Restrictions

Capacity restrictions

The power generation is restricted by the installed capacity c^inst according to Equation (3.5). The power generation is thereby equal to the day-ahead schedul-ing P^day under consideration of up-regulation and down-regulation, P⁺ and P⁻. Capacity margins are kept for the potential use of spinning reserves, R^sp,+, and non-spinning reserves, R^ter,+.

P_u,t^day+P_s,u,t⁺ +P_s,u,t⁻ +R^sp,_u,t⁺+R^ter,_s,u,t⁺ ≤c^inst_u ∀s∈S, u∈U, t∈T (3.5) The power generation is not only restricted by the installed capacity. It must also be smaller than the online capacity, C^online, according to Equation (3.6). In this case, only a margin for the use of spinning reserves is kept as tertiary reserve capacities are defined as being offline (see below).

P_u,t^day +P_s,u,t⁺ +P_s,u,t⁻ +R^sp,_u,t⁺ ≤C_s,u,t^online ∀s∈S, u∈U, t∈T (3.6) The power generation must be higher than the minimal load given by the minimal load factorlf^min and the online capacity according to Equation (3.7). In this case, a margin for the use of negative spinning reserve, R^sp,−, is considered.

P_u,t^day+P_s,u,t⁺ +P_s,u,t⁻ −R^sp,_u,t⁻≥lf_u^min·C_s,u,t^online ∀s∈S, u∈U, t ∈T (3.7) An additional equation, not shown here, assures that the capacity online plus the non-spinning reserves is not higher than the installed capacity. Equation (3.8) assures that the heat generation does not exceed the installed heat generation capacity,c^inst,heat.

P_s,u,t^heat≤ ·c^inst,heat_u ∀s∈S, u∈U, t ∈T (3.8) There are two types of CHP plants in the model: back pressure plants and extrac-tion plants. In back pressure plants the heat generaextrac-tion and the power generaextrac-tion are tightly coupled and there is only one degree of freedom. In extraction plants, the heat generation can partially be chosen independently of the power generation.

The related equations are as given in [86].

Transmission

The transmission between two regions cannot exceed the installed transmission capacity t^reg. The transmission scheduled at the day-ahead market, T^day, intra-day changes, T^intra, and the transmission of tertiary reserves, T^res, are thereby considered according to Equation (3.9). Tertiary reserves can only be transmitted between regions within Germany and the related region pairs are defined by the

set X.

T_r^day_∗,r,t+T_s,r^intra_∗,r,t+T_s,^res_{(r∗,r)∈X,t} ≤t^reg_r∗,r ∀s∈S,∀r∗, r∈R,∀t∈T (3.9) Additional equations, not shown here, assure that the transmission capacities are also not exceeded at the day-ahead market and that the transmissions are symmetrical.

Within the German regions (given by X) the transmission is defined by the regional “voltage” angles Φ and the regional susceptances b^reg. The regional sus-ceptances are connection line parameters that are derived by the approach in Section 3.2.3. The “voltage” angles are variables and there is one angle for each region. The relation between angles and power exchanges on the day-ahead mar-ket is given by Equation (3.10). The intraday transmission changes are realized by changes of the angles according to Equation (3.11).

T_r^day_∗,r,t =b^reg_r∗,r ·(Φ_r∗,t−Φ_r,t) ∀r∗, r∈X, t∈T (3.10)

T_s,r^intra_∗,r,t =b^reg_r∗,r ·(Φ^delta_s,r∗,t−Φ^delta_s,r,t ) ∀s∈S,∀r∗, r ∈X, t∈T (3.11) Both at the day-ahead market and after intraday changes the difference between the angles cannot exceed the maximally allowed angle difference, φ^max, according to Equation (3.12) and Equation (3.13). The maximally allowed angle difference is thereby derived by the approach in Section 3.2.3.

