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

Ahmed S. Alahmed, Audun Botterud, Saurabh Amin

arXiv:2601.02243v3eess.SYcs.SYecon.TH

TL;DR

WDP scheduling must coordinate water production with electricity generation, consumption and grid exchange, while exploiting the complementarities of thermal and membrane-based desalination. The paper develops an analytical threshold-based scheduling framework and a tractable real-time dispatch approach. Its simulations demonstrate performance benefits, and the analysis identifies short-run scope boundaries and omitted value streams.

  • Problem

    WDP scheduling must account for coupled water and electricity decisions, including the complementarities between the plant's generation and controllable loads.

    Source: §I · p. 1

  • Method

    The paper analytically characterizes a threshold-based optimal scheduling policy for WDPs and develops an MPC-based dispatch algorithm for renewable uncertainty.

    Source: §VI · p. 10

  • Results

    Simulations demonstrate performance benefits for efficient dispatch of integrated water-energy systems in renewable-rich, market-responsive environments.

    Source: §VI · p. 10

  • Takeaways & Limitations

    The MPC-based implementation preserves the threshold structure while avoiding the complexity of the underlying multistage stochastic problem.

    Source: §VI · p. 10

    The analysis is short-run and excludes energy storage, long-term investment decisions, and grid value streams such as frequency regulation or generation capacity service.

    Source: §VI · p. 10

Abstract

from arXiv · show

We develop a mathematical framework for the optimal scheduling of flexible water desalination plants (WDPs) as hybrid generator-load resources. WDPs integrate thermal generation, membrane-based controllable loads, and renewable energy sources, offering unique operational flexibility for power system operations. They can simultaneously participate in two markets: selling desalinated water to a water utility, and bidirectionally transacting electricity with the grid based on their net electricity demand. We formulate the scheduling decision problem of a profit-maximizing WDP, capturing operational, technological, and market-based coupling between water and electricity flows. The threshold-based structure we derive provides computationally tractable coordination suitable for large-scale deployment, offering operational and economical insights into how thermal and membrane-based desalination colocated with renewables complementarily provide continuous bidirectional flexibility. The thresholds are analytically characterized in near closed form as explicit functions of technology and tariff parameters. We examine how small changes in the exogenous tariff and technology parameters affect the WDP's profit. Extensive simulations illustrate the optimal WDP's operation, profit, and water-electricity exchange, demonstrating significant improvements relative to benchmark algorithms.

I. INTRODUCTION

WDPs combine thermal generation, membrane-based loads and renewables, creating a setting for jointly scheduling water production and electricity exchange. The paper addresses the limited analytical treatment of these coupled systems with a threshold-based scheduling framework and examines its operating modes and implementation.

A. Related Work

  • Existing approaches largely treat desalination technologies separately or rely on numerical models, leaving limited analytical insight into optimal scheduling structure.Prior RO-renewables studies often treat the power system as exogenous, while other water-power models omit colocated renewables or bidirectional grid transactions.Source: A. Related Work · p. 2

B. Main Contributions

  • The framework analytically characterizes joint water and electricity scheduling for WDPs that combine thermal and RO units with on-site renewables.It models the WDP as a price-taking agent selling water at an exogenous price and importing or exporting electricity according to its net exchange.Source: §I · p. 1, B. Main Contributions · p. 2, §II · p. 3
  • The optimal WDP operates in import, net-zero and export modes, separated by two renewable-generation thresholds.It acts as a controllable load when importing, a self-sufficient microgrid at net zero, and a dispatchable generator when exporting.Source: B. Main Contributions · p. 2, Figure 2 · p. 3
  • As local renewables increase, TDP water and power outputs decrease while RODP water output increases; in net-zero operation, TDP generation follows renewable-adjusted RODP consumption.When RODP water's effective electricity price lies between export and import prices, both technologies adjust to absorb surplus renewables, expanding net-zero operation.Source: B. Main Contributions · pp. 2–3
  • The paper also examines single-technology cases, tariff and technology sensitivities, and simulations comparing optimal schedules, profit and exchanges with benchmarks.It proposes a receding-horizon algorithm for renewable uncertainty whose per-step decision uses the offline threshold policy; suboptimality is governed by forecast error.Source: B. Main Contributions · p. 3

C. Major Mathematical Symbols and Paper Organization

  • The paper provides a notation table and organizes its treatment from system modeling through optimal scheduling, comparative statics, simulations and conclusion.Appendix A presents a receding-horizon algorithm for renewable uncertainty.Source: C. Major Mathematical Symbols and… · p. 3, Table I · p. 3

II. WDP FRAMEWORK AND DISPATCH PROBLEM

  • The WDP model couples fuel-consuming thermal desalination, electricity-consuming reverse osmosis, renewable generation and transactions with water and electric utilities.Prices are announced at the start of each billing period and remain fixed while the WDP schedules across its horizon.Source: §II · pp. 3–4

A. WDP Resources

The WDP combines thermal desalination, which co-produces water and electricity, with reverse osmosis, which uses electricity to produce water. Both technologies are represented with operating limits and conversion relationships.

