An assessment of European electricity arbitrage using storage systems

An assessment of European electricity arbitrage using storage systems Fernando Nú~ nez, David Canca,  Angel Arcos-Vargas * School of Engineering, Department of Industrial Engineering and Management Science, University of Seville, Spain article info Article history: Received 21 October 2020 Received in revised form 5 November 2021 Accepted 12 December 2021 Available online 22 December 2021 Keywords: Arbitrage business with storage European wholesale electricity market International comparison Discount rate calculation Mixed-integer programming Mixed regression models abstract Electricity arbitrage involves the storage of energy at times when prices are low, and offering it on the markets when prices are high. The development of renewable and energy storage technologies may provide a promising business opportunity for electricity arbitrage. In this regard, this study analyses the current viability of the electricity arbitrage business (via Li-Ion batteries) using a sample of European countries in the year 2019; countries where electricity prices (day-ahead market) and financial conditions show a certain degree of heterogeneity. We basically follow a sequence of three analyses in our study. Firstly, a Linear Mixed-Integrated Programming model has been developed to optimize the arbitrage strategy for each country in the sample. Secondly, using the cash-flows from the optimization model, we calculate two financial indicators (Net Present Value (NPV) and Internal Rate of Return) in order to select the optimal inverter size for each country. Tax and discount rates specific to each country have been used with the calculation of this second rate following the methodology proposed by most of the national agencies. Thirdly, a mixed linear regression model is proposed in order to investigate the importance of observed and unobserved heterogeneity (at country level) in explaining the business profitability. The findings show that, in the near future, the most attractive European countries for the electricity arbitrage business should be the United Kingdom and Ireland, with current NPV close to 400,000 V, while Spain and Portugal might show the worst performances, their current NPV are close to 800,000 V. ©2021 Elsevier Ltd. All rights reserved. 1. Introduction Electricity arbitrage involves the storage of energy at times when prices are low, and offering it on the markets when prices are high. This activity is justified because the price of electricity in the wholesale market varies continuously due to the changes in the demand and supply curves. The smoothing effect of arbitration on the price volatility is negligible if the electricity bought and sold to the market is very small in relation to the market volume. Although not in a structured way, the historical need for arbitrage has existed since the origins of the electricity sector at the end of the 19th century. The availability of non-manageable generation sources (flowing water hydroelectric or nuclear) justified the storage of this energy by means of reservoirs that were filled at almost zero cost. In recent years, concern for the environment has led to an increase in renewable generation which has been possible thanks to technological advances and cost reductions in this generation source. In addition, technological advances in battery energy storage have made possible a reduction in storage costs and an improvement in the efficiency, size and weight of the batteries. Despite the many advantages of the clean generation sources, their unmanageable nature increases price volatility. In this context, it might be interesting to develop arbitrage business models that take advantage of both increasing price fluctuations and cheaper batteries. The regulatory and technological changes in this sector pave the way for new business models to be developed, such as the one proposed in this paper. Although there are different storage technologies, lithium-ion batteries can be considered the best performing for arbitration, which is mainly due to their high levels of round trip efficiency, energy density and specific energy esee Arcos-Vargas et al. [1] for a comparison of the different performances of the storage technologies. The present study attempts to show that, in addition to the necessary investment and the technical characteristics of the electrical storage, the effectiveness of electricity arbitrage in Europe is strongly influenced both by daily price variations edifferences between maximum and minimum prices throughout the dayeand by the level of financial risk faced by each national electricity *Corresponding author. E-mail address: [email protected] ( A. Arcos-Vargas). Contents lists available at ScienceDirect Energy journal homepage: www.elsevier.com/locate/energy https://doi.org/10.1016/j.energy.2021.122916 0360-5442/©2021 Elsevier Ltd. All rights reserved. Energy 242 (2022) 122916 market. We follow the research line of Arcos-Vargas et al. [1] who analyse the possibility of doing business through the arbitrage of electrical energy using Li Ion storage systems, and applying it to the Spanish electricity market. Using a mixed-integer linear programming model to optimize energy buying and selling, the authors conclude that if storage technology continues developing at the rate it has done in the last 20 years, the business could make sense in a short period of time. Although Europe should be a single economic entity, there are currently various separate wholesale electricity markets. Despite the fact that most of them are strongly interconnected, prices show different levels and ranges. In addition, the financial risks and fiscal conditions of each country vary, which ultimately implies the existence of different discount rates. In summary, at least from a financial point of view, the European Union is composed of 27 different fiscal regimes, business risks, and financial and electricity markets. 