The computation of the contribution margins for all simulated fuel price scenarios is computational intractable. Consequently, we have to select some representative scenarios for each year of a stage. For this purpose, we use the fast forward selection, a popular method in stochastic programming to select representative scenarios. Before we can apply the fast forward selection, we have to answer two questions:
• Should the simulated prices for the different fuels (coal, gas) and carbon be treated together or independently?
Simulation run Gas price Coal price Carbon price [e/MWh] [e/MWh] [e/t]
1 20 8 20
2 16 7 21
3 17 6 16
Table 3.1: Example of fuel price simulations used to illustrate different possibilities to select representative fuel prices.
• Should the simulated fuel and carbon prices be treated as complete time series, or can we consider each year independently?
In the following, we first discuss these two questions. Then, we define a dis- tance metric, which we use for the selection of representative scenarios. Next, we present the algorithm that we use to select these scenarios based on an explicit rep- resentation of fuel prices. Finally, we briefly outline how the algorithm is adjusted if the climate spread is used instead.
3.2.3.1 Treatment of Different Fuels
The question we face here is whether we can reduce the simulated prices for the different fuels and the carbon price independently of each other. Let us first illus- trate the problem with a simple example.
Example. Assume that there are three different fuel price simulations as shown in Ta-
ble 3.1. Selecting a representative scenario for each fuel (and carbon) separately, we would select for each fuel the scenario that lies between the two other scenarios. The result is a gas price of 17 e/MWh, a coal price of 7 e/MWh, and a carbon price of 20 e/t.
If we consider the coal price, the gas price and the carbon price together, this solution is not feasible as the prices do not belong to the same simulation run. To be able to select representative scenarios, we need a distance metric to determine the distances between the fuel price scenarios of the different simulation runs. Assume that we use the Manhattan distance as distance metric. Then, we would choose the second simulation run. Note that this choice results in a worse approximation of the gas price and the carbon price in this example.
Considering different types of fuels and carbon together reduces the degrees of freedom for the scenario selection. The set of scenarios that can be chosen is a subset of the set of scenarios that can be chosen considering each fuel and carbon
3.2. REPRESENTATION OF FUEL AND CARBON PRICE UNCERTAINTY 129 separately. Consequently, the solution of a combined consideration cannot be a better representation of the fuel prices than a separate consideration.
Nevertheless, we argue that different fuels and carbon must be considered to- gether for the selection of representative scenarios. There are two reasons for this. First, our true objective is to find a good approximation of the expected contribu- tion margin, not of the fuel prices. The contribution margin of a state is obtained by optimizing the power market operation. This power market operation depends on the value of all fuel prices as well as the carbon price.
Secondly, fuel prices are correlated.6 If gas prices are high, coal prices tend to be also high. If we select representative fuel prices independently, we lose this relationship between the different fuel prices. Therefore, we consider complete fuel price scenarios consisting of prices of all fuels and the carbon price for the selection of representative scenarios.
3.2.3.2 Treatment of Different Years
The second decision to take is whether complete time-series should be treated together. If complete time-series are treated together, the selected scenarios of the different years of a state must belong to the same simulation run, e.g., if fuel price scenario 4 is selected for the second year of the stage, it must also be selected for the other years. If we do not consider complete time-series, we may choose for example fuel price scenario 4 for the second year, and scenario 8 for the third year. Considering complete time-series may be appropriate when current fuel prices have an important impact on future fuel prices. This is especially the case, if there is a lagged correlation in prices (as it can be observed between gas and oil prices for example). However, we do not consider such lagged correlations.
