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Markov-Based Modelling for Reservoir Management: Assessing Reliability and Resilience

M. L. Gámiz, N. Limnios, D. Montoro-Cazorla, M. C. Segovia-García

Abstract

This paper develops a comprehensive Markov-based framework for modelling reservoir behaviour and assessing key performance measures such as reliability and resilience. We first formulate a stochastic model for a finite-capacity dam, analysing its long-term storage dynamics under both independent and identically distributed inflows, following the Moran model, and correlated inflows represented by an ergodic Markov chain in the Lloyd formulation. For this finite case, we establish stationary water balance relations and derive asymptotic results, including a central limit theorem for storage levels. The analysis is then extended to an infinite-capacity reservoir, for which normal limit distributions and analogous long-term properties are obtained. A continuous-state formulation is also introduced to represent reservoirs with continuous inflow processes, generalizing the discrete-state framework. On this basis, we define and evaluate reliability and resilience metrics within the proposed Markovian context. The applicability of the methodology is demonstrated through a real-world case study of the Quiebrajano dam, illustrating how the developed models can support efficient and sustainable reservoir management under hydrological uncertainty.

Markov-Based Modelling for Reservoir Management: Assessing Reliability and Resilience

Abstract

This paper develops a comprehensive Markov-based framework for modelling reservoir behaviour and assessing key performance measures such as reliability and resilience. We first formulate a stochastic model for a finite-capacity dam, analysing its long-term storage dynamics under both independent and identically distributed inflows, following the Moran model, and correlated inflows represented by an ergodic Markov chain in the Lloyd formulation. For this finite case, we establish stationary water balance relations and derive asymptotic results, including a central limit theorem for storage levels. The analysis is then extended to an infinite-capacity reservoir, for which normal limit distributions and analogous long-term properties are obtained. A continuous-state formulation is also introduced to represent reservoirs with continuous inflow processes, generalizing the discrete-state framework. On this basis, we define and evaluate reliability and resilience metrics within the proposed Markovian context. The applicability of the methodology is demonstrated through a real-world case study of the Quiebrajano dam, illustrating how the developed models can support efficient and sustainable reservoir management under hydrological uncertainty.
Paper Structure (16 sections, 10 theorems, 34 equations, 5 figures, 2 tables)

This paper contains 16 sections, 10 theorems, 34 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stochastic properties of a reservoir system with discrete input
  3. The finite case
  4. Model description
  5. The long-term performance of the reservoir system
  6. Some limit results for the storage volume in a finite capacity dam
  7. The semi-infinite case
  8. Stochastic properties of a reservoir system based on the Moran model with continuous input
  9. Usual performance measures for reservoir management
  10. Reliability measures
  11. Resilience measures
  12. A real case-study: The Quiebrajano reservoir
  13. The Moran model for the Quiebrajano case
  14. Analysis of stored volume
  15. Performance and resilience metrics
  16. ...and 1 more sections

Key Result

Lemma 1

If $0<p_0$ and $p_0+p_1<1$, then the Markov chain $\{Z_n; n\geq 0\}$ is ergodic, and its stationary distribution is given by $\pi_n= a_n \pi_0$, where where $a_0=1$ and $\pi_0$ is the corresponding normalizing constant.

Figures (5)

  • Figure 1: A Markov model for reservoirs.
  • Figure 2: Water balance during the year 2007 adjusted empirically.
  • Figure 3: Probability that the Quiebrajano reservoir remains non-empty throughout a 25-year period. The initial conditions are set to the ones registered in the dataset, i.e. the initial volume of stored water is at level $I_1=(1/2,3/2]$.
  • Figure 4: Probability that the Quiebrajano reservoir is empty (not necessarily for the first time) at year $n$ regardless of previous empty periods, over a 25-year simulation period. The initial volume of stored water is at level $I_1=(1/2,3/2]$, that is, the initial conditions are set to the ones registered in the dataset.
  • Figure 5: Observed storage volumes (grey dots) and expected values estimated from the Markov chain model (solid line). The model reproduces the long-term mean behaviour of the system, with the expected value stabilizing near the empirical average of the observed data, that is, $\overline{V}= 15.0526\ hm^3$.

Theorems & Definitions (26)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Proposition 1
  • proof
  • Remark 3
  • Proposition 2
  • ...and 16 more