How fast can one overcome the paradox of the energy transition? A physico-economic model for the European power grid
HHow fast can one overcome the paradox of the energy transition? Aphysico-economic model for the European power grid
Laurent Pagnier a,b , Philippe Jacquod a a School of Engineering, University of Applied Sciences of Western Switzerland HES-SO CH-1951 Sion, Switzerland b Institute of Theoretical Physics, EPF Lausanne, CH-1015 Lausanne, Switzerland
Abstract
The paradox of the energy transition is that the low marginal costs of new renewable energy sources (RES) drag elec-tricity prices down and discourage investments in flexible productions that are needed to compensate for the lack ofdispatchability of the new RES. The energy transition thus discourages the investments that are required for its ownharmonious expansion. To investigate how this paradox can be overcome, we argue that, under certain assumptions,future electricity prices are rather accurately modeled from the residual load obtained by subtracting non-flexible pro-ductions from the load. Armed with the resulting economic indicator, we investigate future revenues for European powerplants with various degree of flexibility. We find that, if neither carbon taxes nor fuel prices change, flexible productionswould be financially rewarded better and sooner if the energy transition proceeds faster but at more or less constant totalproduction, i.e. by reducing the production of thermal power plants at the same rate as the RES production increases.Less flexible productions, on the other hand, would see their revenue grow more moderately. Our results indicate thata faster energy transition with a quicker withdrawal of thermal power plants would reward flexible productions faster.
Keywords:
Residual load, Electricity prices, Renewable energy
1. Introduction
The goal of the energy transition is to meet energydemand from human activities in a sustainable way. Inthe electricity sector, the transition currently increasesthe penetration of productions from new renewable en-ergy sources (RES), in particular solar photovoltaic pan-els (PV) and wind turbines (WT). These RES differ fromthe traditional productions they substitute for, in at leasttwo very significant ways. First, they lack dispatchabilityand have little mechanical inertia, second, they have verylow marginal production costs. Their lack of dispatcha-bility and mechanical inertia requires additional flexibleproductions and possibly electrical energy storage (EES)to ensure the stability of the power grid as well as thebalance of demand and supply at all times. Therefore in-creasing penetrations of new RES should be accompaniedby significant investments in new facilities with rather longpayback periods. However, the new RES’s low marginalcost brings spot electricity prices and thus beneficiary mar-gins of electric power companies down, while further ex-tending the payback period for investments in new facil-ities. The energy transition is therefore confronted withthe paradox that it creates economic conditions which, atleast temporarily, strongly discourage the infrastructural
Email addresses: [email protected] (Laurent Pagnier), [email protected] (Philippe Jacquod) investments it needs to progress further. Because of that, anumber of hydroelectric plant projects are currently frozenin Europe. To plan the next steps in the energy transition,to evaluate and anticipate the investments needed for itssafe, steady progress, it is therefore important to get arelatively good quantitative estimate of future electricityprices. The key issue is whether production flexibility willsoon be rewarded well enough that it will motivate in-vestments in fast dispatchable power plants and EES ata level consistent with the rate at which RES penetrationincreases.The traditional way to address such questions is to con-struct economic models for electricity production and con-sumption. Those models are standardly based on a num-ber of assumptions on general economic conditions, demo-graphic evolution, costs of different fuels, maintenance andproduction costs, amount of taxes and subsidies on energyproduction and so forth. Once all these ingredients arefixed, both the electricity demand and the marginal costof different productions can be estimated, which determinethe market price of electricity, from which one finally com-putes expected future revenues. From these revenues, in-vestment decisions can ultimately be made. The accuracyof this procedure relies on the accuracy of each of the as-sumptions on which it is based. Unfortunately, the latterare to a high degree arbitrary – economic growth rates,unemployment rates (indicative of the volume of indus-trial activities), taxation amounts, fuel costs (in particularnatural gas prices), carbon taxes and so forth cannot be
Preprint submitted to Elsevier June 18, 2018 a r X i v : . [ phy s i c s . s o c - ph ] J un on Tue Wed Thu Fri Sat Sun-20-10010203040 R e s i dua l l oad [ G W ] D a y - ahead p r i c e [ E UR / M W h ] Mon Tue Wed Thu Fri Sat Sun-100102030 R e s i dua l l oad [ G W ] D a y - ahead p r i c e [ E UR / M W h ] Figure 1: German residual load (green) and day-ahead electricityprice (blue) for a winter (top) and summer (bottom) week in 2015[data taken from ENTSO-E (2015a)]. Vertical dashed lines indicatenoon time. predicted accurately on time scales of decades, correspond-ing to typical payback times for investments in the energysector. In this manuscript we take a deliberately differentapproach, using as few working hypotheses as possible.
Eurotranselec , our model to be presented below, is mostlybased on physical conditions extracted from the size andproduction types of national power plant fleets as well asthe electricity demand. We argue that it introduces a re-liable though quite simple procedure to evaluate futureelectricity prices in the not too far future, given scenar-ios for the energy transition and the resulting evolutionof power plant fleets. We use it to investigate revenues ofelectric power plants in a time window until 2020.
Our starting observation is that electricity prices re-flect the law of supply and demand. Accordingly they ex-hibit some degree of correlation with what flexible sourcesmust generate to sustain load – the larger the differencebetween demand and non-dispatchable supply, the higherthe electricity price. As a matter of fact, spot marketprices are usually higher at times of larger imbalance be-tween demand and non-dispatchable supply, when the im-balance leans strongly on the demand side. The corre-sponding missing amount of power is quantified by the residual load , which is defined as the difference betweenconsumption and the sum of all non-flexible productions.With the residual load, the new RES are accounted for onthe demand side and not as a production - they are seenas reducing the demand for the rest of the market. Thisis justified by their almost vanishing marginal productioncost.While some degree of correlation between the residual load and spot market prices is expected, a strong correla-tion between them has been reported by von Roon and Hu-ber (2010) for the special case of Germany in 2007–2009,with a coefficient of determination R ∈ [0 . , . r = 0 . Below weinvestigate the correlation between day-ahead prices andresidual load further, for different European countries. Wefind that it always corresponds to a large, positive corre-lation coefficient, r > .