While the fuel price for the next year depends on the current price,7 we do not think that this argument is strong enough to consider complete time-series to- gether. Power market operation in year t+1 is independent of the power market operation in year t in our model. As a consequence, fuel prices in year t have no impact on the power market operation in year t+1. Hence, the problem of approximating the contribution margins of a complete state can be divided into approximating the contribution margins of each year separately. Treating each
6See, e.g., Weber [Web05, p. 58]
7This holds for GBMs and mean-reverting processes, two stochastic processes commonly used
to simulate fuel prices. Other stochastic processes like ARMA models for example, may use a longer price history. As we use either a (correlated) GBM or a (correlated) mean-reverting process, we do not consider processes relying on a longer price history.
year independently of the others allows for a better approximation, as the deci- sion space for considering all time-series together is a proper subset of the decision space for treating each year separately.
3.2.3.3 Distance Metric
As we consider the different fuel prices of a year together for the scenario selec- tion, a distance metric for fuel price scenarios has to be defined. A simple possibil- ity would be to use the Manhattan distance or the Euclidean distance. However, the application of one of these distance metrics without any adjustments has an important drawback in our opinion.
As already pointed out, our objective is to find a good approximation of the contri- bution margins for a state. The impact of the different fuel prices on the contribu- tion margin depends on many factors and cannot be exactly predicted. However, the influence of the coal price, the gas price and the carbon price is not the same. This is due to the different units of fuel and carbon prices as well as the different efficiency of CCGT units and coal units.
Consider the case of coal units: An average coal unit with an efficiency of 40% requires 2.5 MWh of coal to produce 1 MWh electricity. Thereby, it emits about 0.86 t of CO2. The fuel price has a stronger impact on costs compared to the carbon price. Using a distance metric without adjustments does not consider the different impact these prices have on the costs of a unit, and with it on the contribution margin.
We propose to consider these differences in the definition of the fuel price dis- tance function. However, as the efficiencies and the emissions factor of units differ, the choice of the adjustment weights is not obvious. The values we use are based on average efficiencies taken from Caisse des Dépôts [Cai07]. The av- erage efficiency of coal units is assumed to be 0.4, those of CCGT units 0.55. To adjust carbon prices, we take the average of the emission factor for CCGT units (0.36 t CO2/MWh) and the one of coal units (0.86 t CO2/MWh). With these val- ues, we define the distance D(cFuela , cFuelb ) between fuel price scenarios cFuela and
cFuelb as: D(cFuela , cFuelb ):= v u u t (cCoala −cCoalb ) 0.4 !2 + (c Gas a −cGasb ) 0.55 !2 +(cCO2 a −cCOb 2)·0.61 2 . (3.2)
3.2. REPRESENTATION OF FUEL AND CARBON PRICE UNCERTAINTY 131
3.2.3.4 Algorithm
We use Algorithm 1 (which is based on the fast forward selection as described e.g., by Gröwe-Kuska et al. [GK+03]) to determine the representative fuel price scenarios for each state. Note that if two states xaand xbdiffer only in the number of new units xu, but not in the fuel price scenarios xf, then we do run Algorithm 1 only once.
Let us briefly comment the algorithm. To select representative fuel prices for a state, we consider all fuel price scenarios associated to this state, CFuel(x). For each year of the state, the algorithm iteratively selects a fuel price scenario, until the predefined number of selected scenarios per year, R, is reached.
We start by setting the set of non-selected scenariosCtFuel,NonRep(x)for the current year equal to the set of all fuel price scenarios for this year (line 2). The set of se- lected fuel price scenarios CtFuel,Rep(x)is empty at the beginning of the algorithm (line 3). At each iteration, a fuel price scenario cFuels is selected such that it mini- mizes the weighted sum of the distance between all non-selected scenarios to one of the already selected scenarios or to cFuel
s (lines 10, 21).
We define d(cFuelj , cFueli ) as the minimum distance between scenario cFuelj and a
scenario of the set of selected scenarios or scenario cFuel
i . During the first iteration,
d(cFuelj , cFueli )corresponds to the distance between cFueli and cFuelj (line 6), as the set
of selected scenarios is empty.
