Both models presented in the previous subsections are based on the assumption of consecutive timesteps. For our investment module, we are interested in power market results for a whole year. Modeling a whole year is computationally in- tractable. A common approach to circumvent such problems is to consider just a few characteristic days (e.g., Sun et al. [Sun+08], Swider [Swi07], Swider and Weber [SW07]). In [Sun+08; Swi07; SW07], every two months of a year are rep- resented by one workday and one weekend day. Each of these days consists of twelve typical load profiles with a duration of two hours each.
While the representation of the year by typical days significantly reduces the com- putational burden, it requires some model modifications. We discuss these modi- fications in the following for the basic deterministic model.
Objective Function:
As there are more workdays than days of the weekend, the different timesteps in the model do not occur equally often. Hence, they must be accordingly weighted. This can be done by introducing the weighting factor ft.13 If several hours are ag- gregated to one timestep, then an additional parameter dt indicating the duration of each timestep is added. It is important to notice that only the fuel costs and the operating costs have to be multiplied by dt. The start-up costs are indepen- dent of the duration of a timestep, a fact that is often not considered (e.g., [Swi07; SW07]14). The objective function is now the following:
min TC=
∑
u∈U
∑
t∈T
ft· (dt· (FCut+OCut) +SCut).
Besides the modification of the objective function, further modifications are re- quired. To the best of the author’s knowledge, these modifications are not consid- ered in other models. These modifications concern all intertemporal restrictions. In the following, we demonstrate the need for modifications with simple exam- ples, and propose two alternative formulations for the corresponding constraints.
Example(Assumptions). For the following examples, we assume that every two months
are represented by a weekday and a weekend day. Every day is represented by 12 timesteps. Let us further assume that the first day is a weekend day. Timesteps 1–12 are used to represent the first weekend day. Timesteps 13–24 are used for the first workday.
Start-Ups and Shut-Downs:
Start-up restriction (2.7) and shut-down restriction (2.8) are used to determine the capacity started-up and the capacity shut-down at every timestep. These values are required for the minimum up-time and minimum down-time restrictions. Fur- thermore, the variable PStartUp is used to determine start-up costs. Inequalities (2.7) and (2.8) are intertemporal constraints, and need to be adjusted for a model based on typical days. In the following, we briefly demonstrate the need for a modified problem description with a simple example, before we propose two dif- ferent formulations that take these problems into account.
13If every two months are represented by one workday and one weekend day, and there are 41
workdays in January and February, then ftis 41 for each timestep representing a workday in this
period.
2.3. LINEAR PROGRAMMING MODELS 81
Example(Strict start-up constraints). Let us consider how the capacity started at time-
step 13 is determined. Using start-up constraint(2.7), we get:
P13StartUp ≥P13Online−P12Online. (2.15)
This is correct, but only if we assume that the considered workday that is represented is a Monday. If the typical workday is not a Monday, the restriction would be the following:
P13StartUp ≥P13Online−P24Online. (2.16)
As we assumed that the current day is not a Monday, the previous day is also a workday. Since we use just one workday to represent all workdays within two months, the previous day is represented by the same typical day as the current day. Hence, the previous timestep of timestep 13 is timestep 24 in this case. One possibility to cope with this problem is to use both constraints.
Formally written, we use in addition to the “normal” start-up constraint (2.7) the following restriction:
PutStartUp ≥ PutOnline−POnline
u,t+tPerDay−1 ∀u, t mod tPerDay =1, (2.17)
with tPerDay being the number of timesteps per day. Note that this restriction is quite strict. Compared with the original problem (which is not based on represen- tative timesteps), some feasible solutions are excluded. In the following, we refer to this formulation as strict model.
