*2.2. Open Issues*

There are several open issues that required to be tackled if an accurate, objective assessment of V2G technology is going to be done.


All these challenges have been born in mind to design the mathematical model presented in this manuscript, as well as the calculations and results placed in the next sections. Overall, the model can be described as depicted in Figure 1. The main figures that have been taken into account are acquisition and operational costs, externalities inherent to the vehicle, gas consumption, and maintenance, among others.

**Figure 1.** Common variables considered for the mathematical model and their relations.

#### **3. Mathematical Model for V2G Integration**

The costs that have to be faced by an individual (or a small group like a family that will use a single automobile) can be defined as *CtotICE* for the ICE vehicle, and *CtotV*2*G* for the vehicle-to-grid solution, whereas *Capex* figures for the ICE and the V2G solution could represent the cost of acquiring the vehicle by an individual. Finally, *Opex* figures for the ICE and the V2G vehicles represent the mandatory operational expenses needed to have the asset fully functional. Thus, the total costs for each of the transport solutions can be expressed as in (1) and (2):

$$\mathbf{C}\_{\text{totICE}} = \mathbf{C} \mathbf{p} \mathbf{e} \mathbf{x}\_{\text{ICE}} + \mathbf{Q} \mathbf{p} \mathbf{x}\_{\text{ICE}} \tag{1}$$

$$\mathcal{C}\_{\text{tot}\,V2\,G} = \mathcal{C}ap \mathbf{ex}\_{V2\,G} + \mathcal{O}p \mathbf{ex}\_{V2\,G} \tag{2}$$

Note that it has been chosen to consider each of the vehicles as an asset rather than a liability due to V2G potential to generate revenues or at least reduce operational costs. Inflation can also be considered in the *Opex* expenditures by adding its corresponding parameter (represented by *Inf*) into the previous equations, as long as it is defined for a specific amount of time. Typically, inflation will build up as time goes by as a function, or a part of one, where time piles up on an exponential basis. Therefore, inflation-adjusted prices have been added as shown in (3) and (4).

$$\text{C}\_{\text{totICE}} = \text{Capc} \text{x}\_{\text{ICE}} + \sum\_{i=1}^{j} \left(1 + \text{Inf}\right)^{i} \times \text{Oper}\_{\text{ICE}} \tag{3}$$

$$\text{C}\_{\text{tot}\,V2G} = \text{Capex}\_{V2G} + \sum\_{i=1}^{j} \left(1 + \text{Inf}\right)^{i} \times \text{Opex}\_{V2G} \tag{4}$$

Additionally, *Capex* for the EV must be further defined as the costs of installing the required infrastructure to transform the EV into a vehicle with V2G capabilities (represented as *Vconv*) and to charge the EV at home (represented by *Heq*), along with the cost of the vehicle itself ( *CEV*). It has been done in (5). As it can be inferred, these technological needs do not apply for the internal combustion engine vehicle.

$$
\mathbb{C}\text{Capex}\_{V2G} = \mathbb{C}\_{EV} + V\text{conv} + H\text{eq} \tag{5}
$$

The *Opex* for ICE and EV must be further analyzed. It is shown in (6) what the *Opex* value is for an ICE vehicle. During a period of time that ranges from *i* to *j*—considering *i* as a year and *j* as twelve, which has been estimated as the average lifetime of a vehicle, according to a) what is used in [24] and b) the estimation done in [25]—four aspects will add up to the final figure of the expenses: the yearly costs of the externalities of the vehicle (*ExICE*), fuel consumption (*FICE*), and maintenance expenses ( *MICE*). As far as the V2G solution is concerned the equation is represented by the same kind of terms used for the ICE vehicle (7). Nevertheless, contrary to ICE vehicles, in this case, the expenses, maintenance and electricity costs need to be further defined, as it is detailed through the following subsections.

$$\text{Oper}\_{\text{ICE}} = \sum\_{i=1}^{j} \left( \text{Ex}\_{\text{ICE}} + M\_{\text{ICE}} + F\_{\text{ICE}} \right)\_i \tag{6}$$

$$\text{Opex}\_{\text{ICE}} = \sum\_{i=1}^{j} \left( \text{Ex}\_{\text{ICE}} + \text{M}\_{\text{ICE}} + F\_{\text{ICE}} \right)\_i \tag{7}$$

