A Survey Data Approach for Determining the Probability Values of Vehicle-to-Grid Service Provision

Abstract

One of the key aspects of vehicle-to-grid technology (V2G) is the analysis of uncertainty in electric vehicle user behavior. Correct estimation of the amount of available energy from electric vehicles that are expected to provide ancillary services to the electricity system operator or to secure the end user’s demand is essential to design these services in an appropriate way. Therefore, it is necessary to analyze the probabilities of V2G service performance for different scenarios. This paper presents the author’s approach to determining the values of V2G service provision probabilities using survey data. It was found that estimating these values using simulation and forecasting tools makes sense when the coefficients resulting from survey responses are used as initial data. Thus, the paper also presents the results of the surveys that were conducted. As the results from the simulations show, the values of the probabilities of V2G services are not high, which should induce future operators of V2G services to offer a beneficial product for the customer.

1. Introduction

In the 21st century, society is faced with the problem of excessive greenhouse gases emissions, which significantly affect climate change [1,2]. One of the largest emitters of carbon dioxide (CO₂) into the atmosphere is the transportation sector [3,4]. As a result, many government organizations are trying to influence the automotive sector through legislative changes in order to minimize CO₂ emissions into the atmosphere [5]. One of the factors that may influence such action is the promotion of electric vehicles (EVs) [6]. EVs can be divided into three main groups: battery electric vehicles (BEVs), which are vehicles that run solely on electricity from a battery pack, plug-in hybrid electric vehicles (PHEVs), which are hybrid vehicles that use electric motors and an internal combustion engine (ICE) but have the ability to charge the internal batteries from both the internal combustion engine and an electrical socket, and fuel cell electric vehicles (FCEVs), which are run by hydrogen [7]. Because electric vehicles have rechargeable batteries in their design, they can be treated as mobile electricity storage [8]. Therefore, their potential as an auxiliary source of electricity for additional demand should be exploited. These principles are the basis of vehicle-to-grid (V2G) technology or in general vehicle-to-everything (V2X) technology [9]. It involves voluntarily discharging the battery of an electric vehicle into the power grid or any other facility owned by the end-user, which can be called, in simple terms, a V2G service. It is worth mentioning here that in order to release the full potential of V2G technology, electric vehicles should be integrated with local power systems creating microgrids [10,11,12]. Through the development of renewable energy source (RES) technologies, local energy balancing solutions have become popular and there is certainly a place for electric vehicles in them [13]. It is possible to use EVs directly in covering the demand of the household (V2H—vehicle-to-home) or the commercial end user (V2L—vehicle-to-load) [9]. The idea of local energy balancing involves the efficient use of electricity produced from uncertain energy sources, such as photovoltaic plants (PVs) or wind turbines (WTs). This can be achieved by battery energy storage systems (BESS). Knowing that an EV can be treated as a mobile energy storage, its use in accumulating energy from PVs and WTs and possibly discharging it in case of a lack of electricity production due to unfavorable weather conditions seems to be appropriate. Figure 1 shows an example concept for integrating an EV into a microgrid.

With V2G technology also comes a number of economic aspects. Electric vehicles that would participate in V2G technology can also provide ancillary services to the power system operator. There are many references in the literature to such a use of electric vehicles [14,15,16,17,18]. In [14], the use of fuzzy linear programming in forecasting the provision of the most basic ancillary services such as capacity regulation is presented. In [15], the application of V2G technology in primary regulation is presented for a power system located on the island of Bornholm, while in [16] results of a simulation of V2G use in an industrial microgrid are presented. In [17], the capabilities of V2G technology in the context of large-scale power system balancing are presented. In [19,20], various algorithms needed to evaluate the performance of frequency control services are described. There are also mentions of the possibility of providing a supplementary energy source [21]. Electric vehicles are also a means to provide grid flexibility services [13,22,23]. In the literature it is also possible to find examples of creating business models, mainly based on aggregators, i.e., external companies grouping EVs with each other, e.g., from a given region [24]. V2G aggregator models were also presented in [25,26], where the settlement models are based on forecasting the service revenue for day-ahead markets. In [26], solutions similar to auction mechanisms have also been proposed, while in [14] the potential gains from providing system services are presented. In [27], an aggregator concept that can manage electric vehicles for V2H, V2G, and also vehicle-to-vehicle (V2V) technologies is presented. It was also pointed out that such an aggregator should communicate with the grid operator in order to inject energy into the power grid. Typical economic efficiency analyses of V2G service provision concepts have also been performed [28,29]. Following this, the literature also includes discussions on the quantification of revenue from the provision of V2G services. In [9], values of annual service revenue for different types of services are presented. The evaluation of EV battery degradation costs is also an important issue [21,30,31].