1) TDP:

  • TDPs consume fuel to produce both water and electricity, with water output linked to fuel consumption.Their operating costs depend on fuel cost and plant efficiency and are modeled as strictly convex, continuously differentiable, and non-decreasing.Source: 1) TDP · p. 4
  • TDP water and electricity outputs are co-produced according to a water-to-electricity production ratio, and TDP water flow has minimum and maximum operating limits.The limits ensure stable operation.Source: 1) TDP · p. 4

2) RODP:

  • RODPs consume electricity to produce water by forcing pressurized seawater or wastewater through a semipermeable membrane.The conversion function is strictly concave, continuously differentiable, and strictly increasing, with production bounded by minimum and maximum operating limits.Source: 2) RODP · p. 4

3) Renewable Energy Sources:

The WDP schedules thermal and reverse-osmosis units around renewables and fixed billing-period prices to maximize profit while meeting water demand. Its dispatch problem has a concave objective, though the electricity payment makes it non-differentiable.

3) Renewable Energy Sources:

  • Renewable output is revealed sequentially in real time, while each setpoint is committed before the billing-period aggregate is known.A receding-horizon implementation uses the optimal dispatch structure without requiring the renewable-generation probability distribution; renewable installation capital costs are excluded because the analysis concerns short-run operation.Source: 3) Renewable Energy Sources · p. 4

1) Water Revenue:

  • The WDP earns water revenue by selling total thermal and RO desalinated water at the announced water price.The price is fixed at the start of the billing period, and water revenue depends on the two units’ water outputs.Source: 1) Water Revenue · p. 4

2) Electricity Payment:

  • The electricity payment uses fixed import and export prices plus a fixed charge, with import priced no lower than export to avoid risk-free arbitrage.Net exchange is positive for importing, negative for exporting, and zero when power-balanced; the formulation also extends to multiple billing periods with time-varying prices.Source: 2) Electricity Payment · p. 4

C. WDP Profit

  • WDP profit equals water revenue minus the electricity transaction and TDP operating costs.Electricity transactions cost the WDP when it imports and generate revenue when it exports; RODP operating costs are accounted for through electricity procurement.Source: C. WDP Profit · pp. 4–5

D. Profit-Maximizing WDP Dispatch

  • The WDP jointly schedules thermal and RO units to maximize profit subject to production, capacity, and water-demand constraints.The formulation assumes minimum output meets demand and has a global optimum because its concave objective is maximized over a convex, compact set.Source: D. Profit-Maximizing WDP Dispatch · p. 5
  • Although the dispatch objective is concave, its electricity payment makes it non-differentiable.The profit function is concave in RO water output and strictly concave in thermal water output when import price is at least export price.Source: D. Profit-Maximizing WDP Dispatch · p. 5

III. OPTIMAL WDP DISPATCH

The optimal dispatch follows renewable-dependent thresholds that coordinate thermal and reverse-osmosis water production with grid exchange. The resulting policy provides import, net-zero, and export modes with continuous transitions, and its thresholds can be computed offline from tariffs and plant conversion functions.

A. Optimal WDP Dispatch Policy

  • Four thresholds divide optimal dispatch into renewable-dependent regions, jointly determining thermal and reverse-osmosis water output.The stated structure assumes π+ ≥ δNZ1 ≥ δNZ2 ≥ π− and known aggregate renewable generation s.Source: A. Optimal WDP Dispatch Policy · p. 5
  • The thresholds ΓIM and ΓEX separate grid import, net-zero operation, and export as aggregate renewables increase.The net-zero region lies between the import and export thresholds; its two internal thresholds distinguish dispatch regions.Source: A. Optimal WDP Dispatch Policy · pp. 5–6

