Analytical Water Shortage Probabilities and Distributions of Various Lead Times for a Water Supply Reservoir

Reducing the negative impacts of water shortages is the primary concern of water supply reservoir operation. This study aims to propose a theoretical framework of water shortage probabilities and distributions of various future lead times for a water supply reservoir based on the supply–demand relationship. Reservoir supply ability is represented by the water availability, which is the sum of storage and inflow. The water availability distribution is obtained by the convolution of storage and inflow distributions since both are random variables. The water shortage probability is thus the probability that the water availability is insufficient to meet the known demand. The water shortage distribution is a flip of the water availability distribution with a right-shifted amount of demand and truncating the infeasible negative water shortage. The Nanhua Reservoir located in southern Taiwan is used as an example to illustrate the proposed methodology. The water shortage probabilities and distributions of various initial storage amounts in any month for future 1- to 6-month lead times are constructed based on the fitted inflow distribution and known monthly demand. Apparently different water shortage probabilities in various months are attributed to the initial storage and inherently different inflow distributions in low- or high-inflow months. Low initial storage induces greater water shortage probabilities in low-inflow months, but approximate null probabilities are noted in high-inflow months due to abundant inflow. The effects of future lead times on water shortage probabilities and distribution also reflect the characteristics of inflow distributions of the current and lead-time months under consideration.