# Optimization of Water Pressure of a Distribution Network within the Water-Energy Nexus
## 1 Introduction
The beforementioned advancements have led to the widespread utilization of electronic actuators, which are capable of providing better performance and are considered to be an intelligent pressure management method of the WDN in many studies [21]. There are three main categories of electronic control PRV: time-modulated, flow-modulated, and remote node-based modulation. In time-modulated mode, time changes are the basis for applying the pressure adjustment pattern by the operator. In this mode, the flow rate changes are not taken into account. The goal of pressure management in this mode is to supply the anticipated value for upstream pressure [19]. Flow-modulated mode uses the flow through the node for adjusting pressure. In this mode, the flow-pressure relationship adjustment is used to provide the desired amount of flow supplied to the node throughout the pressure management period [22]. In remote node-based modulation mode, one of the network nodes, generally the critical node, is considered as the control node. The pressure is managed in a way that the desired pressure in this control node is provided [23]. Different methods have been employed to specify the best locations for installing PRVs in WDNs. Such methods include probabilistic approaches and deterministic optimization algorithms [24]. For almost all the techniques used for analyzing pressure management, modeling the WDN is the first step. This step is conducted using various software packages, such as EPANET. [PERSON] et al. have also developed a method to evaluate the effects of pressure management in a WDN without the need for WDN modeling [25]. Many articles have realized economic analyses of the operating and maintenance costs associated with WDNs. For example, [PERSON] and [PERSON] [26] proposed a model for calculating the optimum replacement time of pipes in a water network, based on the costs of repairing pipe failures and leaks. They concluded that the optimum replacement time for the city of Athena is 69 years after installation. Their techno-economic model considers many kinds of costs related to the repair and replacement of the trouble-inducing parts of a WDN. [PERSON] and [PERSON] [27] presented a framework that assesses water pipe degradation over time, which increases the operating and maintenance costs of water utility companies throughout the life cycle of pipes. They proposed that the cost-effectiveness of any WDN-related investment could be determined by calculating a risk score for all pipes based on the likelihood of pipe failure. [PERSON] and [PERSON] [28] presented an economic analysis that assesses the consequences of reducing pressure using PRVs or real-time control (RTC) on leakage and burst frequency. They concluded that as long as the operating and maintenance costs are low, there is no need to use PRVs or RTC. When the costs increase, the first option should be to introduce PRVs, then RTC. Energy consumption in water supply systems has led many researchers to look for more integrated approaches to water and energy. In one study, energy consumption in WDNs was evaluated through a life-cycle analysis of water network in three stages (fabrication, use, and disposal), and it was concluded that pipe replacements with a frequency of 50 years result in the lowest amount of energy consumption [29]. In addition, many articles have investigated the water-energy nexus for WDNs, specifically on the energy consumed by pumping stations for water transmission and distribution [30; 31; 32]. The main aim of these studies was to respond to demand through pump scheduling [33]. [PERSON] et al. [34] optimized the performance of a variable-speed pump using an anti-colony algorithm. Using this system, they increased the flexibility of the pumping station, regarding the water demand changes during a day and obtained the pumping schedule that optimized energy costs. Because of the importance of demand variation during daytime hours for pump scheduling, [PERSON] et al.