Note

This page is generated from inline documentation in MESSAGE\model_core.gms.

Mathematical formulation (core model)

The MESSAGEix systems-optimization model minimizes total costs while satisfying given demand levels for commodities/services and considering a broad range of technical/engineering constraints and societal restrictions (e.g. bounds on greenhouse gas emissions, pollutants, system reliability). Demand levels are static (i.e. non-elastic), but the demand response can be integrated by linking MESSAGEix to the single sector general-economy MACRO model included in this framework.

For the complete list of sets, mappings and parameters, refer to the auto-documentation pages Sets and mappings definition and Parameter definition.

Notation declaration

The following short notation is used in the mathematical description relative to the GAMS code:

Mathematical notation of sets

Math notation GAMS set & index notation
\(n \in N\) node (across spatial hierarchy levels)
\(y \in Y\) year (all periods including historical and model horizon)
\(y \in Y^M \subset Y\) time periods included in model horizon
\(y \in Y^H \subset Y\) historical time periods (prior to first model period)
\(c \in C\) commodity
\(l \in L\) level
\(g \in G\) grade
\(t \in T\) technology (a.k.a tec)
\(h \in H\) time (subannual time periods)
\(m \in M\) mode
\(q \in Q\) rating of non-dispatchable technologies relative to aggregate commodity use
\(e \in E\) emission, pollutants
\(s \in S\) scenarios of land use (for land-use model emulator)
\(u \in U\) land-use types
\(r \in R\) set of generic relations (linear constraints)
\(t \in T^{INV} \subseteq T\) all technologies with investment decisions and capacity constraints
\(t \in T^{REN} \subseteq T\) all technologies which draw their input from the renewable level
\(n \in N(\widehat{n})\) all nodes that are subnodes of node \(\widehat{n}\)
\(y \in Y(\widehat{y})\) all years mapped to the category type_year \(\widehat{y}\)
\(t \in T(\widehat{t})\) all technologies mapped to the category type_tec \(\widehat{t}\)
\(e \in E(\widehat{e})\) all emissions mapped to the category type_emission \(\widehat{e}\)

Decision variables

Variable Explanatory text
\(OBJ \in \mathbb{R}\) Objective value of the optimization program
\(EXT_{n,c,g,y} \in \mathbb{R}_+\) Extraction of non-renewable/exhaustible resources from reserves
\(STOCK_{n,c,l,y} \in \mathbb{R}_+\) Quantity in stock (storage) at start of period \(y\)
\(STOCK\_CHG_{n,c,l,y,h} \in \mathbb{R}\) Input or output quantity into intertemporal commodity stock (storage)
\(REN_{n,t,c,g,y,h}\) Activity of renewable technologies per grade
\(CAP\_NEW_{n,t,y} \in \mathbb{R}_+\) Newly installed capacity (yearly average over period duration)
\(CAP_{n,t,y^V,y} \in \mathbb{R}_+\) Maintained capacity in year \(y\) of vintage \(y^V\)
\(CAP\_FIRM_{n,t,c,l,y,q}\) Capacity counting towards firm (dispatchable)
\(ACT_{n,t,y^V,y,m,h} \in \mathbb{R}\) Activity of a technology (by vintage, mode, subannual time)
\(ACT\_RATING_{n,t,y^V,y,c,l,h,q}\) Activity attributed to a particular rating bin [1]
\(CAP\_NEW\_UP_{n,t,y} \in \mathbb{R}_+\) Relaxation of upper dynamic constraint on new capacity
\(CAP\_NEW\_LO_{n,t,y} \in \mathbb{R}_+\) Relaxation of lower dynamic constraint on new capacity
\(ACT\_UP_{n,t,y,h} \in \mathbb{R}_+\) Relaxation of upper dynamic constraint on activity [2]
\(ACT\_LO_{n,t,y,h} \in \mathbb{R}_+\) Relaxation of lower dynamic constraint on activity [2]
\(LAND_{n,s,y} \in [0,1]\) Relative share of land-use scenario (for land-use model emulator)
\(EMISS_{n,e,\widehat{t},y}\) Auxiliary variable for aggregate emissions by technology type
\(REL_{r,n,y} \in \mathbb{R}\) Auxiliary variable for left-hand side of relations (linear constraints)
\(COMMODITY\_USE_{n,c,l,y}\) Auxiliary variable for amount of commodity used at specific level

The index \(y^V\) is the year of construction (vintage) wherever it is necessary to clearly distinguish between year of construction and the year of operation.

All decision variables are by year, not by (multi-year) period, except \(STOCK_{n,c,l,y}\). In particular, the new capacity variable \(CAP\_NEW_{n,t,y}\) has to be multiplied by the number of years in a period \(|y| = duration\_period_{y}\) to determine the available capacity in subsequent periods. This formulation gives more flexibility when it comes to using periods of different duration (more intuitive comparison across different periods).

The current model framework allows both input or output normalized formulation. This will affect the parametrization, see Section Efficiency - output- vs. input defined technologies for more details.

[1]The auxiliary variable \(ACT\_RATING_{n,t,y^V,y,c,l,h,q}\) is defined in terms of input or output of the technology.
[2](1, 2) The dynamic activity constraints are implemented as summed over all modes; therefore, the variables for the relaxation are not indexed over the set mode.

