A two-stage stochastic programming-based planning and layout model for pumped-storage power stations and its efficient solution algorithm

Abstract

Motivated by the need for pumped-storage planning under high renewable energy penetration, this study proposes a two-stage stochastic planning and layout model for pumped-storage power stations. The model considers hydrometeorological factors, source-load uncertainty, candidate-site heterogeneity, and multi-scenario operational verification. First, a meteorology-source-load–regulation-demand mapping mechanism is constructed to convert multi-source uncertainties associated with wind power, photovoltaic generation, and electric vehicles into regional regulation capacity requirements. Second, a candidate-site parameter database is established by considering reservoir capacity, hydraulic head, efficiency, cost, grid-connection conditions, and regional attributes. Furthermore, a compact reformulation and hierarchical warm-start solution strategy is proposed to improve the solution efficiency of the large-scale stochastic mixed-integer model. The strategy integrates single-mode variable reformulation, online-unit-number binding, safe upper-bound tightening, redundant-constraint reduction, and warm-start mechanisms. Case study results show that the proposed model can effectively characterize the planning layout and operational adaptability of pumped-storage power stations under uncertainty, while improving the solution efficiency and stability of large-scale planning models, thereby providing decision support for the scientific allocation of pumped-storage resources in new-type power systems.

1 Introduction

Driven by the dual-carbon goals and energy-structure transition, the installed capacity of renewable energy sources such as wind power and photovoltaics has continued to expand, and power system operation has gradually exhibited strong uncertain-ties on both the source and load sides (Zhao Y. et al., 2024). On the source side, renewable energy output is susceptible to meteorological factors such as wind speed, solar irradiance, temperature, and precipitation, and therefore exhibits pronounced volatility, intermittency, and randomness (Westphal, 2024; Sonyy et al., 2026). Meanwhile, extensive access of new types of loads, particularly electric vehicles, intensifies the uneven spatiotemporal distribution of demand and in-creases load uncertainty, thereby enlarging the peak-to-valley difference (Powell et al., 2022). As renewable energy penetration continues to increase, operational requirements such as short-term ramping and reactive power support rise significantly, imposing higher demands on the overall flexibility and regulation capability of power systems (Zhou et al., 2026; Guo et al., 2021).

Pumped-storage power stations are characterized by large capacity, long service life, and superior regulation performance (Zhou et al., 2024). They can absorb surplus electricity during periods of high renewable generation, release electricity during load peaks or low renewable output, and further provide reserve support and reactive power regulation (Li et al., 2023). Relative to electrochemical storage technologies, pumped-storage hydropower offers clear advantages for long-duration storage and large-scale power system regulation. Scientifically determining the construction locations, installed capacities, and multi-scenario operation strategies of pumped-storage power stations is of great significance for improving renewable energy accommodation, enhancing power system resilience, and promoting the sustainable development of energy systems (Ma et al., 2022).

In recent years, extreme weather events have become more frequent, widespread, and intense, exerting an increasingly prominent impact on the secure operation of power systems (Xu et al., 2025). Extreme weather may induce abnormal load fluctuations, alter renewable energy output characteristics and power flow distributions, and thereby intensify supply–demand balancing pressure and operational uncertainty. On the other hand, hydrometeorological conditions also affect the intrinsic regulation capability of pumped-storage power stations (Brá et al., 2023). Factors such as precipitation, evaporation, reservoir inflow, and temperature may change the available water volume, water levels, and actual hydraulic head of the upper and lower reservoirs, thereby affecting the available energy storage capability of the station (Wang et al., 2022). Under abnormal conditions such as drought or heavy rainfall, the actual dispatchable capability of a pumped-storage station may deviate from its planned capacity, thereby affecting system security and eco-nomic performance under extreme operating conditions. Therefore, pumped-storage planning is not merely a capacity allocation problem (Zhao J. et al., 2024). Since candidate sites vary in terms of head, storage capacity, cost, grid-access conditions, geographical location, and sensitivity to hydrometeorological conditions, the planning decision should jointly determine investment status, construction scale, and multi-scenario operational adaptability, making the problem a joint planning and operation optimization problem considering uncertainty (Yong et al., 2022; Cheng et al., 2023).

In the context of planning flexible regulation resources for power systems with a high proportion of renewable energy, existing research commonly develops representative scenarios from historical wind power output, photovoltaic generation, load profiles, and meteorological data, while using clustering, scenario reduction, and probabilistic modeling to reduce uncertainty dimensions, capture random variations in renewable output and load demand, and support multi-scenario inputs for system planning and operational optimization. Dong et al. proposed a renewable generation scenario generation method based on a controllable generative adversarial network to characterize the uncertainty and variability of wind and photovoltaic output, and generated renewable output scenarios with different statistical characteristics through interpretable features (Dong et al., 2022). In capacity planning of integrated energy systems, Peng et al. used multi-scenario clustering and Gaussian mixture models to represent uncertainty intervals associated with wind, PV, and load forecasting errors, thus alleviating the complexity of multi-source uncertainty modeling in long-term planning (Peng et al., 2023). For the dynamic reconfiguration problem of distribution networks, Sun et al. further con-structed joint representative scenarios involving wind power, PV generation, and load, while employing generative models to maintain the temporal dependence and coupled characteristics of source-load data (Sun et al., 2025). However, most existing studies still mainly focus on generating, compressing, or extracting representative forms of original wind power, photovoltaic generation, and load curves, while planning models typically receive net load curves, typical renewable output curves, or several representative source-load scenarios, without further transforming source-load uncertainty into regulation-demand signals that are truly required for planning decisions.

