The Impact of the Energiewende on the German Day-Ahead Market: a Time Series Analysis

The Impact of the Energiewende on the German Day-Ahead Market: a Time

Series Analysis

Martin Bro Larsen

Master of Economic Theory and Econometrics Department of Economics

University of Oslo

Nov 10 2017

The Impact of the Energiewende on the German Day-Ahead Market: a Time Series Analysis

Martin Bro Larsen

c Martin Bro Larsen, Nov 10 2017

The Impact of the Energiewende on the German Day-Ahead Market: a Time Series Analysis http://www.duo.uio.no/

Publisher: Reprosentralen, University of Oslo

Preface

Acknowledgements

I would first like to thank my supervisor, professor Finn Førsund. My questions were always quickly answered, and he steered me in the right direction, whenever he though I needed it. I would also like to thank my old colleagues at Nord Pool AS, my fellow students, friends and family who were all patient and supportive.

I am forever grateful for their valuable comments on this thesis. Last but not least, would I like to thank the study group- Bærtur. All remaining errors are my own.

Martin Bro Larsen Nov 10 2017

Abstract

It is well established within economic literature that the lower marginal cost of wind -and solar power production, leads to reduced wholesale prices.

These types have very limited storage capacity, causing increased volatility.

The European price coupling aims to balance the wholesale prices, to increase welfare. In this thesis, I fit two suitable time series models for daily German data (2015-2016). I find evidence of strong seasonality and volatility clustering in the wholesale price, which is accounted for. Furthermore, each model shows sufficient forecasting abilities, depending on range. This proves that price predictions are still possible, when the share of stochastic power generation increases. Finally, I confirm that both wind and solar reduce the price, while only wind show a significant increasing impact on price volatility.

Contents

1 Summary 1

2 Literature review 4

3 Market Design 5

3.1 The European Market Coupling . . . 5

3.2 The German Power Market . . . 8

4 Theoretic Background 12 4.1 Time Series Analysis . . . 12

4.2 A Stationary Process . . . 12

4.3 ARMA/ARIMA . . . 13

4.4 ARCH/GARCH . . . 16

5 Data 17 5.1 Sources and evaluation . . . 17

5.2 Visual Inspection . . . 19

6 Methodology 21 6.1 Stationarity . . . 21

6.2 Estimation - The ARMAX model . . . 22

6.2.1 Identification . . . 22

6.2.2 Estimation . . . 24

6.2.3 Diagnosis . . . 25

6.2.4 Fitted seasonal ARMAX model . . . 26

6.3 Estimation - ARMAX GARCH model . . . 26

7 Results 29 7.1 Parameters of Both Models . . . 29

7.2 Forecasting with Both Models . . . 32

8 Volatility Ratio Test 34

9 Discussion and Conclusion 37

10 References 39

11 Appendix 40

List of Tables

1 Summary of endogenous and exogenous variables . . . 20

2 Augmented Dickey-Fuller test for stationarity of Phelix . . . 21

3 Akaike -and Bayesian information criteria - SARMAX . . . 24

4 Estimated SARMAX parameters . . . 30

5 Estimated ARMAX-GARCH parameters . . . 31

6 Summary of actual -and predicted Phelix series . . . 34

7 Variance Test: Prices with High vs. Low Levels of Sun . . . 36

8 Variance Test: Prices with High vs. Low Levels of Wind . . . 36

List of Figures

2 Towards a Single European Market . . . 8

3 Aggregated solar -and wind generation in Germany . . . 19

4 Phelix - Illustrating the German spot price . . . 20

5 The auto -and partial autocorrelation of Phelix . . . 23

6 Bartlett’s test - Periodogram . . . 25

7 SARMAX prediction (dotted line) . . . 32

8 ARMAX-GARCH prediction (dotted line) . . . 33

9 Comparison of in-sample forecasts . . . 34

10 Scatter plot of Generations Volumes to Price . . . 35

11 Correlogram for ARMAX residuals . . . 40

1 Summary

Investments in renewable energy have increased substantially in Germany as well as other countries in recent years. Angela Merkel’s Energiewende started a phase-out of nuclear power, with a goal to rely heavily on renewable energy (particular wind, photovoltaics and hydroelectricity). The transition plan was met with scepticism, when announced.¹ Questions arose as to whether the new renewable energy could grow quickly enough to meet the requirements of German industry. Yet a few years later, the new renewable parks are generating so much that in some cases the state has to pay the generating companies to switch of their production to stop congesting the grid.

Renewable power from solar and wind can be hard to predict and has a very limited storage opportunity. On sunny and windy days are Germany’s grid operators struggling to keep the balance between supply and demand. Negative prices mean that producers must either shut down stations or pay the customers to take the electricity of the grid.² Reduced prices have been benefitial for the consumers, but problematic for the grid operators and the producers. Higher share of fluctuating generation capacities from wind and solar power, leads to more volatile prices. In light of this, I have created a time series analysis in Stata, from the past two years daily data. My models show adequate forecasting capabilities, even with a high share of renewable generation. Furthermore, significant evidence that a higher share reduces the price, while only wind significantly increases the volatility.

In this thesis I start by reviewing various papers discussing the effect of increasing renewable power dependency on prices, and other dynamics of spot-prices. The following section gives an introduction to the European power market, with a focus on the wholesale market. The price coupling initiative after the deregulation, is expected to increase liquidity, efficiency and social welfare by a harmonized

1https://www.theguardian.com/environment/2016/oct/11/germany-takes-steps-to- roll-back-renewable-energy-revolution

2https://www.bloomberg.com/news/articles/2017-10-26/record-wind-will-force- germany-to-pay-power-users-this-weekend

price level between countries. My data for the power exchange EPEX, covers the German wholesale market. This daily data is from the day-ahead auction connected to the European market coupling. Although the price data is from the German power exchange alone, it is said to be a good representation of the overall German power market. It is expected that wholesale market price would be more unstable without the coupling. A part of this thesis is to see how unstable it still is, due to renewable energy. Section 3 also presents the national market structure, by its recent developments and investments. Furthermore, I introduce the main power producers in Germany, their relation to the grid operators and the operators’ role.

Section 4, brings forth some crucial theoretical background for time series analysis.

As of most time series models, a stationary process is necessary to achieve significant estimates. This is to estimate how earlier periods affect the present, without a trend. Seasonality is another factor which must be accounted for, when fitting a model, for the same reason as stationarity. I find that the German wholesale price is stationary, but weekly seasonal (frequency=7). The rest of Section 4 introduces the theory behind the Autoregressive Moving Average (ARMA) model, and Autoregressive Conditional Heteroskedasticity (ARCH) model, with related extensions.

