We attempt to simulate future hydropower revenue which is determined by runoff, wholesale electricity prices, and supply schedules (Gaudard et al. 2013a). Figure 1 highlights the logical structure of our modeling framework, which is detailed in the following sections. The grey areas represent input data we obtained.
Climate Data
We consider three GHG emission scenarios; A1B, A2 and RCP3PD.
-
A1B is considered a ‘reasonable’ scenario, which generates a global warming of + 1.7 ∘C to + 4.4 ∘C between the years 1980–1999 and 2090–2099 (IPCC 2000).
-
A2 is the second-worst scenario in the special report emissions scenarios frame with a temperature increase of + 2 ∘C to 5.4 ∘C (IPCC 2000).
-
RCP3PD, also named RCP2.6, expected global warming is of + 0.3 ∘C to 1.7 ∘C between 1986–2005 and 2081–2100 (IPCC 2013; Meinshausen et al. 2011).
Climate data is provided by the ENSEMBLES project (2013) for 10 combinations of global and regional climate models (GCM-RCM), which have a grid of 25 km × 25 km. This horizontal resolution is insufficient to consider local characteristics especially in mountain areas where local variability is high. For example, the altitude of the nearest grid point may differ by several hundreds of metres in comparison with the actual location. This is the reason why downscaling is necessary
Adapting GCM-RCM data to local situations has been performed by two methods, i.e. the Delta Change Method (DCM) (Bosshard et al. 2011) and the Empirical Distribution Delta Method (EDDM) (Gaudard et al. 2013a). For the whole panel of GHG emission scenarios (A1B, A2 and RCP3PD), we used the data provided by C2SM (CH2011, C2SM (2014)). This team applied the DCM (Bosshard et al. 2011). This consists of moving the mean temperature and precipitation in historical data to reflect climate forecasting. C2SM simulates the meteorological variability by running a random function. It therefore obtains a set of 10 time series for each GCM-RCM, i.e. 100 by GHG emission scenarios.
For the scenario A1B, we also applied EDDM (Gaudard et al. 2013a). It differs from the previous one, because it takes into account the variation in distribution, not only the mean variation. The impact of climate change on interannual variability is therefore considered. In contrast to DCM, EDDM provides only one path per GCM-RCM. This is due to the fact that the variability is not generated by a stochastic variable like with DCM.
Glacio-Hydrological Model
High-mountain catchment areas generally have a long-lasting snow cover and a considerable degree of glacier coverage. This influences the runoff regime and needs to be taken into account in the runoff modeling. For this reason the combined glacio-hydrological model GERM (Farinotti et al. 2012; Huss et al. 2008) is employed to assess the impact of climate change on the evolution of runoff. The model considers the processes of accumulation and the melt of snow and ice masses as well as the glacier evolution. Ablation is basically a function of air temperature and potential solar radiation. Accumulation is modeled by using precipitation lapse rates and air temperature to distinguish between liquid and solid precipitation. Snow redistribution processes (wind, avalanches) are also included. In the last step, the model evaluates the local water balance given by liquid precipitation, melt water and evaporation and routes the water through a couple of linear reservoirs in order to mimic the retention of the water in the catchment area.
The model is forced by daily temperature and precipitation time series from the 1980–2100 time period. Past meteorological data is taken from MeteoSwiss (2013) weather stations in the vicinity of the catchment area. For future time series the downscaled climate scenarios are used (see Section 3.1). The empirical character of the glacier and runoff model requires calibration of various model parameters which is done by means of past ice volume changes, direct mass balance measurements and runoff records. Gabbi et al. (2012) describe the application of the GERM model to our case study. It provides further details about calibration and validation. If it presents the A1B GHG scenario results, the A2 and RCP3PD ones are unpublished so far.
For glacier projections the distribution of ice mass in the modeling domain has to be known. Due to scarce or non existent measurements of ice-thickness, estimation approaches are commonly used (Farinotti et al. 2009). Our own previous estimations showed a total ice volume of 4.41 ± 1.02 km 3. However, recently performed area-wide ice-thickness measurements revealed a smaller ice volume of 3.69 ± 0.31 km 3 (-16 %). We test both initial ice volumes to quantify the sensitivity of our results to this parameter.
