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A new stochastic optimization model for deficit irrigation

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Abstract

Deficit irrigation has been suggested as a way to increase system benefits, at the cost of individual benefits, by decreasing the crop water allocation and increasing the total irrigated land. Deterministic methods are common for determining optimal irrigation schedules with deficit irrigation because considering the inherent uncertainty in crop water demands while including the lower and upper bounds on soil moisture availability is a hard problem. To deal with this, a constraint state formulation for stochastic control of the weekly deficit irrigation strategy is proposed. This stochastic formulation is based on the first and second moment analysis of the stochastic soil moisture state variable, considering soil moisture as bounded between a maximum value and a minimum value. As a result, an optimal deficit irrigation scheduling is determined using this explicit stochastic model that does not require discretization of system variables. According to the results, if irrigation strategy is based on deterministic predictions, achievement of high, long-term expected relative net benefits by decreased crop water allocation and increased irrigated land may have a higher failure probability.

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Authors and Affiliations

  1. Department of Systems Design Engineering, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    A. Ganji & K. Ponnambalam

  2. Water Engineering Department, College of Agriculture, Shiraz University, Shiraz, Iran

    A. Ganji & D. Khalili

  3. School of Civil Engineering, University of Tehran, Tehran, Iran

    M. Karamouz

Authors
  1. A. Ganji
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  2. K. Ponnambalam
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  3. D. Khalili
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  4. M. Karamouz
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Corresponding author

Correspondence to K. Ponnambalam.

Additional information

Communicated by S. Ortega-Farias

Appendix: Second-order derivation for sample terms

Appendix: Second-order derivation for sample terms

Sample term A

Consider the term

$$E\left(\theta_{t}\times 1_{\left(\theta_{\rm pwp} \leq \theta_{t} \leq (1 - p)\left(\theta_{\rm FC} - \theta_{\rm pwp}\right)\right)} \left(\theta_{t}\right)\right),$$

where θpwp is the relative soil moisture unavailable for plant growth or the water content at wilting point (m3/m3), (1−p) the average fraction of total available soil water that can be depleted from the root zone before moisture stress (reduction in ET) occurs and θFC the water content at field capacity. This term can be divided into three individual parts as indicated by Eq. 13:

$$\begin{aligned} \,& nz_{t}E\left(\theta_{t - 1}\times 1_{\left(\theta_{\rm pwp} \leq \theta_{t} \leq (1 - p)\left(\theta_{\rm FC} - \theta_{\rm pwp}\right)\right)} \left(\theta_{t}\right)\right) + E\left(\left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right)\right.\\ &\quad\left. \times 1_{\left(\theta_{\rm pwp} \leq \theta_{t} \leq (1 - p)\left(\theta_{\rm FC} - \theta_{\rm pwp}\right)\right)} \left(\theta_{t}\right)\right) + E\left({\eta_{t}\times 1_{\left(\theta_{\rm pwp} \leq \theta_{t} \leq (1 - p)\left(\theta_{\rm FC} - \theta_{\rm pwp}\right)\right)}} \left(\theta_{t}\right) \right).\\ \end{aligned}$$
(13)

The second-order approximation of the first two parts is calculated next and the third part is determined similarly but not shown here.

AI. Using the property of the indicator function, the first term of Eq. 13, AI, can be expressed as follows:

$$nz_{t} \int\limits_{\theta_{t - 1}^{\min}}^{\theta_{t - 1}^{\max}} \int\limits_{nz_{t}\theta_{t}^{\min} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1}\theta_{t - 1}}^{nz_{t}\theta_{t}^{\max} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1}\theta_{t - 1}} \theta_{t - 1}f_{\eta_{t}} \left(\eta_{t}\right) f_{\theta_{t - 1}} \left(\theta_{t - 1}\right) \hbox{d}_{\eta_{t}} \hbox{d}_{\theta_{t - 1}}.$$
(14)

