Who bears the risk? Analyzing the strategic interaction between regulators and investors when setting incentives for renewable electricity

Who bears the risk? Analyzing the strategic interaction between regulators and investors when setting incentives for renewable electricity Peio Alcorta Iglesias MASTER IN ECONOMICS: EMPIRICAL APPLICATIONS AND POLICIES University of the Basque Country UPV/EHU Faculty of Economics and Business Advisors: Maria Paz Espinosa & Cristina Pizarro-Irizar July 30, 2020 Acknowledgments First of all, I would like to thank professors Maria Paz Espinosa and Cristina Pizarro-Irizar for the various opportunities that they have offered me beyond the writing of this thesis. Now, on a much deeper and personal level, I want to thank you Odra. Your support has been essential during the very complicated and strange months in which this thesis has been carried out. You are the reason why these last four months have been bearable. Thank you from the bottom of my heart. Abstract Energy policies for promoting investment in renewable energy sources have become crucial for deploying different green energy technologies. Depending on their design, the conventional incentive systems assign the risk to either the policymaker or the investor, affecting the strategic interaction between them when setting a price for the subsidy. Moreover, Feed-in Tariffs, which were the principal subsidy scheme used in Spain, were removed in 2013, mainly because their design led to an unbearable deficit. Farrell et al. (2017), combining option pricing theory and game theory, propose an incentive system for Irish Feed-in Tariffs in which both parties would share the risk. Building on this approach, we develop a methodology to evaluate different optimal incentive schemes for Spain and present an application for 2013 and 2019. We perform an extensive numerical analysis to determine how the different proposals would work for Spain. Keywords: Renewable Energy; Feed-in Tariff; Efficient Policy; Green Energy; Energy Policy; Energy Economics; Option Pricing. Contents 1 Introduction 1 2 Mathematical Model 4 2.1 Preliminaries ........................................... 4 2.2 Electricity market price model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Game theoretic approach for wind energy investment model . . . . . . . . . . . . . . . . . 8 2.4 Expected price of different FiTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1 NoSubsidy ........................................ 10 2.4.2 FixedTariff........................................ 11 2.4.3 ConstantPremium.................................... 11 2.4.4 SharedUpside ...................................... 12 2.4.5 Cap&Floor ....................................... 13 3 Numerical Application 14 3.1 Estimation of how the drift and volatility of the VWAP depend on the installed wind power 15 3.2 Calibration ............................................ 19 3.3 Results............................................... 19 3.3.1 Results for the year 2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3.2 Results for the year 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4 SensitivityAnalysis........................................ 28 3.5 Discussion............................................. 29 4 Conclusions 30 References 32 Appendix A: Solving Equation 10 i Appendix B: Construction of the dataset iv Appendix C: Minimum or Maximum? xi 1 Introduction Nowadays, one of the biggest concerns worldwide is to achieve a sustainable and clean energy regime. To a greater or lesser extent, most developed countries worldwide have set the goal of fighting global pollution and stopped relying on fossil fuel reserves, increasingly limited and responsible for the emissions of polluting gases into the atmosphere. Although significant advances have been made in this regard, there is still a long way to reach a fully green and sustainable energy scenario. Since the deployment of Renewable Energy Sources (RES) is still more costly than that of other conventional sources, a public subsidy is often required to create a feasible and appetizing investment environment. There exist many support schemes to incentivize green energy penetration. These schemes include, for example, Renewable Portfolio Standards (i.e., a mechanism that sets an obligation on electricity suppliers to produce a specified fraction of their electricity from renewable energy sources). These Renewable Portfolio Standards might be accompanied by Tradable Green Certificates, which are tradable assets proving that electricity has been generated by renewable energy sources. Green certificates are issued to RES producers, who can trade them to conventional energy suppliers so they can fulfill the quota established in the Renewable Portfolio Standard. In exchange, renewable suppliers receive extra revenue, and the market finds the most efficient way to meet these goals. Another support scheme can be partial, or even full exemption from some taxes and levies to green energy producers. However, Feed-in Tariff (FiT) schemes have become the preferred renewable energy support mechanism in many markets, as they provide greater certainty of remuneration for investors. Nevertheless, their main drawback is the huge costs they usually involve for the regulator, especially if they are not properly designed. That is the main reason why they were abandoned in Spain in 2013 [1]. There are two variants of Feed-in Tariff policy schemes that have been widely used, particularly in Spain: a fixed FiT, and a Feed-in Premium (FiP). A fixed FiT is a mechanism allowing the RES producers to sell the electricity they supply at a fixed price for a specific period. In a FiP scheme, the payment that RES suppliers receive is based on a constant premium offered above the market-clearing price. Until 2013, Spanish RES producers had the option to choose between fixed FiT and FiP subsidy schemes. Indeed, the fixed tariff has been the most widely used FiT design. However, the FiP has been increasingly utilized in Spain, mainly for onshore wind. Since July 2013, the Spanish government suppressed both fixed FiT and FiP subsidies, being renewable energy auctions the current mechanism used for green energy promotion [2]. The benefits of FiT subsidies for renewable energy penetration have been of significant importance, and they may still play a role as long as they are carefully redesigned. Indeed, approximately 64% of today’s global wind power has been promoted through this type of mechanism [3]. Figures 1 and 2 show how both schemes work. The yellow line represents the evolution of the electricity market price, and the green line shows the evolution of the revenue per MWh that a renewable energy investor receives under each policy. Therefore, the green area represents the total cost of the subsidy for the policymaker, whereas the total area (green+yellow), corresponds to the total profits that the investors will receive. As we can see in Figure 1, a fixed tariff regime removes investor exposure to low market prices, being the policymaker the one who bears the risk of market price variability. On the contrary, constant premium policies remove the policymaker’s risk. As shown in Figure 2, under a FiP scheme, the green area is independent of the market price [4]. Since the premium (denoted as Xin the figure) is independent of the stochastic market prices, the policymaker has certainty about the cost per MWh that the public subsidy will entail when designing the policy. However, under a FiP scheme, investors are exposed to the full impact of market price fluctuations. It is a well-documented fact that the effectiveness of the FiTs has been attributed to the reduced risk they entail for investors [5]; therefore, transferring all the risk to them might result counterproductive, leading to the conclusion that adequate management of these risks is of capital importance when designing public subsidy policies to promote RES penetration. 