Example(Weighted start-up constraints). Another possibility is to use a weighted sum
of restrictions(2.15) and (2.16) to define P13StartUp. Therefore, we have to define two addi- tional variables that are, as all variables we use, non-negative. Thereby, P12,13StartUp denotes the capacity that is started from timestep 12 to 13, while P24,13StartUp denotes the capacity started from timestep 24 to 13:
P12,13StartUp ≥P13Online−P12Online
and
We can now define P13StartUp as:
P13StartUp ≥ 15·P12,13StartUp+4
5·P StartUp
24,13 . (2.18)
Note that the auxiliary variables P12,13StartUp and P24,13StartUp in inequality (2.18) cannot be re- placed with the right hand sides of restrictions(2.15) and (2.16). The reason is that the right hand sides of the restrictions may get negative, while the variables are greater or equal to zero. A negative right hand side in one of both restrictions would offset the number of start-ups of the other restriction. Assume for example, P12Online =500, POnline
13 =100 and
P24Online = 0. Without the additional variables, one would get P13StartUp = 1/5· (100−
500) +4/5· (100−0) = 0. As a consequence, no start-up costs are considered in this
case, even though—considering a whole week—in total 400 MW of capacity are started.
To write the weighted start-up constraint (2.18) in a general form, we first have to define mt as:15
mt = (
5 if t belongs to a workday, 2 otherwise.
Thereby, mt counts how often the day to which timestep t belongs is repeated, be- fore the following typical day occurs. It is different from ft defined earlier, which is a weighting factor that counts the total frequency of the hours represented by
t, independent of the fact whether these timesteps occur in a row or with other
timesteps in between.16
The general form of the weighted start-up constraint (2.18) is
PutStartUp ≥ m1
t ·P StartUp
u,t−1,t +mmt−t1·P
StartUp
u,t+tPerDay−1,t ∀u, t mod tPerDay=1, (2.19) with
PuStartUp,t−1,t ≥ PutOnline−PuOnline,t−1 ∀u, t mod tPerDay=1
and
PStartUp
u,t+tPerDay−1,t≥ PutOnline−PuOnline,t+tPerDay−1 ∀u, t mod tPerDay=1.
15We assume here a normal week with five workdays and two weekend days. The definition
can easily be adjusted to consider holidays.
16Assuming there are nine weekends in January and February, ftfor timesteps of a weekend
2.3. LINEAR PROGRAMMING MODELS 83 We refer to this model as weighted model. The shut-down restrictions of the basic model can be extended the same way, either as described in the strict model or the weighted model.
Minimum Up- and Down-Time Constraints:
If just the minimum up-time constraint (2.11) and the minimum down-time con- straint (2.12) from the basic deterministic model are used, a model based on typical days might deliver solutions that are infeasible in reality. This can be illustrated with the following simple example.
Example. Assume that there is a unit with a minimum up-time of two timesteps. As-
sume further that the unit is started on the last timestep of a weekday, while on the last timestep of the previous weekend day, the unit was already started earlier. Then, on the first timestep of the weekday, we might turn off the unit. However, in four out of five cases (namely from Tuesday to Friday), this decision is not allowed in reality, as the unit was just started one timestep before.
To handle this problem, the two different methods already used for the start-up and shut-down variables can be applied. We demonstrate how to use them on the example of the minimum up-time restriction. The first possibility, applied in the
strict model, is to use in addition to the normal minimum up-time restriction (2.11)
the following restriction:
PutOnline ≥
t
∑
τ=t−(tMinUpu −1)
PuStartUp,ˆt(τ)
∀u,⌈(t− (tMinUpu −1))/tPerDay⌉ < ⌈t/tPerDay⌉, (2.20)
with17
ˆt(τ) = τ+⌈t/tPerDay⌉ − ⌈τ/tPerDay⌉·tPerDay.
Thereby, ˆt(τ)simply denotes the preceding timesteps of t that are represented by
the same typical day. For example, if t=14 and τ =13, then ˆt(13) =13. If τ =12, then ˆt(12) = 24.
While the normal minimum up-time constraint (2.11) ensures that the minimum up-time constraint holds if the previous day is represented by another typical day
17We assume that the maximum minimum up-time and the maximum minimum down-time is
0 1000 2000 3000 4000 5000
Mon Tue Wed Thu Fri Sat Sun
Capacity online [MW]
Day
Basic model
Weighted model Strict model
Figure 2.6: Capacity online of a group of coal units using different constraints.
(e.g., the current day is a Monday and the previous day was a Sunday), the ad- ditional minimum up-time constraint (2.20) ensures that the minimum up-time constraint also holds if the previous day is represented by the same typical day (e.g., Tuesday/Monday).