5, and furthermore that r is gen-erally higher in countries with larger penetration of RES(this is shown in Fig. 3 and will be discussed below). Theenergy transition will keep increasing the penetration ofRES, it is therefore reasonable to expect that r will in-crease in the future. If this is confirmed, the residual loadwill reflect the day-ahead price better and better. Day-ahead transactions represent a significant percentage of allelectricity transactions (see Table 1 below), and their shareof the total load is expected to increase with the end oflong-term contracts in the liberalized European electricitymarket. Putting all this together, we propose to introducea synthetic electricity price p da ( t ) as a two-parameter, lin-ear regression of the day-ahead price based on the residualload R ( t ), p da ( t ) = ∆ p da R ( t ) + p da0 . (1.1)In this manuscript, the two parameters ∆ p da and p da0 aredetermined country by country from a least square fit of2015 day-ahead data with Eq (1.1). They are assumed toremain constant in the future, because we consider a rel-atively small time window, until 2020. However differentscenarios can be considered, where ∆ p da and p da0 evolve intime, for instance because economic conditions (fuel prices,subsidies and taxes) vary. Below we discuss in particularhow ∆ p da and p da0 qualitatively vary with varying carbontaxes and natural gas prices.We think that p da ( t ) is a reliable day-ahead electric-ity price because it reproduces qualitatively and even al-most always quantitatively historical time series for thetrue spot market day-ahead price of electricity in all Euro-pean countries we focus on. Not caught by our approachare extreme events, for instance corresponding to recordlow (high) RES productions with simultaneous record high(low) demand, giving unusually high price maxima (unusu-ally low price minima).Armed with p da ( t ) we finally investigate the parallelevolution of the energy transition and the electricity pricesin Germany and Spain and compute expected future rev-enues for various types of power plants, focusing on pumped- Assuming a linear relation between day-ahead prices and residualload, as we do in Eq. (1.1), with coefficients determined by a leastsquare fitting, R is the square of Pearson’s correlation coefficient.Our finding of r = 0 .
89 for Germany in 2015 then correponds to R = 0 .
8, even higher than the largest value reported by von Roonand Huber (2010).
Studies of the impact of increased penetration of newRES on electricity prices abound. Many of them investi-gated historical data vs. the penetration of new RES toempirically express electricity prices as a function of greenin-feed (Cl`o et al., 2015; Paraschiv et al., 2014). In theliberalized European market, electricity prices are deter-mined by a merit order supply curve where production ca-pacities are ordered according to their marginal costs. Theeffect of such merit order on historical electricity pricesunder increased penetration of new RES has been inves-tigated by Cludius et al. (2014), who extrapolated theirfindings to evaluate future revenues of PV and WT. Go-ing further, a number of studies investigated electricitymarkets where prices are determined by simulated meritorders with marginal costs as inputs (Sensfuss et al., 2008;Haas et al., 2013; Auer and Haas, 2016), which often relyon self-consistent optimizations. As interesting as theseworks are, they are based on heavy algorithms as wellas many assumptions (for instance future fuel prices) tobuild the merit order. Schlachtberger et al. (2016) findthat more flexible production is required as the penetra-tion of new RES increases and that flexible sources becomeessential when the penetration of new RES reaches 50%.By definition, the residual load gives indications on pe-riods of surplus or deficit of production of new RES. Ac-cordingly, it has been the focus of many recent investiga-tions evaluating the needed capacity of EES, of thermalstorage and of additional dispatchable productions to helpabsorb large penetrations of new RES (Schill, 2014; Saari-nen et al., 2015; Ueckerdt et al., 2015; Schweiger et al.,2017). In his analysis of negative price regimes, Nicolosi(2012) illustrated a connection between the residual loadand the merit order. To the best of our knowledge, vonRoon and Huber (2010) have been the only ones so far toreport a direct correlation between residual load and spotelectricity prices. Their investigation of the German elec-tricity market before 2010 further assumed that the coaland natural gas price determine the electricity price mostof the time, because the German load back then requiredcoal and gas power plants to produce most of the time.They therefore proposed to model electricity prices as afunction of the natural gas price and of the residual load.Given uncertainties in future gas prices (as well as thoseof other fossil fuels) and the level of CO taxes, we departfrom that analysis and go one step further by modelingprices using the residual load only. In this manuscript we construct a model to investi-gate future economic conditions in the European electric-ity sector. To construct our pricing algorithm, we departfrom von Roon and Huber (2010) in that (i) we modelelectricity prices only as a function of the residual load,(ii) we take changes in production fleet and other scenar-ios into account in our study, (iii) we apply our study tomost European countries, as they also exhibit high de-grees of correlation between day-ahead prices and residualload, and (iv) our pricing procedure may be incorporatedinto an aggregated European grid (Pagnier and Jacquod,2017). We call the resulting model
