Evaluation of a cross-border electricity interconnection: The case of Spain-France

Evaluation of a cross-border electricity interconnection: The case of Spain-France Luis María Abadie a , Jos e Manuel Chamorro b , * a Basque Centre for Climate Change (BC3), Sede Building 1, 1st Floor, Scientific Campus, University of the Basque Country UPV/EHU, 48940, Leioa, Spain b University of the Basque Country: Universidad Del País Vasco/ Euskal Herriko Unibertsitatea, Dpt. Financial Economics II and Institute of Public Economics, Av. Lehendakari Aguirre 83, 48015, Bilbao, Spain article info Article history: Received 17 November 2020 Received in revised form 27 March 2021 Accepted 4 June 2021 Available online 16 June 2021 JEL classification: F18 G31 L94 L98 Q41 Q56 Keywords: Interconnector appraisal Electricity prices Stochastic models Jumps Tobit model abstract This paper focuses on the economics of a cross-border transmission interconnector. The domestic spot electricity price is modelled as a stochastic process with mean reversion and jumps; it also includes a deterministic part that accounts for hourly and daily sasonalities along with non-working days. The two domestic spot prices are assumed to be correlated. As an illustration of the approach, we consider the particular case of the interconnector between Spain (an ‘electric island’) and France. Domestic prices are first calibrated and then used for simulating the stochastic behavior of the price gap between the two countries. In addition, the actual import/export behavior as a function of the price gap is captured by a Tobit model fitted from observed data. This model is then combined with the simulated price gaps to compute a multiple series of hourly prices and exports/imports of electricity through the interconnector. Drawing on these simulations we derive the probability distributions of revenues and expenses from exports and imports, and also some risk measures. According to our results, the economics of this interconector depends on different domestic seasonalities (hourly and daily), the growing trend of the price gap and some stochastic idiosyncrasies. They call for an expanded link. ©2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction Cross-border power interconnections bring about a number of benefits for participating countries and beyond [1], among them: (i) enhance security of electricity supply (SoS) by providing support functions between interconnected electrical systems; (ii) ensure the stability and frequency of the two systems; (iii) exploit price differences through power imports and exports thus increasing economic efficiency; (iv) harness renewable energy sources by allowing the transmission of excess renewable generation; (v) develop the Internal Energy Market in Europe. 1 This paper falls within the literature about power transmission expansion with a special focus on interconnector economics, i.e. item (iii). A number of models have been proposed to address power trade based on price differentials. 2 Many of them are applied to European countries (whether looking backward or forward in time), be it under general or partial equilibrium conditions. Typically, they are optimization models that aim to maximize social welfare or minimize system costs, for instance. They usually consider a single year (or fractions of it) with daily/hourly time steps. Importantly, they tend to be deterministic; the authors account for risks and uncertainties by simulating the models under several scenarios (e.g. without and with a particular expansion of the transmission grid). Besides, the optimization process results in a series of (daily/hourly) power prices, yet their properties are not shown. Thus, whether those optimization-based prices display the usual characteristics in actual power markets is all but impossible *Corresponding author. E-mail addresses: [email protected] (L.M. Abadie), jm.chamorro@ehu. eus (J.M. Chamorro). 1 There can well be conflicts among these goals; see for instance Ref. [32]. 2 Full price convergence is not an objective as such: it would entail over-investing in network infrastructures [33]. Contents lists available at ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy https://doi.org/10.1016/j.energy.2021.121177 0360-5442/©2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/ ). Energy 233 (2021) 121177 for outsiders to tell. Our paper introduces a novel approach that puts electricity prices at the forefront of the analysis. It looks at actual wholesale prices in two specific countries (domestic markets) with a common interconnector. The paper proposes a stochastic model where (correlated) domestic spot prices display seasonality, mean reversion and jumps. The aim here is to evaluate a particular crossborder interconnector from the viewpoint of revenues. Consequently, we also consider actual flows of electricity in both directions. Drawing on the latest price and quantity data publicly available we simulate the