Fossil (31_fossil)

Description

The Fossil Module calculates the costs of a specific amount of fossil resource extraction.

Interfaces

Input

Output

Realizations

(A) exogenous

For this realization exogenous fossil extraction and costs are used. The data are from a baseline run.

Limitations Fossil fuel extraction and costs are fixed to exogenous values.

(B) grades2poly

This realization parametrizes fossil extraction cost curves into 3rd-order polynomials for each fuel (oil, gas and coal) in each region. Input data are taken from REMIND runs with the timeDepGrades fossil realization under various fossil fuel availability assumptions. This approximation of the original cost-grade-based extraction algorithm greatly reduces model runtime.

Uranium extraction costs parameterized as 3rd order polynomial with short-term calibrated adjustment costs which capture inertias, e.g. from infrastructure

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \left( p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCum(ttot,regi,enty,"1") + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCum(ttot,regi,enty,"1")^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCum(ttot,regi,enty,"1")^{3 }\right) \cdot \left(1\$\left(ttot.val eq 2005\right) + \left(\left(1-p31\_fosadjco\_xi5xi6(regi,"xi5",enty)\right) + p31\_fosadjco\_xi5xi6(regi,"xi5",enty) \cdot \left(\frac{\left(vm\_fuExtr(ttot,regi,enty,"1")+1.e-5\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,"1")\$\left(ttot.val gt 2005\right)+1.e-5\right)}\right)^{p31\_fosadjco\_xi5xi6(regi,"xi6",enty) }\right)\$\left(ttot.val gt 2005\right) \right) \cdot vm\_fuExtr(ttot,regi,enty,"1") \end{multline*}\]

Oil, gas and coal extraction costs fitted to existing REMIND output as a 3rd-order polynomial with short-term calibrated adjustment costs which capture inertias, e.g. from infrastructure

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \left( \left( p31\_ffPolyCoeffs(regi,enty,"0") + p31\_ffPolyCoeffs(regi,enty,"1") \cdot v31\_fuExtrCum(ttot,regi,enty,"1") + p31\_ffPolyCoeffs(regi,enty,"2") \cdot v31\_fuExtrCum(ttot,regi,enty,"1")^{2 }+ p31\_ffPolyCoeffs(regi,enty,"3") \cdot v31\_fuExtrCum(ttot,regi,enty,"1")^{3 }\right) \cdot \left(1\$\left(ttot.val eq 2005\right) +\left(\left(\left(1-p31\_fosadjco\_xi5xi6(regi,"xi5",enty)\right) + p31\_fosadjco\_xi5xi6(regi,"xi5",enty) \cdot \left(\frac{\left(vm\_fuExtr(ttot,regi,enty,"1")+1.e-5\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,"1")\$\left(ttot.val gt 2005\right)+1.e-5\right)}\right)^{p31\_fosadjco\_xi5xi6(regi,"xi6",enty) }\right)\$\left(ttot.val gt 2005\right)\right) \right) + \left( p31\_rentdisctot(ttot,enty) \cdot \left( p31\_ffPolyRent(regi,enty,"0") + p31\_ffPolyRent(regi,enty,"1") \cdot v31\_fuExtrCum(ttot,regi,enty,"1") \right)\$\left(v31\_fuExtrCum.l(ttot,regi,enty,"1")\$\left(ttot.val ge 2010 and ttot.val le 2150\right) gt pm\_ffPolyCumEx(regi,enty,"min")\right) \right) \right) \cdot vm\_fuExtr(ttot,regi,enty,"1") \end{multline*}\]

Cumulated fuel extraction (oil, gas and coal) is the sum of extraction in each time step multiplied by the time step length.

\[\begin{multline*} v31\_fuExtrCum(ttot,regi,enty,"1") = v31\_fuExtrCum(ttot-1,regi,enty,"1")\$\left(ttot.val gt 2005\right) + pm\_ts(ttot) \cdot \left(vm\_fuExtr(ttot,regi,enty,"1") \right) \end{multline*}\]

\[\begin{multline*} v31\_fuExtrMC(enty,"1") = p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right) + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{3 } \end{multline*}\]

\[\begin{multline*} v31\_squaredDiff = \left(s31\_max\_disp\_peur - \sum_{regi}\left( v31\_fuExtrCumMax\left(regi, "peur", "1"\right)\right)\right) \cdot \left(s31\_max\_disp\_peur - \sum_{regi}\left( v31\_fuExtrCumMax\left(regi, "peur", "1"\right)\right)\right) \end{multline*}\]

Limitations There are no known limitations.