1.1. Literature review There are numerous papers devoted to analysing the interest of battery arbitrage operations. Practically all of these studies analyse, for a specific country or market, the business opportunities given a storage technology and a required investment. Some of them also investigate the beneficial side effects arising from the deployment of storage systems (renewables and grid upgrading) esee for example Hou et al. [2]; Das et al. [3]. and Jannesar et al. [4]. Likewise, some authors propose the analysis of factors which can affect the business effectiveness (access tariffs, possibility of auctions to provide the service, or level of competition in the market), as is the case of Adebayo et al. [5]; Nasrolahpour et al. [6] and Wu and Lin [7]. Among the papers which mainly analyse different markets and storage technologies, we highlight the following; Walawalkar et al. [8] study the sensitivity of the roundtrip efficiency on energy trading operations where the energy is stored in sodium sulphide batteries (SNa) and flywheels, determining the critical factors which affect the financial yield of each technology in the New York market (NYISO). Subsequently, Berrada et al. [9] analyse the same market (NYISO), while expanding the technologies used to Compressed Air Energy (CAES) and Hydraulic Pumping System (PHS). They also increase the scope of the possible sources of value in the business, analysing the markets of daily, intraday and auxiliary services. An assessment of SNa battery plants and PHS is conducted by Kazempour et al. [10] for the Alberta (Canada) wholesale market. In this case the authors apply a linear programming model which reveals that PHS technology offers a significant and clear alternative to SNa batteries in economic terms. In these three works, although Nomenclature Abbreviations BESS Battery Energy Storage System CAES Compressed Air Energy Storage CES Community Energy Storage D Debt DR i Discount rate DSO Distribution System Operator E Equity EA Energy Arbitrage EA-PS Energy Arbitrage - Peak Shaving EDLC Ultra capacitors EES Electric Storage System GenCo Generation Company IRR Internal Rate of Return LA Lead-acid Li-Ion Lithium - Ion LV Low Voltage MRP Market Risk Premium NMC Lithium Nickel Manganes Cobalt NPV Net Present Value NYISO The New York Independent System Operator OMIE Iberian market operator PHS Pumping Hydraulic System RRF i Risk-Free Rate Sna SodiumeSulfur T i Taxes TSO Transmission System Operator VRFB Vanadium Redox Flow Battery WACC Weighted Average Cost of Capital ZEBRA NickeleSodium Chloride Battery Parameters TSet of periods (Hours) defining the planning horizon u t Market electricity price at hour t2T j o A parameter used to define the initial battery capacity (we will consider a battery of 10 MWh in our experiments) 4Deterioration of the battery capacity per cycle. We consider that after 5000 cycles the initial battery capacity decreases by up to 20% of the initial one j max Conversion rate of the inverter l A parameter that defines the loss of energy in charging and discharging conversion processes. This parameter is measured as a percentage of the energy purchased or sold Variables a t Binary variable. It takes value 1 if the battery is charged at period (hour) t2T, 0 otherwise. b t Binary variable. It takes value 1 if the battery is discharged at period t2T, 0 otherwise. d t Binary variable. It takes value 1 if the battery has finished a charge\discharge cycle at period t2T, 0 otherwise. They are in charge of measuring the battery deterioration with use g t Binary variable. It takes value 1 if the last performed operation prior to period t2Twas a charge and 0 if it was a discharge. These variables will be used as auxiliary variables to enforce cycle occurrences P t Represents the amount of electricity purchased and stored in the battery during period t2T. The purchase decision at period tis conditioned by a t S t Measures the amount of energy discharged from the battery during period t2T. This energy is transformed into electricity and sold at the same period. The sale decision at period tis conditioned by b t C t Integer variable that measures the cumulative number of charging\discharging cycles that has been done just until period t2T K t Represents the remaining capacity of the battery at the end of period t2T LMeasures the level of energy in the battery at the end of period t2T F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 2 the authors search for the optimal markets, technologies and participation strategies, the results are far from the required profitability thresholds. For their part, Sioshansi et al. [11] propose an optimization model for the Independent System Operator PJM. Considering a conservative roundtrip efficiency of 80% and a 12-h storage capacity for large storage technologies, these authors estimate the economic value that arbitrage transmits to society by calculating the overall welfare and the surplus of consumers and producers in the dayahead electricity wholesale market. As a main contribution, the authors present the breakeven cost of storage and describe what regulatory measures could be incorporated in order to make that business more attractive. In our opinion, although their model is meaningful, battery technology has improved significantly in the last ten