A genetic algorithm approach is used for optimum design and operation of water distribution systems. Island ring topology is used to achieve solution.

Optimum design and operation refers to the selection of pump type, capacity, and number of units as well as scheduling the operation of irrigation pumps that results in minimum design and operating cost for a given set of demand curves, minimizing total annual cost which consists of operation and maintenance costs and depreciation cost of initial investment.

A pumping station has N different pumps delivering water to an irrigation district from an assumed infinite water source (Figure 1). Each pump may have its own characteristics.

Figure 1. Irrigation system.

Optimization model

Annual total cost (ATC) for pumping set is:

,

where

;       ;

E_k - consumed energy; P_E - unit energy cost; C_i - cost of the pump; CRF - capital recovery factor; K_i - equivalent cost of pump after construction time; CT - construction time (years); T - project’s useful life after installation (years); r - rate of interest.

Consumed energy is introduced with certain constraints as

 

in which  E_k - total annual consumed energy; Q_i,j - discharge from pump; H_i,j - pumping head of pump; e_i,j - efficiency of pump; dt_j - time step on the demand-duration curve; IQ_j - total demand at time step; p - density of water; g - gravitational acceleration.

Note that pump efficiency is a function of pump discharge and pumping head, which is related to the total discharge. 

Solution of the mathematical model
Application

A genetic algorithm method (GA) is used to solve the problem and to find global optimal solution.
Program WAPIRRA was designed for this problem and has wide range of applications.

On Download Demo Page you may free download WAPIRRA demo version.

For a case study, the main pumping station of the Farabi Agricultural and Industrial Region, Iran, is considered. It consists of a 20,000-hectare agricultural land, which is located in Khoozestan province in southwestern of Iran. In this station, demanded water is supplied for agricultural use from the Karoon River to the main lot. Figure 2 shows the demand duration curve, and its discretized scheme, which must be pumped by main pumping station. 

As a matter of interest, a problem with one discharge duration curve, with 12 divisions representing for each month of a year, and four different types of pumps, just like the real case study, was considered to apply.

On Fig.2 (right picture) the results for optimal solution are shown (sign “$” means only currency unit).
Results are relative to the chosen (real case) Standard = 100% (left picture).

It shows sufficient reduce of cost for optimal solution.

You may download free Demo version of Program with results for this example.
Download Demo Program