Chance-constrained model predictive control for drinking water networks

Abstract

This paper addresses a chance-constrained model predictive control (CC-MPC) strategy for the management of drinking water networks (DWNs) based on a finite horizon stochastic optimisation problem with joint probabilistic (chance) constraints. In this approach, water demands are considered additive stochastic disturbances with non-stationary uncertainty description, unbounded support and known (or approximated) quasi-concave probabilistic distribution. A deterministic equivalent of the stochastic problem is formulated using Boole's inequality to decompose joint chance constraints into single chance constraints and by considering a uniform allocation of risk to bound these later constraints. The resultant deterministic-equivalent optimisation problem is suitable to be solved with tractable quadratic programming (QP) or second order cone programming (SOCP) algorithms. The reformulation allows to explicitly and easily propagate uncertainty over the prediction horizon, and leads to a cost-efficient management of risk that consists in a dynamic back-off to avoid frequent violation of constraints. Results

Reliability–based economic model predictive control for generalised flow–based networks including actuators’ health–aware capabilities

“…To circumvent the aforementioned limitation and determine optimal dynamic safety stocks, the chance-constrained MPC strategy described by Grosso et al (2014) is used here. Such a strategy relaxes the original state constraint (2a) by using probabilistic statements, leading to the form of the so-called (probabilistic) chance constraint, i.e.,…”

“…A lower δ x implies a harder constraint. As discussed by Grosso et al (2014), the constraint (4) is difficult to be addressed since it lacks analytic expressions due to the multivariate probability distributions involved. Nevertheless, there are tractable approximations that can be derived if each element of the demand vector follows a log-concave univariate distribution with a known stochastic description; see the work of Grosso et al (2014, Section 3) for details.…”

“…In fact, the common action to deal with demand uncertainty for such an approach is to heuristically define a conservative constant safety threshold x s,k = βx max for all k, with β ∈ (0, 1), and incorporate a constraint of the form x k ≥ x s,k (or a softened version of it). Grosso et al (2014). It incorporates robustness only for demand uncertainty by replacing the state deterministic constraints with chance constraints.…”

“…Specifically, the two-layer control architecture proposed by Grosso et al (2012) is here simplified and reduced to a less conservative single-layer stochastic approach following the chance-constrained MPC approach presented by Grosso et al (2014). The actuator-health management policy used in this paper follows the one introduced by Pereira et al (2010), but considers stochastic actuator-degradation models and probabilistic actuator lifetime constraints rather than deterministic ones.…”

Non-linear economic model predictive control of water distribution networks

“…Related research for WDNs has been carried out in the past several years [6,7,8,9,10,11,12,13,14,15]. These research works are focused on finding the optimal operation on 35 the WDN in order to achieve the desired control objectives.…”