Eurotranselec . In theperiod 2010–2015, after the work of von Roon and Huber(2010), the new RES penetration in Germany has dra-matically increased, with the WT production more thandoubling and the PV production more than tripling. Point(i) is therefore an important and necessary departure fromvon Roon and Huber (2010), because with this strong in-crease in new RES capacities, electricity prices are lessoften directly determined by natural gas prices. In ana-lyzing other European countries, we moreover observe thatthe correlation between residual load and day-ahead pricesis generally stronger in countries with higher penetrationof RES. Because RES is expected to significantly increasein the future, it is natural to expect that this correlationwill also increase. Additionally, day-ahead markets makea significant part of the total traded electrical energy (seeTable 1 below), a share which will increase with the endof long-term contracts in the liberalized European market.It seems therefore reasonable to expect that the pricingmodel we present in this article will become more andmore accurate as the energy transition proceeds. Whilevon Roon and Huber (2010) should get the credit for un-covering an important correlation between residual loadand spot market prices, the present manuscript uses thefull analytical power behind this correlation for the firsttime, to the best of our knowledge. Our results allowto anticipate how the energy transition should proceed inits next steps in order to overcome the paradox describedabove.The European electricity market is expected to evolvefast with the energy transition. New incentives and taxesmay appear, different financial products related to electric-ity may be introduced, gas and other fuel prices may fluc-tuate. All this will modify the way electricity is both pro-duced and consumed, and will significantly impact powerplant revenues. Our purpose in this manuscript is how-ever to investigate the relatively near future and see howfinancial conditions in the electricity sector will evolve inthe next few years. Accordingly, our investigations delib-erately assume a European electricity market where thepenetration of RES increases and thermal power plantsare withdrawn, all other things remaining constant. Westress that, to extrapolate our investigations to longer timescales, our approach needs to be revisited, for instanceby considering different scenarios and pricing parameters3 p da and p da0 , or different consumption profiles. In Ap-pendix A, we comment on how consumption profiles mod-ified by active demand response could be incorporated inour model.This manuscript is organized as follows. Section 2 de-fines the residual load. In Section 3 we comment brieflyon electricity trading. In Section 4 we show the strongcorrelation between residual loads and day-ahead electric-ity prices in European countries and show that they arestronger in countries with more new RES. In Section 5we construct a synthetic electricity price in each countryconsidered in our model. That price is based on residualload and in Section 6 we use it to investigate future rev-enues of various types of electricity productions. We focuson conventional dam hydroelectricity, as it is one of themost flexible, more easily dispatchable electricity produc-tion and on pumped-hydro, which is to date the dominantEES solution for which a number of projects are howevercurrently put on hold in Europe because of low electricityprices. We finally discuss other productions, dependingon their number of operation hours per year. Conclusionsand future perspectives are given in Section 7.
2. Residual load and must-run
The residual load is defined as the difference betweenthe total consumption and the sum of all non-flexible pro-ductions (Denholm and Hand, 2011; Schill, 2014; Saarinenet al., 2015). Non-flexible productions include new RESand run-of-river hydro. Often neglected as non-flexibleproductions are must-run productions (Nicolosi, 2010), whichare defined as follows. Most thermal power plants faceramping costs to turn their production on and off, and toavoid those costs, they keep producing even when electric-ity prices are below their production costs. That part oftheir production is what is called must-run. It is consis-tent with the definition and meaning of the residual load toinclude must-run productions in non-flexible productionsand treat them as demand reduction. The residual load R i is then defined in each country/region (labeled by anindex i ) in our model as R i ( t ) = L i ( t ) − P PV i ( t ) − P WT i ( t ) − P MR i . (2.1)Here, L i ( t ) is the regional consumption/load, and P PV i ( t ), P WT i ( t ) and P MR i are PV, WT and must-run productionsrespectively. In this manuscript, they are taken at discretetimes t = n ∆ t , with ∆ t = 1 hour. In a given year, P MR i does not depend on time.We take L i ( t ) as the 2015 consumption from ENTSO-E (2015a) without modification, given the relatively shorttime span of our investigations in this manuscript. Forinvestigations further into the future, other consumptionprofiles, and other consumption curves (for instance mod-ified by active demand response, see Appendix A) canbe loaded into Eurotranselec . PV and WT productionsare obtained from ENTSO-E (2015a), which we rescale country by country to take into account planned capacityevolution as given in ENTSO-E (2015b).To obtain R i ( t ), we are left with evaluating the must-run power which is not a uniquely defined procedure (Schill,2014; Denholm and Hand, 2011). To do so, we evaluatethe must-run from duration curves which give the numberof hours in a year that a given load is exceeded. Fig. 2 (a)shows duration curves for the total consumption minus thetotal RES production for four different years in Germany.We extract the must-run as the corresponding power thresh-old exceeded during ”most of the year”, and chose thisto mean 7000 (vertical red dashed line) or 8000 hours(black dashed line). The obtained must-run is plotted inFig. 2 (b) for these two choices (dashed lines). We seethat the two curves mostly differ by a vertical shift of 3-4GW. The must-run is about 30-35 GW in 2010, and keepsdecreasing thereafter, as the penetration of RES increasesand thermal plants are phased out.This is not the only possible procedure to evaluate themust-run but it agrees well with another, altogether differ-ent method. Nicolosi (2010) plots electricity prices hourby hour as a function of the percentage of the used produc-tion capacity for various types of production in Germany,from October 2008 to November 2009. The resulting cloudis rather elongated in all cases – and a linear regression isqualitatively representative of the data. From this linearregression, we may define the must-run as the capacity stillused when this linear regression intersects the horizontalaxis between positive and negative prices. One obtainsa must-run corresponding to 85% of the total nuclear ca-pacity, 70% of the total capacity of lignite power plantsand 10% of