Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Lagrangian-Informed Long-Term Dispatch Policy for Coupled Hydropower and Photovoltaic Systems

Eliza Cohn, Ning Qi, Upmanu Lall, Bolun Xu

TL;DR

The paper tackles long-term coordination of reservoir-based hydropower with floating photovoltaic (FPV) systems under monthly water contracts. It introduces a partial Lagrangian relaxation approach that decomposes the problem into a master water-pricing problem in $\\theta$ and hour-by-hour subproblems, solved efficiently via a bi-section method, with an analytical subproblem solution. A data-driven nonlinear hydraulic head function $\\phi(V)$ is fitted from real data ($\\phi(V)=aV^b$, $R^2=0.99$) to preserve nonlinearity in reservoir dynamics. Numerical results on Hoover–Glen Canyon-scale systems show near-optimal performance (optimality gap <$0.1\%$) and strong robustness under uncertainty, while achieving substantial computational gains compared to full nonlinear solvers. The framework supports non-anticipatory, long-horizon energy management for coupled hydropower and FPV with practical implications for grid reliability and water-resource coordination.

Abstract

This paper presents a long-term dispatch framework for coupled hydropower and floating photovoltaic systems. We introduce a temporal decomposition algorithm based on partial Lagrangian relaxation to address long-term water contract constraints. We derive a real-time, non-anticipatory dispatch policy based on water contract pricing. Our framework is evaluated with a case study using real-world hydrology and power system data from Lake Mead and Lake Powell, on the Colorado River, demonstrating competitive performance against commercial solvers for both linearized and nonlinear reservoir models. We conduct a sensitivity analysis on transmission capacity, electricity price and uncertainty scenarios, showing that the operational performance is significantly impacted by the transmission capacity and electricity prices while remaining relatively robust under uncertainty scenarios.

A Lagrangian-Informed Long-Term Dispatch Policy for Coupled Hydropower and Photovoltaic Systems

TL;DR

The paper tackles long-term coordination of reservoir-based hydropower with floating photovoltaic (FPV) systems under monthly water contracts. It introduces a partial Lagrangian relaxation approach that decomposes the problem into a master water-pricing problem in and hour-by-hour subproblems, solved efficiently via a bi-section method, with an analytical subproblem solution. A data-driven nonlinear hydraulic head function is fitted from real data (, ) to preserve nonlinearity in reservoir dynamics. Numerical results on Hoover–Glen Canyon-scale systems show near-optimal performance (optimality gap <) and strong robustness under uncertainty, while achieving substantial computational gains compared to full nonlinear solvers. The framework supports non-anticipatory, long-horizon energy management for coupled hydropower and FPV with practical implications for grid reliability and water-resource coordination.

Abstract

This paper presents a long-term dispatch framework for coupled hydropower and floating photovoltaic systems. We introduce a temporal decomposition algorithm based on partial Lagrangian relaxation to address long-term water contract constraints. We derive a real-time, non-anticipatory dispatch policy based on water contract pricing. Our framework is evaluated with a case study using real-world hydrology and power system data from Lake Mead and Lake Powell, on the Colorado River, demonstrating competitive performance against commercial solvers for both linearized and nonlinear reservoir models. We conduct a sensitivity analysis on transmission capacity, electricity price and uncertainty scenarios, showing that the operational performance is significantly impacted by the transmission capacity and electricity prices while remaining relatively robust under uncertainty scenarios.

Paper Structure

This paper contains 15 sections, 2 theorems, 8 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Formulation
  4. Fitting the Nonlinear Hydraulic Head Function
  5. Methodology
  6. Partial Lagrangian Relaxation
  7. Bi-section Solution Method of the Master Problem
  8. Analytical Solution to the Subproblem
  9. Full Algorithm
  10. Numerical Studies
  11. Set-up
  12. Comparative Results
  13. Sensitivity Analysis
  14. Sensitivity Study on Uncertainty Incorporation
  15. Discussion and Conclusion

Key Result

Proposition 1

The sum of the water releases, $\sum_{t \in \mathcal{T}} u^*_t$, decreases monotonically with respect to $\theta$ if the hydraulic head function $\phi$ is constant.

Figures (4)

  • Figure 1: Visualization of fitting the nonlinear hydraulic head function.
  • Figure 2: Shadow price of water ($\theta$) and average monthly marginal price of electricity ($\lambda_t$) for varying transmission capacities (P) over 2022-2023.
  • Figure 3: Revenue for varying transmission capacity (P) from 2022 to 2023.
  • Figure 4: Total revenue distribution over varying MAPE from 2022 to 2023.

Theorems & Definitions (3)

  • Proposition 1
  • Remark 2
  • Proposition 3