Modelling systems of reservoirs using structured Markov chains

The management of three connected reservoirs for the capture, storage and supply of urban stormwater is modelled using a pump-to-fill policy that minimises the volume of water lost to overflow. A discrete state Markov model is used with constant daily demand from the supply reservoir and stochastic inflow to the capture reservoir. The pump-to-fill policy is completely deterministic and depends only on the current volume in the supply, storage and capture reservoirs. By judicious ordering of the states the very large transition matrix is shown to possess a nested block upper Hessenberg structure. Standard censoring methods reduce the analysis of the system to a characteristic sequence of full-to-full transitions for the supply reservoir. The nested block structure of the original transition matrix is captured using special recursive algebraic procedures that enable a further reduction to a sequence of simultaneous full-to-full transitions for the supply and storage reservoirs. Capabilities of the model are demonstrated through application to a hypothetical three-reservoir network for the capture and supply of water. The methods proposed in this paper could be used to calculate the steady-state probabilities for three-reservoir storage systems and could assist projections for future water supply capabilities. This paper also provides insight into how the analysis could be extended to systems of more than three reservoirs.