Irrigation scheduling to maximize crop gross margin under limited water availability

Agricultural Water Management 223 (2019) 105678

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Irrigation scheduling to maximize crop gross margin under limited water availability

T

E. López-Mata, J.M. Tarjuelo, J.J. Orengo-Valverde, J.J. Pardo, A. Domínguez

⁎

Universidad de Castilla-La Mancha, Escuela Técnica Superior de Ingenieros Agrónomos y de Montes, Centro Regional de Estudios del Agua (CREA), Campus Universitario s/n, 02071 Albacete Spain

ARTICLE INFO

ABSTRACT

Keywords: MOPECO ORDI Water productivity Irrigation uniformity

MOPECO uses the optimized regulated deficit irrigation (ORDI) methodology to determine the irrigation schedule that maximizes the gross margin (GM) of annual crops when the availability of irrigation water is limited. To this end, it is necessary to determine the functions that relate GM to the gross irrigation water (IG) supplied to the crop. Given that MOPECO is designed for implementation by farmers using online tools, the time required to obtain the “GM vs. IG” function must be low. Thus, the objective was to develop and validate a methodology to generate the most accurate “GM vs. IG” function with a low calculation time that takes into consideration the effect of irrigation uniformity on final yield. Five sub-models were developed and evaluated with barley, onion and maize for the semi-arid conditions of Castilla-La Mancha (Spain). The sub-models included one or more of the following variables: yield (Y); effective precipitation (Pef); soil moisture (S); and its distribution uniformity at plot level (CU). The sub-model “Y-Pef-S-CU” achieved the best fitting. The other submodels (Y, Y-Pef, Y-Pef-S, and Y-Pef-CU) provided solutions that were very close to the optimum, with low calculation time. Nevertheless, the best fit of each sub-model to the optimum function was achieved in different sections of this function. Therefore, the five sub-models are complementary, and it is advisable to use them simultaneously thanks to their low calculation time requirements (3.9 s). The validation with the three crops showed that in farms where irrigation uniformity and water price are low and harvest price is high, it is economically advisable to apply higher irrigation depths than the crop irrigation requirements. Nevertheless, as improving agricultural water productivity is key for sustaining the environment, tools like MOPECO may help to maximize GM and increase water use efficiency by means of deficit irrigation strategies.

1. Introduction

water-scarce conditions (Domínguez et al., 2017). In this sense, some authors have used the previous models to determine the irrigation schedule that optimizes crop yield, and even the profitability of farms, developing their own algorithms or using other pre-existing ones (Akhtar et al., 2013; Geerts et al., 2010; Kloss et al., 2012; Linker et al., 2016). Moreover, ORDI methodology can be improved when determining the optimal irrigation schedule if we take into account the levels of deficit reached in each stage of crop development as a result of irrigation water uniformity, the amount of water available in the root zone or the amount of rain forecast. As with any optimization problem, the time taken to obtain the results and their accuracy depend on the process used. In the case of optimization of irrigation scheduling, most authors resort to metaheuristic techniques (de Paly and Zell, 2009; Garg and Dadhich, 2014; Lalehzari et al., 2015; Perea et al., 2016; Tong and Guo, 2013; Tsang and Jim, 2016). Metaheuristic methods aim to reach a solution that is

In order to improve irrigation management on farms, numerous growth simulation models have been developed and calibrated, such as CROPSYST (Stockle et al., 1994), PILOTE (Mailhol et al., 1997), WOFOST (Boogaard et al., 1998), Daisy (Abrahamsen and Hansen, 2000), APSIM (Keating et al., 2003), MOPECO (Ortega et al., 2004) and AquaCrop (Steduto et al., 2009), among others. All these models are able to simulate the crop yield from a particular irrigation schedule given the climatic conditions with varying degrees of precision. However, MOPECO is the only model that uses the methodology of optimized regulated deficit irrigation by stages (ORDI) (Domínguez et al., 2012a; Leite et al., 2015) to determine the irrigation scheduling that maximizes the gross margin (GM) of the crop with a limited amount of available irrigation water. This considerably increases its usefulness as an irrigation management tool in farms with

⁎

Corresponding author. E-mail address: [email protected] (A. Domínguez).

https://doi.org/10.1016/j.agwat.2019.06.013 Received 25 February 2019; Received in revised form 20 May 2019; Accepted 15 June 2019 0378-3774/ © 2019 Elsevier B.V. All rights reserved.