#### *3.1. Cost of the Externalities of the Vehicle*

The externalities that have been presented will o ffer di fferent values depending on whether an ICE vehicle or a V2G is used. In the first case, as represented in (8), these externalities will be closely linked to the cost of the health impact caused by ICE-based vehicles (*hICE*), the distance run with the vehicle ( *D*), as well as the average consumption of gas (*AvconsICE*), carbon emissions (depicted as *CICE* for the ICE vehicle) and the social cost of carbon (*SCC*) during a certain period of time. These externalities are also reflected for the V2G solution in (9), where the equivalent data has been included. Health impact (*hV*2*G*) and carbon emissions ( *CV*2*G*) are harder to measure in the case of V2G, as they are related to the energy mix from which electricity is coming and, more specifically, the amount of renewable energies present in this energy mix. The cost of the electricity used to move the vehicle (*Econs*) can be determined by the trading operations that can be done by the owner of a V2G automobile (even though it will be usually lower that the cost of oil-based fuels). Overall, the cost of these externalities for society has been represented in a manner resembling the one used in [9], as there were concepts such as SCC or distance that had to be taken into account in the same way as it was done in this related work.

$$\text{Ex}\_{\text{ICE}} = \sum\_{i=1}^{j} \left( h\_{\text{ICE}} \times D + \text{C}\_{\text{ICE}} \times \text{SCC} \right)\_i \tag{8}$$

$$\text{Ext}\_{V2G} = \sum\_{i=1}^{j} \left( h\_{v2g} \times D + \text{C}\_{V2G} \times \text{SCC} \right)\_i \tag{9}$$

#### *3.2. Cost of Yearly Fuel Consumption*

There are several aspects that must be taken into account when including yearly fuel consumption in the mathematical model. For instance, the cost of the energy bought and the price set to sell it back to the market, so that the end user will use arbitrage to their advantage. This feature will be dependent on, among other aspects, two main factors: a) buying and selling actions that take place during energy cost peak or valley hours (while overall an average price for electricity may di ffer during the day depending on the user tari ff, the V2G infrastructure will take advantage of a peak/valley hours setting, as depicted in [35]), and b) the possibility for the end user of the V2G to buy and sell energy at a suitable time

for their interests. The latter implies that due to the usage of the vehicle or their end users' working schedule, they may not be able to charge completely their V2G during valley hours and sell all the electricity during peak hours. All these factors have been taken into account in the mathematical model presented in this manuscript: while energy is bought and sold in the average prices set for valley hours (*Avcbuy*) and peak hours (*Avcsell*), real price of electricity when both purchased and sold is obtained as the combination of *Avcbuy*,*Avcsell* and the addition of four different factors that range from 0 to 1, which effectively describes the percentage of the energy that can be bought and sold during each of the two possible time periods (valley or peak hours). They are used to represent the fact that it will be very difficult for regular end users to buy and sell electricity during all-optimal time periods, so there will be just a majority of power bought and sold when it is best for the end user. They are called *fbb* (for *factor* of energy *bought* during optimal *buying* period), *fbs* (for *factor* of energy *bought* during optimal *selling* period), *fsb* (for *factor* of energy *sold* during optimal *buying* period), and *fss* (for *factor* of energy *sold* during optimal *selling* period). The efficiency to buy and sell electricity at the suitable moment has been estimated at 80% (hence, the 0.8 value of *fbb* and *fss*), so some power will have to be transferred when it is least optimal for them (estimated at 20%, hence the 0.2 value of *fbs* and *fsb*). This has been done so because there are some examples in literature that show how a portion of the EV charge is done in suboptimal periods of time. For example, it is shown [36] that there is some charging done halfway through the day, which is usually the daily time period when electricity prices gone from valley to peak in two-levelled tariffs. Equally, it is shown in [37] how charge estimations done can take place around 6 p.m., a time of the day that is often part of peak hours. These principles have been included in Equations (10) and (11), which represent the final cost of buying (*Crpbuy*) and selling energy (*Crpsell*) when suboptimal intervals are included. These equations are defined like this because it is assumed that there are basically two levels of prices with small fluctuations inside them (as seen in [32]).

$$C\_{rphyy} = Avc\_{bny} \times f\_{bb} + C\_{s\text{ell}} \times f\_{bs} \tag{10}$$

$$\mathbb{C}\_{rphuy} = A \text{vc}\_{buy} \times f\_{b\vartheta} + \mathbb{C}\_{\text{sell}} \times f\_{\text{bs}} \tag{11}$$