(C) MOFEX

This realization is dedicated to the running the standalone version of MOFEX (Model Of Fossil EXtraction), which minimizes the discounted extraction and trade costs of fossils while balancing trade for each time step. This is not to be run within a REMIND run but instead through the standalone architecture or in a soft-linked iteration with REMIND (not yet implemented)

MOFEX (Model of Fossil Extraction) takes fossil demand, imports and exports from a prior REMIND run as inputs, and it calculates fossil extraction and trade as outputs.

\[\begin{multline*} \sum_{regi}\left( vm\_Xport(t,regi,trade) - vm\_Mport(t,regi,trade)\right) = 0 \end{multline*}\]

\[\begin{multline*} v31\_MOFEX\_costMinFuelEx = \sum_{ttot\$\left(ttot.val ge cm\_startyear\right)}\left( \sum_{regi}\left(\frac{ pm\_ts(ttot) }{ \left(\left(1 + pm\_prtp(regi)\right)^{\left(pm\_ttot\_val(ttot)-cm\_startyear\right)}\right) } \cdot \left(\sum_{peFos(enty)} vm\_costFuEx(ttot,regi,enty) + \sum_{peFos(enty)}\left( pm\_costsTradePeFinancial(regi,"Mport",enty) \cdot vm\_Mport(ttot,regi,enty)\right) + \sum_{peFos(enty)}\left( \left(pm\_costsTradePeFinancial(regi,"Xport",enty) \cdot vm\_Xport(ttot,regi,enty)\right) \cdot \left(1 +\frac{ pm\_costsTradePeFinancial(regi,"XportElasticity",enty) }{ sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right) } \cdot \left(\frac{ vm\_Xport(ttot,regi,enty) }{ \left(vm\_Xport(ttot-1,regi,enty) + pm\_costsTradePeFinancial\left(regi, "tradeFloor",enty\right)\right) }- 1\right) \right) \right) \right) \right) \right) \end{multline*}\]

Fossil fuels are represented by discrete grades based on ranges of marginal extraction costs. The total extraction cost for each time step is calculated based on long-term marginal extraction costs and short-term calibrated adjustment costs which capture inertias, e.g. from infrastructure

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \sum_{pe2rlf(enty,rlf)}\left( \left(\left(p31\_grades\left(ttot,regi,"xi1",enty, rlf\right) + pm\_costsTradePeFinancial(regi,"use",enty) + \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ v31\_fuExtrCum\left(ttot-1,regi,enty, rlf\right)\$\left(ttot.val gt 2005\right) }{ p31\_grades\left(ttot,regi,"xi3",enty, rlf\right) }+ \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ \left(v31\_fuExtrCum(ttot,regi,enty,rlf)-v31\_fuExtrCum(ttot-1,regi,enty,rlf)\$\left(ttot.val gt 2005\right)\right) }{ \left(2 \cdot p31\_grades\left(ttot,regi,"xi3",enty, rlf\right)\right) }\right) \cdot \left( \left(1 + \left(p31\_datafosdyn(regi,enty,rlf,"alph") \cdot \frac{ 1}{\left(sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right)\right) } \cdot sqr\left(\frac{\left(vm\_fuExtr(ttot,regi,enty,rlf)-vm\_fuExtr(ttot-1,regi,enty,rlf)\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,rlf)+ 0.001 \cdot p31\_grades(ttot,regi,"xi3",enty,rlf) + p31\_extraseed(ttot,regi,enty,rlf) + 1.e-9\right)}\right) \right)\$\left(ttot.val gt 2005\right) \right) \right) \cdot vm\_fuExtr(ttot,regi,enty,rlf) \right)\$\left(p31\_grades(ttot,regi,"xi3",enty,rlf) gt 0\right) \right) \end{multline*}\]

Cumulated fuel extraction (oil, gas and coal) is the sum of extraction in each time step multiplied by the time step length. If early retirement of oil wells is switched on, any slack capacity from those fields is also added.