Agricultural Water Management 223 (2019) 105678

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close or even equal to the optimal solution by means of progressive improvement in the results obtained by the optimizer, which involves extremely high computational requirements. The use of this methodology is justified by the difficulty of developing a direct solution algorithm that calculates a single result equal to the optimal solution, which must be compatible with the use of crop simulation models that require a large amount of data and variables for their operation (Ioslovich et al., 2012). The importance of the rapid mathematical resolution of optimal irrigation scheduling calculation lies in being able to offer this tool to a large number of online users. In this case, the computational requirements are proportional to the number of users to be served at a given time. Furthermore, the greater the number of variables considered, the more complex the optimization problem to be solved may be and the more accurate might be the results. In this sense, the main variables considered by the MOPECO model that can be used in the optimization process are crop yield (Y), effective precipitation (Pef), quantity of water available in the root zone (S), and uniformity of water distribution in the soil after its application in the plot by the irrigation system (CU) (López-Mata et al., 2010). The aim of this work was to develop a methodology based on ORDI for optimal irrigation scheduling calculation that quickly and efficiently maximizes the “GM vs. IG” function of each crop, while being compatible with the MOPECO model. To achieve this, three specific objectives were proposed: I) To develop five sub-models for a rapid determination of optimal irrigation scheduling based on: a) yield (Y); b) yield and effective precipitation (Y-Pef); c) yield, effective precipitation and the amount of available water in the soil (Y-Pef-S); d) yield, effective precipitation and the water uniformity in the soil (Y-Pef-CU); and e) yield, effective precipitation, and the amount and uniformity of water in the soil (Y-Pef-S-CU); II) To develop a methodology that takes into account the effect on crop yield of irrigation water uniformity in the soil; III) To validate the different sub-models and determine which one obtains the GM vs. IG functions that best fit the Pareto frontier generated through 300,000 random simulations of three crops (barley, maize and onion) and three different climatic conditions (dry, intermediate and wet year).

For the daily calculation of ETm, the model uses the FAO-56 methodology (Allen et al., 1998), which requires the daily values of the crop coefficient (Kc) (Doorenbos and Pruitt, 1977) and the daily reference evapotranspiration (ETo). To calculate ETa, it is necessary to perform a daily water balance in the soil, in which the inputs are Pef (NRCS, 2004) and the net irrigation, while the outputs are the ETa and the deep percolation losses. In this sense, the model considers three soil horizons. The first is occupied by the roots, the second is that being colonized by the roots when they grow during the vegetative development stage, and the third is that which will never be reached by the roots and, therefore, the water that reaches that level is considered as water lost through deep percolation (Domínguez et al., 2011). In the daily soil water balance, the ETa value depends on the atmospheric evaporative demand and the soil water content. If this is higher than the allowable depletion level (ADL) (defined as the fraction of total available water (TAW) that a crop can use without going into water stress), then ETa = ETm. Otherwise, the crop will start to suffer water stress and ETa < ETm, which is calculated according to the equation proposed by Allen et al. (1998).

ETan =

To simulate the effect on crop yield of irrigation uniformity and its subsequent redistribution in the root area, MOPECO divides the irrigated plot into N subplots (25 are sufficient in the cases analyzed) and performs a daily water balance for each of them, obtaining the yield at the end of the crop development cycle for each subplot. By integrating the yields obtained in the set of soil fractions using a combination of Simpson's 1/3 rule and 3/8 rule, the total yield of the irrigated plot is obtained. To calculate the irrigation water distribution in the different fractions of the plot, the model uses Christiansen’s uniformity coefficient (CU) (Christiansen, 1942) together with the irrigation water distribution in the plot fitted to a normal cumulative function (Heermann et al., 1992; Warrick et al., 1989). Consequently, the amount of water received by each of the N considered fractions can be calculated from the variation coefficient (CV) and the average irrigation depth applied (R¯ ) (Fig. 1). The value of CU will be that estimated or measured for the set of irrigation events performed in the plot depending on the irrigation system used and the management and maintenance level implemented (López-Mata et al., 2010).

To estimate the crop yield, MOPECO uses the equation proposed by Stewart et al. (1977).

1 i= 1

K yi 1

ETai ETmi

(2)

2.2. Simulation of the irrigation water uniformity effect on crop yield

2.1. Simulation of crop yield

n

PWP Rdn

where C1n is the water content in the soil occupied by the crop roots at time n, with it being necessary to perform a daily balance of the soil water content; ADL is the permissible depletion level of the crop in relative terms to field capacity (FC) and permanent wilting point (PWP); ADLn is the permissible depletion level in absolute terms considering the crop root depth at time n; and Rdn is the crop root depth at time n.

2. Materials and methods

Y= Ymax

ETmn if C1n ADL n C1n PWP Rdn ETmn if ADL n > C1n ADL n PWP Rdn

(1) −1

where Y is the crop yield (kg ha ); Ymax is the maximum crop yield without water stress and other constraints (kg ha−1); Ky is the crop sensitivity coefficient to water stress for each stage (Doorenbos and Kassam, 1979); and ETa and ETm are the actual and maximum accumulated crop evapotranspiration respectively for each of the n stages considered (mm). The duration of development stages is based on the accumulated growing-degree-day (GDD) method, which is calculated using doubletriangulation methodology (Sevacherian et al., 1977). This methodology requires the daily maximum and minimum temperature, the threshold temperatures between which the normal development of the crop takes place and the duration in GDD of each phenological stage considered.

w V i= 1 i

CU= 1

CV=

1

¯ wV

CU 100 2

¯ V

100

(3)

(4)

where CU is Christiansen’s uniformity coefficient (%); Vi is the water ¯ is the average of all volume collected at the w control points (mm); V

2

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supply uniformity, or when this is close to 100%. However, with lower CU values, the impossibility of obtaining the complete curve arises, since MOPECO performs irrigation programming at the midpoint of the plot, that is, at the plot point corresponding to a value of pa = 50%. This means that 50% of the plot receives a quantity of water lower than the ETm, decreasing the yield in the total plot. It is then impossible to implement an irrigation schedule that applies an irrigation depth greater than the ETm for 50% of the irrigated area. In order to show this limitation, the “Yield vs. Net irrigation” function for sub-model “Y” is shown as an example (Fig. 2A) when this methodology is applied for CU = 75%. To solve this limitation, a methodology was developed that allows the "pa" value to be changed in order to reach values > 50%. The methodology is based on the irrigation scheduling calculated during the same crop stage, using different pa values with the same calculation method, being similar if the average irrigation applied (R¯ ) is the same.