In the field of pumped-storage planning, existing research has primarily addressed system economic performance, capacity configuration, operation dispatch, renewable energy integration, and power grid security. Toufani et al. provided a systematic re-view of pumped-storage optimization under uncertainty, highlighting that pumped-storage hydropower, as a large-scale long-duration storage technology, is characterized by flexible unit commitment, high efficiency, long lifetime, fast load-following capability, and support for voltage stability, while existing optimization models are commonly developed in terms of capacity planning, operational optimization, market benefits, reliability, and multi-objective decision-making (Toufani et al., 2023). Nasir et al. further focused on a coupled system of pumped storage and conventional hydropower, developing a power-energy analysis and capacity optimization model that comprehensively considers site selection, reservoir location, hydraulic head, waterway length, hydrological data, water balance, pumping and generating operation modes, and charge-discharge ratio, indicating that pumped-storage capacity allocation is related not only to system regulation demand but also to the hydrological conditions and engineering layout of specific sites (Nasir et al., 2022). From the perspective of site selection, Ali et al. developed a GIS-AHP-based assessment method that integrates factors including elevation difference, terrain slope, access to water sources, land-use conditions, distances from roads and transmission lines, proximity to wind and photovoltaic resources, precipitation conditions, and levelized cost of energy (Ali et al., 2023). Although the above studies have provided valuable foundations for pumped-storage planning and uncertainty modeling, two limitations still remain. First, most scenario-based planning studies directly use renewable generation, load, or net-load curves as model inputs. The regulation requirements that pumped-storage stations should actually undertake, such as upward and downward power support, ramping support, reactive power support, and regional flexibility demand, are not explicitly extracted from source-load uncertainty. Second, existing pumped-storage planning studies often focus on either site evaluation or capacity optimization, while the hydrometeorological sensitivity and engineering heterogeneity of multiple candidate stations are not fully embedded into a unified regional planning model.

The remainder of this paper is organized as follows. Section 2 presents the overall framework, key modeling methods, and efficient solution strategy of the proposed model. Section 3 analyzes the results of pumped-storage siting and capacity sizing, planning adaptability under different hydrometeorological conditions, and model solution performance through case studies. Section 4 further discusses the mechanism, engineering applicability, and limitations of the proposed method. Section 5 summarizes the conclusions of the paper and provides perspectives on future research directions.

2 Materials and methods

2.1 Overall research framework

For the siting and capacity sizing of pumped-storage power stations in the context of high renewable energy integration, this study constructs a two-stage stochastic planning model that incorporates hydrometeorological factors, along with an efficient solution approach. The overall research framework of this paper is shown in Figure 1.

First, at the data level, this study comprehensively considers multi-source uncertain factors, including wind power, photovoltaic generation, conventional loads, and hydrometeorological states, and constructs a set of typical scenarios that reflect different weather conditions and source-load fluctuation characteristics. Unlike conventional scenario generation approaches, the proposed method transforms representative scenarios into power-balancing requirements, ramping requirements, and regional flexibility demands, thus linking external hydrometeorological disturbances and source-load stochasticity with planning-model input parameters.

At the engineering-parameter level, this study establishes a parameter database for candidate pumped-storage power stations to describe differences among candidate sites in terms of installed-capacity limits, geographic characteristics, reservoir storage capability, investment and construction costs, and regional affiliation. Accordingly, pumped-storage power stations are extended from abstract energy-storage nodes in traditional models to a set of candidate sites with engineering heterogeneity, thereby enhancing the practical engineering significance of the planning results.

This study adopts a two-stage stochastic programming framework to describe the coupling relationship between pumped-storage siting and capacity sizing and multi-scenario operational verification. The first stage is the planning decision stage, which mainly determines whether each candidate station should be constructed and its corresponding installed capacity. The second stage is the operational verification stage, which determines the generating and pumping states of pumped-storage power stations under different typical scenarios and time periods, as well as the satisfaction of various regulation demands. Through two-stage modeling, the planning results can link long-term construction decisions with operational verification under uncertain scenarios, thereby improving the adaptability of the scheme to multi-scenario operational requirements while ensuring planning economy.

For solving the original two-stage stochastic mixed-integer planning model, which suffers from high variable dimensionality, strong integer-variable coupling, and limited solution efficiency, this paper develops a compact-reformulation and hierarchical-warm-start solution method. Through single-mode variable reformulation, online-unit-number binding, safety-based physical upper-bound tightening, redundant-constraint reduction, and a warm-start strategy based on a relaxed dispatch sub-model, this method systematically restructures the model without modifying its core engineering constraints, compresses both the integer search space and the continuous relaxation space, and improves the solution efficiency and stability of large-scale planning models.

2.2 Source-load uncertainty scenario modeling considering extreme weather factors

With the integration of a high proportion of renewable energy, the operating state of power systems is jointly affected by wind and photovoltaic output on the source side, load fluctuations on the demand side, and meteorological factors. To capture uncertainties in system flexibility requirements during the planning stage, this study devel-ops a multi-source stochastic scenario set for source-load variations that incorporates extreme weather factors.

