1. Introduction
Climate change is a proven fact. In the report of 2007 from IPCC [
1], one can read that global warming is an issue to be dealt with urgently. Temperature will raise, and longer and more frequent draught periods will occur. One of the most affected regions will be the Iberian Peninsula. In the south of Iberia, extreme draught periods are already very frequent. Our study considered Portugal, but it could easily be adapted to another part of the world. In this scenario, it is necessary that the decision makers are able to decide on all issues related to water management.
Irrigation of crop fields spends the most water resources in Portugal annually. Thus, it becomes crucial that a proper irrigation plan can maximize the profits of a crop field, while spending the least water resources possible, with the highest efficiency [
2]. In this paper, the authors contribute to achieving such an objective.
The first mathematical model presented in this paper was implemented in MatLab, based on optimal control theory. It allows considering different crops in the same field, and it can plan the percentage of the area to be allocated for each crop, in such a way that the profit is maximum, while minimizing the water spent, and assuring the water requirements of each crop. This could be a valuable tool when a farmer intends to cultivate a field from scratch with different crops. It is not intended that the farmer changes the area of each crop when it is already cultivated. On the other hand, by changing very few parameters, it is possible rethink the crops to have in that field. This model has as inputs the weather variables, the location of the field, the type of soil, the value of the crops, the costs of the crops, and the cost of water.
A second model improves the first one, giving the information on the profit obtained when water from rainfall is collected in a reservoir of a given capacity, allowing to save water while keeping the crop safe.
The models can be improved taking into account a prediction of the weather variables and economic variables using for example time series. It can be easily adapted to a different location or crops depending on the available data.
Optimal control theory emerged as a field of research in the 1950s in response to problems concerning the aerospace exploitation [
3] of the solar system. Nowadays, optimal control is a recognized tool, known for its efficacy, which is applied to different areas, such as robotics [
4], biological systems [
5], health problems [
6], economy problems [
7], oil extraction problems [
8], and agriculture problems [
9], among many others. The goal of optimal control theory is to find a control law for a given system such that a certain optimality criterion is achieved.
In optimal control problems (OCP), it is possible to define decision variables subject to restrictions in the form of differential equations, where these decisions variables are not necessarily smooth (non-differentiable functions). In an OPC, it is also possible to use different tools to solve the problem, to characterize it, to study the sensitivity of its variables, to study the stability of the problem and to apply predictive control to replan the problem [
9,
10,
11]. The nature of the problems addressed in this paper is fit for optimal control theory.
The literature includes publications that have similar objectives, although they use different formulations and techniques to solve them. Osama, Elkholy and Kansoh in [
12] developed a linear programming model for optimal land allocations for different crops in Egypt. Different constraints were incorporated in the model, including the water availability, the land availability in different seasons of the year, self-sufficiency ratios and the areas for each crop under the existing cropping pattern. The authors imposed that the total water requirement for the different crops, should be less or equal than the total water available at the field during the year. However, the irrigation water optimization was not considered.
Kuo, Merkleyb, and Liu [
13] developed a decision support model for an irrigation project plan (in Utah), using a genetic algorithm (GA) optimization method. This model allows optimizing the profits, simulating the water demand and crop yields, and estimating the related crop area percentages with specified water supply and planted area constraints.
Dutta [
14] developed a multi-objective fuzzy stochastic model for determination of optimum cropping patterns for the next crop season guaranteeing the water balance. The objective of the model is to study the effect of various cropping patterns on crop production subject to total water supply in a small farm. The model is implemented using fuzzy stochastic simulation, based on a genetic algorithm, without deriving the deterministic equivalents. However, the solution using a direct method is faster and more accurate than using genetic algorithms (models by Kuo [
13] and Dutta [
14]). In addition, genetic algorithms need an initial solution to start the process which might not be easy to find (see [
15]).