Nevertheless, one of the most important aspects regarding V2G technology is the social and legal aspect. In [32,33], the topic of the functioning of the technology in a legal framework was addressed. In [32,34], it is stated that the basic legal framework for the unrestricted functioning of V2G technology is missing. While analyzing the social aspects of introducing V2G technology into electricity markets, the opinion of future users should also be taken into account. In [33,35], the results of a social survey on V2G technology in the Nordic region are presented. Both the social point of view and potential legislative problems are presented. In [36], the authors present the results of a survey of Australian employers and employees on the use of V2G technology in company vehicles in an economic and operational context. In [37,38], survey results were presented that linked user behavior to proposed economic models of V2G service provision mechanisms. Nevertheless, one of the biggest issues that needs to be addressed relates to the uncertainty of performing V2G services due to the lifestyle of EV users. It should be emphasized that regardless of the energy merits of V2G services, it should be kept in mind that, as a priority, EV users will want to use them for their purposes. Therefore, several papers can be found in the literature on the analysis of travel patterns of EV users [39,40,41]. The literature also addresses the issue of how EV users’ travel patterns affect charging profiles [42]. It is relevant because the times when the vehicle is being charged affect the length of the V2G service provision, as this is the moment when the vehicles are connected to the charging stations. All these aspects should encourage researchers to further expand their knowledge in the area of V2G technology.

This paper presents the consideration of EV user views in the process of calculating the probability of V2G service provision. This paper is also a continuation of the considerations presented in [34]. It should be mentioned that, according to [34], the provision of the V2G service depends not only on the available electric energy in EVs, but also on the probability of its provision. This means that the amount of energy available to the end user or operator will depend on how users behave. It is therefore proposed that surveys should be used to obtain an answer as to how much interest there would be in introducing V2G services into the market. The paper is organized as follows: V2G Program and V2G service are defined in Section 2. A mathematical notation of the provision of a V2G service is also presented. Section 3 presents the description of the survey, which are used to calculate the necessary probabilities later in the paper. Detailed results are depicted in Appendix A. Section 4 develops a methodology for calculating the probabilities of V2G service provision, and in Section 5, exemplary simulation results are presented.

2. Assumptions of V2G Program

2.1. Definitions

This paper is a follow-up to the discussion presented in [34]. Therefore, the most important definitions regarding the designed V2G Program mechanism should be referenced first. In this subsection, the concepts previously presented in [34] will be recalled, and some of them will be supplemented with additional information. The most important definitions are as follows [34]:

• V2G serviceis a defined action, which is undertaken by the V2G Program participant, aimed at the improvement of the power system operation, or ensuring sufficient capacity for the end user.
• V2G Program participant (uEV)is the owner of an electric vehicle or fleet of electric vehicles, who provides services by offering battery capacity to end users or a distribution system operator.
• End user (EndUs)is the energy consumer, who has decided to use electric vehicles for reserve power supply within the V2G Program.
• V2G Programis understood as the activity of a power company involving the use of electric vehicles to improve the operation of the power grid or/and to improve the security (assurance) of supply.
• V2G service provider (V2Gsp)is the party managing the V2G Program in a given area.
• Mandatory modeis a V2G service mode in which the V2G Program participant (uEV) receives a fixed payment in exchange for remaining on standby and unconditionally providing service on demand from the V2G service provider. If the service is not provided, the uEV will incur fines that are proportional to their participation in the V2G Program.
• Optional modeis a V2G service mode in which a V2G Program participant is paid only for the EV discharging action that is completed. The participant will not be remunerated for willingness to perform the service and remain on standby but will also not receive fines for not providing the service.