During scenario generation, this study first normalizes historical source-load and meteorological data and extracts relevant features. On the source side, this study con-siders wind power and photovoltaic output. On the load side, with the rapid development of electric vehicles, their loads exhibit temporal aggregation characteristics that are clearly different from those of conventional loads: vehicles are mostly in travel states during the daytime, resulting in weak charging demand, whereas they return and connect to charging facilities at night, easily forming concentrated loads and thereby changing the system peak-valley difference and local power flow distribution. Therefore, electric vehicle loads are modeled as an independent uncertainty factor to distinguish their differences from conventional loads in terms of temporal distribution and regulation characteristics, thereby enabling more accurate identification of system regulation demand. Then, hydrometeorological states are incorporated into hierarchical scenario clustering. The objective is not simply scenario-number reduction, but rather to preserve the representation of typical weather states, including normal, flood, and drought conditions, in the planning model and to avoid attenuating extreme conditions during scenario reduction.

The set of typical scenarios is defined as , where is the scenario index and is the total number of scenarios. The probability of scenario is denoted by , and the scenario probabilities satisfy Equation 1:

The set of dispatching periods is defined as , where is the time-period index and is the total number of periods in each typical scenario. For each scenario dispatching period , the net load of the system is formulated in Equation 2:where denotes wind power output, denotes photovoltaic output, and represent conventional load and electric vehicle load, respectively. The system net load reflects the overall demand for flexible regulation resources and directly affects the planning and construction decisions of pumped-storage power stations.

To reflect the impact of extreme weather on the uncertainty of source-load scenarios, this paper further introduces a meteorological state set . In this formulation, represents the meteorological state which are categorized into normal, flood, and drought conditions. Different meteorological conditions affect the source-load distribution in typical scenarios, thereby influencing the net-load deficit, ramping demand, energy storage demand, and regional regulation demand. For example, continuous rainy weather can significantly reduce photovoltaic output, whereas high-temperature drought conditions may increase electricity demand and intensify supply-demand balancing pressure under adverse scenarios. This paper couples meteorological states with multi-source stochastic scenarios to fully reflect source-load fluctuation characteristics and variations in regulation demand under extreme meteorological conditions.

2.3 Meteorology-source-load-regulation demand mapping

The inputs required by the pumped-storage siting and capacity sizing planning model are not raw source-load curves, but the regulation demands formed by the system under different scenarios and time periods. This section constructs a meteorology-source-load-regulation demand mapping mechanism to further transform multi-source stochastic scenarios into the power support and ramping support requirements that pumped-storage power stations need to undertake.

2.3.1 Power support requirement

In power systems with a high proportion of renewable energy, active power regulation demand mainly arises from net-load deviations. When the system net load exceeds the baseline operating level or the support capability of conventional resources, pumped storage is required to provide upward regulation support through power generation. Otherwise, it absorbs surplus electricity through pumping operation and provides downward regulation support.

Denote the baseline net load at time period by , the net-load deviation in scenario is then formulated in Equation 3:

Accordingly, the system upward and downward active power regulation demands can be further obtained using Equation 4:

Renewable energy output and load fluctuations affect not only the active power balance of the system but also nodal voltage levels and reactive power support requirements. To reflect the role of pumped-storage power stations in reactive power support and phase-modulation operation, this paper further constructs a reactive pow-er demand indicator.

Let denote the deviation in system reactive power support demand under scenario and time period , which is defined in Equation 5:

After reactive power demand is incorporated into the planning model, the role of pumped-storage power stations is no longer limited to the time-shifting regulation of active energy, but is further extended to operational security support, including voltage support and reactive power coordination.

2.3.2 Ramping requirement

Fluctuations in renewable energy output and load variations can cause rapid changes in net load between adjacent time periods, thereby increasing system ramping demand. Pumped-storage power stations have strong power regulation and ramping response capabilities, enabling them to participate in system peak-valley regulation and flexibility support under certain operating conditions. Accordingly, the planning model not only accounts for the capacity-support function of pumped-storage stations, but al-so further characterizes their ability to respond to system ramping requirements when they are in available operating states.

The net-load ramping amount in scenario is formulated in Equation 6:

Accordingly, the upward and downward ramping requirements of the system are formulated in Equation 7:

Considering that the start-up and shutdown processes and minimum operating time of pumped-storage units are constrained by equipment and hydraulic limitations in actual operation, they cannot be treated as resources with infinitely fast response capability at the planning stage.

2.4 Construction of a candidate site parameter database

Based on the above parameters, candidate stations are no longer treated as homogeneous energy storage nodes, but as planning objects with different engineering boundaries and economic characteristics. According to the site parameters, this method first performs preliminary screening based on hard constraints, preferentially filtering out candidate sites that clearly fail to meet construction conditions, have excessive development costs, or are severely limited by basic conditions, thereby improving the processing efficiency of the subsequent planning model. Subsequently, comprehensive scoring results are obtained through soft evaluation grading, and candidate sites are preliminarily ranked. These results, together with the previously derived system demand-side results, support the subsequent capacity configuration and solution of the planning model.

2.5 Hydrometeorological parameter correction mechanism

The available hydraulic head, reservoir storage state, and operating capability of pumped-storage hydropower plants are influenced by hydrometeorological conditions. To characterize the influence of hydrometeorological states such as rainstorm and drought on the operation and planning outcomes of pumped-storage stations, a hydrometeorological parameter adjustment mechanism shown in Figure 2 is incorporated on the basis of the candidate site parameter database.