the total capacity of hard coal power plant.This estimate sums up to about 30-35 GW for 2010 inGermany, in agreement with our estimate extracted fromduration curves. We therefore validate our procedure forestimating the volume of must-run production and use itto compute residual loads based on the scenario of ENTSO-E (2015b).It is important to realize that the procedure just de-scribed underestimates (overestimates) the must-run forexporting (importing) countries. As a matter of fact, Fig. 2(b) suggests that the German must-run started to decreasealready in 2013, instead, Germany’s thermal productioncapacity has been kept constant in 2013–2016 while its ex-ports have increased significantly. This suggests that Ger-many will keep a large must-run as long as it can export itsproduction when needed. To take this effect into account,we introduce three different scenarios for must-run evolu-tion which we will use in our investigations. These threescenarios are shown in blue, orange and red in Fig. 2 (b).The blue curve corresponds to a must-run that is constantuntil 2015 after which it decreases with the same rate of 3[GW/year] in 2016-2020 as the dashed lines. The red curvecorresponds to the opposite case where thermal capacitiesare withdrawn exactly at the same rate as new RES areinstalled. Finally, the orange curve is a smooth curve in-terpolating somehow arbitrarily between the blue and red4
000 4000 6000 8000Capacity utilization time [hour/year]-10010203040506070
Load - R ES p r odu c t i on s [ G W ] (a) M u s t -r un [ G W ] (b) Figure 2: (a) Duration curves of German load minus RES produc-tions for the years 2000 (solid), 2010 (dashed), 2015 (dotted) and2020 (dash-dotted). (b) Must run power as obtained from the du-ration curves [red and black dashed curves, corresponding to thered/black dashed vertical lines in panel (a)] and our three scenarios:keeping thermal production capacity ”as long as possible” (blue),”exact substitution” of thermal production with new RES produc-tion (red), and smooth, in-between ”interpolated path” (orange).The red circles illustrate the connection between panel (a) and panel(b). scenarios. Below, we dub these three scenarios as long aspossible (blue), interpolated path (orange) and exact substi-tution (red). We use these scenarios to investigate what in-gredient(s) determine(s) the evolution of electricity prices.None of them is actually realized, however investigatingthe three of them allows to understand quantitatively theinfluence of must-run on electricity prices.While we just focused on the German case to describethe procedure for evaluating must-run capacity, the de-scribed method is applied below to other European coun-tries.
3. Electricity day-ahead markets
In the liberalized European electricity market, day-ahead transactions correspond to a significant share of thetotal consumed electricity. This is shown in Table 1. Thisshare is expected to keep increasing in the future as theremaining long-term contracts expire, presumably with atmost partial renewal because of market liberalization. Inthe next section we show the strong correlation betweenthe residual load and the day-ahead price, which allows usto model future day-ahead prices. Given the sizeable shareof day-ahead transactions, a share that will keep increas-ing in the coming years, we argue that this model givesus a faithful, qualitative model for future electricity prices(not only day-ahead).Market zone Traded energy Load percentage[TWh] [%]AT/DE 264 53BE 24 29CH 23 38CZ 20 28ES/PT 259 79FR 106 23IT 195 62NL 43 39NO 133 103PL 24 18SE 128 94UK 47 19Total 1264 49
Table 1: Traded electrical energy and corresponding load percentageof several European day-ahead markets in 2015. Sources: EPEXSPOT (2015), OMIE (2015), OTE (2015) and NordPool (2015).
4. Correlations between residual loads and day-ahead prices
Fig. 1 illustrates the strong correlation between na-tional residual loads R i ( t ) and day-ahead prices p da i ( t ), acorrelation that had been noticed by von Roon and Hu-ber (2010) for Germany in 2007–2009. In this section, wefurther quantify this correlation for other European coun-tries. Statistical correlation between discrete sets of data X = { x k } and Y = { y k } is standardly measured by Pear-son’s correlation coefficient r ( X, Y ) = (cid:80) nk =1 ( x k − ¯ x )( y k − ¯ y ) (cid:112)(cid:80) nk =1 ( x k − ¯ x ) (cid:80) nk =1 ( y k − ¯ y ) , (4.1)where ¯ x and ¯ y are the average values of the two sets. Bydefinition, one has r ∈ [ − , r = 0 indicating theabsence of correlation between the two sets, r = 1 twoperfectly correlated sets and r = − r > . r > .
58, andseem to be constant or perhaps even increasing with time r ( p da i , R i ) 2012 2013 2014 2015FR 0.65 0.74 0.71 0.67DE 0.78 0.86 0.89 0.89IT 0.63 0.58 0.61 0.77ES Table 2: Evolution of the correlation between national day-aheadprices and residual loads.
We further investigate the correlation coefficient for2015 data in a number of continental European countries.Fig. 3 plots the correlation coefficient between nationalresidual load and day-ahead electricity price as a functionof new RES penetration, which we took as the ratio ofyearly RES production to the total electricity production.Data are taken from ENTSO-E (2015a) and have beencrosschecked and completed where necessary with data ob-tained from national grid operators and power markets. The correlation coefficients in Table 2 and Fig. 3 satisfy r ( p da i , R i ) > .
58 in all cases, indicating a strong corre-lation between the residual loads and day-ahead prices.Additionally, r ( p da i , R i ) is larger in countries with largerpenetration of new RES (see Fig. 3), with the exception ofSwitzerland, where the correlation is presumably higherdue to a large penetration of hydroelectricity - an ”old”RES. Given this trend, and the planned increase in newRES penetration in all European countries, it seems nat-ural to expect an even larger correlation between residualloads and day-ahead prices in the future.
5. Present and future electricity prices modeledafter the residual load
We argued in Section 4 that the already large correla-tion factor r between day-ahead prices and residual loads The Italian electricity market has regional prices and we choseto use the Northern Italy price as national price in Fig. 3, giving r = 0 . r = 0 . . r = 0 .
