Adjust calculation of PRICE_EMISSION

Author: OFR-IIASA

message_ix/model/MESSAGE/model_core.gms

@@ -131,21 +131,23 @@ Variables
 *
 * Auxiliary variables
 * ^^^^^^^^^^^^^^^^^^^
-* =========================================================================== ======================================================================================================
-* Variable                                                                    Explanatory text
-* =========================================================================== ======================================================================================================
-* :math:`\text{DEMAND}_{n,c,l,y,h} \in \mathbb{R}`                            Demand level (in equilibrium with MACRO integration)
-* :math:`\text{PRICE_COMMODITY}_{n,c,l,y,h} \in \mathbb{R}`                   Commodity price (undiscounted marginals of :ref:`commodity_balance_gt` and :ref:`commodity_balance_lt`)
-* :math:`\text{PRICE_EMISSION}_{n,\widehat{e},\widehat{t},y} \in \mathbb{R}`  Emission price (undiscounted marginals of :ref:`emission_constraint`)
-* :math:`\text{COST_NODAL_NET}_{n,y} \in \mathbb{R}`                          System costs at the node level net of energy trade revenues/cost
-* :math:`\text{GDP}_{n,y} \in \mathbb{R}`                                     Gross domestic product (GDP) in market exchange rates for MACRO reporting
-* =========================================================================== ======================================================================================================
+* ======================================================================================= =======================================================================================================
+* Variable                                                                                Explanatory text
+* ======================================================================================= =======================================================================================================
+* :math:`\text{DEMAND}_{n,c,l,y,h} \in \mathbb{R}`                                        Demand level (in equilibrium with MACRO integration)
+* :math:`\text{EMISSION_CONSTRAINT_RESCALE}_{n,\widehat{e},\widehat{t},y} \in \mathbb{R}` Annual average marginal of :ref:`emission_constraint`
+* :math:`\text{PRICE_COMMODITY}_{n,c,l,y,h} \in \mathbb{R}`                               Commodity price (undiscounted marginals of :ref:`commodity_balance_gt` and :ref:`commodity_balance_lt`)
+* :math:`\text{PRICE_EMISSION}_{n,\widehat{e},\widehat{t},y} \in \mathbb{R}`              Emission price (undiscounted marginals of :ref:`emission_constraint`)
+* :math:`\text{COST_NODAL_NET}_{n,y} \in \mathbb{R}`                                      System costs at the node level net of energy trade revenues/cost
+* :math:`\text{GDP}_{n,y} \in \mathbb{R}`                                                 Gross domestic product (GDP) in market exchange rates for MACRO reporting
+* ======================================================================================= =======================================================================================================
 *
 ***
 
 Variables
 * auxiliary variables for demand, prices, costs and GDP (for reporting when MESSAGE is run with MACRO)
     DEMAND(node,commodity,level,year_all,time) demand
+    EMISSION_CONSTRAINT_RESCALE(node,type_emission,type_tec,type_year) rescaled of marginals of EMISSION_CONSTRAINT constraint
     PRICE_COMMODITY(node,commodity,level,year_all,time)  commodity price (derived from marginals of COMMODITY_BALANCE constraint)
     PRICE_EMISSION(node,type_emission,type_tec,year_all) emission price (derived from marginals of EMISSION_BOUND constraint)
     COST_NODAL_NET(node,year_all)              system costs at the node level over time including effects of energy trade

message_ix/model/MESSAGE/model_solve.gms

@@ -14,7 +14,7 @@ if (%foresight% = 0,
 * This is the standard option; the GAMS global variable ``%foresight%=0`` by default.
 *
 * .. math::
-*    \min_x \text{OBJ} = \sum_{y \in Y} \text{OBJ}_y(x_y)
+*    \min_x OBJ = \sum_{y \in Y} OBJ_y(x_y)
 ***
 
 * reset year in case it was set by MACRO to include the base year before
@@ -40,11 +40,11 @@ if (%foresight% = 0,
     ) ;
 
 * rescale the dual of the emission constraint to account that the constraint is defined on the average year, not total
-EMISSION_CONSTRAINT.m(node,type_emission,type_tec,type_year)$(
+EMISSION_CONSTRAINT_RESCALE.l(node,type_emission,type_tec,type_year)$(
         EMISSION_CONSTRAINT.m(node,type_emission,type_tec,type_year) ) =
     EMISSION_CONSTRAINT.m(node,type_emission,type_tec,type_year)
         / SUM(year$( cat_year(type_year,year) ), duration_period(year) )
-        * SUM(year$( map_first_period(type_year,year) ), duration_period(year) / df_period(year) * df_year(year) );
+;
 
 
 * assign auxiliary variables DEMAND, PRICE_COMMODITY and PRICE_EMISSION for integration with MACRO and reporting
@@ -54,8 +54,8 @@ EMISSION_CONSTRAINT.m(node,type_emission,type_tec,type_year)$(
             / df_period(year) ;
     PRICE_EMISSION.l(node,type_emission,type_tec,year)$( SUM(type_year$( cat_year(type_year,year) ), 1 ) ) =
         SMAX(type_year$( cat_year(type_year,year) ),
-               - EMISSION_CONSTRAINT.m(node,type_emission,type_tec,type_year) )
-            / df_year(year) ;
+               - EMISSION_CONSTRAINT_RESCALE.l(node,type_emission,type_tec,type_year) )
+            / df_period(year) * duration_period(year);
     PRICE_EMISSION.l(node,type_emission,type_tec,year)$(
         PRICE_EMISSION.l(node,type_emission,type_tec,year) = - inf ) = 0 ;
 
@@ -77,10 +77,10 @@ else
 * Loop over :math:`\hat{y} \in Y`, solving
 *
 * .. math::
-*     \min_x \ \text{OBJ} = \sum_{y \in \hat{Y}(\hat{y})} \text{OBJ}_y(x_y) \\
+*     \min_x \ OBJ = \sum_{y \in \hat{Y}(\hat{y})} OBJ_y(x_y) \\
 *     \text{s.t. } x_{y'} = x_{y'}^* \quad \forall \ y' < y
 *
-* where :math:`\hat{Y}(\hat{y}) = \{y \in Y | \ |\hat{y}| - |y| < \text{optimization_horizon} \}` and
+* where :math:`\hat{Y}(\hat{y}) = \{y \in Y | \ |\hat{y}| - |y| < optimization\_horizon \}` and
 * :math:`x_{y'}^*` is the optimal value of :math:`x_{y'}` in iteration :math:`|y'|` of the iterative loop.
 *
 * The advantage of this implementation is that there is no need to 'store' the optimal values of all decision