Then we calculate for each non-selected scenario cFuel
i the weighted sum of mini- mum distances from all non-selected scenarios to one of the selected scenarios or
cFueli (line 8). Thereby, P(cFuelj ) denotes the occurrence probability for fuel price scenario cFuel
j . The sum of weighted distances, zcFueli , corresponds the sum of min- imum distances between all non-selected scenarios and all selected scenarios un- der the assumption that cFueli is added to the selected scenarios.
Next, we select the scenario with the minimum sum of weighted distances (line 10), remove this scenario from the set of non-selected scenarios (line 11) and add it to the set of selected scenarios (line 12).
In the subsequent iterations, we only have to adjust d(cFuelj , cFueli ). If the scenario chosen during the last iteration, cFuelsr−1, is closer to cFuelj than any of the previously chosen scenarios and cFuel
i , then d(cFuelj , cFueli ) is updated with the distance be- tween cFuel
j and cFuelsr−1. The other steps (line 19 - 23) are the same as the ones in the
Algorithm 1Fast forward selection of representative fuel price scenarios
1: fort=t1s tot1s +Tsdo
2: CtFuel,NonRep(x) :=CtFuel(x)
3: CtFuel,Rep(x) = ∅
4: for allcFueli ∈ CtFuel,NonRep(x) do
5: for allcFuelj ∈ CtFuel,NonRep(x) do
6: d(cFuelj , cFueli )[1] :=D(cFuelj , cFueli )
7: end for
8: z[1]
cFueli := ∑
cFuelj ∈CtFuel,NonRep(x)
P(cFuelj )·d(cFuelj , cFueli )[1]
9: end for
10: cFuels1 = argmin cFueli ∈CtFuel,NonRep(x)
z[1]
cFueli
11: CtFuel,NonRep(x) = CtFuel,NonRep(x)\ {cFuels1 }
12: CtFuel,Rep(x) = CtFuel,Rep(x)∪ {cFuels1 }
13:
14: forr =2 to R do
15: for allcFueli ∈ CtFuel,NonRep(x) do
16: for allcFuelj ∈ CtFuel,NonRep(x) do
17: d(cFuelj , cFueli )[r] :=mind(cFuelj , ciFuel)[r−1], d(cFuelj , cFuelsr−1)[r−1]
18: end for
19: z[r]
cFueli := ∑
cFuelj ∈CFuel,NonRept (x)
P(cFuelj )·d(cFueli , cFuelj )[r]
20: end for
21: cFuelsr = argmin cFueli ∈CFuel,NonRept (x)
z[r]
cFueli
22: CtFuel,NonRep(x) =CtFuel,NonRep(x)\ {cFuelsr }
23: CtFuel,Rep(x) = CtFuel,Rep(x)∪ {cFuelsr }
24: end for
3.2. REPRESENTATION OF FUEL AND CARBON PRICE UNCERTAINTY 133
3.2.3.5 State Representation Based on the Climate Spread
The algorithm just described is based on the assumption of an explicit representa- tion of the stochastic fuel prices. If we use the climate spread to reduce the prob- lem’s dimension, the choice of the representative fuel price scenarios deviates a little bit from the approach outlined before. In principle, we also use Algorithm 1 to select representative climate spread scenarios for each year of the stage. But instead of using the distance metric defined in equation (3.2), we take the absolute difference between two climate spreads values (e.g., for climate spread values of 2.3 and 3.7, we use 1.4 as distance, independent of the concrete fuel prices associ- ated to these climate spreads).
While the use of the climate spread helps us to reduce the complexity of the prob- lem by decreasing the dimensionality, it has its drawbacks. As we have seen in Figure 3.4 on page 125, the same climate spread values obtained with different fuel prices can result in different contribution margins. To account for these prob- lems, we do not only use one fuel price scenario per representative climate spread. Instead, we use Algorithm 1 to select several (for example three or five) represen- tative fuel price scenarios for each representative climate spread scenario. Note that in line 2 of the algorithm, we replace the set of all fuel prices associated to state x with the subset of all fuel prices with the selected climate spread associ- ated to state x.