To determine V2G revenue potentials, we developed an aggregated storage optimization model that covers the use cases of arbitrage trading in the spot markets. The holistic implementation of all V2G use cases facilitates the applicability and avoids building several parallel models.

A simplified representation of the modeling process is displayed in

Figure 1, where the model consists of four different parts. Based on a model of the EV’s battery, optimized charging strategies can be developed for different scenarios depending on charging and discharging restrictions. An uncoupled aggregated EV pool can be modeled to participate in different markets, where marketing strategies are optimized with a rolling forecast of market prices. The optimized strategy depends on electricity prices of the respective energy markets and is based on historical and simulated future market prices. In addition to bidirectionally chargeable EVs, two reference scenarios are considered that cover smart, unidirectionally chargeable EVs and simple, directly chargeable EVs. The entire model is implemented in Matlab, where a CPLEX solver (optimization software package) is used for optimization.

#### 2.1. Modeling of Bidirectionally Chargeable Electric Vehicles

Modeling a bidirectionally chargeable EV consists mainly of a model of the electric battery of the vehicle similar to stationary electricity storage. The EV battery is modeled by the storage equation displayed below, which relates the state of charge (SoC) of the battery to the different amounts of electricity charged into or discharged out of the battery:

here,

t stands for the modeled point in time,

$\u2206t$ is the difference between two points in time and

$\eta $ represents efficiency, which differs for the charging and discharging process.

${P}_{charge}\left(t\right)$ is the charging power and

${P}_{discharge}\left(t\right)$ the discharging power at the modeled time t. The variables are considered on the alternating current side of the charging station and thus correspond to the amounts of energy traded in the market.

${P}_{schedule}$ is the sum of purchases and sales already made at the modeled time

t. Accordingly,

${P}_{counter-purchase}\left(t\right)$ and

${P}_{counter-sale}\left(t\right)$ represent the power which can be purchased or sold in the market to counteract transactions that have already taken place (countertrading). For example, purchased energy in the day-ahead market could be sold in the intraday markets resulting in a countertrade

${P}_{fastcharge}\left(t\right)$ is the power, that can be used to rapidly charge the vehicle if necessary and

${E}_{consumption}\left(t\right)$ is the EV’s energy consumption by driving at time t. The limit values of all time-dependent variables are explained in the following section.

Table A1 in the

Appendix A provides an overview of all these variables and their limit values.

#### 2.1.1. State of Charge

$SoC\left(t-1\right)$ represents the battery’s storage level at the point in time prior to the modeled point in time. The change in capacity of the battery storage corresponds to the difference from $SoC\left(t\right)$ to $SoC\left(t-1\right)$. Due to the limited storage capacity of the electric vehicle and user requirements for a minimum storage level, the variable $SoC\left(t\right)$ can assume a limited range of values.

The maximum storage level of the battery storage is limited by

$So{C}_{max}$. If a value of 100% is set for this parameter, the entire storage capacity is available for the charging process. The minimum value

$So{C}_{min}$ varies depending on the vehicle’s location status, which is known for each point in time. Equation (2) summarizes the values that

$So{C}_{min}$ can assume:

If the vehicle is connected to the electric grid, $So{C}_{min}$ equals $So{C}_{min,safe}$. The storage level must not fall below this value or must load onto it as quickly as possible. If the vehicle is connected to a charging station and is at the point of departure, $So{C}_{min}$ assumes the value $So{C}_{min,dep}$. Before departure, the charging strategy is thus optimized in a manner such that the SoC at the time of departure at least corresponds to $So{C}_{min,dep}$. If the vehicle is not connected to the electric grid, the value $\text{}So{C}_{min,disconnected}$ results in the minimum SoC.

To ensure that

$SoC\left(t\right)$ lies between the minimum and maximum possible storage level, Equation (3) is implemented:

where

C describes the storage capacity of the electric vehicle and

${P}_{supplement}$ stands for additional, theoretical electric power to be charged.

