Integrating Drought Warning Water Level With Analytical Hedging for Reservoir Water Supply Operation

Abstract

The increasing frequency of droughts worldwide, along with their substantial societal impacts, underscore the need for improved drought early warning and enhanced water supply security. While reservoir drought management strategies, including hedging theory and drought warning/limited water level (DWWL), have been extensively studied for their potential to alleviate water supply stress, effective control schemes remain lacking. This study proposes a framework to enhance reservoir drought resistance. The framework starts with the analytical derivation of the optimal hedging rule (HR) for reservoirs supplying multiple competing users, applicable to both real‐time and long‐term optimizations. DWWLs and their seasonal segmentation are then determined based on the optimal water supply process and multivariate time‐series clustering. By utilizing reservoir water level and meteorological drought indicator to monitor drought conditions and characterize reservoir effective storage, a real‐time operational strategy is ultimately proposed that integrates analytical HR with effective storage. We demonstrate the application in the Danjiangkou Reservoir in China, and the proposed operation is testified under both long‐term periods and extreme dry conditions. Results indicate that the analytical HR outperforms conventional rule curves in nearly every aspect, including water supply benefits, reliability, and refill rate. The DWWL proves crucial during extreme drought conditions; when water levels fall below DWWL, the HR considering DWWL restricts water supply earlier, thereby preventing the reservoir from dropping dramatically to critically low levels. This proposed framework is transferable to other water supply reservoirs, offering scientific support, operational relevance, and guidance for drought mitigation and response.

Introduction

Drought is a normal part of climate, occurring regardless of geographic and climatic regions (Wilhite, 2000). Also, as a widespread natural hazard caused by water scarcity, it develops insidiously, often lasting a long time, and could affect a wide range of areas (Maskey & Trambauer, 2023). From 2000 to 2019, drought impacted over 1.4 billion people worldwide and caused economic losses of roughly $124 billion in total (Gao et al., 2019; UNCCD, 2022). The frequency and severity of droughts seem to be increasing globally as a result of climate change and intensive human activities (IPCC, 2023; Wan et al., 2017). Considering the high societal impact of drought, an early warning and hedging for reservoir water supply have proven to be one of the more effective non-structural measurements for drought mitigation (Haro-Monteagudo et al., 2017; Huang & Chou, 2008; Huang & Yuan, 2004). There has been increasing research focusing on the operation of reservoirs to combat drought (Wu et al., 2018; Yang et al., 2019; Zolfagharpour et al., 2021).

The fundamental challenge in managing water supply reservoirs is how to efficiently counterbalance the demands among water use sectors and between water usage and drought defense objectives. Among the various strategies for coping with drought, hedging rules (HR) are particularly favored (Macian Sorribes & Pulido Velazquez, 2020; Wang, Liao, et al., 2024). HRs are effective in mitigating drought impacts because they continuously account for potential future drought, reduce early water consumption, and save water for future use, thereby preventing severe water shortages during drought season and ensuring the necessary water demand during prolonged drought periods (Bayazit & Ünal, 1990; Shih & ReVelle, 1994). Early studies on HR relied on numerical optimization methods to develop hedging policies, representing the relationship between water availability (reservoir storage plus inflow) and reservoir release through curves (Beshavard et al., 2022; Shih & ReVelle, 1994; Tu et al., 2003). Through hydro-economic analysis, Draper and Lund (2004) provided a general economic interpretation of HR for reservoir water supply, followed by Shiau (2011), You and Cai (2008), and Zhao et al. (2011). By shifting water shortages from concentrated, significant deficits to more distributed minor ones, HR can enhance overall economic benefits. The analytical optimal hedging for reservoir water supply has gained increasing attention in recent years, driven by the advancement in hydro-meteorological forecasting (e.g., Ding et al., 2017; Meng et al., 2022; Wan et al., 2016; Wan et al., 2020). However, potential users seem to show litter interest in implementing such methods. Part of this resistance is that these analytical HRs are only designed for real-time operation with forecasting, making them distinct from the conventional rules (and rule curves), which are determined involving continuous water supply regulation or long-term parameterization-simulation-optimization framework. The substantial diversity in these problem-solving approaches poses great challenges for users and decision-makers, including those specializing in reservoir operation, in understanding new ways of analyzing systems and further adopting the analytical-HR-based operation.

In China, drought-induced economic losses account for one-third of the total natural disaster losses, with significant droughts occurring on average every two to 3 years (Han et al., 2021; Yao et al., 2018). To activate drought response and mitigate the impacts of severe droughts, water agencies in China have been promoting various emergency management strategies. One of the most influencing is the introduction of the term “drought-limited water level” by the Ministry of Water Resources and Ministry of Emergency Management of the People's Republic of China (State Flood Control and Drought Relief Headquarters, 2012). This term is later renamed “drought warning water level” (CHES, 2024). In the following, we consider only drought warning water level (DWWL) and refer to both the drought-limited water level and DWWL simply as DWWL.

The DWWL is the indicative reservoir water level(s) that signals the need for drought response measures. However, the absence of a clear definition for DWWL has led to considerable controversy regarding its identification, determination, and implementation (Cao et al., 2021; Chang et al., 2019; Liu et al., 2012; Luo et al., 2023). For example, Peng et al. (2016) defined the DWWL as the end-of-year storage for multi-year regulation reservoirs and proposed an optimization-based DWWL that could minimize the depth of water deficits over consecutive dry years. Chang et al. (2019) defined DWWL as the historical lowest operational water level of a reservoir under specific drought classes and proposed a seasonal determination scheme based on historical hydro-meteorological analysis. Zhang, Kang, et al. (2022) defined DWWL as the critical level required to meet minimum water needs and optimized DWWL that could maximize economic benefits from water supply while ensuring reliability across various sectors.

On the other hand, the DWWL proposed by the Ministry of Water Resources of China is intended for emergency drought response; however, how to support real-time decisions based on this level remains unclear. Some studies interpret it as the minimum water level, marking the point at which the water supply ceases (Chang et al., 2019; Lin et al., 2023), while others take it as a threshold for water rationing (Luo et al., 2023; Zhang, Kang, et al., 2022). However, these interpretations may significantly reduce the effective reservoir capacity or make the DWWL indistinguishable from conventional rule curves, which already account for water rationing based on reservoir storage (Huang & Yuan, 2004; Liu et al., 2023; Wan et al., 2019). Moreover, relying solely on the water level as the drought monitor, without considering the current and future potential hydrological conditions, may undermine the potential effects of DWWL in drought management. Recent studies have attempted to integrate hydrological drought indicators with reservoir drought management (Beshavard et al., 2022; Zolfagharpour et al., 2021), but only the simplest linear HR were adopted and have not incorporated DWWL. We argue that DWWL should essentially be an additional constraint for reservoir regulations during drought periods.