For candidate station under hydrometeorological state , the head correction parameter and storage-capacity correction parameter are denoted by and , respectively. Accordingly, the corrected available hydraulic head and storage capacity are formulated in Equation 8:

Under drought conditions, the available reservoir water volume and effective storage capacity may decrease, typically leading to . For flood conditions, despite improved water replenishment, water-level safety constraints may still limit station operation, requiring adjustments to the constraints on initial storage state, terminal storage state, and available reservoir capacity. For extreme operating conditions, the model can simulate variations in the available capability of pumped-storage power stations under special hydrometeorological states by simultaneously adjusting parameters related to head, water replenishment, evaporation, and terminal energy storage limits.

Further, is defined as the water replenishment amount of candidate station at time period under hydrometeorological state , while represents evaporation or other water losses, so that the following hydrological correction term can be incorporated into the storage-state transition equation, as formulated in Equation 9:where denotes the energy storage state of candidate station and denotes the selected time-period length.

Thus, hydrometeorological states are transformed from external scenario labels into factors that directly shape the operating limits and available capability of pumped-storage stations through parameter corrections and water-balance terms. Relative to conventional approaches based on fixed rated capacity and fixed storage-capacity parameters, the proposed mechanism more accurately captures changes in the practically dispatchable capability of pumped-storage stations under different conditions, and further assesses whether the selected site scheme can retain stable regulation performance and operational adaptability across diverse hydrometeorological conditions, providing support for validating the rationality of the site-selection scheme in terms of both planning outcomes and operation checks.

2.6 Two-stage stochastic programming model

2.6.1 Model decision variables

In this study, the siting and capacity sizing problem of pumped-storage power stations is modeled as a two-stage stochastic planning problem. The first stage makes long-term planning decisions before uncertainty is realized, including candidate pumped-storage site selection, installed capacity configuration, and energy storage scale determination. The second stage performs operational verification after the realization of uncertain scenarios involving source-load variations and hydrometeorological conditions, describing the generation state, pumping state, storage state, and regulation support behavior of each candidate station across different time periods.

The first-stage planning decision variables can be formulated in Equation 10:where is a binary variable indicating whether candidate station is selected for retrofitting or construction, denotes the number of units to be constructed, and represent the planned installed capacity and planned energy storage capacity of station, respectively, and denotes the set of candidate stations.

The second-stage planning decision variables can be formulated in Equation 11:where and denote the generating power and pumping power of candidate station, respectively. denotes the energy storage state. is the operating-mode variable. and represent the upward and downward active power support provided by the candidate station, respectively. denotes the reactive power support capability. and denote the upward and downward ramping support capabilities, respectively.

2.6.2 Objective function

Subject to the planning and operational constraints, the model aims to minimize the comprehensive annual cost by incorporating investment cost, fixed construction cost, operation and maintenance cost, penalty cost for unmet demand, water-spillage penalty cost, and state-transition penalty cost, thereby obtaining an overall optimal planning scheme. The objective function can be formulated in Equation 12:where denotes the first-stage investment cost, denotes the operating cost under scenario , and denotes the penalty cost incurred when regulation demand is not fully satisfied, which can be formulated in Equations 13–15:where denotes the penalty coefficient for the unmet amount of the regulation demand, and is a slack variable. Introducing slack variables can prevent the model from becoming infeasible under extreme scenarios due to insufficient regulation capability, while the penalty cost reflects the extent to which the planning scheme satisfies the system regulation-capability requirements.

2.6.3 First-stage planning constraints

The first-stage planning constraints are used to describe the logical relationships among candidate pumped-storage site selection, installed capacity configuration, unit-number configuration, and investment budget. For candidate station , the site-selection variable and planning variables should satisfy the following constraints, as given in Equations 16, 17:

When , no installed capacity or energy storage capacity is allocated to the candidate station. When , its capacity configuration must not exceed the engineering upper limits of the candidate station.

If the installed capacity is determined by unit configuration, the number of units and the installed capacity are constrained by Equations 18, 19:

This constraint ensures that units can be configured only when a candidate station is selected, and that at least one unit is installed at each selected station.

2.6.4 Second-stage planning constraints

The second-stage planning constraints are used to describe the generation, pumping, and storage-state variations of candidate pumped-storage power stations under source-load stochastic scenario , hydrometeorological state , and time period , as well as their support for system regulation demand. For any unselected candidate station, all second-stage operation variables are constrained to be zero.

First, the generation power of each candidate station is constrained by its con-structed capacity and the number of online units, as formulated in Equation 20:

When the minimum technical output is considered, the generating power should not be lower than the minimum techinical output limit, as formulated in Equation 21:

The constraints for the pumping mode can be formulated analogously. In addition, a pumped-storage power station cannot be in the generating state and pumping state simultaneously in any time period, and should satifsy the following mutual-exclusion constraint, as formulated in Equation 22:where and are binary variables indicating whether candidate station is in the generating state or pumping state, respectively.

The energy storage state recursion constraint is formulated in Equation 23:

The upper and lower bound constraints on the energy storage state are formulated in Equation 24:

To prevent the model from overdrawing reservoir storage in the terminal period, a terminal energy storage constraint is formulated in Equation 25:

The constraints related to active power, reactive power, ramping capability, and regional support in the model are summarized in Equations 26–29:

where the regional support constraint ensures that the selected pumped-storage power stations can not only satisfy regulation demand at the system-wide level, but also cover support requirements across different regions in terms of spatial distribution.