Wholesale Electricity Price Models
For electricity price, we consider one reference, called Ref, which is the repetition of historical prices, and three price models, explained below. They are parameterized with EEX German electricity spot prices (EEX 2013). This index is more relevant for the long-term forecasting, since it is the expression of the leading exchange market in Europe. The period of calibration of all our models is 01-Apr-2001 to 31-Mar-2012. In order to get the real price, we use the Harmonised Index of Consumer Prices (HICP) for Germany provided by Eurostat (2013). Thus all our revenues are year 2005 € equivalent.
We consider an explanatory scenario based on past data (EEA 2000; Kowalski et al. 2009). Our models do not aimed at forecasting electricity prices, which is almost impossible in the long-term. We instead adopt a statistical approach and consider various price models. We instead focus on the comparison of uncertainty created by runoff with electricity prices, rather than price forecasting. Based on our underlying aim, our method provides relevant outcomes.
Common Components to All Models
We provide a detailed explanation of the price models. We thoroughly describe this component since it is not described in our previous papers. This component is new.
We consider the logarithm of the prices in order to keep the variance more stable (Tsay 2010; Wooldridge 2009). The model is therefore as follows:
$$ \log(P_{t}) = {X^{s}_{t}} + \sum\limits_{i=1}^{23} \alpha_{i} D_{i,t}^{hours} + \alpha_{24} D_{24,t}^{we} + \epsilon_{t} $$
(1)
where P
t
is the hourly electricity spot price [€ MWh −1], \({X^{s}_{t}}\) is the seasonal factor, \(D_{i,t}^{hours}\) and \(D_{24,t}^{we}\) are dummy variables that control hourly variation and gaps between weekday and weekends. Short term volatility is denoted as 𝜖
t
.
The volatility is approached with an ARMA(1,1)-GARCH(1,1) model (Bollerslev 1986; Box and Pierce 1970). (2) represents the ARMA(1,1) process while (3) to (4) are the GARCH(1,1) processses. The model is commonly used to forecast short-term electricity prices because it is well suited for market with high volatility (Garcia et al. 2005). We choose an order (1,1,1,1), which means that the error and its variability are determined by error at time t−1.
$$\begin{array}{@{}rcl@{}} \epsilon_{t} &=& \beta^{mod_{tot}}_{1} \, + \beta^{mod_{tot}}_{2} \, \epsilon_{t-1} + \beta^{mod_{tot}}_{3} \, \delta_{t-1} + \delta_{t} \end{array} $$
(2)
$$\begin{array}{@{}rcl@{}} \delta_{t} &=& \sigma \, z_{t} \quad \text{where} \quad z_{t} \sim \mathcal{N}(0,1) \end{array} $$
(3)
$$\begin{array}{@{}rcl@{}} {\sigma_{t}^{2}} &=& \beta^{mod_{tot}}_{4} + \beta^{mod_{tot}}_{5} \sigma_{t-1}^{2}+\beta^{mod_{tot}}_{6} \,\delta_{t-1}^{2} \end{array} $$
(4)
where the set of models is m
o
d
t
o
t
∈{MR
y
,MR
s
,MRJD} and δ
t
is the remaining volatility of the ARMA(1,1) model. At time t, it is normally distributed and simulated with a standard normal random variable, z
t
, and a standard deviation, σ
t
. The latter varies and follows a GARCH(1,1) process.
The Three Electricity Price Models
The difference between the electricity price models lies in the way they treat the seasonal factor, \({X^{s}_{t}}\), in (1). By taking into consideration various price models, it can identify the uncertainty of future prices, in particular linked to seasonality. We also assess whether interannual and intra-annual variability is important for future revenues. Below we briefly describe the models and provide some relevant references.