revenues to the interconnector in the near future. Hence we not only provide average estimates but probability distributions (i.e. risk profiles) as well. We further illustrate our approach by applying it to a singular case study; this is the second contribution of our paper. Right now, Spain is akin to an “electric island”because of its low interconnection ratio of 2.8% in 2019 (computed as the sum of the import capacities divided by the installed generation capacity). This low rate is very far from the EU goal of 10% for 2020 and the minimum of 15% for 2030; ENTSO-e [2]. The Spanish electricity system is connected with France, Portugal, Morocco and, to a lesser extent, Andorra. Anyway, the short AC interconnection with France is very important because it gives access to the vast European electricity market. 3 The French interconnector has a commercial exchange capacity of 2800 MW. 4 After the commissioning of a new project (crossing the Bay of Biscay) this capacity will increase up to 5000 MW; its commercial use is planned to start in 2024 or 2025 [3]. This way the interconnection ratio will rise to about 5%. Thus, Spain will remain an “electric island”for decades to come. This condition leaves it especially vulnerable to low-frequency, highimpact events, whether of natural, accidental or malicious origins. According to our results, Spanish exports would amount to V27 million on average over the three-year period 2020e22, while imports from France would entail average expenses of V964 million. These figures allow justify substantial new investment (taking [4]; and [5]; as benchmarks). 5 One of the drivers behind this result is the stronger upward trend of Spanish power price. As expected, when we impose the restriction of no growth in domestic prices the balance for Spain improves. The paper continues as follows. Section 2reviews the literature on the potential gains from enhanced cross-border interconnectors, preferably (though not exclusively) with a focus on the EU. Then Section 3introduces the theoretical framework, starting from the stochastic model for domestic power price. Section 4focuses on the two countries involved in this paper; it provides background data about domestic spot prices (at the hourly and daily time scales) along with power exports and imports. The price model is calibrated n Section 5. Next in Section 6we draw on the earlier parameter estimates to simulate the price in each country (and the ensuing price gap) over the period 2020e2022. On the other hand, Section 7estimates a model of power flows between France and Spain. Cross-border flows and simulated prices allow simulate the transmission income to the interconnector in Section 8. A sensitivity analysis with respect to the growth rate of power prices is undertaken in Section 9. Section 10 concludes. 2. Literature survey First, we proceed from a ‘macro’to a ‘micro’perspective: EU nations, regions, and industry stakeholders (producers, consumers, and transmission system operators). Then we consider some reasons behind inefficient arbitrage transmission (i.e. flows in the ‘wrong’economic direction) in pairs of neighbouring markets. Abrell and Rausch [6]find considerable scope for two-way cross-border trade in Europe (e.g. between Spain and France).- Power price differentials are far from unidirectional. Further, very frequently there are sizeable price gaps between countries with a cross-border interconnection (e.g. France and Spain). The gaps can certainly arise when transmission constraints are binding. And also when they are not because of: (i) transmission losses and/or ramping restrictions [7]; (ii) inability of the interconnector's owner to take simultaneous long (i.e. purchasing) and short (i.e. selling) positions in the two locations (because market liquidity in at least one of them is too thin; [8]. G€ oransson et al. [9] analyse the European power system at the NUTS-2 level, which results in 50 regions. Their results for 2020 show an annual average marginal cost around 50 V/MWh in the Spanish region ES2 and close to 30 V/MWh in the French one FR2. This ‘congestion’gap implies that the ‘marginal connection capacity value’over 8760 h amounts to some 173 Mill Vper year (rendering this connection one of the five AC interconnections with the highest values). Further, these two regions are a case in so-called ‘all-hour congestion’. Spiecker et al. [10]find utilization rates of the line connecting France and Spain around 90% both with and without grid extension in 2020 (about 2/3 of that rate corresponds to power flowing from France to Spain, and 1/3 to reverse flow). These high rates suggest that bottlenecks occur frequently. Under the expanded grid the average of absolute price differences between these two countries is cut in half. 6 The share of variable wind infeed is significantly higher in Spain than in France, which leads to more frequent reversals in the flow direction. Overall, France is one of the major beneficiaries of new interconnectors; they have a positive impact on producer surplus but a negative one on consumer surplus and congestion rent. Pudjianto et al. [11] consider the period 2010 to 2050. They find that reinforcing the interconnection allows Spanish consumers to access competitive offers from foreign producers, which leads to lower power prices and producer surplus. Yet not all producers suffer; for example, solar PV producers gain while wind producers lose. 7 Instead, French producers will meet a higher demand, which results in an increase of power prices in France (to the detriment of French consumers). At this point, it is worth noting that electricity does not always flow as price arbitrage would suggest. 