Upon these concerns, this article aims to bridge the gap between classical operation theory and practical drought management by providing a new water supply model that integrates the HR with the original intent of DWWL. We expect this model to be applicable in both the planning stage and real-time decision-making, supporting both regular operations and emergency responses. The rest of the paper is structured as follows. Section 2 describes the framework for the proposed reservoir drought-resistance operation, including the HR-based long-term optimization, drought seasonal segmentation, DWWL determination, and DWWL-incorporated operational strategy. Section 3 provides an overview of the case study area, the Danjiangkou Reservoir in central China. Section 4 demonstrates the results of HR-based optimization algorithms, seasonal DWWL, and the effects of incorporating DWWL with HR. In Section 5, we analyze the operational sensitivity to hydrology uncertainty, followed by a discussion on the necessity and potential applications of DWWL. Conclusions are drawn in Section 6.

Methodology

The proposed reservoir drought-resistance operation framework integrates long-term optimization with real-time decision-making through a three-step approach. First, an optimal water supply operation model is developed based on hedging theory, with reservoir storage discretized using dynamic programming (DP) to derive the optimal solution for deterministic long-term operation (see Section 2.1 for details). Second, drought states are defined by analyzing the long-term optimization features, and the corresponding seasonal DWWLs are determined using the typical year hydrological conditions (Section 2.2). Third, real-time water supply decisions are informed by the reservoir's current water level relative to the DWWL and meteorological drought indicator, with a decision rule established based on analytical hedging and effective storage, creating a unified operation model that combines DWWLs with HR (Section 2.3). The key innovation of this framework lies in its consistent application of HR across long-term planning and short-term real-time decision-making, as illustrated schematically in Figure 1.

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Optimal Reservoir Operation for Real-Time and Long-Term Water Supply

To support DWWL (DWWL) determination and enhance the efficiency of both short-term and long-term reservoir operations, we proposed a multi-user water supply model based on the theory of HR. For general water supply reservoir, Draper and Lund (2004) derived the property of optimal hedging for releases from a reservoir, stating that at optimality the marginal benefit of storage must equal the marginal benefit of release: 1 $$\frac{\mathit{\partial }{B}_{R}(R)}{\mathit{\partial }R}=\frac{\mathit{\partial }{B}_{S}(S)}{\mathit{\partial }S}$$ where $$R$$ and $$S$$ are the current decision variables of water release and reservoir carryover storage; $${B}_{R}(\mathit{\cdot })$$ and $${B}_{S}(\mathit{\cdot })$$ are the economic value of water use and storage.

HR for Real-Time Optimization

For reservoirs with multiple competing users such as domestic, agricultural, industrial, and ecological uses, or water users from different districts, denoted by $$n=1,2,\mathit{\dots }N$$, we assume that the loss function due to water deficit in a single-period, non-overlapping water supply is represented by the weighted normalized deviations from the projected water demand (or desired release target), following Shiau (2011) and Meng et al. (2022), with: 2 $${C}_{{R}_{n}}\left({d}_{n,t}\right)={\omega }_{n}{\left(\frac{{D}_{n,t}-{d}_{n,t}}{{D}_{n,t}}\right)}^{m}$$

Similarly, the loss function of reservoir carryover storage target for water supply at a future time, excluding the inactive storage that is restricted from use, is defined as: 3 $${C}_{S}\left({s}_{t}\right)={\omega }_{S}{\left(\frac{{S}_{t}^{\text{Tar}\,}-{s}_{t}}{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}\right)}^{m}$$ where $${D}_{n,t}$$ is the water demand for user $$n$$ in the period $$t$$; $${\omega }_{n}$$ and $${\omega }_{S}$$ are the weighting factors assigned to the loss function of supply and storage, respectively, and satisfy the condition $${\omega }_{S}+\sum \nolimits_{n=1}^{N}{\omega }_{n}=1$$; The exponent parameter $$m > 1$$ defines the convex shape of the utility function $${C}_{{R}_{n}}(\mathit{\cdot })$$ and $${C}_{S}(\mathit{\cdot })$$, as indicated by a positive second derivative. This implies increasing marginal losses associated with water deficits and diminishing marginal returns from water supply or storage decisions; $${S}_{t}^{\text{Tar}}$$ is the desired carryover storage target at the end of period $$t$$, which could be the storage capacity $${S}_{t}^{\max \,}$$ or a desired operational level for flood control or water conservation (Shiau, 2011), with $${S}_{t}^{\text{Tar}\,}\mathit{\le }{S}_{t}^{\max \,}$$; $${S}^{\min }$$ is the minimum storage boundary, usually refers to the dead storage; $${d}_{n,t}$$ and $${s}_{t}$$ are the decision variables for user $$n$$ and reservoir storage.

The objective of single-period optimization multi-user water supply systems is thus to minimize the sum of losses of water supply and reservoir storage, which is formulated as: 4a $$\min \,z=\sum\limits _{n=1}^{N}{C}_{{R}_{n}}\left({d}_{n,t}\right)+{C}_{S}\left({s}_{t}\right)$$ s.t. 4b $${s}_{t-1}+{I}_{t}-{E}_{t}=\sum \nolimits_{n=1}^{N}{d}_{n,t}+{s}_{t}+{l}_{t}$$ 4c $$0\le {d}_{n,t}\le {D}_{n,t}$$ 4d $${S}^{\min }\le {s}_{t}\le {S}_{t}^{\mathrm{m}\text{ax}\,}$$ 4e $${l}_{t}\mathit{\ge }0$$ where $${s}_{t-1}$$ is the known storage at the beginning of period $$t$$; $${I}_{t}$$ is the average inflow; $${E}_{t}$$ is the evaporation, which is a function of reservoir surface area; $${l}_{t}$$ is residual flow after water supply.

According to the Lagrange multiplier approach, the optimality condition for reservoir real-time operation is: 5 $$\frac{\mathit{\partial }{C}_{{R}_{1}}}{\mathit{\partial }{d}_{1}}=\mathit{\cdots }=\frac{\mathit{\partial }{C}_{{R}_{n}}}{\mathit{\partial }{d}_{n}}=\mathit{\cdots }=\frac{\mathit{\partial }{C}_{{R}_{N}}}{\mathit{\partial }{d}_{N}}=\frac{\mathit{\partial }{C}_{S}}{\mathit{\partial }\mathrm{s}}$$

Thus, taking the first derivative of the loss function as defined in Equations 2−3, we have: 6 $\begin{align*}\frac{{\omega }_{n}}{{D}_{n,t}}{\left(\frac{{D}_{n,t}-{d}_{n,t}^{\ast }}{{D}_{n,t}}\right)}^{m-1}=\frac{{\omega }_{S}}{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}{\left(\frac{{S}_{t}^{\text{Tar}\,}-{s}_{t}^{\ast }}{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}\right)}^{m-1}\\ n=1,2,\mathit{\dots }N\end{align*}$

Together with the water balance constraint in Equation 4b, a straightforward calculation yields the following expression of storage decision $${s}_{t}^{\ast }$$, with 7a $${s}_{t}^{\ast }={S}_{t}^{\text{Tar}\,}-{a}_{s,t}(\text{WT}-\text{WA})$$ where 7b $${a}_{s,t}=\frac{\left({S}_{t}^{\text{Tar}\,}-{S}^{\min }\right){\left(\frac{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}{{\omega }_{s}}\right)}^{\tfrac{1}{m-1}}}{\left({S}_{t}^{\text{Tar}\,}-{S}^{\min }\right){\left(\frac{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}{{\omega }_{s}}\right)}^{\tfrac{1}{m-1}}+\sum \nolimits_{n=1}^{N}{D}_{n,t}{\left(\frac{{D}_{n,t}}{{\omega }_{n}}\right)}^{\tfrac{1}{m-1}}}$$ 7c $$WT=\sum \nolimits_{n=1}^{N}{D}_{n,t}+{S}_{t}^{\text{Tar}\,}-{S}^{\min }$$ 7d $$WA={s}_{t-1}+{I}_{t}-{E}_{t}-{S}^{\min }$$ where $$\text{WT}$$ (Water Target) represents the system water required including storage target and supply target; $$\text{WA}$$ (Water Availability) is the total available water volume in the reservoir excluding the minimum storage; $${a}_{s,t}$$ is the shortage allocation index for reservoir storage, reflecting the reservoir's regulation performance.

