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]