Inexact left-hand side two-stage chance-constrained programming for booster optimization in water distribution system

Highlights

• An inexact left-hand-side chance-constrained programming model was proposed.

• Response coefficients were expressed as random variables with normal distribution.

• Effects of decay coefficients and limits on the injection mass were analyzed.

Abstract

In water distribution system (WDS), chlorine is often injected as disinfectant to control the growth of microorganism in WDS. However, the chlorine reacts with organism to form disinfectant byproduct, which can bring risk to human health. As such, the chlorine at nodes in WDS should be kept between acceptable range, which is simulated based on the response at nodes corresponding to unit injection mass at boosters. To deal with the uncertainty in chlorine decay process and lower and upper chlorine concentration limits, an inexact left-hand-side chance-constrained programming (ILCCP) model was proposed in this paper and applied to two WDSs. The response coefficients matrix was expressed as random variables with normal probability distribution in the constraints of lower and upper limits, which was obtained through Monte Carlo simulation by linking with EPANET software. The intervals of injection mass were obtained by solving the ILCCP model with a two-step algorithm. Moreover, the effects of random bulk decay coefficients and interval of chlorine limits on the injection mass were analyzed and compared. The results indicated that the lower bounds of optimal injection mass increased with the rise of probability lever for lower limits, while the upper bounds decreased with the rise of the probability level for upper limits. The results can help managers determine the chlorine injection mass under uncertain scenarios, and can be applied to more complicated WDS to obtain meaningful results.

Introduction

As one of the most important urban facilities, water distribution system (WDS) is designed to meet the customers’ multiple demands including water pressure, flow, and water quality. To keep water quality acceptable in water resources management systems, chlorine was generally applied to inject to WDS as a disinfectant for the inactivation of pathogenic microorganism. During the process of transferring drinking water to consumers in WDS, chlorine acts with organics and inorganics through complex physical, chemical, and microbial actions. The decay of chlorine along WDS leads to less chlorine concentration at nodes, which makes it unable to control the growth of microorganism (Kang and Lansey, 2010). As such, the minimum chlorine concentration is proposed worldwide to satisfy the water quality requirement. On the other hand, a variety of disinfection by-products (DBPs) are formed through the reaction between chlorine and natural organic matter (NOM), which causes adverse health impacts, including bladder cancer, low birth weight, birth defects, stomach and rectal cancers, and miscarriages, etc. (Wang et al., 2018). In addition, surplus chlorine results in unpleasant taste and odor in drinking water (Köker and Altan-Sakarya, 2015). Therefore, maximum chlorine concentration is also regulated worldwide to prevent the formation of DBPs. As such, a trade-off of controlling pathogen and preventing potential long-term health risks should be considered (Boccelli et al., 1998). Generally, minimum and maximum limits of chlorine concentration are set as 0.2 mg/L and 4.0 mg/L, respectively (Goyal and Patel, 2017; Köker and Altan-Sakarya, 2015). The operations of booster disinfection have been investigated by numerous researches with various methods (Xin et al., 2019). To control chlorine concentration at nodes in WDS between the permitted minimum and maximum limits, linear optimization model is usually applied with the objection function of minimizing the injection mass based on superposition theory (Boccelli et al., 1998; Goyal and Patel, 2017). A linear least square (LLS) model was formulated to minimize the sum of the square deviations of residual chlorine at the consumers from a desired target (Propato and Uber, 2004). Multi-objective optimization model was proposed with minimizing the total disinfectant dose and maximizing the volume of water supply within specified residual limits (Prasad et al., 2004). Conjunctive optimal scheduling for minimizing the cost of pumping and chlorine booster design and operation as well as maximizing the injection chlorine dose was extended (Ostfeld and Salomons, 2006). The optimization model can be solved by genetic algorithm (GA) method, non-dominated sorting genetic algorithm (NSGA-II), immune algorithm, which have a wide application in solving WDS problems, including leak detection and calibration, chlorine injection, etc. (Chu et al., 2008; Nouiri, 2017; Ostfeld and Salomons, 2006; Prasad et al., 2004).

Since uncertainty existed in chlorine decay process, it is difficult to simulate accurately chlorine concentration through water quality model. A multitude of stochastic optimization methods have been proposed for water resources decision making under uncertainty. Chance constrained programming (CCP) model is one of the major approaches applied to deal with uncertainty in WDS simulation. The CCP model was applied with consideration of normal and log-normal probability distribution for upper and lower limits (Köker and Altan-Sakarya, 2015). The CCP model can satisfy all the constraints under given reliability level, which solves uncertainty in the right-hand sides of the constraints. However, it cannot deal with the ambiguity in the left-hand-side of the constraints (Guo and Huang, 2008). Therefore, inexact left-hand-side CCP (ILCCP) model was proposed to deal with the uncertainty in the left-hand-side of the constraints, which was applied for stream water quality management, water resources management, etc. (Y. P. Li, Huang and Nie, 2009; Qin and Huang, 2008). An inexact double-sided fuzzy CCP model was formulated to deal with uncertainty in both sides of constraints for agricultural effluent control under uncertainty (Xu and Qin, 2010). The inexact linear programming for both sides of constraints can be solved by scenario-based two-step-method and three-step-method (Cao, 2011; Huang, 2011). To deal with sustainable waste management with two objective functions, a stochastic linear fractional programming was proposed, which can not only deal with two objective functions, but also can deal with uncertain parameters (Zhu and Huang, 2011). An inexact joint-probabilistic CCP with left-hand-side random model was proposed to deal with solid waste management, which can deal with two kinds of uncertainty expressed as intervals and random variables (Sun et al., 2013). An inexact two-stage CCP quadratic programming was developed to deal with uncertainties and nonlinearities in sustainable water quality management (T. Li, Li et al., 2014). A fuzzy-boundary interval programming model was developed to deal with dual uncertainties expressed as crisp intervals and fuzzy-boundary intervals (Liu et al., 2014). A credibility CCP model was developed to manage water resources and control pollution for Heshui River watershed in China (Zhang et al., 2015). However, the uncertainty for booster chlorination in WDS had not been researched sufficiently, especially the chlorine injection mass. The booster injection mass can be determined through water quality simulation, which is closely related to the chlorine decay coefficient. The chlorine decay coefficients are affected by pH value, water temperature, initial chlorine concentration, pipe material, and pipe diameter, etc. The uncertainty in chlorine decay coefficients increases the complexity in optimizing the injection mass in WDS, which makes it more difficult to be solved by conventional optimization models. Thus, the objective of this study is to develop an inexact left-hand-side constrained programming (ILCCP) for handling uncertainties in optimization model effectively, which present chlorine response coefficients as probability distributions and upper and lower limits of chlorine concentration as intervals. The results obtained can provide solutions for water quality management in WDS.

In this paper, an inexact left-hand-side chance constrained programming (ILCCP) model was introduced to minimize the injection mass of WDS in Section 2. The ILCCP model was transformed into sufficient linearization form, which is solved through two-step algorithm, and applied to two WDSs in Section 3. The results obtained through ILCCP model was analyzed and discussed in Section 4. The sensitivity of interval parameters and random variables is performed and compared. In Section 5, the conclusion was drawn.