8 Clements et al. [12]find instances of electricity flowing from Queensland to neighbouring New South Wales despite the former having a higher price. They show that these instances are due to nodal transmission constraints in Queensland only (not to constraints across regional boundaries). On the other hand, Bunn and Zachmann [13] show analytically that a dominant generator in one location, under special circumstances, may choose to export power (to a more competitive neighbouring market) against the direction of efficient arbitrage. Further, as those 3 The interconnector project was first proposed in 1980 (followed by a second proposal in 2003); it started operation 35 years later [34]. 4 The net transfer capacity (NTC) typically sets the commercial (rather than the physical) capacity between two countries. 5 The potential value of an interconnector is much higher. In addition to these revenues from day-ahead coupling it comprises the benefits from intraday coupling, shared balancing resources, avoided undesirable unscheduled flows, and reduced curtailment. According to Ref. [5]; revenues from day-ahead arbitrage make up around 25% of total value. 6 The average of those differences between two regions over a year indicates the welfare effect of a marginal line investment. 7 This can be related to the different levelized cost of electricity (LCOE) of these technologies; Abadie and [35]. 8 Under some circumstances, a flow in the ‘wrong economic direction’may be socially beneficial if its welfare economic cost is smaller than the welfare economic benefit of the congestion relieved by such a flow; [33]. L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 2 special circumstances do apply in the case of the Anglo-French Interconnector, they provide evidence that such flow reversions do occur in reality. The above papers focus on pairs of neighbouring countries for the most part, and we follow suit. Nonetheless, this is a partial perspective. In an AC network, physical electricity flows are hard to control and cannot be directed. Therefore, in highly meshed transmission networks (e.g. continental Europe) the directlyconnected countries are not the only players in determining cross-border flows. Changes in spatial generation/load patterns in non-neighbouring countries reverberate beyond national borders and impact other regions and/or cross-border interconnections in the network. We do not consider these effects beyond immediate neighbours 9 ; ignoring general equilibrium/network effects is not so much of a problem when addressing links to isolated systems like Spain [14]. Further, this is not only a technical issue. As Kunz [15] points out, the identification of flow patterns has important effects on the available cross-border capacity and hence on electricity spot markets. We leave this issue aside. 3. Theoretical framework Our ultimate goal is to simulate the transmission income to the interconnector in the near future (the three-year period 2020e2022). We first introduce a stochastic model of power prices. This model is then to be estimated with publicly available data. Parameter estimates allow simulate power prices in the two countries. Next, it is necessary to estimate a model of power transmission along the interconnector. However, cross-border flows are subject to some constraints; this leads to ‘censoring’ several observations, which in turn calls for abandoning the linear regression model and replacing it with a so-called Tobit model. Upon its estimation (with STATA), it is finally possible to simulate power prices along with flows and derive simulated revenues (with MATLAB). 