Auxiliary variables

Variable Explanatory text
\(DEMAND_{n,c,l,y,h} \in \mathbb{R}\) Demand level (in equilibrium with MACRO integration)
\(PRICE\_COMMODITY_{n,c,l,y,h}\) Commodity price (undiscounted marginals of COMMODITY_BALANCE constraint)
\(PRICE\_EMISSION_{n,e,\widehat{t},y}\) Emission price (undiscounted marginals of EMISSION_BOUND constraint)
\(COST\_NODAL\_NET_{n,y} \in \mathbb{R}\) System costs at the node level net of energy trade revenues/cost
\(GDP_{n,y} \in \mathbb{R}\) gross domestic product (GDP) in market exchange rates for MACRO reporting

Objective function

The objective function of the MESSAGEix core model

Equation OBJECTIVE

The objective function (of the core model) minimizes total discounted systems costs including costs for emissions, relaxations of dynamic constraints

\[OBJ = \sum_{n,y \in Y^{M}} discountfactor_{y} \cdot COST\_NODAL_{n,y}\]

Regional system cost accounting function

Accounting of regional system costs over time

Equation COST_ACCOUNTING_NODAL

Accounting of regional systems costs over time as well as costs for emissions (taxes), land use (from the model land-use model emulator), relaxations of dynamic constraints, and linear relations.

\[\begin{split}COST\_NODAL_{n,y} & = \sum_{c,g} \ resource\_cost_{n,c,g,y} \cdot EXT_{n,c,g,y} \\ & + \sum_{t} \ \bigg( inv\_cost_{n,t,y} \cdot construction\_time\_factor_{n,t,y} \\ & \quad \quad \quad \cdot end\_of\_horizon\_factor_{n,t,y} \cdot CAP\_NEW_{n,t,y} \\[4 pt] & \quad \quad + \sum_{y^V \leq y} \ fix\_cost_{n,t,y^V,y} \cdot CAP_{n,t,y^V,y} \\ & \quad \quad + \sum_{\substack{y^V \leq y \\ m,h}} \ var\_cost_{n,t,y^V,y,m,h} \cdot ACT_{n,t,y^V,y,m,h} \\ & \quad \quad + \Big( abs\_cost\_new\_capacity\_soft\_up_{n,t,y} \\ & \quad \quad \quad + level\_cost\_new\_capacity\_soft\_up_{n,t,y} \cdot\ inv\_cost_{n,t,y} \Big) \cdot CAP\_NEW\_UP_{n,t,y} \\[4pt] & \quad \quad + \Big( abs\_cost\_new\_capacity\_soft\_lo_{n,t,y} \\ & \quad \quad \quad + level\_cost\_new\_capacity\_soft\_lo_{n,t,y} \cdot\ inv\_cost_{n,t,y} \Big) \cdot CAP\_NEW\_LO_{n,t,y} \\[4pt] & \quad \quad + \sum_{m,h} \ \Big( abs\_cost\_activity\_soft\_up_{n,t,y,m,h} \\ & \quad \quad \quad + level\_cost\_activity\_soft\_up_{n,t,y,m,h} \cdot\ levelized\_cost_{n,t,y,m,h} \Big) \cdot ACT\_UP_{n,t,y,h} \\ & \quad \quad + \sum_{m,h} \ \Big( abs\_cost\_activity\_soft\_lo_{n,t,y,m,h} \\ & \quad \quad \quad + level\_cost\_activity\_soft\_lo_{n,t,y,m,h} \cdot\ levelized\_cost_{n,t,y,m,h} \Big) \cdot ACT\_LO_{n,t,y,h} \bigg) \\ & + \sum_{\substack{\widehat{e},\widehat{t} \\ e \in E(\widehat{e})}} emission\_scaling_{\widehat{e},e} \cdot \ emission\_tax_{n,\widehat{e},\widehat{t},y} \cdot EMISS_{n,e,\widehat{t},y} \\ & + \sum_{s} land\_cost_{n,s,y} \cdot LAND_{n,s,y} \\ & + \sum_{r} relation\_cost_{r,n,y} \cdot REL_{r,n,y}\end{split}\]

Here, \(n^L \in N(n)\) are all nodes \(n^L\) that are sub-nodes of node \(n\). The subset of technologies \(t \in T(\widehat{t})\) are all tecs that belong to category \(\widehat{t}\), and similar notation is used for emissions \(e \in E\).

Resource and commodity section

Constraints on resource extraction

Equation EXTRACTION_EQUIVALENCE

This constraint translates the quantity of resources extracted (summed over all grades) to the input used by all technologies (drawing from that node). It is introduced to simplify subsequent notation in input/output relations and nodal balance constraints.

\[\begin{split}\sum_{g} EXT_{n,c,g,y} = \sum_{\substack{n^L,t,m,h,h^{OD} \\ y^V \leq y \\ \ l \in L^{RES} \subseteq L }} input_{n^L,t,y^V,y,m,n,c,l,h,h^{OD}} \cdot ACT_{n^L,t,m,y,h}\end{split}\]

The set \(L^{RES} \subseteq L\) denotes all levels for which the detailed representation of resources applies.