Analyzing the aforementioned definitions, it may be questioned why the terms are limited to vehicle-to-grid technology only and not to the overall service, i.e., vehicle-to-everything (V2X). In principle, there is not much difference in the flow of energy between a vehicle and the electric grid of a facility, household, or a point in the distribution grid. However, the choice of the word “grid” in the definitions is intentional and indicates that the services offered will be delivered to a larger number of customers or even to the power system. The usage of the terms associated with V2G technology is also intended to distinguish these services from the standard vehicle-to-home (V2H) or vehicle-to-building (V2B) technology framework. As included in [27], V2H technology is rather designed for a household’s self-consumption without the intervention of a third-party aggregator. On the other hand, in [34] and in this publication, assumptions are made for services that can also be provided in the household, but nonetheless at the order of a third-party operator—in this case a V2Gsp. Hence, it seems appropriate to use the term “grid”, which suggests third-party interference in the time and place of service provision. However, it is important to emphasize the difference between providing V2G service in one’s own household and providing service in other residential buildings. In the case where a particular uEV would provide the service in other households (e.g., neighbors), this should be considered as providing the service to the end user, to whom it should be moved, and the EndUs should provide a charging space. In [34] this type of service is named as “Emergency work for the energy consumer” and only this type of service will be analyzed in this paper (in Section 4 and Section 5 referring to the V2GL). In case the uEV would provide a V2G service in their household for the purpose of a V2Gsp, it is necessary to refer to a different type of service. In [34], it is indicated that V2G services could be ordered to a V2Gsp by a DSO. Thus, discharging EVs for the household’s own purposes can be treated as a capacity constraint measure (later referred to as V2GH). Hence, it can be considered that this would be a type of grid flexibility services [23]. However, this material will be discussed in a separate paper.

Figure 2 shows the relations between the various members of the V2G Program during the provision of the V2G service. It should also be noted that it was indicated in [34] that it is practically impossible to perform services at the distribution system level and the operation of the V2G Program is limited for now to providing services directly to an EndUs. This is due to the lack of legal regulations regarding the provision of ancillary services by distribution system operators (DSOs).

2.2. Provision of V2G Service—Emergency Work for the Energy Consumer

In the analysis of the V2G service provision involving discharging an EV battery to the electricity grid or to a certain end user, the uncertainty associated with the actual service provision must be taken into account. In [34], it was proposed that the expected value of energy from the EV battery delivered to the end user should be equal to the multiplication of the probability of service provision and the quantity of energy from the required number of vehicles appropriately calculated (Equation (1)).

$$E_{V2G+,t}=P\left(A\right)· {\displaystyle \sum }_{n=1}^{N_{EV}^{REQ}}{e}_{V2G+, n}$$

(1)

where:

• E_d—the end user’s electricity demand from V2G Program, in kWh;
• P(A)—the total probability of providing V2G service;
• e_V2G+,n—the energy flow injected by n-th electric vehicle;
• $N_{EV}^{REQ}$—the required number of electric vehicles for provision of V2G service for an end user;
• E_V2G+,t—the expected energy delivered to the end user in time t.

Equation (1) shows the inputs that need to be calculated in order to determine the expected value of energy delivered to the end user at time t. Firstly, the idea of establishing a V2G service for an EndUs has to be considered. The primary goal of this service has to be covering the electricity demand of the EndUs in the time frame specified by the EndUs. In the literature, methods for determining demand profiles for non-residential users are mainly based on the usage of forecasting techniques [43,44]. That issue is not related to the main subject of this paper; hence, the analysis of method selection for power and energy demand calculation is omitted. However, it is necessary to focus on the calculation of the electrical energy delivered by a single EV. In the literature it is possible to find quite convergent mathematical models describing the charging and discharging processes. In [45], the dependence of the transmitted capacity from EV batteries to the power grid and the efficiency of individual processes is indicated. On the other hand, in [46], the authors focused on modeling the discharging process in relation to the state-of-charge (SOC) and the efficiency of individual processes. Therefore, it is possible to write that the energy injected by an electric vehicle to the grid is equal to [34]:

$$e_{V2G+,\mathrm{n}}=C·\left(SO{C}_t-\left(SO{C}_f\left(1+R\right)-SO{C}_0\right)\right)·{\eta }_d$$

(2)

where:

• C—the battery capacity of the EV, in kWh;
• SOC_t—the current state-of-charge (SOC) at the time of commencing V2G service provision, in %;
• SOC_f—the state-of-charge required for the next journey, in %;
• SOC₀—the minimal state-of-charge limited by technical constraints, in %;
• R—a reserve, which considers the possible lengthening of the route, e.g., to avoid congestion;
• η_d—the efficiency of the discharging process.