1 Figure 1: Fixed Feed-in Tariff scheme Figure 2: Constant Feed-in Premium scheme In 2011 there were approximately 21,000 MW of wind capacity deployed in Spain. That same year, the Spanish government set the target to achieve a total installed capacity of 35,750 MW by the year 2020 (Plan de Energ´ıas Renovables 2011-2020) [6]. As Figure 3 shows, until FiT subsidies were eliminated in 2013, the level of new installed capacity was on the right track, but from then on, it stopped abruptly. Figure 3: Evolution of the installed wind capacity in Spain (source AEE [7]) The research question that we want to answer in this thesis is whether incentive schemes that share market price exposure between regulators and investors can improve upon the usual FiT and FiP. We provide an analytical specification of incentive structures that share market price exposure between regulators and investors and present an application for the Spanish market. Indeed, our results indicate that there are risk-sharing incentive schemes that dominate FiT and FiP: they allow reaching the same investment level at a lower cost. Our results have interesting policy implications for future renewable energy regulation: incentive schemes have to be carefully designed, taking into account risk sharing. Following the approach proposed by Farrell et al. [8], we design subsidy schemes that share out the risks associated with the stochastic nature of electricity price fluctuations among both investors and policymakers. In this context, we analyze schemes based on FiT but incorporating adjustable degrees of market price exposure for investors and regulators. Our analysis will depart from the assumption that the stochastic evolution of the annual electricity price follows a random walk that can be satisfactorily described as a Geometric Brownian Motion (GBM) process. Consequently, the evolution of the payoff that a renewable energy investor receives will follow the Black-Scholes equation but with a different terminal condition, which will depend on the particular design of the FiT in place. Even though FiTs are not financial assets, nor can they be bought or sold, the Black-Scholes equation’s appearance is due to the fact that the payoffs of the corresponding subsidy will depend on the stochastic electricity market price. This fact allows us to obtain analytical solutions for the model, which provides a path of efficient combinations 2 of the FiT parameters. By choosing different efficient combinations of these parameters, we can manipulate both policymaker’s and investor’s risk exposure levels. Accurately quantifying and distributing these risks can be an essential stimulus to incentivize green energy penetration. This is the main contribution of Farrell et al. (2017), who proposed their methodology for a feasible Irish RES deployment scenario. In a nutshell, our methodology can be summarized as follows: •First, for every FiT design under consideration, we solve the stochastic model to find analytical solutions for the evolution of each policy scheme’s expected payoffs and costs. •Then, following Farrel et al. (2017), we model the risk-sharing FiT design problem as a strategic leader game, where the policymaker (leader) takes into account the strategic response of the investors (followers). Considering the expected evolution of payoffs and costs previously calculated, the regulator chooses the optimal risk-sharing FiT parameters that incentivize the desired quantity of RES deployment. •Next, we perform a numerical analysis to determine how these proposals would work in Spain for wind energy deployment. We calibrate the model to the Spanish electricity market in 2013 (the last year with FiTs), and 2019 (the last year with available data). •Since some of the needed parameters for the Spanish market do not appear in the literature, we have to estimate them on our own. In particular, the percentage drift and volatility of the annual volume-weighted average price (VWAP) for wind electricity, which depends on the amount of wind capacity installed. For that purpose, we build a dataset with the hourly energy production by technology and the hourly matching price of electricity, for every hour from 01/01/2014 to 31/12/2019. •With the estimated parameters, we simulate potential electricity prices evolution according to our model, obtaining how the different proposals would work for Spain. •Finally, we perform a sensitivity analysis, changing several parameters and measuring the impact of those variations on the relevant predictions. 3 2 Mathematical Model In this section, we present a model for designing optimal pricing rules for the different subsidy schemes we studied. We draw together the separate fields of FiT policy design, game theory, and stochastic financial calculus. 2.1 Preliminaries One of the cornerstones in the study of financial markets is the so-called Efficient Market Hypothesis. Although there exist several different formulations of this hypothesis, all of them share the following basic features [9]: 1) Markets respond immediately to any new information about an asset price. Corrections of prices are instantaneous, leaving the participants no room for arbitrage. 2) Asset prices reflect all available information. Consequently, it is impossible to “beat the market” consistently on a risk-adjusted basis, since market prices should only react to new information. One of the initial purposes of this hypothesis was to provide arguments in favor of the feasibility of the Random Walk Conjecture, which states that market prices evolve according to a random walk (price changes cannot be predicted). It is easy to see that if the Random Walk Conjecture holds, then, as a direct consequence, the Efficient Market Hypothesis must hold. Suppose that at time t, the asset price is denoted by St. After an infinitesimal time interval dt, the asset price will change by an amount dSt. The most common way to characterize the corresponding change on the asset price is to decompose it into two different contributions:1 dSt=a(St, t)dt +b(St, t)dWt(1) On the one hand, there is a deterministic contribution given by a function a(St, t). On the other hand, we have a random contribution to the asset price change in response to unexpected external effects, given by a deterministic function b(St, t), and the differential dWt. The term dWtcontains the randomness, and it is modeled as a Wiener process. A random process {Wt}T t=0 ={W0, W1..., WT}is a Wiener process when the following properties are met [10]: •i) Wt∼N(0, t)−→ W0= 0 •ii) {Wt}T t=0 has independent increments: P(Wt−Ws|Ws) = P(Wt−Ws) •iii) For 0 ≤s<t: (Wt−Ws)∼N(0, t −s) where N(x, y) represents the normal distribution with mean xand variance y. From these properties, it follows that a Wiener process is also a Markov process.2Since we are usually more interested in the relative change of an asset price than in its absolute change, it may be sometimes more convenient to express Eq.(1) as 1In practice, it is evident that the time intervals we handle when analyzing a stochastic process, are always discrete. However, the continuous formulation is usually made for analytical convenience. Once we have solved the model for continuous time, the discrete approximation can always be made by taking appropriate time intervals. 2A sequence of random variables {Xt}tforms a Markov chain (or process) if: P(Xt+1 =x|X0, ..., Xt) = P(Xt+1 =x|Xt); that is, if given the present, the future and the past of the sequence are independent. In other words, if all the information of the past history of that variable is captured in the present state [11]. 4 dSt St =µ(St, t)dt +σ(St, t)dWt(2) If the functions µ(St, t) = µ∈R, and σ(St, t) = σ > 0 are constant, we obtain a Geometric Brownian Motion process (GBM) [12]: dSt St =µdt +σdWt(3) It is easy to see that the expected value and the variance 3of the LHS in Eq.(3) are given by: E[dSt/St] = µdt, and V ar[dSt/St] = σ2V ar[dWt] = σ2dt. The coefficients µand σ, are usually called the drift and the volatility of the process, respectively. Once we have characterized the GBM, it is time to state both a fundamental theorem and one of its corollaries, which will allow us to handle the randomness in the model. Theorem. (Feynman-Kac): Consider the stochastic differential equation (1): dSt0=a(St0, t0)dt0+b(St0, t0)dWt0(4) Let h(·)be a function of the stochastic variable St0. Let t0∈[0, t]be given, and the final time t > 0 fixed. Define the expected value of h(St)over the period [0, t]as ρ(S, t0) = E[h(St)|St0=S] (5) Then, ρ(S, t0)satisfies the following partial differential equation: ∂ρ ∂t0+a(S, t0)∂ρ ∂S +b(S, t0)2 2 ∂2ρ ∂S2= 0 (6) together with the final condition ρ(S, t) = h(S),∀S(7) Proof: See Shreve (2004) [13]. Equation (6) is known as the Kolmogorov Backward Equation.4When we know for sure that the system will be in a certain state h(St) at some certain time in the future (t0=t), the Kolmogorov backward equation describes the probability of being in a state Sat an earlier time (t0< t) [14]. Corollary. Consider the stochastic differential equation (3) describing the Geometric Brownian Motion: dSt0=µSt0dt0+σSt0dWt0(8) Let h(·)be a function of the stochastic variable St0, and let r to be a given constant. Let t0∈[0, t]be given, and the final time t > 0fixed. Define the expected discounted value of h(St)over the period [0, t] as f(S, t0) = E[e−r(t−t0)h(St)|St0=S] (9) Then, f(S, t0)satisfies the following partial differential equation: 3From the third property of the Wiener process, it follows that: dWt= lim ∆t↓0{Wt+∆t−Wt}=Wt+dt −Wt∼N(0, dt) 4The Kolmogorov Backward Equation is a diffusion type partial differential equation that arises in the theory of continuous-time Markov processes. Even though we arrived at this equation from the Feynman-Kac theorem, it was first studied by the great Russian mathematician Andrey Kolmogorov [15], way before the Feynman-Kac formula was introduced by the mathematician Mark Kac, and the great theoretical physicist Richard Feynman while he was studying path integrals in quantum mechanics [16]. 