Equation EXTRACTION_BOUND_UP

This constraint specifies an upper bound on resource extraction by grade.

\[EXT_{n,c,g,y} \leq bound\_extraction\_up_{n,c,g,y}\]

Equation RESOURCE_CONSTRAINT

This constraint restricts that resource extraction in a year guarantees the “remaining resources” constraint, i.e., only a given fraction of remaining resources can be extracted per year.

\[\begin{split}EXT_{n,c,g,y} \leq resource\_remaining_{n,c,g,y} \cdot \Big( & resource\_volume_{n,c,g} \\ & - \sum_{y' < y} duration\_period_{y'} \cdot EXT_{n,c,g,y'} \Big)\end{split}\]

Equation RESOURCE_HORIZON

This constraint ensures that total resource extraction over the model horizon does not exceed the available resources.

\[\sum_{y} duration\_period_{y} \cdot EXT_{n,c,g,y} \leq resource\_volume_{n,c,g}\]

Constraints on commodities and stocks

Equation COMMODITY_BALANCE

This constraint ensures the nodal and temporal balance of outputs/imports and inputs/exports for all commodities.

\[\begin{split}\sum_{\substack{n^L,t,m,h^A \\ y^V \leq y}} output_{n^L,t,y^V,y,m,n,c,l,h^A,h} \cdot duration\_time\_rel_{h,h^A} \cdot & ACT_{n^L,t,y^V,y,m,h^A} \\ - \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y}} input_{n^L,t,y^V,y,m,n,c,l,h^A,h} \cdot duration\_time\_rel_{h,h^A} \cdot & ACT_{n^L,t,m,y,h^A} \\ + \ STOCK\_CHG_{n,c,l,y,h} & \\[4pt] + \ \sum_s \Big( land\_output_{n,s,y,c,l,h} - land\_input_{n,s,y,c,l,h} \Big) \cdot & LAND_{n,s,y} \\[4pt] - \ demand\_fixed_{n,c,l,y,h} & \geq 0 \quad \forall \ l \notin L^{RES} \subseteq L\end{split}\]

The commodity balance constraint at the resource level is included in the Equation RESOURCE_CONSTRAINT, while at the renewable level, it is included in the Equation RENEWABLES_EQUIVALENCE.

Equation STOCKS_BALANCE

This constraint ensures the inter-temporal balance of commodity stocks. The parameter \(commodity\_stocks_{n,c,l}\) can be used to model exogenous additions to the stock

\[\begin{split}STOCK_{n,c,l,y} + commodity\_stock_{n,c,l,y} = duration\_period_{y} \cdot & \sum_{h} STOCK\_CHG_{n,c,l,y,h} \\ & + STOCK_{n,c,l,y+1}\end{split}\]

Technology section

Technical and engineering constraints

The first set of constraints concern technologies that have explicit investment decisions and where installed/maintained capacity is relevant for operational decisions. The set where \(T^{INV} \subseteq T\) is the set of all these technologies.

Equation CAPACITY_CONSTRAINT

This constraint ensures that the actual activity of a technology at a node cannot exceed available (maintained) capacity summed over all vintages, including the technology capacity factor \(capacity\_factor_{n,t,y,t}\).

\[\sum_{m} ACT_{n,t,y^V,y,m,h} \leq duration\_time_{h} \cdot capacity\_factor_{n,t,y^V,y,h} \cdot CAP_{n,t,y^V,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation CAPACITY_MAINTENANCE_HIST

The following three constraints implement technology capacity maintenance over time to allow early retirment. The optimization problem determines the optimal timing of retirement, when fixed operation-and-maintenance costs exceed the benefit in the objective function.

The first constraint ensures that historical capacity (built prior to the model horizon) is available as installed capacity in the first model period.

\[\begin{split}CAP_{n,t,y^V,'first\_period'} & \leq remaining\_capacity_{n,t,y^V,'first\_period'} \cdot duration\_period_{y^V} \cdot historical\_new\_capacity_{n,t,y^V} \\ & \text{if } y^V < 'first\_period' \text{ and } |y| - |y^V| < technical\_lifetime_{n,t,y^V} \quad \forall \ t \in T^{INV}\end{split}\]

Equation CAPACITY_MAINTENANCE_NEW

The second constraint ensures that capacity is fully maintained throughout the model period in which it was constructed (no early retirement in the period of construction).

\[CAP_{n,t,y^V,y^V} = remaining\_capacity_{n,t,y^V,y^V} \cdot duration\_period_{y^V} \cdot CAP\_NEW{n,t,y^V} \quad \forall \ t \in T^{INV}\]

The current formulation does not account for construction time in the constraints, but only adds a mark-up to the investment costs in the objective function.

Equation CAPACITY_MAINTENANCE

The third constraint implements the dynamics of capacity maintenance throughout the model horizon. Installed capacity can be maintained over time until decommissioning, which is irreversible.

\[\begin{split}CAP_{n,t,y^V,y} & \leq remaining\_capacity_{n,t,y^V,y} \cdot CAP_{n,t,y^V,y-1} \\ \quad & \text{if } y > y^V \text{ and } y^V > 'first\_period' \text{ and } |y| - |y^V| < technical\_lifetime_{n,t,y^V} \quad \forall \ t \in T^{INV}\end{split}\]

Equation OPERATION_CONSTRAINT

This constraint provides an upper bound on the total operation of installed capacity over a year. It can be used to represent reuqired scheduled unavailability of installed capacity.

\[\sum_{m,h} ACT_{n,t,y^V,y,m,h} \leq operation\_factor_{n,t,y^V,y} \cdot capacity\_factor_{n,t,y^V,y,m,\text{'year'}} \cdot CAP_{n,t,y^V,y} \quad \forall \ t \in T^{INV}\]

This constraint is only active if \(operation\_factor_{n,t,y^V,y} < 1\).

