An Optimization Method for Day-Ahead Generation Interval of Cascade Hydropower Adapting to Multi-Source Coordinated Scheduling Requirements

Abstract

Multi-source coordinated scheduling has become the predominant operational paradigm in power systems. However, substantial differences among hydropower, thermal power, wind power, and photovoltaic sources in terms of response speed, regulation capability, and operational constraints—particularly the complex generation characteristics and spatiotemporal hydraulic coupling of large-scale cascade hydropower stations—significantly increase the complexity of coordinated scheduling. Therefore, this study proposes an optimization method for determining the day-ahead generation intervals of cascade hydropower, applicable to multi-source coordinated scheduling scenarios. The method fully accounts for the operational characteristics of hydropower and the requirements of coordinated scheduling. By incorporating stochastic operational processes, such as reservoir levels and power outputs, feasible boundaries are constructed to represent the inherent uncertainties in hydropower operations. A stochastic optimization model is then formulated to determine the generation intervals. To enhance computational tractability and solution accuracy, a linearization technique for stochastic constraints based on duality theory is introduced, enabling efficient and reliable identification of hydropower generation capability intervals under varying system conditions. In practical applications, other energy sources can develop their generation schedules based on the feasible generation intervals provided by hydropower, thereby effectively reducing the complexity of multi-source coordination and fully leveraging the regulation potential of hydropower. Multi-scenario simulations conducted on six downstream cascade reservoirs in a river basin in Southwest China demonstrate that the proposed method significantly enhances system adaptability and scheduling efficiency. The method exhibits strong engineering applicability and provides robust support for multi-source coordinated operation.

1. Introduction

Driven by the goal of building a clean, secure, and efficient energy and power system, the share of renewable energy sources (e.g., wind and photovoltaic power) in the power grid continues to grow [1]. Consequently, the power system is gradually transitioning toward a coordinated scheduling paradigm that integrates multiple energy sources, including hydropower, thermal power, wind power, photovoltaic power, and energy storage [2]. Following this trend, fully exploiting the operational characteristics of diverse energy sources to enhance renewable energy integration and maximize the benefits of source complementarity has become a critical direction in power system scheduling [3,4,5]. However, the differences in generation characteristics and operational constraints across various energy sources—such as the uncertainty of wind and photovoltaic generation [6], the carbon emission limits of thermal and nuclear power [7], the ecological and navigational constraints of hydropower [8], and the high cost of energy storage [9]—have significantly increased the complexity of multi-source coordinated scheduling, posing substantial challenges to the formulation of day-ahead power system schedules [10,11]. In practical scheduling operations, differences among energy sources often lead to conflicts in scheduling plans [12]. As a result, power system control centers are frequently required to engage in multiple rounds of communication and iterative revisions with various planning departments before a feasible multi-source day-ahead schedule can be finalized [13]. This severely limits scheduling decision efficiency. In addition, extreme weather events—such as typhoons and prolonged heatwaves—can cause severe fluctuations in renewable energy output or sudden surges in power demand [14], further increasing the complexity of day-ahead schedule formulation [15]. Therefore, developing scheduling strategies with strong regulation capability to enhance the efficiency of day-ahead scheduling under multi-source coordination has become a critical challenge in modern power system development [16,17].

Within the multi-source coordinated scheduling framework, cascade hydropower has emerged as a critical regulation resource in power systems due to its flexible start-stop capabilities and excellent regulation performance [18]. Fully leveraging the regulation potential of cascade hydropower is therefore essential for satisfying the requirements of multi-source coordinated scheduling and enhancing the stability and security of power system operations [19]. Substantial progress has been made in the study of hydropower participation in multi-source coordinated scheduling [20]. Existing research primarily focuses on the scheduling strategies of hydropower resources under various operational mechanisms [21]. Two major research directions have gradually emerged: one emphasizes multi-source spot market bidding strategies driven by market-based mechanisms, while the other investigates multi-timescale coordinated scheduling optimization under non-market-based mechanisms. Under market-based mechanisms, ref. [22] proposed a joint hydropower–wind bidding strategy that accounts for wind power uncertainty, where the regulation capability of hydropower mitigates the complex impacts of wind variability on market bidding. Leveraging the operational flexibility of hydropower, ref. [23] developed an optimization approach for the joint participation of hydropower and wind power in spot markets, thereby enhancing the efficiency and profitability of multi-source coordinated bidding. In addition, ref. [24] investigated the coordinated scheduling of hydropower, photovoltaic, and thermal power in electricity markets, and established a spot price uncertainty model based on the regulation characteristics of hydropower. Under non-market-based mechanisms, ref. [25] proposed a hydropower regulation strategy based on synchronized peak values to mitigate intra-day load fluctuations by optimizing the coordination of hydro, thermal, wind, and solar power. Ref. [26] developed a medium- and long-term storage scheduling framework for hydro–solar complementarity, ensuring the reliability of cascade hydropower output and proposing coordinated generation strategies under varying storage conditions. Leveraging integrated wind and solar scenarios within a “forecast-as-schedule” paradigm, ref. [27] achieved short-term multi-source coordinated scheduling by exploiting the flexibility of hydropower. Additionally, ref. [28] formulated a multi-energy complementary scheduling model that incorporates the dispatch characteristics of various power sources and proposed a multi-objective solution strategy termed “segmentation–sequencing–feedback” to maximize overall system benefits. In addition, considerable research has been devoted to exploiting the intrinsic characteristics of hydropower, particularly its capabilities in large-scale generation, efficient regulation, and energy storage. For example, some studies have examined structural modifications of cascade hydropower plants, such as integrating pumped storage units, to enhance the efficiency of water resource utilization and significantly increase total system generation [29]. Other works have proposed flexibility evaluation methods for hydropower operation that incorporate complex technical constraints, such as vibration zones of generating units, thereby enabling a more comprehensive utilization of hydropower flexibility [30]. Although substantial progress has been made in the field of multi-source coordinated scheduling, the regulation potential of hydropower has not been fully exploited due to limitations in the adjustable range considered in existing optimization strategies. This shortcoming becomes particularly critical in unexpected situations, such as sudden surges in system load or sharp fluctuations in renewable energy output, where current scheduling schemes lack sufficient flexibility to respond effectively. Consequently, substantial deviations frequently occur between scheduled plans and actual system operations [31]. At the day-ahead scheduling stage, such deviations can severely compromise system stability and security, posing unacceptable risks to power system operations [32].

