In a mathematical model like Temoa, sets are used to index parameters and variables. To enable the modeler to translate between the algebraic formulation, input database, and model code, we provide the mathematical notation used in the model formulation, as well as the associated names in the Python code and database schema in the tables below. The code representation is more verbose than the algebraic version, using full words.

Our discussion of sets is broken down into several logical categories to make it easier to parse. The first group of sets pertains to Temoa’s treatment of time, as shown below. Note: Some entries in the “Database Table” column are comma-separated. In those cases, the first element refers to the name of the database table, and the second refers to specific column within the database table used to identify a specific subset. For example, in the first table below, time_period is the master set and time_exist is a subset that defines the user-defined model time periods to define historical technology vintages that exist prior to the model’s time horizon for optimization.

Set	Database Table	Model Element	Notes
\(\text{P}^e\)	time_period, flag = e	time_exist	model periods prior to the beginning of the optimization time horizon
\(\text{P}^f\)	time_period, flag = f	time_future	model periods within the optimization time horizon
\({}^*\text{P}^o\)	time_period	time_optimize	model time periods to optimize, (\(\text{P}^f - \text{max}(\text{P}^f)\)); the last time period is removed as it represents the end of the final period
\({}^*\text{V}\)	time_period	vintage_exist, vintage_optimize, vintage_all	tech vintages, (\(\text{P}^e \cup \text{P}^o\)), derived from time period set and used to track technology vintages across time periods
\(\text{D}\)	time_of_day	time_of_day	intraday time divisions; each entry carries an hours value (default 1) specifying how many hours it represents
\(\text{S}\)	time_season	time_season	intra-annual divisions (e.g. seasons or representative days); each entry carries a segment_fraction giving its share of the year; ordered by sequence column
	time_season_sequential	time_season_sequential, ordered_season_sequential	superimposed sequential seasons used for seasonal storage and inter-season ramping; only required when time_sequencing = 'representative_periods'

The sets in the table below define Temoa’s representation of regions.

Set	Database Table	Model Element	Notes
\(\text{R}\)	region	regions	distinct geographical regions
	region	regional_indices	set of all the possible combinations of interregional exchanges plus original region indices
		regional_global_indices	set of all used combinations of interregional exchanges plus original region indices and groups

The sets below define how technologies are represented within Temoa. Because technologies can serve many different functions across an energy system, we need to define a large number of technology subsets.

Set	Database Table	Model Element	Notes
\({}^*\text{T}\)	technology	tech_all	all technologies to be modeled (in v4, all technologies are production-type: \(T = T^p\))
\(\text{T}^p\)	technology, flag LIKE 'p%'	tech_production	all production technologies, including baseload and storage (\({T}^b \cup {T}^s \subset {T}^p\))
\(\text{T}^b\)	technology, flag = pb	tech_baseload	baseload electric generators, which have constant output across intraday time segments (\({T}^b \subset T\))
\(\text{T}^s\)	technology, flag = ps	tech_storage	all storage technologies (\({T}^s \subset T\))
\(\text{T}^a\)	technology, annual = 1	tech_annual	technologies that produce constant annual output (\({T}^a \subset T\))
\(\text{T}^{res}\)	technology, reserve = 1	tech_reserve	electric generators contributing to the reserve margin requirement (\({T}^{res} \subset T\))
\(\text{T}^c\)	technology, curtail = 1	tech_curtailment	technologies with curtailable output and no upstream cost (\({T}^c \subset (T - T^{res})\))
\(\text{T}^f\)	technology, flex = 1	tech_flex	technologies producing excess commodity flows (\({T}^f \subset T\))
\(\text{T}^x\)	technology, exchange = 1	tech_exchange	technologies used for interregional commodity flows (\({T}^x \subset T\))
\(\text{T}^{ur}\)	ramp_up_hourly	tech_upramping	electric generators with a ramp up hourly rate limit; derived from ramp_up_hourly table (\({T}^{ur} \subset T\))
\(\text{T}^{dr}\)	ramp_down_hourly	tech_downramping	electric generators with a ramp down hourly rate limit; derived from ramp_down_hourly table (\({T}^{dr} \subset T\))
\(\text{T}^{ret}\)	technology, retire = 1	tech_retirement	technologies allowed to retire before end of life (\({T}^{ret} \subset (T - T^{u})\))
\(\text{T}^u\)	technology, unlim_cap = 1	tech_uncap	technologies that have no bound on capacity (\({T}^u \subset (T - T^{res})\))
\(\text{T}^{ss}\)	technology, flag = 'ps' AND seas_stor = 1	tech_seasonal_storage	seasonal storage technologies; requires both storage flag and seas_stor column (\({T}^{ss} \subset T^s\))
	tech_group	tech_group_names	named groups for use in group constraints
	tech_group_member	tech_group_members	each technology belonging to each group
\(\text{T}^e\)	existing_capacity	tech_exist	existing technologies with a past vintage (\({T}^e \subset T\))

The sets below define commodities that are consumed and produced by different energy technologies.

Set	Database Table	Model Element	Notes
\(\text{C}^d\)	commodity, flag = d	commodity_demand	end-use demand commodities, representing the final consumer demands
\(\text{C}^e\)	commodity, flag = e	commodity_emissions	emission commodities (e.g., \(\text{CO}_\text{2}\) \(\text{NO}_\text{x}\)); filtered by flag
\(\text{C}^p\)	commodity, flag IN (p, wp, s, a, wa)	commodity_physical	superset of physical, source, annual, and their waste variants; includes all non-demand, non-emission commodities
\(\text{C}^w\)	commodity, flag IN (w, wa , wp)	commodity_waste	commodity whose production can be greater than its consumption; can be physical, annual, or neither (not balanced, all wasted)
\(\text{C}^a\)	commodity, flag IN (a, wa)	commodity_annual	commodities whose flows are only balanced over each period, not per-timeslice (\(\text{C}^a \subset \text{C}^p\))
\({}^*\text{C}^l\)		commodity_flex	derived set of commodities produced by a flex technology (\(\text{C}^l \subset \text{C}^p\)); auto-populated from tech_flex outputs
\(\text{C}^s\)	commodity, flag = s	commodity_source	primary source commodities, not balanced by CommodityBalance_constraint
\({}^*\text{C}^c\)		commodity_carrier	union of physical, demand, and waste commodities, (\(\text{C}_p \cup \text{C}_d \cup \text{C}^w\))
\({}^*\text{C}\)		commodity_all	union of all commodity sets; union of carrier and emissions commodities

There is an additional set that defines operators (=, <, >). While not strictly necessary, defining these operators as a set allows modelers to express constraints more efficiently.

Set	Database Table	Model Element	Notes
	operator	operator	constraint operators

As indicated below, there are additional sets that are derived within the model code and thus do not appear in the database schema.

Set	Model Element	Notes
	tech_with_capacity	technologies eligible for capacitization; computed as tech_all - tech_uncap
	tech_or_group	technologies or groups combined; union of tech_group_names | tech_all
\({}^*\text{C}^c\)	commodity_carrier	physical energy carriers and end-use demands; union of physical, demand, and waste commodities
\({}^*\text{C}\)	commodity_all	union of all commodity sets; union of carrier and emissions commodities
\(\text{T}^e\)	tech_exist	technologies with existing capacity; derived from existing_capacity table

There are also python dictionaries and sets used to specify internal data structures during model construction and are not considered formal model elements:

• process_inputs, process_outputs, process_loans

• active_flow_rpsditvo, active_flow_rpitvo

• Various vintage and operational tracking dictionaries

• Time sequencing dictionaries (time_next, time_next_sequential)