Similarly, the optimal expression for the optimal release $${d}_{n,t}^{\ast }$$ is: 8a $${d}_{n,t}^{\ast }={D}_{n,t}-{a}_{n,t}(\text{WT}-\text{WA})$$ where 8b $${a}_{n,t}=\frac{{D}_{n,t}{\left(\frac{{D}_{n,t}}{{\omega }_{n}}\right)}^{\tfrac{1}{m-1}}}{\left({S}_{t}^{\text{Tar}\,}-{S}^{\min }\right){\left(\frac{{S}_{t}^{\text{Tar}\,}-{S}^{\min }}{{\omega }_{s}}\right)}^{\tfrac{1}{m-1}}+\sum \nolimits_{{n}^{\mathit{\prime }}=1}^{N}{D}_{{n}^{\mathit{\prime }},t}{\left(\frac{{D}_{{n}^{\mathit{\prime }},t}}{{\omega }_{{n}^{\mathit{\prime }}}}\right)}^{\tfrac{1}{m-1}}}$$ where $${a}_{n,t}$$ is the shortage allocation index for water user $$n$$: the larger its value, the higher the priority for the reservoir to supply water to user $$n$$. The notation $${n}^{\mathit{\prime }}$$ is introduced to differentiate between the individual user $$n$$ and the complete set of all water users [1, $$N$$], which includes the user $$n$$.

Note that according to Equations 2, 3, and 5, the analytical solution of $${s}_{t}^{\ast }$$ and $${d}_{n,t}^{\ast }$$ would either simultaneously satisfy all constraints, or simultaneously violate either the respective lower bound or the demand target. Let the reservoir constraints govern the solution space, the reservoir storage decision for the current period is: 9 $${s}_{t}^{\ast \ast }=\left\{\begin{array}{@{}lr@{}}{S}^{\min }\hfill & \hfill if\,{s}_{t}^{\ast }< {S}^{\min }\\ {s}_{t}^{\ast }\hfill & \hfill if\,{S}^{\min }\mathit{\le }{s}_{t}^{\ast }\mathit{\le }{S}_{t}^{\mathrm{m}\text{ax}\,}\\ {S}_{t}^{\mathrm{m}\text{ax}\,}& \hfill if\,{s}_{t}^{\ast } > {S}_{t}^{\mathrm{m}\text{ax}\,}\end{array}\right.$$

The corresponding water supply decision $${d}_{n,t}^{\ast \ast }$$ and residual flow $${l}_{t}^{\ast \ast }$$ follow from this adjusted storage become: 10 $${d}_{n,t}^{\ast \ast }=\left\{\begin{array}{@{}lr@{}}0\hfill & \hfill if\,{s}_{t}^{\ast }< {S}^{\min }\\ {d}_{n,t}^{\ast }\hfill & \hfill if\,{S}^{\min }\mathit{\le }{s}_{t}^{\ast }\mathit{\le }{S}_{t}^{\text{Tar}\,}\\ {D}_{n,t}\hfill & \hfill if\,{s}_{t}^{\ast } > {S}_{t}^{\text{Tar}\,}\end{array}\right.$$ 11 $${l}_{t}^{\ast \ast }=\left\{\begin{array}{@{}lr@{}}0\hfill & \hfill if\,{s}_{t}^{\ast }\mathit{\le }{S}_{t}^{\text{Tar}\,}\\ WA-{s}_{t}^{\ast \ast }-{d}_{n,t}^{\ast \ast }\hfill & \hfill if\,{s}_{t}^{\ast } > {S}_{t}^{\text{Tar}\,}\end{array}\right.$$

The standard HR is developed under the assumption that inflows $${I}_{t}$$ are either known or perfectly forecasted. In real-world cases, forecast uncertainty is inevitable and must be taken into account. For simplicity, the actual streamflow $${I}_{t}^{\text{act}}$$ is represented as the sum of the forecasted inflow $${I}_{t}^{\text{pred}}$$ and the forecast error $$\varepsilon $$, as 12 $${I}_{t}^{\text{act}}={I}_{t}^{\text{pred}}+\varepsilon $$

With the introduction of forecast error, the water availability in Equations 7a and 8a is calculated based on the forecasted inflow $${I}_{t}^{\text{pred}}$$. Since the forecast error $$\varepsilon $$ is only known with certainty at the end of the period, after supply decisions have been made, we reasonably assume that $$\varepsilon $$ exclusively influences carryover storage. Consequently, the actual storage at the end of the period, $${s}_{t}^{\mathrm{a}\mathrm{c}\mathrm{t}}$$, is adjusted as follows: 13 $${s}_{t}^{\text{act}}={s}_{t-1}+{I}_{t}^{\text{act}}-{E}_{t}-\sum \nolimits_{n=1}^{N}{d}_{n,t}^{\ast \ast }$$

Again, binding storage constraints will alter the water supply decisions outlined in Equations 10−11, as investigated by Zhao et al. (2011) and Zeng et al. (2021). Specifically, if the ending storage exceeds $${S}_{t}^{\mathrm{m}\text{ax}}$$, the surplus water is released downstream as spillage $${l}_{t}$$; if storage falls below $${S}^{\min }$$ during operation, water supplies are simultaneously suspended.

Constrained Dynamic Programming With Analytical HR for Long-Term Deterministic Optimization

For long-term optimization, historical or predefined hydrologic time series are used as the operation horizon. The mathematical expression of the carryover storage utility function is assumed to be identical to the widely used loss function in Equation 3, representing the minimum accumulated losses from the current period to the final period. The mathematical formulation for long-term multi-user water supply optimization is: 14a $$\min \,f=\sum\limits _{y=1}^{Y}\sum\limits _{t=1}^{T}\sum\limits _{n=1}^{N}{C}_{{R}_{n}}\left({d}_{n,y,t}\right)$$ s.t. 14b $${s}_{y,t-1}+{I}_{y,t}-{E}_{y,t}-\sum \nolimits_{n=1}^{N}{d}_{n,\mathrm{y},t}-{l}_{y,t}={s}_{y,t}$$ 14c $$0\le {d}_{n,y,t}\le {D}_{n,y,t}$$ 14d $${S}^{\min }\le {s}_{y,t}\le {S}_{t}^{\mathrm{m}\text{ax}\,}$$ where symbols $$s$$, $$I$$, $$E$$, $$D$$, $$d$$, $$l$$ hold the same meaning as in Equations 2−4, but substituting the subscript $$t$$ by $$(y,t)$$, which indicates the year and the periodic time step within the year; $$Y$$ and $$T$$ are the number of years and time steps in a year in the given operation horizon.

