2.1.1. Nonlinear Programming Methodology
This study aims to optimize the number of pumping wells and the pumping rate in the study area to achieve the stated economic and ecological goals. The minimization of total cost is taken as the objective function. The number of wells (
N) in the study area and the flow rate of the wells (
Q) are considered as the decision variables. The flow rate is set to be equal to the annual average value. The supply and demand of water, drawdown, groundwater depth, number of wells, and flow rate are regarded as the constraining conditions. Hence, the model of nonlinear planning for the number of wells and the flow rate can be expressed as follows:
where
Rm is the annual management and maintenance cost (RMB) (
, in which
a is the coefficient of well management and maintenance cost,
C is the well construction fee (RMB), and
N is the number of wells in the study area);
Rd is the depreciation cost designed to recognize quantify the loss of services values that an asset has suffered in any year (
, in which
k is the fixed number of years of depreciation);
Rg is the groundwater cost per year (RMB) (
, in which
Q is the well flow rate (m
3/h),
T is the annual average number of operating days,
t is the daily average number of operating hours for each well (h/d), and
rg is the price per cubic meter of groundwater (RMB/m
3); and
Re is the energy cost per year (RMB) (
[
34], in which
h is the net pumping lift (m), and
is the price per kWh of electricity (RMB/(kw·h))).
Subject to:
(1) Constraint of water resources:
where
is the irrigation requirement per year in the study area (m
3/a) (
, in which
is the irrigation water of crop
i throughout the growth period (m
3/ha·a),
is the irrigation area of crop
i in the study area (ha)),
are available surface water resources used for irrigation (m
3/a), which mainly includes river water and rainwater, and
refers to the water resources for irrigation (m
3/a) (
, in which
is the extractable amount of groundwater (m
3/a),
is the return water demand of irrigation and ecology (m
3/a),
is the industrial water demand (m
3/a),
is the domestic water demand (m
3/a), and
is the required amount of ecological water, which is the total amount of water to maintain the ecological environment (m
3/a)).
(2) Constraint of drawdown:
where
is the drawdown (m), and
is the allowed maximum drawdown (m).
(3) Constraint of groundwater depth:
where
refers to the dynamic change of the groundwater depth from the start moment to the end moment (m);
refers to the groundwater depth at the initial time (m);
refers to the allowed minimum groundwater depth (m); and
refers to the allowed maximum groundwater depth (m).
(4) Constraint of the number of wells:
where
is the number of existing wells in the study area.
(5) Constraint of the well flow rate:
where
is the maximum value of the flow rate of individual wells.
2.1.2. Nonlinear Programming Application
In this study, the pumping-well number optimization model of all equations are represented as follows:
Minimum total consumption:
Subject to the following constraints:
(2) Drawdown:
In Northwest China, one of the significant irrigation water sources is the groundwater in the phreatic water zone. The stability of groundwater resources in a short period can be determined based on the balance between groundwater exploitation and recharge.
After a long period of constant pumping rate, a relatively stable con of depression is formed near the well. Based on the Dupuit assumption, which is commonly used in the modeling of groundwater, we assumed that the phreatic water to the well is approximately horizontal and the discharge of groundwater in different cross-sectional area is equal to the flow rate of the well. The equations for the relationship between the flow-rate and drawdown of pumping-wells can be written as follows [
5,
35,
36,
37,
38], and the detailed derivation process of Equation (12) is demonstrated in the
Appendix A:
Based on Equations (12) and (13), the relationship between
and
can be denoted as:
where
is the well flow-rate without interference from other wells (m
3/h);
is the permeability coefficient (m/h);
is the distance from the bottom elevation of the unconfined aquifer to the phreatic free surface (m);
is the radius of influence of the pumping well (m);
is the radius of the pumping well (m); and
is the abatement coefficient of total discharge.