B. Optimal Dispatch Properties and Economical Insights

  • Thresholds depend on tariffs and plant conversion functions, can be computed offline, and are independent of the realized renewable generation.A real-time receding-horizon policy that accounts for renewable uncertainty is described in Appendix A.Source: B. Optimal Dispatch Properties and… · p. 6
  • Under the standard tariff conditions, RODP dispatch is independent of grid prices, while TDP output decreases and RODP output increases with renewables.In net-zero operation, the two technologies interchangeably use available renewables to maintain energy balance.Source: B. Optimal Dispatch Properties and… · p. 6
  • In the net-zero region, TDP and RODP adjust continuously to balance generation and load, avoiding discrete mode switches.As renewables increase, the RODP remains at its minimum, then adjusts alongside the TDP, and finally reaches its maximum while TDP output continues to fall.Source: B. Optimal Dispatch Properties and… · p. 6, A. Optimal WDP Dispatch Policy · p. 6

C. Special Tariff Cases

When grid tariffs fall outside the RODP’s marginal water-value range, the RODP remains at a bound and the TDP follows a two-threshold rule. Single-technology plants also have simpler dispatch policies, with their operation depending on available technology and the relevant tariffs.

C. Special Tariff Cases

  • When both grid prices exceed the RODP’s marginal water value, it idles; when both are below that value, it saturates at its maximum.In either case, RODP operation is independent of renewable generation, leaving the TDP to follow a two-threshold rule.Source: C. Special Tariff Cases · p. 6
  • If a grid price lies inside the RODP’s marginal water-value range, the RODP can operate at an interior setpoint and the dispatch generalizes to two or three thresholds.The threshold count depends on which of the import and export prices lies inside that range.Source: C. Special Tariff Cases · p. 6

A. Special Technology Cases: RODP-Only and TDP-Only

  • An RODP-only WDP follows a two-threshold policy, while a TDP-only WDP’s optimal dispatch is independent of renewables.The RODP-only plant acts as a controllable load under net metering; the TDP-only setpoint depends on export and water prices.Source: A. Special Technology Cases: RODP-Only… · pp. 6–7

B. WDP Profits under Optimal Dispatch

The paper characterizes how optimal profit changes with renewable generation, tariffs, and technology parameters, then illustrates dispatch and profit under a realistic plant setup. It also compares the optimal policy with max-RODP and passive-TDP benchmarks, although the supplied passages do not report comparative numerical outcomes.

B. WDP Profits under Optimal Dispatch

  • Optimal profit increases with renewables at a non-increasing rate, with its marginal increase ranging from the import price in import mode to the export price in export mode.Within net-zero operation, the marginal rate decreases continuously across boundary and interior regions; profit is non-increasing in π+ and non-decreasing in π−, πw, and the specified conversion functions.Source: B. WDP Profits under Optimal Dispatch · p. 7

V. NUMERICAL RESULTS

  • The study models a natural-gas thermal plant, reverse-osmosis units, and solar PV, with hourly tariffs and 10-minute dispatch.The 45 MW Jubail PV array produced 77,276 MWh in 2019, with peak hourly output of 37.6 MWh.Source: §V · p. 7

A. WDP Dispatch Algorithms

  • The benchmark comparison uses the same resources and water and electricity tariffs for all dispatch algorithms.The algorithms are optimal joint dispatch, max-RODP, and passive-TDP; Figure 5 plots hourly dispatch and profits for each.Source: A. WDP Dispatch Algorithms · p. 7, Figure 5 · p. 8

1) Max-RODP Algorithm:

  • The max-RODP algorithm fixes RODP at maximum and co-optimizes renewables with the TDP.It treats the RODP as saturated throughout the dispatch horizon.Source: 1) Max-RODP Algorithm · p. 7

2) Passive-TDP Algorithm:

  • The passive-TDP algorithm dispatches the TDP at its export-price setpoint, leaving the RODP to follow a two-threshold policy.The TDP is scheduled as if the plant were an independent water and power producer and always faced π−.Source: 2) Passive-TDP Algorithm · p. 8

B. WDP Daily Dispatch

The optimal dispatch coordinates thermal desalination and reverse osmosis (RO) with renewable availability, shifting between imports, net-zero operation, and exports. Its advantage over max-RODP depends strongly on water price, while the framework also supports tractable renewable-uncertainty dispatch.