The linkage between the two stages is established through non-anticipative planning decisions and scenario-dependent operational recourse decisions. In the first stage, the site-selection variable, number of units, installed capacity, and energy storage capacity are determined before the realization of source-load and hydrometeorological uncertainties. These decisions are shared by all scenarios and cannot be adjusted after a specific scenario is observed. In the second stage, the operation of each selected pumped-storage station is verified under each joint scenario and time period. The second-stage variables, such as generating power, pumping power, and ramping support, are constrained by the first-stage capacity decisions and hydrometeorological correction parameters. Therefore, the first-stage decisions determine the feasible operating boundaries of the second-stage dispatch problem, while the expected second-stage operating cost and shortage penalty feed back to the first-stage objective function. This structure enables the model to balance long-term construction economy and multi-scenario operational adaptability.

2.7 Efficient solution method based on compact reformulation and hierarchical warm-start strategy

The pumped-storage siting and capacity sizing model constructed in this paper is a large-scale two-stage stochastic mixed-integer programming problem, involving constraints related to candidate-station construction decisions, multi-scenario operation scheduling, pumped-storage state transitions, reservoir energy balance, and regional regulation demand. As the numbers of candidate sites, typical scenarios, and time periods increase, the scale of binary variables and coupling constraints in the original model expands rapidly, which may lead to an inflated integer search space, weak root-node relaxation bounds, and unstable solution convergence. Therefore, this paper performs a compact reformulation of the original model structure and combines it with a hierarchical warm-start strategy to improve the solution efficiency of the complete MILP model.

2.7.1 Single-mode variable reformulation

Traditional models usually use two binary variables to represent the generating state and pumping state, respectively, and impose a mutual-exclusion constraint to prevent both variables from taking the value of 1 simultaneously. Although this meth-od is intuitive, it significantly increases the number of integer variables under large-scale scenario settings. This paper adopts a single-mode variable to uniformly represent the operating mode, which is formulated in Equation 30:

The essence of this reformulation is to replace separate judgments of generation and pumping states with a direct representation of the unique operating mode of each station under every scenario and time period, thus reducing redundant binary variables and mutual-exclusion constraints.

2.7.2 Coupling between the number of online units and power output capability

Let and denote the numbers of online generating units and online pumping units of candidate station under scenario and time period , respectively. Then, the generating power and pumping power can be directly constrained by the number of online units, as formulated in Equation 31:

With this reformulation, the power output capability of each station is directly linked to the number of units allocated in the first stage, which prevents the second-stage model from introducing too many unit-status variables.

2.7.3 Safety-based tightening of physical upper bounds

In the original formulation, assigning the same loose upper bound to all generation, pumping, and support variables weakens the LP relaxation and consequently de-creases the computational efficiency of the MIP. This paper tightens variable upper bounds scenario by scenario and period by period according to the installed capacity of candidate sites, the number of units, scenario-specific demand, and hydrometeorological correction factors. First, the tightened upper bounds of the generating and pumping power are formulated in Equation 32:where denotes the region to which candidate station belongs, is the maximum installed capacity allowed by the engineering conditions of the candidate station, and represent the corrected maximum generation capability and maximum pumping capability under hydrometeorological state , respectively. Accordingly, the constraints on generating and pumping power are formulated in Equation 33:

Meanwhile, to ensure that the operating power does not exceed the first-stage planned capacity and that unselected sites do not participate in operation, the capacity-binding constraints are retained as given in Equation 34:

This constraint guarantees that the storage-state variable remains within the practically available storage-capacity range under each hydrometeorological condition. Without altering the valid optimal solutions of the original model, the above tightening process replaces the unified large upper bounds with safe physical upper bounds, re-duces invalid search space, and effectively improves the solution efficiency and con-vergence stability of the large-scale two-stage stochastic MILP model.

2.7.4 Dominated constraint reduction

In the original model, simultaneously imposing regional coverage constraints and a system-wide coverage constraint may lead to redundant constraints. Since this paper has decomposed the system regulation demand into different regions and imposed regional demand satisfaction constraints separately, as formulated in Equation 35:where represents upward regulation, downward regulation, reactive power, upward ramping, and downward ramping demands, respectively.

When the total system demand is obtained by summing the demands of all regions, the system-level demand satisfaction relationship can be derived from the regional coverage constraints, as formulated in Equation 36:

Therefore, the system-wide coverage constraint is dominated by the regional coverage constraints and can be removed as a redundant constraint. This procedure retains regional-level checks on regulation capability, reduces the constraint count, and enhances the preprocessing performance of the solver.

2.7.5 Hierarchical warm-start solution strategy

In large-scale two-stage stochastic mixed-integer programming models, the quality of the initial feasible solution has an important impact on model solution efficiency. On the basis of compact reformulation, a layered warm-start strategy is developed to enhance the stability of model solution, with its detailed workflow illustrated in Figure 3.