The data is presented with a critical view in Section 5, followed by a visual inspection. Note that all the data was manually formatted in Excel before imported to Stata for estimation. I find a summer-winter cycle in solar data, a more stochastic series in wind, and a stationary price. The price show evidence of volatility clustering and multiple days with negative prices. For the first model, I follow the Box-Jenkins method to fit an ARMA model with wind and solar as exogenous variables. Such an extension is called an ARMAX model, and is more carefully explained in Section 6. The model is further fitted to a SARMAX (seasonal) model, to counter inn the weekly seasonality in the dependent variable.

Both an ARCH and a Generalized ARCH (GARCH) model is later constructed, for a conditional variance. This is due to the evidence of volatility clustering.

My thesis finds statistical evidence of reduced prices and affected volatility, by increasing renewable generation. The last two sections show that the price is to

some extend possible to predict, even with more renewable energy. While the SARMAX model forecast is most reliable for the first couple of months, the model countering in conditional variance (with GARCH), is preferred for a longer range.

Finally, my model fails to find a significant influence in increased volatility from more solar power. A shock from wind power on the other hand, has a significant increasing impact on volatility. An argued reason is the peak-demand correlated solar generation, which is explained in the concluding remarks of Section 7.

2 Literature review

The deregulation of power markets worldwide during the last four decades has attracted a lot of attention in the field of Economics. New technological development of power generation, as well as increasingly integrated power markets are just a few of major changes still in play. Woo et al.(2011) emphasized that an extensive research literature on spot electricity price behaviour and dynamics already exists, but only with limited evidence based on actual market data. They suggest (based on empirical data) that an increase in intermittent wind generation offers two economic benefits; in a competitive market will wind power producers, win market share and push the prices down with their lower marginal cost. However, a higher dependency may come with an increased volatility. Gianfreda (2010) reflected on the driving forces behind the volatility in the electricity prices, and it’s linkage to risk derivatives. She finds evidence of an ARCH effect in multiple European price areas and conduct various ARCH/GARCH alternatives, for her diction models.

Bunn and Gianfreda (2009) found that a crucial stylised fact for electricity is, once it is produced it cannot be stored in any significant amount, so forward prices are generally seen as expectations of spot, adjusted perhaps with a risk premium. They provided evidence of a greater market integration following higher interconnection capacities. Escribano et al. (2011) analysed the spot price characteristics for various price areas on multiple continents in the 1990s and early 2000, after the deregulation of worldwide power markets. Again, finding strong characteristics of both seasonality and time-varying volatility. Ragwitz et al. (2008), studied the feed-in tariff and the merit-order for renewable energy producers in Germany, and estimated how it has led to high growth rate of supported technologies and investments. The merit-order refers to the policy of prioritising the power producers of lowest marginal costs to the grid. Data prior to 2006 showed that the merit-order effect has reduced prices, especially peak-hour prices with the help of increased solar power feed during mid-day. Other effects were increased international trade of power, as well as increased grid costs for new transmissions.

3 Market Design

3.1 The European Market Coupling

The deregulation of the European power market started after observing success in the pioneer countries as Chile, Argentina, Norway and the UK in the 1980s and the 1990s. Deregulation, removed the government’s control over the prices, by introducing market players in the sector to compete. The liberalization, privatization and restructuring of the energy supply and and its distribution, are the main drivers behind the reform, and were motivated by cost efficiency through market competition among participants.

The power price is determined by the balance between supply and demand, whereas factors such as the weather and limitations to power plant’s production capacity may impact the power prices. European power exchanges such as Nord Pool and EPEX provides a trading area for the physical electricity market in their price regions.³

A hydro-based generation owner offers his production to the power exchange EPEX. EPEX can then connect an electricity suppliers who need to cover a certain demand in the retail market. The generator may offer a higher amount of power on a rainy day. A hydro plant, with a dam or reservoir, has the advantage of a flexible production. This is due to its stored capacity. A similar solution is still very limited for solar -or wind power producer. Both supply and demand often varies over time and areas, and can be hard to predict. The balancing market ensures that demand equals supply at all time. See the market structure illustration in Figure 1. This figure and history from the development of the European electricity market, is gathered from Meesus et al. (2005). The power exchange collects all willingness-to-buy bids and willingness-to-sell offers for delivery the following day, and runs a calculation for a social optimal allocation. This process is a part of the European balancing and wholesale market, but not all generators and suppliers are trading in the this market. Other trades in the wholesale market are based on

3http://www.epexspot.com/en/market-coupling/pcr

bilateral forward contracts or over-the-counter agreements. An increasing share in balancing market participants has been observed during the last decade, but still minor compared to the wholesale market volume. I will focus on the numbers from EPEX in the wholesale market, in this thesis.

Figure 1: The Wholesale Power Market Structure

The transmission system operators (TSO) are responsible for electrical stability and security of supply in their areas. These operators are non-commercial organizations, neutral and independent with regards to market members. I will be referring to the cable capacities of the TSOs, as well as the limited production capacities of the power generators, when mentioning capacity later in this thesis.

The balancing market can often be divided into the day-ahead market and the intraday market, where the day-ahead market is the main area for power trading today. In the day-ahead market, the equilibriums by participants are set per

hour for the following day. The Price Coupling of Regions (PCR) initiative started in 2009, as a step towards full market integration in Europe, and has since then expanded. The PCR is a cooperative method between European power exchanges, to share their order books and the TSO capacities of the included areas, before running a common algorithm for coordinated results (”Euphemia” - name of algorithm in use for the daily PCR). The Market Coupling uses so-called implicit auctions in which players do not actually receive allocations of cross-border capacity themselves, but just bid for energy on their Exchange. The Exchanges then use the available cross-border transmission capacity to minimize the price difference between two or more areas. In doing so, market coupling maximizes the social welfare, avoids any artificial splitting of the markets, and sends the most relevant price signal for investment in cross-border transmission capacities.

