Symbols list

All symbols can be obtained from the model here.

Sets and subsets

Type	Symbol	Description
Set	tech	Generation technologies
Subset	dis(tech)	Dispatchable generation technologies
Subset	nondis(tech)	Nondispatchable generation technologies
Subset	con(tech)	Conventional generation technologies
Subset	res(tech)	Renewable generation technologies
Set	sto	Storage technologies
Set	rsvr	Reservoir technologies
Set	h	Hours
Set	n	Nodes
Set	l	Lines
Set	ev	EV types
Set	bu	Building archtypes for heating
Set	ch	Heating combination type
Subset	hst(ch)	Heating technology that feeds to storage
Subset	hp(ch)	Heat pump technologies
Subset	hel(ch)	Hybrid electric heating technologies - electric part
Subset	hfo(ch)	Hybrid electric heating technologies - fossil part
Set	dsm	DSM technologies
Subset	dsm_shift(dsm)	DSM load shifting technologies
Subset	dsm_curt(dsm)	DSM load curtailment technologies
Set	reserves	Reserve qualities
Subset	reserves_up(reserves)	Positive reserves
Subset	reserves_do(reserves)	Negative reserves
Subset	reserves_spin(reserves)	Spinning reserves
Subset	reserves_nonspin(reserves)	Nonspinning reserves
Subset	reserves_prim(reserves)	Primary reserves
Subset	reserves_nonprim(reserves)	Nonprimary reserves
Subset	reserves_prim_up(reserves)	Primary positive reserves
Subset	reserves_nonprim_up(reserves)	Nonprimary positive reserves
Subset	reserves_prim_do(reserves)	Primary negative reserves
Subset	reserves_nonprim_do(reserves)	Nonprimary negative reserves

Parameters

Category	Type	Symbol & domain	Description and units
Balance	Time-series	d(n,h)	Demand hour h [MWh]
Generation	Costs	c_fix(n,tech)	Annual fixed costs [EUR per MW per year]
Generation	Costs	c_inv_overnight(n,tech)	Investment costs: Overnight [EUR per MW]
Generation	Costs	c_do(n,tech)	Load change costs DOWN [EUR per MWh]
Generation	Costs	c_up(n,tech)	Load change costs UP [EUR per MWh]
Generation	Costs	co2price(n,tech)	CO2 price in [EUR per ton]
Generation	Costs	fuelprice(n,tech)	Fuel price conventionals [EUR per MWh thermal]
Generation	Costs	interest_rate_tech(n,tech)	Investment costs: Interest rate [%]
Generation	Technology	carbon_content(n,tech)	CO2 emissions per fuel unit used [tons per MWh thermal]
Generation	Technology	co2_cap	Yearly CO2 overall cap
Generation	Technology	co2_cap_exog(n)	Yearly CO2 cap per node
Generation	Technology	eta(n,tech)	Efficiency of conventional technologies [0 1]
Generation	Technology	grad_per_min(n,tech)	Maximum load change relative to installed capacity [% of installed capacity per minute]
Generation	Technology	lifetime_tech(n,tech)	Investment costs: technical lifetime [a]
Generation	Technology	m_e(n,tech)	Investment: maximum installable energy [TWh per a]
Generation	Technology	m_p(n,tech)	Investment: maximum installable capacity per technology [MW]
Generation	Technology	phi_min_res(n)	Minimum renewables share per node [0 1]
Generation	Time-series	phi_res(n,tech,h)	Renewables availability technology res in hour h [0 1]
Storage	Costs	c_fix_sto(n,sto)	Annual fixed costs [EUR per MW]
Storage	Costs	c_inv_overnight_sto_e(n,sto)	Investment costs for storage energy: Overnight [EUR per MWh]
Storage	Costs	c_inv_overnight_sto_p(n,sto)	Investment costs for storage capacity: Overnight [EUR per MW]
Storage	Costs	c_m_sto(n,sto)	Marginal costs of storing in or out [EUR per MWh]