At the same time, the nonlinear, high-dimensional, and strongly coupled nature of hydropower systems makes the design and solution of optimization models a fundamental research challenge [33]. Stochastic modeling has been extensively applied to capture the uncertainty of inflows and renewable generation through the generation of multiple scenarios, thereby enabling the characterization of probabilistic operational risks [34]. However, its effectiveness is often constrained by the exponential growth of scenarios, which may lead to computational intractability for large-scale systems [35]. To address this, robust optimization methods have been developed, focusing on worst-case realizations of uncertainty within predefined uncertainty sets [36]. While these approaches improve solution robustness, they may yield overly conservative dispatch strategies that reduce economic efficiency [37]. More recently, machine learning (ML)-based scheduling has emerged as a promising alternative, leveraging data-driven techniques to learn complex patterns of renewable generation and system dynamics [38]. These methods demonstrate advantages in scalability and real-time adaptability, but their interpretability and generalizability across different hydrological or power system contexts remain open research questions [39].

To address the above challenges, this study proposes an optimization method for the day-ahead generation interval of cascade hydropower, designed to meet the requirements of multi-source coordinated scheduling. The main innovations and challenges of this study can be summarized as follows:

(1)

Explicitly formulating stochastic generation adjustment constraints;

(2)

Characterizing the maximum feasible generation interval as the optimization objective;

(3)

Employing a linearization approach based on duality theory to achieve the efficient solution of the stochastic optimization model.

The research framework is organized as follows. First, feasible boundaries for key operational parameters of cascade hydropower stations are constructed to reduce the computational burden. Then, with the objective of maximizing the total generation interval of the cascade system, the model is reformulated using a stochastic constraint linearization technique based on duality theory. This enables the efficient solution and rational allocation of generation capacity among cascade stations with varying regulation capabilities. Ultimately, the method provides a maximally adjustable day-ahead generation interval for cascade hydropower, significantly reducing the complexity of schedule formulation in multi-source power systems. To validate the effectiveness of the proposed method, a case study is conducted on six downstream reservoirs in a river basin in southwest China under various inflow conditions. The results demonstrate that the method not only satisfies the feasibility boundaries of day-ahead scheduling but also effectively enhances the regulation capability of cascade hydropower under multi-source coordinated scheduling. In addition, a comprehensive sensitivity analysis on reservoir level limits, regulation performance, and generation capacity boundaries further improves the interpretability of the model.

2. Optimization Model for the Generation Interval of Cascade Hydropower

To fully accommodate the requirements of multi-source coordinated scheduling, this section develops an optimization model for the generation interval of cascade hydropower, aiming to maximize the feasible generation range. The model incorporates both the conventional operational constraints of cascade hydropower stations and the constraints associated with the stochastic adjustment process of generation.

2.1. Objective Function

Let the feasible generation interval of cascade hydropower during the $t$-th period be denoted by $\left[S_t^{\min },S_t^{\max }\right]$, where $S_t^{\max }\ge S_t^{\min }\ge 0$. Here, $S_t^{\max }$ and $S_t^{\min }$ represent the accumulated upper and lower feasible generation boundaries of all cascade hydropower stations during the $t$-th period, corresponding to the up-regulation and down-regulation limits $S_{h,t}^{\max }$ and $S_{h,t}^{\min }$, respectively.

To alleviate operational pressure on the power grid and enhance adaptability to various scheduling scenarios, it is desirable to maintain a sufficiently large feasible generation interval across the entire scheduling horizon. Accordingly, the objective is to maximize the total feasible generation interval of cascade hydropower throughout the scheduling horizon, which is formulated as follows:

$$\max {\displaystyle {\displaystyle \sum _{t=1}^T}(S_t^{\max }-S_t^{\min })}$$

(1)

where $T$ denotes the total number of scheduling periods, which is set to 24 in this study.

2.2. Operational Constraints of Cascade Hydropower During the Planning Stage

To support multi-source coordinated scheduling, cascade hydropower needs to comply with operational constraints to ensure secure and reliable operation during the planning stage.

2.2.1. Water Balance Constraint of Hydropower

$$V_{h,t}=V_{h,t-1}+(I_{h,t}+Q_{h-1,t}+F_{h-1,t}-Q_{h,t}-F_{h,t})\Delta t$$

(2)

where $V_{h,t}$ denotes the reservoir storage of hydropower station $h$ at the end of the $t$-th period (m³); $I_{h,t}$, $Q_{h,t}$, and $F_{h,t}$ represent the interval inflow, generation discharge, and spill discharge of hydropower station $h$ during the $t$-th period (m³/s), respectively. $\Delta t$ is the time step of the dispatch calculation, which is set to 1 h in this study. In short-term hydropower scheduling, the spill discharge is typically restricted to 0 m³/s. Incorporating occasional spill would slightly reduce the feasible generation range, but it would not change the overall structure of the optimization model [35].

2.2.2. Operational Boundary Constraints of Hydropower

The constraints Equations (3)–(5) represent the generation discharge, power output, and reservoir storage boundaries of hydropower station $h$ during the $t$-th period, respectively.

$$Q_h^{\min }\le Q_{h,t}\le Q_h^{\max }$$

(3)

$$P_h^{\min }\le P_{h,t}\le P_h^{\max }$$

(4)

$$V_h^{\min }\le V_{h,t}\le V_h^{\max }$$

(5)

where $Q_h^{\max }$ and $Q_h^{\min }$ denote the lower and upper bounds of generation discharge (m³/s); $P_h^{\max }$ and $P_h^{\min }$ denote the lower and upper bounds of power output (MW); $V_h^{\max }$ and $V_h^{\min }$ denote the lower and upper bounds of reservoir storage (m³), all corresponding to hydropower station $h$ during the $t$-th period.

2.2.3. Initial and Terminal Reservoir Storage Constraints of Hydropower

$$V_{h,0}=V_h^{begin}$$

(6)

$$V_{h,T}=V_h^{End}$$

(7)

where $V_h^{Beg}$ and $V_h^{End}$ denote the prescribed initial and terminal reservoir storage values for hydropower station $h$ during the scheduling horizon (m³). The subscripts $t=0$ and $t=T$ of $V_{h,t}$ represent the reservoir storage of hydropower station $h$ at the beginning and end of the scheduling horizon, respectively (m³).

2.2.4. Hydropower Generation Function

In short-term scheduling, intra-day water level fluctuations have a limited impact on the water consumption rate of hydropower stations. Based on the actual operating characteristics of the study case, a fixed water consumption rate is adopted in this paper to calculate the power output. The hydropower generation function is expressed as follows:

$$Q_{h,t}=P_{h,t}{\displaystyle \frac{r_{h,t}}{3.6}}$$

(8)

where $r_{h,t}$ denotes the water consumption rate of hydropower station $h$ during the $t$-th period (m³/kWh).