The efficiency of the mechanism is furthermore revealed by an increasing price convergence between market areas. Market coupling mechanisms are based on the reference prices emerging from liquid markets such as the one managed by EPEX SPOT.⁴

Figure 2 is from one of Nord Pool’s presentations, and shows the PCR process of Europe so far. Some other factors affecting the spot prices are weather forecasts, price speculation with new trading products and ”Bottlenecks”. ”Bottleneck”- A bottleneck arises when the transmission grid is not capable of transmitting sufficient power, i.e. when the desired consumption in an area exceeds possible generation and import capacity, and correspondingly when the desired generation in an area exceeds consumption and export capacity. A bottleneck occurs as a consequence of too little available generation capacity in conjunction with limited possibilities for import, or as a consequence of a generation surplus in conjunction with limited export possibilities.⁵ Thus, when a price difference may occur, say between the areas of SE4 and DE (”The Baltic Cable” between Sweden and Germany) will an import to the area with higher price happen, depending on available capacity on the cable. However, the TSO only accepts a flow as long as the bottleneck income is larger than the losses on the cable. This can naturally

4http://www.epexspot.com/en/market-coupling

5http://2014.statnett.no/en/power-terminology

Figure 2: Towards a Single European Market

occur within a country too, for countries with multiple price areas. The reason for multiple bidding areas within a country, is related to the national power distribution and the flow capacity. Say, in the case of an expected energy shortage in a geographically restricted area, in order to handle large and prolonged bottlenecks in the regional and central grid grid.⁶

3.2 The German Power Market

Germany’s electricity market is leading in terms of production and consumption, compared to the rest of the continent. Heavily invested in renewable energy over the last several years, as well as convergence toward a heavily integrated market through the European market coupling. Much of the same goes for France, another part of the power exchange EPEX. France has a large share of nuclear power

6http://www.statnett.no/en/Market-and-operations/the-power-market/Elspot- areas--historical/Elspot-areas/

generation, while Germany relies heavier on renewable energy from wind and solar. Especially since the introduction of Energiewende, a transition targeting greenhouse gas (GHG) and promoting lower carbon emissions through renewable power production. This has built up a divergent price relationship between the two areas, as shown by Keppler et al. (2016). Germany’s renewable power is reliable on metrological terms at a minimal marginal cost (solely operational and maintenance). France’s nuclear power supply is rather close to constant, but at a higher cost. To avoid high volatility in German power prices, they rely on wholesale power trade and the balancing market. This trade is of course limited to the connecting power cable capacities. The power from PV-generation naturally clusters during daytime, when conveniently the demand for power is higher. The daily peak demand is usually in the morning or afternoon. Keppler et al. (2016) showed that the divergence between French and German prices, as a result of increasing renewable production in Germany, was lower with the market coupling then if it would have been without one. The incentive for market coupling is price balance over price areas, to maximize welfare. Higher divergence results in reduced welfare in the wholesale market. Thus, trade to balance the divergence, is reliable on inter-connective power lines. The ELIX index, EPEX’s calculated price for a market with no capacity constraints shows the possible welfare equilibrium. The same study estimates the welfare losses of an average price increase in the German market with a real price at ELIX (”infinite” capacity) to be much lower than the increased welfare by reduced French prices. (F +2.29B EUR vs. G -265M EUR).

The cost of new cables for higher capacities must be included in a cost benefit analysis, where the optimal allocation would be at an equilibrium of marginal costs equal the marginal benefits of the market (in this case, for both France and Germany together). The same study showed an overall increased welfare constrained to an increased capacity cost.

The German energy market lies geographically in the heart of Europe, with the largest annual electricity demand, generation capacity and with ten interconnected

network cables to neighbouring countries.⁷ A power decision made in Germany naturally affects the European connected power market. A change in demand, generation and capacities for the German power market will likely affect the other participants in the market, and vice versa. The questions are how the market will be affected, and by how much? The same report shows that Germany has Europe’s largest power system, with Europe’s leading capacity from renewable power production, and 3rd largest renewable capacity (excluding hydro power) in the world, with the four major electricity utility companies E.ON, Vattenfall, RWE and EnBW.

Energiewende was initiated in 2010, with the goal to decarbonize the economy, transition to renewable energy and phase out nuclear power by 2022. This would be done in addition to seek improved energy efficiency in all types of generation, increase the share of renewable energy production, phase out carbon-intensive production like coal, and strengthen cross-border interconnectivity and capacity.

The capacity of today’s batteries, makes a stable feed from wind and solar alone difficult. The practically zero marginal costs from wind and solar production, force others to shut down, if not subsidized. Thus, relying even more on interconnected capacities and market structure. The renewable energy based electricity generation was almost at 30% by the beginning of 2016,⁸ making it the most important energy sources for the third year in a row, mainly from solar, biomass and wind power. The major upscaling in renewable energy production in Germany during the last few years has brought them into a top 5 in terms of capacity, generation and investment in renewables. In 2015 alone, Germany installed 26,772 wind turbines on -and offshore for the value of 9.7 billion Euros. This made them Europe’s leader on wind capacity and new installations, with 10% of all installed wind turbine capacity. The same report shows that the last month of December 2015 set a new national record of wind generation of 85.4 billion kWh. Photovotalic systems on the other hand, range around 1.5 million instalments, with a capacity of above 38GW. The peak of

7https://www.agora-energiewende.de/fileadmin/downloads/publikationen/

CountryProfiles/Agora\_CP\_Germany\_web.pdf

8https://1-stromvergleich.com/strom-report/renewable-energy-germany/

investments in PV systems was in 2010, with 18.4 billion Euros, in everything from rooftop panels to solar parks. The total generation has been increasing gradually ever since the Renewable Energy Source Act (EEG), which guaranteed a fixed feed-in payment (20- year term feed-in tariff) to increase incentives for sustainable generation instalments. The German government later reduced the feed-in tariff (FIT) probably explaining the reduced annual investments since 2010. Although Germany witnessed reduced investments and build-outs of solar power production, was 2015 the all-time-high year in solar power generation with 36.8 billion kWh, thanks to a sunny spring and summer. Numbers from 2015 showed that Germany was ranked second per capita installed solar capacity, in the world. Since the introduction of the EEG in 2000, the share of green power consumption in Germany has more than quadrupled (from 6% to 27%). This increase is expected to reach the goal of 40% by the year 2025.

The demand for electricity, from both private households and industry, has seen a reduction for the past few years. Non-renewable based production from coal and nuclear has had a decline, mainly due to the Energiewende, but the falling natural gas price gave a boost in its power production. A survey done in 2015 by PWC showed that 92% of the German consumers support the energy transition, Energiewende. The nuclear power phase-out was shown to be the main motivator in the survey report, followed by the issue of scarcity of fossil fuel resources and the reduction of carbon dioxide (CO2) emissions. The German private consumer has become more aware of the importance of renewable energy, highlighted by 16-28%

of households choosing green energy suppliers.

4 Theoretic Background

4.1 Time Series Analysis

The majority of the theoretical background builds on the course ”Time Series Analysis” by Prof. Ju at Fudan University (2016), and the book ”Introduction to Time Series Using Stata”, by Becketti (2013).

