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Dam Management in the Era of Climate Change

This paper formulates and numerically solves an optimal switching control problem for dam management under climate change by modeling clustered rainfall and dry spells with a Hawkes process, revealing that increased event clustering shifts the optimal strategy toward prioritizing flood protection over water storage.

Original authors: Cristina Di Girolami, M'hamed Gaïgi, Vathana Ly Vath, Simone Scotti

Published 2026-07-17

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CC BY 4.0

Original authors: Cristina Di Girolami, M'hamed Gaïgi, Vathana Ly Vath, Simone Scotti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world's weather as a chaotic DJ spinning records. For centuries, the playlist was predictable: a steady beat of rain in spring, a slow fade in summer, and a reliable rhythm that farmers and dam managers could dance to. But lately, the DJ has gone rogue. The music has stopped being a steady beat and turned into a jarring mix of sudden, deafening bass drops (massive storms) followed by agonizingly long periods of silence (droughts). This is the reality of climate change: it's not just about the temperature getting hotter; it's about the rhythm of water becoming dangerously erratic.

To understand how to manage this chaos, we need to look at two main ideas. First, think of a dam not just as a wall holding back water, but as a giant, high-stakes battery. It needs to store enough water to generate electricity and keep rivers flowing during dry spells, but it also has to be ready to dump water instantly to avoid bursting. If the water gets too high, the dam could fail; if it gets too low, the turbines stop and the river dies. Second, we need a new way to predict the weather. Old models assumed rain fell like a gentle, steady drizzle. But nature is now throwing "water bombs"—clumps of rain that happen in clusters, where one storm makes another more likely to happen right after. Scientists use a special math tool called a Hawkes process to model this. Think of it like a snowball rolling down a hill: one snowflake hits, and the impact makes it easier for the next one to stick, creating a massive avalanche. This self-exciting behavior is exactly what happens with extreme weather today.

This paper, titled "Dam Management in the Era of Climate Change," tackles the question: How do we run a dam when the weather is acting like a manic DJ? The authors, a team of mathematicians and economists, propose a new strategy. Instead of guessing, they use a complex computer simulation to find the "perfect" way to operate the dam's turbines and spillways. They treat the problem like a game of chess where the board is shaking. The goal is to maximize electricity production while keeping the water level in a "Goldilocks zone"—not too high, not too low.

The researchers built a model where the water level jumps up and down based on these "water bombs" (modeled by the Hawkes process) rather than a smooth, predictable flow. They asked the computer to figure out the best time to turn the turbines on or off and when to open the spillways to let water out. The results are surprising. As the "self-exciting" nature of the weather gets stronger (meaning storms are more likely to cluster and droughts are more likely to drag on), the optimal strategy changes. The simulation suggests that dam managers should keep the water level lower than they used to.

Why lower? It comes down to a scary trade-off. A higher water level helps during a drought, but if a massive "water bomb" hits, a full dam is a ticking time bomb waiting to overflow. The authors' simulations show that the risk of the dam overflowing (overtopping) is currently a bigger threat than the risk of running out of water. So, the smartest move is to keep the reservoir emptier, ready to catch the next storm, even if it means having less water for a dry spell. The paper concludes that in this new, chaotic climate, dams might need to stop acting primarily as water storage tanks and start acting more like flood shields. While this is a strong finding based on their specific computer models, the authors note that real-world application will depend on better data to estimate these weather patterns accurately. For now, the math suggests that when the weather goes crazy, the safest bet is to keep the dam's belly empty.

Technical Summary: Dam Management in the Era of Climate Change

Problem Formulation
This paper addresses the optimal control of a dam under the dual pressures of climate change and operational constraints. The core challenge lies in managing water inflows that exhibit extreme fluctuations: prolonged dry spells interspersed with intense, clustered rainfall events ("water bombs"). Traditional models relying on Brownian motion fail to capture these discontinuous, self-exciting dynamics. Consequently, the authors model the dam management problem as a stochastic optimal switching control problem.

The state of the system is defined by two variables:

Water Level ( $H_t$ ): Must remain within safety bounds ( $h^-$ to $h^+$ ) to prevent dam failure (overtopping) and ensure functionality (touristic and ecological requirements).
Rainfall Intensity ( $\lambda_t$ ): Modeled as a Hawkes process intensity, capturing the self-exciting nature of rainfall where past events increase the probability of future events.

The control variables include:

Turbine Status ( $I_t$ ): A binary switching control (0 for closed, 1 for open) to generate electricity.
Spillway Flow ( $\beta_t$ ): A continuous control variable regulating water release without power generation.