¯ = Ri1 = Ri2 = Ri Mi1 Mi2 1+

the volumes (mm); and CV is the coefficient of variation of the volume of irrigation water that reaches the ground for a normal distribution. The percentage of the area adequately irrigated (pa) describes the proportion of the plot that receives at least the amount of predetermined irrigation water (R). The rest of the plot will receive an irrigation depth of less than R. Thus:

R M= R¯

M 1 CV

Ri1 = (1 pa 1) CVi 1+

1

Ri2 (1 pa 2) CVi

(7)

¯ is the average irrigation applied to the plot; Ri1 is the amount where Ri ¯ by its of irrigation water received by pa1 area, which will be equal to Ri multiplier Mi1 (Ri2 is obtained for pa2 area in the same way); and CVi is the variation coefficient of the irrigation depth applied in the stage in question, with this value depending on the uniformity coefficient (Eq. 5). To obtain an irrigation schedule for any desired water amount (Rtotal) higher than the irrigation level obtained to satisfy the crop requirements (ETa/ETm = 1) for pa = 50%, irrigation depth denominated (Rmax_50), the model calculates the pa value that achieves ETa/ ETm = 1 by applying the amount of water Rtotal.

Fig. 1. Surface distribution of irrigation water in plot.

pa= 1

1

(5)

M=

Rmax_ 50 R total

(8)

Using multiplier M from Eq. (8) in Eq. (5), we obtain the pa value for an average irrigation depth equal to the total irrigation desired (Rtotal) with ETa/ETm = 1 in all the crop stages. Hence, the complete response curve of the crop for each sub-model is obtained (Fig. 2B).

(6)

where (Φ) is the probability distribution function of normal distribution, which has no analytical expression, so it is necessary to resort to a numerical approximation such as that of Abramowitz and Stegun (1966), with a maximum error of 7.5 × 10−8; CV is the variation coefficient (Eq. 4); M is the multiplier that allows the average irrigation depth applied to the whole plot (R¯ ) to be transformed into the irrigation depth received by each of the subplots into which the plot has been divided. Both Eq. (1) and the ORDI methodology establish that the maximum ETa/ETm ratio that can be reached in a particular stage of crop development is 1, which is equivalent to the maximum crop requirements. Therefore, higher irrigation depths should not be applied since the result would be a loss in irrigation water productivity due to leaching. In any event, this is not a problem for the model when calculating “Yield vs. Net Irrigation” (Y vs. IN) without taking into account the irrigation

2.3. Optimal distribution of irrigation water available at the farm to maximize yield The last step of the MOPECO model is to determine the crop distribution on the farm that maximizes the GM, by calculating the irrigable area and irrigation volume to be applied to each crop. To do this, it is previously necessary to determine the “Y vs. IN” function that achieves the maximum yield for each net irrigation volume applied to each crop (Fig. 2B), as well as the irrigation schedule that achieves the maximum yield.

Rtotal = R1 + R2 + …+Rm

1

+ Rm

(9)

Fig. 2. Result of 300,000 simulations with the optimization sub-model “Y” for CU = 75%, A) with an area adequately irrigated pa = 50%; B) increasing the value of pa up to 100%. 3

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In the equation used by MOPECO to estimate the crop yield (Eq. 1), this depends on the ETa/ETm ratio reached in each development stage (Ky), which is significantly conditioned by irrigation water distribution during the season. Therefore, the function to be optimized is a yield function that accepts each of the irrigations applied to the crop as parameters and which also determines the amount of irrigation water to be provided in each stage of crop development to achieve combinations of ETa/ETm that obtain the maximum yield. This concept is the basis of the ORDI methodology (Domínguez et al., 2012a).

Y(R1, R2, ….,Rm 1, Rm) = f1 (R1, …, R a) f2 (R1, …, R b) … fn (R1, …, Rm)

regardless of the number of stages. In this way, the computational complexity is greatly reduced. Therefore, the optimization algorithm is based on solving Eq (15). n i= 1

where the MdMi functions are the inverse functions of the MdMi (Ri) function for each of the n crop stages. When the irrigation uniformity effect is considered, the MdM algorithm generates a distribution of the total desired irrigation (Rtotal) for each stage. However, when assigning the irrigation that obtains a ratio of ETa/ETm = 1, which would be the maximum irrigation for pa = 50%, the optimizer cannot assign more irrigation than that established by the upper limits of each of the stages. For this reason, the methodology described in section 2.2 for changing the pa value was used (Eq. 5 and 8).

