. Consequently, while maintaining solution set quality, HEA significantly enhances computational efficiency.

To further validate the competitiveness of the HEA algorithm, we conducted a performance comparison between HEA and four other benchmark algorithms on the WFG series test functions. Specifically, we applied the Wilcoxon signed-rank test to the HV and IGDp metrics obtained from 20 independent runs of each algorithm on each test function. If the resulting p-value was less than 0.05, HEA was considered to have a statistically significant advantage over the comparator on that metric; otherwise, the difference was deemed not significant. The test results are presented in

Table 4. Wilcoxon test results of HEA algorithm on HV index relative to other algorithms.

Function	MOPSO	MOBCA	NSGA-III	LSMOF
WFG1	1.91 × 10−6 (Y)	1.91 × 10−6 (Y)	7.84 × 10−3 (Y)	2.33 × 10−5 (Y)
WFG2	4.00 × 10−2 (Y)	1.53 × 10−2 (Y)	8.12 × 10−3 (Y)	4.09 × 10−2 (Y)
WFG3	1.68 × 10−4 (Y)	3.81 × 10−6 (Y)	3.12 × 10−4 (Y)	4.75 × 10−2 (Y)
WFG4	1.91 × 10−6 (Y)	1.34 × 10−5 (Y)	8.69 × 10−4 (Y)	2.02 × 10−2 (Y)
WFG5	1.91 × 10−6 (Y)	7.08 × 10−4 (Y)	8.69 × 10−4 (Y)	1.05 × 10−2 (Y)
WFG6	4.77 × 10−5 (Y)	4.86 × 10−3 (Y)	8.12 × 10−4 (Y)	7.84 × 10−3 (Y)
WFG7	1.34 × 10−5 (Y)	1.91× 10−5 (Y)	8.12 × 10−4 (Y)	1.77 10−2 (Y)
WFG8	1.91× 10−5 (Y)	9.54 × 10−6 (Y)	6.48 × 10−3 (Y)	1.05 10−2 (Y)
WFG9	1.21 × 10−3 (Y)	4.86 × 10−3 (Y)	6.56 10−2 (N)	7.29 × 10−3 (Y)

and

Table 5. Wilcoxon test results of HEA algorithm on IGDp index relative to other algorithms.

Function	MOPSO	MOBCA	NSGA-III	LSMOF
WFG1	1.91 × 10−6 (Y)	1.91 × 10−6 (Y)	7.01 × 10−4 (Y)	2.67× 10−5 (Y)
WFG2	8.31 × 10−3 (Y)	1.99 × 10−3 (Y)	5.22 × 10−4 (Y)	3.28 10−2 (Y)
WFG3	4.77× 10−5 (Y)	3.81 × 10−6 (Y)	4.75 × 10−4 (Y)	3.30 10−2 (Y)
WFG4	1.91 × 10−6 (Y)	4.77× 10−5 (Y)	9.56 × 10−4 (Y)	1.89 10−2 (Y)
WFG5	1.91 × 10−6 (Y)	4.83 × 10−4 (Y)	6.48 × 10−4 (Y)	6.37 × 10−3 (Y)
WFG6	9.54 × 10−6 (Y)	7.08 × 10−4 (Y)	1.00 × 10−3 (Y)	9.85 × 10−4 (Y)
WFG7	5.72 × 10−6 (Y)	1.91× 10−5 (Y)	9.56 × 10−4 (Y)	1.14 10−2 (Y)
WFG8	1.91 × 10−6 (Y)	9.54 × 10−6 (Y)	7.29 × 10−4 (Y)	1.89 × 10−3 (Y)
WFG9	2.67× 10−5 (Y)	1.21 × 10−3 (Y)	7.56 10−2 (N)	6.74 × 10−3 (Y)

, where “Y” indicates that the significance test was passed, HEA demonstrates a significant advantage over the corresponding algorithm on that test function—and “N” indicates that the test was not passed, HEA does not show a significant advantage over the corresponding algorithm on that test function.

The results of the Wilcoxon test indicate that, across all WFG benchmark functions, the HEA algorithm demonstrates a statistically significant advantage over the other algorithms in both the HV and IGDp metrics, further confirming its competitive strength.

4. Case Study

To assess the hydropower station’s regulation performance under different hydrological conditions with respect to renewable energy output and grid load, two representative months were selected: January 2022 (dry season, weak solar and strong wind) as typical month A and August 2022 (wet season, strong solar and weak wind) as typical month B. The station’s inflow, initial water level, dispatch horizon, and upper and lower water level limits were all set according to real-world operational data.

Table 6. Operational data for typical months of the hydroelectric power station.

