Robust Optimal Booster Disinfectant Injection in Water Systems under Uncertainty

Abstract

Water distribution systems (WDSs) require high-quality water for safe consumption. To achieve this, disinfectants such as chlorine are often added to the water in the system. However, it is important to regulate the levels of chlorine to ensure they fall within acceptable limits. The higher limit is to control disinfection by-products, while the lower limit is established to guarantee that the water is free of organic contaminants. The rate at which chlorine reacts within the pipes is affected by various factors, such as the type of pipe, its age, the pH level of the water, the temperature, and others. This variability makes it challenging to accurately model water quality in WDSs, which can impact the optimal rate of booster injection. To address the uncertainty in the chlorine reaction rate, the current research proposes a robust counterpart reformulation of the booster chlorination scheduling problem, which considers the chlorination reaction rate as uncertain. The proposed reformulation was tested on two benchmark WDSs and analyzed with a thorough sensitivity analysis. The results showed that as the size of the uncertainty set increased, the injection mass also increased. This reformulated approach can be applied to any WDS and provides a way to obtain optimal scheduling within the desired protection levels.

Booster Disinfection Scheduling under Uncertainty in Water Distribution Systems: Approximate Robust Reformulation Approach

“…The procedure for sampling the boundary points in the ellipsoidal set is explained in the appendix section. The response matrices are estimated from EPANET simulations, the methodology for estimation of the response matrices through extended period water quality simulations is explained in detail in (Boccelli et al, 1998;Boindala et al, 2023a;Köker and Altan-Sakarya, 2015;Tryby et al, 1999). After the estimation of the response matrices, the problem defined in (Equations 13-18) is solved using a non-linear solver 'fmincon' and the robust booster schedule for booster disinfection injection is estimated that ensures minimal deviations from the desired residual disinfectant concentration.…”

“…This robustness is especially advantageous when managing noisy input data or grappling with substantial, difficult-to-quantify uncertainties in the system. RO's versatility has been demonstrated in addressing uncertainty within water distribution systems (Boindala et al, 2023a;Pankaj et al, 2022;Perelman et al, 2013;Schwartz et al, 2016).…”

“…A box uncertainty set is the ∞-norm unit circle in n-dimensions. For the box uncertainty set the extreme uncertain parameter values k b , which occur at the corners of the box, correspond to extreme values of the response matrices (Boindala et al, 2023a). Furthermore, the different components of uncertain parameter can take values independent of each other.…”

“…The intricate nature of WDS is characterized by an underlying uncertainty which also poses a substantial obstacle to conventional deterministic optimization approaches used for both the design and operation of these systems (Boindala et al, 2023a;Dandy et al, 2022). This uncertainty stems from a multitude of factors, encompassing various dimensions, such as fluctuations in demand patterns, supply dynamics, and the inherent variability of parameters within water quality models (Pandey et al, 2020).…”