q31_fuExtrCum(ttot,regi,pe2rlf(peEx(enty),rlf))$(ttot.val ge cm_startyear)..
        v31_fuExtrCum(ttot,regi,enty,rlf) 
        =e= 
        v31_fuExtrCum(ttot-1,regi,enty,rlf)$(ttot.val gt 2005) + pm_ts(ttot)*(vm_fuExtr(ttot,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot,regi,enty,rlf)
        );

These dynamic constraints on the decline and increase rates of production reflect physical and technical inertias of oil, gas and coal Dynamic constraint on decline rate

q31_fuExtrDec(ttot+1,regi,enty2rlf_dec(enty,rlf))$(pm_ttot_val(ttot+1) ge max(2010,cm_startyear))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot+1,regi,enty,rlf)
         =g=
          p31_datafosdyn(regi,enty,rlf,"decoffset") * p31_grades(ttot,regi,"xi3",enty,rlf) * 0.5 * ( pm_ts(ttot+1) + pm_ts(ttot) )
$if not setglobal test_TS   + (1-p31_datafosdyn(regi,enty,rlf,"dec"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * vm_fuExtr(ttot,regi,enty,rlf);

\[\begin{multline*} v31\_fuSlack(ttot+1,regi,enty,rlf) \leq v31\_fuSlack(ttot,regi,enty,rlf) + \left(pm\_ttot\_val(ttot+1)-pm\_ttot\_val(ttot)\right) \cdot cm\_earlyreti\_rate \cdot 0.3 \cdot p31\_max\_oil\_extraction(regi,enty,rlf) \end{multline*}\]

Dynamic constraint on increase rate

q31_fuExtrInc(ttot+1,regi,enty2rlf_inc(enty,rlf))$((p31_grades(ttot,regi,"xi3",enty,rlf) gt 0) AND (pm_ttot_val(ttot+1) ge max(2010,cm_startyear)))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
         =l=
         (1 + p31_datafosdyn(regi,enty,rlf,"inc"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * (vm_fuExtr(ttot,regi,enty,rlf) + p31_datafosdyn(regi,enty,rlf,"incoffset"))
$ifthen.cm_oil_scen %cm_oil_scen% == "highOil"    +(10)$(cm_startyear eq 2015 AND pm_ttot_val(ttot+1) eq 2015 AND ((sameas(enty,"peoil") AND sameas(rlf,"7")) OR (sameas(enty,"pegas") AND sameas(rlf,"6"))))
$elseif.cm_oil_scen %cm_oil_scen% == "6"          +(10)$(cm_startyear eq 2015 AND pm_ttot_val(ttot+1) eq 2015 AND (sameas(enty,"peoil") AND sameas(rlf,"1") AND sameas(regi,"REF")));

option nlp = conopt4;  !! Greatly speed up convergence process (x3~x4)

Lower bounds on fossil fuel extraction for all time steps To make the model “see” all grades

loop(regi,
  loop(peFos(enty),
    loop(rlf,
      loop(t,
        if (p31_grades(t,regi,"xi3",enty,rlf) gt 0,
          vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf)) = 1e-9;
        );
      );
    );
  );
);

Special case for grades declining to a zero value Set lower and upper bounds to 0.0 to make the model converge. p31_grades declines linearly whereas vm_fuExtr declines exponentially This particular situation prevent the model from finding a solution [TODO] In the future a small amount should be added to p31_grades to allow for extraction from these grades

loop(regi,
  loop(peFos(enty),
    loop(rlf,
      if (p31_grades("2005",regi,"xi3",enty,rlf) gt 0.0 and p31_grades("2035",regi,"xi3",enty,rlf) eq 0.0,
        vm_fuExtr.up(t,regi,pe2rlf(enty,rlf))$(rlf.val ge 6) = 0.0;
        vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf))$(rlf.val ge 6) = 0.0;
        vm_fuExtr.up(t,regi,pe2rlf(enty,rlf))$(rlf.val lt 6) = 0.0;
        vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf))$(rlf.val lt 6) = 0.0;
      );
    );
  );
);
if (s31_debug eq 1,
  display vm_fuExtr.lo, vm_fuExtr.up;
);
v31_fuExtrCum.up(t,regi,peExGrade(enty),rlf) = p31_grades(t,regi,"xi3",enty,rlf);