Whether a time series is stationary or not, is perhaps its most important property.

A stationary process (or time series) has the same unconditional distribution (same mean and autocovariance) at any point in time. In other words, the dependent variable of a stationary process ”y_t”, will never depend on time ”t”. This property is required to estimate dependency of variables over time, explained more formally in this section.

4.2 A Stationary Process

Consider the process (y_t)^+∞_−∞ where the the autocovariance between y_t and y_t−j is defined by

γ_jt ={E[y_t−E(y_t)][yt−j−E(yt−j)]} (1) If neither E(y_t) nor γ_jt depends on t, then the process satisfies the conditions to be covariance-stationaryor weakly stationary. Thus,

E(y_t) = µ,∀t (2)

and

E[(y_t−µ)(yt−j−µ)] =γ_j,∀t∀j (3)

Weak stationarity implies that the value of the data in y_t fluctuate with constant variation around a fixed level. It is common in Finance literature to assume that an asset’s return series is weak-sense stationary. This assumption can be checked empirically, provided that a sufficient number of observations in the historical data is available. I will later analyse the stationary properties of electricity prices in the Energy market, for comparison. The covariance-stationary process is, for the j−th autocorrelation, defined as

ρ_j = γj

γ₀ (4)

4.3 ARMA/ARIMA

The Autoregressive Moving Average (ARMA) model is one of the most commonly applied in time series analysis. Xie (1993) explains how any stationary process can be approximated by stationary ARMA models, thanks to ”Wold’s Decomposition Theorem”. The ARMA model includes time correlations of the studied phenomenon in the dependent variable and random nature over time. The current value of the process with the ARMA(p,q) model is dependent on its own past values ”φ”

through the autoregressive part, and on past noise values ”θ” through the moving average part, where “p” and “q” are the number of lags of each, respectively. The model can be written:

y_t =φ₁yt−1+φ₂yt−2+...+φ_pyt−p +u_t+θ₁ut−1+...+θ_qut−q (5) Where the error term is assumed to have the properties {u_t} ∼WN(0,σ²_u).

”WN” denotes ”White Noise”, with the properties of mean zeroE(u_t) = 0, variance σ² and zero autocorrelationE(u_tu_v) = 0 when t6= 0 and i.i.d.

If all the θ’s are equal to zero, the ARMA(p,q) model reduces to the an AR(p) model:

y_t=φ₁yt−1+φ₂yt−2+...+φ_pyt−p+u_t (6)

If all theφ’s are equal to zero, the ARMA(p,q) model reduces to a MA(q) model:

y_t =u_t+θ₁ut−1+...+θ_qut−q (7) To be stationary, the ARMA(p,q) process or the AR(p) should satisfy the condition of all roots to lie outside the unit circle of the following equation:

1−φ₁ζ−φ₂ζ² −...−φ_pζ^p = 0 (8) The backward shift (lag) operator may be defined as L by L(y_t) = yt−1. Thus, the p-th order lag operator by L^p(y_t) = yt−p. A lag is referred to as the order of previous time periods. The ARMA(p,q) can be rewritten as:

(1−φ₁L−φ₂L²−...−φ_pL^p)y_t= (1 +θ₁L+θ₂L²+...+θ_qL^q)u_t (9) Lets defineφ(L)≡1−φ₁L−φ₂L²−...−φ_pL^p andθ(L)≡1+θ₁L+θ₂L²+...+θ_qL^q, allowing us to simplify the representation of the ARMA(p,q) model to:

φ(L)y_t =θ(L)u_t (10)

In general can the process be rewritten:

y_t= θ(L)

φ(L)u_t= θ(L)

(1−λ₁L)(1−λ₂L)...(1−λ_pL)u_t =

∞

X

i=0

ψ_iut−i (11) Where all the coefficientsλ’s lie outside of the unit circle. The mean and variance of the process can easily be obtained by using the independence of{u_t}asE(y_t) = 0 andγ₀ =var(y_t) =σ_u^2P∞_i=0ψ²_i whereψ₀ = 1. Becausevar(y_t)<∞, must{ψ_i²}be a convergent sequence. In another way must ψ_i² →0 as i→ ∞. Thus, the impact of the remote shock u_t−1 on the stationary process of y_t will vanish as i increases.

The lag-l autocovariance of the process yt is computed as

γl =cov(y_tyt−l) =E[(

∞

X

i=0

ψiut−i)(

∞

X

j=0

ψjut−j−l)]

=E(^X

i

X

j

ψiψjut−iut−j−l)

=σ_u²

∞

X

j=0

ψ_jψ_j+l

(12)

Consequently, the lag-l autocorrelation of the process is:

ρ_l=

P∞

i=0ψ_iψ_i+l

P∞

i=0ψ_i² (13)

A weakly stationary/covariance-stationary times series will have the properties ψ_i → 0 as i → ∞, hence the autocorrelation ρ_l will converge to zero as lag l increases.

An extension of the ARMA(p,q) model, is the ARIMA(p,d,q) model, where the

”I” stands for ”Integrated”. The ”d” is not the number of lag operators, but rather the degree of differencing applied to reach a stationary process. A non-stationary time series can be stabilized by removing its trend or seasonality by differencing, thus making it stationary. Multiple orders of differencing may be needed to reach stationarity, referring to the chosen degree of ”d”. Many time series are non-stationary. Suppose that was the case for the initial process y_t in equation (5). An ARIMA(p,d,q) could make the process stationary by differencing ”d”

times. The extension can be written:

φ(L)∆^dy_t =θ(L)u_t (14)

Where ∆^dyt =yt−yt−d is the process made stationary by differencing, with the ARIMA(p,d,q) model.

Seasonality may as well occur in a time series. A further extension of the ARIMA(p,d,q)

has been made to deal with this: The Seasonal-ARIMA (SARIMA) model,

y_t∼ARIM A(p, d, q)(P, D, Q)_Swhere the residuals{u_t}of the initial ARIMA(p,d,q) model are ARIMA(P,D,Q) at a seasonal lag ”S”. Hence, for quarterly seasonality in monthly data;S = 4. In case of weekly seasonality in daily data; S = 7, and so on and so forth.

The SARIMA model may be rewritten in the following way

Φ(L^S)φ(L)∆^D_S∆^dyt= Θ(L^S)θ(L)u_t (15) Whereu_tis white noise, ”S” is the seasonal frequency, and Φ and Θ are the seasonal related autoregression and moving average parameters.