years thereby justifying updating it to current values. From an international perspective, Connolly et al. [12] analyse a number of strategies to optimize electricity arbitrage in the day-ahead market in eleven different countries, but without discriminating on the basis of fiscal and financial risk features; in general, the results obtained show that the most favourable strategy is the one that focuses on short-term trade (i.e., next 24 h). The contribution of Yucekaya [13] consists of the application of a mixed integer optimization model based on Markov chains, with which it simulates arbitrage using exclusively CAES (Compressed Air Energy Storage) technology in the Turkish electricity market, and analyses the influence that each of the technical parameters of CAES has on business profitability. The same technology is being analysed by Das et al. [3] for the Californian market, though using 1min resolution demand data and allowing participation in the ancillary services market, thus making the operation even more profitable and enhancing it as wind power penetration increases. There is more literature in this field; for example, we have found works that compare technical and financial pros and cons for different technologies, such as PHS, CAES, ZEBRA (nickel-sodium chloride battery), EDLC (ultracapacitors), lead-acid battery (LA), and Li-Ion batteries esee for instance Refs. [14,15]; and [16]. Other studies analyse different business models derived from the participation of storage agents in the electricity markets e [5,17e19]; and [4]. For their part, Nasrolahpour et al. [6] analyse how the more or less competitive character of the market affects the arbitrage business. Finally, in the case of China, since the electricity market is not competitive, Wu and Lin [7] study the impact that the use of storage systems can have on the efficiency of the electrical system. 1.2. Research objectives The main contributions of our research are the following: We have developed an optimization model to determine the optimal energy purchase and sale strategy considering hourly electricity market prices during a full year. The model supports different pricing schemes, so it can be used in a multi-country framework. In order to deal with a long term analysis, we have designed a sequential optimization procedure where outputs for one year (effective battery capacity and number of performed chargedischarge cycles) are used as initial conditions for the next year. This sequential procedure enables us to analyse the battery performance while the operation remains profitable and the battery has sufficient capacity. A risk model has been developed for the electricity sector of each selected European country. Based on public information, we obtain their respective discount rates, which allows an international comparison of the arbitration business. A random effect model has been estimated to measure the sensitivity of the optimal financial indicators to changes in the cost and capacity of the storage system, taking into account the different financial and electricity price conditions in the European countries analysed. As far as we know, there are no studies which make a comparative analysis of several countries while also including in that analysis information on the characteristics of the wholesale electricity markets, financial risks and fiscal issues at country level. Our research assumes Li-Ion as the optimal storage technology, due to the great technological and economic advances registered in this type of battery in recent years, and assumes a competitive nature for the electricity wholesale market; consequently, the contribution of the storage player will not affect the equilibrium price (they are price taker). Likewise, we make an international comparison, adopting a common procedure to measure risk and tax levels of every country in the sample. These three aspects considered together give a distinctive character to our work within the existing literature in this field. Our research raises some specific questions: 1) Will the different European wholesale electricity markets be equally interested in using battery arbitrage? If not, 2) What are the determinants of these differences? And finally, 3) In which country (and at what moment) would it be more profitable for a company to undertake the arbitrage business with storage? In order to address them, the following steps have been taken. First, the hourly prices of electricity distribution (daily market) in 24 European countries have been analysed, evaluating both their levels and daily gaps. Once we know how electricity markets perform, the discount rates of each country are needed; in our case, these rates are taken from a report developed by the Spanish regulator [20]. The fact that there are different tax rates and risk premiums in each country implies that the values of the discount rates are consequently different. Parallel to the calculation of the discount rates, we use price data from the daily electricity market and BESS technological information to develop an optimization model which determines the best buy and sell strategies for different inverter sizes (from 1 to 10 MW) given a battery of size 10 MWh, 5,000 cycles and 92% round trip efficiency. The considered inverters are bidirectional, i.e., they transform both DC to AC and AC to DC. Finally, with the results of the purchase and sale of energy coming from the optimization model, and with the financial and fiscal information of each country, the financial viability of each BESS configuration, in each of the 24 countries considered, is analysed. In addition to a broad international