3.1. A stochastic model of electricity prices As Weron [16] points out, the European convention is to refer to the day-ahead electricity price as the ‘spot price’. We use spot prices because of their greater informational content and liquidity. Besides, they reflect market fundamentals (as opposed to expectations about future market fundamentals, which are reflected in the prices of futures and forward contracts on electricity); Hirth [17]. Several approaches have been developed for analyzing and predicting electricity prices; see Weron [16]. So-called reducedform (quantitative, stochastic) models characterize the statistical properties of power prices over time. We refer in particular to Escribano et al. [18]; Lucía and Schwartz [19]; Seifert and UhrigHomburg [20]; and Villaplana [21]. We use a modified version of the stochastic model in MathWorks [22] to account for the effects of non-working days. Specifically, we describe the (natural logarithm of) daily spot price p t in a given country i¼{S (Spain), F (France)}, under the statistical measure, as the sum of two components: lnpi t¼fiðtÞþXi t(1) The first part, f i ðtÞ, is deterministic. It includes annual and semiannual seasonalities (through sine and cosine functions), a trend (t), and a dummy variable (D i t ) for weekends and public holidays (we consider only official national holidays, not regional ones): D i t ¼1 on weekends and non-working days, D i t ¼0 otherwise. It also includes a constant ( b i 7 ) along with 24 parameters ( b i j ) that correspond to the hourly seasonality (H j7;t ;j¼8;…31) in each country: fiðtÞ¼ b i 1sinð2 p tÞþ b i 2cosð2 p tÞþ b i 3sinð4 p tÞþ b i 4cosð4 p tÞ þ b i 5tþ b i 6Di tþ b i 7þX 31 j¼8 b i jHj7;t (2) The second part, X i t , is modelled as a stochastic equation 10 dXS t¼ a S k SXS tdt þ s SdWS tþJS m S j; s S jdqS j(3) dXF t¼ a F k FXF tdt þ s FdWF tþJF m F j; s F jdqF j(4) EdWS tdWF t¼ r dt (5) Specifically, Equations (3) and (4) are Ornstein-Uhlenbeck (OU) mean-reverting processes with jumps. They include three terms on the right hand; the first one is a function of X i t , while the other two are stochastic. Leaving the latter aside for a moment, the equation can be rewritten as dX i t ¼ð a i  k i X i t Þdt ¼ k i a i k i X i t !dt. Thus, the (log) stochastic part of the electricity price in country itends toward a i = k i in the long term, with a reversion speed k i .IfX i t falls below its long-run quilibrium value the parenthesis will be positive, which induces an increase in its value (dX i t >0); and conversely: if X i t rises above a i = k i the parenthesis will be negative, pushing X i t downwards (dX i t <0). In sum, when X i t departs from its long-term equilibrium (due to the impact of stochastic shocks, namely OU and jumps), the first term tends to restore the equilibrium (always subject to shocks). Besides, the higher the speed of reversion k i , the sooner X i t approaches its equlibrium value. Now, the second term generates a random behaviour without jumps. The volatility of the mean-reverting process is s i ;dW i t is the increment to a standard Wiener process. The third term is a Poisson process with intensity l i (the mean rate of event occurrence); if time is measured in years then l i jumps are expected per year. The jump size is normally distributed with mean m i j and volatility s i j . Here dq i j is a Poisson process such that dq i j ¼1 with probability l i dt, and dq i j ¼0 with probability 1  l i dt. We assume that dW i t and dq i j are independent. Note that Equations (3) and (4) allow negative values (the logarithm of some low electricity prices can be negative). On the other hand, sometimes both French and Spanish prices can move stochastically for common reasons. Equation (5) shows that these processes are correlated as measured by r . In this regard, the higher the price correlation, the lower the ability to benefit from the price gap between countries and hence from the interconnector. 3.2. Calibration of the price model Calibrating the above jump-diffusion model is related to the 9 [36] adopt this broader view but aim at a different goal. 10 This second part can be interpreted as a special case of the general stochastic differential equation for the increment of the (deseasonalized and detrended) spot electricity price in Ref. [16]. L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 3 more general problem of estimating the parameters of continuoustime jump processes from discretely sampled data; Cont and Tankov [23] offer an excellent review. Estimation procedures that involve the characteristic function, such as maximum likelihood (ML) estimation, are of particular interest from the viewpoint of statistical soundness. Below we will proceed in two steps. First we address the deterministic part of the price processes, and then their stochastic part. We stick with daily power prices in both countries. 3.3. Monte Carlo simulation of power prices We simulate the stochastic part of the log prices by means of an Euler discretization of Equations (3) and (4): Xi tþ1¼Xi tþ a i k iXi t D tþ s iffiffiffiffiffiffi D t pviþ D qi j m i jþ s i jxi j;(6) where D q i j ¼1 with probability l i D t, and D q i j ¼0 with probability 1 l i D t(here D t¼1=365). The Poisson behaviour is simulated with random numbers sampled from a binary distribution with jump probability l i D t. When there is a jump its size is m i j þ s i j x i j , which is simulated with random numbers x i j from independent N(0,1) samples. This amounts to extracting the jump size from a normal distribution Nð m i j ; s i j Þ. Regarding the OU component, we generate random samples of correlated daily log prices for France according to this scheme: vS¼xS;vF¼ r xSþxFffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 r 2 q(7) Here v S denotes samples of the third term in Equation (6) for Spain, while v F does so for France. Instead, x S and x F are two independent N(0; 1) samples. r is the correlation coefficient between the two stochastic parts, X S t and X F t . 