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Real-time Coordination of Cascaded Hydroelectric Generation under Decision-Dependent Uncertainties

Eliza Cohn, Ning Qi, Upmanu Lall, Bolun Xu

Abstract

This paper proposes a real-time control policy for cascaded hydropower systems that incorporates decision-dependent uncertainty (DDU) to capture the coupling of streamflow uncertainties across the network. The framework jointly models exogenous forecast errors and endogenous uncertainty propagation, explicitly characterizing the dependence between upstream releases and downstream inflow variability through a heteroskedastic variance model conditioned on past errors, variance, and control actions. We formulate a joint chance-constrained optimization problem to ensure reliable system operation under uncertainty, and develop a tractable supporting hyperplane algorithm that enables explicit and adaptive risk allocation under DDU. We establish convergence of the proposed method and show that it recovers the Bonferroni approximation under steady-state conditions. A randomized case study based on Columbia River data demonstrates that the proposed framework improves both energy generation and reservoir reliability by accounting for DDU. Sensitivity analyses on drought severity and model parameters further highlight the value of adaptive risk allocation for resilient hydropower operations.

Real-time Coordination of Cascaded Hydroelectric Generation under Decision-Dependent Uncertainties

Abstract

This paper proposes a real-time control policy for cascaded hydropower systems that incorporates decision-dependent uncertainty (DDU) to capture the coupling of streamflow uncertainties across the network. The framework jointly models exogenous forecast errors and endogenous uncertainty propagation, explicitly characterizing the dependence between upstream releases and downstream inflow variability through a heteroskedastic variance model conditioned on past errors, variance, and control actions. We formulate a joint chance-constrained optimization problem to ensure reliable system operation under uncertainty, and develop a tractable supporting hyperplane algorithm that enables explicit and adaptive risk allocation under DDU. We establish convergence of the proposed method and show that it recovers the Bonferroni approximation under steady-state conditions. A randomized case study based on Columbia River data demonstrates that the proposed framework improves both energy generation and reservoir reliability by accounting for DDU. Sensitivity analyses on drought severity and model parameters further highlight the value of adaptive risk allocation for resilient hydropower operations.
Paper Structure (19 sections, 2 theorems, 35 equations, 10 figures, 2 tables, 1 algorithm)

This paper contains 19 sections, 2 theorems, 35 equations, 10 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. Formulation
  4. Inflow and Uncertainty Characterization
  5. Decision-Independent Uncertainty
  6. Decision-Dependent Uncertainty
  7. Solution Methodology
  8. Bonferroni Solution Algorithm
  9. Supporting Hyperplane Solution Algorithm
  10. Explicit Risk Attribution of SSH
  11. Numerical Studies
  12. Setup
  13. Policy Learning under Forecasted Droughts
  14. Policy Testing under Stochastic Drought Scenarios
  15. Sensitivity Analysis
  16. ...and 4 more sections

Key Result

Theorem 1

The Sequential Supporting Hyperplane (SSH) algorithm either finds a global optimizer to the joint chance-constrained problem in a finite number of iterations or it generates a sequence $\{x^{k_i}_t\}_{i=1}^\infty$ converging to the global optimizer with tolerance $\varepsilon_{LP}$.

Figures (10)

  • Figure 1: Cascaded hydropower network with correlated inflows
  • Figure 2: Empirical characterization of DDU. (a) Residual magnitude increases with upstream release, demonstrating heteroskedastic forecast error. (b) Corresponding conditional normal distributions showing how DDU produces state-dependent variance to DIU.
  • Figure 3: Block diagram of online and state-dependent variance estimation
  • Figure 4: Seasonal inflow behavior and dry-season baseline definition. (a) Seasonal hydrograph with interquartile variability and identified dry-season window. (b) Empirical distribution of dry-season inflows with kernel density estimate centered near the baseline flow
  • Figure 5: Optimal policy trajectories for water release and reservoir state under DET, DIU, and DDU uncertainty frameworks using the supporting hyperplane solution algorithm.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2