comparison, our study is novel in another sense. Once the profitability obtained from the arbitrage business has been determined for each country, we analyse the influence that each of the input variables (economic and technical) has on that profitability by estimating a two-level mixed regression model. Specifically, the model is estimated for both the Net Present Value (NPV) and the Internal Rate of Return (IRR) indicators. As we will see, the need to control for the existence of unobservable heterogeneity is important in these types of international comparisons. In our opinion, the conclusions and business implications provided by this study can be used by entrepreneurs to determine in which countries the business of battery arbitrage can be more or less profitable, as well as to know what factors affect the economic returns. This information can also be valuable for regulators and technology research centres. After this introduction, the information used in this paper (at market, technical and financial level) is presented and comparatively analysed (by country) in section 2. Subsequently, section 3 develops the formulation of the mathematical model of optimal trading with storage. Section 4calculates the financial results by F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 3 country and investment configuration, using as input information the optimal cash-flows provided by the previous optimization model. Moreover, an econometric analysis is proposed in order to analyse the main determinants of the different financial returns observed, thus providing answers to the research questions proposed above in this introduction. Finally, section 5concludes. 2. Data and materials 2.1. Market information The transparency platform ENTSO-E collects data on generation, transmission and consumption for the pan-European market. This data collection has allowed us to observe the hourly wholesale electricity prices for the year 2019 in a sample of 24 European countries. Table 1 orders those countries from highest to lowest average hourly price (V/MWh). The table also shows the average daily differential between the maximum and minimum intraday hourly prices. Prices are relatively high in Southeast Europe and relatively low in Northern Europe, where the generation mix costs are lower. Note also that countries such as Spain and Portugal or the United Kingdom and Ireland have integrated markets, so they share the same prices for most of the year eif the interconnections were unlimited, the prices would be the same in both countries; only in those cases where the interconnections are saturated, different prices may appear. To better visualize the table, Fig.1 represents the average hourly price, its standard deviation and the average differential price for each country in the sample for the year 2019. As can be seen, the figure shows some positive correlation between the average price and the differential price. The United Kingdom and Ireland stand out for their high differential prices (even higher than their average prices) while Sweden and Norway stand out for their small differentials. It is also remarkable that Portugal and Spain show mid-level prices with relatively small standard deviations, which determines that their price daily gaps are relatively small in their price quartile. 2.2. Technical information As stated in the introduction, Li-Ion batteries have been selected as the preferred storage technology due to their better level of performance and recent technological advancement. An analysis of the different storage technologies considered can be found in Arcos-Vargas et al. [1]. Table 2 shows the technical specifications of Li-Ion batteries, which have been obtained from various sources: Gomez-Exp osito et al. [22]; Segui [23]; Bardo-C aceres [24]; Hern andez-Romero [25]; Battery University [26]; Jofemar Energy [27]; V elez-Moreno [28]; CleanTechnica [29] and IRENA [30]. The current performance of this type of storage technology has improved significantly in recent years. In particular, the cost has been reduced by an average of 20% per year during the period 2010e2019, and given the intensity of research focused on this field, it is expected that the trend of improvement will continue. If cost reduction continues at this rate, they would be halved in 3.5 years. 2.3. Financial information Although the European Union should be a common business area, the reality is still far off. The existence of different wholesale electricity markets, taxation regimes, and financial risks, motivated on several occasions by the decisions of their governments, means that different discount rates have to be applied depending on the country in which the installation is operated. The Weighted Average Cost of Capital (WACC) represents the minimum return that a company must earn, on an existing asset volume, to satisfy its creditors, owners, and other providers of capital, or they will invest in other more profitable activities [31]. Subsequently, a reasonable differential must be added to this minimum return in order to obtain the discount rate [20]. The proposed model for evaluating the BESS arbitrage in different European countries is based on the following two hypotheses: 1) there is a unicity of capital goods and technology, 2) there are country-specific capital market conditions determined by their fiscal conditions and business risks. From the Table 1 Wholesale electricity prices in Europe (Source [21]]. Country Hourly electricity prices (V/MWh) Daily price differential (V/MWh) Mean price Price Std. Dev. Min Max Mean gap Std. Dev. Min Max Greece 63.8 11.8 0.0 145.0 28.2 17.6 3.3 84.6 Italiy 