5 FB,t(St) = X(31) We find that the expected discounted profits and costs at time tare given by e−rtE[PB,t] = e−rt(X+S0eµt) (32) e−rtE[FB,t] = Xe−rt (33) Following the same procedure, we find the optimal solution for the constant premium X: X(opt)= C− TF X t=1 S0e(µ−r)tt∂µ ∂QGt+∂Gt ∂Q Q=QI T1 X t=1 e−rt ∂Gt ∂Q Q=QI (34) 2.4.4 Shared Upside The payoffs that an investor receives at (t≤T1) are described by PC,t(St) = max{KC, ω(St−KC) + KC}=(KCSt< KC ω(St−KC) + KCKC≤St (35) where ω∈[0,1] represents the share of the market upside received by the investor, and KCrepresents the price floor. Under this scheme, the policymaker’s costs will be FC,t(St) = max{KC−St,0}+ (ω−1) max{St−KC,0}=(KC−StSt< KC (ω−1)(St−KC)KC≤St (36) If ωhappened to be equal to one, then we would have a FiT under which the investor charges the market price, but having a guaranteed minimum price, in case the market price is lower than this floor. On the contrary, if ωhappened to be zero, we would obtain a fixed FiT with a fixed price KC. As before, we find the expected discounted profits and costs at each period. e−rtE[PC,t] = e−rt KC(1 −ωΦ(d2)) + ωS0eµtΦ(d1)(37) e−rtE[FC,t] = e−rt KC−S0eµt +ω(S0eµtΦ(d1)−KCΦ(d2))(38) where Φ is the cumulative distribution function of the standard normal distribution, and d1and d2 are defined as follows: d1(K, t) = log S0 K+µ+σ2 2t σ√td2(K, t) = log S0 K+µ−σ2 2t σ√t(39) 12 we find the condition for a feasible optimal combination of KCand ω: K(opt) C(ω) = C− T1 X t=1 ωΦ(d1(K(opt) C, t))S0e(µ−r)tt∂µ ∂QGt+∂Gt ∂Q Q=QI− TF X t=T1+1 S0e(µ−r)tt∂µ ∂QGt+∂Gt ∂Q Q=QI T1 X t=1 1−ωΦ(d2(K(opt) C, t))e−rt ∂Gt ∂Q Q=QI (40) As it can be seen in equation (40), since both d1and d2depend on KC, we cannot solve the equation for KCexplicitly. Given a certain value for ω, the previous implicit equation may be solved by numerical methods. By performing an iterative procedure, the solution finally converges to the desired optimal value K(opt) C. Conversely, given a value of KC, we could redo the procedure for obtaining an optimal value for ω(opt). It could be proved, and we will see it later in our practical study, that equation (40) describes a unique locus of efficient pairs of KCand ω, with a single efficient ωfor each value of KC(and the other way around). Moreover, it could be proved that there is an inverse relationship between these optimal values of ωand KC. 2.4.5 Cap & Floor The payoff that an investor receives at (t≤T1) are described by PD,t(St) = max{KD,min{St,C}} =     KDSt< KD StKD≤St< C C C ≤St (41) where KDand Crepresent the price floor and cap, respectively. Under this scheme, the policymaker’s costs will be FD,t(St) = max{K−St,0}+ max{C−St,0}=     KD−StSt< KD 0KD≤St< C C−StC≤St (42) As always, we find the expected discounted profits and costs at each period:9 e−rtE[PD,t] = e−rt KD(1 −Φ(d2)) + S0eµt(Φ(d1)−Φ(d3)) + CΦ(d4)(43) e−rtE[FD,t] = e−rt KD(1 −Φ(d2)) −S0eµt(1 −Φ(d1)) −S0eµtΦ(d3) + CΦ(d4)(44) where d3and d4are defined as follows: d3(C, t) = log S0 C+µ+σ2 2t σ√td4(C, t) = log S0 C+µ−σ2 2t σ√t(45) Proceeding as in the previous case, the condition for a feasible optimal combination of KDand C: 9For simplicity, when computing all the derivatives for both Shared Upside and Cap & Floor schemes, we neglect the variation of Φ (di(µ(Q))) with respect to Q, where i∈ {1,2,3,4}. 13 K(opt) D(C) = C− T1 X t=1 S0e(µ−r)tΦ(d1(K(opt) D, t)) −Φ(d3(C, t))t∂µ ∂QGt+∂Gt ∂Q Q=QI T1 X t=1 1−Φ(d2(K(opt) D, t))e−rt ∂Gt ∂Q Q=QI + + − T1 X t=1 Ce−rtΦ(d4(C, t))∂Gt ∂Q Q=QI− TF X t=T1+1 S0e(µ−r)tt∂µ ∂QGt+∂Gt ∂Q Q=QI T1 X t=1 1−Φ(d2(K(opt) D, t))e−rt ∂Gt ∂Q Q=QI (46) Again, it is not possible to explicitly solve the previous equation for KD. Instead, for a given value of the cap C, we can solve the implicit equation (46) iteratively until it converges to the optimal solution K(opt) D. Conversely, given a value for KD, we could solve it to obtain an optimal value of C(opt). In any case, equation (46) describes the locus of efficient combinations of floor and cap. As in the shared upside regime, there is an inverse relationship between these two policy parameters. A lower floor implies a higher efficient cap, and the other way around. As a consequence, there will be an inverse relationship in investor’s and policymaker’s exposure to market price variability. 3 Numerical Application We apply the mathematical model developed in the previous section to analyze how each of the tariffs under consideration would work in Spain in the years 2013 and 2019. We have chosen these two years because, on the one hand, 2013 was the last year where the government supported RES deployment with FiTs, and on the other hand, 2019 is the last year with available data. First of all, in order to analyze each of the two different scenarios, we have to calibrate the several parameters of the model. In some cases, we find the corresponding values in the literature, but in many other cases, we have to estimate them ourselves. The price we must pay for obtaining analytical solutions is that we must ignore some technical issues. For instance, as we mentioned in section 2.2, electricity prices have to be lower than e180.3/MWh by law. However, this requirement cannot be included in the analytical model, and because of that, and due to the high value of the drift µwe will use, there is a good chance that in the last periods, this requirement does not hold. Thus, we must clarify that given some of the assumptions and technical limitations present in the methodology, the objective of the following numerical analysis is not to obtain exact figures regarding the benefits and costs that each tariff would entail; this exercise would require a much more thorough and meticulous analysis, and above all, with much more available data. Rather, this exercise’s primary purpose is to show how the intermediate schemes behave compared to the usual FiT in the scenario that we try, within the limitations, to be similar to the Spanish market. In the case of µ, we are not only interested in a particular value of the annual percentage drift of the VWAP, but rather on its functional form depending on the total amount of capacity deployed. As we mentioned earlier, this is a key feature on our model, since µand thus, also its derivative ∂µ ∂Q  are endogenous to the investor’s optimization problem. In their study, Farrel et al. estimated the corresponding functional form in the case of Ireland using the study previously done by Doherty and O’Malley (2011) [22], in which they estimated the projected future wind weighted market prices in Ireland as a function of the deployed wind capacity. Since there is no such study done for Spain, we devise a different methodology in order to estimate µ(Q). It is worth mentioning that Farrel et al. treated σas a constant; however, in order to be more rigorous, we also estimate how the volatility depends on wind energy penetration. As we will see, the estimated functions µ(Q) and σ(Q) are almost, but not perfectly, linear, at least in the range of wind deployment we are interested in. For that reason, we estimate both functions assuming a second-order polynomial functional form in Q=Q0+QI. 14 3.1 Estimation of how the drift and volatility of the VWAP depend on the installed wind power For the estimation of these two functions, as well as the estimation of the VWAP, we use a methodology devised by ourselves collecting vast quantities of data and performing some regressions in R. We include the R code used to construct the dataset in Appendix B. We use data from Red El´ectrica de Espa˜na (REE) and OMIE. The data from REE includes the amount of energy generated by each technology in MWh: wind, solar, hydraulic, nuclear, coal, combined cycle, rest of special regime (biomass, renewable thermal, etc.), fuel/gas, international exchanges and Balearic bond, for every single hour ranging from 0101-2014 to 12-31-2019 (one file per day) [23]. Our dataset also includes the matching price of electricity at every single hour (pricei), which is taken from OMIE [24] (one file per day), and the nominal wind power installed up to a given year (poweri), which is taken from