CAPRI Online Manual (update)

Files:

%curdir%/capdis/disyield.gms
%curdir%/capdis/disyield_sets.gms
%curdir%/capdis/m_hpdYieldIrri_init.gms
%curdir%/capdis/m_hpdYieldIrri.gms
%datdir%/capdishsu/pesetagrid_fractionfsu.gdx
%datdir%/capdishsu/irriShare2000fsu.gdx

CAPREG disaggregation: simulation model m_hpdYieldIrri.gms

Crop yield and irrigation shares are distributed synchronously under the assumption that realized yield on a certain spatial unit is increasing linearly from the simulated water-limited yield (PESETA) to the simulated potential yield.

\begin{equation} y_{h,c}=y_{h,c}^{wly}+f_{h,c}^{irri}\cdot (y_{h,c}^{py}-y_{h,c}^{wly}) \quad \forall h,c \end{equation}

\(y_{h,c}\) = Yield [variable, kg/ha] of crop c in spatial unit h under irrigation share \(f_{h,c}^{irri}\)
\(y_{h,c}^{wly}\) = Water limited yield [parameter, kg/ha] from PESETA simulations for PESETA, remapped to spatial unit h and crop c
\(f_{h,c}^{irri}\) = Irrigation share [variable, dimensionless] of crops c in spatial unit h.
\(y_{h,c}^{py}\) = Potential yield [parameter, kg/ha] from PESETA simulations for PESETA, remapped to spatial unit h and crop c

As the total production in the region must be recovered, but the water is limited and potential yields obtained from PESETA simulations are based on assumptions and parameters that are independent from – and therefore not necessarily consistent with – statistical yields available in CAPRI, a scaling factor is defined which links the observed average yield in the region with the average simulated water limited yield from PESETA. The water limited yield of a crop in each spatial unit is introduced as well relative to the average simulated water limited yield from PESETA.

\begin{equation} y_{h,c}^{wly}=Y_{r,c} \cdot s_{r,c}^Y \cdot r_{h,c}^{wly} \quad \forall h,c \end{equation}

\(y_{h,c}^{wly}\) = as defined above
\(Y_{r,c}\) = Observed yield [parameter, kg/ha] for crop c in region r from CAPRI simulations (ex post or ex ante)
\(s_{r,c}^Y\) = Scaling factor [variable, dimensionless] linking observed yield in region r to the average water limited yield \(Y_{r,c}^{wly} = \sum_h (a_{h,c}⋅y_{h,c}^{wly} )/ \sum_h(a_{h,c} )\) reflecting differences in the assumptions and underlying data between PESETA and CAPRI, and technological progress
\(r_{h,c}^{wly}\) = Relative water limited yield [parameter, dimensionless] of crop c in spatial unit h.

In practice, the product of the observed regional yield and the scaling factor \(Y_{r,c} \cdot s_{r,c}^Y\) represents the regional average water limited yield that is consistent with observed (or simulated) irrigation shares and regional observed yields.

With the above, we can formulate the constraints of recovering total regional production (Equation TOTYIELD_) and reformulate the equation for the crop yield per spatial unit MCACTYIELD_

\begin{equation} Y_{r,c} \cdot A_{r,c}=\sum_h \{y_{h,c} \cdot a_{h,c} \} \quad \forall r,c \end{equation}

\(y_{h,c}, Y_{r,c}\) = as defined above
\(a_{h,c}\) = Cultivated area [parameter, 1000 ha] of crop c in spatial unit h
\(A_{r,c}\) = Cultivated area [parameter, 1000 ha] of crop c in region r, with \(A_{r,c}=\sum_h\{a_{h,c}\}\)

\begin{equation} y_{h,c}=Y_{r,c}\cdot s_{r,c}^Y \cdot r_{h,c}^{wly}\cdot \left( 1 + f_{h,c}^{irri}\cdot (y_{h,c}^{relpy}-1)\right) \quad \forall h,c \end{equation}

\(y_{h,c}, Y_{r,c}, s_{r,c}^Y,r_{h,c}^{wly}, f_{h,c}^{irri}\) = as defined above
\(y_{h,c}^{relpy}\) = Relative yield increase [parameter, dimensionless] for crop c in spatial unit h if removing water limitation by full irrigation as from PESETA simulations. \(y_{h,c}^{relpy}=\frac{y_{h,c}^{py}}{y_{h,c}^{wly }\), and \(y_{h,c}^{relpy}-1=\frac{y_{h,c}^{py}-y_{h,c}^{wly}}{y_{h,c}^{wly}}\)

In principle, there are endless possibilities to adjust the irrigation shares to satisfy the equations TOTYIELD_ an MCACTYIELD_ so constraints in the irrigation shares itself are required. Those are found in statistical data on the irrigation shares by crop from the Survey on Agricultural Production Methods (SAPM, 2010) and the FAO irrigation map.