52.2 12.7 1.0 113.1 29.8 9.2 9.2 58.5 Serbia 50.5 18.0 0.5 153.5 37.0 13.0 0.0 92.1 Romania 50.4 21.2 0.0 158.0 47.8 20.0 16.2 127.1 Hungary 50.4 18.8 0.0 138.8 42.0 17.2 14.1 103.0 Ireland 50.3 23.7 11.9 365.0 54.7 37.5 17.5 298.1 UK 50.2 23.8 11.9 365.0 54.6 37.6 0.0 298.1 Poland 49.4 17.5 1.2 114.0 19.5 9.6 1.5 63.2 Croatia 49.2 18.9 20.2 200.0 40.4 20.9 11.2 161.7 Slovenia 48.7 18.2 20.2 200.0 38.3 20.4 10.5 161.7 Portugal 47.9 10.8 0.0 74.7 16.3 7.6 2.2 50.4 Spain 47.7 10.9 0.0 74.7 16.9 8.3 2.2 55.2 Latvia 46.3 15.8 0.1 200.0 32.7 26.8 4.3 197.0 Lithuania 46.1 15.8 0.1 200.0 32.7 26.6 4.3 197.0 Finland 44.0 15.3 0.1 200.0 28.3 21.4 3.4 170.9 Netherlands 41.2 11.3 9.0 121.5 26.6 9.5 6.7 65.3 Switzerland 40.9 12.3 39.5 108.1 19.8 8.1 3.9 70.5 Czech 40.2 13.5 48.1 109.3 27.7 10.4 7.9 80.8 Austria 40.1 13.1 59.8 121.5 26.7 11.3 4.0 87.6 France 39.5 14.0 24.9 121.5 27.2 9.4 6.4 64.9 Norway 39.3 8.3 5.9 109.5 7.7 6.7 1.1 54.1 Denmark 38.5 13.2 48.3 109.5 26.3 13.8 4.1 105.2 Sweden 37.9 9.9 0.1 107.7 13.5 9.5 1.9 52.3 Germany&Lux. 37.7 15.5 90.0 121.5 30.1 15.6 6.7 117.3 F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 4 first hypothesis, it is clear that the same technology and the same investment cost can be accessed in all European countries. Furthermore, given the principle of free capital flow, all markets will have the same Market Risk Premium (MRP) and the same b coefficient for the business associated with the operation in the electricity markets. The second assumption implies different fiscal systems and, therefore, different profit Taxes (T i ). To estimate the Discount Rate for each European country i(DR i ), the first step consists in calculating the WACC value for each country efor those companies which participate in their respective wholesale electricity markets. For this purpose, the well-known formula of the WACC is used: WACCi¼E EþDrEiþD EþDrDð1TiÞ(1) Where E=ðEþDÞand D=ðEþDÞare respectively the equity and debt fractions of total capital employed, r D is the cost of debt, and T i and r Ei are the profit tax and the cost of equity of each country i respectively. To obtain the cost of equity (r Ei ), the CAPM (Capital Asset Pricing Model) method is applied, as it is the most widely used in the financial context: rEi¼RRFiþ b elec PMR (2) Substituting (2) in (1), we obtain: WACCi¼E EþDðRRFiþ b elec PMRÞþ D EþDrDð1TiÞ(3) where RRF i represents the risk-free rate of return in country i, b elec is the beta coefficient of European companies in the electricity market, and PMR is the Premium Market Risk. Two sets of parameters can be differentiated in equation (3): those that present a common value for all the countries of the sample {E=ðEþDÞ,D=ðEþDÞ, b elec ,PMR,r D }, and those whose values will depend on the country in which the activity is performed {RRF i ,T i }. In order to assign values to the first group of parameters (common information), we take representative values from a group of European listed companies which are mainly active in the wholesale electricity market. These values are taken from the Spanish regulator [20,32], which, in turn, uses data from Bloomberg [33] and Dimson et al. [34]. Specifically, the estimated values are: { E EþD ¼0:45, D EþD ¼0:55, b elec ¼0:77, PMR ¼4:75%, r D ¼4:49%}. For the second set of variables, the country-varying variables {RRF i ,T i }, the risk-free rates for each country ðRRF i Þhave been calculated by subtracting the 10-year sovereign bond yields of each country from that of Germany [35]; and [36], while the tax profit (T i ) has been obtained from the “Deloitte International Tax Source Report”[37]. Finally, for the determination of the Discount Rate (DR i ) we apply equation (4), which is frequently used by national agencies to establish discount rates for regulated activities. The only parameter which has not been specified so far is the spread for additional risks, which, for the sake of simplicity, and since the peculiarities of each country are not known in depth, is maintained for all countries at the 0.5% proposed by the CNMC for the Spanish electricity distribution, although any other fair value could have been chosen. DRi¼WACCiTiþSpread 1Ti (4) Table 3 shows all the variables that change depending on the country considered ecountries are ranked from highest to lowest DR i . The discount rate and the profit tax will be used in the following sections to carry out our financial analysis. Fig. 2 depicts these two variables. As can be observed, southern European countries have high discount rates compared to central and northern European countries. Moreover, there is a certain positive correlation between the discount rate (left axis) and the profit tax (right axis), with Spain and France standing out for having relatively high tax rates. 3. Trading model formulation This section describes the optimization trading model designed for determining the optimal purchase and sale strategy for a given battery capacity. The specific battery size does not influence the optimal results, since the operation policy is conditioned by the ratio ‘battery capacity/inverter transformation’. The objective of the proposed trading model consists of the maximization of cash-flows obtained from energy purchase and sale operations during certain planning horizons. Later, we will explain how the proposed Fig. 1. Electricity prices in Europe. Mean, standard deviation and mean gap (max. emin.). Table 2 Main features of Li-Ion batteries. Li-Ion Specific energy (Wh/kg) 130e147 Energy density (Wh/L) 250e730 Specific power (W/kg) 250e340 Nominal voltage (V) 3,6 Charge/discharge cycles 5000 Monthly self-discharge (%) 3% Round trip efficiency (%) 92% CAPEX (V/kWh) 100 F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 5 optimization framework deals with multi-year planning intervals. As its main input, the model considers hour-by-hour electricity prices. Fig. 3 shows a plausible buying and selling schema, and it is used to explain the model characteristics more clearly. At the top of the figure, series A depicts a possible hourly pattern of the electricity price, each interval corresponding to 1 h. A rational policy would consist of buying energy when market prices are low and selling it when prices are high. The middle part of the figure illustrates a feasible strategy. Filled and dashed rectangles represent purchasing and sales decisions respectively. The height of each rectangle corresponds to the amount of energy charged or uncharged. Numbers (1) and (2) are related to charging operations whereas numbers (3) and (4) depict discharging ones. The line C represents the battery charge level. As shown, a purchase operation turns into an increment in the battery charge level. The dashed line B measures the maximum battery capacity. As illustrated, each time a new charge/discharge cycle takes place, the maximum battery capacity must be decreased, representing in this way the battery deterioration (depreciation in economic terms) with use. Next we explain in detail the model parameters, variables and the set of constraints defining the set of feasible solutions. Parameters. TSet of periods (Hours) defining the planning horizon. u t Market electricity price at hour t2T. j o A parameter used to define the initial battery capacity (we will consider a battery of 10 MWh in our experiments). 4Deterioration of the battery capacity per cycle. We consider that after 5000 cycles the initial battery capacity decreases up to a 20% of the initial one. j max Conversion rate of the inverter. l A parameter that defines the loss of energy in charging and discharging conversion processes. This parameter is measured as a percentage of the energy purchased or sale. As previously defined, the model parameters are determined by the battery and inverter technologies. Specifically, we consider: the round-trip efficiency in the charge\discharge process ( l ), the initial capacity of battery ( j o ), the conversion capacity ( j max ) and the useful life of the battery (given by the deterioration factor 4, i.e., the number of charging\discharging cycles until the effective battery capacity decreases to 20% of the initial one). It is assumed that the battery capacity diminishes in a linear way with the number of charging\discharging cycles and that the operation and maintenance battery costs are 1% of the equipment value. Variables. The mathematical formulation is as follows: Max X t2T u tðStPtÞ(5) PtðKt1Lt1Þ=ð1 l Þ;t2T(6) StLt1ð1 l Þ;t2T(7) Pt j max a t;t2T(8) St j max b t;t2T(9) a tþ b t1;t2T(10) Lt1þPtð1 l ÞSt=ð1 l Þ¼Lt;t2T(11) Kt¼Kt14: j o: a t1;t2T(12) g 1¼0 (13) Table 3 Financial variables by country. Country T i (%) RRF i (%) r Ei (%) WACC i (%) DR i (%) Greece 28.0% 2.10% 5.76% 4.37% 7.07% Croatia 18.0% 2.75% 6.41% 4.91% 6.99% Italiy 24.0% 1.79% 5.45% 4.33% 6.69% Romania 16.0% 2.25% 5.91% 4.73% 6.63% Spain 30.0% 0.81% 4.47% 3.74% 6.34% Serbia 15.0% 1.75% 5.41% 4.53% 6.33% Czech 19.0% 1.25% 4.91% 4.21% 6.20% Portugal 21.0% 0.96% 4.62% 4.03% 6.10% Slovenia 19.0% 1.01% 4.67% 4.10% 6.06% Norway 22.0% 0.74% 4.40% 3.91% 6.01% Latvia 20.0% 0.80% 4.46% 3.98% 5.98% France 33.3% 0.03% 3.69% 3.31% 5.96% Poland 19.0% 0.75% 4.41% 3.98% 5.92% Hungary 9.0% 1.15% 4.81% 4.41% 5.85% Lithuania 15.0% 0.75% 4.41% 4.08% 5.80% UK 19.0% 0.30% 3.96% 3.78% 5.67% Austria 25.0% 0.01% 3.65% 3.49% 5.66% Netherlands 25.0% 0.17% 3.49% 3.42% 5.56% Sweden 21.4% 0.13% 3.53% 3.53% 5.49% Ireland 12.5% 0.16% 3.82% 3.88% 5.43% Finland 20.0% 0.23% 3.43% 3.52% 5.40% Denmark 22.0% 0.44% 3.22% 3.37% 5.33% Germany&Lux. 15.0% 0.47% 3.19% 3.53% 5.16% Switzerland 8.5% 0.49% 3.17% 3.68% 5.03% a) Binary variables to simulate the behaviour depicted in Fig. 3 a t Binary variable. It takes value 1 if the battery is charged at period (hour) t2T, 0 otherwise. As illustrated in Fig. 3, these variables are activated each time a new charging process is performed, for instance, numbers (1) and (2). Note that in the table at the bottom of Fig. 3, for clarity, 0 values are not represented. b t Binary variable. It takes value 1 if the battery is discharged at period t2T, 0 otherwise. They correspond, for instance, to numbers (3) and (4) in Fig. 3. d t Binary variable. It takes value 1 if the battery has finished a charge\discharge cycle at period t2T, 0 otherwise. For illustration purposes, these variables are represented in the last row of the table at the bottom of Fig. 3. They are in charge of measuring the battery deterioration with use. g t Binary variable. It takes value 1 if the last performed operation prior to period t2Twas a charge and 0 if it was a discharge. These variables will be used as auxiliary variables to enforce cycle occurrences. b) Real variables to measure amounts of stored energy, purchases and sales P t Represents the amount of electricity purchased and stored in the battery during period t2T. Graphically, P t are represented by the height of the rectangles (1) and (2) in Fig. 3. The purchase decision at period tis conditioned by a t . S t Measures the amount of energy discharged from the battery during period t2T. This energy is transformed into electricity and sold at the same period. Variables S t are depicted as the height of the rectangles (3) and (4) in Fig. 3. The sale decision at period tis conditioned by b t . C t Integer variable that measures the cumulative number