Theoretically, the optimization algorithm proposed in Section 2.1.1 could be directly applied to long-term optimization with zero forecast error. However, this approach faces challenges in determining appropriate carry-over storage targets $${S}_{t}^{\text{Tar}}$$. A typical simplification sets $${S}_{t}^{\text{Tar}\,}={S}_{t}^{\mathrm{m}\text{ax}}$$, implying that maintaining maximum storage is economically optimal and any carryover storage $${s}_{t}< {S}_{t}^{\mathrm{m}\text{ax}\,}$$ incurs a loss. While this may hold during refill seasons (when targeting $${S}_{t}^{\mathrm{m}\text{ax}\,}$$ is reasonable), it becomes inappropriate afterward. There is no economic loss if the reservoir operates below capacity, as long as future demands can be met over the remaining horizon. To address this limitation, we employ DP for long-term deterministic optimization. The DP framework enables a time-recursive search of storage trajectories (Dobson et al., 2019; Yakowitz, 1982) and thus fundamentally restructuring the problem formulation: rather than treating carryover storages as decision variables (as in real-time operation), we now consider them as predetermined state variables, with water allocations to multiple users serving as the sole decision variables. This reformulation allows efficient solution through analytical HR while eliminating the necessity to consider $${S}_{t}^{\text{Tar}}$$, leading to our proposed DP-HR that combines the algorithmic strengths of constrained DP with the computational efficiency of analytical HR for long-term multi-user water supply optimization.

DP discretizes reservoir storage into $$k$$ intervals. For each period, all possible combinations of discretized storage states at the beginning and end of the period are tested. Specifically, for certain discretized storage states, $${s}_{y,t-1}(\mathrm{i})$$ and $${s}_{y,t}(j)$$, $$i,j=1,2,\mathit{\dots }k$$, the total supply availability $${SA}_{i,j}$$ is obtained. The analytical solution similar to Equations 7−8 can be applied to optimize water supply allocations: 15 $${d}_{n,y,t}^{\ast }={D}_{n,y,t}-\frac{{D}_{n,y,t}\left(\sum \nolimits_{{n}^{\mathit{\prime }}=1}^{N}{D}_{{n}^{\mathit{\prime }},y,t}-S{A}_{i,j}\right)}{\sum \nolimits_{{n}^{\mathit{\prime }}=1}^{N}{D}_{{n}^{\mathit{\prime }},y,t}{\left(\frac{{\omega }_{n}{D}_{{n}^{\mathit{\prime }},y,t}}{{\omega }_{{n}^{\mathit{\prime }}}{D}_{n,y,t}}\right)}^{\tfrac{1}{m-1}}}$$ where $${SA}_{i,j}={s}_{y,t-1}(i)+{I}_{y,t}-{E}_{y,t}-{s}_{y,t}(j)$$

Considering constraints of water supply (Equation 14c), the water supply decision becomes: 16 $${d}_{n,y,t}^{\ast \ast }=\left\{\begin{array}{@{}lr@{}}0\hfill & \hfill if\,{d}_{n,y,t}^{\ast }< 0\,\\ {d}_{n,y,t}^{\ast }\hfill & \hfill if\,0\mathit{\le }{d}_{n,y,t}^{\ast }\mathit{\le }{D}_{n,y,t}\\ {D}_{n,y,t}\hfill & \hfill if\,{d}_{n,y,t}^{\ast } > {D}_{n,y,t}\end{array}\right.$$

The optimal benefit losses starting from storage states $${s}_{y,t-1}(i)$$ is thus the sum of loss from the period $$(y,t)$$ to the end of operation horizon $$Y\times T$$, which can be formulated as a recursive function: 17 $${L}_{i,j,y,t}=\sum\limits _{n=1}^{N}{\omega }_{n}{\left(\frac{{D}_{n,y,t}-{d}_{n,y,t}^{\ast \ast }}{{D}_{n,y,t}}\right)}^{m}+{L}_{i,j,y,t+1}$$

A further iterative and recursive computation of DP should be performed, starting from the last operational period and moving backward to the first, to find the optimal reservoir storage trajectory $${s}_{y,t}^{\ast }$$. Once the storage trajectory for each period is established, the optimal long-term water supply solutions are determined again using Equations 15 and 16.

Segmentation of Drought Seasons and Determination of DWWL

Research has widely acknowledged the limitations of a single, unified DWWL as initially proposed by the Ministry of Water Resources (State Flood Control and Drought Relief Headquarters, 2012). Drawing inspiration from the specification of seasonal flood control water level (Liu et al., 2015), studies have developed various methods for determining DWWL based on hydrological states. While high water levels in flood season are typically driven by flood characteristics, low water levels in reservoirs can be attributed to a combination of factors, including low inflow, fluctuating water demand, continuous water supply, and shifts in reservoir management priorities. This may lead to an out-of-sync with the timing of low inflow and low water levels. Therefore, for large reservoirs that serve multiple purposes of flood control, water supply, and drought resistance, which usually correspond to the variation of storage constraints along with the dry-wet cycle, we propose a drought season segmentation method using multi-dimensional time series clustering (Sections 2.2.1 and 2.2.2) and a method for determining DWWL (Section 2.2.3).

Metrics for Drought Season Segmentation

Streamflow and reservoir water levels are generally regarded as closely related to drought management strategies (Chang et al., 2019; Li et al., 2023; Lin et al., 2023). Management of DWWL may become critical during dry spells with very low reservoir inflow or during peak irrigation periods characterized by high water demand. Beyond the immediate concerns of water supply, fluctuations in reservoir water levels may also result from pre-flood drawdowns or post-flood refilling activities; such operational aspects are often overlooked. We suggest that the segmentation of drought season should prioritize (a) instances of insufficient water supply, as captured by the periodic water deficit metric (Equation 18), rather than solely on variations in reservoir inflow; and (b) the depletion of reservoir storage, as indicated by the periodic storage balance metric (Equation 19), rather than the current storage or water level. The formulations are as follows: 18 $${x}_{y,t}^{1}=\max \left(0,\sum \nolimits_{n=1}^{N}{D}_{n,y,t}-{I}_{y,t}\right)$$ 19 $${x}_{y,t}^{2}={S}_{t}^{\max }-{s}_{y,t}^{\ast }$$

Segmentation of Drought Season Using Multi-Dimensional Time Series Clustering

Drought season clustering relying on the mean monthly feature of drought metrics can lead to subjective biases along with the selection of statistical features. To enhance the objectivity and robustness of drought season segmentation, we propose a method that accounts for the interannual variability of drought metrics by using a wide range of possible hydrological conditions.

To eliminate the influence of different scales, the extracted drought metrics, as defined in Equations 18 and 19, are standardized into dimensionless values using the following equation: 20 $${{x}^{\mathit{\prime }}}_{y,t}^{{j}_{0}}=\frac{{x}_{y,t}^{{j}_{0}}-\mu \left({x}_{y,t}^{{j}_{0}}\right)}{\sigma \left({x}_{y,t}^{{j}_{0}}\right)}$$ where $${x}_{y,t}^{{j}_{0}}$$ is the original metric, with $${j}_{0}$$ denoting a specific drought metric; $$\mu $$ and $$\sigma $$ are the mean and standard deviation of the sequence, respectively.