where n is the number of crop stages considered and m the number of irrigations events applied. The irrigation and rainfall distribution means that the daily ETa/ ETm value is not constant over time, since it depends on the soil water content and the climatic conditions on each day. However, in order to simplify the optimization process, in this work it was assumed to be possible to implement an irrigation distribution in such a way that the daily ETa/ETm ratio remains constant throughout each development stage (average value for the stage) and is equal to the target ETa/ETm ratio corresponding to the entire stage. With this assumption, the irrigations established to achieve the target ETa/ETm relationship in each stage can be grouped into a single irrigation. It was also assumed that the stages are independent of each other, in order to avoid the effect on ETa/ETm of the amount of soil available water at the beginning of the next development stage. In this way, in the yield function to be maximized, the factor value of each stage only depends on the total amount of irrigation water applied to each one (Eq. 11). (11)

Rtotal = R1 + R2 + …+Rn

(12)

2.4. Optimization sub-models To determine the daily value of ETa (Eq. 2) in each subplot when the irrigation uniformity effect is considered (Fig. 1), it is necessary to perform a daily water balance in each subplot. For this, the inputs are the water content in the root zone, the irrigation water applied in each subplot (Eqs. 3–6), and effective precipitation. The balance outputs are the ETa and the deep percolation losses (Domínguez et al., 2011). The greater the number of these factors taken into account when determining the ETai (Ri) value in Eq. (13), the more complex are the calculation requirements of the algorithm to be developed. Thus, five sub-models were developed, with an increasing level of complexity, in order to determine whether greater calculation requirements achieve greater precision:

where n is the number of growing stages considered. For the MOPECO model, each of the factors corresponding to the Ky stages in Eq. (11) can be expressed as according to Eq. (1).

fi (Ri) = 1

K yi 1

ETai (Ri) ETmi

- Sub-model “Y”, in which, to calculate the soil water balance and determine the ETa, only the irrigation water applied to the crop is considered. - Sub-model “Y-Pef”, which takes into account the irrigation applied to the crop and the effective precipitation. - Sub-model “Y-Pef-S”, which takes into account the irrigation applied to the crop, the effective precipitation and the initial water amount present in the root zone. - Sub-model “Y-Pef-CU”, which takes into account the irrigation applied to the crop, the effective precipitation, and the deep percolation in the different zones of the plot due to CU. - Sub-model “Y-Pef-S-CU”, which takes into account the irrigation applied to the crop, the effective precipitation, the initial water amount present in the root zone, and the deep percolation in the different zones of the plot due to CU.

(13)

During the optimization process, the only unknown variable in Eq. (13) is irrigation (R), since a previously calibrated value of Ky is chosen for the crop, and a priori known climatic data are used. This allows the duration of each development stage and the corresponding value of ETm to be determined. ETa (R) is the function that calculates the actual crop evapotranspiration in each development stage according to the irrigation applied, being considered in this work as an optimization submodel. In Annex A, it is mathematically demonstrated that the maximum crop yield for a limited irrigation water volume (Rtotal) is obtained when the value of fi (Ri) is the same for each of the development Ri

2.4.1. Sub-model Y This is the simplest optimization sub-model, in which the ETa is assumed to be equal to the amount of irrigation water applied, and ETa = R.

fi (Ri)

stages. This value is called "Multiplier divided by the derivative of the Multiplier" (MdM).

MdMi (Ri) =

fi (Ri) f (Ri) R i i

=

1

(

K yi 1 Kyi ETmi

ETai (Ri) ETmi

ETai (Ri) R i

(15)

1 (MdM)

(10)

Y(R1, R2, ….,Rn 1, Rn) = f1 (R1) f2 (R2) … fn 1 (Rn 1) fn (Rn)

MdMi 1 (MdM)

fRtotal (MdM) =

)

ETai (Ri) = Ri (14)

(16)

where Ri is the irrigation depth applied in stage i (mm). This option can achieve good results in areas or crops in which the contribution of rain is practically negligible, the soils are shallow and irrigation systems with a high water allocation uniformity are used.

If the functions MdMi (Ri) of each stage are injective, an optimization algorithm based on the search for the MdM value can be created, where the sum of the irrigation events obtained for each of the stages (Ri) is equal to the total desired irrigation (Rtotal). Thus, the optimization problem, instead of searching for n optimal values, which are the irrigations for each stage, involves finding only an MdM value,

2.4.2. Sub-model Y-Pef This model is somewhat more complex than the previous one. The

4

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level of precision increases, facilitating a more exact distribution of the available irrigation water, taking into account the precipitation contributions in each stage.

2.4.4. Sub-model Y-Pef-CU This sub-model is derived from the P-Pef sub-model, with the exception that the irrigation water is not uniformly distributed in the plot as the Pef, but according to uniformity coefficient CU (Fig. 3). The average irrigation depth (R¯i ) is distributed in the plot according to the CU value (Eq 3). The evapotranspiration on the whole plot is equal to the effective precipitation plus the area defined by the function representing the normal distribution of irrigation water in the plot. In the subplot where Ri + Pef > ETm, the crop is considered to only evapotranspirate ETm, with excess water being eliminated by leaching (Fig. 3). To solve the irrigation integral, GNU Maxima computer algebra software was used.

(17)

ETai (Ri) = Pefi + Ri

where Pefi is the effective precipitation throughout stage i (mm); Ri is the irrigation depth applied in stage i (mm). In rainy areas and/or with crops that show significant differences in the amount of precipitation received in the different development stages, this option can provide significant improvements with respect to Eq. (16).