of charging\discharging cycles that has been done just until period t2T. K t Represents the remaining capacity of the battery at the end of period t2T. It is represented by line B in Fig. 3. LMeasures the level of energy in the battery at the end of period t2T. It is represented by line C in Fig. 3. F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 6 g t1 g t d t;t2T\f1g(14) a t g t;t2T(15) g t1e b tt2T(16) g t1 b t a t g tt2T\f1g(17) g t1þ b tþ a t g tt2T\f1g(18) Co¼0 (19) Ct¼Ct1þ d tt2T\f1g(20) a t; b t; g t; d t2f0;1g;t2T Pt;St;Kt;Lt0;t2T(21) Ct2Nþ;t2T The objective function (5) maximizes the cash-flows obtained from the battery operation during the planning horizon. At each period t, the amount of electricity purchased, when a t ¼1, which is represented by P t , must be less than or equal to the available battery capacity, as stated in constraints set (6), which is unknown and measurable by the term (K t1 L t1 )=ð1 l Þ. Also, the amount of energy will be constrained by the conversion capacity (8). These two constraints are represented by numbers (6) and (5) respectively in Fig. 3. As previously mentioned, l measures the loss of energy as a consequence of the conversion process which takes place at the inverter. Then, ð1 l Þstands for the roundtrip efficiency of the electricity conversion process. Concerning the discharge process, constraints (7) and (9) limit the electricity that can be discharged and sold at period t,S t . In the first case, the discharged energy must be less than or equal to the level of charge of the battery plus the energy lost in the conversion process. Then, at each period t, the maximum possible discharge is L t1 ð1 l Þ. In the second case, electricity sales are also bounded by the inverter capacity trough constraint set (9). These constraints incorporate binary variables a t and b t into the right-hand side, so that, when they are 0, P t and S t are also 0. Since charging and discharging operations cannot be performed in the same period, constraints set (10) are Fig. 3. Illustration of buying and selling operations. Fig. 2. Discount rates and profit taxes by country. F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 7 incorporated into the model. The temporal evolution of the battery level (L t )esee line C in Fig. 3e, which depends on whether a purchase or sale of electricity has been done at period t1, is modelled by constraints set (11). At each period t, the battery level is measured by adding or subtracting the charge (P t ) or discharge (S t ) of energy to the previous battery level L t1 . The battery behaviour also takes into account the decrement of the battery capacity with time, represented by line B in Fig. 3, which is due to the increment in the number of charging\discharging cycles. This deterioration is modelled by constraints set (12). Note that binary variables d t control the occurrence of a new charging/discharging cycle. In order to properly model the behaviour of variables d t a new set of binary variables is needed. To this end, binary step state variables g t are introduced. They represent the last battery performed operation before period t. Suppose, for instance, that the battery was charged at period t¼4 and no other battery operation occurs until period t¼7, where a discharge operation takes place. In this example a 4 must be equal to 1 (a charge), then g 4 ¼ g 5 ¼ g 6 ¼1 (no operation has been done from periods 5 to 4) and finally g 7 ¼1 (since a discharge has taken place). At this point, d 7 must be equal 1, since a new charging\discharging cycle must be computed and the battery capacity must be decreased in 4, which is done by constraint set (20). Constraints set (13)e(18) are used to model this behaviour in a general way. In the initial period, the step state variable is set to zero (13). For the rest of the periods, as stated in (14),if g t1 is 1 (the last operation before twas a charge one) and g t ¼0 (the battery has been discharged at period t), the variable d t must be set to 1. The step state variables are also related to the binary charging\discharging variables, a t and b t respectively. On one hand, if a t ¼1, a charging operation has been done at period t, then g t must be one, as enforced by constraints (15). On the other hand, if a discharge operation has been done at period t, b t ¼1, the battery step state variable g t must be set to zero, as imposed by constraints (16). When the last performed battery operation at time t1 was a charge ( g t1 ¼1) and no charge or discharge has been done at time t, g t must be 1, according to constraints set (17). Finally, when at time t1 the last performed battery operation was a discharge ( g t1 ¼0) and no charge or discharge operation has been done at period t( a t ¼ b t ¼0), the variable g t must be equal to zero, as imposed by constraints (18). Constraints (21) define the domain of variables. The next table summarizes these constraints (Table 4): Constraints set (21) are used to specify the domain of the model variables. - Optimization framework implementation issues. First, since the purchase and sale of electricity per hour depends on the inverter conversion rate, the optimal trading strategy must be obtained by varying the inverter size, using the hourly prices of electricity as an input. Second, at the first glance the length of the planning horizon is unknown since it represents the number of operation periods (hours) until the battery capacity reaches a low enough residual value (in our case 20% of the initial battery capacity) which cannot be determined before running the optimization model enote that the battery deterioration is a consequence of the