3.4. Estimation of power flows along the interconnector Drawing on historical hourly data (2016e2019, Table 1) we assume a maximum transmission capacity of 3500 MWh. Therefore, in our computations below, exports and imports are left-censored (i.e. censored from below) at a value of zero, and right-censored (i.e. censored from above) at 3500 MWh. Censoring’means that we observe the independent variables for all cases, but the dependent variable is observed only over a restricted range of values (not its entire range). Censoring does not change the sample, but involves loss of information in a systematic way. In our case, left-censored data are aggregated and included as 0s, and rightcensored ones as 3,500s. Consequently, the standard Linear Regression Model provides inconsistent estimates of the parameters. Instead, the Tobit model provides consistent estimates (assuming, as usual, that the errors are normal and homoscedastic); it uses all of the information, including information about the censoring. Thus, ordinary least squares (OLS) must be replaced by ML estimation; see Long [24]. 4. Data Our data set includes daily and hourly information on domestic electricity spot prices (V/MWh) along with imports/exports (MWh) between Spain and France; it can be downloaded from the e-sios database (https://www.esios.ree.es/). The sample period is 2016e2019, i.e. four years. In particular, we have 1461 daily prices and 35,064 hourly prices. During this period the commercial interconnection capacity remained constant at 2800 MW. The upper block in Table 1 shows descriptive statistics of hourly power prices. Spanish prices are 6.38 V/MWh higher than the French ones on average (¼49.21e42.84) 11 ; the latter are more volatile than the former (20.32 V/MWh vs 14.35). Besides, the price gap between Spain and France shows negative skewness (14.83), i.e. the left tail of the distribution is longer/fatter than the right one (in other words, the probability mass is concentrated on the right of the distribution). It also displays positive excess kurtosis (741.26), that is, extreme values are, well, more extreme than in a Normal distribution (whose kurtosis is 3); this is confirmed by the maximum (68.50), minimum (810.96), and the 5% and 95% percentiles (11.07 and 25.45, respectively). 12 Sizeable positive or negative price gaps contribute positively to the economic value of the interconnection with France. The lower block provides information about actual power flows. For instance, maximum exports from Spain to France reach 3632.08 MWh, and 3755.34 the other way round. Thus, the maximum capacity of the interconnector is somewhat higher than 3500 MW, above its commercial capacity (2800 MW as already stated), because of an additional capacity devoted to SoS. Anyway, commercial capacity is not exactly constant (see Figure A4); it is periodically reset by Red El ectrica de Espa~ na (REE, the Spanish transmission system operator). During the sample period the net balance shows electricity flowing from France to Spain (at a rate of 1597.84e399.74 ¼1198.10 MWh on average). Table 2 shows some hourly price and quantity correlations. The Table 1 Hourly prices and power flows: Descriptive statistics (2016e2019). Mean Minimum Maximum Standard Deviation Skewness Excess Kurtosis Percentile 5% Percentile 95% Electricity Price Spain (V/MWh) 49.21 0.03 101.99 14.35 0.45 0.77 23.04 70.67 Electricity Price France (V/MWh) 42.84 31.82 874.01 20.32 6.64 222.74 17.27 74.74 Price gap Spain-France (V/MWh) 6.38 810.96 68.50 14.75 14.83 741.26 11.07 25.45 Exports (Spain- >France) (MWh) 399.74 0.00 3632.08 774.97 1.91 2.25 0.10 2286.46 Imports (France- >Spain) (MWh) 1597.84 0.00 3755.34 1053.14 0.26 1.21 0.00 3091.84 France net imports-exports (MWh) 1198.10 3632.08 3636.90 1692.51 0.97 0.27 2250.88 3057.39 France total imports þexports (MWh) 1997.58 127.75 4291.60 744.84 0.18 0.64 698.41 3153.28 11 ACER (2020, Table 5) shows the average gap across the Pyrenees in 2016 (2.9 V/MWh), 2017 (7.3), 2018 (7.1), and 2019 (8.2). This price differential is not the same as the ‘marginal value of transmission capacity’in Spiecker et al. (2017), which corresponds to the sum (or average) of absolute price differences between two regions over a year. In our sample period, the average absolute gap on this interconnector has been 9.78 V/MWh. As a reference, it was 11 V/MWh across the England-France interconnector for 2011e12; [14]. In the case of Spain-France, ACER (2020) provides yearly estimates in 2016 (8 V/MWh), 2017 (10.2), 2018 (10.8), and 2019 (10.1). 