Equation MIN_UTILIZATION_CONSTRAINT

This constraint provides a lower bound on the total utilization of installed capacity over a year.

\[\sum_{m,h} ACT_{n,t,y^V,y,m,h} \geq min\_utilization\_factor_{n,t,y^V,y} \cdot CAP_{n,t,y^V,y} \quad \forall \ t \in T^{INV}\]

This constraint is only active if \(min\_utilization\_factor_{n,t,y^V,y}\) is defined.

Constraints representing renewable integration

Equation RENEWABLES_EQUIVALENCE

This constraint defines the auxiliary variables \(REN\) to be equal to the output of renewable technologies (summed over grades).

\[\begin{split}\sum_{g} REN_{n,t,c,g,y,h} \leq \sum_{\substack{n,t,m,l,h,h^{OD} \\ y^V \leq y \\ \ l \in L^{REN} \subseteq L }} input_{n^L,t,y^V,y,m,n,c,l,h,h^{OD}} \cdot ACT_{n^L,t,m,y,h}\end{split}\]

The set \(L^{REN} \subseteq L\) denotes all levels for which the detailed representation of renewables applies.

Equation RENEWABLES_POTENTIAL_CONSTRAINT

This constraint sets the potential potential by grade as the upper bound for the auxiliary variable \(REN\).

\[\begin{split}\sum_{\substack{t,h \\ \ t \in T^{R} \subseteq t }} REN_{n,t,c,g,y,h} \leq \sum_{\substack{l \\ l \in L^{R} \subseteq L }} renewable\_potential_{n,c,g,l,y}\end{split}\]

Equation RENEWABLES_CAPACITY_REQUIREMENT

This constraint connects the capacity factor of a renewable grade to the installed capacity of a technology. It sets the lower limit for the capacity of a renewable technology to the summed activity over all grades (REN) devided by the capactiy factor of this grade. It represents the fact that different renewable grades require different installed capacities to provide their full potential.

\[\begin{split}\sum_{y^V, h} & CAP_{n,t,y^V,y} \cdot operation\_factor_{n,t,y^V,y} \cdot capacity\_factor_{n,t,y^V,y,h} \\ & \quad \geq \sum_{g,h,l} \frac{1}{renewable\_capacity\_factor_{n,c,g,l,y}} \cdot REN_{n,t,c,g,y,h}\end{split}\]

This constraint is only active if \(renewable\_capacity\_factor_{n,c,g,l,y}\) is defined.

Constraints for addon technologies

Equation ADDON_ACTIVITY_UP

This constraint provides an upper bound on the activity of an addon technology that can only be operated jointly with a parent technology (e.g., abatement option, SO2 scrubber, power plant cooling technology).

\[\sum_{\substack{t' \sim t^A, y^V \leq y}} ACT_{n,t',y^V,y,m,h} \leq \sum_{\substack{t, y^V \leq y}} addon\_up_{n,t^a,y,m,h,t^A} \cdot addon\_conversion_{n,t',y^V,y,m,h} \cdot ACT_{n,t,y^V,y,m,h}\]

Equation ADDON_ACTIVITY_LO

This constraint provides a lower bound on the activity of an addon technology that has to be operated jointly with a parent technology (e.g., power plant cooling technology). The parameter addon_lo allows to define a minimum level of operation of addon technologies relative to the activity of the parent technology. If addon_minimum = 1, this means that it is mandatory to operate the addon technology at the same level as the parent technology (i.e., full mitigation).

\[\sum_{\substack{t' \sim t^A, y^V \leq y}} ACT_{n,t',y^V,y,m,h} \geq \sum_{\substack{t, y^V \leq y}} addon\_lo_{n,t^a,y,m,h,t^A} \cdot addon\_conversion_{n,t',y^V,y,m,h} \cdot ACT_{n,t,y^V,y,m,h}\]

System reliability and flexibility requirements

This section followi allows to include system-wide reliability and flexility considerations. The current formulation is based on Sullivan et al., 2013 [6].

Aggregate use of a commodity

The system reliability and flexibility constraints are implemented using an auxiliary variable representing the total use (i.e., input of each commodity per level).

Equation COMMODITY_USE_LEVEL

This constraint defines the auxiliary variable \(COMMODITY\_USE_{n,c,l,y}\), which is used to define the rating bins and the peak-load that needs to be offset with firm (dispatchable) capacity.

\[\begin{split}COMMODITY\_USE_{n,c,l,y} = & \sum_{n^L,t,y^V,m,h} input_{n^L,t,y^V,y,m,n,c,l,h,h} \\ & \quad \cdot duration\_time\_rel_{h,h} \cdot ACT_{n^L,t,y^V,y,m,h}\end{split}\]

This constraint and the auxiliary variable is only active if \(peak\_load\_factor_{n,c,l,y,h}\) or \(flexibility\_factor_{n,t,y^V,y,m,c,l,h,r}\) is defined.

Auxilary variables for technology activity by “rating bins”

The capacity and activity of certain (usually non-dispatchable) technologies can be assumed to only partially contribute to the system reliability and flexibility requirements.