Storage	Costs	interest_rate_sto(n,sto)	Investment costs: for storage Interest rate [%]
Storage	Technology	eta_sto(n,sto)	Storage efficiency [0 1]
Storage	Technology	etop_max(n,sto)	Maximum E to P ratio of storage types [#]
Storage	Technology	lifetime_sto(n,sto)	Investment costs: for storage technical lifetime [a]
Storage	Technology	m_sto_e(n,sto)	Investment into storage: maximum installable energy [MWh]
Storage	Technology	m_sto_p(n,sto)	Investment into storage: maximum installable power [MW]
Storage	Technology	phi_sto_ini(n,sto)	Initial storage level [0 1]
Reservoir	Costs	c_fix_rsvr(n,rsvr)	Annual fixed costs [EUR per MW per a]
Reservoir	Costs	c_inv_overnight_rsvr_e(n,rsvr)	Investment costs for reservoir energy: Overnight [EUR per MWh]
Reservoir	Costs	c_inv_overnight_rsvr_p(n,rsvr)	Investment costs for reservoir capacity: Overnight [EUR per MW]
Reservoir	Costs	inv_interest_rsvr(n,rsvr)	Investment costs for reservoir: Interest rate [%]
Reservoir	Technology	eta_rsvr(n,rsvr)	Generation efficiency [0 1]
Reservoir	Technology	inv_lifetime_rsvr(n,rsvr)	Investment costs for reservoir: technical lifetime [a]
Reservoir	Technology	m_rsvr_e(n,rsvr)	Investment into reservoir: maximum installable energy [MWh]
Reservoir	Technology	m_rsvr_p(n,rsvr)	Investment into reservoir: maximum installable capacity [MW]
Reservoir	Technology	min_flh(n,rsvr)	Min flh per node and reservoir
Reservoir	Technology	phi_rsvr_ini(n,rsvr)	Initial reservoir level [0 1]
Reservoir	Technology	phi_rsvr_lev_min(n,rsvr)	Minimum filling level [0 1]
Reservoir	Technology	phi_rsvr_maxout(n,rsvr)	per node and reservoir
Reservoir	Technology	phi_rsvr_min(n)	Minimum hourly reservoir outflow as fraction of annual energy [0 1]
Reservoir	Technology	phi_rsvr_minout(n,rsvr)	per node and reservoir
Reservoir	Time-series	rsvr_in(n,rsvr,h)	Reservoir inflow in hour h [0 1]
Transmission	Costs	c_fix_ntc(l)	Fixed costs [EUR per MW per a]
Transmission	Costs	c_inv_overnight_ntc(l)	Investment costs in: overnight [EUR per MW]
Transmission	Costs	inv_interest_ntc(l)	Investment costs: Interest rate [%]
Transmission	Technology	dist(l)	Distance covered by link l [km]
Transmission	Technology	inc(l,n)	Incidence index of link l on node n
Transmission	Technology	inv_lifetime_ntc(l)	Investment costs: technical lifetime [a]
Transmission	Technology	inv_recovery_ntc(l)	Investment costs: Recovery period in [a]
Transmission	Technology	m_ntc(l)	Investment into NTC: maximum installable capacity [MW]
Electric vehicles	Costs	c_m_ev(n,ev)	Marginal costs of discharging V2G [EUR per MWh]
Electric vehicles	Costs	pen_phevfuel(n,ev)	Penalty for non-electric PHEV operation mode [EUR per MWh]
Electric vehicles	Technology	eta_ev_in(n,ev)	Electric vehicle efficiency of charging (G2V) [0 1]
Electric vehicles	Technology	eta_ev_out(n,ev)	Electric vehicle efficiency of discharging (V2G) [0 1]
Electric vehicles	Technology	ev_phev(n,ev)	Defines whether an electric vehicle is a PHEV REEV [1 if yes 0 otherwise]
Electric vehicles	Technology	ev_quant(n)	Overall number of electirc vehicles [#]
Electric vehicles	Technology	n_ev_e(n,ev)	Electric vehicle battery capacity [MWh]
Electric vehicles	Technology	phi_ev_ini(n,ev)	Electric vehicle charging level in initial period [0 1]
Electric vehicles	Technology	phi_ev(n,ev)	Share of electric vehicles per load profile in actual scenario [0 1]