4.4 ARCH/GARCH

Other than conditional mean, conditional variance is an important factor of interest in time series estimation, when the variance estimate is no longer constant. The autoregressive conditional heteroskedasticity (ARCH) model was first introduced by Engle (1982), considering ”volatility clustering”. A high variance is often followed by high variance, and low variance often followed by low. In other words, the persistence in volatility is captured by modelling the conditional variance as a stochastic process with a serial process.

The lagged autoregressive terms of previous error terms in an ARCH(q) model, where ”q” is the lag order, can be written in the following way:

u_t=σ_tt, _t∼W N ARCH(q) :σ_t² =π₀ +

q

X

i=1

π_iu²_t−i, π₀ >0, π_i ≥0(i= 1, ..., q),

q

X

i=1

π_i <1 (16)

Bolleslev (1986) extended the ARCH model in 1986 to the Generalized ARCH model (GARCH model), applied for cases requiring a larger number of included lags.

The GARCH(p,q) model can then be written as:

u_t=σ_tt, _t∼W N GARCH(p, q) :σ²_t =π₀+

p

X

i=1

τ_iσ²_t−i+

q

X

i=1

π_iu²_t−1,

π₀ >0, τ_i ≥0(i= 1, ..., p), π_i ≥0(i= 1, ..., q),

p

X

i=1

τ_i+

q

X

i=1

π_i <1

(17)

5 Data

5.1 Sources and evaluation

The German high voltage market is mainly divided by four large market players.

EnBW, E.ON, RWE and Vattenfall are the leading distributors, generators and suppliers based on volume and market share. It is interesting to note, in the report by Bayer (2015), that these four agents are also the owners of the four TSOs in Germany. 50Hertz Transmission GmBH (owned by Elia, formerly owned by Vattenfall), Amprion GmBH (owned by RWE), Tennet TSO GmBH (owned by TenneT, formerly owned by E.ON) and TransnetBW (owned by EnBW). These four mainly collected and offered market data separately, prior to the EU initiative ENTSO-E. ENTSO-E works to merge the TSO network of Europe, and offers access to collective historical data from 2009.⁹ Another relatively new open source platform called Open Power System Data (OPSD),¹⁰supported by several German private and public organizations, offers national power generation data as well. I collected the nationwide power generation data for solar, onshore wind and offshore wind in 15 minutes intervals, for the period 1 January 2015 to 31 December 2016.

9https://www.entsoe.eu/data/data-portal/Pages/default.aspx

10https://open-power-system-data.org/

I compared and verified the data from the two sources, before adjusting the data in Excel. I aggregated the 15 minute data, to hourly data, and from hourly to daily data. Note, the data set was complete (no missing values). The onshore and offshore wind generation values were added up to a total wind generation variable. I chose this certain time interval for a couple of reasons. Firstly, I am interested to see the effect on recent periods, thus data from the last two full years is a close fit. Secondly, alternative data sets had missing values, in manner of full year intervals. Lastly, I found no similar studies examining the same market for the same time interval elsewhere. There are of course many other possible explanatory variables which can be included to estimate the effect on the German spot price, like generation capacities, temperatures, air density, economical activity (like GDP), and so on. I have chosen to look away from further expansion, due to simplicity of the model, and due to the fact that my main goal is to examine an effect of increased reliability of renewable energy on spot prices, as a result of multiple public policies.

The wholesale spot price for the German electricity market may be represented through the day-ahead market index, ”Phelix”. Phelix is the Austrian/German physical electricity index, a registered trademark of the European Energy Exchange AG (EEX), under EPEX SPOT.¹¹ EnerginetDK offers hourly data on several spot prices in Europe, among others, the Phelix day-ahead spot price per hour.

The market data is free-of-charge and updated twice a week.¹² I downloaded the Phelix hourly spot for the two year period, calculated the day base in Excel, and compared it to EPEX own website publications. The day base of a day-ahead index is a widely used term for the arithmetic mean of the market clearing prices for the delivery periods (starting) between 0h00 CET (including) and 24h00 CET (excluding).

11https://www.epexspot.com/document/36857/EPEX\%20SPOT\%20Indices

12https://en.energinet.dk/Electricity/Energy-data

5.2 Visual Inspection

Figure 3: Aggregated solar -and wind generation in Germany

As shown in Figure 3, the data for solar generation is intuitively much higher in the summer than during the winter, due to the extended length of daytime periods. Therefore, shifting weather during the summer, has a stronger impact on volatility. The wind generation, on the other hand, has a more unpredictable pattern, as shown in Figure 3. Naturally due to its independence of daytime.

As mentioned, data is collected in megawatts, but I have converted it to terawatt (1M W = 1.0∗10⁻⁶T W) for clearer illustrations. I hypothesize the rising wind -and solar generation to reduce the market price, which translates to the hypothesis of negative coefficients later on.

Figure 4 depicts the evolvement of the Phelix process (y_t). The series shows no clear sign of a trend, and a visual inspection may suggest stationarity, although unstable volatility over time. Moreover, the figures do not suggest any data issues.

Fore example, ’jumps’ or ’breaks’ - which might suggest anomalies in data that do not arise from major disturbances in the power market - are absent, among others.

Whether the index is indeed stationary, is tested in the next section. The variables

Figure 4: Phelix - Illustrating the German spot price

are summarized in Table 1, where ”ger” is the day base Phelix measured in Euros, and ”solar” and ”wind” are the accumulated daily power generation measured in Mega Watts (MW). Note that the mean of wind generation is more than double of solar, with a threefold maximum level. The average price is about 30 Euros, with a minimum value of -12.89. Figure 4 confirms that the period had more than one day with negative prices, indicating an excess supply from solar and wind power.

I assume both energy sources to be independent on each other in this thesis, as both stochastic. Thus, no interaction term of the two sources will be included in the model specification of next section.

Table 1: Summary of endogenous and exogenous variables Variable Obs Mean Std. Dev. Min Max

ger 731 30.30179 9.392159 -12.89 60.05833 solar 731 375979.7 241524.9 0 918848 wind 731 830221.5 615559.5 0 2975163

6 Methodology

6.1 Stationarity

An Augmented Dickey-Fuller test for stationarity was executed on the Phelix process, to check if the variables follow a unit-root process, as mentioned in the theoretical section. The null hypothesis is that the variables lie within the unit root, and the alternative is that they lie without the unit root and is thus generated by a stationary process. I ran the test all the way up to 15 lags (more than two weeks), and was still able to reject the null hypothesis at a 1% level, as shown in Table 2

Table 2: Augmented Dickey-Fuller test for stationarity of Phelix

———- Interpolated Dickey-Fuller ———

Test Stat 1% Crit. Value 5% Crit. Value 10% Crit. Value

Z(c) -3.761 -3.432 -2.860 -2.570

Stationarity is a necessity in estimation of ARMA models and most time series, but first-differencing won’t be necessary as the process is already shown to be stationary. I recognize that there have been disagreements in earlier studies to whether or not to take the logarithm of the electricity price. I have chosen not to do so. First of all, negative prices in the dataset are then excluded, which removes empirical evidence already observed in Table 1. Secondly, I want to include all evidence of grid congestion from excess supply. In financial time series, logarithm is used as a tool to easier obtain stationarity and to even-out heavy spikes in intervals of high volatility. An exclusion of these negative values and heavy spikes would have made my predictions less realistic.