${P}_{supplement}\left(t\right)$ is incorporated to meet the storage level restriction in Equation (2) at any time. The possibility that the value of the storage level is below

$So{C}_{min}$ exists even if the EV is connected to the electric grid. This might occur if the storage level is lower than the minimum SoC when the vehicle arrives at a charging station or if it is not possible to charge to the minimum SoC before departure because the vehicle has not been connected for long enough. In the case that the minimum storage level cannot be reached with the charging strategy,

${P}_{supplement}\left(t\right)$ takes on a value greater than 0 to simulate a hypothetical charging process. The variable thus can be interpreted as penalty costs that arise from the driving behavior of a user who disregards the requirements for a minimal SoC. By introducing

${P}_{supplement}$(t), the model can optimize all driving profiles regardless of driving behavior or consumption, so that a selection of unsuitable driving profiles does not have to be made in advance.

${P}_{supplement}$(t) is not taken into account in the storage equation and does not change the actual storage level.

#### 2.1.2. Charging/ Discharging Power and Already Traded Energy

In the storage Equation (1), ${P}_{charge}\left(t\right)$ and ${P}_{discharge}\left(t\right)$ describe the purchase and sale of power and determine the change in storage capacity for each time step. Due to the limited power of any EV charging station, the variables are limited to the maximum charging and discharging power ${P}_{charge,max}\left(t\right)$ and ${P}_{discharge,max}\left(t\right)$ and to the minimum charging and discharging power ${P}_{charge,min}\left(t\right)$ and ${P}_{discharge,min}\left(t\right)$. If the vehicle is connected to a charging station, the EV battery can be charged with ${P}_{charge,max}\left(t\right)$ or discharged with ${P}_{discharge,max}\left(t\right)$. If the vehicle is not connected to the electric grid, both maximum and minimum charging and discharging power become 0.

The boolean variables

${b}_{charge}$ and

${b}_{discharge}$ describe the state of the battery during charging and discharging processes. If a charging process takes place,

${b}_{charge}$ is

true. Analogously, the variable

${b}_{discharge}$ becomes

true during discharging. Due to the fact that it is not possible to purchase and feed electricity into the electric grid at the same time, only one of the boolean variables can assume the value 1 (=

true) at the modeled point in time Equation (4). A simultaneous purchase and sale in the market is, therefore, excluded.

The resulting constraints regarding these variables are shown in Equations (5) and (6):

There is a possibility that the storage capacity of the electric vehicle has already been marketed through previous trading on the electricity markets, for example through consecutive trading on different spot markets. Such traded power must be taken into account in subsequent storage optimizations. In Equations (5) and (6),

${P}_{schedule,purchase}$ and

${P}_{schedule,sale}$ correspond to already made purchases or sales at the modeled time t. These amounts of electricity reduce the maximum charging or discharging power in such a way that only the capacity that has not yet been traded can be marketed.

${P}_{schedule}$ can be defined as the difference between

${P}_{schedule,purchase}$ and

${P}_{schedule,sale}$ Equation (7) and is included in the storage Equation (1).

#### 2.1.3. Countertrades

Electricity spot markets in Germany include consecutive day-ahead and intraday trading resulting in the opportunity to countertrade day-ahead purchases or sells in the intraday market. The model not only accounts for already marketed storage capacities, it also includes the possibility of compensation transactions (countertrades) that compensate for the previous trade in the opposite direction. In this regard, the variable

${P}_{counter-purchase}\left(t\right)$ corresponds to a buyback (counter purchase), the variable

${P}_{counter-sale}\left(t\right)$ to a sellback (counter sale) of already traded storage capacities. The volume of a countertrade can at most assume the previously inversely traded amount of energy. Countertrades do not describe a physical loading or unloading process. The resulting constraints are shown in Equations (8) and (9):

Similar to the charging and discharging processes, the boolean variables

${b}_{counter-purchase}$ and

${b}_{counter-sale}$ describe the state of countertrades. If a counter-purchase takes place at time t (

${b}_{counter-purchase}=1$), the EV battery cannot be discharged at the same time. Conversely, no charging process can be conducted during a counter-sale (

${b}_{counter-sale}=1$). Equations (10) and (11) show these constraints:

#### 2.1.4. Electricity Consumption and Fast Charging

The battery of a vehicle has an electric energy consumption

${E}_{consumption}\left(t\right)$ at the modeled time t, which is considered in the storage equation. Due to the foresight of the driving profiles (explained in

Section 2.4), EV consumption is known at all times. Each driving phase of the vehicle results in a capacity reduction.