2.2.5. Water Level–Storage Relationship Function of Hydropower

$$Z_{h,t}=f_{h,t}(V_{h,t})$$

(9)

where $Z_{h,t}$ denotes the water level of hydropower station $h$ at the end of the $t$-th period (m) and $f_{h,t}(V_{h,t})$ represents the water level–storage relationship function of hydropower station $h$.

2.3. Operational Constraints in the Stochastic Generation Adjustment Process of Cascade Hydropower

During operation, the feasible generation interval represents the maximum allowable range within which the hydropower station’s power output can be rapidly adjusted around the scheduled value. In the adjustment process, reservoir outflow, generation discharge, and water level are modeled as stochastic variables and are required to satisfy the corresponding operational constraints.

2.3.1. Stochastic Generation of Cascade Hydropower

To meet multi-source scheduling requirements, the generation of cascade hydropower stations often needs to be randomly adjusted based on the scheduled generation, and the adjusted generation is referred to as the stochastic generation. Meanwhile, to ensure the safe and stable operation of cascade hydropower stations, the stochastic generation should satisfy the feasible generation interval constraints. Let the stochastic generation of cascade hydropower in each time period be denoted as ${\tilde{S}}_t^{all}$, which is subject to the following constraint:

$$S_t^{\min }\le {\tilde{S}}_t^{all}\le S_t^{\max }$$

(10)

At each period, the generation of a cascade hydropower station is adjusted relative to the scheduled generation, where the adjustment represents the deviation between the stochastic generation and the scheduled generation. Therefore, let ${\tilde{S}}_t^{\mathrm{dev}}$ denote the relative deviation between the stochastic generation and the scheduled generation of cascade hydropower at the $t$-th period, which can be expressed as follows:

$${\tilde{S}}_t^{\mathrm{dev}}={\tilde{S}}_t^{all}-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}$$

(11)

2.3.2. Stochastic Power Output and Generation Discharge of Hydropower

The total generation adjustment ${\tilde{S}}_t^{all}$ of the cascade hydropower system is implemented at the level of individual power stations, requiring the allocation of ${\tilde{S}}_t^{all}$ to each hydropower station. Based on the relative deviation between stochastic generation and scheduled generation at period $t$, the overall power output adjustment of the cascade hydropower system is denoted as ${\tilde{S}}_t^{\mathrm{dev}}/\Delta t$. By introducing ${\alpha }_{h,t}$ as the adjustment coefficient that allocates the generation deviation to individual stations, the stochastic adjustment process of hydropower generation can be expressed in terms of the total adjustment ${\tilde{S}}_t^{\mathrm{dev}}/\Delta t$, the coefficient ${\alpha }_{h,t}$, and the scheduled output $P_{h,t}$ of each station, as shown in Equation (12). By combining this with Equation (8), the stochastic adjustment process of hydropower generation can further be represented in terms of generation discharge, as shown in Equation (13).

$${\tilde{P}}_{h,t}=P_{h,t}+{\displaystyle \frac{{\alpha }_{h,t}{\tilde{S}}_t^{\mathrm{dev}}}{\Delta t}},{\displaystyle {\displaystyle \sum _{h=1}^H}{\alpha }_{h,t}}=1,0\le {\alpha }_{h,t}\le 1,\forall t$$

(12)

$${\tilde{Q}}_{h,t}^P=Q_{h,t}^P+{\displaystyle \frac{{\alpha }_{h,t}{\tilde{S}}_t^{\mathrm{dev}}{r}_{h,t}}{3.6\Delta t}},{\displaystyle {\displaystyle \sum _{h=1}^H}{\alpha }_{h,t}}=1,0\le {\alpha }_{h,t}\le 1,\forall t$$

(13)

where ${\tilde{P}}_{h,t}$ and ${\tilde{Q}}_{h,t}^P$ represent the power output (MW) and generation discharge (m³/s) of hydropower station $h$ during the stochastic adjustment process at the $t$-th period, respectively. ${\alpha }_{h,t}$ denotes the adjustment coefficient used to allocate generation deviations to each hydropower station, which is an optimization variable.

2.3.3. Water Balance Equation in the Stochastic Generation Adjustment Process of Hydropower

$${\tilde{V}}_{h,t}={\tilde{V}}_{h,t-1}+(I_{h,t}+{\tilde{Q}}_{h-1,t}-{\tilde{Q}}_{h,t})\Delta t$$

(14)

where ${\tilde{V}}_{h,t}$ denotes the reservoir storage (m³) of hydropower station $h$ during the stochastic adjustment process at the $t$-th period. To meet the requirements of multi-source coordinated scheduling while ensuring efficient hydropower operation, the discharge of spilled water is constrained to zero (0 m³/s) during the stochastic adjustment process.

Owing to the temporal and spatial coupling inherent in cascade hydropower, the reservoir storage at any given period is influenced by the storage conditions of preceding periods and, in turn, affects subsequent periods. Accordingly, by substituting Equation (13) into Equation (14) and iteratively expanding over multiple time steps, the water balance equation incorporating generation deviations is derived as Equation (15).

$${\tilde{V}}_{h,t}=V_{h,0}+{\displaystyle {\displaystyle \sum }_{\tau =1}^t}(I_{h,\tau }+Q_{h-1,\tau }-Q_{h,\tau }+{\displaystyle \frac{{\alpha }_{h-1,\tau }{\tilde{S}}_{\tau }^{\mathrm{dev}}{r}_{h-1,\tau }}{3.6\Delta \tau }}-{\displaystyle \frac{{\alpha }_{h,\tau }{\tilde{S}}_{\tau }^{\mathrm{dev}}{r}_{h-1,\tau }}{3.6\Delta \tau }})\Delta \tau$$

(15)

2.3.4. Operational Boundary Constraints in the Stochastic Generation Adjustment Process of Hydropower

$$P_h^{\min }\le {\tilde{P}}_{h,t}\le P_h^{\max }$$

(16)

$$Q_h^{\min }\le {\tilde{Q}}_{h,t}\le Q_h^{\max }$$

(17)

$$V_h^{\min }\le {\tilde{V}}_{h,t}\le V_h^{\max }$$

(18)

$$V_h^{spot,\min }\le {\tilde{V}}_{h,T}\le V_h^{spot,\max }$$

(19)

where $V_h^{spot,\max }$ and $V_h^{spot,\min }$ represent the lower and upper bounds of the generation control reservoir storage for hydropower station $h$ at the end of the scheduling horizon, respectively, measured in cubic meters (m³).