(Ri + Pefi

2.4.3. Sub-model Y-Pef-S Including the initial soil water content in the daily balance complicates the ETa (R) calculation as this depends on irrigation (R) and, consequently, the amount of irrigation water to be applied in each stage. From Eq. (2) and from the daily water balance performed by MOPECO, the ETa (R) value is determined as:

ETai (Ri) =

C1Ii + VolIncrRdi + Ri + Pefi 1+

ADL i

PWP Rdi ETmi

ETmi)

Ri + Pefi ETmi CVi Ri

(19)

Where N(x) is the normal function. 2.4.5. Sub-model Y-Pef-S-CU In this optimization sub-model, the integral to be solved is more complex than in the previous sub-model Y-Pef-CU. It is necessary to consider the soil water content at the beginning of each development stage, which will be different in each of the N subplots into which the irrigated area is divided (Section 2.2). This means that the algebraic resolution of the irrigation integral is not viable and thus its resolution must be resolved numerically, using the Y-Pef-S sub-model as a basis. This sub-model is solved by determining the ETa (R) value for each of the subplots to take into account the lack of uniformity of the irrigation system. The methodology used in the Y-Pef-S sub-model is then applied to each one. The value of ETa (R) is obtained by integrating the ETa (R) value of each subplot (Fig. 4). Consequently, this algorithm is the slowest of all those proposed but is able to obtain the most precise results, given that it considers the greatest number of variables.

PWP Rdi (18)

where C1Ii is the water content in the soil occupied by the crop roots at the beginning of stage i (mm); VolIncrRdi is the water volume stored in the soil which the crop can access by increasing root depth throughout stage i; Ri is the irrigation applied across stage i (mm); Pefi is the effective precipitation during stage i (mm); PWP is the soil water content at the permanent wilting point (mm/mm); Rdi is the root depth at the end of stage i; and ADLi is the allowable depletion level in absolute terms at the end of stage i. The equations used by this sub-model depend on the soil moisture content in the previous stages. This aspect is not considered in the MdM optimization method because, for the sake of simplicity, the stages are considered as being independent of each other. Thus, a first optimization of irrigation distribution must be carried out with the sub-model YPef to have an idea of the total irrigation applied in each stage. Having obtained the irrigation for each stage, the initial soil state for each stage is then calculated, and the irrigation schedule is optimized using the MdM optimization algorithm. The separation between the calculation of the soil condition and the optimization process is necessary since the soil moisture content must be calculated for each stage starting from a valid solution, that is, from a distribution of irrigation whose sum is the total irrigation. The optimization process acts independently in each of the stages, and if the soil water content is updated for another stage, the results may not be valid.

2.5. Determination of irrigation scheduling To calculate irrigation scheduling based on desired total irrigation, the MdM optimizer solution was used, which indicates the target ETa/ ETm ratios for each stage referring to a properly irrigated area. This determines the daily irrigation required to maintain the crop evapotranspiration within the target ETa/ETm ratios for each crop stage. To do this, the daily soil water balance is performed in each subplot to take CU into account, converting these irrigations to pa = 50% using the multiplicative factor for that area (Eq. 8). If sub-models obtained by simplifying the MOPECO simulator are used in the optimization process, when simulating the irrigation scheduling obtained by the MdM optimizer, there will evidently be deviations between the total water used by the simulator and the total desired irrigation. Generally, the more complex the sub-model of optimization used, the smaller this deviation will be. Nevertheless, the total irrigation objective can easily be obtained with successive approximations. To do this, the total irrigation included in the MdM optimizer can be modified until the total irrigation distributed by the simulator reaches the target level. Thus, using a modification of Muller's method (Muller, 1956), the optimal calendar of irrigations with an error of 0.001 mm can be obtained using between only 5 and 8 iterations. To obtain a complete crop response curve using 100 points, performing the iterations is unnecessary, since the crop response is obtained by optimizing different target total irrigations using the total irrigation obtained by the simulator. 2.5.1. Evaluation of optimization sub-models To evaluate the methodologies developed for the generation of optimum irrigation scheduling during the crop cycle, three crops were chosen: barley, maize and onion, which were calibrated for the

Fig. 3. Distribution of irrigation and rainfall inputs in the Y-Pef-CU optimization sub-model. 5

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Fig. 4. Flowchart for calculation of the ETai (Ri) function for the Y-Pef-S-CU sub-model.

MOPECO model under conditions in Castilla-La Mancha (Spain) (Domínguez et al., 2012b, c; Pardo, 2018) (Table 1). The necessary economic data for the transformation of the “Y vs. IN” functions in “GM vs. IG” (Table 2) are also available (Domínguez et al., 2017).

GM= Y SP

Vc

0.999. Therefore, many of these irrigation schedules do not make agronomic sense, since it is not possible for a crop to achieve a particular yield if the ETa/ETm ratio is very close to 0 in any of its stages, or with very small pa values. Therefore, for the evaluation of the 5 submodels only the irrigation schedules that comply with the restrictions imposed by the ORDI methodology were used (Domínguez et al., 2012a).