Typical Month	Initial Water Level (m)	Final Water Level (m)	Average Water Level (m)	Average Inflow Rate (m3/s)
Typical Month A (2022.1)	798.86	799.73	799.47	991.03
Typical Month B (2022.8)	786.87	794.04	790.26	1968.29

shows the operating data for the typical months of the hydroelectric power station.

The case study adopts MOPSO, MOBCA, NSGA-III, LSMOF, and HEA to solve the optimization model. The algorithm settings were as follows: population size is 50, and the maximum number of generations is 5000.

Table 7. Detailed Pareto scheme under typical month A.

Plan	Power Generation (104 kWh)	Residual Load RMSE (MW)	Plan	Power Generation (104 kWh)	Residual Load RMSE (MW)
1	416,344.28	463.74	26	412,314.02	384.50
2	416,010.67	453.81	27	412,076.98	382.49
3	415,888.75	447.78	28	411,914.92	379.64
4	415,848.36	446.39	29	411,763.88	378.12
5	415,679.12	444.53	30	411,524.64	373.76
6	415,550.18	441.96	31	411,307.55	371.25
7	415,406.48	437.98	32	410,959.66	370.53
8	415,277.39	430.77	33	410,878.92	367.78
9	415,072.83	426.39	34	410,838.36	367.46
10	415,035.55	425.60	35	410,551.68	366.34
11	414,756.23	421.88	36	410,277.54	363.34
12	414,580.23	419.28	37	410,064.30	361.54
13	414,378.99	417.04	38	409,662.54	360.50
14	414,180.22	412.23	39	409,478.16	357.82
15	413,976.49	409.87	40	408,871.21	357.16
16	413,636.97	404.55	41	408,764.70	357.03
17	413,459.39	402.35	42	408,489.79	356.18
18	413,372.69	399.79	43	408,487.44	354.40
19	413,280.96	397.41	44	408,247.55	352.48
20	413,116.55	395.52	45	407,960.19	350.87
21	412,920.76	392.40	46	407,922.82	347.08
22	412,772.16	391.11	47	407,537.81	346.42
23	412,679.92	388.80	48	407,300.25	346.16
24	412,576.01	387.41	49	406,623.96	345.54
25	412,370.26	385.91	50	406,242.87	341.43

and

Table 8. Detailed Pareto scheme under typical month B.

Plan	Power Generation (104 kWh)	Residual Load RMSE (MW)	Plan	Power Generation (104 kWh)	Residual Load RMSE (MW)
1	453,333.35	331.34	26	450,748.44	288.33
2	453,333.35	331.34	27	450,602.83	288.22
3	453,126.01	326.81	28	450,590.59	286.21
4	453,025.42	324.90	29	450,425.00	285.79
5	452,910.52	322.58	30	450,413.06	284.37
6	452,869.67	321.89	31	450,318.61	282.89
7	452,786.27	320.31	32	450,174.36	281.16
8	452,668.31	319.25	33	450,045.24	279.69
9	452,616.87	317.11	34	449,872.27	278.56
10	452,559.84	316.28	35	449,738.92	276.69
11	452,450.73	314.33	36	449,621.09	275.53
12	452,338.56	312.55	37	449,442.88	274.00
13	452,218.22	311.52	38	449,319.81	272.13
14	452,163.55	310.13	39	449,173.52	270.23
15	452,106.23	308.24	40	449,037.13	270.13
16	451,934.42	307.24	41	448,779.56	267.47
17	451,809.55	304.89	42	448,687.26	266.58
18	451,674.91	302.48	43	448,460.86	265.27
19	451,598.68	300.62	44	448,204.94	264.38
20	451,406.91	298.57	45	448,079.62	263.42
21	451,207.73	298.19	46	447,921.43	263.35
22	451,182.45	295.16	47	447,815.45	261.40
23	451,026.72	294.53	48	447,625.50	260.64
24	450,968.19	291.20	49	447,341.23	260.16
25	450,900.91	291.08	50	447,114.70	258.91

list the 50 Pareto solutions obtained by the HEA in typical months A and B. Figure 10 presents the Pareto front results of all algorithms for typical months. Figure 11 displays the Pareto outcome distributions obtained by the HEA for typical months.

Figure 10 demonstrates that MOPSO, constrained by its fixed inertia weight, fails to balance exploration and exploitation, leading to premature convergence. This results in the poorest performance during typical month A, where its Pareto front exhibits significant rightward deviation and contraction—indicating higher mean square error of the residual load at equivalent power generation levels.