3. Solution Method

During the stochastic generation adjustment process of cascade hydropower, both the stochastic generation and the relative generation deviation are considered random variables. Consequently, constraints Equations (16)–(19) involve numerous stochastic variables, which substantially reduce the tractability of the optimization problem. This section linearizes constraints Equations (16)–(19) to enable efficient model solution.

3.1. Linearization of Power Output and Generation Discharge Boundary Constraints

$${\underset{\_}{P}}_{h,t}\le P_{h,t}+{\displaystyle \frac{{\alpha }_{h,t}{\tilde{S}}_t^{\mathrm{dev}}}{\Delta t}}\le {\overline{P}}_{h,t}$$

(20)

Equation (21) is equivalent to the condition that, for any case where ${\tilde{S}}_t^{all}\in \left[S_t^{\min },S_t^{\max }\right]$, the maximum possible power output is less than the upper power limit, and the minimum possible power output is greater than the lower power limit. By combining Equations (11) and (12), the generation deviation ${\tilde{S}}_t^{\mathrm{dev}}$ is required to satisfy the following condition:

$$\left(S_t^{\min }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\le {\tilde{S}}_t^{\mathrm{dev}}\le \left(S_t^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)$$

(21)

Furthermore, based on Equation (20), the power output boundary constraint can be reformulated as follows:

$$\left\{\begin{array}{l}{P}_{h,t}+{\displaystyle \frac{{\alpha }_{h,t}}{\Delta t}}\left(S_t^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\le {\overline{P}}_{h,t}\\ P_{h,t}+{\displaystyle \frac{{\alpha }_{h,t}}{\Delta t}}\left(S_t^{\min }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\ge {\underset{\_}{P}}_{h,t}\end{array}\right.$$

(22)

Similarly, the generation discharge boundary constraint Equation (17) can be reformulated as follows:

$$\left\{\begin{array}{l}{Q}_{h,t}^P+{\displaystyle \frac{{\alpha }_{h,t}{r}_{h,t}}{3.6\Delta t}}\left(S_t^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\le {\overline{Q}}_{h,t}\\ Q_{h,t}^P+{\displaystyle \frac{{\alpha }_{h,t}{r}_{h,t}}{3.6\Delta t}}\left(S_t^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\ge {\underset{\_}{Q}}_{h,t}\end{array}\right.$$

(23)

3.2. Reformulation of Reservoir Storage Boundary Constraints

3.2.1. Reservoir Storage Boundary Constraints

Since the generation deviation of cascade hydropower is treated as a stochastic variable and satisfies constraint Equation (21), Equation (16) implies that the reservoir storage of hydropower station $h$ is bounded during the $t$-th period. Let ${\overline{V}}_{h,t}$ and ${\underset{\_}{V}}_{h,t}$ denote the maximum and minimum reservoir storage of hydropower station $h$ at the $t$-th period, respectively. When ${\overline{V}}_{h,t}=\max \left({\tilde{V}}_{h,t}\right)$ and ${\underset{\_}{V}}_{h,t}=\min \left({\tilde{V}}_{h,t}\right)$, both values simultaneously satisfy constraints Equations (18) and (19).

$${\overline{V}}_{h,t}\le V_{h,t}^{con,\max }=\left\{\begin{array}{l}{V}_{h,t}^{\max },t<T\\ \min (V_{h,T}^{\max },V_h^{spot,\max }),t=T\end{array}\right.$$

(24)

$${\underset{\_}{V}}_{h,t}\ge V_{h,t}^{con,\min }=\left\{\begin{array}{l}{V}_{h,t}^{\min },t<T\\ \max (V_{h,T}^{\min },V_h^{spot,\min }),t=T\end{array}\right.$$

(25)

3.2.2. Reformulation Method for Reservoir Storage Boundary Constraints

The variables ${\overline{V}}_{h,t}$ and ${\underset{\_}{V}}_{h,t}$, as decision variables in a linear optimization problem involving stochastic variables, can be solved using duality theory. Taking a minimization problem as an example, the general procedure for solving such optimization problems using the Lagrangian dual function is described as follows:

The general form of the minimization problem is given as follows:

$$\left\{\begin{array}{l}\underset{x}{\min }f\left(x\right)\\ s.t.g_i\left(x\right)\le 0,i=1,\dots ,m\\ \hspace{1em} h_j\left(x\right)\le 0,j=1,\dots ,p\end{array}\right.$$

(26)

where $x\in R^n$ is the optimization variable, $f:R^n\to R$ is the objective function, and $g_i:R^n\to R$ and $h_j:R^n\to R$ represent $m$ inequality constraints and $p$ equality constraints, respectively.

The Lagrangian function is constructed based on Equation (27) as follows:

$$L\left(x,\lambda ,\upsilon \right)=f\left(x\right)+{\displaystyle {\displaystyle \sum _{i=1}^m}{\lambda }_i{g}_i\left(x\right)}+{\displaystyle {\displaystyle \sum _{j=1}^p}{\upsilon }_j{h}_j\left(x\right)}$$

(27)

where $\lambda =\left({\lambda }_1,\dots ,{\lambda }_m\right)\in R_{+}^m$ is the non-negative Lagrangian multiplier (corresponding to ${\lambda }_i\ge 0$), and $\upsilon =\left({\upsilon }_1,\dots ,{\upsilon }_p\right)\in R^p$ is the unconstrained Lagrangian multiplier.

The Lagrangian dual problem is then formulated as Equation (28), and the optimal value $f{\left(x\right)}^{\ast }$ of the primal problem and the optimal value $D^{\ast }$ of the dual problem satisfy $f{\left(x\right)}^{\ast }\ge D^{\ast }$.

$$\underset{\lambda \ge 0,\upsilon }{\max }\underset{x}{ \inf } L\left(x,\lambda ,\upsilon \right)$$

(28)

3.2.3. Reformulated Result of the Reservoir Storage Boundary Constraint

Based on constraint Equation (21), the feasible domain of the generation deviation ${\tilde{S}}_t^{\mathrm{dev}}$ is reformulated in the form of a constraint as follows:

$$\left\{\begin{array}{l}{\tilde{S}}_t^{\mathrm{dev}}-\left(S_t^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)\le 0:{\overline{\rho }}_{h,t}^{\max },{\underset{\_}{\rho }}_{h,t}^{\max }\\ \left(S_t^{\min }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,t}\Delta t}\right)-{\tilde{S}}_t^{\mathrm{dev}}\le 0:{\overline{\rho }}_{h,t}^{\min },{\underset{\_}{\rho }}_{h,t}^{\min }\end{array}\right.$$