The standardized drought metrics, originally organized as a matrix of dimensions $$(Y\times T)$$ rows with $$J$$ columns (representing years, time periods, and metrics, respectively), are restructured into a multivariate sequence. This transformation results in a matrix $$R=\left({r}_{t}^{y,j}\right)$$ of dimensions $$T$$ rows with $$(Y\times J)$$ columns, where each row corresponds to a time period within a single year, and each column represents the metrics across multiple years: 21 $${r}_{t}^{y,j}=\text{Trans}\left({{x}^{\mathit{\prime }}}_{y,t}^{j}\right)$$

This transformation preserves the original values but reorganizes the multi-year time series into a single-year data set with multiple metrics per period.

Using this restructured matrix $$R$$ as the clustering object, a multi-dimensional clustering method, such as K-means clustering (MacQueen, 1967), spectral cluster (Dunn, 1973), or hierarchical clustering (Kaufman & Rousseeuw, 2009), is applied to segment the drought season. Given the ordered nature of time series data, the initial clusters containing adjacent periods are identified and removed from the data set. The remaining data undergoes iterative clustering until no clusters with non-adjacent periods remain. This iterative process ensures that all non-adjacent periods are separated into distinct clusters, thus achieving the final grouping of drought seasons.

Determination of DWWL

Like the determination of flood-limited water levels (e.g., Liu et al., 2015), we define the DWWL as the initial reservoir water level or storage volume that is just sufficient to meet water demand during pessimistic hydrological conditions. Analogous to the selection of a typical flood event (Parkes & Demeritt, 2016), typical dry years are identified by analyzing historical inflow frequencies. Common thresholds include the 25th, 50th, and 75th percentiles exceedance, corresponding to wet, average, and dry years, respectively (Şen et al., 2020). We select years with an annual mean inflow close to the 75th percentile exceedance probability as our typical dry years. Following this selection, the periodic water supply regulation in reverse order, starting from the minimum storage, is initiated: 22 $${S}_{T+1,\text{ini}}={S}^{\min }$$ 23 $${S}_{t,\text{ini}}=\max \left({S}^{\min },\sum \nolimits_{n=1}^{N}{D}_{n,t}^{\text{dry}}-{I}_{t}^{\text{dry}}+{E}_{t}^{\text{dry}}+{S}_{t+1,\text{ini}}\right)$$ where $${S}_{t,\text{ini}}$$ is reservoir storage at the beginning of each period within the representative drought year, $$t$$ = 1,2,…, $$T$$; $${D}_{n,t}^{\text{dry}}$$ is the water demand in the typical dry year; $${I}_{t}^{\text{dry}}$$ and $${E}_{t}^{\text{dry}}$$ are the inflow and evaporation losses from the reservoir.

Finally, for manageability, the obtained multi-period storage volumes are grouped seasonally according to the drought season segmentation results. We denote by $${\mathrm{D}\mathrm{W}\mathrm{W}\mathrm{L}}_{t\mathit{\in }seg}$$ the seasonal DWWL, which is the minimum water level or storage within the same drought season: 24 $${\mathrm{D}\mathrm{W}\mathrm{W}\mathrm{L}}_{t\mathit{\in }seg}=\min \left(f\left({S}_{t,ini}\right),t\mathit{\in }seg\right)$$ where $$f(\mathit{\cdot })$$ is the function describing the relationship between reservoir water storage and level.

Utilization of DWWL Incorporating Drought Trends and HR

Drought is a long-lasting disaster, which is largely driven and influenced by meteorological factors. Water supply operations considering DWWL should explicitly involve the development trends of drought. Drought signals often exhibit temporal and spatial delays during the propagation of different types of drought (Ahmadalipour et al., 2017; Apurv & Cai, 2020; Zhang, Hao, et al., 2022). Among various drought types, meteorological drought manifests the most directly and often serves as a precursor to other drought types. In China, strong correlations have been observed in regions like the Huai River Basin (Li et al., 2021) and the Yangtze River Basin (Wang, Wang, et al., 2024). It is estimated that about half of the severe meteorological droughts nationwide can potentially lead to hydrological droughts (Yang et al., 2024), further threatening the reservoir water supply. Considering that water demand from the reservoir has remained relatively stable over the years, we characterize meteorological conditions as the indicator for the reservoir's future replenishment capability. The Meteorological drought Composite Index (Meterological drought Composite Index (MCI)) is selected as the meteorological drought indicator according to the national standards (QX/T 597–2021, Liao et al., 2021; GB/T 20481-2017, Zhang et al., 2017). The drought index MCI has demonstrated its effectiveness in various regions across China (Han et al., 2021; Ruina et al., 2021; Zhang, Duan, et al., 2022). The calculation formulas for MCI can be found in Section S1 in Supporting Information S1. At the same time, the reservoir water level/storage relative to the DWWL serves as an additional indicator of the current water scarcity. Together, the current MCI and initial water level make up the drought watch system, which is used to guide real-time drought management for large water supply reservoirs.

Given the robust performance of HR in real-time reservoir water supply (Ahmadianfar & Zamani, 2020; Wan et al., 2016), we advocate for the integration of DWWL and HR as the real-time drought-resistance operation strategy. Specifically, if the water level at the beginning of a time step is higher than the DWWL, all reservoir storage (excluding minimum storage) and incoming streamflow should be utilized through a standard HR operation. In this case, the effective reservoir storage is equivalent to the current storage: 25 $${S}_{t-1}^{\text{eff}}={f}^{\mathit{\prime }}\left({Z}_{t-1}\right)$$ where $${Z}_{t-1}$$ is the reservoir water level at the beginning of time step $$t$$; $${f}^{\mathit{\prime }}(\mathit{\cdot })$$ is the inverse function of the reservoir water storage-level relationship.

Should the water level drop below the DWWL, the use of water storage becomes contingent, and a rationing policy for reservoir storage is enforced based on the MCI value: 26 $${S}_{t-1}^{\text{eff}}=(1-\alpha )\left({f}^{\mathit{\prime }}\left({Z}_{t-1}\right)-{S}_{t}^{\min }\right)+{S}_{t}^{\min }$$ where $$\alpha =\left\{\begin{array}{@{}cr@{}}\frac{{a}_{\mathrm{m}\text{ax}}-{\mathrm{M}\mathrm{C}\mathrm{I}}_{t}}{\,{a}_{\max }-{a}_{\min }}& \hfill if\,{a}_{\min }< \,{\mathrm{M}\mathrm{C}\mathrm{I}}_{t}\mathit{\le }{a}_{\max }\\ \,1& if\,{\mathrm{M}\mathrm{C}\mathrm{I}}_{t}\mathit{\le }{a}_{\min }\\ 0& if\,{\mathrm{M}\mathrm{C}\mathrm{I}}_{t} > {a}_{\max }\end{array}\right.$$

Here, the rationing factor $$\alpha $$ is a normalized MCI value. An MCI value below $${a}_{\min }$$ indicates an exceptional drought, necessitating the retention of all water storage in the reservoir to avert further depletion, with water allocation to consumers sourced exclusively from the inflow. Conversely, an MCI value exceeding $${a}_{\max }$$ suggests the potential for moist hydrological conditions, which may mitigate the impacts of ongoing water scarcity, allowing for the full utilization of available water for supply regulation. The parameters $${a}_{\min }$$ and $${a}_{\max }$$ can be assigned empirically by analyzing historical MCI ranges and their correlation with anticipated streamflow, or refined through parametric optimization.