The objective function seeks to minimize the deviation of the irrigation shares by crop as given in SAPM (2010) base at regional level, and the total irrigation share as given in the Irrigation map (FAO), while trying to minimize the deviation of the yield in the spatial units from the average yield in the region. The first two terms in the objective function shall ensure that deviations from observed irrigation shares per unit and per crop are kept small, while the third term shall ensure that irrigation shares are higher where water limited yield is low.

\begin{align} \begin{split} hdp_{irri} \cdot \left( \sum_c A_{r,c} \right)^2 = \left\{ \sum_h \left( \sum_c (a_{h,c}) \cdot \frac{f_h^{irri} -\hat{f_h^{irri}}} {s_{f_h^{irri}}} \right)^2 + \sum_{h,c} \left( A_{r,c} \cdot \frac{r_{r,c}^{irri}-\hat{r_{r,c}^{irri}}} {s_{r_{r,c}^{irri}}} \right)^2 \\ + \sum_{h,c} \left( a_{h,c} \cdot \frac {y_{h,c}-\hat{Y_{r,c}}}{s_{y_{h,c}}} \right)^2 \right\} \end{split} \end{align}

\(a_{h,c}, y_{h,c}, Y_{r,c}\) = as defined above
\(f_h^{irri}\) = Irrigation share [variable, dimensionless] over crops in spatial unit h. This is calculated as weighted average of irrigation shares of individual crops, using the cultivated area as weights

\begin{equation} f_h^{irri} \cdot \sum_c \{a_{h,c} \} =\sum_c \{a_{h,c} \cdot f_{h,c}^{irri} \} \quad \forall h \end{equation}

\(r_{r,c}^{irri}\) = Relative irrigation share [variable, dimensionless] of crop c in region r. The relative irrigation share \(r_{r,c}^{irri}\) is calculated from average irrigation share of crop c over all spatial units in a region \(\sum_h \{a_{h,c} \cdot f_{h,c}^{irri} \} ⁄ \sum_h \{a_{h,c} \}\), divided by the average irrigation share of all crops \(c^\prime\) over all spatial units in a region \(\sum_{h,c^\prime}\{a_{h,c^\prime} \cdot f_{h,c^\prime}^{irri} \} ⁄ \sum_{h,c^\prime} \{ a_{h,c^\prime} \} \), or re-arranged:

\begin{equation} r_{r,c}^{irri} \cdot \sum_{h,c^\prime}\{a_{h,c^\prime} \cdot f_{h,c^\prime}^{irri} \} \cdot \sum_h \{a_{h,c} \} = \sum_h \{a_{h,c} \cdot f_{h,c}^{irri} \} \cdot \sum_{h,c^\prime} \{ a_{h,c^\prime} \} \quad \forall h \end{equation}

\(s_{f_h^{irri}}\), Standard deviation of the irrigation share of crop c in spatial unit h. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{f_h^{irri}}=max⁡(0.001,0.025 \cdot \hat{f_h^{irri}}) \)
\(s_{r_{r,c}^{irri}}\) Standard deviation of the relative irrigation share of crop c in region r. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{r_{r,c}^{irri}} =max⁡(0.001,0.025 \cdot \hat{r_{r,c}^{irri}})\)
\(s_{y_{h,c}}\) Standard deviation of the yield of crop c in spatial units h. As this is unknown it is calculated from a coefficient of variation of 2.5% according to \(s_{r_{r,c}^{irri}} = max⁡(0.001,0.025 \cdot Y_{r,c})\)

The ‘hats’ \(\hat{x} in Equation YIELDHPD_ indicate prior values of the variables, determined as explained below.

This completes the set of equations required to set up the model minimizing the objective function under the given constraints:

\begin{equation} m_{hpdYiedIrri} : min [ hdp_{irri} ] cf. TOTYIELD\_ ,MCACTYIELD\_ ,UAARIRRI\_ ,IRRISHRE\_ \end{equation}