0 5 10 15 20 25 new RES peneration [%] r( p da i , R i ) ATBECH CZ DEESFR ITNLPL
Figure 3: Pearson’s correlation coefficient r between national residualloads R i and 2015 national day-ahead prices p da i as a function of thepenetration of new RES in several European countries. (see Table 2) is expected to become even larger as the pen-etration of new RES increases. This is so, because, as isillustrated in Fig. 3, r is larger in countries with largeramounts of new RES. Simultaneously, day-ahead marketsrepresent a significant part of all electricity transactions asis shown in Table 1, a share that is likely to keep increasingin Europe as the liberalization of the market becomes com-plete. It therefore makes sense to model future electricityprices from residual loads. The latter are based solely onscenarios for the evolution of the consumption, the futureRES productions and the must-run. The economic feasi-bility and the future of different production types undergiven scenarios can then be checked quantitatively. In thissection we construct such a price and show that it repro-duces historical prices with very good accuracy, except forrare extreme events.We construct a synthetic electricity price as a linearregression of the residual load, p da ( t ) = ∆ p da R ( t ) + p da0 , (5.1)where we rewrote Eq. (1.1). For the sake of simplicity,we drop the country index i here. We focus on electric-ity prices and revenues for various productions in Spainand Germany, two large European countries that are al-ready well engaged in their energy transition in the elec-tric sector, with large penetration of new RES. The RESmix has proportionally less PV in Spain than in Germany,which allows us to identify differences in the evolution ofprices from different choices of RES mixes. Based on 2015data, we obtain ∆ p da ≈ . e /MWh · GW − ] and p da0 ≈
20 and 30 [ e /MWh] in Germany and Spain re-spectively for the parameters in Eq. (5.1). We found verylittle change in these parameters during the years 2013–2015 in Germany and therefore assume these parametersto be constant in time in each country, for a time windowranging from 2015 to 2020. That this is reasonable is illus-6rated in Fig. 4 which shows that the effective price p da ( t )of Eq. (5.1) reproduces historical day-ahead prices quitewell. The agreement is already good in 2006 and becomeseven better in 2013. Exceptional price spikes and troughsare not totally captured, which correspond however to un-usual situations beyond the reach of our modeling. Mon Tue Wed Thu Fri Sat Sun020406080 p da [ E UR / M W h ] Tue Wed Thu Fri Sat Sun Mon020406080 p da [ E UR / M W h ] Mon Tue Wed Thu Fri Sat Sun020406080 p da [ E UR / M W h ] Mon Tue Wed Thu Fri Sat Sun020406080 p da [ E UR / M W h ] Figure 4: Electricity price (5.1) built on the residual load (greenline) and actual day-ahead electricity price (blue line) during a weekin winter and summer 2006 (top two panels) and 2013 (bottom twopanels) in Germany. Dashed lines show the monthly average priceand the dotted-dashed lines prices exceeded 10 and 90 % of the timeduring that month. Vertical dashed lines indicate noon time.
Eq. (5.1) allows us to qualitatively forecast electric-ity prices and price fluctuations in the framework of theenergy transition. The latter substitutes thermal produc-tions with new RES. Doing so, it reduces the must-run andchanges fluctuations in the residual load. The way thesefluctuations change depends on the chosen RES mix: PVproduces more around noon, it therefore is correlated withthe main load peak; WT production on the other hand iseffectively random in time on time scales of the order offew hours to few days, and therefore uncorrelated with con-sumption on such time scales. Consequently, fluctuationsin residual loads will always increase if the substitutionmix is made of WT only, while they will first decrease be-fore increasing again if the mix is dominated by PV. Thisis illustrated in Fig. 5 which sketches the behavior of theresidual load at three different stages of the energy tran-sition. Panel (a) shows the situation at the very initialstage of the energy transition, with low RES penetration.The shape of the residual load is very similar to the loaditself and the must-run is high. Panel (b) illustrates the transition period with increased RES penetration with asignificant fraction of PV, corresponding to the Germanmix. PV significantly decreases the load peak during officehours, which reduces fluctuations of the residual load. Themust-run is still high. In our model, this reduces fluctua-tions in electricity prices, therefore there are less financialopportunities for flexible productions. In the final stagesof the energy transition, the large RES penetration com-pletely changes the shape of the residual load, which looksnow very different from the load, see Fig. 5 (c). The must-run power is lower, bringing average prices higher. Mostimportantly, fluctuations in the residual load are compa-rable to and even higher than those at the early stages ofthe energy transition.
Day 1 Day 2 Day 3 Day 40204060 R e s i dua l l oad Day 1 Day 2 Day 3 Day 40204060 R e s i dua l l oad Day 1 Day 2 Day 3 Day 40204060 R e s i dua l l oad (a)(b)(c) Figure 5: Sketch of the residual load (green area) and must-run (lightred band) at three stages of the energy transition: (a) Initial, (b)intermediate and (c) late stages of the transition. PV (yellow) andWT (light blue) production profiles are superimposed. Red arrowsindicate the magnitude of fluctuations of the residual load. Verticaldashed lines indicate noon time.