AEE [7]. The aim is to get an estimation of the annual drift (µ) and volatility (σ) as a function of the installed wind power. Let us denote for each year (t) the predicted annual VWAP by pt. We are considering discrete annual timesteps, thus: dt ≈∆t= 1. Hence, recalling the properties followed from equation 3 for the expected value and variance, we get: E[dpt/pt] = µ, and V ar[dpt/pt] = σ2, respectively. Thus, the GBM parameter estimation is made as follows [25]: µ(Q) = 1 5 2019 X t=2015 pt(Q)−pt−1(Q) pt−1(Q)(47) σ(Q) = v u u t1 4 2019 X t=2015 pt(Q)−pt−1(Q) pt−1(Q)−µ(Q)2 (48) First, we need to estimate the VWAP (pt) as a function of the total wind power (Q). As a first step, we perform a simple linear regression model for the hourly electricity price using the 52,584 observations in our dataset (one for each hour). pricei=β0+β1windi+β2solari+β3hydraulici+β4nucleari+β5coali +β6combinedi+β7speciali+β8fuelgasi+β9exchangesi+β10baleari + 24 X j=2 αjhour(j) i+ 7 X m=2 χmday(m) i+ 12 X l=2 ζlmonth(l) i+ 2019 X k=2015 ξkyear(k) i+εi (49) where {hour(j)}24 j=1 is a set of dummy variables indicating the hour of the day corresponding to the observation: for example, hour(2) itakes the value one if the observation icorresponds to the hour 2:00, and zero otherwise. In the same way, {year(k)}2019 k=2014 indicates the corresponding year: for example, year(2019) itakes the value one if the observation ibelongs to the year 2019 and zero otherwise, and so on. Similarly, {month(l)}12 l=1 indicates the month, and {day(m)}7 m=1 indicates the day of the week (1= Monday, ... , 7=Sunday) corresponding to each observation. The obtained results are shown in Table 2. As we can see, almost every variable is statistically significant, and in most cases, with p-values lower than the working precision of the machine. Another thing to note is that our OLS model explains 77.3% of the variation of our dependent variable (R2= 0.773). As we can tell from the results, there is an inverse relationship between the hourly price and the wind energy generated. That is not surprising since we already discussed the merit-order effect [26]. The same applies to solar production. On the other hand, at peak hours, production from conventional sources is higher, and so are electricity prices. With the estimated coefficients, we can get a prediction of the hourly prices (\ pricei). 15 Now, let us suppose that for a given observation, instead of having the corresponding wind power installed at that moment (poweri), we have a different given power (Q1), while everything else remains unchanged. Then, the only difference would be in the wind energy production at that hour (windi). For the sake of simplicity, and because we are not trying huge wind power differences for the new capacity, assuming a linear relationship with the corresponding new generation might be justified: windi(Q1)≈Q1 poweriwindi(poweri). That is, it seems reasonable to expect that if the wind capacity Q1 had been, for example, a 10% higher than the actual capacity poweri, then the wind production would also have been a 10% higher. Using the parameters we estimated with the regression model in Eq.(49), we predict for an arbitrary installed wind capacity Q1, an hourly price for every observation in our dataset: \ pricei(Q1) = b β0+b β1 Q1 poweri windi+b β2solari+b β3hydraulici+b β4nucleari+b β5coali +b β6combinedi+b β7speciali+b β8fuelgasi+b β9exchangesi+b β10baleari + 24 X j=2 bαjhour(j) i+ 7 X m=2 bχmday(m) i+ 12 X l=2 b ζlmonth(l) i+ 2019 X k=2015 b ξkyear(k) i (50) Once we obtain the predicted hourly prices \ pricei(Q1)for a given capacity of our choice, the next step is to obtain the corresponding annual volume-weighted average price pt(Q1): pt(Q1) = X year(t) i=1 \ pricei(Q1)·windi X year(t) i=1 windi (51) Q µ(Q)σ(Q) 10000 0.07128004 0.2756571 12500 0.07185035 0.2770892 15000 0.07242749 0.2785360 17500 0.07301158 0.2799975 20000 0.07360274 0.2814742 22500 0.07420110 0.2829661 25000 0.07480679 0.2844736 27500 0.07541993 0.2859969 30000 0.07604066 0.2875363 32500 0.07666911 0.2890919 35000 0.07730542 0.2906641 37500 0.07794975 0.2922531 40000 0.07860222 0.2938592 Table 1 Then, we obtain µ(Q1) and σ(Q1) applying equations (47) and (48), respectively. If we repeat the procedure trying different values for the new wind capacity, we can obtain lists of values {µ(Qλ)}λ, and {σ(Qλ)}λfrom which we can make a nonlinear fit to estimate its functional form, at least, in the range of Qin which we are interested. For example, the interval Q∈[10000,40000] might be reasonable as an interval of interest. We try different values belonging to this interval. The values Qthat we used and the results for µ(Q) and σ(Q) are shown in Table 1. Since the values we obtain in both cases suggest functional forms approximately linear, following Taylor’s theorem, including terms up to second order may be enough to successfully approximate both functions. From these results, we estimate µ(Q) and σ(Q) as second order polynomials in Q=Q0+QI: µ(Q) = 0.069125 + 2.09513 ·10−7Q+ 6.83117 ·10−13Q2(52) σ(Q)=0.27021 + 5.31872 ·10−7Q+ 1.47851 ·10−12Q2(53) Plotting the obtained values shown in Table 1, together with the functional forms that we estimated in Eq.(52) and Eq.(53), we see that we got a very nice fit in both cases. 16 Figure 6: Adjustment for µ(Q) Figure 7: Adjustment for σ(Q) As we can observe in Figure 6, although the hourly prices, and hence the annual VWAP decrease as the installed wind capacity increases (being consistent with the merit-order effect), the annual drift of the VWAP is an increasing function of the deployed wind capacity. The same thing applies to the annual volatility, as shown in Figure 7. •Note: It is important to clarify that the development of the methodology we have just discussed has several significant limitations. The most notable one might be the small number of years used to determine the parameters. This is due to the fact that the available data in which the generation by technology is completely broken down begins in 2014. On the other hand, our initial purpose of determining how the amount of installed wind energy affects wholesale electricity prices was to simulate the daily market matching. As we have described in section 2.2, the matching of the market price follows a deterministic method, knowing all the daily sale and purchase offers, and identifying which of those offers come from wind units; we would have a method that would allow us to reliably determine the price of the daily market depending on the amount of the installed wind capacity. Unfortunately, the codes that identify the wind units in OMIE and REE do not coincide at all. Therefore, this alternative methodology cannot be carried out. If this changes in the near future, the parameter determination process would improve substantially. 17 Table 2: Results of the linear regression model Dependent variable: price wind −0.000233∗∗∗ (0.000019) solar −0.000563∗∗∗ (0.000046) hydraulic 0.000920∗∗∗ (0.000023) nuclear 0.002260∗∗∗ (0.000054) coal 0.004138∗∗∗ (0.000024) combined 0.001001∗∗∗ (0.000024) special 0.001450∗∗∗ (0.000040) fuelgas 0.013194∗∗∗ (0.000535) exchanges 0.001011∗∗∗ (0.000029) balear 0.002703∗∗∗ (0.000858) year2015 4.399542∗∗∗ (0.140081) year2016 0.602026∗∗∗ (0.132652) year2017 9.986042∗∗∗ (0.138793) year2018 17.950020∗∗∗ (0.130002) year2019 18.105060∗∗∗ (0.163003) hour2 −1.955861∗∗∗ (0.219181) hour3 −3.481786∗∗∗ (0.222104) hour4 −4.102226∗∗∗ (0.223917) hour5 −4.699547∗∗∗ (0.224572) hour6 −4.336743∗∗∗ (0.223236) hour7 −2.927000∗∗∗ (0.218837) hour8 −1.196431∗∗∗ (0.218881) hour9 −0.788441∗∗∗ (0.224914) hour10 0.290636 (0.238500) hour11 0.813615∗∗∗ (0.257489) hour12 0.787607∗∗∗ (0.271186) hour13 0.892097∗∗∗ (0.279696) hour14 0.860672∗∗∗ (0.281255) Observations 52,584 R20.773808 Adjusted R20.773572 Residual Std. Error 7.193956 (df = 52528) F Statistic 3,267.271000∗∗∗ (df = 55; 52528) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 Dependent variable: price hour15 0.300588 (0.275495) hour16 −0.664138∗∗ (0.267760) hour17 −1.039772∗∗∗ (0.258639) hour18 −0.270835 (0.248545) hour19 0.871813∗∗∗ (0.244286) hour20 1.696003∗∗∗ (0.244304) hour21 1.849687∗∗∗ (0.246245) hour22 2.021346∗∗∗ (0.243832) hour23 1.257100∗∗∗ (0.230010) hour24 −0.197932 (0.220333) day2 −1.125314∗∗∗ (0.118240) day3 −1.367910∗∗∗ (0.118420) day4 −1.163725∗∗∗ (0.118053) day5 −0.814275∗∗∗ (0.117670) day6 0.324673∗∗∗ (0.123201) day7 −0.290607∗∗ (0.134090) month2 −5.677778∗∗∗ (0.160314) month3 −2.855130∗∗∗ (0.169959) month4 −1.656027∗∗∗ (0.177717) month5 2.699987∗∗∗ (0.190722) month6 1.451458∗∗∗ (0.182082) month7 −1.613603∗∗∗ (0.182219) month8 −0.050674 (0.182902) month9 −0.143878 (0.169652) month10 4.706360∗∗∗ (0.169246) month11 3.971814∗∗∗ (0.176132) month12 2.020612∗∗∗ (0.158199) Constant −1.410498∗∗ (0.597689) Observations 52,584 R20.773808 Adjusted R20.773572 Residual Std. Error 7.193956 (df = 52528) F Statistic 3,267.271000∗∗∗ (df = 55; 52528) Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01 18 3.2 Calibration The installation target (QI) is taken as the difference between the objective set by the Spanish government and the total power installed up to that year (Q0): in 2011, they established the objective of 35.75 GW for 2020 in Plan de Energias Renovables (PER) [6], while the objective of 50.33 GW for 2030 is defined in Plan Nacional Integrado de Energ´ıa y Clima (PNIEC) [27]. On average, wind turbines are operational for 20 years [28]. The duration of the fixed-FiT and constant-FiP given by the Spanish government was 20 years as well [29]. We obtained the VWAP of the initial year (S0) with the dataset we used for the methodology we have just explained in 3.1, but expanding it, in order to include 2013 too (indeed, we included every hour from 2007 to 2019). For a given year, we took the amount of wind energy generated at every single hour of that year, and the market matching price of that hour. Then, each hour’s price is weighted accordingly to the amount of wind electricity generated at that same hour. The capital and operative costs of onshore and offshore wind energy are different. In general offshore wind-farms are much more expensive both to build and to maintain than the onshore ones. For that reason, we estimate Aas a weighted average of the onshore and offshore wind capital costs, weighted according to the respective proportions of onshore and offshore deployment on the installation target QI. We follow the same procedure for estimating the operative costs. For the discount rate (r), we use the Weighted Average Cost of Capital (WACC). Generally, the risk associated with energy projects which are financed by the government is considerably lower than those which are entirely promoted by the private sector. Thus, the discount rate used in each of the two cases is usually different. WACC, which is a weighted average cost of the firm’s equity and debt [30], does not directly consider the risk involved in the investment. This is why public subsidized RES projects often use this method to discount future cash flows and consider the financial feasibility of the investment. All the calibrated parameters and their respective sources are outlined in Table 3. Table 3: Calibrated parameters and sources Parameter Value 2013 Sources 2013 Value 2019 Sources 2019 TF20 IDAE (2011) [28] 20 IDAE (2011) [28] T120 BOE [29] 20 BOE [29] Ae1,400,000/MW Own estimation (data: PER 2011-2020 [6]) e1,150,000/MW Own estimation (data: PNIEC 2021-2030 [27]) O0.035AOwn estimation (data: PER 2011-2020 [6]) 0.035AOwn estimation (data: PER 2011-2020 [6]) Q022,960 MW AEE [7] 25,704 MW AEE [7] QI12,790 MW PER 2011-2020 [6] 24,626 MW PNIEC 2021-2030 [27] u·v·h2100 PER 2011-2020 [6] 2355 PNIEC 2021-2030 [27] Qmax 332,000 MW IDAE [31] 332,000 MW IDAE [31] γ4.05 ·10−6Own estimation (data: REE [23]) 3.6·10−6Own estimation (data: REE [23]) a1.0207 Own estimation (data: REE [23]) 1.0185 Own estimation (data: REE [23]) µ00.0691 Own estimation (data: REE [23] & OMIE [24]) 0.0691 Own estimation (data: REE [23] & OMIE [24]) µ12.095 ·10−7Own estimation (data: REE [23] & OMIE [24]) 2.095 ·10−7Own estimation (data: REE [23] & OMIE [24]) µ26.831 ·10−13 Own estimation (data: REE [23] & OMIE [24]) 6.831 ·10−13 Own estimation (data: REE [23] & OMIE [24]) σ00.270 Own estimation (data: REE [23] & OMIE [24]) 0.270 Own estimation (data: REE [23] & OMIE [24]) σ15.319 ·10−7Own estimation (data: REE [23] & OMIE [24]) 5.319 ·10−7Own estimation (data: REE [23] & OMIE [24]) σ21.479 ·10−12 Own estimation (data: REE [23] & OMIE [24]) 1.479 ·10−12 Own estimation (data: REE [23] & OMIE [24]) r0.10 Noothout et al. [32] 0.071 CNMC [33] S0e39.80/MWh Own estimation (data: REE [23] & OMIE [24]) e45.63/MWh Own estimation (data: REE [23] & OMIE [24]) 3.3 Results 3.3.1 Results for the year 2013 First of all, we check for each of the four subsidy schemes, whether the singular points defined by the equations (29), (34), (40) and (46), are maximum points of investor’s problem in Eq.(19) for the given installation target QI. As we mentioned in section 2.4, these equations characterize only those points in which first-order necessary conditions are satisfied. Therefore, for concluding whether those points are 19 maximum or not, we have to check second-order conditions as well. As shown in appendix C, for the parameter values corresponding to the year 2013, the analyzed points in the cases of Fixed Tariff, Shared Upside, and Cap & Floor policies are maximum points for the installation target Q=QI. On the contrary, in the case of Constant Premium, the value given by equation (34) for the current parametrization corresponds to a minimum point. No subsidy: Without any subsidy, the expected profits that investors will make according to equation (23) are e-2,287.8M. That is, investors are expected to lose more than 2 billion euros if they only retrieve the market price. Fixed Tariff (FT): In the case of a fixed-FiT, the risk-neutral decision maker will find it optimal to invest in the deployment of exactly QIunits of wind energy, if, according to the solution given by equation (29), the policymaker sets a FT of: K(opt) A=e87.838/MWh, which leads to the following expected benefits and costs: ΠF T =e607.52 M ΛF T =e2,895.29 M Constant Premium (CP): As we advanced above, for the given value of the parameters, it does not exist a premium Xfor which investors will find optimal to deploy exactly QInew units in this unrestricted optimization problem.10 In this case, equation (34) corresponds to a minimum point. Therefore, we must set another criterion to establish an adequate premium Xboth for investors and regulators. Since in Spain, before the removal of the FiTs, the Spanish government was indifferent between offering to a RES investor the option of a FT and of a CP, by choosing an adequate value of the premium (X), both the expected investor’s profits and policymaker’s costs are identical to those given by the optimal FT we just set above. The value for that chosen CP would be X∗=e10.746/MWh, leading to benefits and costs identical to those of the fixed tariff (e607.52M and e2,895.29M, respectively). Even though both benefits and costs are identical under the two previous policies, as we will see, the risk exposure of investor and policymaker will be totally asymmetric. Shared Upside (SU) and Cap & Floor (C&F): In these two cases, since we have two degrees of freedom when designing each of the FiT, the optimal configuration is now defined by a locus of efficient combinations (KC, ω) for the SU, and (KD, C) for the C&F regimes. These locus are represented in Figures 8 and 9. In both cases, for each possible value of a price floor K, there exists an optimal value of the other parameter (and the other way around). For any point belonging to this locus, under the corresponding FiT scheme, the investor will find optimal to invest in the installation of exactly QInew units. As it can be seen in Figures 8 and 9, there is a negative relationship between KCand ω, as well as between KDand C. This is not a surprise since it is quite intuitive that there must be some trade-off from the policymaker’s perspective: if he or she is willing to offer a higher guaranteed minimum price to the investor, we can expect that he or she will demand a higher share of the potential upside (1 −ω) for the SU regime, and a lower cap Cfor the C&F, which would imply more frequent remuneration for him of her (as everything exceeding the cap goes for the regulator). For the given values of the parameters, the feasible values for the price floor ranges from approximately e47 to e87 per MWh under the Shared Upside and from 64 to 87 for the Cap & Floor.11 10It could be possible to find an optimal point if we add specific restrictions to the problem. For example, we could bound policymaker’s budget. 11The feasible values are those fulfilling some sensible requirements, first of all, they must correspond to maximum points. In addition, for the SU tariff, they must fulfill (0 < ω < 1), whereas for the C&F regime: (K < C). 