We consider two mean-reversing models, called MR
y
and MR
s
(Uhlenbeck and Ornstein 1930). These processes are commonly used in raw commodity investment analyses because under certain hypotheses prices are attracted by production costs. They are defined as follows:
$$\begin{array}{@{}rcl@{}} {\Delta} X^{mod}_{t} &=& \kappa^{mod}(\mu^{mod}-{X_{t}^{y}}){\Delta} t+\eta^{mod} {\Delta} W_{t} \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} {\Delta} W_{t} &=& z_{t} \sqrt{\Delta t} \end{array} $$
(6)
where the set of models is m
o
d∈{MR
y
,MR
s
}, κ is the reversion speed and μ the long-term mean. η is the Brownian motion term, which explains why ΔW
t
follows the Wiener process represented in (6). As in (3), z
t
is a standard normal random variable.
MR
y
is realitively stable. Its mean yearly price may change, but the intra-annual profile do not change, as in:
$$ {X^{s}_{t}} =\gamma^{\text{MR}_{y}}_{0} + X^{\text{MR}_{y}}_{t}+ \sum\limits_{j=1}^{3} \gamma^{\text{MR}_{y}}_{j} D_{j,t}^{season} $$
(7)
where \(X^{\text {MR}_{y}}_{t}\) is an annual mean [€ MWh −1] that follows a mean reversion process.
MR
s
considers that the mean price of each season, \(X^{\text {MR}_{s}}_{t}\), follows a mean reversion process. In contrast to MR
y
, the season with the mean highest prices may change from the historical pattern. This price model is then formulated as:
$$ {X^{s}_{t}} =\gamma^{\text{MR}_{s}}_{0} + X^{\text{MR}_{s}}_{t} $$
(8)
where \(X^{\text {MR}_{s}}_{t}\) is a seasonal mean [€ MWh −1], i.e. average over three months, that follows a mean reversion process.
The mean-reversing jump diffusion model, called MRJD, allows to consider that price is attracted by production cost, like the two previous ones (Kaminski 1997). However, some shocks may perturb the price. One may observe a jump over a certain period of time, which may be formalized as:
$$\begin{array}{@{}rcl@{}} {X^{s}_{t}} &=&\gamma^{\text{MRJD}}_{0} + \sum\limits_{j=1}^{3} \gamma^{\text{MRJD}}_{j} D_{j,t}^{season} + X^{\text{MRJD}}_{t} \end{array} $$
(9)
$$\begin{array}{@{}rcl@{}} {\Delta} X^{\text{MRJD}}_{t} &=&\underbrace{ \kappa^{\text{MRJD}}(\mu^{\text{MRJD}}-{X_{t}^{y}}){\Delta} t+\eta^{\text{MRJD}} {\Delta} W_{t}}_{\text{mean-reversion process}} + \underbrace{J{\Delta} q_{t}}_{\text{jump}} \quad \text{where} \quad J \sim \mathcal{N}(\nu,\theta) \end{array} $$
(10)
$$\begin{array}{@{}rcl@{}} {\Delta} q_{t} &=& \{ \begin{array}{ll} 1 & \text{with probability} \,\, \lambda {\Delta} t \\ 0 & \text{with probability} \,\, (1-\lambda) {\Delta} t \end{array} \end{array} $$
(11)
where \(X^{\text {MRJD}}_{t}\) is the daily mean price [€ MWh −1] and q
t
is a poisson process that produces infrequent jump of size J.
Hydropower Plant Management
The operator of the hydropower installation aims to maximize profit. Because the variable costs are small for this technology, they can be ignored in this analysis. The objective function is given by Forsund (2007):
$$ \mathit{OF}(b) = g \, \rho \, \eta \, f \, \, {\Delta}_{t} \left( \sum\limits_{t=1}^{t=T}h_{t} \, b_{t} \, P_{t} \right) + R_{T} $$
(12)
where g is the acceleration due to gravity [m s −2], ρ the water density [kg m −3], η the plant efficiency, f the water flow through the turbine [m 3 s −1], h
t
the hydraulic head [m], P
t
the electricity spot price [€ Wh −1]. The objective function is defined on the time horizon T and for a binary variable b
t
which indicates whether one produces or not.