Depending on the user’s driving behavior, it might occur that the electricity consumption of an EV is so high at one point in time that the current storage level is not sufficient to meet the energy demand, for example if the EV has not been connected to a charging station for too long. In this case, the fast charging power ${P}_{fastcharge}\left(t\right)$ is utilized to comply with the restrictions of $So{C}_{min}\left(t\right)$ and to avoid a supposed negative storage level. The employment of the fast charging process is accompanied by an increase in storage capacity. ${P}_{fastcharge}\left(t\right)$ represents the charging power at a public charging station rather than at a bidirectional charging station. The vehicle user is thus given the opportunity to charge the vehicle on the road.

#### 2.2. Formulation of Optimization Model

The developed model of a bidirectionally chargeable EVs allows for the implementation of different optimized charging and discharging strategies, which differ in particular in the structure of the objective function. For the assessment of the use cases of arbitrage trading on the day-ahead market as well as on the intraday market, three different charging strategies are implemented: a strategy for bidirectional charging, a strategy for smart charging, and a strategy for unmanaged charging.

First, the bidirectional charging strategy, which allows for charging and discharging of the EV, is restricted to the storage equation and its aforementioned constraints. The objective of this charging strategy is to charge at minimum costs while discharging at maximum revenue. To do so, the objective function of the optimization model aims at minimizing all costs considered:

where

T corresponds to the number of time steps of the optimization. Depending on the respective market, traded energy quantities per time step as well as corresponding market prices

${p}_{market,i}\left(t\right)$ are considered. In this regard,

${p}_{market,buy}$ is the price at which electricity is bought and

${p}_{market,sell}$ is the price at which electricity is sold, where both prices can include respective transaction costs and possibly additional electricity price components. Charged power corresponds to a purchase transaction and is associated with costs as is each counter purchase of power. In contrast, discharged power and counter sales are traded with corresponding revenues, which is why

${P}_{discharge}$ and

${P}_{counter-sale}$ are subtracted.

Fast charging power and supplement power are also included in the objective function. Both fast charging costs

${p}_{fastcharge}\left(t\right)$ and penalty costs

${p}_{supplement}\left(t\right)$ are fixed to be a relatively high value, so that only the minimum necessary power is charged to meet the requirements for minimum storage level. As

${P}_{fastcharge}\left(t\right)$ should only be utilized to the extent that a negative

SoC is avoided,

${p}_{fastcharge}\left(t\right)$ should be selected sufficiently larger than

${p}_{supplement}\left(t\right)$. Thus, the following condition must also be fulfilled to guarantee a functioning bidirectional charging strategy:

Second, the smart charging strategy is implemented as a reference scenario to simulate already existing smart charging stations. The objective is to minimize electricity purchase costs by intelligent charging of the EV. Since discharging the EV battery is impossible in this scenario, Equation (14) is implemented. By eliminating the discharge power, the objective function already defined via Equation (12) for the bidirectional charging strategy can also be used for the smart charging strategy.

Third, the unmanaged charging strategy accounts for today’s most commonly installed simple charging stations as a second reference scenario, where the EV battery is charged as soon as the vehicle is connected without an optimized charging control. As with the smart charging strategy, discharging is not possible Equation (14). The aim of the unmanaged charging strategy is, therefore, to maximize the storage level that is equal to minimizing the negative value of

$\mathrm{SoC}\left(\mathrm{t}\right)$ at all times. The battery is accordingly charged at maximum charging power until the battery’s storage level corresponds to

$So{C}_{max}\left(t\right)$ or until the EV leaves the location. In addition, as for the previously explained strategies, fast charging costs and penalty costs for an insufficient SoC with regard to the location-based limit values are included. The resulting objective function is expressed as follows:

#### 2.3. Optimization with Limited Forecast in Consecutive Spot Markets

To investigate the influence of different characteristics and requirements of the considered markets on revenue potentials of bidirectional charging, optimized trading strategies based on price forecasts are simulated by a rolling optimization model, where realistic trading behavior results from a limited foresight of market prices.

Acting in the market under uncertainty is modeled in the same manner as described in [

17], where each day is divided into 8 time slices of three hours each. The model regards real trading times in the spot markets.