20 Figure 8: Optimal Locus (Shared Upside, 2013) Figure 9: Optimal Locus (Cap & Floor, 2013) It is remarkable that in both cases if the price floor is set at a value e87.838/MWh, under the SU scheme the value of ω(the efficient share that the investor would perceive of the upside exceeding the floor) becomes zero, and under the C&F, the efficient cap Cbecomes e87.838/MWh as well. In both cases, we would obtain the efficient FT we discussed in the first place, which is a result that supports the consistency of our model. We wish to represent the expected benefits and costs for the four policies we discussed, as well as the risk exposure. In the case of SU and C&F, the expected benefits and costs depend on the particular point of their respective locus that we use to design the tariff. In other words, depending on the price floor K (as setting the value of Kuniquely determines the efficient paired parameter ω/C). In order to quantify the different risk exposures of each tariff, we will define the Value at Risk (VaR) as a certain percentile of the distribution of potential benefits(costs) that investors(policymakers) might obtain(incur) under the unpredictable electricity price evolution. In our case, we will set the VaR at the 10th percentile of investor’s benefits as a measure of exposure to under-remuneration, and at the 90th percentile of regulator’s costs as a measure of exposure to cost overrun. Therefore, investors will prefer subsidies offering higher values of VaR, whereas investors will prefer policies offering lower VaR values. We say that investors are exposed to risk if their VaR is lower than their expected profits. In the same way, regulators are exposed to risk when their VaR is higher than the expected costs. In order to obtain a value for the VaR, we first have to estimate the probability distribution for each tariff of potential profits and costs. For achieving that goal, with the parameters given in Table 3, we simulate the stochastic GBM described in Eq.(15) a great number of times (in our particular case, we performed 500,000 trials). All the simulations are carried out using Wolfram Mathematica. For each trial, we obtain a particular stochastic evolution of the VWAP (St), and then, we use that value to calculate the profits and costs each FiT would imply, and in the case of SU and C&F, we do it for every feasible price floor. Finally, with the obtained sample of results, and the frequency of each result, we can easily estimate the VaR for investor’s benefits and policymaker’s costs. Next, we show for each tariff, the expected benefits (Figure 10), the expected costs (Figure 11), the investor’s VaR (Figure 12), and the policymaker’s VaR (Figure 13). 21 3.4 Sensitivity Analysis Given the values of the parameters that we have been able to predict or estimate, the solutions that we have obtained for the benefits and costs under each subsidy scheme are optimal in expectation. However, the possibility that some parameters that depend on the market might not evolve as expected should be regarded. Knowing the sensitivity of each policy’s expected benefits and costs with respect to deviations in the values of these market-depending parameters can be a fact to consider for both investors and regulators when designing their optimal policies. Let us consider that once each tariff is designed, with the police parameters (K, X, ω, C) set, it is revealed that some of our model’s parameters did not evolve as we anticipated. Thus, the expected profits and costs will be compromised. We show below in Table 6 the elasticities (E) of expected profits (Π) and costs (Λ) with respect to some market-depending parameters. Specifically, we analyze deviations in the drift (µ) and volatility (σ) of the GBM, in the discount rate (r), in the total investment cost per installed MW (C), and in the electrical generation in each period (Gt). Where, as usually, elasticities of a variable αwith respect to other variable β, are defined as: Eα,β =∂α/α ∂β/β . This means that if the variable βincreases by 1%, then the variable αincreases approximately by an amount of Eα,β%. Therefore, lower exposure to unexpected changes of a given parameter will be given by elasticities lower in absolute value. Large elasticities mean that if unexpected changes occur, the expected results can either increase or decrease dramatically. Thus, the closer the elasticities to zero, the safer the investor or regulator will be. For illustration, we briefly discuss some of the obtained results. 2013 FT CP SU (K=55) C&F (K=55) SU (K=67.5) C&F (K=67.5) SU (K=80) C&F (K=80) Profits EΠ,µ 0 25.97 140.19 74.80 30.67 17.31 6.99 3.99 EΠ,σ 0 0 43.42 6.51 11.50 -1.36 3.00 -0.88 EΠ,r -22.35 -30.40 -204.13 -120.28 -62.26 -45.19 -31.46 -27.61 EΠ,C -37.95 -37.96 -269.30 -161.61 -88.54 -66.60 -49.47 -44.56 EΠ,G 38.95 38.96 270.30 162.61 89.54 67.60 50.47 45.56 Costs EΛ,µ -5.45 0 -1.59 -2.10 -3.06 -3.72 -4.55 -4.89 EΛ,σ 0 0 1.57 0.38 1.18 -0.18 0.51 -0.16 EΛ,r 0.95 -0.73 -0.48 -0.34 0.05 0.27 0.61 0.74 EΛ,C 0 0 0 0 0 0 0 0 EΛ,G 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2019 FT NS SU (K=42) C&F (K=42) SU (K=55) C&F (K=55) SU (K=62) C&F (K=62) Profits EΠ,µ 0 2.30 49.44 7.30 6.06 2.19 0.75 0.33 EΠ,σ 0 0 7.24 -7.22 1.32 -2.03 0.19 -0.30 EΠ,r -9.92 -1.73 -61.43 -17.06 -16.27 -12.08 -10.71 -10.24 EΠ,C -21.90 -1.58 -116.61 -35.98 -33.51 -26.12 -23.34 -22.53 EΠ,G 22.90 2.58 117.61 36.98 34.51 27.12 24.34 23.53 Costs EΛ,µ 2.47 - 1.65 2.06 2.11 2.30 2.41 2.44 EΛ,σ 0 - -0.10 0.33 -0.07 0.13 -0.01 0.02 EΛ,r -1.09 - -0.91 -1.03 -1.01 -1.07 -1.08 -1.09 EΛ,C 0 - 0 0 0 0 0 0 EΛ,G 1.00 - 1.00 1.00 1.00 1.00 1.00 1.00 Table 6: Elasticities of expected profits and costs with respect to different parameters Regarding unexpected changes in µ, investors under a Constant Premium bear the entire degree of uncertainty compared to those under a Fixed Tariff. For the intermediate schemes, the higher the value of K, the lower the elasticities with respect to the VWAP drift parameter. On the contrary, policy costs are insensitive to such changes under CP schemes, and regulators under FT are the ones who bear the risk, while for the SU and C&F schemes, the higher the value of the price floor, the higher the sensitivity (in absolute value). Expected profits and policy costs are insensitive to changes in volatility under both FT and CP regimes, as the expected value of the GBM (S0eµt) does not depend on σ. Under a SU 28 or C&F schemes, a higher value of K(and thus a lower value of ωor C) decreases both investor’s and regulator’s sensitivity to changes in the value of σ. As expected, regulators are insensitive to changes in total installation costs per MW (C). As can be seen, the profit’s elasticities with respect to Gand C differ exactly in one unit, but with the opposite sign. For both parameters, the higher the value of K, the lower (in magnitude) the investor’s sensibility under SU or C&F regimes. For any policy, regulators share the same exposure to changes in the electricity generation at each period G. 3.5 Discussion Our methodology relies on the assumption that investors are risk-neutral, which might be justifiable. When investors are risk-neutral, it is because they can diversify their investments in other sectors. In the best-case scenario, they will have investments negatively correlated with the one we are discussing, so that, if they lose some of the investments in renewables, they could still make money. This assumption allows us to obtain analytical solutions for the optimal expected profits and costs associated with each subsidy scheme. Nevertheless, in light of the results obtained previously, we can push further our analysis. Let us suppose that in the first place, investors seek to maximize expected profits, and policymakers seek to minimize expected regulatory costs. Then, once the optimum is determined for each FiT scheme, risk can be taken into consideration by choosing the most appropriate subsidy type, together with an adequate value of the FiT parameters for SU and C&F designs (i.e., by choosing the proper price floor K). From the results obtained in section 3.3.1, we can conclude that it is unlikely that investors in 2013 decided to invest in the installation of the wind capacity required to meet the government’s objective set in 2011 for the year 2020, without any public subsidy support. Furthermore, the fact that wind energy investment froze after subsidies were eliminated in 2013 strongly supports our findings. Our results suggest that from investors’ perspective, the best way of creating a feasible appetizing investment environment in 2013 was to offer them a Fixed Tariff support scheme, or instead, in order to allow some risk-bearing from investor’s part and relieving some risk exposure from the regulator, offering a Shared Upside or Cap & Floor schemes as close as necessary to the Fixed Tariff (i.e., with the floor Kclose to the optimal FiT price e87.838/MWh). As we have seen in Figures 10 and 11, the higher the value of the price floor, the higher the expected benefits and the lower the risk exposure to under remuneration the investor has to bear. If investors