Upper bound on fossil fuel costs

vm_costFuEx.up(t,regi,peExGrade(enty)) = 10.0;

option nlp = conopt4;  !! Greatly speed up convergence process (x3~x4)
o_modelstat = 100;
option solprint=on
if(o_modelstat ne 2,
   solve m31_MOFEX using nlp minimizing v31_MOFEX_costMinFuelEx;
   o_modelstat = m31_MOFEX.modelstat;
);
p31_MOFEX_fuelex_costMin(ttot,regi,enty,rlf)  = vm_fuExtr.l(ttot,regi,enty,rlf);
p31_MOFEX_cumfex_costMin(ttot,regi,enty,rlf)  = v31_fuExtrCum.l(ttot,regi,enty,rlf);
p31_MOFEX_Mport_costMin(ttot,regi,trade)      = vm_Mport.l(ttot,regi,trade);
p31_MOFEX_Xport_costMin(ttot,regi,trade)      = vm_Xport.l(ttot,regi,trade);
if(m31_MOFEX.modelstat ne 2,
  Execute_Unload 'mofex';
  abort "MOFEX did not find an optimal solution. Stopping job...";
);
vm_prodPe.lo(ttot,regi,peExGrade(enty)) = 1.e-9;
vm_prodPe.up(ttot,regi,peExGrade(enty)) = 1.e+2;
display p31_MOFEX_fuelex_costMin;
option nlp = %cm_conoptv%;

Limitations There are no known limitations.

(D) timeDepGrades

This realization represents fossil fuel resources as a time-dependent cost grade structure. The grades are each defined
by a minimum and a maximum price, and these cost brackets change over time based on the rate of technological change as prescribed exogenously by the socioeconomic scenario. Each model region contains different volumes of each cost grade of each fuel (oil, gas and coal) based on data compiled from IIASA’s 2012 Global Energy Assessment and the German Federal Institute for Geosciences and
Natural Resources (BGR) The model extracts resources in a Hotelling fashion, such that lower cost grades are depleted in each region
before shifting production to higher-cost grades.

Uranium extraction is represented as a 3rd order polynomial parametrized by long-term marginal extraction costs.

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \left( p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right) + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right)^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right)^{3 }\right) \cdot \cdot ' Short term decline and increase rates are determined by adjustment costs. \left(1\$\left(ttot.val eq 2005\right) +\left( \left(1-p31\_fosadjco\_xi5xi6(regi,"xi5",enty)\right) + p31\_fosadjco\_xi5xi6(regi,"xi5",enty) \cdot \left(\frac{\left(vm\_fuExtr(ttot,regi,enty,"1")+1.e-5\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,"1")\$\left(ttot.val gt 2005\right)+1.e-5\right)}\right)^{p31\_fosadjco\_xi5xi6(regi,"xi6",enty) }\right)\$\left(ttot.val gt 2005\right) \right) \cdot vm\_fuExtr(ttot,regi,enty,"1") \end{multline*}\]

Two dummy equations further determine regional uranium extraction bounds.

\[\begin{multline*} v31\_fuExtrMC(enty,"1") = p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right) + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{3 } \end{multline*}\]

\[\begin{multline*} s31\_max\_disp\_peur = \sum_{regi}\left( v31\_fuExtrCumMax\left(regi,enty, "1"\right)\right) \end{multline*}\]

MOFEX (Model of Fossil Extraction) minimizes the discounted extraction and trade costs of fossils while balancing trade for each time step. The model takes fossil demand, imports and exports from a prior REMIND run as inputs, and it calculates fossil extraction and trade as outputs. In this realization, the output variables are returned to the current REMIND run as starting points for the hybrid optimization.