6.2 Estimation - The ARMAX model

An ARMAX model¹³is an extension of the ARMA model. I try to fit an ARMAX(p,q) model for the Phelix, including solar -and wind generation, based on the practical step-by-step approach, suggested by Box and Jenkins (1970), widely known as the Box-Jenkins method.

• Identification stage: Visual inspection of the time series, Autocorrelation Functions, and Partial Autocorrelation Functions to determine lag orders

• Estimation stage: Once the model has been identified, the values for mean, variance and ARMA parameters must be estimated

• Diagnostic stage: Statistical test are then applied for our identified and estimated model to determine the model’s adequacy. The cycle either starts over, or ends when we have a model- when there are no further diagnosed problems

The Box-Jenkins method has since then been widely applied in modelling real-world time series. As Becketti (2013) explains, the method is neither foolproof (hence, the iterative cycle of identification, estimation and diagnostic checking) nor definitive (you still have to exercise judgement), but it provides structured guidance for a challenging problem.

6.2.1 Identification

Visual inspection of the Phelix process and the Augmented Dickey-Fuller test has so far confirmed it to be a stationary time series, but there is an alternative way to confirm it. The Autocorrelation Function (ACF) is based on the autocorrelation of equation (13). The function is a measure of the correlation between a time

13Autoregressive Moving Average with exogenous inputs model (ARMAX), where the ”X”

represents the exogenous variables, wind and solar.

series at time ”t”, and all its lagged values (t-1, t-2, ..). The ACF presents not only the direct effect of the previous period, but also all the ”indirect” effects from all previous ones. The Partial Autocorrelation Function (PACF) eliminates all the

”indirect” effects, in turn only presenting the direct one. The ACF and PACF of the Phelix variable ”ger”, is put forward in Figure 5

Figure 5: The auto -and partial autocorrelation of Phelix

The ACF gradually dies out towards zero, but with clear swings. The PACF cuts off after the first or maybe second lag (outside the 95% confidence band – grey area), but clearly returns outside again after every 6-8 lags. The gradually decrease in the ACF and the early cut-off for the PACF suggests stationarity, but continues to swing after every 6-8 lags. This does not seem to be random, but rather seasonal. Seasonality may cause similar problems for our estimates as non-stationarity, so I further explore a way to factor this in. It is not uncommon that the spot prices follow day-of-the-week patterns, where expectations to, say the price for Monday, weighs heavily on what the price was last Monday, and so on.

It is also intuitive that weekdays have on an average higher prices than weekends, as the overall demand for power is generally higher during the working days.

6.2.2 Estimation

The ACF and PACF suggested at first glance an ARMA(p,q) for the combinations p = 1 and q ∈ {6,8}, where the comparison of the AIC and BIC values for all combinations suggested ARIMAX(1,0,7) to be best fit. I avoid to include the results here, as clear evidence of seasonality strongly suggest another model specification.

Stata provides simple commands for ARMA estimations, using maximum likelihood functions. Remembering the three steps of the Box-Jenkins method, I start over again with the identification of ACF, PACF for initial chosen lags, followed by AIC and BIC for those with significant estimates, but this time with a seasonal ARMA specification. Multiple variations of lag orders and differencing are based on the ACF and PACF for the seasonal specificationSARIM A(p, d, q)(P, D, Q)_S. Due to an already stationary process, was it little reason to include differencing.

I chose to do it either way as an illustration to compare values. I based the best fitted candidate on the AIC and BIC values, as well as a significance level of 5%.

I suggest SARIM A(1,0,1)(1,0,1,7) as the best candidate since it presents the smallest value of AIC and BIC with all variables significant at the critical value.

The marked (stars) of Table 3 are the specifications with all estimated variables within 95% confidence interval, where the best fit is marked with two stars.

Table 3: Akaike -and Bayesian information criteria - SARMAX ARIMAX-SARIMA

arima ger tsol twin, arima(p,d,q) sarima(P,D,Q,S) (p,d,q) x (P,D,Q) S AIC BIC

0,0,1-0,0,1,7 4549.809* 4577.375*

1,0,1-1,0,1,6 4638.645 4675.401 1,0,1-1,0,1,7 4126.899** 4163.654**

1,0,1-1,0,1,8 4653.729 4690.484 1,1,1-1,0,1,8 4604.958 4641.703 1,0,1-1,1,1,7 4081.484 4118.162 1,1,1-1,1,1,7 4061.500 4098.167 1,0,2-1,1,1,7 4065.824 4107.087

6.2.3 Diagnosis

Finally, one needs to verify whether the estimated residuals of the

SARIM A(p, d, q)(P, D, Q)_S model follow White Noise, as specified in Section 4.

If this turns out not to be the case, one needs to reconsider the model. There are several tools to test this, like the Portmanteau-Q test, comparison of the ACF and PACF of the predicted residuals, or Bartlett’s Cumulative Periodogram. I have chosen to include the latter. The periodogram presented in Figure 6, show that the values never appear outside of the default confidence bands of a 95% level. Thus, concluding that the process of the residuals is not different from White Noise.

Figure 11 in the Appendix depicts that the ACF and PACF of the residuals are within the error bands in a correlogram, implying that the p-values for those lags to be below 5%, thus White Noise.

Figure 6: Bartlett’s test - Periodogram

As mentioned in Subsection 4.2, the Phelix does visualize some signs of unstable volatility, and these signs seem to continue vaguely throughout the previous estimation.

My quest to find a better suited model, implementing this ’clustering of volatility’

phenomenon, continues in Section 6.3.