Electric vehicles	Time-series	ev_ed(n,ev,h)	Driving electricity consumption per ev in hour h [MW]
Electric vehicles	Time-series	ev_ged_exog(n,ev,h)	Grid electricity demand in case of uncontrolled charging (exogenous) per ev in hour h [MW]
Electric vehicles	Time-series	n_ev_p(n,ev,h)	Power rating of the charging station in hour h [MW]
Prosumage	Technology	m_res_pro(n,tech)	Maximum installable: renewables capacity [MW]
Prosumage	Technology	m_sto_pro_e(n,sto)	Maximum installable: storage energy [MWh]
Prosumage	Technology	m_sto_pro_p(n,sto)	Maximum installable: storage capacity [MW]
Prosumage	Technology	phi_pro_load(n)	Share of prosumagers among total load [0 1]
Prosumage	Technology	phi_pro_self	Minimum self-generation shares for prosumagers [0 1]
Prosumage	Technology	phi_sto_pro_ini(n,sto)	Prosumagers initial storage loading [0 1]
Heat	Costs	pen_heat_fuel(n,bu,ch)	Penalty term for non-electric fuel usage for hybrid heating technologies [EUR per MWh]
Heat	Technology	area_floor(n,bu,ch)	Floor area subject to specific heating technology in specific building type [m2]
Heat	Technology	eta_dhw_aux_stat(n,bu,ch)	Static efficiency for auxiliary DHW technologies [0 1]
Heat	Technology	eta_heat_dyn(n,bu,ch)	Static efficiency for heating technologies [0 1]
Heat	Technology	eta_heat_stat(n,bu,ch)	Static efficiency for heating technologies [0 1]
Heat	Technology	n_heat_e(n,bu,ch)	Maximum energy level of heating storage technologies [MWh]
Heat	Technology	n_heat_p_in(n,bu,ch)	Maximum power inflow into heating technologies [MW]
Heat	Technology	n_heat_p_out(n,bu,ch)	Maximum power outflow from heating technologies [MW]
Heat	Technology	n_sets_dhw_e(n,bu,ch)	SETS auxiliary DHW module - energy storage capacity [MWh]
Heat	Technology	n_sets_dhw_p_in(n,bu,ch)	SETS auxiliary DHW module - power rating - electricity intake [MW]
Heat	Technology	n_sets_dhw_p_out(n,bu,ch)	SETS auxiliary DHW module - power rating - DHW output [MW]
Heat	Technology	n_sets_e(n,bu,ch)	SETS - Energy storage capacity [MWh]
Heat	Technology	n_sets_p_in(n,bu,ch)	SETS - Power rating - electricity intake [MW]
Heat	Technology	n_sets_p_out(n,bu,ch)	SETS - Power rating - heat output [MW]
Heat	Technology	phi_heat_ini(n,bu,ch)	Inititial storage level of heating technologies [0 1]
Heat	Technology	phi_heat_type(n,bu,ch)	Share of heating type ch per building archetype bu [0 1]
Heat	Technology	temp_sink(n,bu,ch)	Heat pumps - sink temperature [Celsius degrees]
Heat	Time-series	d_dhw(n,bu,ch,h)	Hourly DHW demand [MWh per m2]
Heat	Time-series	dh(n,bu,ch,h)	Hourly heat demand [MWh per m2]
Heat	Time-series	temp_source(n,bu,ch,h)	Heat pumps - source temperature [degree Celsius]
DSM	Costs	c_fix_dsm_cu(n,dsm)	Annual fixed costs load curtailment capacity [EUR per MW per a]
DSM	Costs	c_fix_dsm_shift(n,dsm)	Annual fixed costs load shifting capacity [EUR per MW per a]
DSM	Costs	c_inv_overnight_dsm_cu(n,dsm)	Investment costs for DSM load curtailment: Overnight [EUR per MW]
DSM	Costs	c_inv_overnight_dsm_shift(n,dsm)	Investment costs for DSM load shifting: Overnight [EUR per MW]
DSM	Costs	c_m_dsm_cu(n,dsm)	DSM: hourly costs of load curtailment [EUR per MWh]
DSM	Costs	c_m_dsm_shift(n,dsm)	DSM: costs for load shifting [EUR per MWh]
DSM	Costs	inv_interest_dsm_cu(n,dsm)	Investment costs for DSM load curtailment: Interest rate [%]