B. WDP Daily Dispatch

  • Daily profits were $142,158 under joint dispatch, $135,886 under max-RODP, and $132,996 under passive-TDP.Under max-RODP, only the TDP adjusted to renewable availability, leading to imports of 9.6 MWh in every solar-free offpeak hour.Source: B. WDP Daily Dispatch · p. 8

C. Net Electricity Exchange

  • Under optimal dispatch, 41% of annual hours were net-zero and imports totaled 2,027 MWh, versus 35,049 MWh of imports under max-RODP.Both policies exported roughly 17,594 MWh annually during hours 9–15; the optimal policy's import/net-zero/export pattern held in every month.Source: C. Net Electricity Exchange · p. 9, Figure 7 · p. 9

D. Sensitivity of Profit to Tariff and Technology Parameters

  • A 50% increase in water price raised daily profit by approximately $123.5k, while a 50% decrease reduced it by about $97.4k.The increase in water revenue (+$160.2k) dominated the other effects despite higher TDP cost.Source: D. Sensitivity of Profit to Tariff and… · p. 9, Figure 8 · p. 9
  • A 50% increase in the RO factor raised profit by $54.9k, while reducing the TDP co-production ratio by 50% raised it by $70.3k.The RO-factor gain reflected higher water sales and lower TDP and grid-import costs; the co-production-ratio decrease mainly raised electricity revenue by $57.5k.Source: D. Sensitivity of Profit to Tariff and… · p. 9

E. Optimality Gap of the Max-RODP Algorithm

  • The optimal policy's profit advantage over max-RODP stayed below about 13% as renewable capacity varied, but depended much more strongly on water price.The gap reached zero near $1/m3, where maximizing RO output was optimal, and increased sharply as water price fell.Source: E. Optimality Gap of the Max-RODP… · p. 9, Figure 9 · p. 9

VI. CONCLUSION

  • The analytical dispatch has five renewable-dependent regions, with intermediate net-zero modes allocating surplus renewables first to reduced thermal output, then increased RO output, and finally further thermal displacement.A receding-horizon implementation accommodates renewable uncertainty while preserving the threshold structure.Source: §VI · p. 10

APPENDIX A WDP DISPATCH UNDER RENEWABLE UNCERTAINTY

The appendix adapts the one-shot threshold policy for sequential renewable observations by using forecasts in a receding-horizon dispatch. Its per-step problems retain a shifted threshold rule, and simulations show very small losses that fall with more frequent decisions.

1) Information structure and per-step problem:

  • Renewables arrive sequentially, so period setpoints must be chosen before future generation is observed; the exact dispatch is a multistage stochastic program that is generally intractable.The plant uses forecasts for the remaining periods and exercises only the current period's setpoints before observing new generation and re-solving.Source: Appendix A · p. 11, 1) Information structure and per-step… · p. 11

2) Shifted threshold policy:

  • Each receding-horizon step solves its remaining-horizon problem exactly using a shifted four-threshold rule.The rule replaces the one-shot policy's aggregate renewable value with its projection for the remaining horizon and shifts the thresholds accordingly.Source: Appendix A · p. 11, 2) Shifted threshold policy · p. 11

3) Properties and implementation:

  • With exact forecasts, the receding-horizon policy reproduces the one-shot optimum; its suboptimality is governed entirely by renewable forecast error.The thresholds retain their offline character, with only the running offset and horizon length changing across periods.Source: 3) Properties and implementation · p. 11

4) Numerical illustration:

  • At 20% forecast error, Algorithm 1 achieved 99.6% of optimal profit with hourly commitment and 99.97% with six decisions per hour.The representative-day illustration uses 10-minute decisions and shows deviations from the perfect-information trajectory only when forecast error moves projected renewables across a threshold.Source: 4) Numerical illustration · p. 12, Figure 10 · p. 12
  • At 20% forecast error, mean daily regret fell from about $550/day with hourly commitment to $40/day at 10-minute decisions and $2/day at 1-minute decisions.Moving from hourly to 10-minute decisions removed over 90% of the loss; the further refinement saved another $38/day.Source: 4) Numerical illustration · p. 12

APPENDIX B MATHEMATICAL PROOFS

The proofs establish concavity of the scheduling objective and derive the threshold-based dispatch by solving convex subproblems for import, export, and net-zero operation. The regional solutions are then combined into a globally optimal policy.

A. Proof of Lemma 1

  • The profit is concave in thermal and RO water outputs: water revenue is affine, electricity payment is convex, and negative fuel cost is concave.The argument uses the assumed conversion-function properties; strict concavity in thermal output follows from strict convexity of its fuel cost.Source: A. Proof of Lemma 1 · p. 12

B. Proof of Theorem 1

  • Theorem 1's proof partitions dispatch into import, export, and net-zero subproblems, whose KKT conditions are necessary and sufficient under Slater's condition.The import and export cases apply the corresponding price conditions and project setpoints onto technology bounds.Source: B. Proof of Theorem 1 · pp. 12–13

Solution to PEX:

The proof obtains the dispatch by applying stationarity and bound conditions across export and net-zero regimes, then selecting among feasible regions using the renewable thresholds.

