Note

This page is generated from inline documentation in MESSAGE/scaling_investment_costs.gms.

Auxiliary investment parameters

Levelized costs excluding fuel costs

For the ‘soft’ relaxations of the dynamic constraints and the associated penalty factor in the objective function, we need to compute the parameter levelized_costn,t,y.

levelized_costn,t,m,y,h:= inv_costn,t,yinterestratey(1+interestratey)|y|(1+interestratey)|y|1+fix_costn,t,y,y1hduration_timehcapacity_factorn,t,y,y,h+var_costn,t,y,y,m,h

where |y|=technical_lifetimen,t,y. This formulation implicitly assumes constant fixed and variable costs over time.

Warning: Levelized capital costs do not include fuel-related costs. All soft relaxations of the dynamic activity constraint are disabled if the levelized costs are negative!

Construction time accounting

If the construction of new capacity takes a significant amount of time, investment costs have to be scaled up accordingly to account for the higher capital costs.

construction_time_factorn,t,y=(1+interestratey)|y|

where |y|=construction_timen,t,y. If no construction time is specified, the default value of the investment cost scaling factor defaults to 1. The model assumes that the construction time only plays a role for the investment costs, i.e., each unit of new-built capacity is available instantaneously.

Comment: This formulation applies the discount rate of the vintage year (i.e., the year in which the new capacity becomes operational).

Investment costs beyond the model horizon

If the technical lifetime of a technology exceeds the model horizon Ymodel, the model has to add a scaling factor to the investment costs (end_of_horizon_factorn,t,y). Assuming a constant stream of revenue (marginal value of the capacity constraint), this can be computed by annualizing investment costs from the condition that in an optimal solution, the investment costs must equal the discounted future revenues, if the investment variable CAP_NEWn,t,y>0:

inv_costn,t,yV=yYn,t,yVlifetimedf_yearyβn,t,

Here, βn,t>0 is the dual variable to the capacity constraint (assumed constant over future periods) and Yn,t,yVlifetime is the set of periods in the lifetime of a plant built in period yV. Then, the scaling factor end_of_horizon_factorn,t,yV can be derived as follows:

end_of_horizon_factorn,t,yV:=yYn,t,yVlifetimeYmodeldf_yearyyYn,t,yVlifetimedf_yeary+beyond_horizon_factorn,t,yV(0,1],

where the parameter beyond_horizon_factorn,t,yV accounts for the discount factor beyond the overall model horizon (the set Y in contrast to the set YmodelY of the periods included in the current model iteration (see the page on the recursive-dynamic model solution approach).

beyond_horizon_lifetimen,t,yV:=max{0,economic_lifetimen,t,yVyyVduration_periody}
beyond_horizon_factorn,t,yV:=df_yeary^1(1+interestratey^)|y^|1(11+interestratey^)|y~|111+interestratey^

where y^ is the last period included in the overall model horizon, |y^|=duration_periody^ and |y~|=beyond_horizon_lifetimen,t,yV.

If the interest rate is zero, i.e., interestratey^=0, the parameter beyond_horizon_factorn,t,yV equals the remaining technical lifetime beyond the model horizon and the parameter end_of_horizon_factorn,t,yV equals the share of technical lifetime within the model horizon.

Possible extension: Instead of assuming βn,t to be constant over time, one could include a simple (linear) projection of βn,t,y beyond the end of the model horizon. However, this would likely require to include the computation of dual variables endogenously, and thus a mixed-complementarity formulation of the model.

Remaining installed capacity

The model has to take into account that the technical lifetime of a technology may not coincide with the cumulative period duration. Therefore, the model introduces the parameter remaining_capacityn,t,yV,y as a factor of remaining technical lifetime in the last period of operation divided by the duration of that period.