12 Just to put these figures into context [8], analyse five pairs of European neighbouring countries. Absolute average hourly spreads range between 0.27 V/MWh and 15.56 V/MWh, with standard deviations from 17.76 to 40.75. The maximum spread is 915 V/MWh (between The Netherlands and UK), and the minimum spread is 901 V/MWh (between Germany and the Netherlands), both during peak hours. L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 4 correlation between Spanish and French prices is 0.6878. 13 As expected, Spanish imports from France are positively correlated with the price gap between these countries (0.4639). At the same time, Spanish exports to France are negatively correlated with the price gap (0.5261). 14 Thus, the price gap is a major driver of power flow along the interconnection with France. Now, Table 3 shows descriptive statistics of daily electricity prices. The average price gap with France remains similar as with hourly prices (6.47 V/MWh). Not surprisingly, price gap volatility (10.79 V/MWh) is lower than with hourly ones (14.75). Daily prices also show positive excess kurtosis. The average daily net import from France is 28,754.37 MWh. As seen in Table 4, the correlation between Spanish and French daily prices (0.7549) is a bit higher than with hourly prices; this lower hourly correlation can be explained by different seasonality in these countries (e.g. different hourly habits in consumer behaviour). The correlation between the price gap and power flows is 0.6709 for Spanish imports and 0.7500 for exports to France, both stronger than with hourly prices. Fig. 1 displays daily prices in both countries. Most of the time French prices are cheaper than Spanish ones. Besides, in both countries price volatility is high. In Spain, the minimum price is lower than the average less three times the standard deviation (1.94 <49.21e312.86 ¼10.63), while the maximum price in France is higher than the average plus five times the volatility (125.67 >42.83 þ516.31 ¼124.38). Usually, whenever there is an abnormal peak (or the opposite) the starting price is more or less normal and then returns toward a normal level in the following day. Fig. 2 shows the daily price gap between these countries. There is a seasonal pattern, with wider gaps in the summer and narrower ones in winter. Further information extracted from our sample data is available in Appendix A. 5. Estimation of the price model Regarding the first seven parameters of the deterministic part, from Equation (2) and applying OLS we derive the estimates in Table 5. Some estimates are relevant for the value of the deterministic component. b i 1 and b i 2 in particular reveal a greater impact of annual seasonality for France. Others, such as b S 3 , have little influence.The estimates of the trend coefficients, b S 5 ¼0.0728 and b F 5 ¼0:0454, suggest that the price gap has been widening over time. This in turn translates into an increase in the economic value of the interconnection. We have also derived numerical estimates of the parameters involved in hourly seasonality: b i 8 ,…, b i 32 , with i¼S,F. For this purpose we calculate the difference between the (log) price in each hour of a day and the (log) price in that day. Thus, since the sample comprises 1461 days, we have 1461 differences for each of the 24 h. The average of those 1461 differences for, say, the first hour of the day, is the seasonality for that hour. The process is repeated for each of the remaining hours and separately for France and Spain. Thus, the numbers in Table 6 are to be interpreted with respect to the daily price (in a given day): a positive (respectively, negative) figure means an hourly price above (resp. below) the overall daily price (note that we use log prices). As can be seen in Fig. 3, hourly seasonality shows wider variation in France, with peaks and troughs further away from each other than in Spain. Maximum hourly prices tend to happen around 20:00 in France and 22:00 in Spain; the minimum prices are usually reached about 5:00 in both countries. In France below-average prices are found from 1:00 till 7:00; in Spain they run until 8:00. These different hourly patterns can impact both export and import power flows between the two countries. Upon estimation of f i ðtÞwe can break the price process into its two components: deterministic and stochastic. The upper panel in Fig. 4 shows the (natural logarithm of) power price in Spain, lnðp S t Þ, alongside its deterministic part, f S ðtÞ. The lower panel, instead, displays lnðp S t Þwith f S ðtÞremoved, i.e. the stochastic part, X S t . Similarly for France, the upper panel in Fig. 5 shows lnðp F t Þ, alongside f F ðtÞ. Instead, the lower panel displays the stochastic component, X F t . Concerning the stochastic part of the (natural logarithm of) power prices, X i t , we follow maximum likelihood estimation (see Appendix B) and obtain the parameter estimates in Table 7. On the other hand, the correlation coefficient between X S t and X F t is r ¼ 0:6570. This is somewhat different from the one obtained with daily prices (0.7549), because it refers only to the stochastic parts of the (log) prices. In Spain an average of 32.15 jumps are expected per year and 37.88 in France; thus, the daily ( D t¼1=365) jump probabilities are l i dt ¼0.0881 and 0.1038, respectively. Jumps in Spain follow a normal distribution Nð m S j ; s S j Þ¼Nð-0.1347; 0.5462). In France they behave according to Nð-0.0973, 0.4431); this suggests negative, less pronounced, and less volatile jumps. However, in the absence of jumps, the log price in France is more volatile (2.4374) and tends to return faster (67.4638) to its long-term equilibrium value. 