Technical Summary: Optimal Scheduling of Electricity and Water in Renewable-Colocated Desalination Plants

Problem Statement

The paper addresses the operational challenge of scheduling Water Desalination Plants (WDPs) that function as hybrid generator-load resources. These facilities integrate thermal desalination plants (TDPs), reverse osmosis desalination plants (RODPs), and on-site renewable energy sources (e.g., solar PV). Unlike conventional resources that are either pure generators or pure loads, these WDPs possess continuous bidirectional flexibility, capable of simultaneously selling desalinated water to a utility and transacting electricity bidirectionally with the grid (importing when demand exceeds local generation, exporting when surplus exists).

The core problem is to determine the optimal joint scheduling of water production and electricity transactions to maximize the WDP's profit. This requires navigating complex couplings between:

1. Technological constraints: TDPs co-generate water and electricity from fuel, while RODPs consume electricity to produce water.
2. Market dynamics: The plant faces exogenous water prices and a net energy metering (NEM) electricity tariff with distinct import ($\pi^+$) and export ($\pi^-$) rates.
3. Renewable variability: The plant must adapt to fluctuating renewable generation ($g$) without relying on energy storage (in the base model).

Existing literature often treats these systems via simulation or numerical optimization, lacking analytical insights into the structural properties of the optimal solution, particularly regarding how thermal and membrane-based loads complement each other.

Methodology

The authors develop a mathematical framework for a profit-maximizing WDP operating as a price-taking agent.

1. System Modeling:

   • TDP: Modeled as a fuel-consuming unit producing water ($w_h$) and electricity ($q_h$) with linear conversion factors ($\alpha_h, \beta_h$) and a strictly convex operating cost function $C_h(p_h)$.
   • RODP: Modeled as an electricity-consuming unit producing water ($w_r$) with a linear conversion factor ($\alpha_r$) and bounded flow rates.
   • Renewables: Treated as a known, perfect forecast ($g$) for the scheduling period.
   • Markets: Water is sold at a fixed price $\pi_w$. Electricity transactions follow a net metering tariff where net import ($z>0$) is charged at $\pi^+$ and net export ($z<0$) is credited at $\pi^-$.

2. Optimization Formulation:
   The problem is formulated as a concave maximization of the profit function $\Pi$, defined as water revenue minus electricity payments and TDP fuel costs, subject to energy balance and capacity constraints. The objective function is non-differentiable due to the piecewise nature of the electricity payment function.

3. Analytical Derivation:
   Instead of relying solely on numerical solvers, the authors derive closed-form analytical solutions for the optimal dispatch. They characterize the solution using a threshold-based policy. The optimal setpoints for TDP and RODP water production are determined by comparing the renewable generation $g$ against specific thresholds ($\Gamma_{IM}, \Gamma_{NZ1}, \Gamma_{NZ2}, \Gamma_{EX}$). These thresholds are explicit functions of technology parameters and tariff rates, computable offline.

Key Contributions

The paper's primary contributions are analytical and structural:

• Threshold-Based Optimal Dispatch: The authors prove that the optimal scheduling policy follows a four-threshold structure for the TDP and a two-threshold structure for the RODP when the marginal value of RODP water falls within the import/export price spread ($f'_r \pi_w \in [\pi^-, \pi^+]$).
  • Import Mode (IM): Low renewable generation; the plant imports power, RODP runs at minimum, TDP runs at a level determined by the sum of water and import electricity values.
  • Net-Zero Mode (NZ): Intermediate generation; the plant is self-sufficient. Surplus renewables first displace TDP generation, then increase RODP output.
  • Export Mode (EX): High renewable generation; the plant exports power, RODP runs at maximum, and TDP runs at minimum.
• Closed-Form Characterization: The optimal setpoints ($w^*_h, w^*_r$) are derived as explicit functions of $g$ and the thresholds, enabling computationally tractable coordination suitable for large-scale deployment.
• Comparative Statics: The paper analyzes how exogenous parameters (tariffs, conversion factors) affect the optimal dispatch and maximum profit. It demonstrates that profit is monotonically increasing with renewable generation and water prices, and decreasing with import electricity prices.
• Special Cases: The framework extends to scenarios with single desalination technologies (RODP-only or TDP-only) and special tariff structures (e.g., single price LMPs).

Results

Extensive simulations using semi-synthetic data from Saudi Arabia (50 MW PV capacity) validate the analytical framework:

• Performance vs. Benchmarks: The optimal joint dispatch significantly outperforms benchmark algorithms:
  • Max-RODP: Keeps RODP at maximum regardless of grid prices, leading to suboptimal profits during low-renewable hours due to high import costs.
  • Passive-TDP: Treats the plant as a traditional Independent Water and Power Producer (IWPP), failing to dynamically adjust to the bidirectional electricity market.
  • Optimal Joint Dispatch: Achieved a daily profit of $119,180, compared to $99,872 (Max-RODP) and $112,850 (Passive-TDP).
• Operational Dynamics: The simulations illustrate smooth transitions between modes. For instance, as solar generation increases, the plant shifts from importing power to self-sufficiency, and finally to exporting, while dynamically reallocating energy between TDP and RODP to maximize the marginal value of water production.
• Sensitivity Analysis:
  • Water Price ($\pi_w$): Had the strongest impact; a 50% increase in water price raised daily profit by approximately $1.2M.
  • Electricity Tariffs: The profit sensitivity to import prices ($\pi^+$) saturates once the price exceeds the marginal value of RODP water, at which point the plant stops importing for RODP operation.
  • Technology Parameters: Improvements in RODP efficiency ($\alpha_r$) and TDP conversion factors ($\alpha_h, \beta_h$) consistently increased profitability.

Significance and Claims

The paper claims to provide a tractable analytical solution to a complex water-energy nexus problem, moving beyond purely numerical approaches. Its significance lies in:

1. Operational Insights: It reveals how thermal generation and membrane-based loads can complementarily provide continuous bidirectional flexibility without the need for energy storage. The WDP acts as a "soft" resource that can smoothly transition between import, balanced, and export states.
2. Scalability: The derived threshold policies are computationally efficient and independent of real-time renewable forecasts (thresholds are computed offline), making them suitable for large-scale deployment.
3. Grid Integration: The framework demonstrates how WDPs can support grid stability by absorbing renewable variability through continuous adjustment of water production, offering predictable dispatch and smooth zero-crossing transitions.

The authors conclude that while the current model focuses on short-run operations with simplified conversion functions, the structural insights offer a foundation for guiding efficient dispatch in increasingly renewable-rich and market-responsive water-energy systems. Future work is suggested to incorporate energy storage, more complex environmental dependencies, and long-term investment planning.