It is worth noting that the moment when the vehicle records the SOC_t refers to the moment when the service provision commences. If the vehicle is not present at the place where the service is provided, the energy required to get it from the location where the service is accepted to the place where the service is provided must also be considered. This is denoted by Equation (3).

$$SO{C}_t=C·\left(SO{C}_{act}-\frac{u{s}_{EV}·d}{C}\right)$$

(3)

where:

• SOC_act—the state-of-charge at the time the service proposal is accepted by uEV, in %;
• us_EV—the average electricity consumption of an electric vehicle, in kWh/km;
• d—the distance between the uEV and the service provision point, in km.

Equations (2) and (3) were used to determine the energy injected into the power grid by the EV; however, the transmission capacity of this process must also be considered. It is known that both the on-board charger and the charging station have their capacity limitations. Therefore, it can be written that

$$\{\begin{array}{c}{P}_{MIN}^{EVSE}\le P_{V2G+}\le P_{MAX}^{EVSE}\\ P_{MIN}^{CHARG}\le P_{V2G+}\le P_{MAX}^{CHARG}\end{array}$$

(4)

where:

• $P_{MIN}^{EVSE}$—the minimum capacity transmitted from the V2G charging station to the power grid, in kW;
• $P_{MAX}^{EVSE}$—the maximum capacity transmitted from the V2G charging station to the power grid, in kW;
• $P_{MIN}^{CHARG}$—the minimum capacity transmitted from the EV on-board charger to the power grid, in kW;
• $P_{MAX}^{CHARG}$—the maximum capacity transmitted from the EV on-board charger to the power grid, in kW;
• $P_{V2G+}$—the discharging capacity of the EV, in kW.

For each EndUs facility, the maximum capacity resulting from the injection of energy in the V2G service can be equal to the sum of the EVSE capacity (Equation (5)).

$$P_d\le {\displaystyle \sum }_{i=1}^{n_{CS}}{P}_{MAX,i}^{EVSE}$$

(5)

where:

• $P_d$—the maximum demand that can be covered by the V2G service, in kW;
• n_CS—the number of bi-directional charging points owned by the EndUs;

Thus, the maximum electricity demand of the EndUs in a given time frame of service provision can be determined as follows:

$$E_d^{MAX}={{\displaystyle \int }}_{\tau =1}^T{P}_{d,\tau } d\tau$$

(6)

where:

• $E_d^{MAX}$—the maximum electricity demand of the EndUs in a given time frame of service provision;
• $\tau$—a single hour of V2G service provision;
• $T$—the length of the time frame, in hours.

By analogy, the energy injected into the EndUs system can be determined considering the transmission capacity:

$$e_{V2G+,\mathrm{n}}={{\displaystyle \int }}_{\tau =1}^T{P}_{V2G+, \tau } d\tau$$

(7)

Due to the fact that the probability of service P(A) cannot be greater than 1, it is necessary to find a sufficiently larger number of electric vehicles, $N_{EV}^{EST}$, which in a way covers this uncertainty. This can be represented by Equations (8)–(10) [34].

$$E_{V2G+,t}^{\prime }\ge E_d$$

(8)

$$E_{V2G+,t}^{\prime }=P\left(A\right)·{\displaystyle \sum }_{n=1}^{N_{EV}^{EST}}{e}_{V2G+, \mathrm{n}}$$

(9)

$$N_{EV}^{EST}=\frac{N_{EV}^{REQ}}{P\left(A\right)}$$

(10)

where:

• $N_{EV}^{EST}$—the estimated number of electric vehicles expected to be involved in establishing V2G service, considering the probability of service provision;
• E’_V2G+,t—the expected energy delivered to the end user in time t, which covers the reserve resulting from the probability of service provision P(A) < 1.