Although NSGA-III approaches HEA’s performance on WFG series test functions, it proves markedly inferior in actual scheduling scenarios. In typical month A, NSGA-III’s solution set coverage is narrower than HEA’s, attributed to its static reference vectors’ inability to handle objective conflicts. During Typical Month B, NSGA-III further underperforms in minimizing the mean square error of the residual load due to inadequate diversity preservation. Its Pareto front shows sparse solution distribution in the high-power-generation–low-variance region, yielding significantly worse solution set continuity compared to HEA.

And the HEA consistently achieves the best Pareto front solutions under both dry and wet season conditions. Specifically, for a given power generation target, HEA delivers higher peak-shaving benefits; conversely, for the same level of peak-shaving performance, it achieves greater power output. This demonstrates HEA’s strong capability in balancing multiple objectives and highlights its superior overall optimization performance.

During typical month A, where water availability is limited and scheduling constraints are tighter, the differences between algorithms become more pronounced. The Pareto front obtained by HEA is clearly superior to those of the comparative algorithms—shifting upward overall and exhibiting a broader distribution. This indicates that HEA effectively harnesses scheduling potential under resource-constrained conditions by adaptively regulating reservoir levels to maximize comprehensive benefits. And the optimized solution with the best peak shifting decreased by 22% relative to the original residual load mean square error, and the solution with the worst peak shifting decreased by 12% relative to the original residual load mean squared error.

During typical month B, with abundant inflows, all algorithms tend to perform well in high-generation regions, and the resulting Pareto fronts are relatively close. Nonetheless, HEA still demonstrates stronger global exploration capabilities. It not only secures the highest generation efficiency but also achieves greater peak-shaving capability within the high-generation domain. The solution set it produces is more diverse and better distributed across the objective space, further validating its robustness and global optimization capacity. And the optimized solution with the best peak shifting decreased by 21% relative to the original residual load mean square error, and the solution with the worst peak shifting decreased by 6% relative to the original residual load mean squared error.

As shown in Figure 11, the solution set is uniformly distributed along the Pareto front, and the front exhibits a smooth shape without noticeable discontinuities or clustering. This indicates that the HEA not only ensures strong convergence performance but also maintains excellent solution diversity, effectively avoiding entrapment in local optima.

The 1st, 25th, and 50th Pareto solutions were selected as representative schemes for analyzing the reservoir water level trajectory and were renamed as Scheme I, Scheme II, and Scheme III, respectively—corresponding to the highest total generation, best trade-off solution, and optimal peak-shaving benefit.

Figure 12 and Figure 13 show the reservoir level trajectories under different algorithms for January and August, respectively. In the January dry season with strong wind and weak solar conditions, the HEA algorithm lowers the water level to approximately 795.8 m during low-demand periods—more aggressively than other algorithms—to maximize early cumulative generation; during peak demand, it raises the level to around 801.1 m to provide greater instantaneous output. Crucially, HEA’s level adjustments follow a “gradual descent–stable–gradual ascent–stable” pattern without abrupt oscillations, avoiding sudden releases or impoundments that could induce load spikes or troughs on the grid while maintaining smooth generation output. Thus, HEA effectively achieves the dual objectives of maximizing generation and minimizing the mean square error of the residual load through its “deep-valley to high-peak” strategy with smooth transitions. Under the flood-season scenario of strong solar and weak wind, HEA (red line) again demonstrates precise balancing of the dual objectives: from days 1–5 (high-insolation period), it moderately raises the level to about 789 m to capture early peak generation opportunities without over-releasing and depleting storage; during days 6–18, it holds the level between 785 and 787 m, effecting small oscillations to achieve stable discharge that sustains continuous generation and smooths grid load; when demand rises again on days 19–32, it gently lifts the level to 792–794 m, providing higher head and generation potential while avoiding sudden jumps that could trigger grid load spikes. In contrast, other algorithms either release water too early (e.g., MOBCA, MOPSO) or exhibit plateaued stagnation or abrupt surges in mid-to-late stages (e.g., NSGA-III, LSMOF), failing to reconcile full-period generation efficiency with peak-regulation smoothness. HEA’s “timely peak raising–stable delivery–re-peak raising” strategy thus achieves optimal coordination for multi-objective scheduling in the flood season.

Next, we will further analyze the characteristics of the reservoir level trajectories produced by the HEA algorithm under different scheduling periods and schemes. As shown in the Figure 14 and Figure 15.

Under the background of the dry season, where actual water levels tend to rise slowly, all three optimized water level trajectories exhibit a scheduling pattern characterized by “early-stage drawdown–mid-term stabilization–late-stage recovery”. This strategy is designed to precisely match peak generation needs and minimize residual load variance, targeting two objectives: First, in early January, water levels are lowered to accelerate the release of inflows and meet high demand during the early peak period; second, from days 6–14, each scheme maintains the water level through small, gradual adjustments to preserve sufficient reservoir volume for mid- to late-month hydropower generation and avoid sharp drops that could destabilize the system; from day 15 onward, water levels rise again to meet end-of-month load demands.