Thus, the drawdown in this study can be demonstrated as:
(3) Groundwater depth:
Because of the overexploitation of groundwater in the most northwestern areas of China, the constraints are too strict for the groundwater depth. In future planning, the quantity of groundwater exploitation should be lower than the value of groundwater recharge to achieve a gradual rise in the groundwater level. Thus, the maximum value of the difference in groundwater depth between the end moment and start moment must be smaller than zero for several decades to meet the constraint. Thus, Equations (5) and (6) can be changed to the constraint
The balance of groundwater can be presented as
where
is the groundwater discharge (m
3) (
, in which
is the lateral discharge (m
3) and
is the discharge from evaporation (m
3)); and
is the groundwater recharge (m
3) (
, in which
is the lateral recharge (m
3) and
is the recharge due to precipitation (m
3));
is the extent of groundwater recharge; and
is the area of the irrigated district (m
2).
The above equation can be converted to
Thus, the constraining condition of groundwater depth can be expressed as:
2.1.3. Genetic Algorithm
This is a nonlinear programming problem with multivariate and multi-constraint conditions, and cannot be solved using the standard mathematical methods. To solve such problems, it can be converted into a linear problem, or dynamic programming, or the application of adaptive search algorithm, such as genetic algorithm, etc. Due to the robustness of genetic algorithm and the speed of computer operations, genetic algorithms are often the preferred algorithms for nonlinear optimization problems. The genetic algorithms take the objective function as the fitness function to carry out the optimization calculation until it satisfies the convergence condition or reaches evolutionary generation.
The conceptual framework of a genetic algorithm was originally developed by Holland [
20]. It is a highly parallel, stochastic and adaptive search method that implements natural selection and the natural genetic mechanism of biology for reference. Different from traditional optimization algorithms, which propagate a single solution to the optimum one, the genetic algorithms search for optimum solutions by evaluating and successively improving a set of problem solutions, also denoted as generation. Each individual in the generation of solution is independently evaluated, which allows for the parallelization process.
Figure 2 displays the flow diagram of a genetic algorithm. The objective function can be
;
n is the number of decision variables;
is one of the decision variables, which contains a vector of
; and the equation of
is generated, in which
T represents the transpose operator; the number of constraint conditions about inequality and equality are
d1 and
d2, respectively;
and
are the inequality and equality constraint conditions (
i = 1, 2, …,
d1;
j = 1, 2, …,
d2), respectively; and
and
are the lower and upper bounds of
(
k = 1, 2, …,
n), respectively. In this study,
represents minimum total consumption,
respectively represent the number of wells (
N) in the study area and the flow rate of the wells (
Q); there is no equality constraints, so
respectively represent the constraints water resources, drawdown and groundwater depth for
i = 1, 2, 3; corresponding,
and
are the lower and upper bounds of
N,Q (
k = 1, 2), respectively. After defining the optimization model, the operation steps of the algorithm are as follows:
(1) Encoding the target problem
The internal expression (genotype) of chromosome is a genes combination, which determines the external expression of individuals. Binary coding, as one of the encoding works to map from phenotype to genotype, is implemented. Based on the upper bounds (200 m3/h) and lower bounds (0 m3/h) of the independent variable of the wells flow rate, we determined the code length of the binary code (, the accuracy level is 10−2). Then, a set of binary codes were randomly generated, such as b0, b1, …, b10, which were converted to decimal numbers (), and the numbers were mapped to real numbers in the interval ().
(2) Population initialization
To ensure the feasibility and diversity of the individuals in the first generation, the initial individuals are randomly produced according to the constraint conditions.
(3) Determination of fitness
The fitness function is transformed by the object function, and the individuals are evaluated in terms of its fitness value. In the search process, on the basis of the fitness function, the genetic algorithm utilizes the fitness value of each individual in the generation; however, it does not use external information.
(4) Genetic manipulation
The evolution of the population depends on the genetic operators acting on the current population and producing a new generation of the population. The common genetic operators encompass selection, crossover and mutation and play a decisive role in the algorithmic performance. Implicit parallelism and effective utilization of global information are two salient features of genetic algorithms.
(5) Optimal saving strategy
The idea behind an optimal preservation strategy is to retain the good traits of individuals in the parent generation in those of the offspring. In other words, the individual with the greatest fitness in the current population is used to replace the individual with the least fitness after crossover, mutation and other genetic manipulations in the population, and it does not carry through the crossover and mutation processes.
Through the above schemes, the optimal solution set can be obtained.