The residual load quantifies the balance between non-flexible supply and demand, and accordingly, the corre-lation between electricity prices and residual load can beunderstood as a logical consequence of the economic lawof supply and demand. This correlation may vary in thefuture, however, from fundamental laws of economics, it islikely to remain sizeable in any event. It therefore makessense to introduce an electricity price as in Eq. (5.1). Vary-ing economic conditions may however impact the pricingparameters ∆ p da and p da0 and there is no reason a priori toconsider them constant in time (the scenario we discuss inthis manuscript). Other scenarios with varying ∆ p da and p da0 can be investigated. Qualitatively, one anticipatesthat p da0 is determined by the marginal cost of must-run7roduction. In Europe this is essentially the marginal priceof electricity from coal-fired plants, and therefore p da0 in-creases if carbon taxes increase. The parameter ∆ p da onthe other hand is more directly related to the order ofmerit, and thus to the marginal cost of electriciy fromgas-fired power plants. As such it will follow the evolu-tion of both carbon taxes and natural gas prices. Howmuch these parameters vary for given variations in gasprices and carbon taxes needs to be calibrated. Perform-ing this calibration goes beyond the purpose of the presentmanuscript and is left to future works. Here, we considerconstant ∆ p da and p da0 and restrict our investigations toa relatively short time window, until 2020.With these considerations, and under the assumptionsdescribed above, it is easy to qualitatively predict the evo-lution of electric revenues. Consider for instance a high-power pumped-storage (PS) hydroelectric plant. Its rev-enues directly depend on the difference between highestand lowest prices. From the discussion above, a PS plantsees its revenue first decrease in the initial stages of the en-ergy transition, where increased RES penetration reducesprice fluctuations. The revenue however increases later,once the RES penetration is such that it restores largefluctuations in residual loads and thus in electricity prices.In the upcoming sections we show that the intermediateperiod of reduced revenues for flexible productions dependson (i) the rate at which RES penetration increases, (ii) thechosen RES mix, and (iii) the rate at which must-run isreduced. To overcome the paradox of the energy transi-tion, one needs to chose scenarios such that these threeingredients, when combined, reduce the duration of theintermediate period with low revenues.
6. Future revenues by electricity production type
We investigate the future revenues of different electric-ity productions in Europe with the synthetic electricityprice of Eq. (5.1). We initially focus on the hydroelec-tric sector, which can provide flexibility of production andstorage capacities needed to integrate new RES into theelectric grid. We next turn our attention to general powerplants characterized by their annual number of operationhours.
The revenue G of a PS plant over a time interval t ∈ [ t i , t f ] is given by G = (cid:90) t f t i p da ( t ) · P PS ( t )d t, (6.1)where P PS ( t ) is the electric power produced ( P PS ( t ) > P PS ( t ) <
0) by the plant. Optimizing therevenue of PS plants means producing when p da ( t ) is largeand consuming when p da ( t ) is low. It therefore makes senseto assume that P PS and p da are strongly correlated. This Mon Tue Wed Thu Fri Sat Sun Mon Tue Wed-300-200-1000100200300 P r odu c t i on [ M W ] D a y - ahead p r i c e [ E UR / M W h ] Figure 6: The actual (solid black curve) and computed (dotted blackcurve) production of a typical Swiss PS plant and the day-aheadprice (blue curve). Negative production means pump load. Sources:Swissgrid and ENTSO-E (2015a). assumption is confirmed in Fig. 6, which shows the pro-duction of a Swiss PS plant and the day-ahead price for10 consecutive days in 2015. Neglecting losses for the timebeing, we write P PS ( t ) ∼ = π PS · [ p da ( t ) − ¯ p da ] , (6.2)where ¯ p da is the average price over the considered timeperiod and π PS is a prefactor linking prices to production.Eq. (6.2) guarantees that (cid:82) t f t i P PS ( t )d t = 0 as should befor a PS plant without loss. Inserting Eq. (6.2) into (6.1)and using Eq. (5.1), we get G ∼ = π PS ∆ p T Var[ R ] , (6.3)for the annual revenue with T = 8760 hours. This resultshows that the revenue of a lossless PS plant is propor-tional to the variance of the residual load. Eq. (6.3) for-malizes the relation qualitatively discussed at the end ofSection 5 between revenues of PS plants and fluctuationsof the residual load. We next investigate numerically the revenue of PS plantsin Germany and Spain in the period 2005-2020. Residualloads are calculated from 2015 data for the load and RESprofiles, the latter being scaled up from year to year tointerpolate linearly between the 2015 realized annual pro-duction and the planned 2020 annual production (ENTSO-E, 2015b). The evolution of the annual RES productionsis given in Figs. 7 (b) and 8 (b). The revenue is given by G = (cid:88) k p da ,k P PS ,k ∆ t = (cid:88) k p da ,k [ P t ,k − P p ,k ] ∆ t , (6.4)with the time step ∆ t used in the calculation – one hourin our case – and where we introduced P PS ,k = − P p ,k when P PS ,k < P PS ,k = P t ,k when P PS ,k >
0. Thepower profile is related to the evolution of the reservoirlevel S PS ,k , and for a PS plant with pump/turbine effi-ciency 0 ≤ η ≤ S PS ,k +1 = S PS ,k + [ ηP p ,k − η − P t ,k ]∆ t . (6.5)8
005 2007 2009 2011 2013 2015 2017 20190.70.80.911.11.21.3 N o r m a li z ed r e v enue R ES p r odu c t i on [ T W h ] (a)(b) Figure 7: (a) Normalized revenue (divided by the revenue of 2005in the scenario ”as long as possible”) of a German PS plant with η = 0 .
9, for different scenarios of must-run withdrawal (see Sec-tion 2) : ”as long as possible” (blue), ”interpolated path” (orange)and ”exact substitution” (red). (b) Evolution of WT (light blue)and PV (yellow) annual production in Germany. Dashed rectanglescorrespond to planned future evolution (ENTSO-E, 2015b).