have to choose between the two intermediate schemes, they will prefer the Cap & Floor subsidy regime with the highest possible price floor. From the policymaker’s perspective, the policy configuration that leads to the lowest expected costs and lower exposure to cost overrun would be the Shared Upside scheme with a price floor as low as possible. Indeed, not only the Shared Upside, but the Cap & Floor scheme also provides better expected results for the policymaker than the traditional subsidies. We cannot stress this finding enough: it is possible to design intermediate schemes that share the risk exposure between regulators and investors, and more importantly, they do it more efficiently, leading to the same level of investment but involving lower expected government costs. On the contrary, regarding the results obtained in section 3.3.2, we can conclude that an investor in 2019 would decide to invest in the installation of the wind capacity required to meet the government’s objective in 2030 without the need of any public subsidy. For instance, our analysis suggests that a riskneutral investor seeking to maximize expected profits, will prefer to receive precisely the market price rather than any supporting policy we have considered, contrary to what happened in 2013. This finding is consistent with the fact that in all the energy auctions held in the last couple of years, the discount rate for which the auction winners bid was zero. That is, they were all willing to invest in the installation of wind energy, receiving only the market price as payment. Three main reasons can explain this phenomenon as far as we are concerned. Firstly, the costs associated with the installation of a wind farm have plummeted compared to 2013. It is increasingly cheaper to invest in wind energy deployment. Second, since the annual drift of the market price is endogenous to the installed capacity target and an increasing function of it, as this installation target is so large for the next decade, this parameter increases considerably, and therefore, the market price tends to be higher as well. Moreover, the price of the initial period (S0) is 29 considerably higher in 2019 than in 2013. Finally, in recent years since the end of the financial crisis, the WACC has been decreasing, and therefore, future payments to investors are discounted at a lower rate. As we have seen, any of the three schemes analyzed involve negative expected costs (i.e., earnings for the regulator) due to the high market prices. Although this result may seem unrealistic, we must remember that this thesis’s objective is not to obtain completely reliable and accurate results for the future Spanish energy market; rather, our objective is to compare how intermediate schemes behave with respect to the usual FiT. Therefore, it is crucial to point out that, as happened in 2013, the policymaker’s expected costs of the intermediate schemes are again lower than those corresponding to the Fixed-Tariff or to the absence of any subsidy. The same level of investment (QInew units of wind power) is encouraged more efficiently, evidence of an indisputable gain compared to the usual FiT schemes. We have seen how the interests of the policymaker and those of the investor tend to be opposed. Both parties not only have to consider protecting themselves against the risk of low electricity market prices, but they must also consider the risk posed by the possibility that the market-depending parameters for which public subsidies are designed, do not evolve as expected. Thus, the existing trade-off between higher protection against low market prices and higher(lower) expected benefits(costs) that investors(governments) have to face, requires an exhaustive and in-depth analysis which should be handled carefully, and will eventually depend on the particular interest, budget, and the risk that each of the two parties is willing to assume. 4 Conclusions Until they were eliminated in 2013 mainly due to the high costs that they implied for the government, commonly used Feed-in Tariffs have been the main instrument to support investment in renewable energy in Spain. Since then, wind energy investment has been frozen for several years, making it impossible to achieve the installation target set for that regulatory period ending in 2020. Another drawback of the typical FiT schemes is that only one of the parties must assume all the risk. Inspired by the innovation introduced by Farrel et al., by combining sequential game theory, FiT design, and stochastic calculus, we have been able to design and to test intermediate policies that allow risk-sharing between investors and governments, and more importantly, doing it more efficiently. In this thesis, we have had to devise our own methodology to predict the drift and volatility of the volume-weighted average price of electricity as a function of the amount of installed wind capacity. Even though in our particular case the result can be compromised with a considerable error, due to the fact that in Spain there is not enough openly available data, we believe that this method could be useful in other studies beyond the particular use that we have given to it in this project. Moreover, it could be used for different renewable technologies. Next, we have run stochastic simulations to determine how these schemes would behave in a scenario that tries to approximate the Spanish environment for the years 2013 and 2019. We fully characterized the distribution of potential benefits and costs under each scheme. Finally, we carried out a sensitivity analysis to analyze the effects on the expected benefits and costs under each policy if different market-depending parameters do not evolve as expected. According to the results we obtained, it is possible to design Shared Upside and Cap & Floor schemes that allow achieving the same level of investment in renewables at a lower cost for the government. Then, investors and regulators can adjust these more efficient schemes to modify the value of the expected costs and benefits, as well as the risk each of the parties has to assume. Besides, our results reflect that there is an unavoidable conflict between the interests of investors and governments, which tend to be opposed. Furthermore, each of the parties faces their own trade-off: in the case of investors, they might have to choose between higher expected profits and lower risk of under remuneration. In turn, regulators might be forced to choose between lower expected costs and lower risk of public cost overruns. Furthermore, we have shown how the sensitivity analysis can offer some interesting insight to both investors and regulators when designing each policy. 30 Our results have interesting policy implications for future renewable energy regulation. Although the methodology performed may seem complex to manage at first glance, the irruption of technologies such as artificial intelligence, machine learning, and big data, allow this type of analysis to be automated and extended. It would be feasible that such tools that could be programmed nowadays without too many difficulties and outlay, carry out those simulations for all the possible values of the efficient parameters of the FiT, and that given the preferences of the investors, and some technical restrictions, such as, the budget of the government for renewable energy support, would return to the policymaker the optimal tariff and parameter choice that facilitates the desired investment. Also, unlike in Spain, where the parameters of the standard FiT and FiP were reviewed every several years, this automatable procedure would allow updates of the parameters of the FiTs in place much more frequently, which would be an undoubted improvement in efficiency for both parties. As we have pointed out on several occasions, the purpose of this thesis is not to characterize or predict with accuracy the financial results that a particular policy would have, but rather, to test how each of the analyzed policies behaves compared to the rest. This is because the methodology that we have developed has several limitations in order to consider the obtained figures as reliable. Firstly, the limitations mentioned above in estimating the parameters µand σ. Secondly, the obtainment of analytical solutions to efficient configurations of each tariff does not come without a cost, the high value obtained for the drift of the GBM has the consequence that when we carry out the simulations a large number of times, in some of those trials the legal maximum e/180.3 MWh is exceeded. Finally, the various approximations made to obtain closed-form solutions, such as taking the identical costs per MW (C) for the entire