6.2.4 Fitted seasonal ARMAX model

Finally, theSARIM AX(1,0,1)(1,0,1)₇model can be written as equation (18)

(1−φ₁L)(1−Φ₁L⁷)y_t= (1 + Θ₁L⁷)(1 +θ₁L)u_t (18) Or

yt=α+γ₁Xt+ω1Zt+φ1yt−1+ Φ₁yt−7−φ1Φ₁yt−8+ Θ₁ut−7+θ₁ut−1+ Θ₁θ1ut−8+ut

(19) Where φ and θ are the non-seasonal parameters of autoregressive and moving average, respectively, and Φ and Θ the seasonal parameters in the equivalent order.

The actual estimates of this model’s parameters are shown in the as results in Section 7.

6.3 Estimation - ARMAX GARCH model

Various ARMA models have assumed that the random contribution to their process, the error term, has a constant variance. This assumption has in practice been violated various times, especially in financial and economic data. Quiet time intervals are at times followed by more turbulent periods. In other words, the volatility of a process can have a time-series process on its own. Time-varying volatility may be a phenomenon related to the Phelix index as well. This is examined in the following section.

The ARMA model was built on the observable Phelix process, ”y_t”. The ACF, PACF and other summaries, have helped determine a good prediction to fit a

model based on historical data. Estimated volatility of a prediction model on the other hand, is based on unobserved data, ”σ_y²”, making it a bit more complicated.

I ran an ARCH-LM test, as a post-estimation tool in Stata. The prognosis tool confirmed such an effect in my estimates. The ARCH effect is by definition present when the squared residuals of a time series model exhibit autocorrelation. I ran the test all the way up to the first 15 lags, and could reject the null hypothesis of no ARCH effect at a 1% significance level for all.

Stata offers a broad spectre of ARMA and ARCH specification, with and without constant variance. Unfortunately, the seasonal ARMA specification does not apply for the ARCH model in Stata, so I had to adjust for seasonality in an alternative way. The seasonality is instead accounted for by seasonal dummy variables for each day of the week ”D”, as well as each quarter of the year ”Q”, where the last “i” of each dummy variable is excluded to prevent perfect multicollinearity.

This method is often referred to as a ”seasonal dummy model”. I ran multiple variations of lags,to fit the seasonal dummy model, and found the GARCH(1,1) to be the best fit. ACF and PACF can not be applied to fit this model type, so old-fashioned approach of eliminating insignificant coefficients down to the best fit, was instead applied. A large number of lags in the initial ARCH model were significant at a 5% level. In cases like this is a GARCH model desired. Not only is the result still significant, but as mentioned in Becketti (2013), does fewer terms (as in this generalized model - GARCH) reduce the chance of unnecessary over-fitting. Equation (20), is the new seasonal dummy model:

y_t=α+γ₁X_t+ω₁Z_t+δ_t⁰D+κ⁰_tQ+φ₁y_t−1 +θ₁u_t−1+u_t (20) With againαas the constant term,γ andωas the coefficient for TW generation of solar and wind power, respectively. The vectorλ⁰ represent the coefficients for six out of the seven days in a week, with the seasonal dummy variable D, in the same way as the vector κ⁰ represents three out of four yearly quarters to the seasonal dummy Q. The rest of the terms in equation (20) are defined in the familiar same way as in equation (19), but this time with a conditional variance for GARCH(1,1).

This is shown in equation (21) where π is ARCH effect of first order and τ the GARCH effect of first order, for the error term.

u_t=σ_tt, _t∼W N

σ_t² =π₀+τ₁σ_t−i² +π₁u²_t−1 (21)

Becketti (2013) lists some of the more important empirical regularities, which may be captured by other extensions of the initial ARCH model:

• The uneven, intermittent, or random arrival of “news” (impactful additional information) is commonly cited as cause of time-varying volatility. However, many series appear to react asymmetrically to positive and negative news.

• The conditional mean of y_t, the observable time series, often appears to depend on the current level of volatility. For example, the level of stock prices often declines during periods of unusually high uncertainty.

• Asset prices, such as stock prices, tend to have distribution with “fat tails”.

In other words, extreme events (unusually large price increases or decreases) occur more frequently than in normal distribution. The ARCH model introduces some leptokurtosis (a distribution with positive excess kurtosis, compared to a normal distribution with zero excess kurtosis) in the conditional variance, but less than typically observed in financial time series.

I recognize that various agents in the balancing and wholesale markets may speculate, or merely react to news in different ways. Weather forecasts, production problems or cable capacity limitations are factors which may create shocks. Further ARCH model extensions can possibly factor in similar effects, but I have for simplicity chosen to not further extend my model.

7 Results

7.1 Parameters of Both Models

Now that I have settled on two alternative time series models, how do I choose between the two specifications? I this section, I compare the estimates of the two models, make an in-sample prediction for a given period, before ideally settling on one specification based on a comparison of the two. Let me stress that there is rarely just one suitable model for a data set. There may be several models with significant estimates, catching the possible relation between variables, as well as multiple specifications suited for forecasting. The flexibility of time series models makes it difficult to definitively choose a single ”best” fit specification.

The Box-Jenkins method for the ARIMA model, and careful elimination of seasonal dummies and lags variables, nailed down these final two. Table 4 shows the estimated parameters of theSARIM A(1,0,1)(1,0,1)₇ model, with all parameters significant at a 5% level. The external variables for renewable energy- solar and wind, both have a negative impact on the spot price. The estimated effect from these two covariates are not the direct effect on Phelix when generation increases by one unit (as in simple regression theory), but conditioned on previous values as well.¹⁴ Although, this may not be completely intuitive, it still gives a pretty clear understanding of the total effect on the price index by increasing wind -or solar power supply. The effects from solar (TW) and wind (TW) are estimated to -8.58682 and -9.072535, respectively. Increased feed-in of wind and solar power, reduces the day-ahead price for the German area. These negative coefficients are consistent with recent studies introduced in Section 2. Both the seasonal and non-seasonal coefficients of the moving average and the autoregressive term seem to follow a typical form. Note that the seasonal autocorrelation (0.996462) is higher than the first order autocorrelation. These findings show that today’s price is more correlated with the same day last week, than with yesterday’s. The

14https://robjhyndman.com/hyndsight/arimax/

estimated standard deviation of the white noise disturbance is ∼3.58.