\[\begin{multline*} \sum_{regi}\left( vm\_Xport(t,regi,trade) - vm\_Mport(t,regi,trade)\right) = 0 \end{multline*}\]

\[\begin{multline*} v31\_MOFEX\_costMinFuelEx = \sum_{ttot\$\left(ttot.val ge cm\_startyear\right)}\left( \sum_{regi}\left(\frac{ pm\_ts(ttot) }{ \left(\left(1 + pm\_prtp(regi)\right)^{\left(pm\_ttot\_val(ttot)-cm\_startyear\right)}\right) } \cdot \left(\sum_{peFos(enty)} vm\_costFuEx(ttot,regi,enty) + \sum_{peFos(enty)}\left( pm\_costsTradePeFinancial(regi,"Mport",enty) \cdot vm\_Mport(ttot,regi,enty)\right) + \sum_{peFos(enty)}\left( \left(pm\_costsTradePeFinancial(regi,"Xport",enty) \cdot vm\_Xport(ttot,regi,enty)\right) \cdot \left(1 +\frac{ pm\_costsTradePeFinancial(regi,"XportElasticity",enty) }{ sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right) } \cdot \left(\frac{ vm\_Xport(ttot,regi,enty) }{ \left(vm\_Xport(ttot-1,regi,enty) + pm\_costsTradePeFinancial\left(regi, "tradeFloor",enty\right)\right) }- 1\right) \right) \right) \right) \right) \right) \end{multline*}\]

Fossil fuels are represented by discrete grades based on ranges of marginal extraction costs. The total extraction cost for each time step is calculated based on long-term marginal extraction costs and short-term calibrated adjustment costs which capture inertias, e.g. from infrastructure

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \sum_{pe2rlf(enty,rlf)}\left( \left(\left(p31\_grades\left(ttot,regi,"xi1",enty, rlf\right) + pm\_costsTradePeFinancial(regi,"use",enty) + \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ v31\_fuExtrCum\left(ttot-1,regi,enty, rlf\right)\$\left(ttot.val gt 2005\right) }{ p31\_grades\left(ttot,regi,"xi3",enty, rlf\right) }+ \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ \left(v31\_fuExtrCum(ttot,regi,enty,rlf)-v31\_fuExtrCum(ttot-1,regi,enty,rlf)\$\left(ttot.val gt 2005\right)\right) }{ \left(2 \cdot p31\_grades\left(ttot,regi,"xi3",enty, rlf\right)\right) }\right) \cdot \left( \left(1 + \left(p31\_datafosdyn(regi,enty,rlf,"alph") \cdot \frac{ 1}{\left(sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right)\right) } \cdot sqr\left(\frac{\left(vm\_fuExtr(ttot,regi,enty,rlf)-vm\_fuExtr(ttot-1,regi,enty,rlf)\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,rlf)+ 0.001 \cdot p31\_grades(ttot,regi,"xi3",enty,rlf) + p31\_extraseed(ttot,regi,enty,rlf) + 1.e-9\right)}\right) \right)\$\left(ttot.val gt 2005\right) \right) \right) \cdot vm\_fuExtr(ttot,regi,enty,rlf) \right)\$\left(p31\_grades(ttot,regi,"xi3",enty,rlf) gt 0\right) \right) \end{multline*}\]

Cumulated fuel extraction (oil, gas and coal) is the sum of extraction in each time step multiplied by the time step length. If early retirement of oil wells is switched on, any slack capacity from those fields is also added.

q31_fuExtrCum(ttot,regi,pe2rlf(peEx(enty),rlf))$(ttot.val ge cm_startyear)..
        v31_fuExtrCum(ttot,regi,enty,rlf) 
        =e= 
        v31_fuExtrCum(ttot-1,regi,enty,rlf)$(ttot.val gt 2005) + pm_ts(ttot)*(vm_fuExtr(ttot,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot,regi,enty,rlf)
        );

These dynamic constraints on the decline and increase rates of production reflect physical and technical inertias of oil, gas and coal Dynamic constraint on decline rate

q31_fuExtrDec(ttot+1,regi,enty2rlf_dec(enty,rlf))$(pm_ttot_val(ttot+1) ge max(2010,cm_startyear))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot+1,regi,enty,rlf)
         =g=
          p31_datafosdyn(regi,enty,rlf,"decoffset") * p31_grades(ttot,regi,"xi3",enty,rlf) * 0.5 * ( pm_ts(ttot+1) + pm_ts(ttot) )
$if not setglobal test_TS   + (1-p31_datafosdyn(regi,enty,rlf,"dec"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * vm_fuExtr(ttot,regi,enty,rlf);

\[\begin{multline*} v31\_fuSlack(ttot+1,regi,enty,rlf) \leq v31\_fuSlack(ttot,regi,enty,rlf) + \left(pm\_ttot\_val(ttot+1)-pm\_ttot\_val(ttot)\right) \cdot cm\_earlyreti\_rate \cdot 0.3 \cdot p31\_max\_oil\_extraction(regi,enty,rlf) \end{multline*}\]