(20)

IG Wc+ Subs −1

−1

where GM is the gross margin (€ ha ); Y is the yield (kg ha ); SP is the sale price of the harvest (€ kg−1); Vc is the variable costs associated with the crop (€ ha−1); IG is the gross irrigation applied to the crop (m3 ha−1); Wc is the water cost (€ m-3); and Subs. are the CAP (Common Agricultural Policy of the European Union) subsidies (€ ha−1). In this work, a permanent sprinkler irrigation system was considered, with evaporation and drift losses of 10% (Ortiz et al., 2010), which are added to IN to calculate IG. The climatic data used in the simulations correspond to the typical humid, intermediate and dry meteorological years (TMY) (Domínguez et al., 2013; Hall et al., 1978) based on 54 years of climate series recorded at Albacete weather station (Leite et al., 2015) (Fig. 5). The simulation of 300,000 random irrigation schedules with MOPECO for the economic (Table 2) and climatic conditions (Fig. 5) considered in this study allows the Pareto frontier of each crop to be determined. When establishing the random irrigation schedule, the ETa/ETm ratios to be reached in each stage were progressively increased from 0 to 1 and the area adequately irrigated pa from 0.001 to

- The minimum ETa/ETm ratio to be reached at each crop development stage should be 0.5, and the maximum 1. - During the establishment stage (Ky i), ETa/ETm ≥ 0.8 is set in order to favor crop emergence. - The maximum difference in deficit between consecutive development stages |(ETa/ETm)i (ETa/ETm)i+1| must be limited to prevent an excessive deficit in one stage conditioning crop development in the next. In the case of barley, maize and onion, these values are 0.3, 0.3 and 0.4 respectively. In addition, to analyze the results obtained with the different optimization sub-models using a variable deficit in the different crop stages, two aspects must be considered: a) the sub-model achieving the highest GM for the irrigation water available; and b) the sub-model achieving a better fit to the Pareto boundary in Section B, which is located between the point of tangency of the GM curve with the straight line passing through the origin (P2) until the point of maximum GM (P3) (Fig. 6).

6

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Table 1 Parameters for barley, maize and onion simulation with MOPECO in Castilla-La Mancha. Stage

Kc

GDD (ºC)

Stage

Ky

GDD (ºC)

Other parameters

Value

Barley I II

0.30 0.30-1.15

290.3 744.5

i ii

0.20 0.55

645.3 981.2

3 9,100

III IV Maize I II

1.15 1.15-0.45

1,087.2 1,449.5

iii iv

0.30 0.15

1,186.1 1,449.5

ET group Ym (kg ha−1) TL (ºC) TU (ºC)

0.30 0.30-1.10

353.4 902.3

i ii

0.35 1.05

789.3 1,206.6

4 19,500

III IV Onion I II

1.10 1.10-0.55

1,381.2 1,802.8

iii iv

0.40 0.20

1,519.3 1,802.8

ET grupo Ym (kg ha−1) TL (ºC) TU (ºC)

0.65 0.65-1.20

458.5 926.5

i ii

0.45 —

926.5 —

1 95,000

III IV

1.20 1.20-0.75

1,805.2 2,283.4

iii iv

0.80 0.20

1,805.2 2,283.4

ET group Ym (kg ha−1) TL (ºC) TU (ºC)

2 28

8 30

Fig. 5. Daily evolution of the accumulation of reference evapotranspiration (ETo) and precipitation (P) in the typical dry, intermediate and humid meteorological years (TMY) in Albacete (Spain). The table shows the typical meteorological year (from 1951 to 2004) corresponding to each month.

5 45

Kc: growth stages associated with FAO 56 crop coefficients; Kc (I): initial; Kc (II): crop development; Kc (III): middle stage; Kc (IV): late stage; GDD: accumulated growing degree days; Ky yield formation coefficient; Ky (i): vegetative period. This stage is divided into two sub-stages: Ky (i') "establishment", which coincides with Kc (I) and Ky (i”) "vegetative development" from the end of Kc (I) to the beginning of the next stage Ky; Ky (ii): flowering (not applicable to onion during the crop’s commercial development); Ky (iii): formation of crop yield; Ky (iv): maturation; ET group: conditions the daily value of TAW that the crop can extract without water stress (Danuso et al., 1995); Ym: potential crop yield for the varieties used in this study; TU: upper threshold temperature from which the growth rate begins to decrease; TL: lower threshold temperature from which the growth ratio stops. Table 2 Economic data associated with barley, maize and onion. Yield (kg ha−1) Barley 9,100 8,498 7,218 6,055 Rainfed Maize 19,500 18,155 15,499 13,186 Rainfed Onion 95,000 85,981 75,967 65,647 Rainfed

Sale price (€ ha−1)

Variable costs (€ ha−1)

Water cost (€ m−3)

Subsidies (€ ha−1)

0.164 0.164 0.164 0.164 0.164

1,017.03 980.74 903.59 833.48 700,38

0.12 0.12 0.12 0.12 —

200.00 200.00 200.00 200.00 200.00

0.173 0.173 0.173 0.173 0.173

1,948.97 1,853.52 1,665.00 1,500.84 500,00

0.12 0.12 0.12 0.12 —

200.00 200.00 200.00 200.00 200.00

0.115 0.115 0.115 0.115 0.115

7,665.00 7,112.42 6,498.89 5,866.56 3,356.90

0.12 0.12 0.12 0.12 —

200.00 200.00 200.00 200.00 200.00

Fig. 6. Sections in which the crop response function is divided to evaluate the different optimization sub-models.