Equation ACTIVITY_RATING_BIN

The auxiliary variable for rating-specific activity of each technology cannot exceed the share of the rating bin in relation to the total commodity use.

\[ACT\_RATING_{n,t,y^V,y,c,l,h,q} \leq rating\_bin_{n,t,y,c,l,h,q} \cdot COMMODITY\_USE_{n,c,l,y}\]

Equation ACTIVITY_SHARE_TOTAL

The sum of the auxiliary rating-specific activity variables need to equal the total input and/or output of the technology.

\[\begin{split}\sum_q ACT\_RATING_{n,t,y^V,y,c,l,h,q} = \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y}} & ( input_{n^L,t,y^V,y,m,n,c,l,h^A,h} + output_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n^L,t,y^V,y,m,h^A} \\\end{split}\]

Reliability of installed capacity

The “firm capacity” that a technology can contribute to system reliability depends on its dispatch characteristics. For dispatchable technologies, the total installed capacity counts toward the firm capacity constraint. This is active if the parameter is defined over \(reliability\_factor_{n,t,y,c,l,h,'firm'}\). For non-dispatchable technologies, or those that do not have explicit investment decisions, the contribution to system reliability is calculated by using the auxiliary variable \(ACT\_RATING_{n,t,y^V,y,c,l,h,q}\) as a proxy, with the \(reliability\_factor_{n,t,y,c,l,h,q}\) defined per rating bin \(q\).

Equation FIRM_CAPACITY_PROVISION

Technologies where the reliability factor is defined with the rating firm have an auxiliary variable \(CAP\_FIRM_{n,t,c,l,y,q}\), defined in terms of output.

\[\begin{split}\sum_q CAP\_FIRM_{n,t,c,l,y,q} = \sum_{y^V \leq y} & output_{n^L,t,y^V,y,m,n,c,l,h^A,h} \cdot duration\_time_h \\ & \quad \cdot capacity\_factor_{n,t,y^V,y,h} \cdot CAP_{n,t,y^Y,y} \quad \forall \ t \in T^{INV}\end{split}\]

Equation SYSTEM_RELIABILITY_CONSTRAINT

This constraint ensures that there is sufficient firm (dispatchable) capacity in each period. The formulation is based on Sullivan et al., 2013 [6].

\[\begin{split}\sum_{t, q \substack{t \in T^{INV} \\ y^V \leq y} } & reliability\_factor_{n,t,y,c,l,h,'firm'} \cdot CAP\_FIRM_{n,t,c,l,y} \\ + \sum_{t,q,y^V \leq y} & reliability\_factor_{n,t,y,c,l,h,q} \cdot ACT\_RATING_{n,t,y^V,y,c,l,h,q} \\ & \quad \geq peak\_load\_factor_{n,c,l,y,h} \cdot COMMODITY\_USE_{n,c,l,y}\end{split}\]

This constraint is only active if \(peak\_load\_factor_{n,c,l,y,h}\) is defined.

Equation SYSTEM_FLEXIBILITY_CONSTRAINT

This constraint ensures that, in each sub-annual time slice, there is a sufficient contribution from flexible technologies to ensure smooth system operation.

\[\begin{split}\sum_{\substack{n^L,t,m,h^A \\ y^V \leq y}} & flexibility\_factor_{n^L,t,y^V,y,m,c,l,h,'unrated'} \\ & \quad \cdot ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n,t,y^V,y,m,h} \\ + \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y}} & flexibility\_factor_{n^L,t,y^V,y,m,c,l,h,1} \\ & \quad \cdot ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT\_RATING_{n,t,y^V,y,c,l,h,q} \geq 0\end{split}\]

Bounds on capacity and activity

Equation NEW_CAPACITY_BOUND_UP

This constraint provides upper bounds on new capacity installation.

\[CAP\_NEW_{n,t,y} \leq bound\_new\_capacity\_up_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation NEW_CAPACITY_BOUND_LO

This constraint provides lower bounds on new capacity installation.

\[CAP\_NEW_{n,t,y} \geq bound\_new\_capacity\_lo_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation TOTAL_CAPACITY_BOUND_UP

This constraint gives upper bounds on the total installed capacity of a technology in a specific year of operation summed over all vintages.

\[\sum_{y^V \leq y} CAP_{n,t,y,y^V} \leq bound\_total\_capacity\_up_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation TOTAL_CAPACITY_BOUND_LO

This constraint gives lower bounds on the total installed capacity of a technology.

\[\sum_{y^V \leq y} CAP_{n,t,y,y^V} \geq bound\_total\_capacity\_lo_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation ACTIVITY_BOUND_UP

This constraint provides upper bounds by mode of a technology activity, summed over all vintages.

\[\sum_{y^V \leq y} ACT_{n,t,y^V,y,m,h} \leq bound\_activity\_up_{n,t,m,y,h}\]

Equation ACTIVITY_BOUND_ALL_MODES_UP

This constraint provides upper bounds of a technology activity across all modes and vintages.

\[\sum_{y^V \leq y, m} ACT_{n,t,y^V,y,m,h} \leq bound\_activity\_up_{n,t,y,'all',h}\]

Equation ACTIVITY_BOUND_LO

This constraint provides lower bounds by mode of a technology activity, summed over all vintages.

\[\sum_{y^V \leq y} ACT_{n,t,y^V,y,m,h} \geq bound\_activity\_lo_{n,t,y,m,h}\]

We assume that \(bound\_activity\_lo_{n,t,y,m,h} = 0\) unless explicitly stated otherwise.