Thus, it can be concluded that finding the probability of service P(A) is crucial to correctly estimate the number of vehicles needed to provide the service and, consequently, it is necessary to secure the demand of the end user using electric vehicles for the emergency service. The algorithm for selecting the vehicles to provide the service is described in detail in [34]. Also in [34], the probability of service provision is defined. It depends on three simultaneous random events from three points of view:

• End user—P(EndUs);
• V2G service provider—P(V2Gsp);
• V2G Program participant—P(uEV).

The probability of service provision P(A) can be described by Equation (11) [34]:

$$P\left(A\right)=P\left(EndUs\right)·P\left(V2Gsp\right)·P\left(uEV\right)$$

(11)

Moreover, in [34], each of the random events was defined according to Formulas (12)–(14):

$$P\left(EndUs\right)=\left(1-P\left(F_{EVSE}\right)\right)·P\left(D_{PL}\right)$$

(12)

$$P\left(V2Gsp\right)=1-P\left(F_S\right)$$

(13)

$$P\left(uEV\right)=P\left(EC\right)·P\left(U{S}_{EV}\right)·\left(1-P\left(INT\right)\right)$$

(14)

where:

• P(F_EVSE)—the probability of failure of a bi-directional charging point;P(D_PL)—the probability of the availability of a V2G charging point at the place of service delivery.
• P(F_S)—the probability of failure of a metering and billing system.
• P(EC)—the probability of the user’s reaction to providing the service at a given time, e.g., receiving an economic incentive or the willingness to perform a service due to the lifestyle of the uEV;
• P(US_EV)—the probability of electric vehicle availability due to its lack of use;
• P(INT)—the probability of service interruption.

A minor modification from the publication [34] should be highlighted at this point. Previously, P(US_EV) was described as the probability of using the vehicle. A minor adjustment had to be made here due to the fact that service provision will only be possible when the vehicle is not in use for its own uEV’s purposes, which will be superior to service provision.

2.3. Novelty

As pointed out in [34], there is a lack of widely available data to calculate the probability of V2G service delivery, mainly for the level related to participant decisions. This problem is also highlighted in [47], where the authors point out that the absence of an EV at the discharge location can be caused by a human decision. In [47], a single two-state model was used for the process of service interruption and the absence of an EV. In this paper, it was decided to separate these probabilities. The probability of service interruption will be determined according to the methodology presented in [47], while the concept of EV absence was decided to be redefined to depend on the vehicle usage at a given moment—the probability P(US_EV) described in Section 2.2. Furthermore, according to [34], the probability of service provision will also depend on the participant’s decision. In order to determine these values, it was decided to use the public’s voice to design metrics that will serve as starting data in the V2Gsp business model. This approach seems to be appropriate since it will be the EV users who will ultimately use the V2G Program. The problem of a lack of social analysis in the technology area was also mentioned in [48]. As the authors note in [48], most research papers focus on the technical layer, and those that consider the social layer of the problem of implementing V2G technology are single percentages. In particular, they note that the description of user behavior was described in only 2.1% of the articles they analyzed [48]. The research survey exemplar papers use advanced statistical models, which most often illustrate the decision-making models that a V2G Program operator would need to make to ensure the viability of the service [36,38]. However, from the point of view of V2Gsp, both the statistical parameters on which the economic model will be built and, in the later real implementation stage, a tool to select the vehicles to provide these services need to be determined. In [34], such an algorithm is presented, for the operation of which the probabilities of events that make up the performance of the service should be used. These probabilities are described in detail in Section 2.2. It is therefore proposed to determine them experimentally based on results from surveys. Their purpose is to be used as starting values in the aforementioned algorithm [34]. In case of a real implementation, these indicators should be updated by the V2Gsp in real-time with current user behavior. This paper also considers the author’s approach to calculating probability values associated with the decisions made by V2G Program participants. It should be kept in mind that the fact of examining the available energy from a certain number of EVs does not mean that all this energy will be delivered to the grid or end user. By using a probabilistic approach in evaluating the provision of V2G services, EVs can be used more efficiently in their integration into the grid. Both the V2Gsp and DSO will be offered an indicator to estimate the approximate real value of the energy distributed by EVs to the grid or for end users’ own consumption. In addition to the mathematical description of the indicators determining the probability, a methodology for determining them for V2Gsp will be proposed, and example calculations based on realized surveys will be presented.