(29)

where ${\overline{\rho }}_{h,t}^{\max }$ and ${\overline{\rho }}_{h,t}^{\min }$ denote the dual variables associated with the maximum reservoir storage constraint ${\overline{V}}_{h,t}$ of hydropower station $h$ at the $t$-th period;${\underset{\_}{\rho }}_{h,t,\tau }^{\max }$ and ${\underset{\_}{\rho }}_{h,t,\tau }^{\min }$ correspond to the dual variables of the minimum reservoir storage constraint ${\underset{\_}{V}}_{h,t}$ of hydropower station $h$ at the $t$-th period. This can be further expressed as follows:

$$\left\{\begin{array}{l}{\overline{\rho }}_{h,t}^{\max }={\overline{\rho }}_{h,t,1}^{\max },\dots ,{\overline{\rho }}_{h,t,t}^{\max }\\ {\underset{\_}{\rho }}_{h,t}^{\max }={\overline{\rho }}_{h,t,1}^{\max },\dots ,{\overline{\rho }}_{h,t,t}^{\max }\\ {\overline{\rho }}_{h,t}^{\min }={\overline{\rho }}_{h,t,1}^{\min },\dots ,{\overline{\rho }}_{h,t,t}^{\min }\\ {\underset{\_}{\rho }}_{h,t}^{\min }={\underset{\_}{\rho }}_{h,t,1}^{\min },\dots ,{\underset{\_}{\rho }}_{h,t,t}^{\min }\end{array}\right.$$

(30)

By integrating the methods described in the previous subsection, the reservoir storage boundary constraints Equations (18) and (19) are reformulated. The resulting expressions for the upper and lower bounds of reservoir storage are as follows:

$$\left\{\begin{array}{l}{V}_{h,0}+{\displaystyle {\displaystyle \sum }_{\tau =1}^t}{I}_{h,\tau }+{\displaystyle {\displaystyle \sum }_{\tau =1}^t}(Q_{h-1,\tau }-Q_{h,\tau })\Delta \tau -\\ {\displaystyle {\displaystyle \sum }_{\tau =1}^t}\left({\overline{\rho }}_{h,t,\tau }^{\max }\left(S_{\tau }^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,\tau }\Delta \tau }\right)-{\overline{\rho }}_{h,t,\tau }^{\min }\left(S_{\tau }^{\min }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,\tau }\Delta \tau }\right)\right)\le V_{h,t}^{con,\max },\\ {\displaystyle \frac{{\alpha }_{h-1,\tau }{r}_{h-1,\tau }}{3.6}}-{\displaystyle \frac{{\alpha }_{h,t}{r}_{h,t}}{3.6}}-{\overline{\rho }}_{h,t,\tau }^{\max }+{\overline{\rho }}_{h,t,\tau }^{\min }=0,{\overline{\rho }}_{h,t,\tau }^{\max }\ge 0,{\overline{\rho }}_{h,t,\tau }^{\min }\ge 0,\tau =1,2\dots t\end{array}\right.$$

(31)

$$\left\{\begin{array}{l}{V}_{h,0}+{\displaystyle {\displaystyle \sum }_{\tau =1}^t}{I}_{h,\tau }+{\displaystyle {\displaystyle \sum }_{\tau =1}^t}(Q_{h-1,\tau }-Q_{h,\tau })\Delta \tau -\\ {\displaystyle {\displaystyle \sum }_{\tau =1}^t}\left({\underset{\_}{\rho }}_{h,t,\tau }^{\max }\left(S_{\tau }^{\max }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,\tau }\Delta \tau }\right)-{\underset{\_}{\rho }}_{h,t,\tau }^{\min }\left(S_{\tau }^{\min }-{\displaystyle {\displaystyle \sum _{h=1}^H}{P}_{h,\tau }\Delta \tau }\right)\right)\ge V_{h,t}^{con,\min },\\ {\displaystyle \frac{{\alpha }_{h-1,\tau }{r}_{h-1,\tau }}{3.6}}-{\displaystyle \frac{{\alpha }_{h,t}{r}_{h,t}}{3.6}}+{\underset{\_}{\rho }}_{h,t,\tau }^{\max }-{\underset{\_}{\rho }}_{h,t,\tau }^{\min }=0,{\underset{\_}{\rho }}_{h,t,\tau }^{\max }\ge 0,{\underset{\_}{\rho }}_{h,t,\tau }^{\min }\ge 0,\tau =1,2\dots t\end{array}\right.$$

(32)

3.3. Solution Strategy for the Optimization Objective

Since the objective function Equation (1) involves the optimization of the boundaries of stochastic variables ${\tilde{S}}_t^{all}$, it is difficult to solve directly. This section introduces a generation baseline value $S^{base}$ to reformulate the objective function. The relationships between the baseline value, the stochastic generation at each time period, and the upper and lower bounds of the feasible generation interval are defined as $\left(S_t^{\max }-{\tilde{S}}_t^{all}\right)={\kappa }_t^{\max }{S}^{base},\forall t$ and $\left({\tilde{S}}_t^{all}-S_t^{\min }\right)={\kappa }_t^{\min }{S}^{base},\forall t$, where ${\kappa }_t^{\max }\ge 0,\forall t$ and ${\kappa }_t^{\min }\ge 0,\forall t$ are determined based on the power system’s generation requirements. Accordingly, the objective function in Equation (1) can be reformulated as follows:

$$\max {\displaystyle {\displaystyle \sum _{t=1}^T}\left({\kappa }_t^{\max }+{\kappa }_t^{\min }\right)}{S}^{base}$$

(33)

where ${\kappa }_t^{\max }\ge 0,\forall t$ and ${\kappa }_t^{\min }\ge 0,\forall t$ are predefined parameters.