Equation ACTIVITY_BOUND_ALL_MODES_LO

This constraint provides lower bounds of a technology activity across all modes and vintages.

\[\sum_{y^V \leq y, m} ACT_{n,t,y^V,y,m,h} \geq bound\_activity\_lo_{n,t,y,'all',h}\]

We assume that \(bound\_activity\_lo_{n,t,y,'all',h} = 0\) unless explicitly stated otherwise.

Constraints on shares of technologies and commodities

This section allows to include upper and lower bounds on the shares of modes used by a technology or the shares of commodities produced or consumed by groups of technologies.

Share constraints on activity by mode

Equation SHARES_MODE_UP

This constraint provides upper bounds of the share of the activity of one mode of a technology. For example, it could limit the share of heat that can be produced in a combined heat and electricity power plant.

\[ACT_{n^L,t,y^V,y,m,h^A} \leq share\_mode\_up_{s,n,y,m,h} \cdot \sum_{m\prime} ACT_{n^L,t,y^V,y,m\prime,h^A}\]

Equation SHARES_MODE_LO

This constraint provides lower bounds of the share of the activity of one mode of a technology.

\[ACT_{n^L,t,y^V,y,m,h^A} \geq share\_mode\_lo_{s,n,y,m,h} \cdot \sum_{m\prime} ACT_{n^L,t,y^V,y,m\prime,h^A}\]

Share constraints on commodities

These constraints allow to set upper and lower bound on the quantity of commodities produced or consumed by a group of technologies relative to the commodities produced or consumed by another group.

The implementation is generic and flexible, so that any combination of commodities, levels, technologies and nodes can be put in relation to any other combination.

The notation \(S^{share}\) represents the mapping set map_shares_commodity_share denoting all technology types, nodes, commodities and levels to be included in the numerator, and \(S^{total}\) is the equivalent mapping set map_shares_commodity_total for the denominator.

Equation SHARE_CONSTRAINT_COMMODITY_UP

\[\begin{split}& \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y, (n,\widehat{t},m,c,l) \sim S^{share}}} ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n^L,t,y^V,y,m,h^A} \\ & \geq share\_commodity\_up_{s,n,y,h} \cdot \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y, (n,\widehat{t},m,c,l) \sim S^{total}}} ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n^L,t,y^V,y,m,h^A}\end{split}\]

This constraint is only active if \(share\_commodity\_up_{s,n,y,h}\) is defined.

Equation SHARE_CONSTRAINT_COMMODITY_LO

\[\begin{split}& \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y, (n,\widehat{t},m,c,l) \sim S^{share}}} ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n^L,t,y^V,y,m,h^A} \\ & \leq share\_commodity\_lo_{s,n,y,h} \cdot \sum_{\substack{n^L,t,m,h^A \\ y^V \leq y, (n,\widehat{t},m,c,l) \sim S^{total}}} ( output_{n^L,t,y^V,y,m,n,c,l,h^A,h} + input_{n^L,t,y^V,y,m,n,c,l,h^A,h} ) \\ & \quad \cdot duration\_time\_rel_{h,h^A} \cdot ACT_{n^L,t,y^V,y,m,h^A}\end{split}\]

This constraint is only active if \(share\_commodity\_lo_{s,n,y,h}\) is defined.

Dynamic constraints on market penetration

The constraints in this section specify dynamic upper and lower bounds on new capacity and activity, i.e., constraints on market penetration and rate of expansion or phase-out of a technology.

The formulation directly includes the option for ‘soft’ relaxations of dynamic constraints (cf. Keppo and Strubegger, 2010 [3]).

Equation NEW_CAPACITY_CONSTRAINT_UP

The level of new capacity additions cannot be greater than an initial value (compounded over the period duration), annual growth of the existing ‘capital stock’, and a “soft” relaxation of the upper bound.

\[\begin{split}CAP\_NEW_{n,t,y} \leq & ~ initial\_new\_capacity\_up_{n,t,y} \cdot \frac{ \Big( 1 + growth\_new\_capacity\_up_{n,t,y} \Big)^{|y|} - 1 } { growth\_new\_capacity\_up_{n,t,y} } \\ & + \Big( CAP\_NEW_{n,t,y-1} + historical\_new\_capacity_{n,t,y-1} \Big) \\ & \hspace{2 cm} \cdot \Big( 1 + growth\_new\_capacity\_up_{n,t,y} \Big)^{|y|} \\ & + CAP\_NEW\_UP_{n,t,y} \cdot \Bigg( \Big( 1 + soft\_new\_capacity\_up_{n,t,y}\Big)^{|y|} - 1 \Bigg) \\ & \quad \forall \ t \ \in \ T^{INV}\end{split}\]

Here, \(|y|\) is the number of years in period \(y\), i.e., \(duration\_period_{y}\).