Table 4: Estimated SARMAX parameters SARMAX

Model (A)

Parameter Value Std. Error T-Statistic

α 42.737080 5.878848 7.27

γ -8.586820 1.386766 -6.19 ω -9.072435 0.265583 -34.16 φ₁ 0.861241 0.022005 39.14 θ₁ -0.403069 0.032884 -12.26 Φ₁ 0.996462 0.001756 567.43 Θ₁ -1.114391 0.022101 -50.42

σ 3.575129 0.075446 47.39

The estimated results of the seasonal dummy model ARIMAX(1,0,1)-GARCH(1,1) can be found under ”(B)” in Table 5. All variables are significant at a 5% level, except the seasonal dummy for third quarter (significant at 10%). The constant term is considerably lower in this model, due to the clear variation in average price on weekdays (peaks midweek, Wednesday) to weekends (especially to the base value of Sunday), as intuitively expected. The seasonal quarter dummies seem to have the greatest negative impact during the first and second quarter, before increasing gradually throughout the year. The autoregressive and moving average terms are again following a reasonable magnitude and are significant. First-lag autocorrelation is now larger than in last model, while adjusted by a lower moving average.

The GARCH estimates look reasonable, where the ARCH effect measures today’s volatility shock on the following day’s volatility. The constant term of the GARCH estimation is 1.890858. The two remaining coefficients are estimated to 0.205622 and 0.686447. The combination of these is the rate at which the effect dies out over time. Both the ARCH and GARCH terms have positive coefficients, confirming that an increased volatility in one period is followed by increased volatility in the next, and vice versa.

Table 5: Estimated ARMAX-GARCH parameters ARMAX-GARCH

Model (B) Model (C)

Parameter Value Std. Err T-Stat Value Std. Err T-Stat α 34.229670 1.427806 23.97 34.719590 1.509474 23.00 γ -8.036480 1.037106 -7.75 -7.404551 1.053662 -7.03 ω -8.160738 0.236636 -34.49 -8.180961 0.306818 -26.66 δ₁ 10.300040 0.352796 29.20 10.231860 0.336248 30.43 δ₂ 11.593520 0.375914 30.84 11.670600 0.379717 30.74 δ₃ 11.655050 0.441621 26.39 11.595850 0.449216 25.81 δ₄ 11.335580 0.388812 29.15 11.186610 0.406224 27.54 δ₅ 10.360570 0.377099 27.47 10.067760 0.413525 24.35 δ₆ 5.056160 0.317196 15.94 4.954713 0.343632 14.42 κ₁ -6.503132 1.626840 -4.00 -6.788705 1.736690 -3.91 κ₂ -6.524590 1.720006 -3.79 -7.658529 1.909270 -4.01 κ₃ -3.193775 1.664930 -1.92 -3.528413 1.746297 -2.02 φ₁ 0.914497 0.019358 47.24 0.910962 0.021001 43.38 θ₁ -0.479940 0.052287 -9.18 -0.458367 0.054037 -8.48 π₀ 1.890858 0.355886 5.31 0.524572 0.264796 1.98 π₁ 0.205622 0.034972 5.88 0.185327 0.045342 4.09 τ₁ 0.686447 0.041169 16.67 0.595208 0.056252 10.58

γ -0.089528 0.235204 -0.38

ω 0.734609 0.117670 6.24

A significance level of 10% instead of 5%, as for the latter model, may indicate a case of ”overfitting”. According to Box-Jenkins (1970), is an all-significant model of fewer variables usually preferred. However, the Box-Jenkins method was made for ARMA specifications, long before the introduction of Engle’s ARCH model, Therefore, are both models kept rolling until a comparison of forecasts is complete.

7.2 Forecasting with Both Models

An in-sample forecast is executed for both models, based on each model’s estimates to July 1st 2016, and predicted throughout the sample (December 31st 2016).

Figure 7 shows the prediction (dotted line) based on the estimated SARMAX model. The actual Phelix movement is included for comparison,in the same time horizon.

Figure 7: SARMAX prediction (dotted line)

The SARMAX model succeeds to predict the Phelix movement in a reasonable pattern for a month or two, before continuing to underestimate both the overall price increase and the shifting volatility. Figure 8 illustrate the ARMAX-GARCH model prediction (dotted line) compared to the actual Phelix curve. The fat tail behaviour of clustering volatility as observed for the Phelix in the last quarter of 2016, is naturally better accounted for by the latter model. The actual spikes seem to be even heavier than estimated, indicating some heavy outliers. Maybe not so

surprising, the SARMAX model does seem to be the closest fit to the actual spot price, and thus the preferred fit, in the short run.

Figure 8: ARMAX-GARCH prediction (dotted line)

However, in the long run, volatility clustering seem more likely to occur within the horizon. Therefore, the ARMAX-GARCH model is better suited to adjust for and further predict this. The ARMAX-GARCH model could be a better fit to the Phelix data, if I had chosen to remove all large outliers. As explained in Section 5, they were included to make the model more realistic for all time intervals. It is possible to see clearly an evidence of actual clustering in a time with increased renewable generations. This could argue for further expansion of the balancing and wholesale market. The graphs in Figure 9 combine both predictions to the actual process, for comparison.

I have included a summary of the three functions in Table 6. The ARMAX-GARCH

Figure 9: Comparison of in-sample forecasts

model seems to be a better model for the overall horizon, even though both models fail to estimate negative minimum prices for any days.

Table 6: Summary of actual -and predicted Phelix series

Variable Obs Mean Std. Dev. Min Max

ger 244 31.02869 10.14631 -12.89 60.05833 trend1 244 26.75895 6.385271 3.579622 38.3774 trend2 224 31.64272 7.468284 5.347721 45.35629

8 Volatility Ratio Test

The effects on the German spot price from increasing renewable generation and clustering volatility has until now been investigated and accounted for, before running forecast comparisons with the goal to find the best fitted model. Even

though there is significant evidence of an ARCH effect in the data, has the change in volatility due to increasing renewable power yet to be estimated. Studies from Paraschiv et al. (2014) and from Ketterer (2014) investigate the changes in volatility of the spot price, when wind -and solar feed-in in Europe increases.

The scatter plots in Figure 10 shows the relation of the price to the two energy sources. A first visual inspection of the two plots shows a relationship between generation and price, but no clear evidence of increasing price volatility.

Figure 10: Scatter plot of Generations Volumes to Price

A variance ratio test is assigned to the price based on each source of energy, to investigate this further. I created a dummy variable, based on the mean of solar, from Table 1. Dummy=1 when generation is larger than the mean, and dummy=0 otherwise. By multiplying the new dummy to the actual Phelix, I could compare the prices of ”sunny” days (p hsol) to the prices on days with less sun (p lsol). The same procedure was done for the wind variable, with the explanatory variables

”p hwin” and ”p lwin”. The variance test runs a two-sided hypothesis test for equal standard deviations. As shown in the Table 7 can I reject at a 5% level. The table further indicate a lower mean price and a lower variance on days of higher feed-in. The test is executed for wind in stead of solar in Table 8, with the same