(Φ_r∗,t−Φ_r,t)≤φ^max ∀r∗, r∈X, t∈T (3.12)

(Φ_r∗,t+ Φ^delta_s,r∗,t)−(Φ_r,t+ Φ^delta_s,r,t )≤φ^max ∀s∈S,∀r∗, r ∈X, t∈T (3.13)

Inter-temporal restrictions

There are several inter-temporal equations in the model. In order to calculate the start-up costs in the objective function, the started capacity, P^start, is required. It is defined by the change of the online capacity according to Equation (3.14).

P_s,u,t^start C_s,u,t^online−C_s,u,t^online₋₁ ∀s ∈S, u∈U, t∈T (3.14) The power plant operation is constraint by minimal operation times, t^minop, ac-cording to Equation (3.15). The online capacity at (t−1) minus the online capacity at t gives the capacity that is shut down at t. The condition for shutting down capacity at t is that the capacity has been online before t for the time period

t^minop at least.

C_s,u,t^online_op C_s,u,t^online₋₁−C_s,u,t^online (3.15)

∀s∈S, u∈U, t∈T, t_op∈T|t−t^minop_u ≤t_op≤t−1 In a similar manner minimal shutdown times, t^minsd, are considered by Equa-tion (3.16). The online capacity att minus the online capacity at t−1 gives the capacity that is started at t. The condition for starting capacity at t is that the capacity has been offline before t for the time period t^minsd at least.

C_s,u,t^online

sd c^inst_u −(C_s,u,t^online−C_s,u,t^online₋₁) (3.16)

∀s ∈S, u∈U, t∈T, t_sd ∈T|t−t^minsd_u ≤t_sd ≤t−1

The consideration of start-up times has two aspects. On the one hand, the pro-cess to start-up a power plant takes some hours and the power plant can not be immediately online again after having turned it off. This is considered by the minimal shutdown (offline) times. On the other hand, the decision to start-up a power plant may be spontaneous and not be planned in forehand. A start-up time should then be respected even if the power plant has been offline for many hours.

In the first hours of each optimization, the online capacity can therefore not be higher than the capacity that was planned to be online by the last optimization, given by c^onl,prev in Equation (3.18). The number of the affected hours is thereby defined by the start-up time, t^minst, of the power plant.

C_s,u,t^online c^onl,prev_u,t (3.17)

∀s ∈S, u∈U, t∈T|t≤t^minst_u

Hydro and storage power plants

Some extra equations are given for hydro power plants next to the constraints that apply to all power plants. There are three types of hydro power plants.

Pump storage power plants, hydro power plants with a reservoir but without pumping capacities and run of river hydro power plants. Pump storage power plants are defined by U_st. The storing of electricity, L^day, is scheduled at the day-market but rescheduling can take place leading to an up-regulation, L⁺, or down-regulation,L⁻, of the storing. Storage power plants can also provide reserve capacities by a potential decrease or increase of the storing. A potential increase of storing stands for negative spinning reserve, R^sp,stor,−. The total storing plus the reserve can not exceed the total storing capacity, c^inst,load.

L^day_u,t +L⁺_s,u,t−L⁻_s,u,t+R^sp,stor,_u,t ⁻c^inst,load_u ∀s∈S, u∈U_st, t∈T (3.18)

Positive spinning reserve, R^sp,stor,+, can be provided by a potential decrease of storing. The provision of positive spinning reserve respectively a decrease of storing is only possible to the extent storing is in process.

R^sp,stor,_u,t ⁺L^day_u,t +L⁺_s,u,t−L⁻_s,u,t ∀s∈S, u∈U_st, t∈T (3.19) The storage content, F^storage, is subject to the following equation. The con-tent is increased by pumping and decreased by generation under consideration of the losses l^loss. The storage content at the beginning is thereby set to the value resulting from the previous optimization.

F_s,u,t^storage−F_s,u,t^storage₋₁ = (1−l^loss_u )·(L^day_u,t +L⁺_s,u,t−L⁻_s,u,t)−(P_u,t^day +P_s,u,t⁺ −P_s,u,t⁻ ) (3.20)

∀s∈S, u∈U_st, t∈T Further restrictions, not shown here, assure that the storage level respects the capacity of the storage reservoir and that there is a margin for the potential use of negative or positive reserve.