Equation NEW_CAPACITY_SOFT_CONSTRAINT_UP

This constraint ensures that the relaxation of the dynamic constraint on new capacity (investment) does not exceed the level of the investment in the same period (cf. Keppo and Strubegger, 2010 [3]).

\[CAP\_NEW\_UP_{n,t,y} \leq CAP\_NEW_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation NEW_CAPACITY_CONSTRAINT_LO

This constraint gives dynamic lower bounds on new capacity.

\[\begin{split}CAP\_NEW_{n,t,y} \geq & - initial\_new\_capacity\_lo_{n,t,y} \cdot \frac{ \Big( 1 + growth\_new\_capacity\_lo_{n,t,y} \Big)^{|y|} } { growth\_new\_capacity\_lo_{n,t,y} } \\ & + \Big( CAP\_NEW_{n,t,y-1} + historical\_new\_capacity_{n,t,y-1} \Big) \\ & \hspace{2 cm} \cdot \Big( 1 + growth\_new\_capacity\_lo_{n,t,y} \Big)^{|y|} \\ & - CAP\_NEW\_LO_{n,t,y} \cdot \Bigg( \Big( 1 + soft\_new\_capacity\_lo_{n,t,y}\Big)^{|y|} - 1 \Bigg) \\ & \quad \forall \ t \ \in \ T^{INV}\end{split}\]

Equation NEW_CAPACITY_SOFT_CONSTRAINT_LO

This constraint ensures that the relaxation of the dynamic constraint on new capacity does not exceed level of the investment in the same year.

\[CAP\_NEW\_LO_{n,t,y} \leq CAP\_NEW_{n,t,y} \quad \forall \ t \ \in \ T^{INV}\]

Equation ACTIVITY_CONSTRAINT_UP

This constraint gives dynamic upper bounds on the market penetration of a technology activity.

\[\begin{split}\sum_{y^V \leq y,m} ACT_{n,t,y^V,y,m,h} \leq & ~ initial\_activity\_up_{n,t,y,h} \cdot \frac{ \Big( 1 + growth\_activity\_up_{n,t,y,h} \Big)^{|y|} - 1 } { growth\_activity\_up_{n,t,y,h} } \\ & + \bigg( \sum_{y^V \leq y-1,m} ACT_{n,t,y^V,y-1,m,h} + \sum_{m} historical\_activity_{n,t,y-1,m,h} \bigg) \\ & \hspace{2 cm} \cdot \Big( 1 + growth\_activity\_up_{n,t,y,h} \Big)^{|y|} \\ & + ACT\_UP_{n,t,y,h} \cdot \Bigg( \Big( 1 + soft\_activity\_up_{n,t,y,h} \Big)^{|y|} - 1 \Bigg)\end{split}\]

Equation ACTIVITY_SOFT_CONSTRAINT_UP

This constraint ensures that the relaxation of the dynamic activity constraint does not exceed the level of the activity.

\[ACT\_UP_{n,t,y,h} \leq \sum_{y^V \leq y,m} ACT_{n^L,t,y^V,y,m,h}\]

Equation ACTIVITY_CONSTRAINT_LO

This constraint gives dynamic lower bounds on the market penetration of a technology activity.

\[\begin{split}\sum_{y^V \leq y,m} ACT_{n,t,y^V,y,m,h} \geq & - initial\_activity\_lo_{n,t,y,h} \cdot \frac{ \Big( 1 + growth\_activity\_lo_{n,t,y,h} \Big)^{|y|} - 1 } { growth\_activity\_lo_{n,t,y,h} } \\ & + \bigg( \sum_{y^V \leq y-1,m} ACT_{n,t,y^V,y-1,m,h} + \sum_{m} historical\_activity_{n,t,y-1,m,h} \bigg) \\ & \hspace{2 cm} \cdot \Big( 1 + growth\_activity\_lo_{n,t,y,h} \Big)^{|y|} \\ & - ACT\_LO_{n,t,y,h} \cdot \Bigg( \Big( 1 + soft\_activity\_lo_{n,t,y,h} \Big)^{|y|} - 1 \Bigg)\end{split}\]

Equation ACTIVITY_SOFT_CONSTRAINT_LO

This constraint ensures that the relaxation of the dynamic activity constraint does not exceed the level of the activity.

\[ACT\_LO_{n,t,y,h} \leq \sum_{y^V \leq y,m} ACT_{n,t,y^V,y,m,h}\]

Emission section

Auxiliary variable for aggregate emissions

Equation EMISSION_EQUIVALENCE

This constraint simplifies the notation of emissions aggregated over different technology types and the land-use model emulator. The formulation includes emissions from all sub-nodes \(n^L\) of \(n\).

\[\begin{split}EMISS_{n,e,\widehat{t},y} = \sum_{n^L \in N(n)} \Bigg( \sum_{t \in T(\widehat{t}),y^V \leq y,m,h } emission\_factor_{n^L,t,y^V,y,m,e} \cdot ACT_{n^L,t,y^V,y,m,h} \\ + \sum_{s} \ land\_emission_{n^L,s,y,e} \cdot LAND_{n^L,s,y} \text{ if } \widehat{t} \in \widehat{T}^{LAND} \Bigg)\end{split}\]

Bound on emissions

Equation EMISSION_CONSTRAINT

This constraint enforces upper bounds on emissions (by emission type). For all bounds that include multiple periods, the parameter \(bound\_emission_{n,\widehat{e},\widehat{t},\widehat{y}}\) is scaled to represent average annual emissions over all years included in the year-set \(\widehat{y}\).