Dynamic constraint on increase rate

q31_fuExtrInc(ttot+1,regi,enty2rlf_inc(enty,rlf))$((p31_grades(ttot,regi,"xi3",enty,rlf) gt 0) AND (pm_ttot_val(ttot+1) ge max(2010,cm_startyear)))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
         =l=
         (1 + p31_datafosdyn(regi,enty,rlf,"inc"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * (vm_fuExtr(ttot,regi,enty,rlf) + p31_datafosdyn(regi,enty,rlf,"incoffset"))
$ifthen.cm_oil_scen %cm_oil_scen% == "highOil"    +(10)$(cm_startyear eq 2015 AND pm_ttot_val(ttot+1) eq 2015 AND ((sameas(enty,"peoil") AND sameas(rlf,"7")) OR (sameas(enty,"pegas") AND sameas(rlf,"6"))))
                                                  +(10)$(cm_startyear eq 2015 AND cm_oil_scen eq 6 AND pm_ttot_val(ttot+1) eq 2015 AND (sameas(enty,"peoil") AND sameas(rlf,"1") AND sameas(regi,"REF")));

  abort "MOFEX did not find an optimal solution. Stopping job...";

Uranium extraction is represented as a 3rd order polynomial parametrized by long-term marginal extraction costs.

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \left( p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right) + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right)^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCum\left(ttot,regi,enty, "1"\right)^{3 }\right) \cdot \cdot ' Short term decline and increase rates are determined by adjustment costs. \left(1\$\left(ttot.val eq 2005\right) +\left( \left(1-p31\_fosadjco\_xi5xi6(regi,"xi5",enty)\right) + p31\_fosadjco\_xi5xi6(regi,"xi5",enty) \cdot \left(\frac{\left(vm\_fuExtr(ttot,regi,enty,"1")+1.e-5\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,"1")\$\left(ttot.val gt 2005\right)+1.e-5\right)}\right)^{p31\_fosadjco\_xi5xi6(regi,"xi6",enty) }\right)\$\left(ttot.val gt 2005\right) \right) \cdot vm\_fuExtr(ttot,regi,enty,"1") \end{multline*}\]

Two dummy equations further determine regional uranium extraction bounds.

\[\begin{multline*} v31\_fuExtrMC(enty,"1") = p31\_costExPoly(regi,"xi1",enty) + p31\_costExPoly(regi,"xi2",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right) + p31\_costExPoly(regi,"xi3",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{2 }+ p31\_costExPoly(regi,"xi4",enty) \cdot v31\_fuExtrCumMax\left(regi,enty, "1"\right)^{3 } \end{multline*}\]

\[\begin{multline*} s31\_max\_disp\_peur = \sum_{regi}\left( v31\_fuExtrCumMax\left(regi,enty, "1"\right)\right) \end{multline*}\]

MOFEX (Model of Fossil Extraction) minimizes the discounted extraction and trade costs of fossils while balancing trade for each time step. The model takes fossil demand, imports and exports from a prior REMIND run as inputs, and it calculates fossil extraction and trade as outputs. In this realization, the output variables are returned to the current REMIND run as starting points for the hybrid optimization.

\[\begin{multline*} \sum_{regi}\left( vm\_Xport(t,regi,trade) - vm\_Mport(t,regi,trade)\right) = 0 \end{multline*}\]