Solution to PEX:

  • In the export regime, the TDP setpoint is determined at the export price π−, with the RODP at its upper bound.The export solution saturates the RODP upper bound under the stated price condition.Source: Solution to PEX · p. 13
  • For net-zero dispatch, the TDP setpoints depend on the dual price, while the RODP setpoint and dual are jointly fixed by stationarity and net-zero balance.The proof distinguishes interior, lower-bound, and upper-bound net-zero subsegments.Source: Solution to PEX · p. 13
  • The lower net-zero boundary holds the RODP at its lower bound and chooses the dual to satisfy net-zero balance; the upper boundary analogously holds it at its upper bound.The TDP setpoints in both boundary cases retain the threshold-rule form.Source: Solution to PEX · p. 14
  • The optimal regime is selected by five aggregate-renewable ranges: import, lower net-zero, interior net-zero, upper net-zero, and export.Their intervals are separated by ΓIM, ΓNZ1, ΓNZ2, and ΓEX.Source: Solution to PEX · p. 14

C. Proof of Corollary 1

The proofs characterize dispatch through import, net-zero, and export regimes. In the TDP-only problem, the same three-region analysis yields price-dependent TDP setpoints and thresholds.

C. Proof of Corollary 1

  • In the NZ regime, aggregate net consumption is zero by construction.The proof distinguishes this from the IM and EX regimes, where the RODP water setpoint equals its bound.Source: C. Proof of Corollary 1 · p. 14

D. Proof of Proposition 1

  • When the faced-price terms force the RODP to a bound, its setpoint saturates the upper or lower bound throughout the operating range.The proof establishes the two cases using strict monotonicity of f′_r and non-negative bound multipliers.Source: D. Proof of Proposition 1 · p. 14
  • With the RODP setpoint fixed, the residual TDP problem applies the same IM/NZ/EX analysis, with no RODP degrees of freedom.In NZ, the dual lies in [π−, π+] and is pinned by the net-zero balance.Source: D. Proof of Proposition 1 · p. 14
  • The TDP setpoints are ewh_t(π+) in IM, ewh_t(π−) in EX, and ewh_t(δσ) across the regimes.The corresponding thresholds match those in Proposition 1.Source: D. Proof of Proposition 1 · p. 14

E. Extended Proposition 1: Interior RODP

The extended proposition characterizes dispatch when a grid price falls inside the RODP price band. The accompanying proofs give special-case dispatch results and establish how optimal profit varies with technology.

E. Extended Proposition 1: Interior RODP

  • When a grid price lies inside (δNZ2, δNZ1), the RODP setpoint is clamped to its bounds or is interior, depending on the faced price.The setpoint is interior between the thresholds; the threshold locations determine which regime segments remain.Source: E. Extended Proposition 1: Interior RODP · p. 15
  • Case A permits imports while producing RO water, Case B is its export-side mirror, and Case C has an interior RODP throughout NZ.Case C reduces to a two-threshold policy because both faced prices are interior.Source: E. Extended Proposition 1: Interior RODP · p. 15

F. Proof of Corollary 2

  • With only the RODP, strict concavity makes per-period setpoints coincide; in the linear conversion case, any feasible allocation summing to aggregate renewable input is optimal.The uniform optimum is f_r(1⊤g/T); allocating q^r_t=g_t is generally sub-optimal under strict concavity.Source: F. Proof of Corollary 2 · p. 15
  • With only the TDP, the WDP always operates in EX and sets wh*_t to ewh_t(π−) for every period.Source: F. Proof of Corollary 2 · p. 15

G. Proof of Theorem 2

  • Optimal profit is non-decreasing under pointwise improvements in either conversion function, and the increase is strict when the corresponding plant operates within its bounds.Improved conversion relaxes the relevant input-output relationship without tightening constraints.Source: G. Proof of Theorem 2 · p. 16

H. Proof of Proposition 2

The proof reduces the remaining-horizon problem to the original scheduling form using effective aggregate renewables. Its net-zero dual has a unique root, and the demand floor remains slack.

H. Proof of Proposition 2

  • The reduced problem over the remaining horizon has unchanged prices, conversion functions, and bounds, with effective aggregate renewables.Theorem 1 applies to this reduced problem.Source: H. Proof of Proposition 2 · p. 16
  • The demand floor stays slack in the remaining-horizon problem under the sizing condition.The proof bounds remaining water production using water already exercised and the sizing condition.Source: H. Proof of Proposition 2 · p. 16
  • The net-zero balance pins a unique dual root because its left-hand side is strictly decreasing in μ.Source: H. Proof of Proposition 2 · p. 16
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