6. Monte Carlo simulation of power prices Our numerical application of the scheme in Equation (7) generates 10,000 correlated random samples with r ¼0.6574, very close to the estimated value of 0.6570. We run 10,000 simulations for three years (2020, 2021 and 2022), i.e.1096 days, under the realworld probability measure. Table 2 Hourly prices and power flows: Correlation coefficients (2016e2019). Electricity Price Spain Electricity Price France Price gap Spain-France Exports (Spain- >France) Imports (France- >Spain) Electricity Price Spain 1.0000 Electricity Price France 0.6878 1.0000 Price gap Spain-France 0.0251 0.7084 1.0000 Exports (Spain- >France) 0.0104 0.3747 0.5261 1.0000 Imports (France- >Spain) 0.0874 0.2751 0.4639 0.7075 1.0000 13 [29] estimate a correlation of 0.6524 based on 81 monthly price observations between 2004 and 2011. Both figures are similar to the correlation (0.67) between hourly prices in France and the UK from November 2001 through June 2009 found by Ref. [8]. 14 These signs are consistent with results in Ref. [36]. Drawing on monthly data for 29 European countries they find that power price (as an explanatory variable of net exports) is statistically significant in most of their specifications; it has a negative impact, i.e. decreasing domestic prices make net exports more appealing. Interestingly for our case, they also find that, on average, neighbouring countries (Spain) of “large”countries (France) are positive net importers (admittedly, at low orders of magnitude). L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 5 Starting with Spain, initially we simulate the stochastic daily part, X S t using Equation (7). In a second step we add the deterministic daily part, f S ðtÞ, according to Equation (1), with the annual and semi-annual seasonalities, trend, effects of weekend and nonworking days, and a constant. Finally, we transform the log prices into absolute prices (in V/MWh). Fig. 6 shows the historical path of the daily log price over 2016e2019 along with a simulated path for 2020e2022 and the deterministic part alone. As before, we add the deterministic part to X F t and finally come up with simulated paths of daily power prices for France. Thus, we have 10,000 simulated paths of future daily prices in each country. Hence we can compute 10,000 daily price gaps between these countries for every single day over the period 2020e2022. Fig. 7 displays the average of 10,000 daily gaps in any day during this period. The gap shows a seasonal behaviour; the same applies to the observed price gap (see Fig. 2). Next, we transform the simulated daily log price series, lnðp i t Þ, into hourly series by applying the hourly seasonality coefficients (Table 6) to each of the former series (thus obtaining 24 log prices for each day); the log prices are further translated into absolute prices (V/MWh). Finally, we compute the hourly price gaps over 2020e22, namely 24 (366 þ365þ365) ¼26,304 hourly gaps for each of our 10,000 simulations, i.e. 10,000 hourly paths of 26,304 values each. Fig. 8 shows the resulting probability distribution. The 10% percentile is 13.63 V/MWh while the 90% percentile is 39.19 V/MWh. The average is 13.65 and the median a bit higher, namely 13.88 V/MWh. The distribution displays negative skewness. 7. Power flows along the Spain-France interconnector Based on the 35,064 hourly observations for Spain, we estimate a Tobit model for exports to France and another one for imports Table 3 Daily prices and power flows: Descriptive statistics (2016e2019). Mean Minimum Maximum Standard Deviation Skewness Excess Kurtosis Percentile 5% Percentile 95% Electricity Price Spain (V/MWh) 49.21 1.94 91.88 12.86 0.52 0.87 25.41 68.01 Electricity Price France (V/MWh) 42.83 2.66 125.67 16.31 0.99 1.98 20.86 70.89 Price gap Spain-France (V/MWh) 6.47 68.04 50.87 10.79 0.99 5.41 9.53 21.90 Exports (Spain- >France) (MWh) 9593.75 0.00 71,261.77 15,472.64 1.80 2.22 2.60 47,360.50 Imports (France- >Spain) (MWh) 38,348.12 0.00 78,997.68 21,106.50 0.23 0.99 1369.82 70,036.63 France net imports-exports (MWh) 28,754.37 70,673.92 78,994.02 34,564.26 0.95 0.08 44,301.98 69,326.31 France total imports þexports (MWh) 47,941.86 15,596.54 81,699.32 13,231.98 0.19 0.63 27,997.47 