DSM	Costs	inv_interest_dsm_shift(n,dsm)	Investment costs for DSM load shifting: Interest rate [%]
DSM	Technology	eta_dsm_shift(n,dsm)	DSM: Efficiency of load shifting technologies [0 1]
DSM	Technology	inv_recovery_dsm_cu(n,dsm)	Investment costs for DSM load curtailment: Recovery period [a]
DSM	Technology	inv_recovery_dsm_shift(n,dsm)	Investment costs for DSM load shifting: Recovery period [a]
DSM	Technology	m_dsm_cu(n,dsm)	DSM: Maximum installable capacity load curtailment [MW]
DSM	Technology	m_dsm_shift(n,dsm)	DSM: Maximum installable capacity load shifting [MW]
DSM	Technology	t_dur_dsm_cu(n,dsm)	DSM: Maximum duration load curtailment [h]
DSM	Technology	t_dur_dsm_shift(n,dsm)	DSM: Maximum duration load shifting [h]
DSM	Technology	t_off_dsm_cu(n,dsm)	DSM: Minimum recovery time between two load curtailment instances [h]
DSM	Technology	t_off_dsm_shift(n,dsm)	DSM: Minimum recovery time between two granular load upshift instances [h]
Reserves	Technology	reserves_reaction(n,reserves)	Activation reaction time for reserves qualities [min]
Reserves	Time-series	phi_reserves_call(n,reserves,h)	Hourly share of reserve provision that is actually activated [0 1]
Reserves	Time-series	reserves_exogenous(n,reserves,h)	Hourly reserve provision [MW]

Derived parameters

Type	Symbol & domain	Description and units
Costs	c_m(n,tech)	Marginal production costs for conventional plants [EUR per MWh]
Costs	c_i(n,tech)	Annualized investment costs by conventioanl plant [EUR per MW]
Costs	c_i_res(n,tech)	Annualized investment costs by renewable plant [EUR per MW]
Costs	c_i_sto_e(n,sto)	Annualized investment costs storage energy [EUR per MWh]
Costs	c_i_sto_p(n,sto)	Annualized investment costs storage capacity [EUR per MW]
Costs	c_i_rsvr_e(n,rsvr)	Annualized investment costs storage energy [EUR per MWh]
Costs	c_i_rsvr_p(n,rsvr)	Annualized investment costs storage capacity [EUR per MW]
Costs	c_i_dsm_cu(n,dsm)	DSM: Annualized investment costs load curtailment [EUR per MW]
Costs	c_i_dsm_shift(n,dsm)	DSM: Annualized investment costs load shifting [EUR per MW]
Costs	c_i_ntc(l)	Investment for net transfer capacity [EUR per MW and km]

Variables

Type	Symbol & domain	Description and units
Variable	Z	Value objective function [Euro]
Non-negative variable	G_L(n,tech,h)	Generation level in hour h [MWh]
Non-negative variable	G_UP(n,tech,h)	Generation upshift in hour h [MWh]
Non-negative variable	G_DO(n,tech,h)	Generation downshift in hour h [MWh]
Non-negative variable	G_RES(n,tech,h)	Generation renewables type res in hour h [MWh]
Non-negative variable	CU(n,tech,h)	Renewables curtailment technology res in hour h [MWh]
Non-negative variable	STO_IN(n,sto,h)	Storage inflow technology sto hour h [MWh]
Non-negative variable	STO_OUT(n,sto,h)	Storage outflow technology sto hour h [MWh]
Non-negative variable	STO_L(n,sto,h)	Storage level technology sto hour h [MWh]
Non-negative variable	N_TECH(n,tech)	Technology tech built [MW]
Non-negative variable	N_STO_E(n,sto)	Storage technology built - Energy [MWh]
Non-negative variable	N_STO_P(n,sto)	Storage loading and discharging capacity built - Capacity [MW]
Non-negative variable	RSVR_OUT(n,rsvr,h)	Reservoirs: outflow in hour h [MWh]
Non-negative variable	RSVR_L(n,rsvr,h)	Reservoirs: level in hour h [MWh]
Non-negative variable	N_RSVR_E(n,rsvr)	Reservoirs: installed energy capacity [MWh]