\(f_{{\eta}_{t}} \left(\eta_{t}\right)\) is the probability density function of the random noise term, η t ∼ N[ 0, Var(η t ) ], for the current work. \(f_{\theta_{t - 1}} \left(\theta_{t - 1}\right)\) is the density function of the soil moisture state variable in the previous period. (θ t - 1) and η t are assumed independent; thus it can be written:

$$nz_{t} {\int\limits_{\theta^{\min}_{t - 1}}^{\theta^{\max}_{t - 1}} {{\left[ {{\int\limits_{nz_{t} \theta^{\min}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}}^{nz_{t} \theta^{\max}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}} {f_{\eta_{t}} \left(\eta_{t}\right)\hbox{d}_{\eta_{t}}}}} \right]}}}\theta_{t - 1} f_{\theta_{t - 1}} \left(\theta_{t - 1}\right) \hbox{d}_{\theta_{t-1}},$$
(15)

which is the expectation of a function of (θ t - 1) and can be shown as

$$ = nz_{t} E{\left\{{\theta_{t - 1} {\left[ {{\int\limits_{nz_{t} \theta^{\min}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}}^{nz_{t} \theta^{\max}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}} {f_{\eta_{t}} \left(\eta_{t}\right)\hbox{d}_{\eta_{t}}}}} \right]}} \right\}},$$
(16)

where \(f_{\eta_{t}} \left(\eta_{t}\right)\) is presented as follows:

$$f_{\eta_{t}} \left(\eta_{t}\right) = \frac{1}{{{\sqrt {2\pi \hbox{Var}\left(\eta_{t}\right)}}}}\exp {\left[{- \frac{1}{2}{\left({\frac{{\left(\eta_{t}\right)^{2}}}{{\hbox{Var}\left( \eta_{t}\right)}}} \right)}} \right]}.$$
(17)

For the current study, the integral in Eq. 16 is represented as follows:

$$\begin{aligned} & E\left(nz_{t - 1} \theta_{t - 1} \Pr \left\{ nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} \left(\theta_{t - 1}\right) < \eta_{t}\right.\right. \\ &\quad \left.\left. \leq nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} \left(\theta_{t - 1}\right)\right\} \right), \\ \end{aligned} $$
(18)
$$nz_{t - 1} E\left(\theta_{t - 1} \Pr \left\{\eta_{t} \leq - nz_{t - 1} \left(\theta_{t - 1}\right)\right\} \right)^{{nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right)}}_{{nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right)}}.$$
(19)

The second-order Taylor’s series approximation of the expectation of Eq. 16 is thus given as:

$$\begin{aligned} &\left\{nz_{t - 1} E\left(\theta_{t - 1}\right)\left\{ \frac{1}{2}\left(\hbox{erf}\left(\frac{{nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t} E\left(\theta_{t - 1}\right)}}{{\sqrt{2\hbox{Var }\eta_{t}}}}\right) \right.\right.\right.\\ &\quad \left.\left.\left. - \hbox{erf}\left(\frac{{nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t} E\left(\theta_{t - 1}\right)}}{{{\sqrt{2\hbox{Var }\eta_{t}}}}}\right)\right) \right\}\right\}\\ &\quad - \hbox{exp} \left(-\frac{1}{2}\frac{{\left(nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1}E\left(\theta_{t - 1}\right)\right)^{2}}}{{\hbox{Var }\eta_{t}}}\right) \\ &\quad\times \left[ \frac{{{\left({nz_{{t - 1}}} \right)}^{2}}}{{{\sqrt {2\pi \hbox{Var }\eta_{t}}}}}\left(2 + \frac{{\left(nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)\right)nz_{t - 1} E\left(\theta_{t - 1}\right)}}{{\hbox{Var }\eta_{t}}}\right) \right] \\ &\quad \times \frac{{\hbox{Var}\left(\theta_{t - 1}\right)}}{2} + \hbox{exp} {\left({- \frac{1}{2}\frac{{\left(nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)\right)^{2}}}{{\hbox{Var }\eta_{t}}}} \right)}\\ &\quad\times \left[ \frac{{{\left({nz_{t - 1}} \right)}^{2}}}{{{\sqrt {2\pi\hbox{Var }\eta_{t}}}}}\left(2 + \frac{{\left(nz_{t} \theta ^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)\right)nz_{{t - 1}} E\left(\theta_{t - 1}\right)}}{{\hbox{Var }\eta_{t}}}\right) \right]\\ &\quad\times \frac{{\hbox{Var}\left(\theta_{t - 1}\right)}}{2}. \\ \end{aligned}$$
(20)