The formulation includes historical emissions and allows to model constraints ranging over both the model horizon and historical periods.

\[\begin{split}\frac{ \sum_{y' \in Y(\widehat{y}), e \in E(\widehat{e})} \begin{array}{l} duration\_period_{y'} \cdot emission\_scaling_{\widehat{e},e} \cdot \\ \Big( EMISS_{n,e,\widehat{t},y'} + \sum_{m} historical\_emission_{n,e,\widehat{t},y'} \Big) \end{array} } { \sum_{y' \in Y(\widehat{y})} duration\_period_{y'} } \leq bound\_emission_{n,\widehat{e},\widehat{t},\widehat{y}}\end{split}\]

Land-use model emulator section

Bounds on total land use

Equation LAND_CONSTRAINT

This constraint enforces a meaningful result of the land-use model emulator, in particular a bound on the total land used in MESSAGEix. The linear combination of land scenarios must be equal to 1.

\[\sum_{s \in S} LAND_{n,s,y} = 1\]

Dynamic constraints on land use

These constraints enforces upper and lower bounds on the change rate per land scenario.

Equation DYNAMIC_LAND_SCEN_CONSTRAINT_UP

\[\begin{split}LAND_{n,s,y} \leq & initial\_land\_scen\_up_{n,s,y} \cdot \frac{ \Big( 1 + growth\_land\_scen\_up_{n,s,y} \Big)^{|y|} - 1 } { growth\_land\_scen\_up_{n,s,y} } \\ & + \big( LAND_{n,s,y-1} + historical\_land_{n,s,y-1} \big) \cdot \Big( 1 + growth\_land\_scen\_up_{n,s,y} \Big)^{|y|}\end{split}\]

Equation DYNAMIC_LAND_SCEN_CONSTRAINT_LO

\[\begin{split}LAND_{n,s,y} \geq & - initial\_land\_scen\_lo_{n,s,y} \cdot \frac{ \Big( 1 + growth\_land\_scen\_lo_{n,s,y} \Big)^{|y|} - 1 } { growth\_land\_scen\_lo_{n,s,y} } \\ & + \big( LAND_{n,s,y-1} + historical\_land_{n,s,y-1} \big) \cdot \Big( 1 + growth\_land\_scen\_lo_{n,s,y} \Big)^{|y|}\end{split}\]

These constraints enforces upper and lower bounds on the change rate per land type determined as a linear combination of land use scenarios.

Equation DYNAMIC_LAND_TYPE_CONSTRAINT_UP

\[\begin{split}\sum_{s \in S} land\_use_{n,s,y,u} &\cdot LAND_{n,s,y} \leq initial\_land\_up_{n,y,u} \cdot \frac{ \Big( 1 + growth\_land\_up_{n,y,u} \Big)^{|y|} - 1 } { growth\_land\_up_{n,y,u} } \\ & + \Big( \sum_{s \in S} \big( land\_use_{n,s,y-1,u} + dynamic\_land\_up_{n,s,y-1,u} \big) \\ & \quad \quad \cdot \big( LAND_{n,s,y-1} + historical\_land_{n,s,y-1} \big) \Big) \\ & \quad \cdot \Big( 1 + growth\_land\_up_{n,y,u} \Big)^{|y|}\end{split}\]

Equation DYNAMIC_LAND_TYPE_CONSTRAINT_LO

\[\begin{split}\sum_{s \in S} land\_use_{n,s,y,u} &\cdot LAND_{n,s,y} \geq - initial\_land\_lo_{n,y,u} \cdot \frac{ \Big( 1 + growth\_land\_lo_{n,y,u} \Big)^{|y|} - 1 } { growth\_land\_lo_{n,y,u} } \\ & + \Big( \sum_{s \in S} \big( land\_use_{n,s,y-1,u} + dynamic\_land\_lo_{n,s,y-1,u} \big) \\ & \quad \quad \cdot \big( LAND_{n,s,y-1} + historical\_land_{n,s,y-1} \big) \Big) \\ & \quad \cdot \Big( 1 + growth\_land\_lo_{n,y,u} \Big)^{|y|}\end{split}\]

Section of generic relations (linear constraints)

This feature is intended for development and testing only - all new features should be implemented as specific new mathematical formulations and associated sets & parameters!

Auxiliary variable for left-hand side

Equation RELATION_EQUIVALENCE

\[\begin{split}REL_{r,n,y} = \sum_{t} \Bigg( & \ relation\_new\_capacity_{r,n,y,t} \cdot CAP\_NEW_{n,t,y} \\[4 pt] & + relation\_total\_capacity_{r,n,y,t} \cdot \sum_{y^V \leq y} \ CAP_{n,t,y^V,y} \\ & + \sum_{n^L,y',m,h} \ relation\_activity_{r,n,y,n^L,t,y',m} \\ & \quad \quad \cdot \Big( \sum_{y^V \leq y'} ACT_{n^L,t,y^V,y',m,h} + historical\_activity_{n^L,t,y',m,h} \Big) \Bigg)\end{split}\]

The parameter \(historical\_new\_capacity_{r,n,y}\) is not included here, because relations can only be active in periods included in the model horizon and there is no “writing” of capacity relation factors across periods.

Upper and lower bounds on user-defined relations

Equation RELATION_CONSTRAINT_UP

\[REL_{r,n,y} \leq relation\_upper_{r,n,y}\]

Equation RELATION_CONSTRAINT_LO

\[REL_{r,n,y} \geq relation\_lower_{r,n,y}\]