The proposed strategy includes two levels, namely, the reuse of external historical solutions and the correction of internal relaxed dispatch solutions. Specifically, if similarities exist in case data, candidate station sets, or scenario structures, external historical feasible solutions are first reused and adjusted for feasibility based on current capacity limits, hydrometeorological corrections, and regulation requirements. Mean-while, a relaxed dispatch submodel is developed by relaxing selected integer decisions or fixing selected planning variables to efficiently produce high-quality dispatch solutions, which are then mapped to initial solutions of the full MILP model. By coordinating the external and internal warm-start layers, the full model is provided with a high-er-quality initial point that lies closer to the feasible region at the beginning of the solution process, which accelerates convergence and improves solution stability for large-scale scenarios.

2.8 Solution workflow of the two-stage stochastic programming model

The solution workflow of the proposed two-stage stochastic programming model for pumped-storage siting and capacity sizing is shown in Figure 4. The proposed method links the data interface, model formulation, structural tightening, and warm-start strategy within a unified optimization framework, providing effective support for large-scale case analyses of pumped-storage siting and capacity sizing.

3 Case study

3.1 System and parameter settings

To verify the effectiveness of the proposed two-stage stochastic programming method for pumped-storage siting and capacity sizing considering hydrometeorological factors, this paper selects a regional power system as the case study object and con-ducts pumped-storage siting and capacity sizing planning for candidate sites. The region has a relatively high penetration of renewable energy sources such as photovoltaic and wind power, while also featuring concentrated electric vehicle loads and several candidate sites with potential for pumped-storage retrofitting. The proposed optimization model is implemented in MATLAB and solved using Gurobi, with the detailed parameter settings shown in Table 1.

In the case study, the five typical scenarios are obtained through a hierarchical scenario construction procedure. First, historical wind power, photovoltaic generation, conventional load, electric vehicle load, and hydrometeorological variables are normalized to eliminate differences in units and magnitudes. Then, the hydrometeorological state is identified according to precipitation, evaporation, temperature, and reservoir-related indicators, and the samples are classified into normal, rainstorm, and drought categories. Within each category, source-load features are further clustered. The representative scenario of each cluster is selected according to the cluster center or the closest historical sample, and its probability is determined by the proportion of samples belonging to the corresponding cluster.

3.2 Planning results under the base case

This section applies the complete proposed model to conduct siting and capacity sizing planning for candidate pumped-storage power stations under baseline hydro-meteorological conditions. In this case, the regional coverage rate requirement is set to 90%, while the SOC range is constrained to ±5%. The results are shown in Table 2.

The planning results show that the proposed model can comprehensively deter-mine the pumped-storage layout scheme based on regional regulation demand, engineering conditions of candidate stations, hydrometeorological correction parameters, and investment costs, rather than simply prioritizing candidate stations with the largest capacity upper limits or the lowest costs. Meanwhile, the model favors candidate stations with sound engineering conditions, strong regional support capability, and superior economic efficiency. For regions with higher regulation demand or greater renew-able energy fluctuations, the model prioritizes pumped-storage stations with larger available capacity and stronger energy storage capability, so as to enhance regional active power support, downward regulation and absorption, and ramping response capability. The capacity allocation results show that the planned capacities differ among the selected stations. This indicates that the model does not treat all candidate stations as homogeneous energy storage resources, but performs differentiated configuration according to site capacity limits, reservoir storage capability, head conditions, efficiency, regional affiliation, and operational constraints.

3.3 Planning results under multiple operating conditions

To verify the planning adaptability of the model under different operating conditions, this paper compares pumped-storage planning schemes by considering different hydrometeorological conditions and constraint intensities, and examines the practical operational adaptability of the planning results through changes in candidate site se-lection, installed capacity configuration, and regulation-demand satisfaction capability.

3.3.1 Different hydrometeorological conditions

This paper sets three types of hydrometeorological conditions: the normal condition, the off condition, and the stress condition. The Off case is designed to examine how neglecting hydrometeorological conditions affects the planning outcomes and computational difficulty. The stress condition is used to test the adaptability and robustness of the planning scheme under extreme hydrometeorological scenarios, and the planning results are shown in Table 3. Figure 5 shows the three types of costs that exhibit relatively significant differences among the planning results under different operating conditions.

In the figure, CAPEX denotes investment cost, GAP denotes shortage penalty cost, and O&M denotes operation and maintenance cost. The results show that, after hydrometeorological correction is turned off, the number of constructed stations decreases from 19 under the base case to 16, making the planning scheme appear simpler. However, the operational verification results do not improve accordingly. At this point, the upward active power shortage of the system increases significantly, the penalty cost nearly doubles, and the final total cost rises markedly instead. This indicates that ignoring hydrometeorological correction does not truly simplify the planning problem. Instead, it may reduce the adaptability of the site combination to actual operational requirements.

Under the stress condition with intensified hydrometeorological impacts, the model exhibits a clear defensive capacity expansion characteristic. Relative to the base-line case, the model maintains the capacity scale of major stations and further expands the capacity of several auxiliary stations. However, due to the obvious deterioration of basic conditions, both the upward active power shortage and regulation penalty cost increase significantly, resulting in a final total cost higher than that of the base case.

3.3.2 Different Constraint Intensities

To further evaluate the adaptability of the proposed model under different security requirements, this study imposes a 95% regional coverage requirement and a ±2% SOC range constraint, and analyzes their effects on the planning outcomes. The corresponding planning results are shown in Table 4 and Figure 6.