Fuel costs are modelled differently depending on the vehicle that is used as a private transport mean. If the ICE-based solution is used, gas costs will be as shown in (12). It is basically the same way that diesel fuel costs are described in [9] (average fuel consumption *Avcons* and gas price *C fICE* have been used as variables), with the exception that figures used in this case correspond to private automobiles. The equation in (13) shows how yearly costs would be for the V2G solution. Unlike an ICE automobile, it relies heavily on the trading activities that are done with the energy stored in the battery of the V2G vehicle, which imply buying and selling energy (represented in the formula as *Ebuy* and *Esell*) to different costs: one to buy it—*Crpbuy*—and a different one to sell it—*Crpsell*. Buying prices are expected to be lower than selling ones; otherwise, the opportunity to make up for some of the expenditures will be lost). As it can be inferred, if during a certain period of time there is more energy sold than the one consumed, electricity cost will result negative for the V2G, which means that the owner of the vehicle will be obtaining a profit from trading with the electricity, rather than just reducing its costs via V2G usage. Both equations have included the inflation rates for ICE fuel and electricity (*In*ff ). Lastly, since according to [17] there will be 95% efficiency when charging a vehicle via plug-in charging mode, an efficiency factor (*ef*) has been introduced to reflect the small loss of charge when energy is transferred in and out of the electric vehicle.

$$F\_{\rm ICE} = \sum\_{i=1}^{j} \left( 1 + \ln f f \right)^{i} \times \left( A v\_{\rm consICE} \times \mathbb{C} f\_{\rm ICE} \right) \tag{12}$$

$$F\_{V2G} = \sum\_{i=1}^{j} \left(1 + Inff\right)^{i} \times \left(E\_{buy} \times \mathbb{C}\_{rpbuy} \times \varepsilon\_{f} - E\_{\text{sell}} \times \mathbb{C}\_{rpsell} \times \varepsilon\_{f}\right) \tag{13}$$

The cornerstone of the vehicle-to-grid technology is the capability to sell electricity to the power grid where it is installed, since it offers a unique selling point that cannot be found in other regular vehicles. Thus, the energy that can be sold back to the system has been accounted in (14). If a yearly period is considered regarding the energy that can be sold (*Esell*), then the overall available energy to trade—that is to say, energy that can be sold during peak hours, as opposed to the most advisable time to purchase it, which would be valley hours—will be the remaining energy after considering two variables from all the energy that has been bought for charging the battery (*Ebuy*): a) the energy consumed to move the automobile (*Econs*) and b) the passive discharge of the battery when it is idle (*pdis*). Yearly amount of energy sold and bought from the power grid can be considered after learning past patterns in energy pricing and consumption. Information for a long-time span can be found from the transport system operator if required [38].

$$E\_{\rm scl} = E\_{\rm buy} - E\_{\rm cons} - p \text{dis} \tag{14}$$

In order to understand the previous equation, *Ebuy* and *Econs* must be defined too. The energy that is bought for the battery of the V2G will result from calculating the amount of power (*Pw*) purchased during a certain period of time (*t*). However, the degradation of the battery will take its toll during the battery lifetime, resulting in declining energy storage capabilities. In addition, the passive discharge of the battery must also be born in mind. While it is negligible in the short term, its effects are more noticeable during the whole lifespan of the battery. Lastly, the difference between the nominal and the actual battery charge values must also be considered. These two latter variables are hard to quantify and no work from the literature seems to portray them in an accurate manner in mathematical models for V2G technology. As far as the V2G model is concerned, they have been included as *Dg* (degradation factor for the battery). When numerical values are used to evaluate the model, the maximum discharge speed of the battery will also have to be considered as a non-functional requirement, as no battery can provide an immediate amount of limitless energy. Due to this, *Dg* will have a role in the model, even though differences may not be that significant according to D. Wang et al. [39] or H. Ribberink et al. [40]. Battery degradation for purchased energy has been included in (15).

$$E\_{hyy} = \sum\_{i=1}^{j} (Pw \times t)\_i \times (1 - Dg) \tag{15}$$

Battery degradation has been estimated by the authors of this manuscript to be at 1.25% of its total capacity per year so it can be included with more accuracy in the mathematical model. The reviewed literature shows extreme disparity regarding this value, with some sources claiming that it will degrade up to 10% after 160,000 miles for an electric vehicle [41]. However, battery degradation considered for this scenario has been regarded as significantly higher, as a) suboptimal charge and discharge behavior patterns from the vehicle owners must be taken into account, and b) V2G usage of an electric vehicle implies a heavier utilization of the vehicle battery. A more realistic approach is found in [42], where a thorough V2G-based experiment was run with experimental lithium batteries showing that they would reach their end of life (EOL), regarded to be the point when the battery has lost 20% of its original maximum capacity retention, after 3000 cycles of charge and discharge. For the purpose of this mathematical model, it has been estimated that, on average, 1000 cycles will take place every year for the V2G solution (as described in [43]), and after eight years the battery total capacity will be depleted a 20% and have to be replaced with a new one. Thus, battery degradation is defined as represented in (16).