Non-negative variable	N_RSVR_P(n,rsvr)	Reservoirs: installed power capacity [MW]
Variable	F(l,h)	Energy flow over link l in hour h [MWh]
Non-negative variable	NTC(l)	Trade: installed NTC on line l [MW]
Non-negative variable	EV_CHARGE(n,ev,h)	Electric vehicle charging vehicle profile ev hour h [MWh]
Non-negative variable	EV_DISCHARGE(n,ev,h)	Electric vehicle discharging vehicle profile ev hour h [MWh]
Non-negative variable	EV_L(n,ev,h)	Electric vehicle charging level vehicle profile ev hour h [MWh]
Non-negative variable	EV_PHEVFUEL(n,ev,h)	Plug in hybrid electric vehicle conventional fuel use vehicle profile ev hour h [MWh]
Non-negative variable	EV_GED(n,ev,h)	Grid electricity demand for mobility vehicle profile ev hour h [MWh]
Non-negative variable	CU_PRO(n,tech,h)	Prosumage: curtailment of renewable generation in hour h [MWh]
Non-negative variable	G_MARKET_PRO2M(n,tech,h)	Prosumage. energy sent to market in hour h [MWh]
Non-negative variable	G_MARKET_M2PRO(n,h)	Prosumage: withdrawal of energy from market in hour h [MWh]
Non-negative variable	G_RES_PRO(n,tech,h)	Prosumage: hourly renewables generation in hour h [MWh]
Non-negative variable	STO_IN_PRO2PRO(n,tech,sto,h)	Prosumage: storage loading from generation for discharging to consumption in hour h [MWh]
Non-negative variable	STO_IN_PRO2M(n,tech,sto,h)	Prosumage: storage loading from generation for discharging to market in hour h [MWh]
Non-negative variable	STO_IN_M2PRO(n,sto,h)	Prosumage: storage loading from market for discharging to consumption in hour h [MWh]
Non-negative variable	STO_IN_M2M(n,sto,h)	Prosumage: storage loading from market for discharging to market in hour h [MWh]
Non-negative variable	STO_OUT_PRO2PRO(n,sto,h)	Prosumage: storage discharging to consumption from generation in hour h [MWh]
Non-negative variable	STO_OUT_PRO2M(n,sto,h)	Prosumage: storage discharging to market from generation in hour h [MWh]
Non-negative variable	STO_OUT_M2PRO(n,sto,h)	Prosumage: storage discharging to consumption from market in hour h [MWh]
Non-negative variable	STO_OUT_M2M(n,sto,h)	Prosumage: storage discharging to market from market in hour h [MWh]
Non-negative variable	STO_L_PRO2PRO(n,sto,h)	Prosumage: storage level generation to consumption in hour h [MWh]
Non-negative variable	STO_L_PRO2M(n,sto,h)	Prosumage: storage level generation to market in hour h [MWh]
Non-negative variable	STO_L_M2PRO(n,sto,h)	Prosumage: storage level market to consumotion in hour h [MWh]
Non-negative variable	STO_L_M2M(n,sto,h)	Prosumage: storage level market to market in hour h [MWh]
Non-negative variable	N_STO_E_PRO(n,sto)	Prosumage: installed storage energy [MWh]
Non-negative variable	N_STO_P_PRO(n,sto)	Prosumage: installed storage power [MW]
Non-negative variable	STO_L_PRO(n,sto,h)	Prosumage: overall storage level in hour h [MWh]
Non-negative variable	N_RES_PRO(n,tech)	Prosumage: installed renewables capacities [MW]
Non-negative variable	H_DIR(n,bu,ch,h)	Heating: direct heating in hour h for building type bu with haeting technology ch [MWh]
Non-negative variable	H_SETS_LEV(n,bu,ch,h)	Heating: storage level SETS technologies [MWh]
Non-negative variable	H_SETS_IN(n,bu,ch,h)	Heating: storage inflow SETS technologies [MWh]
Non-negative variable	H_SETS_OUT(n,bu,ch,h)	Heating: storage outflow SETS technologies [MWh]