\[\begin{multline*} v31\_MOFEX\_costMinFuelEx = \sum_{ttot\$\left(ttot.val ge cm\_startyear\right)}\left( \sum_{regi}\left(\frac{ pm\_ts(ttot) }{ \left(\left(1 + pm\_prtp(regi)\right)^{\left(pm\_ttot\_val(ttot)-cm\_startyear\right)}\right) } \cdot \left(\sum_{peFos(enty)} vm\_costFuEx(ttot,regi,enty) + \sum_{peFos(enty)}\left( pm\_costsTradePeFinancial(regi,"Mport",enty) \cdot vm\_Mport(ttot,regi,enty)\right) + \sum_{peFos(enty)}\left( \left(pm\_costsTradePeFinancial(regi,"Xport",enty) \cdot vm\_Xport(ttot,regi,enty)\right) \cdot \left(1 +\frac{ pm\_costsTradePeFinancial(regi,"XportElasticity",enty) }{ sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right) } \cdot \left(\frac{ vm\_Xport(ttot,regi,enty) }{ \left(vm\_Xport(ttot-1,regi,enty) + pm\_costsTradePeFinancial\left(regi, "tradeFloor",enty\right)\right) }- 1\right) \right) \right) \right) \right) \right) \end{multline*}\]

Fossil fuels are represented by discrete grades based on ranges of marginal extraction costs. The total extraction cost for each time step is calculated based on long-term marginal extraction costs and short-term calibrated adjustment costs which capture inertias, e.g. from infrastructure

\[\begin{multline*} vm\_costFuEx(ttot,regi,enty) = \sum_{pe2rlf(enty,rlf)}\left( \left(\left(p31\_grades\left(ttot,regi,"xi1",enty, rlf\right) + pm\_costsTradePeFinancial(regi,"use",enty) + \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ v31\_fuExtrCum\left(ttot-1,regi,enty, rlf\right)\$\left(ttot.val gt 2005\right) }{ p31\_grades\left(ttot,regi,"xi3",enty, rlf\right) }+ \left(p31\_grades(ttot,regi,"xi2",enty,rlf)-p31\_grades\left(ttot,regi,"xi1",enty, rlf\right)\right) \cdot \frac{ \left(v31\_fuExtrCum(ttot,regi,enty,rlf)-v31\_fuExtrCum(ttot-1,regi,enty,rlf)\$\left(ttot.val gt 2005\right)\right) }{ \left(2 \cdot p31\_grades\left(ttot,regi,"xi3",enty, rlf\right)\right) }\right) \cdot \left( \left(1 + \left(p31\_datafosdyn(regi,enty,rlf,"alph") \cdot \frac{ 1}{\left(sqr\left(pm\_ttot\_val(ttot)-pm\_ttot\_val(ttot-1)\right)\right) } \cdot sqr\left(\frac{\left(vm\_fuExtr(ttot,regi,enty,rlf)-vm\_fuExtr(ttot-1,regi,enty,rlf)\right)}{\left(vm\_fuExtr(ttot-1,regi,enty,rlf)+ 0.001 \cdot p31\_grades(ttot,regi,"xi3",enty,rlf) + p31\_extraseed(ttot,regi,enty,rlf) + 1.e-9\right)}\right) \right)\$\left(ttot.val gt 2005\right) \right) \right) \cdot vm\_fuExtr(ttot,regi,enty,rlf) \right)\$\left(p31\_grades(ttot,regi,"xi3",enty,rlf) gt 0\right) \right) \end{multline*}\]

Cumulated fuel extraction (oil, gas and coal) is the sum of extraction in each time step multiplied by the time step length. If early retirement of oil wells is switched on, any slack capacity from those fields is also added.

q31_fuExtrCum(ttot,regi,pe2rlf(peEx(enty),rlf))$(ttot.val ge cm_startyear)..
        v31_fuExtrCum(ttot,regi,enty,rlf) 
        =e= 
        v31_fuExtrCum(ttot-1,regi,enty,rlf)$(ttot.val gt 2005) + pm_ts(ttot)*(vm_fuExtr(ttot,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot,regi,enty,rlf)
        );