When the SOC fluctuation range is tightened, the number of configured units de-creases by two and the total installed capacity of the system declines compared with the base scheme, while the overall site-allocation principle remains unchanged. In this case, the model still prioritizes retaining core stations that provide major support for regional regulation demand, while appropriately reducing dependence on a single main station and strengthening the supplementary role of some distributed stations. This result indicates that, after the energy storage state constraint is tightened, the model does not simply increase installed capacity to compensate for insufficient regulation capability. Instead, it rebalances investment cost and the satisfaction level of regulation demand by adjusting the site combination and capacity distribution.

When the regulation coverage target is increased, the total installed capacity de-creases to some extent instead. With the configuration of the main stations remaining basically unchanged, the model reduces the configured capacity of two auxiliary stations. Although the regulation shortage increases slightly under this scheme, the total cost decreases. Under the current case data and constraint settings, the baseline scheme already provides a relatively high level of regulation coverage. Therefore, further in-creasing the coverage target does not significantly change the configuration of core stations, but prompts the model to reassess the necessity of marginal auxiliary capacity.

3.4 Comparison of solution performance of different models

To verify the effectiveness of the proposed compact reformulation and hierarchical warm-start strategy, this paper conducts an ablation comparison experiment using three models: the original two-stage stochastic programming model, the structurally tightened model incorporating compact reformulation and safe upper-bound tightening, and the complete methodological model combining compact reformulation with hierarchical warm start, and compares their differences in terms of objective function value, candidate site selection, capacity configuration, regulation-demand satisfaction, and solution time. The solution performance differences of the different models are shown in Table 5.

The results show that the objective function values and all cost indicators of the three schemes are completely identical, indicating that the acceleration strategies do not affect the quality of the optimal solution. In terms of solution efficiency, compared with conventional solution, the constraint-optimization scheme reduces the computational time by approximately 71.3%, while the complete efficient solution scheme re-duces it by approximately 96.7%. Meanwhile, compared with constraint optimization alone, the introduction of hierarchical warm start further reduces the computational time by approximately 88.5%. This result demonstrates that safe physical upper-bound tightening and redundant constraint reduction can effectively compress the model search space, while hierarchical warm start can significantly improve the quality of the initial solution and the efficiency of branch-and-bound search. Their combination pro-duces a more pronounced acceleration effect.

4 Discussion

4.1 Significance of the meteorology-source-load-regulation demand mapping mechanism

The meteorology-source-load-regulation demand mapping mechanism construct-ed in this paper incorporates wind power, photovoltaic generation, conventional load, electric vehicle load, and hydrometeorological states into unified multi-source scenario modeling, and further transforms typical source-load scenarios into regulation demand signals that can be directly used by the planning model. This mechanism does not re-main at the level of typical scenario extraction. Instead, it forms inputs such as upward active power demand, downward active power demand, reactive power support demand, ramping demand, and regional regulation demand through net-load deviation, reactive power shortage, ramping variation, and regional demand decomposition, thereby establishing an effective connection between source-load stochastic processes and planning decision variables. By avoiding the information loss associated with relying solely on net-load profiles, this mechanism allows pumped-storage stations to be modeled not only as energy-shifting resources, but also as providers of active-power support, reactive power support, ramping response, and regional security support. It also improves the interpretability of planning outcomes and enables representative scenario results to serve subsequent siting, sizing, and operation checks. This mechanism provides more targeted scenario inputs for the two-stage stochastic programming model developed in this paper and serves as the basis for improving the engineering interpretability and adaptability of the planning results.

4.2 Analysis of method adaptability

4.2.1 Mechanism of the impact of hydrometeorological intensity

Hydrometeorological factors exert a dual impact on pumped-storage power stations. Extreme weather changes the source-load characteristics of the system, thereby altering system regulation demand. On the other hand, hydrometeorological states change the available head, reservoir-based storage capability, water replenishment conditions, and evaporation losses of pumped-storage power stations themselves, thereby affecting their operational feasible region.

Under the condition in which hydrometeorological correction is disabled, the model reduces the number of constructed stations but does not achieve better overall performance. The reason is that disabling hydrometeorological correction attenuates the differences among candidate stations in hydraulic head, storage capacity, and replenishment conditions, causing the model to favor a few apparently cost-effective sites according to static capacity and investment cost. However, such site combinations cannot sufficiently match regional regulation demand in multi-scenario operational verification, especially showing large shortages in upward active power support, which leads to a significant increase in penalty costs. This indicates that hydrometeorological correction does more than increase model complexity, it also enables the model to recognize the true differences in the available capabilities of candidate stations.

Under extreme conditions, the defensive capacity expansion exhibited by the model reflects the adaptation mechanism of siting and capacity sizing decisions to extreme conditions. When drought conditions lead to reduced water replenishment, in-creased evaporation, and contracted available reservoir capacity, the energy space available for intertemporal regulation by pumped-storage power stations is compressed. Although the main stations still have large installed capacity, their deliverable upward support capability decreases. Therefore, the model instead compensates for regional regulation capability by increasing the capacity of auxiliary stations and adopting a more distributed configuration. This indicates that the proposed model can automatically adjust the capacity configuration structure under extreme hydrometeorological conditions, rather than simply expanding the capacity of main stations, and therefore has good adaptability to extreme operating conditions.