Figure 2 illustrates the methodical procedure of consecutive trading in the day-ahead and intraday markets with rolling price forecast horizons, where each horizontal bar displays the prices known in the respective optimization run of three hours. At 12 noon of day

$d$, for instance, a market participant sees averaged continuous intraday prices of the following 12 quarter-hourly products. At the same time, less precise quarter-hourly prices of the continuous intraday are assumed for the interval from 3 pm to midnight. For day

$d+1$, day-ahead market prices are known and for

$d+2$ a forecast of the day-ahead market prices is presented. The participant’s trading decision, which is the optimized marketing strategy, is based on this limited foresight.

As described by the example, foresight of market prices varies for the individual markets. Since the auction on the day-ahead market takes place daily at 12 noon for the respective following day

$d+1$, precise prices forecasts for

$d+1$ are known shortly before 12 noon on day

$d$. To prevent unrealistic trading behavior at the end of day

$d+1$, such as discharging all batteries to maximize revenues, estimated prices for day

$d+2$ are included in the forecast horizon, where prices are also presented at 12 noon of day

$d$. The forecast period

$d+2$ can be one or more days representing a worse or better foresight and is evaluated in

Appendix B. The length of the optimization time steps for day-ahead trading is 1 h.

For the intraday auction, precise price forecasts of day $d+1$ are known shortly before 3 pm of day $d$, since the auction takes place at 3 pm. The length of the optimization time steps is 0.25 h.

Following the intraday auction, continuous intraday trading starts at 4 pm with quarter-hourly products, which defines the length of the optimization time steps. Here, a first forecast horizon of relatively precise prices is set to three hours covering the following 12 quarter-hourly products, where prices are based on trading transactions of these three hours. For the period following the three-hour time window, all continuous intraday transactions of this interval are used to calculate a second forecast price, thereby reflecting the uncertainty of market prices.

Hence, optimization runs before noon include market price information of the remaining day

$d$ and the following day

$d+1$. The optimization runs from 12 noon on the trading day also include day

$d+2$. The total revenue of the marketed EV battery corresponds to the summed costs and revenues of all traded products (filled areas). The cross-hatched areas in

Figure 2 are not regarded as revenues, since these are only price forecasts serving as reference points for the trading strategy.

If consecutive trading takes place in several markets, storage capacities already marketed must be taken into account in subsequent optimization runs and can be countertraded as described before. The storage level at the end of real continuous intraday trading (filled blue area) of each optimization run determines the actual charging and discharging behavior of the vehicle. This storage level is applied as the starting value for the subsequent optimization run.

#### 2.4. Input Data and Parameterization of Electric Vehicle (EV) Pool Scenarios

In the model, parameters related to the EV are the battery’s storage capacity

$C$, charging and discharging power

${P}_{charge/discharge}$, and different efficiency parameters. To investigate the range of revenue potentials in detail, three different sets of EV parameters are implemented: First, a currently common-sized EV is modeled (EV1), comparable to a 2018 BMW i3 [

18] and a 2018 Renault Zoe [

19], using realistic values regarding storage capacity, charging and discharging power and efficiencies. Second, a relatively large EV and a highly efficient charging station are defined representing a future EV (EV2). Third, a set of ambitious, yet plausible future values is selected to model maximum revenue potentials (EV3). These parameter sets were discussed and agreed upon within the research project BCM.

Table 1 summarizes the chosen parameter values for the three sets of EV models.

All losses and efficiencies considered in the model are based on discussions and on the consultation with experts from the BCM project. Other studies assume roundtrip efficiencies that are similar to EV1 [

13] or slightly lower [

14]. Constant values are set for the efficiencies in order to allow for a linear optimization problem, which results in much faster optimization times and thus enables many more optimization runs, i.e., more results. In real operation, however, efficiencies follow a declining, non-linear course for decreasing charging power. Hence, resulting revenue potentials of the presented model overestimate real revenues of bidirectional charging.

The user parameters result from characteristics, requirements and behavior of the vehicle user. In contrast to a large-scale stationary storage system, the battery of an EV is not continuously connected to the grid. The availability of an EV battery for V2G use cases largely depends on the individual driving profile of the user, the location of an appropriate charging station and the probability that the user has connected the vehicle to this charging station.

