Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network

Abstract

In this paper, we have investigated minimization of total cost to pump a given flow rate from any number (n) of wells up to a water tank, under steady-state flow conditions. Regarding groundwater flow, we have considered infinite or semi-infinite aquifers, to which the method of images applies. Additional regional groundwater flow can be taken into account, too. The pipe network connecting the wells to the tank can include junctions at the locations of the wells only. Moreover, all pumps have equal efficiency. We have derived a new analytical formula, which holds at the critical points of the total cost function. Based on this formula, we derived a system of n equations and n unknowns, to calculate the well flow rate combinations which correspond to the critical points of the total cost function. The n-1 equations are 2nd degree polynomials, while the remaining one is linear, expressing the constraint that the sum of well flow rates must be equal the required total flow rate. The solution of the system can be achieved using commercial solvers. Moreover, we have concluded that there is one feasible solution that minimizes the total cost. Finally, we present a tabulation process to facilitate the use of solvers and we provide and discuss two illustrative examples.

Pumping Station Design in Water Distribution Networks Considering the Optimal Flow Distribution between Sources and Capital and Operating Costs

“…9) are exposed. The OPEX costs are shown according to Equation (13). The last row details the annualized costs according to Equation (2) for each PS.…”

“…Finally, to calculate the OPEX, the cost of the total electrical power consumed by all pumps running in the WDN during time step N t is determined using Equation (13).…”

“…The differences between these models lie in the decision variables used to assemble their object functions and constraints. In this regard, some previous researchers have focused their efforts on optimizing the location of each PS, maximizing energy production [12,13], minimizing losses by installing turbines to recover energy [14], or using the position and setting of a PS, in addition to the working statuses of VSPs, as the decision variables [15]. Other researchers have focused on analyzing the PS capacity to optimize the design and operational costs of stations; the authors of [16] introduced a model with binary variables for each design and operation option available in the face of a small number of demand scenarios.…”

Cost minimization of groundwater supply to a central tank

“…In a previous work Nagkoulis & Katsifarakis (2020) have proved that the operational cost K oper (K oper ¼ K dr þ K trans ) to pump a given total flow rate q tot from n wells up to a tank through an existing pipe network is minimized, when the following conditions hold:…”

“…Compared with that case, K constr will be always smaller, but K dr may be larger. To find the minimum cost we calculate the optimal flowrate distribution, as described by Nagkoulis & Katsifarakis (2020) and briefly outlined in the previous section, in order to find the respective K dr . Comparison for different data sets appear in a following section.…”