The transformed objective function in Equation (34) enables the determination of the feasible generation interval of hydropower by optimizing the baseline value $S^{base}$. To solve the model, an iterative method based on the Fibonacci search algorithm is adopted to identify the maximum value of the parameter that satisfies the constraints, given the decision variables and relevant parameters. The solution procedure is as follows:

• Step 1: Select an initial interval $[{\theta }^{\min },{\theta }^{\max }]$, ensuring that when $S^{base}$ takes the two endpoints of the interval, the model yields a feasible solution and an infeasible solution, respectively. This guarantees that the maximum value of $S^{base}$ satisfying the constraints lies within the interval.
• Step 2: Determine the number of iterations $n$ such that the interval length meets the required accuracy $e$.

  $${\displaystyle \frac{{\theta }^{\max }-{\theta }^{\min }}{e}}\le F_n$$

  (34)

  where $F_n$ denotes the $n$-th Fibonacci number.
• Step 3: Initialize two internal points ${\theta }_1$ and ${\theta }_2$, assign $S^{base}$ the values of ${\theta }_1$ and ${\theta }_2$, respectively, solve the model for each case, and determine whether a feasible solution exists.

  $${\theta }_1={\theta }_{\min }+{\displaystyle \frac{F_{n-2}}{F_n}}\left({\theta }_{\max }-{\theta }_{\min }\right)$$

  (35)

  $${\theta }_2={\theta }_{\min }+{\displaystyle \frac{F_{n-1}}{F_n}}\left({\theta }_{\max }-{\theta }_{\min }\right)$$

  (36)
• Step 4: If the model is infeasible at ${\theta }_1$ but feasible at ${\theta }_2$, it indicates that the maximum feasible value of $S^{base}$ lies within the interval $[{\theta }_1,{\theta }^{\max }]$. In this case, update the search interval to $[{\theta }_1,{\theta }^{\max }]$ and recalculate the internal points. Conversely, if the model is feasible at ${\theta }^{\min }$ and infeasible at ${\theta }_2$, update the interval to $[{\theta }^{\min },{\theta }_2]$. If both points are infeasible, it implies that no feasible solution exists within the current interval, and a larger initial interval should be selected, or the model constraints should be relaxed.
• Step 5: Update the Fibonacci index to $n-1$ and repeat the above procedure until the desired accuracy is achieved. The final result is the maximum feasible value $\overline{\theta }$ of $S^{base}$.
• Step 6: Set $S^{base}=\overline{\theta }$, substitute it into the model, and calculate the feasible generation interval boundaries for cascade hydropower at each time period, thereby obtaining the complete generation scheduling results.

4. Case Study

4.1. Engineering Background

A clean energy base composed of a cascade hydropower complex with six downstream hydropower stations in a river basin in Yunnan Province, China, is selected as the research case. The hydropower resources of this cascade are fully utilized, with a total installed capacity accounting for approximately 19% of the total hydropower installed capacity of Yunnan’s power grid. Hydropower stations with different regulation capabilities are distributed alternately along the cascade. The significant hydraulic and electric coupling relationships pose great challenges to unified cascade regulation, making this case suitable for effectively validating the rationality and effectiveness of the proposed model.

Table 1. Basic parameters of cascade hydropower stations.

Hydropower Stations	Regulated Capacity	Installed Capacity (MW)	Normal Water Level (m)	Normal Storage (104 m3)	Dead Water Level (m)	Dead Storage (104 m3)	Fixed Water Consumption Rate
GGQ	daily	900	1307	31,600	1303	26,680	8.00
XW	annual	4200	1240	1,455,700	1166	466,200	1.71
MW	seasonal	1670	994	37,159	988	24,899	5.24
DCS	seasonal	1350	899	70,809	887	44,610	5.24
NZD	multi-year	5850	812	2,177,760	765	1,044,250	1.94
JH	weekly	1750	602	87,090	591	56,200	6.61

lists the basic data of the hydropower stations required by the model, and Figure 1 illustrates the geographical topology of the cascade. In the study, 24 h simulation analysis is conducted using operational data from a typical dry day, normal day, and wet day in 2024. The settings of ${\kappa }_1^{\max }={\kappa }_2^{\max }=\dots ={\kappa }_t^{\max }$ and ${\kappa }_1^{\min }={\kappa }_2^{\min }=\dots ={\kappa }_t^{\min }$ are applied in this study.

The model was implemented in Python 3.8 and solved using GUROBI 10.0.1 (Gurobi Optimization, LLC, Beaverton, OR, USA). Computations were performed on a Windows 11 workstation (Microsoft Corporation, Redmond, WA, USA) equipped with an Intel^® Core™ i7-12700H CPU (Intel Corporation, Santa Clara, CA, USA) and 16 GB RAM.

4.2. Analysis of Optimization Results for Power Generation

In the day-ahead scheduling process of cascade hydropower participating in multi-energy coordinated dispatch, to avoid excessive solution dimensionality, overly broad feasible domains, and poor interpretability of the optimization results, the maximum adjustable operational boundaries for the day-ahead schedule are usually determined according to the actual operating conditions of the stations. The boundary quantification covers indicators such as water level, output, and power generation flow, to ensure the rationality and validity of the model results. Taking the dry day in the study year as an example,

Table 2. Initial boundary data on a typical dry day.

Hydropower Stations	Initial Water Level (m)	Final Water Level (m)		Interval Flow Rate (m3/s)	Output (MW)		Power Generation Flow (m3/s)
		Upper	Lower		Upper	Lower	Upper	Lower
GGQ	1303.59	1305.79	1303.50	31	675	110	464	462
XW	1232.57	1239.50	1166.50	60	4200	120	854	804
MW	992.03	993.00	990.00	30	1420	86	735	733
DCS	892.80	895.97	891.97	55	1025	104	730	680
NZD	810.68	811.50	765.50	165	5200	120	1531	1349
JH	599.14	601.00	598.97	30	810	750	1455	1452

shows the boundary input factors for the optimization model on the typical dry day.

The operation mode of hydropower stations is significantly influenced by inflow conditions. To validate the effectiveness of the proposed method, this study selected the dry day, normal day, and wet day of 2024 under different inflow frequencies as typical cases for computational analysis. Figure 2 illustrates the optimized planned power generation of the cascade, adjustable power range, and specific power generation process of each hydropower station under different inflow conditions. The results indicate that the daily planned electricity generation fluctuates considerably across the dry, normal, and wet days, yet all exhibit a similar generation process. During the early morning low-demand period, to reserve sufficient accommodation space for renewable energy sources such as wind and solar, the cascade hydropower system minimizes generation. During midday and evening peak demand periods, to assist the grid in peak shaving, the cascade system maintains peak generation. Meanwhile, the adjustable power generation ranges under different inflow conditions are ±748 MWh, ±1456 MWh, and ±2099 MWh for the dry, normal, and wet days, respectively. This suggests that inflow conditions have a significant impact on the adjustable power generation capacity of cascade systems and the specific power generation processes of individual stations. For instance, the maximum generation of the NZD decreases significantly from 3024 MWh in the dry day to 2304 MWh in the normal day, while the generation of the XW and MW increases markedly by 470 MWh and 787 MWh, respectively, to support the feasible generation range of the cascade system. Additionally, driven by the objective of maximizing cascade generation, to ensure the cascade hydropower stations maintain the maximum adjustable generation range during multi-energy coordinated scheduling to address sudden power regulation needs, the proposed model maximizes the generation range while consistently maintaining symmetrical adjustment boundaries for upward and downward regulation.