4.2.2 Mechanism of the impact of constraint strength

From the perspective of constraint mechanisms, tightening the SOC fluctuation range essentially limits the energy storage space that pumped-storage power stations can flexibly utilize within the dispatching cycle. When the SOC fluctuation range is tightened, the storage state must remain within a narrower interval, and the model can no longer fully utilize the initial or terminal storage margins to support regulation demand in the current period. Therefore, even if some major stations have large installed capacity, their additional capacity can hardly be fully converted into effective upward regulation capability.

The results after increasing the coverage target indicate that, in the current case, the regulation coverage constraint is not the main binding constraint. Since the baseline scheme already achieves a relatively high coverage level, raising the coverage target further does not compel the model to substantially expand capacity. Instead, when a small regulation shortage is allowed and penalty costs are considered, the model identifies some auxiliary stations as marginal redundant capacity. These stations can mitigate local shortages, but their contribution to improving the overall system regulation capability is limited, while their investment costs are relatively high. Therefore, the optimization result reduces two auxiliary stations and obtains a lower total cost at the expense of a slight increase in regulation shortage. The above results show that the pro-posed model can not only provide a single capacity configuration result, but also identify the functional positioning of different candidate sites through changes in constraint strength, thereby offering stronger engineering significance.

4.3 Contributions of compact reformulation and hierarchical warm start

As the numbers of candidate stations, scenarios, and dispatching periods increase, the scale of the two-stage stochastic programming model constructed in this paper expands rapidly. The first stage includes integer planning variables such as site selection, installed capacity, energy storage capacity, and the number of units. The second stage involves operational variables and constraints for generation, pumping, SOC variation, active power support, reactive power support, and regional coverage under multiple scenarios and time periods. If the original MILP model is solved directly, problems such as a large number of integer variables, weak LP relaxation, and excessive solution time are likely to arise. Therefore, this paper constructs an efficient solution framework for large-scale pumped-storage siting and capacity sizing models, whose advantages are mainly reflected in model compactness, constraint tightening, and hierarchical warm start.

First, the model replaces the traditional two binary variables for generation and pumping with a single-mode variable, thereby reducing a large number of second-stage integer variables and mutual-exclusion constraints. Second, safe physical upper-bound tightening makes the value ranges of generation, pumping, SOC, and regulation sup-port variables closer to the actual feasible region, thereby improving the quality of LP relaxation. Meanwhile, deleting the system coverage constraint dominated by regional coverage constraints further reduces the constraint scale. On this basis, the hierarchical warm-start strategy uses external historical results and an internal relaxed submodel to generate high-quality initial solutions, thereby improving the initial feasible solution quality and branch-and-bound search efficiency of the complete MILP model. This method is particularly suitable for optimization under extreme operating conditions, as it can overcome the solution difficulties caused by stronger hydrological constraints and a tighter feasible region, while significantly shortening solution time.

4.4 Method limitations and future work

The two-stage stochastic programming-based planning and layout model for pumped-storage power stations and its efficient solution algorithm developed in this paper still have room for further improvement.

This paper mainly focuses on regional regulation demand and pumped-storage operational verification, while network constraints such as power flow, voltage and short-circuit capacity have not yet been systematically considered. Future research may further couple DC power flow or linearized AC power flow models to make pumped-storage planning results more consistent with the requirements of secure grid operation. Second, this paper adopts a two-stage stochastic programming method based on typical scenarios, which mainly reflects planning performance in an expected-value sense. For low-probability, high-impact extreme weather events, future research may introduce risk-measurement methods such as robust optimization and distributionally robust optimization to enhance the risk resistance of planning schemes under extreme operating conditions. Finally, further enrichment is needed in terms of engineering data support and algorithmic validation. Future research may incorporate real terrain, elevation difference, grid-connection distance, ecological constraints, historical precipitation-runoff data, and renewable energy output data to improve the engineering applicability and generalization value of the method.

5 Conclusion

This paper proposes an efficient two-stage stochastic programming model for the siting and capacity sizing of pumped-storage power stations under high renewable energy penetration. The model integrates source-load uncertainty, hydrometeorological impacts, candidate-site heterogeneity, and multi-scenario operational verification into a unified siting and capacity-sizing framework.

The main findings can be summarized as follows. First, the proposed meteorology-source-load-regulation-demand mapping mechanism converts multi-source uncertainties into model-ready regulation requirements, improving the interpretability of pumped-storage planning. Second, the candidate-site parameter database and hydrometeorological correction mechanism enable the model to capture differences in regulation capability among stations and generate differentiated siting and capacity-sizing schemes. Third, the proposed compact reformulation and hierarchical warm-start strategy significantly improve solution efficiency without affecting optimality, reducing the solution time from 507.6 s to 16.68 s while maintaining a 99.71% upward active power coverage rate. These results demonstrate the effectiveness of the proposed model and solution approach.

Case study results verify the effectiveness of the proposed model and algorithm in pumped-storage resource planning and layout, and can provide decision support for the scientific allocation and operational adaptability assessment of pumped-storage power stations in high-renewable-penetration systems. Future work will couple the proposed planning model with DC power flow and further extend it to include voltage and reactive-power security requirements. On this basis, refined hydrological processes, including runoff evolution, reservoir inflow uncertainty, and water-level constraints, will be incorporated to improve the physical representation of pumped-storage operation. In addition, risk-measurement methods will be introduced to enhance the resistance of planning schemes to low-probability high-impact extreme weather events. These extensions will make the proposed method more applicable to practical regional grid planning and engineering decision-making.

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