project, or assuming constant wind generation (Gt) over time, will compromise the results. This thesis demonstrates how the FiT schemes used in Spain until 2013 are improvable in numerous aspects. However, the design of the optimal support scheme for a specific year has not been carried out, given that such a task requires a much longer time and extension than what is understood to comprise this master’s thesis, and especially, a very accurate calibration. Nevertheless, we believe that this exercise can be carried out in the foreseeable future. We propose that the next step in the direction of this research is to give more weight to numerical methods and to play down the importance of obtaining closed-form solutions. By doing so, we believe that some of the mentioned limitations could be overcome, such as the requirement of bounded market prices, and allowing the possibility of relaxing some of the assumptions in which we have based our work. The most remarkable improvement could be to characterize investors’ preferences by a constant relative risk aversion (CRRA) utility function, which would make the obtained results much more general. Another advantage of a fully computational approach could be to include non-constant generation over time (Gt) and variable inversion costs (C). Furthermore, we believe that a way to overcome the limitations imposed by the obtained values of µand σcould be instead of considering the VWAP as the market price, to use the wholesale price. Since the VWAP is usually cheaper than the pool price but following similar trends, the wholesale price could be used as an approximation when studying the prices received by investors in renewables. A similar approach is followed by Blazquez et al. (2018) [19]. Finally, the numerically found optimal scheme could be compared to some of the other public support systems for renewables that are or have been commonly used, such as Tradable Green Certificates and Energy Auctions [35]. These tasks are left as open problems for future research. 31 References [1] Ciarreta, A., Espinosa, M. P., & Pizarro-Irizar, C. (2014). Is green energy expensive? Empirical evidence from the Spanish electricity market. Energy Policy, 69, 205-215. [2] Duffield, J. (2020). The politics of renewable power in Spain. European Journal of Government and Economics, 9(1), 5-25. [3] Colombo, E., Bologna, S., & Masera, D. (Eds.). (2013). Renewable energy for unleashing sustainable development. [4] Couture, T. D., Cory, K., Kreycik, C., & Williams, E. (2010). Policymaker’s guide to feed-in tariff policy design (No. NREL/TP-6A2-44849). National Renewable Energy Lab.(NREL), Golden, CO (United States). 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Mathematical methods for physics and engineering: a comprehensive guide. Cambridge university press. [35] Ciarreta, A., Espinosa, M. P., & Pizarro-Irizar, C. (2014). Switching from feed-in tariffs to a Tradable Green Certificate market. In The Interrelationship Between Financial and Energy Markets (pp. 261280). Springer, Berlin, Heidelberg. 33 Appendix A: Solving Equation 10 ∂f ∂t0+µS ∂f ∂S +σ2S2 2 ∂2f ∂S2−rf = 0 f(S, 0) = h(S) (54) First, we define the time to maturity τ=t−t0. Then we have: ∂ ∂τ =−∂ ∂t0. Substituting in Eq.54 and multiplying the whole equation by 2 σ2: −2 σ2 ∂f ∂τ +2µS σ2 ∂f ∂S +S2∂2f ∂S2−2r σ2f= 0 ∂2f ∂S2−rf = 0 f(S, τ =t) = h(S) (55) In order to replace variables rand µ, we define a=2r σ2, and b=2µ σ2, which yields to −2 σ2 ∂f ∂τ +bS ∂f ∂S +S2∂2f ∂S2−af = 0 f(S, τ =t) = h(S) (56) We now replace Sand τin the above equation by defining the variables uand v: v=σ2(b−1)2τ 2u= (b−1) log S K+v(57) According to the chain rule to u(S, τ) and v(S, τ): ∂ ∂S =∂v ∂S ∂ ∂v +∂u ∂S ∂ ∂u (58) ∂ ∂τ =∂v ∂τ ∂ ∂v +∂u ∂τ ∂ ∂u (59) From equation 57, solving for Sand τ, it is straightforward to obtain: S(u, v) = Ke(u−v b−1)(60) τ(u, v) = 2v σ2(b−1)2(61) We find the Jacobi matrix (J) for the coordinate transformation (S(u, v), τ(u, v)): J=     ∂S ∂u ∂S ∂v ∂τ ∂u ∂τ ∂v     =     S (b−1) −S (b−1) 02 σ2(b−1)2     (62) From the general properties of the Jacobi matrix, it is easy to prove that the partial derivatives of the inverse transformation (u(S, τ), v(S, τ)) are given by J−1. That is: i      ∂u ∂S ∂u ∂τ ∂v ∂S ∂v ∂τ     =J−1=     b−1 S σ2(b−1)2 2 0σ2(b−1)2 2     (63) which yields to ∂ ∂S =b−1 S∂ ∂u (64) ∂ ∂τ =σ2(b−1)2 2∂ ∂u +∂ ∂v (65) In addition, from Eq.64, we obtain: ∂2 ∂S2=∂ ∂S b−1 S∂ ∂u=−(b−1) S2 ∂ ∂u +(b−1)2 S2 ∂2 ∂u2(66) Substituting these three last expressions in equation 56, we obtain: −∂f ∂v +∂2f ∂u2−af (b−1)2= 0 (67) With the additional change of function f(S, t0) = f(S, t −τ)−→ g(u, v), defined by: f(S, t −τ) = e−rτ g(u, v) = e−av (b−1)2g(u, v) (68) which yields to: ∂f ∂u =e−av (b−1)2∂g ∂u ∂f ∂v =e−av (b−1)2∂g(u, v) ∂v −ag(u, v) (b−1)2(69) Substituting f,∂f ∂u and ∂f ∂v in 67, finally yields to a very well known PDE: the heat equation. ∂2g(u, v) ∂u2=∂g(u, v) ∂v g(u, 0) = g0(u) (70) In order to solve this equation, we use the Fourier Transform method to solve PDEs. The Fourier transform (F) of the function g(u) for −∞ < u < ∞is given by F[g(u)] = F(z) = 1 √2πZ∞ −∞ g(u)e−izudu (71) Thus, it is easy to verify that the transformation of the partial derivatives ∂2g(u,v) ∂u2, and ∂g(u,v) ∂v are given by the following formulas F∂2g ∂u2=−z2F[g] (72) F∂g ∂v =∂ ∂v F[g] (73) ii Hence, denoting F[g] = G(v), and F[g0] = G0(Z), equation 70 becomes dG(v) dv =−z2G(v)G(0) = G0(z),(74) Thus, the solution to this problem is obviously given by: G(v) = G0(z)e−z2v(75) Now, finding the inverse transform (F−1), it can be verified that the solution to equation 70 is defined by the following expression: g(u, v) = F−1[G0(z)e−z2v] = 1 2√πv Z∞ −∞ g0(z)e−(u−z)2 4vdz (76) Finally, replacing u(S, τ), v(S, τ), g0and substituting z= (b−1) log x K, we obtain the original terminal condition h(x): f(S0, t) = e−rτ σ√2πτ Z∞ 0 h(x) xexp −(log x S0−(µ−σ2 2)τ 2σ2τ!dx (77) •All the methods applied during these calculations can be checked out in: Riley, K. F., Hobson, M. P., & Bence, S. J. (2006). Mathematical methods for physics and engineering: a comprehensive guide. Cambridge university press. iii Appendix B Construction of the dataset Peio Alcorta Iglesias 2020-07-23 rm(list=ls()) years=2014:2019 for (a in 1:length(years)){ year=years[a] setwd("/Users/peio/desktop/TFM_ordenado/R") n=365 if (year==2008|year==2012|year==2016|year==2020){n=366} h=n*24 origindate=paste(as.character(year),"-01-01",sep="") file.1=paste("/Users/peio/desktop/DATOS/REE/",as.character(year),"/Custom-Report-", as.Date(0,origin = origindate),"-Estructura de generación (MW).csv",sep="") datafile.1 =read.csv(file.1,header=F, sep=",") datafile.1=datafile.1[-c(1:21),] datafile.1=datafile.1[-c(146:163),] m1=length(datafile.1[,2]) m2=length(datafile.1[10,]) for (j in 2:m2) { datafile.1[,j]=as.numeric(paste(datafile.1[,j])) } if (year>2015){ aux8to9=datafile.1[,8] aux9to10=datafile.1[,9] datafile.1[,8]=datafile.1[,12]+datafile.1[,13] datafile.1[,11]=datafile.1[,10]+datafile.1[,11] datafile.1[,9]=aux8to9 datafile.1[,10]=aux9to10 } colnames(datafile.1)=c("Hora","Eolica","Nuclear","Fuel/Gas","Carbón","Ciclo Combinado", "Hidraulica","Resto.Reg.Especial","Intercambios","Enlace Balear","Solar") datafile.1=datafile.1[,c(1:11)] data_horario.1=data.frame("Hora" = c(1:24), "Eolica"=c(1:24), "Nuclear"=c(1:24), "Fuel/Gas"=c(1:24), "Carbón"=c(1:24), "Ciclo Combinado"=c(1:24), "Hidraulica"=c(1:24), "Resto.Reg.Especial"=c(1:24), "Intercambios"=c(1:24), 1 Appendix B: Construction of the dataset (R code) iv Appendix C: Minimum or Maximum? We check for each of the four subsidy schemes, whether the singular points defined by the equations (29), (34), (40) and (46), are maximum points of (19) for the installation target QI. For both 2013 and 2019, the analyzed points in Fixed Tariff, Shared-Upside, and Cap & Floor policies are maximum points for the installation target Q=QI. On the contrary, in the case of Constant Premium, the values given by equation (34) for the parametrization of both years, correspond to minimum points. Furthermore, we see how for SU and C&F schemes, the singular points can either be maximum or minimum depending on the particular value of the efficient price floor (K) chosen. 2013 Fixed Tariff Solution of Eq.(29): KA= 87.838 Figure A1: FT 2013 Constant Premium Solution of Eq.(34): X= 8.1963 xi Figure A2: CP 2013 Shared Upside KCranging from 55 to 85 Figure A3: SU 2013 Cap & Floor KDranging from 55 to 85 xii Figure A4: C&F 2013 2019 Fixed Tariff Solution of Eq.(29): KA= 63.4575 Figure A5: FT 2019 xiii Constant Premium Solution of Eq.(34): X=−43.2159 Figure A6: CP 2019 Shared Upside KCranging from 25 to 65 Figure A7: SU 2019 xiv Cap & Floor KDranging from 5 to 65 Figure A8: C&F 2019 xv