We propose HR, integrated with DWWL through effective storage, for real-time operation. By replacing $$WA$$ (water availability in Equations 7−8 in Section 2.1.1) with the following effective water availability $${WA}_{t}^{\text{eff}}$$ as: 27 $${WA}_{t}^{\text{eff}}={S}_{t-1}^{\text{eff}}+{I}_{t}^{\text{pred}}-{E}_{t}-{S}^{\min }$$

Substituting $${WA}_{t}^{\text{eff}}$$ into Equations 7−8 yields the optimal solution for water supply that accounts for potential drought conditions.

Study Area and Data

The Danjiangkou Reservoir (32$$\mathit{{}^{\circ}}$$36–33$$\mathit{{}^{\circ}}$$48′N; 110$$\mathit{{}^{\circ}}$$59′-111$$\mathit{{}^{\circ}}$$49′E, Figure 2) is situated on the middle reaches of the Hanjiang River and serves as a water source for the South-to-North Water Diversion Project in China. Its drainage area spans across the provinces of Shaanxi, Hubei, and Henan. The primary functions of the reservoir are water supply and flood control. Key water supply sectors include transfers to the South-to-North Water Diversion through the Taocha Channel, irrigation for districts in Hubei and Henan provinces via the Qingquan Canal, and ecological flow requirements for downstream reaches, as indicated by the red arrows in Figure 2. The reservoir's feasible water level ranges from 145 m (corresponding to a storage of 10.0 billion m³) to 170 m (29.05 billion m³). During the main flood season (21st June−20th August), the flood-limited water level is 160 m (19.82 billion m³), while in the post-flood season (21st August−15th October), it is adjusted to 163.5 m (22.858 billion m³).

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The operation time step is set to 1 month ($$T=$$ 12 for 12 months), and the available historical hydro-meteorological data spans from 1956 to 2020 ($$Y=$$ 65 for 65 years). The collected data include meteorological observations (daily precipitation, temperature, atmospheric pressure, sunshine hours, relative humidity, as shown by the red points in Figure 2 for their distribution) within the drainage area, reservoir inflow time series, monthly water demand targets for three sectors, reservoir water level-area-storage relationships, the current operational rule curves (Figure S1 in Supporting Information S1), and seasonal evaporation coefficients. The primary objective for the Danjiangkou Reservoir is to minimize the total benefit losses across all three water use sectors. The exponent parameter $$m$$ in the loss function (Equation 2) is set to 3. For real-time operation, the weight parameter for reservoir storage is set to $${\omega }_{S}=\frac{1}{2}$$, while the weights for the three water use sectors ($$n=\left\{1,2,3\right\}$$) are set to $${\omega }_{n}=\frac{1}{6}$$ (Equations 2 and 3). Due to the absence of an annual scheduling plan, the periodic storage target is simply set equal to the maximum storage for the subsequent period, that is, $${S}_{t}^{\text{Tar}\,}={S}_{t+1}^{\mathrm{m}\text{ax}\,}$$. Given that the release decision is based on forecasted inflow, a simulation-based forecasting method is employed to generate monthly inflow forecasts, assuming a forecast accuracy of $${R}^{2}=0.8$$ (see Section S5 in Supporting Information S1). For long-term operation, only the weights for water use sectors need to be considered, with each assigned an equal weight of $${\omega }_{n}=\frac{1}{3}$$ (Equation 15).

Results

Drought Season Segmentation of the Danjiangkou Reservoir

The historical long-term operation process of a reservoir is often not available or not applicable due to major alterations in operating targets. To address this, the long-term deterministic optimization method using DP-HR algorithm, as proposed in Section 2.1.2, is applied to obtain the optimal water level process for the period from 1956 to 2020 (Figure 3). The mean annual inflow of the Danjiangkou Reservoir is 1135 m³/s. Comparing the mean monthly inflow and water level (blue lines in Figure 3), it is evident that there is an asynchronous variation from January to September. This is primarily due to the flood control and streamflow regulation of large reservoirs. Despite the abundant inflow between May and October, the reservoir water level remains low from May to August mainly due to flood control measures, followed by a gradual refill in September and October. From November onward, the water level steadily decreases as demand surpasses inflow. Additionally, since flood control storage accounts for 48% of the reservoir's capacity, the storage constraints during flood seasons significantly restrict reservoir water supply. Therefore, it is crucial to incorporate these storage constraints into drought season segmentation for large reservoirs.

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After calculating drought metrics of water deficit (Equation 18) and storage balance (Equation 19), a hierarchical clustering analysis (Section S2 in Supporting Information S1) is adopted for the formatted multi-dimensional time series (Sections 2.2.1 and 2.2.2). Four groups are obtained as {January–March, April–May, June–September, October–December}. The clustering dendrogram is shown in Figure 4a. The method accurately distinguishes between wet and dry phases and identifies the early dry season (October–December), main dry season (January–March), and late dry season (April–May). This allows the Danjiangkou Reservoir to set different DWWLs during dry periods and thus maximize its drought resistance potential.

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For comparison, Figures 4b and 4c also show the results using mean monthly inflow and water level as drought metrics with Fisher's optimal partitioning method (see Section S3 in Supporting Information S1). The clustering varies clearly with different metrics. If only mean monthly inflow is considered, the clustering results only bring detailed segmentation for wet periods, leaving the entire dry periods as one group (Figure 4b). With additional consideration of reservoir water level, the equal-weighted sum of both variables is adopted as the clustering metric. However, the inherent asynchronous between inflow and water level results in unreasonable groups: the months from February to June are clustered into one group. The proposed method considering storage constraints and long-term time series enhances the rationality of drought season segmentation.

Seasonal Drought Warning Water Level

According to the drought season segmentation result (Figure 4a), the hydrological year is the 12 months from the beginning of June to the end of May the following year. This aligns with the water level patterns depicted in Figure 3b, which decrease from January to April and continue to decline until early June in two-thirds of the historical years. Therefore, it is logical to consider June to September as the reservoir refill periods and October to May as the drawdown periods.

To avoid bias in selecting pessimistic representative drought years, four hydrological dry years are chosen based on their mean annual inflow, which is close to the empirical frequency of 75%. These years are the hydrological years 1962 (June to May of the following year), 1972, 1976, and 1994. The corresponding annual inflows for these years are 831.2, 829.8, 842.8, and 828.3 m³/s, respectively (see Figure S2 in Supporting Information S1 for their monthly inflows). Figure 5 (dashed lines) depicts the required initial storage for these typical dry years, obtained by applying the recursive water supply regulation (Equations 18−19). The variation among these curves results from differences in the intra-annual distribution of reservoir inflows. For example, the hydrological year 1994 (June to May of 1995, purple dashed line in Figure 5) exhibits a more evenly distributed streamflow, resulting in a relatively stable water level process. In contrast, the scarce inflow during dry periods in 1972 (orange dashed line) lead to higher required reservoir water levels. The highest required monthly water level among the representative years, indicated by the gray dashed line, reflects the reservoir's necessary storage capacity under consistently low flows throughout the year. We, therefore, refer to it as the worst-case scenario. Finally, for the convenience of drought management, the seasonal DWWL is set to the lowest water level of the worst-case scenario within the same drought season (solid black line).