71,082.50 Table 4 Daily prices and power flows: Correlation coefficients (2016e2019). Electricity Price Spain Electricity Price France Price gap Spain-France Exports (Spain- >France) Imports (France- >Spain) Electricity Price Spain 1.0000 Electricity Price France 0.7549 1.0000 Price gap Spain-France 0.0465 0.6171 1.0000 Exports (Spain- >France) 0.0605 0.4457 0.7500 1.0000 Imports (France- >Spain) 0.1420 0.3266 0.6709 0.7805 1.0000 Fig. 1. Daily spot electricity prices in France and Spain, 2016e2019. L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 6 from France. Tables 8 and 9 show the results from STATA, respectively. Regarding Spanish exports, the likelihood ratio (LR) chi-square with p-value ¼0.0000 informs us that the Tobit model is significantly better than an empty model. The coefficients are statistically significant. The/sigma statistic (671.7724) is the estimated standard error of the regression. With the numerical estimate of the price gap coefficient (28.06604) we simulate power exports to France under uncertainty: Exports (S/F) ¼564.46e28.06 simulated gap þN(0; 671.77). As for Spanish imports, again the LR tells us that the Tobit model is significantly better than an empty model. The coefficients are statistically different from zero. The estimated standard error of the regression in this case is 958.8273. We use the gap coefficient (53.39729) to simulate power imports from France under uncertainty. 8. Simulated transmission income to the interconnector Power flows from one country to the other give rise to a monetary income (congestion rent) the size of which depends on the price gap between them. At the same time, we assume transmission costs of 5 V/MWh (as in Ref. [25]; or [8]. Thus, sometimes the net income can be negative because of this transmission cost. Nonetheless, it can also be negative because there can be exports when the gap price is positive (i.e. power flows from Spain to France despite its higher price in Spain), the same way that there can be imports when the gap is negative (that is, Spain purchases power Fig. 2. Daily power price gap between Spain and France, 2016e2019. Table 5 Parameter estimates of price processes (2016e2019, daily data): deterministic part f i ðtÞ, as shown in Eq. (2). Parameter Spain (i¼S) France (i¼F) b i 1 0.1288 0.1597 b i 2 0.0350 0.2081 b i 3 0.0007 0.0329 b i 4 0.0662 0.0206 b i 5 0.0728 0.0454 b i 6 0.1770 0.3233 b i 7 3.7569 3.6939 Table 6 Parameter estimates of price processes (2016e2019, daily data): deterministic part f i ðtÞ, hourly seasonality in Eq. (2). Spain France Hour b S h Hour b S h Hour b F h Hour b F h 1:00 0.0237 13:00 0.0471 1:00 0.0906 13:00 0.0551 2:00 0.1154 14:00 0.0401 2:00 0.1990 14:00 0.0056 3:00 0.1838 15:00 0.0084 3:00 0.2747 15:00 0.0675 4:00 0.2274 16:00 0.0324 4:00 0.3868 16:00 0.0941 5:00 0.2463 17:00 0.0438 5:00 0.4277 17:00 0.0735 6:00 0.2075 18:00 0.0090 6:00 0.3143 18:00 0.0455 7:00 0.1163 19:00 0.0385 7:00 0.1271 19:00 0.1802 8:00 0.0177 20:00 0.0887 8:00 0.0432 20:00 0.2281 9:00 0.0334 21:00 0.1206 9:00 0.1187 21:00 0.1640 10:00 0.0693 22:00 0.1324 10:00 0.1315 22:00 0.0804 11:00 0.0713 23:00 0.0872 11:00 0.1014 23:00 0.0884 12:00 0.0575 24:00 0.0153 12:00 0.0861 24:00 0.0132 L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 7 from France in spite of higher prices there). This behaviour emanates from the actual behaviour reflected in the Tobit model (excluded 1073 þ3433 negative observations; the 96 observations above 3500 are censored). Fig. 9 displays the probability distribution of cumulative revenues and expenses (from the viewpoint of Spain) over the simulation horizon. Table 10 shows a few basic descriptive statistics of the 3-year transmission income. Power exports generate an average of V27 million over 2020-22 while imports entail expenses of V964 million on average during the same period. The value of bilateral Fig. 3. Log price of electricity in France and Spain (2016e2019, hourly data): hourly seasonality (the average of the difference between the log price in each hour of a day and the log price in that day). Fig. 4. Log electricity price in Spain (2016e2019, daily data): deterministic and stochastic parts. L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 8 Fig. 5. Log electricity price in France (2016e2019, daily data): deterministic and stochastic parts. Table 7 Parameter estimates of price processes (2016e2019, daily data): stochastic part X i t , Eqs. (3) and (4). Parameter Spain France Value 95% confidence interval Value 95% confidence interval a 4.3161 2.2018e6.4304 3.7019 0.8469e6.5568 k 56.3486 46.2128e66.4844 67.4638 56.9641e77.9635 m j 0.1347 0.2378e0.0316 0.0973 0.1842e0.0105 s 1.8261 1.7266e1.9205 2.4374 2.2788e2.5862 s j 0.5462 0.4547e0.6244 0.4431 0.357e0.515 l 32.1534 23.7531e40.5537 37.8801 24.1242e51.636 Fig. 6. Actual daily log price of electricity in Spain (2016e2019) and a simulated stochastic path (2020e2022). L.M. Abadie and J.M. Chamorro Energy 233 (2021) 121177 9