Objective Function

The objective function, which is minimized by the model, consists of several cost components which are explained subsequently.

Variable costs


Z = E =
          sum( (h,map_n_tech(n,dis)) ,            c_m(n,dis)     * G_L(n,dis,h) )
        + sum( (h,map_n_tech(n,dis))$(ord(h)>1) , c_up(n,dis)    * G_UP(n,dis,h) )
        + sum( (h,map_n_tech(n,dis)) ,            c_do(n,dis)    * G_DO(n,dis,h) )
        + sum( (h,map_n_tech(n,nondis)) ,         c_cu(n,nondis) * CU(n,nondis,h) )
        + sum( (h,map_n_sto(n,sto)) ,             c_m_sto(n,sto) * ( STO_OUT(n,sto,h) + STO_IN(n,sto,h) ) )

The function value Z, which reflects overall system costs, is made up of various additive terms.

For the following terms, sums are formed as products of a cost parameter and a variable and run over every hour h as well as all countries n. The objective function includes:

  • the sum of variable costs of conventional power plants (sum( (h,map_n_tech(n,dis)) , c_m(n,dis)*G_L(n,dis,h) )), where c_m(n,dis) is the variable cost parameter of dispatchable technology dis in country n and G_L(n,dis,h) the generation of that technology in that country in hour h. To reduce the model size, the function map_n_tech(n, dis) makes sure that only those generation technologies are considered that are actually available in the respective country. Similar mapping functions are used in the following in numerous instances.

  • the costs related to changing the (aggregate) generation of dispatchable power plants (G_UP and G_DO)

  • the costs attached to curtailment CU (variable) and c_cu (parameter) for non-dispatchable technologies nondis.



+ sum( (h,map_n_dsm(n,dsm_curt)) , c_m_dsm_cu(n,dsm_curt) * DSM_CU(n,dsm_curt,h) )
+ sum( (h,map_n_dsm(n,dsm_shift)) , c_m_dsm_shift(n,dsm_shift) * DSM_UP_DEMAND(n,dsm_shift,h) )
+ sum( (h,map_n_dsm(n,dsm_shift)) , c_m_dsm_shift(n,dsm_shift) * DSM_DO_DEMAND(n,dsm_shift,h) )


In case the demand-side management (DSM) module is switched on, the objective function also includes the variable costs of load curtailment as well as of upward and downward load shifting.

Endogenous electric vehicles


+ sum( (h,map_n_ev(n,ev)) , c_m_ev(n,ev) * EV_DISCHARGE(n,ev,h) )
+ sum( (h,map_n_ev(n,ev)) , pen_phevfuel(n,ev) * EV_PHEVFUEL(n,ev,h) )


If the electric vehicle module with endogenous vehicle charging and discharging decisions is switched on, variable costs of discharching electricity from vehicles to the grid are added to the objective function. A penalty for fuel use in plug-in hybrid electric vehicles may also be included, which can minimize the non-electric use of these vehicles.



+ sum( map_n_tech(n,tech) , c_i(n,tech)*N_TECH(n,tech) )
+ sum( map_n_tech(n,tech) , c_fix(n,tech)*N_TECH(n,tech) )
+ sum( map_n_sto(n,sto) , c_i_sto_e(n,sto)*N_STO_E(n,sto) )
+ sum( map_n_sto(n,sto) , c_fix_sto(n,sto)/2*(N_STO_P(n,sto)+N_STO_E(n,sto)) )
+ sum( map_n_sto(n,sto) , c_i_sto_p(n,sto)*N_STO_P(n,sto) )

Here, annualized investment costs and annual fixed costs of all electricity generation and storage technologies are summed up. Annual fixed costs of storage are equally distributed to storage energy and storage power capacity.



+ sum( map_n_dsm(n,dsm_curt) , c_i_dsm_cu(n,dsm_curt)*N_DSM_CU(n,dsm_curt) )
+ sum( map_n_dsm(n,dsm_curt) , c_fix_dsm_cu(n,dsm_curt)*N_DSM_CU(n,dsm_curt) )
+ sum( map_n_dsm(n,dsm_shift) , c_i_dsm_shift(n,dsm_shift)*N_DSM_SHIFT(n,dsm_shift) )
+ sum( map_n_dsm(n,dsm_shift) , c_fix_dsm_shift(n,dsm_shift)*N_DSM_SHIFT(n,dsm_shift) )


If the demand-side management module is switched on, annualized investment costs and annual fixed costs of both load curtailment and load shifting technologies are added to the objective function.