After obtaining the optimal generation interval for the cascade hydropower system, the rational allocation of adjustable power based on the operational status and boundary conditions of each station is particularly critical. Figure 3 illustrates the optimal allocation coefficients and corresponding adjustable power for each hydropower station, optimized by the proposed algorithm under different inflow conditions. The results show that, due to the relatively narrow feasible boundary conditions during the normal day, the algorithm significantly reduces the adjustable power of the NZD, which serves as the primary regulating station during the dry and wet days, with an average allocation coefficient of only 0.14, compared to 0.29 and 0.21 for the wet and dry days, respectively. This result is closely related to the aforementioned analysis results of power generation, further enhancing the model’s interpretability. Additionally, to maximize the adjustable power capacity of the cascade system in multi-energy coordinated scheduling, the algorithm minimizes the number of primary regulating stations while ensuring feasible boundary conditions. For instance, during the dry day, the XW and NZD primarily provide regulation capacity, whereas on the normal and wet days, regulation is mainly handled by XW (as the primary station), MW, and NZD. Notably, the leading station GGQ, with poor regulation performance, struggles to cope with the larger inflow variations during the normal and wet days, requiring significant adjustments in its generation process based on natural inflow, which introduces considerable uncertainty to the cascade’s generation range. In this context, while ensuring the cascade’s adjustable power remains constant, the proposed model leverages the downstream daily regulating stations MW and DCS (as XW is constrained by boundary conditions and unable to provide adjustable resources) to absorb the fluctuating regulation process of GGQ, thereby significantly enhancing the cascade system’s operational stability.

Figure 4 presents the operational processes corresponding to the adjustable power generation of each hydropower station during the typical dry day. It can be observed that the water level processes corresponding to the upper and lower limits of power generation, optimized by the proposed model, fall within the daily water level boundaries specified in Table 2, meeting the practical engineering requirements. As shown in Figure 3a, XW and NZD undertake the primary adjustable power during the dry day, resulting in a continuous decline in their water levels in Figure 4, with larger adjustable power corresponding to more significant water level reductions. Conversely, although MW, DCS, and JH do not contribute to power supply, their water level processes are significantly affected by the generation processes of the upstream XW and NZD. Additionally, the poor regulation performance of these stations leads to a wider range of water level fluctuations. In contrast, since GGQ does not undertake adjustable power during the dry day and is unaffected by upstream stations, its water level processes for the upper and lower limits remain nearly identical to the planned power generation process, demonstrating high stability.

In summary, the proposed model can optimize the maximum adjustable power range and corresponding planned generation processes for cascade hydropower stations under different inflow boundary conditions. Furthermore, the model optimizes the allocation of cascade power based on the regulation capabilities of individual hydropower stations and inflow processes, yielding optimal allocation coefficients for each hydropower station. Meanwhile, the water level processes corresponding to the upper and lower limits of the adjustable power ranges for each hydropower station, as calculated by the model, satisfy the daily water level boundary requirements. This model provides efficient and specific guidance for cascade hydropower participation in multi-energy coordinated scheduling, meeting the engineering demand for interpretable scheduling results and effectively addressing the high-dimensional complexity challenges in formulating feasible generation range plans.

4.3. Sensitivity Analysis of Water Level Boundary

To investigate the impact of day-ahead water level boundaries and hydropower station regulation performance on the adjustable power capacity of the cascade system, this section conducts a sensitivity analysis focusing on the dry day as a case study. The analysis adjusts the day-ahead final water level ranges for hydropower stations with different regulation capabilities to examine the coupled effects of water level and regulation performance.

Table 3. Boundary conditions for final water level.

Hydropower Stations	Final Water Level Boundary
GGQ	(1303.8, 1305.4)
XW	(1166.8, 1239.2)
MW	(990.3, 992.7)
DCS	(892.27, 895.67)
NZD	(765.8, 811.2)
JH	(599.27, 600.7)

presents the new final water level boundary conditions for the typical dry day after uniformly reducing the extreme final water level ranges of each station by 0.5 m. With the reduction in boundary conditions, the originally constrained feasible solution space is further compressed, significantly increasing the dimensionality of the optimization problem. Despite this challenge, the proposed model successfully derives effective and engineering-interpretable feasible solutions, providing valuable insights for practical engineering applications. Figure 5 illustrates the optimized results under these boundary conditions, including the power generation range of the cascade system (Figure 5a), the generation processes of each hydropower station (Figure 5b), the distribution result of the adjustable power generation range (Figure 5c), and the water level variation processes of the main reservoirs (Figure 5d–f). The results show that after reducing the adjustable range of final water levels, the cascade’s power generation range is significantly reduced, with the adjustable power range decreasing from ±748 MWh to ±371.6 MWh. This indicates a direct correlation between the adjustable final water level range of each hydropower station and the feasible power generation range of the cascade system, where a larger water level variation range corresponds to a broader adjustable power generation range. Additionally, the ±371.6 MWh power generation range of the cascade system is primarily supported by hydropower stations with better regulation performance, namely XW, DCS, and NZD, with average contributions of ±95.5 MWh, ±35.5 MWh, and ±170.3 MWh, respectively. Compared to the typical dry day scenario, the reduction in the final water level range leads to a significant decrease in the adjustable power generation, particularly for the main regulating hydropower station XW, where the adjustable power generation decreases from ±225.1 MWh to ±95.5 MWh. The corresponding water level variation amplitude and range are markedly reduced, nearly aligning with the planned power generation water level process, forming a stark contrast with water level process in Figure 4. Meanwhile, the adjustable power provided by NZD increases sharply, with a corresponding expansion in its water level adjustable range, contrasting with XW’s behavior.

Additionally, this section conducts a sensitivity analysis on the water level boundaries of hydropower stations with varying regulation performance to evaluate their impact on the adjustable power generation of the cascade hydropower system. Figure 6 presents the sensitivity analysis results for the reduction in final water levels across different regulation performance hydropower stations Among them, Figure 6a shows the reduction effects for hydropower stations with annual and multi-year regulation capabilities (XW and NZD), Figure 6b illustrates the reduction effects for seasonal regulation hydropower stations (MW and DCS), and Figure 6c depicts the reduction effects for hydropower stations with daily and weekly regulation capabilities (GGQ and JH). The results indicate that hydropower stations with stronger regulation performance contribute more significantly to the feasible power generation range of the cascade system. When the final water levels of XW and NZD are reduced by 0.5 m, the cascade’s adjustable power generation range drops sharply to ±412.2 MWh, nearly equivalent to the ±371.6 MWh observed when all hydropower stations’ final water levels are reduced in Figure 5a, highlighting that the primary adjustable power generation of the cascade system is borne by the main reservoirs XW and NZD. In contrast, when only the final water levels of the weaker regulation hydropower stations GGQ and JH are reduced, the cascade’s adjustable power generation range remains relatively high at ±609.6 MWh, close to the ±748 MWh shown in Figure 2a, indicating that it has a relatively small contribution to the feasible power generation range of the cascade system.

