Sets and mappings definition¶

This file contains the definition of all sets and mappings used in MESSAGEix.

Sets in the MESSAGEix implementation¶

Set name	Notation	Explanatory comments
node [1]	\(n \in N\)	regions, countries, grid cells
commodity	\(c \in C\)	resources, electricity, water, land availability, etc.
level	\(l \in L\)	levels of the reference energy system or supply chain (primary, secondary, … , useful)
grade	\(g \in G\)	grades of resource quality in the extraction & mining sector
technology [tec]	\(t \in T\)	technologies that use input commodities to produce outputs; the short-hand notation “tec” is used in the GAMS implementation
mode [2]	\(m \in M\)	modes of operation for specific technologies
emission	\(e \in E\)	greenhouse gases, pollutants, etc.
land_scenario	\(s \in S\)	scenarios of land use (for land-use model emulator)
land_type	\(u \in U\)	land-use types (e.g., field, forest, pasture)
year [year_all] [3] [4]	\(y \in Y\)	model horizon (including historical periods for vintage structure of installed capacity and dynamic constraints)
time [5]	\(h \in H\)	subannual time periods (seasons, days, hours)
relation [6]	\(r \in R\)	set of generic linear constraints
rating	\(q \in Q\)	identifies the ‘quality’ of the renewable energy potential
lvl_spatial		set of spatial hierarchy levels (global, region, country, grid cell)
lvl_temporal		set of temporal hierarchy levels (year, season, day, hour)

[1] The set node includes spatial units across all levels of spatial disaggregation (global, regions, countries, basins, grid cells). The hierarchical mapping is implemented via the mapping set map_spatial_hierarchy. This set always includes an element ‘World’ when initializing a MESSAGE-scheme message_ix.Scenario.

[2]	For example, high electricity or high heat production modes of operation for combined heat and power plants.

[3] In the MESSAGEix implementation in GAMS, the set year_all denotes the “superset” of the entire horizon (historical and model horizon), and the set year is a dynamic subset of year_all. This facilitates an efficient implementation of the historical capacity build-up and the (optional) recursive-dynamic solution approach. When working with a message_ix.Scenario via the scientific programming API, the set of all periods is called year for a more concise notation. The specification of the model horizon is implemented using the mapping set cat_year and the type “firstmodelyear”.

[4]	In MESSAGEix, the key of an element in set `year` identifies the last year of the period, i.e., in a set \(year = [2000, 2005, 2010, 2015]\), the period ‘2010’ comprises the years \([2006, .. ,2010]\).

[5]	The set `time` collects all sub-annual temporal units across all levels of temporal disaggregation. In a `MESSAGE`-scheme `ixmp`.Scenario, this set always includes an element “year”, and the duration of that element is 1 (\(duration\_time_{'year'} = 1\)).

[6]	A generic formulation of linear constraints is implemented in MESSAGEix, see Section of generic relations (linear constraints). These constraints can be used for testing and development, but specific new features should be implemented by specific equations and parameters.

Category types and mappings¶

This feature is used to easily implement aggregation across groups of set elements. For example, by setting an upper bound over an emission type, the constraint enforces that the sum over all emission species mapped to that type via the mapping set cat_emission satisfies that upper bound.

Set name	Notation	Explanatory comments
level_resource (level) [7]	\(l \in L^{RES} \subseteq L\)	levels related to fossil resources representation
level_renewable (level) [7]	\(l \in L^{REN} \subseteq L\)	levels related to renewables representation
type_node [8]	\(\widehat{n} \in \widehat{N}\)	Category types for nodes
cat_node (type_node,node)	\(n \in N(\widehat{n})\)	Category mapping between node types and nodes
type_tec [9]	\(\widehat{t} \in \widehat{T}\)	Category types for technologies
cat_tec (type_tec,tec)	\(t \in T(\widehat{t})\)	Category mapping between tec types and technologies
inv_tec (tec) [10]	\(t \in T^{INV} \subseteq T\)	Specific subset of investment technologies
renewable_tec (tec) [11]	\(t \in T^{REN} \subseteq T\)	Specific subset of renewable-energy technologies
type_emission	\(\widehat{e} \in \widehat{E}\)	Category types for emissions (greenhouse gases, pollutants, etc.)
cat_emission (type_emission,emission)	\(e \in E(\widehat{e})\)	Category mapping between emission types and emissions
type_tec_land (type_tec) [12]	\(\widehat{t} \in \widehat{T}^{LAND} \subseteq \widehat{T}\)	Mapping set of technology types and land use
balance_equality (commodity,level)	\(c \in C, l \in L\)	Commodities and level related to Equation COMMODITY_BALANCE_LT

[7]	(1, 2) The constraint `EXTRACTION_EQUIVALENCE` is active only for the levels included in this set, and the constraint `COMMODITY_BALANCE` is deactivated for these levels.

[8]	The element “economy” is added by default as part of the `MESSAGE`-scheme `ixmp`.Scenario.

[9]	The element “all” in `type_tec` and the associated mapping to all technologies in the set `cat_tec` are added by default as part of the `MESSAGE`-scheme `message_ix`.Scenario.

[10] The auxiliary set inv_tec (subset of technology) is a short-hand notation for all technologies with defined investment costs. This activates the investment cost part in the objective function and the constraints for all technologies where investment decisions are relevant. It is added by default when exporting MESSAGE-scheme message_ix.Scenario to gdx.

[11]	The auxiliary set `renewable_tec` (subset of `technology`) is a short-hand notation for all technologies with defined parameters relevant for the equations in the “Renewable” section. It is added by default when exporting `MESSAGE`-scheme `message_ix`.Scenario to gdx.

[12]	The mapping set `type_tec_land` is a dynamic subset of `type_tec` and specifies whether emissions from the land-use model emulator module are included when aggregrating over a specific technology type. The element “all” is added by default in a `MESSAGE`-scheme `message_ix`.Scenario.

Mappings sets¶

These sets are generated automatically when exporting a MESSAGE-scheme ixmp.Scenario to gdx using the API. They are used in the GAMS model to reduce model size by excluding non-relevant variables and equations (e.g., actitivity of a technology outside of its technical lifetime).

Set name	Notation	Explanatory comments
map_node(node,location)		mapping of nodes across hierarchy levels (location is in node)

Mapping sets (flags) for bounds¶

There are a number of mappings sets generated when exporting a message_ix.Scenario to gdx. They are used as ‘flags’ to indicate whether a constraint is active. The names of these sets follow the format is_<constraint>_<dir>.

Such mapping sets are necessary because GAMS does not distinguish between 0 and ‘no value assigned’, i.e., it cannot differentiate between a bound of 0 and ‘no bound assigned’.

Mapping sets (flags) for fixed variables¶

Similar to the mapping sets for bounds, there are mapping sets to indicate whether decision variables are pre-defined to a specific value, usually taken from a solution of another model instance. This can be used to represent imperfect foresight where a policy shift or parameter change is introduced in later years. The names of these sets follow the format is_fixed_<variable>.