Linear Versus Nonlinear (Convex and Concave) Hedging Rules for Reservoir Optimization Operation

Xiang Zeng, Jay R. Lund, Ximing Cai

Research output: Contribution to journalArticlepeer-review


Hedging rules of reservoir operations decide the timing and magnitude of current water delivery reduction and consequent carryover water storage conservation based on the tradeoff between the utility of current and future water use. Different forms of hedging can be used for reservoir release decision. This study presents general optimality conditions for hedging with convex, concave, and linear relationships between reservoir water availability and release, using criteria based on the relative value of marginal value of risk tolerance (MVRT) with respect to current reservoir release versus carryover storage. MVRT is quantified to measure reservoir operators' risk tolerance change (sensitivity) to the change of reservoir release or carryover storage, reflecting their attitudes to the risk with current release and that with the carryover storage (i.e., future release), as well as their perception of future water availability uncertainty. Higher, equal, or lower MVRT with current release than that with carryover storage for future release corresponds to convex, linear or concave hedging. Various levels of risk tolerance toward hydrologic uncertainty underlie different hedging types for reservoir release decisions. Specifically, negative, positive, and null risk premiums (or risk seeking, risk aversion, and risk neutrality) result from convex, concave, and linear hedging rules, respectively. In general, high uncertainty will move the convex or concave curve closer to the linear curve. The MVRT-based criteria for optimal hedging policies are illustrated through a real-world case study.

Original languageEnglish (US)
Article numbere2020WR029160
JournalWater Resources Research
Issue number12
StatePublished - Dec 2021


  • convex and concave
  • linear
  • marginal value of risk tolerance (MVRT)
  • nonlinear
  • risk premium

ASJC Scopus subject areas

  • Water Science and Technology


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