specific policy followed. Since we are dealing with hourly prices, there are 8,670 time periods per year (24 h per day times 365 days per year). For only one year the model size rises 100,765 constraints, 30,654 continuous variables and 35,033 binary variables. Moreover, the length of Twill be an unknown multiple of 8,670 time periods. To reasonably resolve these shortcomings, we apply a sequential strategy in order to solve the problem; this is depicted in Fig. 4.As shown in the figure, we opt for solving the problem in a sequential way, year by year, beginning each year of the battery operation by using the residual battery capacity obtained after solving the previous year (the upper index in the initial battery levels corresponds to the year of operation). This process is repeated a certain number of times (depending on the inverter size considered) until the battery capacity is exhausted. Without loss of generality, we consider an initial battery capacity of j o ¼10 MWh. The specific battery size considered to solve the problem does not affect the results since they are completely scalable. Which is really important is the rate ‘battery capacity/ inverter size’. Consequently, we solve the optimization model by varying the inverter size from 1 to j o ¼10 MWh, covering rates from 10/1 to 1. For each inverter size, we repeat the optimization year by year, until the maximum battery deterioration is reached. We are assuming that, in any case, the battery size is not enough to influence the market's operation. 4. Financial results and econometric analysis In this section, we analyse the financial aspects of the optimal buying/selling activity determined in the previous section for the 10 MWh battery with 92% of round-trip efficiency and an expected life of 5000 cycles. The following figure (Fig. 5) shows the hours when, during the first operating year and for the different countries, the battery is either buying electricity, selling it, or staying inactive. To simplify the figure, we represent only four of the ten inverter sizes considered (1 MW, 4 MW, 7 MW, and 10 MW) and rank the countries in each plot according to the hours of battery inactivity. Several aspects stand out in the figure. On the one hand, the higher the inverter, the lower the number of hours the battery is active (buying or selling). On the other hand, the total hours buying and the total hours selling are quite similar in each country, which would indicate that the battery charges or discharges using all the capacity of the inverter. Finally, different behaviour is observed between countries and inverters. For example, for the 4 MW inverter, the most active countries (France, Netherlands, Ireland and the UK) spend approximately 2000 h buying and 2000 h selling, while the rest of the year (4760 h) the battery remains inactive; these values contrast with those observed in countries like Norway or Sweden, where the battery is approximately 1000 h buying and 1000 h selling during the first year of operation. Note that these operational differences between countries imply that the battery lasts longer in some countries than others. Therefore, in each country, the business will last until the battery runs out or until it begins to generate negative cash flows (if this occurs); in this last case the residual value of the equipment (battery and inverter) would be recovered. To further investigate the heterogeneity among countries, we represent in Fig. 6 the two countries with the greatest difference in trading strategy when the inverter size is 7 MW, Ireland and Norway. As can be seen, prices in Ireland are on average somewhat higher than in Norway and show a much greater volatility. These two facts help explain one of the results of our financial analysis: the BESS arbitrage is more profitable in Ireland than in Norway. Our optimization model guarantees optimal arbitrage with a Table 4 Relationships among binary decisions variables. g t1 a t b t g t Constraints set 000/0 (18) 010/1 (15) 001/0 (16) 100/1 (17) 110/1 (15) 101/0 (16) F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 8 10 MWh battery for each country and inverter (from 1 MW to 10 MW), given the hourly electricity prices in each country (during the year 2019) and the costs of the battery (100,000 V/MWh) and the inverter (30,000 V/MW). With this optimal trading information, we can generate annual cash-flows by country and inverter, and calculate for each case two financial indicators, the Net Present Value (NPV) and the Internal Rate of Return (IRR). We restrict the coming financial and econometric analysis to the most profitable inverter sizes in each country according to those financial indicators eas represented by Fig. 7. The respective inverter sizes which maximize NPV and IRR indicators do not have to coincide within each country or to be the same across countries eas can be seen in the table attached to Fig. 7. All countries move in negative values of both indicators (the business is not profitable yet), and there is a high positive correlation between the two indicators. Another interesting fact is that the best inverters in terms of NPV do not exceed 6 MW of capacity, while in IRR terms they are usually equal to or greater than 6 MW, with some exceptions such as Sweden or Norway. The economic and technological information dealt with in this article allows us to study the main determinants of the NPV and IRR indicators. For this purpose, we propose the estimation of two Fig. 4. Optimization framework structure inverter. Fig. 5. First year of trading activity by country and inverter. F. Nú~ nez, D. Canca and  A. Arcos-Vargas Energy 242 (2022) 122916 9