Finally, the reservoir level must be positive but smallerthan its maximal level at all times, giving the constraint0 ≤ S PS ,k ≤ S maxPS , ∀ k. (6.6)Eqs. (6.4)–(6.6) govern the production of a PS plant. Wegenerate PS power profiles by maximizing the revenue G of Eq. (6.4) under the constraints of Eqs. (6.5) and (6.6).Using the actual day-ahead price, this procedure generatesa fictitious PS production profile given by the dotted blackcurve in Fig. 6, which is very close to the actual one (solidblack curve). We attribute the few discrepancies to thefact that our maximization of G in Eq. (6.4) is made withperfect advance knowledge of the load and RES produc-tions. This test substantiates our procedure for calculatingpower profiles and evaluating revenues of PS plants.We calculate revenues of PS plants from Eqs. (6.4)–(6.6) with the synthetic price of Eq. (1.1) and the usuallyreported efficiency of η = 0 . η <
1, cost less at higher must-run,which increases revenues. N o r m a li z ed r e v enue R ES p r odu c t i on [ T W h ] (a)(b) Figure 8: (a) Normalized revenue (divided by the revenue of 2005 inthe scenario ”as long as possible”) of a Spanish PS plant with η = 0 . The striking feature in Fig. 7 (a) is that, as expectedfrom the discussion in Section 5 together with Eq. (6.3),the drop in revenues corresponds to the acceleration ofthe penetration of PV, which reduces the mid-day resid-ual load peak. The fluctuations of the residual load godown, leading to reduced revenues through Eq. (6.3). Asthe penetration of PV further increases, so do the fluctu-ations of the residual load – one enters the stage depictedin Fig. 5 (c) and the revenues increase again. The drop inrevenues does not last long. The importance of PV in thisphenomenon becomes clear when comparing Fig. 7 (a) andFig. 8 (a). The latter figure displays no significant dropfor a Spanish PS plant. This is so, because the mix ofnew RES is clearly dominated by WT in Spain. Fluctu-ations in residual load are increased at all stages of thetransition, regardless of the chosen scenario for must-runreduction. Thus, from Eq. (6.3), revenues also always tendto increase.
We next consider conventional dam hydroelectric powerplants. The main difference with PS plants is that (i) con-ventional dam hydroelectric plants only produce and (ii)their reservoir is filled by natural water inflow. We modelthem slightly differently from PS plants. Their revenue isgiven by G = (cid:88) k p da ,k P D ,k . (6.7) There are of course also water inflows for PS plants, howeverthey are negligible against normal operation which typically fills andempties the reservoir in a matter of few days.
9e use the same synthetic price for p da ,k in Eq. (6.7) asfor the analysis of PS plants. The power profile P D ,k isrelated to the evolution of the reservoir level S D ,k , S D ,k +1 = S D ,k + [ I k − P D ,k ] ∆ t , (6.8)where I k is the power corresponding to water inflow intothe dam (rain- and snowfall, snow- and icemelt) at thetime interval k . The reservoir level must be positive butsmaller than the maximal storage capacity at all times,giving a condition similar to (6.6),0 ≤ S D ,k ≤ S maxD , ∀ k . (6.9)As for PS power plants, we determine the power profile P D ,k by maximizing the gain G in Eq. (6.7) for a typi-cal dam hydro power plant in the Alps. We take { I k } as the water inflow averaged over all Swiss dams, as ex-tracted from weekly dam energy content and production(Swiss Federal Office of Energy, 2016). A conventionaldam hydroelectric plant is characterized by its rated power P maxD , its storage capacity S maxD and the annual energy in-flow E D = (cid:80) k I k ∆ t . Relative revenue evolution thereforedepends on only two dimensionless parameters which wetake as S maxD /P maxD ∆ t ≡ N empty and E D /P maxD ∆ t ≡ N op ,giving the number of hours of operation at full power toempty the reservoir and to use all the annual energy in-flow respectively. We found that revenues depend onlyvery weakly on N empty , and therefore focus on the evolu-tion of revenues vs. N op . In multiannual average, damsannually produce their energy inflow E D , and in continen-tal Europe, this usually corresponds to N op ∈ [1000 , N o r m a li z ed r e v enue Figure 9: Normalized revenue of conventional dam hydroelectricplants with high ( N op = 1000 hours; solid lines) and low ( N op = 3000hours; dashed lines) power capacity in the Alps, with N empty = 1000hours and for the scenarios ”as long as possible” (blue), ”expectedpath” (orange) and ”exact substitution” (red) of must-run with-drawal. Fig. 9 shows the revenues of conventional dam hydro-electric plants with N empty = 1000 and N op = 1000 , P maxD , i.e. lowernumber N op of annual operation hours, see their revenuedecrease less than those with lower power, because thehigher the rated power, the easier it is to produce almostonly during peaks of financial opportunities. To understand better the trends discussed above, wefinally investigate different types of productions charac-terized only by the number of hours N op they operate atmaximal power P maxD per year with no further constraint.Accordingly, we consider four classes of power plants whichare (i) super-peaking plants, functioning N op = 1000 hoursper year at peak power, (ii) peaking plants N op = 2000,(iii) load-following plants N op = 5000 and (iv) base-loadplants N op = 8000. We calculate their revenues usingEq. (6.7), with the constraint (cid:80) k P D ,k = P maxD N op . Fig. 10shows the evolution of these revenues as the energy transi-tion unfolds in Germany, for our three scenarios for must-run reduction. We see first, that regardless of must-runreduction, peak production plants always have higher rev-enues and second, that faster must-run withdrawal leadsto smaller reductions in revenues. In particular, there isno significant decrease in revenue in the exact substitutionscenario, for which thermal plants are retired in direct pro-portion to the penetration increase of new RES.These results indicate that the currently very low elec-tricity prices in Europe (and the poor revenues of the hy-droelectricity sector) are due to an overcapacity of electric-ity production more than anything else. Thermal plantsare currently retired too slowly compared to the rate atwhich the penetration of new RES increases. We thereforeconjecture that the paradox of the energy transition canbe overcome by retiring thermal plants faster.
