A multi-state Markov chain model for rainfall to be used in optimal operation of rainwater harvesting systems

Abstract

Consideration of rainfall dynamics can rationalize the operation of rainwater harvesting systems. This study aims at establishing a methodology to construct a Markov chain model for time series of rainfall in temperate climates that optimal operation of rainwater harvesting systems is cast in the framework of stochastic dynamic programming. Due to the seasonal climate, the model is time-varying but stationary in each month. The states of the Markov chain are ranges of rainfall depths in 10 min. Transition probabilities determining the dynamics of the Markov chain are to be estimated for each month. The occurrence of dry states is frequent enough to estimate the transition probabilities from a dry state empirically. While, on the condition that a wet state is observed, the rainfall depth in the next 10 min is assumed to obey to the gamma distribution with two parameters. New formulae, including two exponent parameters, are proposed to relate the conditional mean and variance with the parameters of the gamma distribution. The values of the exponent parameters are identified from sequential searches so that the monthly average rainfall depths in 10 min become consistent with the observed ones. Then, a complete set of transition probabilities achieving the mean-reverting property is obtained to establish the Markov chain model. Examples of operating a hypothetical rainwater harvesting system are presented to demonstrate the utility of the Markov chain model in application to optimal management of water resources and stormwater retention in the framework of stochastic dynamic programming. The mean-reverting smooth transition probabilities also contribute to stabilizing the optimal policy.