In our study, direct methods in optimal control (such as Interior Point OPTimizer (IPOPT), Sequential quadratic programming (SQP), Active Set, etc), which guarantee that a feasible solution is obtained, were used. In similar mathematical models (to the ones presented in this article), it is proven that the obtained solution is a local extrema [
16]. The first model presented in this article is able to plan the percentage of area for each crop in such a way that the profit is maximum, while minimizing the water spent and assuring the water requirements of each crop. The optimal control theory allows defining an objective function that is the sum of the profits obtained for each crop, taking into account the value of the crop, the costs of production, the costs of water, and the type of weather conditions.The dynamic equation considered guarantees the water balance (taking into account rainfall, irrigation, humidity in the soil, evapotranspiration and losses due to infiltration), the constraints considered guarantee that the crops have their needs of water fulfilled. Therefore, this model is better than others based on GA, and assures that water balance is never broken. On the other hand, optimal control is able to guarantee a smoother solution, not present in “on–off” type irrigation systems, since the weather and an economic forecast are used.
The possibility for the farmer to build a reservoir (also considered in [
17]) of a given capacity to collect rainwater was considered, and it was the basis for the second mathematical model proposed in this paper. Since climate change is showing us that water availability is going to drop, planning a proper reservoir to collect rainfall is crucial to preserve the crops and increase profit. It is possible to use the second model to solve the problem where there is no reservoir. However, the number of variables involved double, and the CPU time to solve it is greatly increased.
This paper is divided into five sections. The Introduction is presented in the
Section 1. In
Section 2, two models for optimized management of a field with several crops are presented. Data for the numerical model are introduced in
Section 3. Results for each model are shown and discussed in
Section 4. Finally, conclusions are presented in
Section 5. Two appendixes are also included at the end of the paper.
3. Data for the Numerical Model
The models presented in the previous section allow considering many crops. Since we chose Portugal as the study site, we had to use crops that are widely cultivated there, and for which we had enough data. We considered only two crops so that the results are easier to present. The crops considered in our study were olive trees and vines. Using known data from these cultures in Portugal, including cost of labour, machines, energy and water, as well as the estimated value of the products obtained per hectare, it was possible to determine the cost function of each crop per hectare. The cost function used, as explained in previous sections, takes into account these data.
In the rural region of the Lisbon district, the values of olives and grapes are in the interval of [750, 1760] euro/ hectare and [2700, 3300] euro /hectare, respectively. The price of water is 0.07 euro/m
, the cost of labour, machines, energy and services are on average 700 euro/hectare for olive trees and 2780/hectare for vines [
18,
19,
20].
We also considered the rainfall to be the average of the rainfall in the last 10 years for each month of the year, as shown in
Table 1 [
23].
To create the possibility of different weather scenarios, the above table is multiplied by a
precipitation factor:
where the precipitation factor allowed us to consider a typical year if this factor is 1, a drought year if it is less that 1 and a rainy year if it is above 1 [
24].
The Pennman–Monteith methodology [
25] was used to calculate evapotranspiration of the cultures along the year, from the following equation
where
is the culture coefficient for the evapotranspiration and
is the tabulated reference value of evapotranspiration from [
26] for the Lisbon region. The evapotranspiration of the cultures in Lisbon is given in
Table 2.
The following data were also considered to perform simulations in
Section 4:
5. Conclusions
Climate change and global warming are facts and are here to stay. In many parts of the world, the estimated rise of temperature for a very near future is significant, and water shortages will happen. One of the most affected regions is the Iberian Peninsula. In Portugal, most water is spent in agriculture, thus a proper irrigation plan is crucial. It is necessary that the crops chosen are the best fit for the farm field, the least amount of water (keeping the crops safe) is spent, the economic results are good and the efficiency in the use of water is as high as possible.
A first mathematical model, based on optimal control theory was implemented in Matlab. Its inputs were the weather variables, the location (the results presented consider the area of Lisbon in Portugal), the type of soil, the value of the crops, the costs of the crops, and the cost of water. The outputs were profit, the percentage of the area to be allocated to each crop, and an irrigation plan for the farm that guarantees the crops are safe.
A second model (also implemented in MatLab and based on optimal control theory) improved the first one, by giving the farmer the possibility of building a reservoir of a given capacity to collect rainwater, and therefore improving the profits and spending less water. The outputs were the profit, the percentage of the area to be allocated to each crop, the amount of water in the tank, and an irrigation plan for the farm that guarantees the crops are safe.
We believe the models developed can help to make the best management decisions when designing (partially designing) a crop field. The developed Matlab application can be easily adapted to a different location or crops.
The presented mathematical models can be improved, taking into account a prediction of the weather variables and economic variables, using for example time series.