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Stochastic Co-design of Storage and Control for Water Distribution Systems

Ye Wang, Erik Weyer, Chris Manzie, Angus R. Simpson, Lisa Blinco

TL;DR

This work develops a tractable stochastic co-design framework for water distribution systems that jointly optimizes storage tank size and a price-threshold control policy under stochastic demands and electricity prices. By discretizing demands, flows, and volumes into a finite-state Markov chain, the authors derive a long-run operating cost that converges with probability 1 to an expected cost, enabling efficient optimization via SPSA. The method is demonstrated through three illustrative examples and a real-world South Australia case study, showing substantial potential savings and the ability to improve operation of existing infrastructure. The approach offers practical benefits for greenfield design and brownfield operations, while highlighting robustness to certain distributional assumptions and outlining directions for future refinements of the stochastic models.

Abstract

Water distribution systems (WDSs) are typically designed with a conservative estimate of the ability of a control system to utilize the available infrastructure. The controller is designed and tuned after a WDS has been laid out, a methodology that may introduce unnecessary conservativeness in both system design and control, adversely impacting operational efficiency and increasing economic costs. To address these limitations, we introduce a method to simultaneously design infrastructure and develop control parameters, the co-design problem, with the aim of improving the overall efficiency of the system. Nevertheless, the co-design of a WDS is a challenging task given the presence of stochastic variables (e.g. water demands and electricity prices). In this paper, we propose a tractable stochastic co-design method to design the best tank size and optimal control parameters for WDS, where the expected operating costs are established based on Markov chain theory. We also give a theoretical result showing that the average long-run operating cost converges to the expected operating cost with probability~1. Furthermore, this method is not only applicable to greenfield projects for the co-design of WDSs but can also be utilized to improve the operations of existing WDSs in brownfield projects. The effectiveness and applicability of the co-design method are validated through three illustrative examples and a real-world case study in South Australia.

Stochastic Co-design of Storage and Control for Water Distribution Systems

TL;DR

This work develops a tractable stochastic co-design framework for water distribution systems that jointly optimizes storage tank size and a price-threshold control policy under stochastic demands and electricity prices. By discretizing demands, flows, and volumes into a finite-state Markov chain, the authors derive a long-run operating cost that converges with probability 1 to an expected cost, enabling efficient optimization via SPSA. The method is demonstrated through three illustrative examples and a real-world South Australia case study, showing substantial potential savings and the ability to improve operation of existing infrastructure. The approach offers practical benefits for greenfield design and brownfield operations, while highlighting robustness to certain distributional assumptions and outlining directions for future refinements of the stochastic models.

Abstract

Water distribution systems (WDSs) are typically designed with a conservative estimate of the ability of a control system to utilize the available infrastructure. The controller is designed and tuned after a WDS has been laid out, a methodology that may introduce unnecessary conservativeness in both system design and control, adversely impacting operational efficiency and increasing economic costs. To address these limitations, we introduce a method to simultaneously design infrastructure and develop control parameters, the co-design problem, with the aim of improving the overall efficiency of the system. Nevertheless, the co-design of a WDS is a challenging task given the presence of stochastic variables (e.g. water demands and electricity prices). In this paper, we propose a tractable stochastic co-design method to design the best tank size and optimal control parameters for WDS, where the expected operating costs are established based on Markov chain theory. We also give a theoretical result showing that the average long-run operating cost converges to the expected operating cost with probability~1. Furthermore, this method is not only applicable to greenfield projects for the co-design of WDSs but can also be utilized to improve the operations of existing WDSs in brownfield projects. The effectiveness and applicability of the co-design method are validated through three illustrative examples and a real-world case study in South Australia.
Paper Structure (24 sections, 1 theorem, 49 equations, 16 figures, 5 tables, 1 algorithm)

This paper contains 24 sections, 1 theorem, 49 equations, 16 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Stochastic Co-design Optimizations based on Finite-state Markov Chain
  4. Quantized Demands, Pumping Flows and Volumes
  5. Finite-state Markov Chain of Volume Evolution
  6. States with no pumping
  7. States with pumping based on $\alpha_{\kappa}(i\Delta x)$
  8. States with enforced pumping
  9. Stochastic Co-design Formulation
  10. Capital Cost
  11. Operating Cost
  12. Stochastic Co-design Algorithm
  13. Illustrative Examples
  14. State-independent Price Threshold and Constant Demand
  15. State-dependent Price Thresholds and Constant Demand
  16. ...and 9 more sections

Key Result

Theorem 1

For a given state $(i,\kappa)$, assume that the cost function $\ell_\kappa(\cdot, \alpha_{\kappa}(i \Delta x), i \Delta x)$ has finite second-order moment. Let Then, using the control policy eq:pumping flow under Assumptions assump:demands-assump:irr, it holds that for any distribution of initial state,

Figures (16)

  • Figure 1: The topology of an aggregated water distribution system.
  • Figure 2: Labeled water volumes in the tank and associated control actions.
  • Figure 3: The states in the Markov chain.
  • Figure 4: States and transition probabilities with no pumping.
  • Figure 5: States and transition probabilities with pumping based on $\alpha_{\kappa}(i \Delta x)$.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2