Hydro power plants with large natural reservoirs but without pumping capacity (U_rs) are refilled by natural inflows, i^res. Their power generation and reservoir content are related by Equation (3.21).

F_s,a,t^reservoir−F_s,a,t^reservoir₋₁ =i^res_a,t −V_s,a,t^spill−

u∈(U_a∩U_rs)

P_u,t^day +P_s,u,t⁺ −P_s,u,t⁻ (3.21)

∀s ∈S, a∈A, t∈T

The variableV^spill gives the possibility to spill water without producing electricity.

The reservoir content F^reservoir is also restricted by a maximal fill level and a minimal fill level (not shown here). The power generation of run of river hydro power plants can not be scheduled. It is directly considered in Equation (3.2).

Reserves

Different kind of reserve capacities have to be kept in the system. Primary and secondary reserves in positive and negative direction are considered by Equa-tions (3.22) and (3.23). The reserve requirements are defined per region and the reserves can only be provided by power plants that are capable of it, defined by the unit setU_sp, and that are located in the region, defined by the unit setU_r. An equal distribution of the reserve capacities over all regions can so be considered, see Section 2.2.2. ; It was shown in Section 2.2.2 that these reserve types can normally be provided by the same power plant types. The positive primary and secondary reserve requirements, d^prim,+andd^sec,+, are thereby summed up. The same applies to the negative primary and secondary reserve,d^prim,−andd^sec,−. The positive and

negative reserve capacities are denoted by R^sp,+ and R^sp,−. Pump storage plants, defined by Ust, can also provide reserves by changing the storing process. The potential decrease of storing represents positive spinning reserve, R^sp,stor,+, and the potential increase represents negative spinning reserve, R^sp,stor,−.

u∈(U_r∩U_sp)

R^sp,_u,t⁺+

u∈(U_r∩U_sp∩U_st)

R^sp,stor,_u,t ⁺d^prim,_r ⁺+d^sec,_r ⁺ ∀t ∈T,∀r ∈R (3.22)

u∈(U_r∩U_sp)

R^sp,_u,t⁻+

u∈(U_r∩U_sp∩U_st)

R^sp,stor,_u,t ⁻d^prim,_r ⁻+d^sec,_r ⁻ ∀t ∈T,∀r ∈R (3.23) The reserve capacities do not depend on the forecast scenarios s as they are only scheduled at the day-ahead market. In the following intraday optimizations they are fixed to the day-ahead schedule and changes are not possible. Thus, it is taken into account that primary and secondary reserves are normally tendered in constant shares for a longer period in the future.

Tertiary reserves can be provided both by fast activating off-line plants such as gas turbines and by spinning power plants. In general, the use of off-line plants will be preferred as there are no direct costs involved. Nevertheless, capacities that are normally used for primary and secondary reserves can also provide ter-tiary reserves. The sum of primary and secondary reserve capacities, R^sp,+, and off-line tertiary reserve capacities, R^ter,+, must therefore cover the sum of pri-mary, d^prim,+, secondary, d^sec,+, and tertiary, d^ter,+, reserve requirements accord-ing to Equation (3.24). Off-line tertiary reserve capacities can only be provided by appropriate power plants like gas turbines or hydro plants, defined by U_tr. The power plants must also be located in the right region, so belonging to U_r, but it is possible to exchange tertiary reserves between the German regions by T^res. Negative tertiary reserves are not considered in the model.

u∈(U_r∩U_sp)

R^sp,_u,t⁺+

u∈(U_r∩U_tr)

R^ter,_s,u,t⁺+

u∈(U_r∩U_sp∩U_st)

R^sp,stor,_u,t ⁺+

r∗|(r∗,r∈X)

T_s,r^res_∗,r,t d^prim,_r ⁺+d^sec,_r ⁺+d^ter,_r ⁺ ∀s∈S, r ∈R, t∈T (3.24) The reserve requirements are calculated in the second part of the following section.