7. Conclusions
Our interest in this manuscript has been to investi-gate the conditions under which the energy transition inthe electric sector can proceed in Europe, without finan-cially jeopardizing flexible productions. This is a key is-sue, since increasing the penetration of new RES in the10
005 2007 2009 2011 2013 2015 2017 20190.80.911.1 N o r m a li z ed r e v enue N o r m a li z ed r e v enue N o r m a li z ed r e v enue (c)(b)(a) Figure 10: Normalized revenues of super-peaking plants (functioning1000 hours/year at maximal power; solid lines), peaking plants (2000hours/year; dashed lines), load-following plants (5000 hours/year;dotted-dashed lines) and base-load plants (8000 hours/year; dottedlines) under the three scenarios ”as long as possible” (panel a), ”ex-pected path” (panel b), ”exact substitution” (panel c) of must-runwithdrawal. continental European grid will eventually require sizeablepower reserves that can be mobilized fast and often tocompensate unavoidable, stochastic fluctuations in RESproductions. Such reserves already exist (an example ishydroelectricity), however, they are currently under strongfinancial stress because of very low electricity prices. It isof paramount importance to figure out how long this stresswill last and how it can be reduced in order to secure thefuture of flexible productions and with them, the harmo-nious unfolding of the energy transition. This is what wecalled ”overcoming the paradox of the energy transition”.To investigate this issue, we constructed a physico-economic model of the European grid. We observed astrong correlation between day-ahead electricity prices andresidual loads all over Europe, and from that correlation,we constructed an electricity price solely based on theresidual load, Eqs. (1.1) and (5.1). This price enabledus to investigate future revenues of various types of powerplants. We found that three ingredients determine the oc-currence, magnitude and duration of the paradox of theenergy transition: (i) the rate at which the energy tran-sition proceeds and RES penetration is increased, (ii) themix of new RES and (iii) the rate of must-run withdrawal.In particular, if the must-run is kept high as RES pen-etration increases, electricity prices go down with baseprices well below the marginal cost of any flexible pro-duction. The hydroelectric sector in continental Europe is currently suffering from very low electricity prices, andour results indicate that this situation is mostly due to sur-plus must-run capacity. The duration of this paradoxicalsituation will decisively depend on how fast surplus ther-mal production capacities are withdrawn to compensatefor the increased production from new RES. We advocatea faster withdrawal, for instance by substituting flexiblegas-powered plants for coal-fired plants, which would ad-ditionally reduce greenhouse gas emissions faster.The energy transition is accompanied by an increasedneed of production flexibility as it proceeds. Plants withlarge power will be needed more frequently, and it is ex-pectable that they will operate on a peak mode with 2000hours of operation per year or less. It may well be thatdifferent business plans will be developed for such plants,with different financial tools to reward not the energy pro-duced, but the ancillary services provided. How such in-centives will be introduced remains speculative. Our re-sults suggest that, even without them, peak and super-peak power plants should soon benefit again from im-proved financial conditions.As a final comment we note that different scenarioswith different consumption curves can be investigated withthe model presented above. In particular, active demandresponse can be incorporated into the model, as we discussin Appendix A.
Acknowledgment
This work has been supported by the Swiss NationalScience Foundation. We thank T. Coletta, R. Delabaysand M. Tyloo for their useful comments on the manuscript,M. Emery and M. Schmid for comments and Swissgriddata and R. Whitney for proofreading the manuscript.
Appendix A. Incorporating active demand response
We sketch how active demand response (ADR) can beincorporated into our model. With ADR, the residual loadis given by R ( t ) = L ( t )+ δL ( t ) − P PV ( t ) − P W T ( t ) − P MR = R ( t )+ δL ( t ) , (A.1)where changes in the load profile due to ADR are includedin δL ( t ) and R is the residual load without ADR, as inEq. (2.1). ADR can be deployed for various reasons, forinstance to reduce electricity costs of end users or to miti-gate load fluctuations on the distribution network. In bothinstances, ADR tries to reduce variations in the residualload, and we incorporate this goal in an optimizing pro-cedure which we briefly describe. For the sake of simplic-ity, we do not incorporate specific load constraints such ascomfort temperature intervals for ADR with thermostat-ically controlled loads. The only constraint on the ADRprofile is (cid:12)(cid:12) δL ( t ) (cid:12)(cid:12) < δL max , ∀ t , (A.2)11here δL max is the maximal ADR power. We further as-sume that the annual consumption remains unchanged, t f (cid:90) t i δL ( t )d t = 0 . (A.3)Recent estimates of the potential of ADR indicate thatonly a fraction of the total consumption can be shifted, t f (cid:90) t i (cid:12)(cid:12) δL ( t ) (cid:12)(cid:12) d t ≤ σ t f (cid:90) t i L ( t )d t , (A.4)with σ (cid:39) .
01 giving the maximal fraction of the totalconsumption that can been shifted, while the maximalADR power δL max is about 10 % of the maximal load L max (roughly corresponding to 7 GW in Germany) (Gils,2016). These numbers may seem rather small, howeverthey have been obtained assuming a broad load participa-tion in ADR (Gils, 2016).Our procedure is to compute the ADR profile that min-imizes the fluctuations of the residual load,min δL (cid:2) Var( R ) (cid:3) = min δL (cid:2) Var( L + − L − + R ) (cid:3) , (A.5)where we defined L ± ( t ) = max[0 , ± δL ( t )]. We linearlyincrease σ and δL max /L max from 0 in 2015 to σ = 0 .
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