The objective is to maximize the expected discounted revenue from power production while minimizing penalties associated with water levels falling below a touristic threshold ( $h^-$ ), exceeding a safety threshold ( $h^+$ ), or failing to meet minimum outflow requirements ( $\mu$ ) during low-intensity periods. The problem terminates if the water level reaches a critical failure point ( $h_{max}$ ), incurring a terminal penalty.

Methodology
The authors propose a mathematical framework that departs from continuous diffusion models by utilizing a marked Hawkes process to drive the water level dynamics. The inflow is modeled as a compound jump process where the intensity $\lambda_t$ evolves according to a mean-reverting process with jumps triggered by rainfall events.

Stochastic Differential Equations (SDEs): The water level dynamics are governed by a discontinuous SDE driven exclusively by the marked Hawkes process, without a Brownian component. This choice is justified by the nature of mountain dams with small catchment areas, where local rainfall causes abrupt level changes.
Optimal Switching Formulation: The problem is cast as a mixed regular and switching control problem. The value functions $v_0$ and $v_1$ (corresponding to the turbine being initially off or on) are defined over the state space $D = [0, h_{max}] \times [b, +\infty)$ .
Viscosity Solutions: Due to the lack of a priori regularity in the value functions (a consequence of the jump dynamics and switching controls), the authors characterize the solution using viscosity solutions of a system of Hamilton-Jacobi-Bellman (HJB) partial integro-differential inequalities. They establish a weak Dynamic Programming Principle (DPP) and prove that the value functions are the unique viscosity solutions to the associated HJB system.
Numerical Approximation: To solve the HJB equations, the authors employ a finite difference scheme adapted from controlled Markov chain problems (Kushner and Dupuis). This involves discretizing the state space and iterating to convergence to approximate the value functions and optimal policies.

Key Contributions

Novel Modeling of Inflow: The paper introduces the use of marked Hawkes processes to model dam inflows, explicitly capturing the clustering of extreme rainfall events and the duration of dry spells. This addresses a gap in existing literature where Brownian motion or standard Poisson processes are often used, failing to reproduce the "self-exciting" nature of climate-driven weather patterns.
Discontinuous Dynamics: Unlike previous works (e.g., Chevalier et al. [16]) that modeled water levels as diffusion processes, this study utilizes a pure jump process driven by Hawkes intensity. This better reflects the dynamics of small, alpine dams where inflow is highly discontinuous.
Viscosity Solution Characterization: The authors provide a rigorous characterization of the optimal value functions as viscosity solutions to integro-differential inequalities, handling the complexities introduced by the jump process and the switching control.
Sensitivity Analysis: The study investigates the impact of the self-exciting parameter ( $c$ ), which quantifies the degree of clustering in rainfall events. This parameter serves as a proxy for climate change intensity in the model.

Numerical Results
Numerical experiments were conducted using parameters representative of mountain dams (e.g., height < 100m). Key findings include:

Optimal Policy Structure: The optimal spillway control is of a "bang-bang" type, switching between minimum and maximum flow rates.
Switching Behavior: For the specific parameter configurations tested (notably high switching costs), it is never optimal to stop electricity production once the turbine is running. However, the decision to start production depends on the water level and intensity.
Sensitivity to Self-Excitation: A critical finding is that as the self-exciting parameter $c$ $c$ increases (simulating more extreme climate variability), the optimal water level inside the dam decreases.
- Interpretation: While higher $c$ increases both the risk of drought (requiring higher water levels) and the risk of overtopping (requiring lower water levels), the numerical results indicate that the overtopping risk dominates. Managers respond by keeping reservoir levels lower to create buffer capacity for sudden "water bombs," even at the cost of increased drought risk.
Role Shift: The results suggest a paradigm shift in dam management: under high climate variability, dams may increasingly function as flood protection mechanisms rather than primary water reserves.

Significance and Claims
The paper claims to be the first study to explicitly integrate physical constraints and climate change impacts (modeled via Hawkes processes) into a dam management optimization framework. The authors emphasize that standard diffusion models are no longer valid for temperate zones experiencing extreme weather fluctuations.

The significance of the work lies in:

Providing a mathematical tool to analyze the trade-offs between flood protection and water supply in a changing climate.
Demonstrating that the optimal management strategy is highly sensitive to the clustering nature of rainfall, suggesting that traditional "rule curves" may need revision.
Highlighting that the self-exciting nature of extreme events forces a conservative management stance (lower reservoir levels) to mitigate catastrophic failure risks, potentially altering the primary economic role of dams from energy/water storage to flood mitigation.

The authors acknowledge limitations, noting that Hawkes intensity is not directly observable and requires estimation from hydrological data, which remains a challenge in a non-stationary climate. They also note that the sensitivity analysis focused primarily on the self-excitation parameter, leaving broader economic and regulatory sensitivity studies for future research.

Technical Summary: Dam Management in the Era of Climate Change

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