3. Results and discussion 3.1. Validation of the methodologies developed MOPECO simulated 300,000 random irrigation schedules for barley, maize, and onion (Fig. 7A, B, and C) under wet, intermediate, and dry typical conditions (Fig. 7a, b, and c). According to the results, only 12.4%, on average, meet the ORDI requirements for barley, 6.7% for maize and 3.4% for onions. Since climatology conditions the minimum irrigation water that must be applied to each crop, it was not possible to find irrigation schedules for an intermediate TMY that are compatible with ORDI when the irrigation water applied is less than about 125 mm in the case of barley, 250 mm in maize and 450 mm in onion (Fig. 7Ab, Bb and Cb). The Pareto frontier is clearly defined (black line in Fig. 7), showing that for CU = 85% and pa = 50% it is possible to increase the plot yield applying an irrigation depth higher than the crop requirements (black dot in Fig. 7). This result justifies why, in conditions of low water price and/or high water availability, farmers usually uses this irrigation strategy. Nevertheless, it must be borne in mind that once the point corresponding to pa = 50% has been exceeded, the slope of the curve

This part of the function is then used to determine optimal crop distribution on the farm. Section A is not relevant given that it is preferable to leave part of the farm unirrigated and apply an irrigation depth equal to that determined by P2 to the irrigated area (López-Mata et al., 2016). Section C (P3-P4) is not relevant either, since it will never be beneficial to apply greater water volume than that necessary to achieve a particular GM.

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Fig. 7. Yield of barley (A), maize (B) and onion (C) as a function of the net irrigation water received by the crop after the 300,000 random irrigation schedules simulated under wet (a), intermediate (b) and dry (c) TMY conditions, considering a uniformity coefficient of water in soil of 85%. The black dot corresponds to a properly irrigated area pa = 50%.

usually tends rapidly to 0. Consequently, the agronomic water productivity also decreases at that rate. Therefore, the improvement in yield involves unjustifiable water expenditure from both the economic perspective and that of the sustainability of the agricultural systems, given that increasingly less water is available for agriculture due to climate change and the priority for other uses such as urban supply and the environment. The best way to reduce this effect is to raise farmers' awareness of the importance of properly maintaining and managing their irrigation systems in order to achieve high CU values. Nascimento et al. (2019) carried out an experiment with irrigated maize in Albacete (Spain) in order to compare the traditional irrigation schedule applied by a farmer with the theoretical one, aiming to fulfill the irrigation requirements of the crop without considering the uniformity of the irrigation system. Their results showed that the traditional schedule increased the amount of water supplied to the crop (17%), the yield (8%) and the profitability of the plot (22%), while the water productivity of the theoretical schedule was 7.8% higher. These results were conditioned by both the low CU value (83.5%) and the irrigation water cost (0.07 € m−3). The economic data associated with each crop (Table 2) allowed the “Y vs. IN” functions (Fig. 7) to be transformed into “GM vs. IG” (Fig. 8) (Eq. 20). As expected, the wet TMYs achieve the highest economic profitability, because the maximum yield is reached by applying less

irrigation water (Fig. 7Aa, Ba and Ca). The loss of agronomic productivity as a result of the lack of uniformity in irrigation water applied means maximum economic profitability (Fig. 8) is reached before achieving the potential yield for the whole plot (Fig. 7). In the case of barley, the maximum GM is achieved even before applying the maximum crop irrigation requirements for pa = 50% (black dot in Fig. 8A). In contrast, for maize, the maximum GM is obtained for a value of pa ≈ 70% and for onion of pa ≈ 77% (Fig. 8B, and C). These results are due to the different effect of irrigation water cost on the total profitability of each crop. Consequently, in the most profitable crops, it may be economically advisable to apply an irrigation depth higher than the maximum crop requirements when the CU is not high. However, this may be contrary to the sustainability of agricultural systems in areas with water scarcity, since the increase in irrigation water generates very little economic productivity. Famers in Castilla-La Mancha who use deficit irrigation in barley are empirically aware of this behavior. However, in order to achieve maximum profitability for the irrigation volume applied, it is necessary for farmers to use ORDI methodology. To this end, it would be necessary to provide the productive sector with models and tools to advise on how to schedule optimized regulated deficit irrigation. When comparing the “GM vs. IG” function formed by the ORDI Pareto frontier (Fig. 9) with the “GM vs. IG” functions of the different

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Fig. 8. Gross margin of barley (A), maize (B) and onion (C) as a function of the gross irrigation received by the crop after the simulation of 300,000 random irrigation schedules under the conditions of wet (a), intermediate (b) and dry (c) TMY, considering CU = 85%. The black dot corresponds to pa = 50%.