While this approach excludes some solutions that are feasible in the original prob- lem, it guarantees that no solutions that violate the minimum up-time restrictions in reality are allowed in the model. In the weighted model, we combine the two minimum up-time restrictions (2.11) and (2.20) to
PutOnline ≥ m1 t · t ∑ τ=t−(tMinUpu −1) PuStartUp,τ +mt−1 mt · t ∑ τ=t−(tMinUpu −1) PStartUp u,ˆt(τ) ∀u, t. (2.21)
If ⌈(t− (tMinUpu −1))/tPerDay⌉ = ⌈t/tPerDay⌉, then the weighted minimum up-
time constraint (2.21) corresponds to normal minimum up-time constraint (2.11). Note that the weighted formulation of the minimum up-time constraint allows infeasible solutions as well as it excludes feasible solutions, but both to a lower extend compared with the basic model and the strict model, respectively.
Figure 2.6 shows an example of the operation of a group of coal units for a week applying the basic model, the weighted model and the strict model. In this ex- ample, both the weighted model as well as the strict model lead to a feasible unit commitment. The unit commitment of the basic model is infeasible due to a vio- lation of the minimum up-time constraint during the night of Saturday.
2.3. LINEAR PROGRAMMING MODELS 85
Storage Constraints:
It is very important to adjust the storage content constraint accordingly. The nor- mal storage content constraint (2.13) ignores that timesteps have a different fre- quency. If just constraint (2.13) is used,18 a very monotone storage operation as the one depicted in Figure 2.7(a) can be observed. During workdays, the storage produces electricity at its maximum level, while during weekends, the storage gets charged. Storage operation seems to be independent of electricity prices. At the moment of the highest prices, the storage is charged at the maximum charging level.
This behavior is caused by the non-consideration of the different frequencies of the timesteps. For the storage operation, the model assumes equal frequencies for all timesteps, which is not true. As there are more workdays than weekend days, much more electricity is produced compared to the electricity used to charge the storage. The obtained solution is infeasible in reality.
To get a more realistic storage operation, we use two additional constraints. For the last timestep of each representative day (i.e., t+1 mod tPerDay=0), we replace the normal storage content restriction (2.13) with the following equation:
Vu,t+tPerDay =Vu,t+
t+tPerDay
∑
τ=t+1
(mτ ·dτ· (ηu·Wuτ−Puτ))
∀u∈ UStorage, t mod tPerDay =0. (2.22)
Equation (2.22) ensures that the storage content at the end of a representative day corresponds to the the storage content at the end of the previous representative day adjusted by the electricity usage of the storage during the period represented by the representative day. For example, constraint (2.22) guarantees that the stor- age content at the end of Friday corresponds to the storage content at the end of Sunday adjusted by the production and charging during the whole week.
The second constraint we add ensures that the storage content at the end of the two months period represented by one weekday and one weekend day corre- sponds to the storage content at the last timestep of the previous period adjusted
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000 20 30 40 50 60 70 80 90 100 Storage operation [MW]
Electricity price [EUR/MWh]
Timestep
Storage operation Electricity price
(a) Using only constraint (2.13)
-8000 -6000 -4000 -2000 0 2000 4000 6000 8000 20 30 40 50 60 70 80 90 100 Storage operation [MW]
Electricity price [EUR/MWh]
Timestep
Storage operation Electricity price
(b) Using constraints (2.13), (2.22) and (2.23)
Figure 2.7: Storage operation (left axis) and electricity prices (right axis). Negative values for the storage operation correspond to storage charging, positive values represent electricity production.
by the storage production and charging within this period:
Vu,t+2·tPerDay =Vu,t+
t+2·tPerDay
∑
τ=t+1
(fτ ·dτ· (ηu ·Wuτ−Puτ))
∀u∈ UStorage, t mod 2·tPerDay=0. (2.23)
Examining Figure 2.7(b), it can be seen that considering constraints (2.22) and (2.23) in addition to constraint (2.13) gives a much more realistic storage opera- tion. The storage operation is now strongly correlated with the electricity prices. During high price periods, electricity is produced, while during low price peri- ods, the storage is charged. Compared with Figure 2.7(a), one can notice that the price peaks are now lower—a result of the more appropriate storage opera- tion.19 However, as the storage produces more electricity in the version depicted in Figure 2.7(a) than it can produce in reality, average electricity prices rise from 60.59 e/MWh to 62.49 e/MWh with the new constraints.