+ sum( (h,map_n_sto(n,sto),reserves_up) , phi_reserves_call(n,reserves_up,h) * c_m_sto(n,sto) * (RP_STO_OUT(n,reserves_up,sto,h) - RP_STO_IN(n,reserves_up,sto,h)))
- sum( (h,map_n_sto(n,sto),reserves_do) , phi_reserves_call(n,reserves_do,h) * c_m_sto(n,sto) * (RP_STO_OUT(n,reserves_do,sto,h) - RP_STO_IN(n,reserves_do,sto,h)))
+ sum( (h,map_n_rsvr(n,rsvr),reserves_up) , RP_RSVR(n,reserves_up,rsvr,h) * phi_reserves_call(n,reserves_up,h) * c_m_rsvr(n,rsvr) )
- sum( (h,map_n_rsvr(n,rsvr),reserves_do) , RP_RSVR(n,reserves_do,rsvr,h) * phi_reserves_call(n,reserves_do,h) * c_m_rsvr(n,rsvr) )

%EV_exogenous%        + sum( (h,map_n_ev(n,ev),reserves_up) , RP_EV_V2G(n,reserves_up,ev,h) * phi_reserves_call(n,reserves_up,h) * c_m_ev(n,ev) )
%EV_exogenous%        - sum( (h,map_n_ev(n,ev),reserves_do) , RP_EV_V2G(n,reserves_do,ev,h) * phi_reserves_call(n,reserves_do,h) * c_m_ev(n,ev) )

+ sum( (h,map_n_dsm(n,dsm_curt),reserves_up) , RP_DSM_CU(n,reserves_up,dsm_curt,h) * phi_reserves_call(n,reserves_up,h) * c_m_dsm_cu(n,dsm_curt) )
+ sum( (h,map_n_dsm(n,dsm_shift),reserves) , RP_DSM_SHIFT(n,reserves,dsm_shift,h) * phi_reserves_call(n,reserves,h) * c_m_dsm_shift(n,dsm_shift) )


If the reserve module is switched on, variable costs of reserve provision via electricity storage, hydro reservoirs, electric vehicles, and demand-side management are added to the objective function. Respective variable costs of dispatchable generators are not added here, as these are already included in the variable costs shown above; variable renewables are assumed not to incur variable costs for reserve provision.



+ sum( map_n_res_pro(n,res) , c_i(n,res)*N_RES_PRO(n,res) )
+ sum( map_n_res_pro(n,res) , c_fix(n,res)*N_RES_PRO(n,res) )
+ sum( map_n_sto_pro(n,sto) , c_i_sto_e(n,sto)*N_STO_E_PRO(n,sto) )
+ sum( map_n_sto_pro(n,sto) , c_fix_sto(n,sto)/2*(N_STO_P_PRO(n,sto) + N_STO_E_PRO(n,sto)) )
+ sum( map_n_sto_pro(n,sto) , c_i_sto_p(n,sto)*N_STO_P_PRO(n,sto) )
+ sum( (h,map_n_sto_pro(n,sto)) , c_m_sto(n,sto) * ( STO_OUT_PRO2PRO(n,sto,h) + STO_OUT_M2PRO(n,sto,h) + STO_OUT_PRO2M(n,sto,h) + STO_OUT_M2M(n,sto,h)
+ sum( res , STO_IN_PRO2PRO(n,res,sto,h) + STO_IN_PRO2M(n,res,sto,h)) + STO_OUT_PRO2M(n,sto,h) + STO_OUT_M2M(n,sto,h) ) )


If the prosumage module is switched on, annualized investment costs, annual fixed costs, and variable cost of decentralized solar PV and battery storage plants are added.


+ sum( map_l(l) , c_i_ntc(l) * NTC(l)*dist(l) )

This term reflects the costs of expanding net transfer capacities between model nodes.


+ sum( (h,map_n_rsvr(n,rsvr)), c_m_rsvr(n,rsvr) * RSVR_OUT(n,rsvr,h) )
+ sum( map_n_rsvr(n,rsvr) , c_i_rsvr_e(n,rsvr) * N_RSVR_E(n,rsvr) )
+ sum( map_n_rsvr(n,rsvr) , c_i_rsvr_p(n,rsvr) * N_RSVR_P(n,rsvr) )
+ sum( map_n_rsvr(n,rsvr) , c_fix_rsvr(n,rsvr) * N_RSVR_P(n,rsvr) )

Here, the annualized investment costs, annual fixed costs, and variable costs of hydro reservoirs are added.



+ sum( (h,n,bu,hfo) , pen_heat_fuel(n,bu,hfo) * H_STO_IN_FOSSIL(n,bu,hfo,h))


If the residential heating module is switched on, this term may be used to penalize the use of fossil fuels in hybrid heating systems, and thus ensure a high share of electricity used in such installations.


+ sum( (h,n) , c_infes * G_INFES(n,h) )

The model also includes an infeasibility variable (also referred to as slack variable) and a respective penalty factor, which may be used to ensure feasible solutions in capacity-constrained settings. Usually, this infeasibility variable is not used.