As a detailed representation of the driving behavior of the user, vehicle-specific driving profiles describe the EV’s whereabouts as well as its energy consumption while driving in a chronological sequence. Based on data regarding household and route information as well as individual user logbooks from the “Mobility in Germany 2017” study [

20] and a methodology first developed in the MOS 2030 [

21] project, annual driving profiles of various EVs are created that are available as

Supplementary Materials (see

Section 6). Each profile meets the following standards:

A change of location is always accompanied by a driving phase.

During each driving phase, the EV has discrete consumption, which leads to a reduction of the storage level.

The EV can be located and connected either at the place of residence, the place of work or the public space

The temporal resolution of the driving profiles is quarter-hourly intervals. For each profile, the energy consumption is calculated based on information regarding driving speed, outside temperature and vehicle type.

The basic data is additionally used to cluster these driving profiles into user groups to further analyze the influence of user behavior on revenue potentials resulting in a set of commuter groups which display typical commuter behavior, and a set of non-commuter groups with homogeneous behavior different to commuter behavior. The commuter set consists of 12 commuter groups. These are defined by the time of arrival of the vehicle at the place of work and the distance traveled from the place of residence to the place of work. The non-commuter set is made up of three user groups, which are determined by age and number of persons in a household. The number of created commuter and non-commuter profiles per group reflects the real distribution within the German vehicle fleet [

22]. Since revenue potentials are strongly related to driving behavior, these two different pools of driving profiles are defined as input for the model:

Table 2 summarizes the characteristics of the two pools of driving profiles including the probability of the EVs’ whereabouts, which is the averaged probability of the EV’s location at any given point in time. The sum of probability of all three locations apart from the driving phase is 94.5% for the commuter pool and 96.8% for the non-commuter pool, which represents the theoretical availability for bidirectional charging management if an appropriate charging station is installed at their location. To analyze the influence of possible charging station locations on revenue potentials of the discussed V2G use cases, the charging point location parameter can be flexibly selected in the model for each individual EV, where the distinguished three locations can be individually defined as available for bidirectional charging or not available.

The probability of each individual EV user to plug the vehicle into an available bidirectional charging station upon arrival determines the plug-in probability, where the expected value of a normally distributed probability is defined as a parameter. A higher plug-in probability results in a greater availability of the EV for V2G use cases. The parameter can be set flexibly to any value between 0% and 100%. As users will most likely be rewarded in some way for plugging in their EV, the plug-in probability is expected to be very high, up to a 100% certainty.

The parameter $So{C}_{min,\mathrm{safe}}$ states the minimum storage level not to be undercut when the EV is connected to the electric grid, which guarantees a certain safety range in the event of an unscheduled departure. This parameter can be set flexibly to meet the requirements of users. The storage level that must be reached at the time of a scheduled departure, $So{C}_{min,dep}$, should be adjustable by the user according to his/her preferences in a real implementation. In the model, the parameter can be set between 0% and 100%.

The charging and discharging behavior model is determined in particular by the time series of market prices. For the use cases of arbitrage trading, actual price time series of day-ahead and intraday markets from 2019 are used to represent price forecasts of a maximum of two and a half days [

23,

24]. For trading in the day-ahead and the intraday markets, corresponding auction prices of 2019 are used. Regarding the continuous intraday market, real prices are bilaterally determined for each transaction, where buy and sell orders are constantly matched. Thus, two representative forecast prices are determined. For the relatively precise forecast of the next three hours after modeling time t, ID3 is calculated, which is the volume-weighted quarter-hourly price of all transactions in the market for the last three hours, where market liquidity is sufficient to determine a representative market price. For the more uncertain time beyond three hours after modeling time, ID

_{Avg} is used, which is the volume-weighted quarter-hourly price of all transactions for this forecasted time horizon.

The regulatory framework for bidirectional charging applications is not yet fully defined to the point that simulated revenue potentials might determine what kind of regulatory incentive or obstacle enables or respectively prevents the considered V2G use cases. A market design with a reduction of different electricity price components such as grid fees and taxes would decrease the marginal costs of the EV accordingly and thus lead to an increased discharging behavior. To incorporate this highly important role of the market design in the model, various values are assigned to the additional charges on purchased energy parameter and resulting differences in revenues are assessed. The applied additional charges on purchased energy range from 0 €/MWh, which corresponds to a complete exemption from all additional electricity price components, to 234 €/MWh, which reflects the amount of all electricity price components for households in Germany in 2019, excluding electricity purchase prices [

25].