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The derived DWWL is high at the beginning of dry periods, reaching 157.4 m, and the lowest at 146.5 m at the end of dry periods. Raising the DWWL during the dry periods, particularly at the outset, has the potential to prevent severe water shortages if timely drought measures are not implemented. Conversely, lowering the DWWL at the end of the dry season helps avoid unnecessary frequent drought warnings in periods when low water levels are acceptable. According to the optimal water level process, there are 89 instances (constituting 10.6% of the total time span) between 1956 and 2020 where the reservoir water level fell below the determined DWWL. These occurrences are distributed across different seasons, with 31 instances occurring from June to September, 24 from October to December, 16 from January to March, and 12 from April to May. This evenly distributed pattern demonstrates that the specified DWWL is relevant for guiding reservoir drought management throughout the entire year. The consistency between the pattern of DWWL and the occurrences of lower water levels supports the validity of the derived DWWL.

Water Supply Management Considering Drought Characteristics

Reservoir Drought Characteristics

The 65-year historical meteorological data (precipitation, temperature, evaporation, etc.) from the upstream of the Danjiangkou Reservoir, monthly MCI values are calculated, which range from $${-}$$3.02 to +3.38. About 74% of the periods fall within ±1, and less than 5% of the periods exceed ±2 (extreme wet/drought), indicating an overall stable historical meteorological condition.

The 3-month scale standardized streamflow index values are calculated to represent the hydrological drought conditions of the Danjiangkou Reservoir. Visual observations indicate a roughly 2-month lag between meteorological drought and hydrological drought (Figure S3 in Supporting Information S1). The MCI effectively reflects the characteristics of inflow for the current month (with a Pearson correlation of 0.67) and for the next 3 months (correlation of 0.52). These correlations are higher in the earlier decades and decreased in recent decades, partly due to the impact of upstream reservoir regulation and water withdrawals, which have weakened the response between meteorological and hydrological conditions. The overall correlation of over 0.5 suggests an acceptable level of predictability for incoming inflows and a consequential relationship with reservoir water resources management.

HR Incorporating DWWL in Real-Time Operation

To evaluate the effectiveness of operations with and without considering DWWL, a rolling horizon approach (Zhao, Yang, et al., 2012) is employed, using dynamically updated meteorological and hydrological observations from 1956 to 2020. The time series for the forecasted monthly inflow is presented in Figure S4a in Supporting Information S1. Operational decisions are made every month, following conventional rule curves and comparing HR operations with and without DWWL (Section 2.1.1 and Section 2.2.3). Additionally, the DP-HR long-term deterministic optimization is also included for comparison as the upper boundary of what could be achieved.

According to the operational indexes listed in Table 1, HR outperforms conventional rule curves in almost every aspect. HRs (both with and without DWWL) result in only half the total losses of water supply deficits compared to operations using rule curves. HRs not only achieve lower frequencies of low levels (water level below critical level of 150 m) and higher rates of end-of-flood-season refill probability but also make better use of the water supply. Specifically, the guaranteed rate for the South-to-North Water Diversion increases by 12.3%, and the water intake for irrigation districts rises by 10.8%. The DP operation, with perfect foresight of long-term inflows, achieves better water allocation without considering the refill rate while maintaining nearly identical guaranteed water supply rates for all three users. In real-time decision-making, the HR method accounts for reservoir storage targets at each time step, leading to a relatively higher storage rate at the end of the refill season. In cases of water supply deficits, the hierarchy hedging prioritizes local water use first, followed by ensuring water transfer, resulting in varying guaranteed water supply rates among users. However, the difference in HR operations with and without DWWL is minimal. While the total benefit loss is slightly higher with DWWL than without, the frequency of reservoir levels dropping below the critical low level is reduced. These limited changes are attributed to the historically stable and abundant inflow to the Danjiangkou Reservoir over the past half-century, meaning DWWL has rarely activated—only 18 months out of 780 months in real-time operations.

Table 1 Comparison of the Real-Time Operation Results Using Hedging Rule Coupled With and Without Drought Warning Water Level, Conventional Rule Curves, and Long-Term Optimization Using DP-HR for the Period 1956 to 2020

Operation method indexes	DP-HR	HR with DWWL	HR without DWWL	Conventional rule curve
Total lossa	1.52	2.58	2.49	5.70
Low-level monthb	1.08	0.16	0.18	0.75
Maximum lossc	0.68	0.36	0.36	0.40
Refill rated	81.0%	84.5%	84.5%	82.9%
Transfer reliabilitye	96.9%	78.5%	78.5%	66.2%
Irrigation reliabilityf	96.9%	95.4%	95.4%	84.6%
Ecology reliabilityg	99.4%	99.6%	99.6%	99.5%

Further typical real-time operations are conducted under extreme drought periods, specifically the extreme drought year of 1959 (June 1959 to May 1960, with an annual mean inflow of 587.3 m³/s) and the consecutive drought years from June 1997 to May 2000 (with a mean annual inflow of 618.6 m³/s). The runoff forecasts for these two drought periods are provided in Figure S4 in Supporting Information S1, with mean forecasted inflows of 538.9 and 648.7 m³/s, respectively. The starting regulation water level is set to the minimum water level of 145 m. The operation results are shown in Figure 6.

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HR generally maintains a higher water level compared to rule curves, this is because HR allows for sustained minor water shortages to guard against severe ones. However, this effect is limited under extreme dry conditions. The DWWL serves as a safeguard for reservoir water retention. When water levels fall below the DWWL, HR incorporating DWWL restricts water supply earlier (as shown by the orange line in Figure 6). This early restriction helps prevent the reservoir from depleting to extremely low levels rapidly. Nonetheless, under relatively abundant inflow, even if the water level is below the DWWL (October–December 1999 in Figure 6b), the substantial inflow provides ample water availability, and the impact of DWWL-induced storage rationing on the final hedging decision is minimal.

Discussion

Sensitivity of Hydrology Variation and Forecast Uncertainty

The previous results are based on historical streamflow. Considering that history does not repeat itself, the stochastic streamflow generation method (Kirsch et al., 2013) is adopted to produce 100 sets of 65-year monthly inflow series and MCI values (see Section S4 in Supporting Information S1). Accordingly, to support real-time operations based on forecast and drought indices, we also simulated streamflow forecasts assuming varying forecast accuracy ($${R}^{2}$$) from 0 to 1 (see Section S5 in Supporting Information S1). A forecast accuracy $${R}^{2}=0$$ represents forecast uncertainty equivalent to the variance of historical monthly inflows, while $${R}^{2}=1$$ signifies a perfect prediction of the incoming period.