The optimization problem and its constraints are formalized as:
$$\begin{array}{@{}rcl@{}} &&\max\limits_{b} \; \mathit{OF}(b) \end{array} $$
(13)
$$\begin{array}{@{}rcl@{}} &&V_{t} = V_{t-1} + I_{t} \, {\Delta}_{t} - f \, b_{t} \, {\Delta}_{t} \end{array} $$
(14)
$$\begin{array}{@{}rcl@{}} &&h_{t} = {\Phi}(V_{t}) \end{array} $$
(15)
$$\begin{array}{@{}rcl@{}} &&V_{min} < V_{t} < V_{max} \end{array} $$
(16)
$$\begin{array}{@{}rcl@{}} &&b_{t} \in \{0,1\} \end{array} $$
(17)
where V
t
is the reservoir content at time t [m 3], I
t
the water intake [m 3 s −1], h
t
is a function of V
t
, and V
min, V
max are the capacity limits of the reservoir.
We optimize over a time horizon of two years and validate the first one. Then, we move the windows for the following optimization one year later. To tackle the residual water volume, R
T
, we fix the final volume after two years at the initial level.
The objective function is maximized with a local search method, called Threshold Accepting (Dueck and Scheuer 1990; Moscato and Fontanari 1990). The algorithm starts with a random turbine schedule, it evaluates the corresponding objective function value. Then it considers a solution in the neighborhood by introducing a small random perturbation to the current schedule and evaluates the new corresponding objective function value. The new solution is accepted if it is no worse than a given threshold. This means that the algorithm also allows down steps in order to escape local minima. Of course, the threshold decreases gradually to zero which means that in the final steps only solutions which improve the objective functions are accepted. A data driven procedure determines the threshold sequence as described by Winker and Fang (1997) and adapted by Gilli et al. (2006). This algorithm as a whole has proven its effectiveness to find an acceptable optimum. We do however acknowledge that other algorithms also exist as reviewed by Ahmad et al. (2014). For instance, some papers consider a genetic algorithm (Hincal et al. 2011; Cheng et al. 2008) or a simulated annealing one (Teegavarapu and Simonovic 2002). Some papers also investigate multi-objective optimization rather than one objective function (Liao et al. 2014; Kougias et al. 2012).
We assume a clear-sighted manager, i.e. he knows in advance the price and runoff. Even if this hypothesis is optimistic (Tanaka et al. 2006), it is commonly used in papers for computational efficiency (Francois et al. 2015; Maran et al. 2014; Hendrickx and Sauquet 2013; Schaefli et al. 2007). Since we look at trends and do not aim to forecasting exact future revenue, this hypothesis does not affect our conclusions.
We validated our model by using historical data. Once the optimization performed with past runoff and electricity prices, we have compared the hourly, weekly and seasonal production with the actual production. We also verified that the lake level followed the same pattern in our simulation as in the real life. Gaudard et al. (2013a) provides more details about the management model.
Field Site
The case study is based on the hydropower plant of Mauvoisin, situated in the Swiss Alps (7o35′E, 46∘00′N). A reservoir gathers the runoff coming from nine glaciers (Petit Combin, Corbassière, Tsessette, Mont Durand, Fenêtre, Crête Sèche, Otemma, Brenay, Giètro), which covered in 2009 40 % of the catchment area (Gabbi et al. 2012).
The reservoir volume is 192 × 106 m 3, which represents 624 GWh. The mean multi-year production is 1040 GWh (years 1999–2009), 53 % in winter and 47 % in summer (FMM 2009). The reservoirs in the Alps transfer energy from summer, when snow and glaciers melt, to winter, when electricity demand is high.
Several studies were carried out on the impact of climate change in the Bagne Valley where the Mauvoisin power plant is situated (Gaudard 2015; Gaudard et al. 2013a; Gabbi et al. 2012; Terrier et al. 2011; Schaefli et al. 2007). So far, the operators have taken advantage of the glacier melting, which increases the annual volume of water. According to Gabbi et al. (2012), this trend will persist for the next 20 years. The annual runoff will subsequently decrease in the second half of the century. By 2100, glaciers are expected to disappear in this catchment area. Thanks to the reservoir, the power plant operator may, however, mitigate the effects resulting from the runoff seasonality (Gaudard et al. 2013a). A new pumped-storage may even be built from 2040 on the surface released by the glacier and therefore increases energy production on the site (Gaudard 2015; Terrier et al. 2011)