Dam Management in the Era of Climate Change
Climate change has a dramatic impact, particularly by concentrating rainfall into a few short periods, interspersed with long dry spells. In this context, the role of dams is crucial. We consider the optimal control of a dam, where the water level must neither exceed a designated safety threshold nor fall below a minimum level to ensure functionality and sustainability for the downstream river. To model dry spells and intense rainfall events, commonly referred to as water bombs, we introduce a Hawkes process, a well-known example of a self-exciting process characterized by time-correlated intensity, which endogenously reproduces the clustering of events. The problem is formulated as an optimal switching problem with constraints. We establish existence results and propose numerical methods for approximating the solution. Finally, we illustrate the main achievements of this approach through numerical examples focusing in particular on the sensitivity of the self-exciting parameter describing the importance of both water bombs and dry-spells. For the parameter configurations considered in this paper, the optimal water level inside the dam decreases as the self-exciting parameter increases. This numerical finding suggests that, under the considered calibration, the management response is driven more strongly by overtopping risk than by drought risk. In conclusion, dams will increasingly lose their role as water reserves and take on a greater role in flood protection.