State-transition matrices as an analysis and forecasting tool applied to water quality in reservoirs

João Marcos Carvalho,Tobias Bleninger

doi:10.1590/2318-0331.262120210072

Abstract

ABSTRACT Water reservoirs have the function to control the temporal variability of the water availability, thus bringing greater security over these resources. The water quality of these systems must be adequate for their multiple uses, and one of the main tools to understand it, is mathematical modelling. Given the importance of the water quality, the goal of this paper is to develop an analysis that takes into account the randomness of the variables that affect the thermal and/or biochemical regimes of a reservoir. For this, it is proposed a combination of deterministic and statistical analysis, where the probabilities of occurrence of a given event are considered. Difficult factors, such as the lack of data on the water quality and other variables, were considered, which increases the replicability of the method. The research method is divided into three groups: Modelling, Scenarios and Compilation of these scenarios. Through modelling, a base layout is created, enabling the use of scenarios, which are statistically analysed, and compiled into a state-transition matrix. With this, a more robust tool to understand the dynamics of water quality in a system is obtained, since it is not heavily dependent on field measurements and is easily adaptable and replicable.

Highlights

Reservoirs in general are fundamental systems for the development of society, since they increase the availability of a given resource, facilitating its management in the face of natural variability
It is possible to use n different inflows to n volume scenarios, in order to allow a view of the general behavior of the system. When organizing these n scenarios and their probabilities in a matrix, a matrix that can indicate the probability that the reservoir is with a volume i and transits to a volume j is obtained, and this new matrix is the state-transition matrix of volumes for a given reservoir (Figure 1)
The interpretation can be done as follows: The probability that the reservoir will have an average temperature within the range

Summary

Introduction

Reservoirs in general are fundamental systems for the development of society, since they increase the availability of a given resource, facilitating its management in the face of natural variability. Two of the various methods for determining the capacity of regularization and size of a reservoir are those of Moran (1954) and Gomide (1975) Both have similar bases and simplifications, and can be summarized as follows: The chance that a reservoir will fail (“dry up”) depends on three factors - the regulated outflow, the probability distribution of the inflows and the volume of the reservoir. As the only random variable considered in the method is the inflow, the probability of the reservoir’s response to a certain inflow, is equal to the probability of the inflow Through these hypotheses, it is possible to use n different inflows to n volume scenarios, in order to allow a view of the general behavior of the system. When organizing these n scenarios and their probabilities in a matrix, a matrix that can indicate the probability that the reservoir is with a volume i and transits to a volume j is obtained, and this new matrix is the state-transition matrix of volumes for a given reservoir (Figure 1)

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