These dynamic constraints on the decline and increase rates of production reflect physical and technical inertias of oil, gas and coal Dynamic constraint on decline rate

q31_fuExtrDec(ttot+1,regi,enty2rlf_dec(enty,rlf))$(pm_ttot_val(ttot+1) ge max(2010,cm_startyear))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
$if %cm_OILRETIRE% == "on"    + v31_fuSlack(ttot+1,regi,enty,rlf)
         =g=
          p31_datafosdyn(regi,enty,rlf,"decoffset") * p31_grades(ttot,regi,"xi3",enty,rlf) * 0.5 * ( pm_ts(ttot+1) + pm_ts(ttot) )
$if not setglobal test_TS   + (1-p31_datafosdyn(regi,enty,rlf,"dec"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * vm_fuExtr(ttot,regi,enty,rlf);

\[\begin{multline*} v31\_fuSlack(ttot+1,regi,enty,rlf) \leq v31\_fuSlack(ttot,regi,enty,rlf) + \left(pm\_ttot\_val(ttot+1)-pm\_ttot\_val(ttot)\right) \cdot cm\_earlyreti\_rate \cdot 0.3 \cdot p31\_max\_oil\_extraction(regi,enty,rlf) \end{multline*}\]

Dynamic constraint on increase rate

q31_fuExtrInc(ttot+1,regi,enty2rlf_inc(enty,rlf))$((p31_grades(ttot,regi,"xi3",enty,rlf) gt 0) AND (pm_ttot_val(ttot+1) ge max(2010,cm_startyear)))..
         vm_fuExtr(ttot+1,regi,enty,rlf)
         =l=
         (1 + p31_datafosdyn(regi,enty,rlf,"inc"))**(pm_ttot_val(ttot+1)-pm_ttot_val(ttot)) * (vm_fuExtr(ttot,regi,enty,rlf) + p31_datafosdyn(regi,enty,rlf,"incoffset"))
$ifthen.cm_oil_scen %cm_oil_scen% == "highOil"    +(10)$(cm_startyear eq 2015 AND pm_ttot_val(ttot+1) eq 2015 AND ((sameas(enty,"peoil") AND sameas(rlf,"7")) OR (sameas(enty,"pegas") AND sameas(rlf,"6"))))
                                                  +(10)$(cm_startyear eq 2015 AND cm_oil_scen eq 6 AND pm_ttot_val(ttot+1) eq 2015 AND (sameas(enty,"peoil") AND sameas(rlf,"1") AND sameas(regi,"REF")));

  abort "MOFEX did not find an optimal solution. Stopping job...";

Lower bounds on fossil fuel extraction for all time steps To make the model “see” all grades

if(ord(iteration) eq 1,
  loop(regi,
    loop(peFos(enty),
      loop(rlf,
        loop(t,
          if (p31_grades(t,regi,"xi3",enty,rlf) gt 0,
            vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf)) = 1e-9;
          );
        );
      );
    );
  );
);

Special case for grades declining to a zero value Set lower and upper bounds to 0.0 to make the model converge. p31_grades declines linearly whereas vm_fuExtr declines exponentially This particular situation prevent the model from finding a solution [TODO] In the future a small amount should be added to p31_grades to allow for extraction from these grades

if(ord(iteration) eq 1,
  loop(regi,
    loop(peFos(enty),
      loop(rlf,
        if (p31_grades("2005",regi,"xi3",enty,rlf) gt 0.0 and p31_grades("2035",regi,"xi3",enty,rlf) eq 0.0,
          vm_fuExtr.up(t,regi,pe2rlf(enty,rlf))$(rlf.val ge 6) = 0.0;
          vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf))$(rlf.val ge 6) = 0.0;
          vm_fuExtr.up(t,regi,pe2rlf(enty,rlf))$(rlf.val lt 6) = 0.0;
          vm_fuExtr.lo(t,regi,pe2rlf(enty,rlf))$(rlf.val lt 6) = 0.0;
        );
      );
    );
  );
);
if (s31_debug eq 1,
  display vm_fuExtr.lo, vm_fuExtr.up;
);
v31_fuExtrCum.up(t,regi,peExGrade(enty),rlf) = p31_grades(t,regi,"xi3",enty,rlf);

Upper bound on fossil fuel costs

vm_costFuEx.up(t,regi,peExGrade(enty)) = 10.0;

Limitations p31_grades used in equation emissengregi in core/equations.gms, calculation of p31_prod_ini, p31_prod_share in grades/preloop.gms uses additional interfaces (pm_fosadjco_xi7xi8 and vm_edemini) not used

Definitions

Objects

Sets

Authors

Nico Bauer, Jerome Hilaire, Robert Pietzcker, Lavinia Baumstark

See Also

02_welfare, 24_trade, core

References