$$D\mathbf{g} = 0.0667 \times \sum\_{i=1}^{j} i \tag{16}$$

The energy that is consumed by the V2G solution can be described as the average energy consumption of the vehicle during a specific distance (*Econs*). As explained before, passive energy losses have been included as the *pdis* parameter.

#### *3.3. Cost of Maintenance*

Although it is not bound to happen inevitably, the battery used in the EV-V2G is very likely to eventually have to be replaced. However, it does not necessarily mean that the vehicle owner will pay for the full replacement if the vehicle has been acquired under a battery leasing agreement. Therefore, there are two possible options: if the vehicle and the battery are purchased, battery replacement costs will have to be considered; with the technology available today in commercial products, it is unlikely that a vehicle battery will outlive the vehicle itself. The other option, though, is that the vehicle manufacturer leases the batteries to the vehicle owner during a certain time period. In this way, battery reposition could be regarded as a periodic paymen<sup>t</sup> (*Bleasi*) done during the lifetime of the vehicle. This latter scenario is modelled in (17) as *Bleasi*; while this is not the default choice for consumers buying an electric vehicle, it is a feature usually overlooked in other models for V2G, so it has been included in this analysis. When price data are used to estimate the cost differences between acquiring and leasing the battery in the V2G, *Bleastot* would be used as the maintenance cost for rented batteries, whereas *Batr*, added in (19), will be used as the parameter representing the cost of a battery replacement when the battery is purchased with the EV.

$$Blcas\_{tot} = \sum\_{i=1}^{j} Blcas\_i \tag{17}$$

Lastly, maintenance costs have been included in the model as a way to evaluate the differences between the two kinds of vehicles. The ICE vehicle (18) makes use of a maintenance rate (*DrateICE*), in a way that resembles the one presented in [9], but using private transport rather than a school bus. Labor costs of refilling the fuel (*Lab*) and distance (*D*) have also been included. Furthermore, the equation conceived for the V2G solution (19) is making use of an equivalent rate (*DrateV*2*G*) and a distance *D* and the cost of one battery replacement (*Batr*). Taking into account the average lifetime of EV vehicles and of their batteries before a replacement (which can be estimated at roughly eight years according to the period warranty used in most car manufacturers [43,44]), it is more likely that a new vehicle will be acquired rather than a new whole battery is bought more than once. The equation that has been added as (19) can be modified to consider how battery costs impact the maintenance of a V2G solution when the battery is leased instead of purchased (20). Note that both kinds of vehicles will require the paymen<sup>t</sup> of yearly insurance costs (which has been represented by *Ins*). However, according to [45], their paymen<sup>t</sup> can be regarded to be the same for them.

$$M\_{\rm kCE} = \sum\_{i=1}^{j} \left( Data\_{\rm lCE} \times D + Lab + lns \right)\_i \tag{18}$$

$$M\_{V2G} = \sum\_{i=1}^{j} \left( True\_{V2G} \times D + Ins \right)\_i + Butr \tag{19}$$

$$M\_{V2G} = \sum\_{i=1}^{j} \left( True\_{V2G} \times D + Ins + Blens\right)\_i \tag{20}$$

#### **4. Numerical Assessment**

The equations of the mathematical model described previously have been put to use for three different use cases, namely, professional drivers (that is to say, people that drive as a way to make their living), frequent drivers (people that drive on a usual basis), and occasional drivers (people that drive rarely), under certain considerations and assumptions as described in the following subsections. Most references and subsidy figures that have been used are relative to the United States of America, due to the fact that it is one of the places where the amount of information was plentiful enough to obtain the data used in this study. Specifically, data for professional drivers was very reliable as it was based on statistics from taxi drivers that are o ffered online freely. The definition of these use cases is pivotal for the study that has been carried out, as the usage that is done of the V2G solution di ffers greatly in each of them. Depending on the usability of the vehicle for travelling, V2G capabilities will become prominent. For example, the greater amount of distance that a V2G solution works, the lower energy will be left to trade it when it is suitable.
