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Stochastic control of reservoir systems using indicator functions: New enhancements
Author(s) -
Fletcher S. G.,
Ponnambalam K.
Publication year - 2008
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2006wr005269
Subject(s) - discretization , moment (physics) , inflow , quadratic equation , mathematical optimization , gaussian , nonlinear system , current (fluid) , variable (mathematics) , random variable , mathematics , state variable , work (physics) , statistics , mathematical analysis , engineering , mechanical engineering , physics , geometry , classical mechanics , quantum mechanics , mechanics , electrical engineering , thermodynamics
In our previous works, deterministic release policies were considered for the development of approximations of the two lower moments of the storage volume defined by the dynamic equation of the reservoir in discrete time but in continuous state space. Important innovation in that work was the incorporation of the lower and upper bounds of reservoir systems into the dynamic equation for the storage volume using indicator functions. The current work, which also does not use discretization, looks at an extension of previous developments that incorporates standard operating policies, and also a new randomized release policy, both of which make the moments calculations exact under the assumptions that (1) the sum of current random inflow and the previous storage volume can be described by just the two lower moments and (2) only the means and variances of the inflows are known. First‐ and second‐moment expressions are derived for the stochastic storage state variable and include terms for the failure probabilities (probabilities of spills or deficits). Expected values of the storage state, variances of storage, release policies, and failure probabilities are obtained by solving the optimal reservoir operations problem using nonlinear programming. The various statistics thus obtained from this optimization compare extremely well with those obtained from simulation for the single‐reservoir monthly operations problem studied. The exact characterization of the mean and variance of the storage state variable is derived, which is a difficulty in existing formulations based on linear quadratic Gaussian methods. For example, the latter methods have been unable to estimate these moments reasonably accurately, especially for long‐term operations, whereas the traditional storage theory based on discretization brings on the “curse of dimensionality.” The presentation herein is directed to both traditional reservoir storage theorists who are interested in the design of a reservoir where estimating the probabilities of spills and deficits is important and modern reservoir analysts who are interested in multiperiod optimal release decisions in the operations of reservoirs.