submodels, together with the results of using a constant deficit in all crop stages, it can be observed that crop type and climatic conditions affect the fit achieved by each sub-model. Bearing in mind that the objective of each of the optimization sub-models is for the GM to be as close as possible to the maximum obtained by the ORDI Pareto frontier for each applied irrigation depth, the results of the different submodels in Section B were analyzed due to their being the most beneficial from an economic point of view (López-Mata et al., 2016) (Fig. 6). In general, the most complex submodel (Y-Pef-S-CU) achieves the best fit, with barley for the wet TMY being the only exception within the studied cases. However, in other sections of the function, the best fits are achieved by other sub-models. Therefore, since no optimization submodel stands out clearly from the rest in all sections of “GM vs. IG”, it is proposed to calculate all of them for use in the direct solution algorithm that determines the optimal crop distribution (López-Mata et al., 2016). The results also show the highest GM achieved by the ORDI methodology with respect to the constant deficit in all crop stages. The average calculation time needed to generate the “GM vs. IG” curves for each sub-model by using 100 points is very low (between 0.6´´ for the simplest sub-model and 1.3´´ for the most complex). If all sub-models are calculated, the time is 3.9´´ on average, using a single core of an Q9550 intel processor at 2.83 Ghz, which is perfectly acceptable for use in a web application with a large number of users. Evidently, only simulating 100 irrigation schedules per sub-model to

obtain the “Y vs. IN” function considerably reduces the calculation time with respect to the option of calculating 300,000 irrigation schedules (Fig. 9). Despite being able to obtain slightly higher GM, this latter option is not acceptable for web applications due to its high calculation time (30´). It would cause a significant slowdown in the system when used massively by a large number of users. 4. Conclusions The 5 sub-models developed in this work are capable of generating “gross margin vs. gross irrigation” (GM vs. IG) functions with a highly similar level of precision to the function obtained with the Pareto frontier determined by the simulation of 300,000 random irrigation schedules. Calculation times are low (between 0.6´´ and 1.3´´), and acceptable for application in online tools for large-scale use. Sub-model Y-Pef-S-CU, which takes into account the yield, effective precipitation, and the uniformity of water in soil, achieves the best fit to the “GM vs. IG” function. However, the best solution is to use all the sub-models together to determine the “GM vs. IG” function, due to the low calculation time (3.9´´) of this option reaches a better solution. In farms where irrigation uniformity and the price of water are low and the harvest price is high, farmers may increase the profitability of their plots by supplying a greater irrigation depth than the maximum requirements of the crop, thus compensating for the effect of the lack of

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Fig. 9. Comparison between the “GM vs. IG” functions obtained with the ORDI Pareto frontier and the optimization sub-models for barley (A), maize (B) and onion (C) under the conditions of wet (a), intermediate (b) and dry (c) TMY, considering CU = 85%. The black dot corresponds to pa = 50%.

uniformity on yield. Nevertheless, this option increases the negative impact of agriculture on the environment and must be avoided. Consequently, it is necessary to establish awareness policies and carry out training programs to promote the adequate management and maintenance of irrigation systems in order to achieve high uniformity and efficiency when applying irrigation water. For crops whose profitability is significantly affected by the amount of irrigation water applied, it is recommended to use tools as MOPECO

to advise farmers on establishing irrigation schedules that maximize crops GM by using regulated deficit irrigation strategies as ORDI. Acknowledgements This paper has been developed within the framework of the project AGL2017-82927-C3-3-R (MINECO, Spain), financed with European Union FEDER funds.

Annex A Mathematical demonstration of the MdM optimization algorithm Proposition: Let the yield function be Y (R1, R2 , …., Rn) = f1 (R1) f2 (R1) … fn (Rn ) where the fi functions are real functions of a real variable [0, + [, and Rtotal = R1 + R2 + …+ Rn , being Rtotal a known value. Thus, the necessary condition for function P to reach the optimal value in a certain point (R1, R2 , …, Rn) can be expressed as:

f1 (R1) R1

f1 (R1)

=

f2 (R2) R2

f2 (R2)

= …=

fn (Rn ) f (Rn) Rn n

(A.1)

Demonstration: As the aim is to optimize the Y (R1, R2 , …., Rn) function subjected to the Rtotal = R1 + R2 + …+ Rn condition, Lagrange’s multiplication method is used. Considering L ( , R1, R2 , …., Rn) = Y (R1, R2 , …., Rn) + (R1 + R2 +…+ Rn Rtotal ) and equaling its gradient to zero

0=

L ( , R1 , R2 , …., Rn) = where

Y (R1, R2 , …., Rn )

P (R1, R2 , …., Rn) +

(R1 + R2 +…+Rn

(R1 + R2 +…+ Rn

Rtotal )

Rtotal ) , thus 10

(A.2)

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E. López-Mata, et al.

Y Y Y = = …= R1 R2 Rn

(A.3)

Because for any i = 1, …, n , we obtain Particularly, because for any i

fi (Ri ) Ri

Y Ri

=

f j (Rj )

Ri

f1 (R1 ) … fi (Ri ) … fn (Rn) =

f j (Rj ) =

Rj

fi (Ri )

f1 (R1 ) … fi (Ri ) … fn (Rn) , where fi (Ri ) means it does not have that factor.

Rj

f j (Rj) f (Rj) R j

Y Ri

=

Y , Rj

thus:

¯

f1 (R1 ) … f j (Rj ) … fn (Rn )

And cancelling the common factors (if fi (Ri )

fi (Ri )

Ri

j and according to Eq. (A.3) we obtain

¯

f j (Rj )

fi (Ri )

¯

¯

=

j

(A.4)

0 , for any i )

fi (Ri) f (Ri) Ri i

(A.5)

which was the aim of this demonstration.

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