Non-negative variable	H_HP_IN(n,bu,ch,hh)	Heating: electricity demand heat pump technologies [MWh]
Non-negative variable	H_STO_LEV(n,bu,ch,h)	Heating: storage level storage technologies [MWh]
Non-negative variable	H_STO_IN_HP(n,bu,ch,h)	Heating: storage inflow from heat pumps to storage technologies [MWh]
Non-negative variable	H_STO_IN_ELECTRIC(n,bu,ch,h)	Heating: storage inflow from electric heating to storage technologies [MWh]
Non-negative variable	H_ELECTRIC_IN(n,bu,ch,h)	Heating: hybrid electric heaters electricity demand [MWh]
Non-negative variable	H_STO_IN_FOSSIL(n,bu,ch,h)	Heating: storage inflow from nonelectric heating to storage technologies [MWh]
Non-negative variable	H_STO_OUT(n,bu,ch,h)	Heating: storage outflow from storage technologies [MWh]
Non-negative variable	H_DHW_DIR(n,bu,ch,h)	Heating - domestic hot water: provision in case of direct electric heating [MWh]
Non-negative variable	H_DHW_STO_OUT(n,bu,ch,h)	Heating - domestic hot water: DHW storage outflow [MWh]
Non-negative variable	H_DHW_AUX_ELEC_IN(n,bu,ch,h)	Heating - domestic hot water: electrical energy input of auxiliary hot water tank for SETS [MWh]
Non-negative variable	H_DHW_AUX_LEV(n,bu,ch,h)	Heating - domestic hot water: level of auxiliary hot water tank for SETS [MWh]
Non-negative variable	H_DHW_AUX_OUT(n,bu,ch,h)	Heating - domestic hot water: auxiliary DHW provision for SETS [MWh]
Non-negative variable	DSM_CU(n,dsm,h)	DSM: Load curtailment hour h [MWh]
Non-negative variable	DSM_UP(n,dsm,h)	DSM: Load shifting up hour h technology dsm [MWh]
Non-negative variable	DSM_DO(n,dsm,h,hh)	DSM: Load shifting down in hour hh to account for upshifts in hour h technology dsm [MWh]
Non-negative variable	DSM_UP_DEMAND(n,dsm,h)	DSM: Load shifting up active for wholesale demand in hour h of technology dsm [MWh]
Non-negative variable	DSM_DO_DEMAND(n,dsm,h)	DSM: Load shifting down active for wholesale demand in hour h of technology dsm [MWh]
Non-negative variable	N_DSM_CU(n,dsm)	DSM: Load curtailment capacity [MW]
Non-negative variable	N_DSM_SHIFT(n,dsm)	DSM: Load shifting capacity [MWh]
Non-negative variable	RP_DIS(n,reserves,tech,h)	Reserve provision by conventionals in hour h [MW]
Non-negative variable	RP_NONDIS(n,reserves,tech,h)	Reserve provision by renewables in hour h [MW]
Non-negative variable	RP_STO_IN(n,reserves,sto,h)	Reserve provision by storage in in hour h [MW]
Non-negative variable	RP_STO_OUT(n,reserves,sto,h)	Reserve provision by storage out in hour h [MW]
Non-negative variable	RP_EV_V2G(n,reserves,ev,h)	Reserve provision by electric vehicles V2G hour h [MW]
Non-negative variable	RP_EV_G2V(n,reserves,ev,h)	Reserve provision by electric vehicles G2V hour h [MW]
Non-negative variable	RP_DSM_CU(n,reserves,dsm,h)	Reserve provision by DSM load curtailment in hour h [MW]
Non-negative variable	RP_DSM_SHIFT(n,reserves,dsm,h)	Reserve provision by DSM load shifting in hour h [MW]
Non-negative variable	RP_RSVR(n,reserves,rsvr,h)	Reserve provision by reservoirs h [MW]
Non-negative variable	RP_SETS(n,reserves,bu,ch,h)	Reserve provision by SETS [MW]
Non-negative variable	RP_SETS_AUX(n,reserves,bu,ch,h)	Reserve provision by SETS auxiliary DHW modules [MW]
Non-negative variable	RP_HP(n,reserves,bu,ch,h)	Reserve provision by heat pumps [MW]
Non-negative variable	RP_H_ELEC(n,reserves,bu,ch,h)	Reserve provision by hybrid electric heaters [MW]