The distribution of 100-set synthetic monthly inflows closely resembles historical inflows but with a broader range for both extremes (Figure 7a). Under a forecast accuracy $${R}^{2}=0.8$$, the HR operation considers DWWL for the Danjiangkou Reservoir outperforms the rule curves operations in terms of water supply rates and frequency of low-level months (Figure 7b). The overall performances align with the historical results (refer to Table 1 in Section 4.3.2), indicating that HR considering DWWL is relatively stable and can effectively adapt to runoff variability.

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The operational performance index, exemplified by total loss (Figure 7c), consistently surpasses rule curves across different forecast accuracies. This might be due to the adopted monthly decision time step, whereas rule curves were originally designed for 10-day intervals, potentially causing mismatches in curves and parameters. Additionally, unlike rule curves, which reduce water supply in fixed proportions based on water level intervals, HR adjusts water supply uniformly based on user demand, offering greater flexibility. As a result, even with less accurate forecasts, HR still yields superior outcomes compared to rule curves.

The improvement in water supply benefits with increasing forecast accuracy is limited and far smaller than the difference between the adopted operation methods. This suggests that accurate forecasting for a single period does not substantially enhance operational benefits, underscoring the importance of incorporating multi-period forecasts.

Integrating DWWL with MCI enables the reservoir to strategically allocate its storage during low levels and unfavorable inflow forecasts. The MCI thresholds, $${a}_{\min }$$ and $${a}_{\max }$$, play a pivotal role in defining the effective storage (refer to Equation 26). Assuming that $${a}_{\max }$$ equals to the absolute value of $${a}_{\min }$$, we compare the operation results under varying $${a}_{\min }$$ from $${-}1$$ to $${-}4$$, as compared to the standard HR without DWWL.

Figure 8 illustrates the implementation results across 100 sets of inflow scenarios. The rationing policy takes effect in 2–68 months out of 780 operational steps. Compared to HR without DWWL, the inclusion of DWWL tends to increase total water supply loss by 0.2–1 (see panels a and c in Figure 8), while concurrently diminishing the frequency of months per year with water levels below the threshold 150 m by 0–0.2 months/yr (Figures 8b and 8d). A decrease in forecast accuracy, with $${R}^{2}$$ dropping from 0.8 to 0.2, increases the number of periods in which the rationing policy is activated, as indicated by the rightward shift of the density plot at the top of Figures 8a–8d. Lowering the MCI threshold $${a}_{\min }$$ from $${-}1$$ to $${-}4$$ results in more consistent operational outcomes, as observed by the higher peaks of the density plot on the right in Figure 8. However, the changes are not pronounced, indicating that the proposed HR operation, integrated with DWWL, demonstrates low sensitivity to the thresholds of the meteorological drought indicator.

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The Necessity of DWWL and DWWL-Based Operation

The results from Section 5.1 indicate that considering DWWL does not significantly improve overall operational benefits and can sometimes even be detrimental to water supply, as compared to HR without DWWL. Given this, the question arises as to whether it is necessary to adopt drought resistance operations that incorporate DWWL. In the context of global climate change and the increasing frequency of heat and rainfall extremes, it is indeed essential to strengthen and implement regional drought management measures. However, similar to the management of flood-limited water levels, the concept of DWWL may remain controversial. This is particularly the case because a rule extracted from historical information could, in theory, be superseded by a sufficiently optimal real-time operation algorithm, which could enhance water supply efficiency in most aspects (as implied by the comparison between HR and the rule curves in Table 1 in Section 4.3.2). Nonetheless, until a widely accepted and universally applicable real-time operation method is developed and validated, DWWL may persist as a simplified yet indispensable management tool for coping with complex climate extremes. Operations based on DWWL may offer advantages in the following fields.

• Drought Pre-warning: From the perspective of water use sectors, early warnings of potential drought can trigger water conservation measures. Gradually reducing water supply in response to these warnings allows for proactive management of the situation, thus reducing water supply while maintaining water use efficiency. Compared to conventual rules or HR, incorporating DWWL enables a more prompt response to drought and initiates quantitative rationing earlier.

• Managerial Insight: DWWL provides reservoir managers with an intuitive indicator of current water level severity, signaling the need for reducing water supply or drought response measures. This facilitates better decision-making and more effective management strategies.

• Drought Backup Plan: DWWL can act as a trigger for activating emergency water resources. For instance, the Shahe Reservoir in Jiangsu Province, China (Cui et al., 2016), is designed to supplement water from downstream large rivers when the reservoir water level drops below 17.5 m. However, the determination of this critical water level and volume of supplemented water are experimental and somewhat arbitrary. DWWL in this case can establish more quantifiable and precise rules for such emergency measures.

Conclusions

Within the realm of DWWL (DWWL) research in China, this study has proposed a framework to integrate DWWL with analytical HR for reservoir water supply operations, which can be applied to both real-time and long-term operations. Starting from the identification of optimal HR, we developed a long-term HR combined with a DP algorithm for historical deterministic optimization. DWWLs and their seasonal segmentation are determined through optimal water supply process and multivariate time-series clustering. From a real-time perspective, we recommend using reservoir water level and meteorological drought indicator to monitor drought conditions and characterize reservoir effective storage. A real-time operation strategy is ultimately proposed that integrates analytical HR with effective storage. The application of this framework to the Danjiangkou Reservoir has demonstrated a marked enhancement in water supply benefits and reliability for various users as compared to the existing rule curves. The sensitivity of this HR operation considering DWWL to the precision of single-period forecasts and the choice of drought index threshold is found to be low.

The analytical HR for water supply was originally derived with the objective expressed as the weighted sum of current release and carryover storage. We extended this approach to long-term multi-user water supply and real-time drought resistance operations. For long-term optimization with multiple water users, conventional DP requires discretizing both water storage and allocated water volumes. In this study, we propose an innovative method that explicitly incorporates optimal analytical HR, enabling precise allocation of water supply across users and periods. This reduces the dimensionality of the search domain and decreases computational time. While the improvement for single-reservoir systems is limited, this framework offers a promising solution for long-term multi-reservoir optimization.

Regarding DWWL operations, unlike previous studies that make supply decisions based on predetermined rule curves (Chang et al., 2019; Luo et al., 2023), we, for the first time, integrated DWWL into a forecast-decision framework. This brings to a unified operation that smoothly transitions between regular water supply and drought response while explicitly incorporating forecasts. However, a limitation of this model is that the current model only considers the forecasting inflow for a single period and simply sets the full storage as the reservoir storage target, overlooking the discrepancy between decision and forecast horizons within the operation horizon. Typically, an algorithm applicable to deterministic optimization can be adapted for stochastic environments (e.g., Zeng et al., 2019; Zhao, Cai, et al., 2012). By implementing multi-stage forecasting, which often involves incorporating various sets of possible inflow scenarios, a more rational setting of the storage targets can be attained. This will be the focus of our subsequent research.

Acknowledgments

We appreciate the inspiration for the DP-HR algorithm provided by Dr. Weisa Meng from Tsinghua University. This research was financially supported by the National Key Research and Development Program of China (2023YFC3006601), the National Natural Science Foundation of China (U2240223, 52409040), and the Water Conservancy Science and Technology Project of Jiangsu Province (No. 2023014).

Data Availability Statement

Data essential for evaluating the findings of this study and the compiled codes are publicly available at .

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