The three schemes present trade-offs in terms of valley depth, regulation range, and ramping rate. Scheme I is more aggressive, featuring the highest peak load rate. Scheme III is more balanced, with smoother transitions and a relatively flatter water level trajectory. While Scheme III slightly compromises peak generation, it achieves the critical goal of ‘controlled drawdown, concentrated ramp-up’.

In summary, this “early drawdown–stable mid-stage–late-stage recovery” regulation strategy, under dry-season conditions, effectively enhances peak capacity and generation efficiency. Meanwhile, the three schemes provide differentiated advantages in balancing energy generation and residual load fluctuations, offering sound economic efficiency and rational operational outcomes.

By comparing the water level trajectories of the representative schemes with actual observations, it is evident that, under the context of abundant inflows during the flood season and the need to balance power generation and peak shaving, the three optimized schemes exhibit a smoother water level rise compared to the natural trajectory. All follow the scheduling strategy of “early-stage ramp-up–mid-stage drawdown–late-stage high water level.” Specifically, they start slightly above the actual water level to leverage early-stage runoff and elevated head for power generation. From days 4–16, the water level is gradually reduced to reach its minimum—Scheme III achieves the lowest level on days 8–12—to prepare for subsequent peak-shaving operations and head restoration. Starting on day 17, the water level begins to recover and reaches its peak during days 22–26, aligning with the system’s daily peak demand to ensure maximum power output.

Among the schemes, Scheme I (maximum generation) hasthe fastest recovery, highest peak level, and effective utilization of maximum water head; Scheme III (smallest peak-load variance) features the flattest and most stable profile, minimizing peak deviation and smoothing grid-side residual load; and Scheme II represents a compromise between the two, achieving a balanced trade-off.

In summary, the “early ramp-up–mid drawdown–late replenishment” strategy not only enhances high-head power generation efficiency but also ensures better alignment with grid-side load requirements, providing a rational and effective operational solution.

5. Conclusions

To address the grid regulation challenges posed by the uncertainty of wind and solar power output, this study adopts four evaluation indicators to quantify the complementarity and synergy between wind and solar energy. A multi-objective optimization model is developed with the goals of maximizing total system power generation and minimizing the residual load variance. To overcome the limitations of existing algorithms for hydropower multi-objective scheduling, a novel Hyper-dominance Evolutionary Algorithm (HEA) is introduced to solve the mid- to long-term optimization problem for wind–solar–hydropower integrated systems. Through benchmark function testing and application to a real hydropower station in Yunnan, the effectiveness and superiority of the HEA are validated. The main conclusions are as follows:

(1)

Based on the station’s wind and solar output data, the variation patterns of renewable energy generation were thoroughly analyzed. Four complementarity indicators—Pearson correlation coefficient, standard deviation complementarity ratio, RBF complementarity ratio, and first-order difference complementarity ratio—were employed to quantify the wind–solar complementarity. The results show that intraday complementarity varies significantly under different conditions.

(2)

For the optimal scheduling of wind–solar–hydropower hybrid systems, a bi-objective model was established to maximize power generation and minimize residual load variance. The model fully incorporates critical operational constraints such as water balance, reservoir level limits, and output restrictions. To address the drawbacks of traditional algorithms—such as susceptibility to local optima, slow convergence, and lack of robustness—the HEA was introduced.

(3)

Benchmark and case study results demonstrate that the HEA outperforms MOPSO, MOBCA, NSGA-III, and LSMOF in terms of convergence speed and solution accuracy. Under different typical-month scenarios, HEA consistently achieved the best Pareto solutions, effectively smoothing the volatility of wind and solar output, enhancing system operational stability, and meeting the practical requirements of integrated wind–solar–hydropower scheduling.

In summary, the study confirms that the HEA provides an effective solution for multi-objective optimization in wind–solar–hydropower hybrid systems and offers a superior scheduling strategy for clean energy integration under complex operating environments. While the current model focuses on multi-objective complementarity within a single hydro-wind-solar power station, future research will expand to cascaded hydropower systems with integrated energy storage and thermal power. This extension will establish a large-scale, multi-energy synergistic dispatch framework to enhance operational flexibility and security across regional power grids. In addition, it can be applied to different other climate zones by modifying the constraint and tuning parameters, e.g., adding snowmelt uncertainty constraints applied to snowmelt basins.