In summary, the proposed model can still derive feasible solutions for the cascade hydropower system under tightened final water level boundaries and increased solution dimensionality. The results also demonstrate a strong correlation between the cascade’s adjustable power generation and the adjustable final water level ranges of each hydropower station, with higher correlation for hydropower stations with stronger regulation performance. This provides an effective reference for achieving unified cascade scheduling.

4.4. Sensitivity Analysis of Adjustable Power Generation Boundary

The previous section has demonstrated that the operational status of hydropower stations significantly affects the adjustable power generation range of the cascade system in multi-energy coordinated scheduling. Similarly, the requirements for adjustable power generation boundaries in the day-ahead planning of multi-energy coordinated scheduling also have a notable impact on the overall operational status of the cascade system. To investigate the changes in the operational status of the cascade system under different adjustable power generation boundaries, this section conducts a sensitivity analysis of the adjustable power generation boundaries during multi-energy coordinated scheduling, using a typical dry day as the case study.

From the previous analysis, it is evident that under the influence of the objective of maximizing the power generation range, the upper and lower adjustable power generation boundaries of the cascade hydropower system are typically evenly distributed to accommodate sudden power adjustments in the electricity system. However, in practical scheduling, due to the influence of various factors, the multi-energy coordinated scheduling process imposes different proportional requirements for the upper and lower adjustable power generation ranges during different periods. Figure 7 presents the sensitivity analysis results for three boundary conditions set in this study. Firstly, regardless of the boundary proportion, the proposed model consistently optimizes the overall upper and lower adjustable power generation ranges for the cascade system and the power generation support ranges for each hydropower station, according to the specified requirements. However, since the initial input conditions, such as inflow and water levels, remain the same, the total adjustable power generation of the cascade system remains constant at ±1496 MWh, determined by the day-ahead planning of the cascade system. Additionally, it can be observed that under different power generation boundary conditions, the adjustable power generation of the cascade system is still primarily supported by the hydropower stations with the best regulation performance, namely NZD and XW. Finally, it is also evident that after determining the adjustable power generation range of the cascade, the power generation allocation results for each hydropower station continue to adhere to the set power generation boundary proportions, ensuring coordinated scheduling across the entire cascade system.

Figure 8 illustrates the boundary and planned water level variation processes of the main hydropower stations XW and NZD under different adjustable power generation boundaries. Due to identical initial conditions for the cascade system under the three boundary proportions, the planned water level variation processes for XW and NZD are entirely consistent. Meanwhile, under the influence of different power generation boundary proportions, the water level variations for the upper and lower power generation limits also exhibit the same proportional relationships, corresponding to the power generation boundaries. Furthermore, the water level processes under different boundaries align perfectly with the generation processes shown in Figure 7. In addition, under the condition where the upper boundary is 1.2 times the lower boundary, XW hydropower station undertakes the primary regulation capacity for the cascade system, resulting in larger upper and lower adjustable water level boundaries in Figure 8, thereby enhancing the interpretability of the power generation plan.

4.5. Analysis of Engineering Applicability

Computational efficiency and scalability directly determine the practical applicability of the proposed method. This section validates the method’s performance in real engineering contexts. To evaluate computational efficiency and scalability, the original engineering case was duplicated and expanded to construct a comparative scenario. In this scenario, the project’s scale was doubled by adding another set of cascade hydropower stations downstream and establishing hydraulic connections.

The computational details for both the original and comparative scenarios using the proposed method are presented in

Table 4. The computational details for both the original and comparative scenarios using the proposed method.

Computational Information	Original Scenario	Comparative Scenario
Modeling time (s)	36	39
Solution time (s)	32	38
Total time (s)	68	74
Number of iterations	8	11
Gap	0	0
e	<1	<11

. In the original case with four cascade hydropower stations, the modeling, solution, and total computation times were 36 s, 32 s, and 68 s, respectively. In the comparative case with eight cascade hydropower stations, the modeling, solution, and total computation times were 39 s, 38 s, and 74 s, respectively. The number of iterations required in the solution process was 8 for the original case and 11 for the comparative case, with both achieving a gap of zero, indicating that the maximum feasible generation interval was obtained within the required accuracy. Moreover, the increases in modeling, solution, and total computation times from the original to the comparative case were relatively small, suggesting that the proposed method is only marginally affected by system size and remains applicable to more complex engineering scenarios. These results demonstrate that the proposed method performs well in terms of computational efficiency and scalability, thereby satisfying the requirements of practical scheduling and offering notable advantages for engineering implementation.

5. Conclusions

This study addresses the flexibility requirements of cascade hydropower systems in adapting to multi-energy coordinated scheduling during the day-ahead grid scheduling process. An optimization model for the regulation capacity of cascade hydropower systems, incorporating fixed parameter boundary constraints, was developed and validated through various typical practical cases, leading to the following four conclusions:

The proposed model and solution method effectively reduce the computational complexity of determining the feasible power generation range by specifying the boundary of primary constraints, successfully deriving the optimal generation plans and corresponding maximum adjustable power generation ranges for the cascade system during the dry, normal, and wet day.

As the regulation performance of the hydropower station improves, its contribution to the feasible power generation range of the cascade system increases, and with more abundant inflow resources, the allocated adjustable power generation of the cascade system also rises.

The final water level boundary conditions significantly influence the adjustable power generation range of the cascade system; a larger adjustable final water level range corresponds to a broader adjustable power generation range, with this effect being more pronounced in hydropower stations with better regulation performance.

Under the given day-ahead boundary, the total adjustable power generation range of the cascade system remains fixed, but the proportional relationship between the upper and lower boundaries can be fully adjusted according to the actual demands of the power system.

By expanding the engineering scale for comparative analysis, it is demonstrated that the proposed method is only marginally affected by system size, while both computational efficiency and accuracy satisfy practical scheduling requirements, confirming its applicability to large-scale hydropower systems.

Future work will further focus on characterizing the feasible generation intervals of cascade hydropower systems under complex hydroelectric coupling conditions, such as water delay time, unit maintenance, and multi-grid transmission. In addition, greater attention will be devoted to renewable energy uncertainties and market-oriented challenges, to advance a hydropower operation theory framework that addresses the scheduling needs of multi-source systems.