AII. Using the property of the indicator function and assuming (θ t - 1) and η t are independent, the second term of Eq. 13, AII, can be expressed as follows:

$$nz_{t} {\int\limits_{\theta^{\min}_{t - 1}}^{\theta^{\max}_{t - 1}} {{\left[ {{\int\limits_{nz_{t} \theta^{\min}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}}^{nz_{t} \theta^{\max}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} \theta_{t - 1}} {\eta_{t} f_{\eta_{t}} \left(\eta_{t}\right)\hbox{d}_{\eta_{t}}}}} \right]}}}f_{\theta_{t - 1}} \left(\theta_{t - 1}\right)\hbox{d}_{\theta_{t - 1}},$$
(21)

which is the expectation of a function of (θ t - 1) and can be shown as:

$$ = nz_{t} E{\left\{{{\left[ {{\int\limits_{nz_{t} \theta^{\min}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right) \theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)}^{nz_{t} \theta^{\max}_{t} - \left({\rm Ir}_{t} + {\rm Ra}_{t} + n\left(z_{t} - z_{t - 1}\right) \theta_{\rm r} - {\rm ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)} {\eta_{t} f_{\eta_{t}} \left(\eta_{t}\right)\hbox{d}_{\eta_{t}}}}} \right]}} \right\}},$$
(22)

where \(f_{\eta_{t}} \left(\eta_{t}\right)\) is presented as follows:

$$f_{\eta_{t}} \left(\eta_{t}\right) = \frac{1}{{{\sqrt {2\pi \hbox{Var}\left(\eta_{t}\right)}}}}\exp {\left[ {- \frac{1}{2}{\left({\frac{{\left(\eta_{t}\right)^{2}}}{{\hbox{Var}\left(\eta_{t}\right)}}} \right)}} \right]}.$$
(23)

The expectation of Eq. 22 is thus given as

$$E{\left\{{- \frac{{{\left({nz_{t - 1}} \right)}{\sqrt{\hbox{Var }\eta _{t}}}}}{{{\sqrt {2\pi}}}}\exp {\left({- \frac{1}{2}\frac{{\left(k - nz_{t - 1} \theta_{t - 1}\right)^{2}}}{{\hbox{Var }\eta_{t}}}} \right)}^{{nz_{t} \theta ^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right) \theta_{\rm r} - \hbox{ET}_{t}\right)}}_{{nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right)}}} \right\}}.$$
(24)

The second-order Taylor’s series approximation of the expectation of Eq. 24 is thus given as

$$\begin{aligned} &\frac{{{\sqrt{\hbox{Var }\eta_{t}}}}}{{{\sqrt{2\pi}}}}\hbox{exp} {\left({- \frac{1}{2}\frac{{\left(nz_{t} \theta^{\max}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t} E\left(\theta_{t - 1}\right)\right)^{2}}}{{\hbox{Var }\eta_{t}}}} \right)}\\ &\quad \times \left[ 1 + \frac{\hbox{Var}\left(\theta_{t - 1} \right)}{2} \left[ \frac{-\left(nz_{t - 1}\right)^{2}}{\hbox{Var }\eta_{t}} + \left( \frac{nz_{t}\theta_{t}^{\max} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t} \right) - nz_{t-1}E\left(\theta_{t - 1}\right)}{\hbox{Var }\eta_{t}}\left(nz_{t - 1}\right) \right)^{2}\right]\right]\\ &\quad - \frac{{{\sqrt{\hbox{Var }\eta_{t}}}}}{{{\sqrt{2\pi}}}}\hbox{exp} {\left({- \frac{1}{2}\frac{{\left(nz_{t} \theta^{\min}_{t} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t} E\left(\theta_{t - 1}\right)\right)^{2}}}{{\hbox{Var }\eta_{t}}}} \right)}\\ &\quad\times \left[ 1 + \frac{{\hbox{Var}\left(\theta_{t - 1}\right)}}{2}\left[ \frac{{- {\left({nz_{t - 1}} \right)}^{2}}}{{\hbox{Var }\eta_{t}}} + \left(\frac{{nz_{t} \theta^{t}_{\min} - \left(\hbox{Ir}_{t} + \hbox{Ra}_{t} + n\left(z_{t} - z_{t - 1}\right)\theta_{\rm r} - \hbox{ET}_{t}\right) - nz_{t - 1} E\left(\theta_{t - 1}\right)}}{{\hbox{Var }\eta_{t}}}{\left({nz_{t - 1}} \right)} \right)^{2} \right] \right].\\ \end{aligned}$$
(25)

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Ganji, A., Ponnambalam, K., Khalili, D. et al. A new stochastic optimization model for deficit irrigation. Irrig Sci 25, 63–73 (2006). https://doi.org/10.1007/s00271-006-0035-y

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