In summary, this article includes the following novel contributions:

• The inclusion of the results of social surveys in the calculation process of the initial values of the V2G service provision probabilities and the calculation of the number of participants in Mandatory and optional mode (as assumed in Section 2.2).
• A proposed methodology for dealing with the calculation of the values of the probabilities of V2G service provision by V2Gsp.
• A proposed targeted methodology for calculating the probability values of the uEV reaction.
• The use of probabilistic methods to enhance the process of EV integration into the electricity grid within the scope of V2X technology.

3. Survey Research

Previous surveys have primarily relied on asking respondents what benefits the use of V2G technology might bring, what their concerns are, and what potential benefits they might be able to receive in exchange for discharging their vehicles. Another type of survey is the widely available survey on the use of electric vehicles and their charging strategies. The purpose of this survey is to combine the behaviors of EV and ICE users along with the assumptions of the V2G Program. Matching the right group of respondents is an essential issue. The peculiarity of the V2G technology, i.e., a relatively narrow group of consumers, which is additionally imposed by the Polish conditions of electromobility development, should make one think whether it is reasonable to carry out the survey on a representative group. Due to the aforementioned facts, it has been suggested that the survey should focus on people interested in electromobility, in particular the users of electric vehicles or people connected with the power sector. Due to the still limited possibilities of people-to-people communication caused by the COVID-19 pandemic, it was decided to use an online form of the survey using the Microsoft Forms tool. The suitable form was made available on electric vehicle user forums. The anonymous survey was conducted between 6 July 2021 and 19 July 2021 and consisted of 23 closed-ended single- or multiple-choice questions. The survey resulted in 130 responses. Additionally, to strengthen the tone of the survey, 5 responders were asked to elaborate on their feelings about the proposed V2G Program. Interviews were conducted in a manner that ensured anonymity. The selection process was based on their reaction to a request for contact after the survey was completed to clarify trip behaviors. It should be emphasized that survey research is not the main research problem of this paper. In fact, it is only a supplement to begin the discussion of determining service probability metrics.

Table 1. Basic information about the survey.

Duration of the Survey	6–19 July 2021
Form of the Survey	Internet—MS Forms
Number of Answers Received	130, including 5 interviews with EV users

provides background information on the survey.

A description of the survey results is included in Appendix A, due to the very extensive analysis of the responses.

4. Determination of the Parameters Needed to Provide the V2G Service

4.1. Method Based on a Two-State Model

This section of the paper is divided into three parts. In the first one, the probability values of some of the events described in Section 2.2 are calculated based on a two-state reliability model, which is commonly used in power system reliability analyses [49]. In the second part of this section, a novel method for calculating the initial probability values from the V2G Program participant perspective and the ratio of the number of vehicles in the mandatory mode to the number of vehicles in the optional mode are proposed. Finally, the last part of the section presents the author’s approach to calculate the probability value P(EC) after the first month of V2G Program settlement, as it is one of the most difficult to estimate due to the random nature of user behavior. Also, the deliberations are summarized and a graphical diagram of the developed methodology is presented.

The first probability values determined concern events from the perspective of the end user (EndUs). As can be seen from Equation (12), the probability of V2G service from the EndUs perspective is determined by two separable events—charging station operation (1-P(F_EVSE)) and the charging station availability at the service location (P(D_PL)). The values to calculate the first one can be found in [50]. The authors in [50] used a two-state model in their calculations. It should be noted that the transition intensities from state “0” to “1” and “1” to “0” are unique for each parameter analyzed and will be identified when discussing the individual probabilities. A “0” state may indicate a non-functioning charging station or supervisory system but may also indicate a lack of available space to discharge an EV or an interruption of V2G service. State “1” refers to events where the V2G service is provided in an uninterrupted and fault-free manner.

Using the two-state model, it can be assumed that the probability of charging station failure is determined according to Equation (15).

$$q_{EVSE}=\frac{{\lambda }_{EVSE}}{{\lambda }_{EVSE}+{\mu }_{EVSE}}$$

(15)

where:

• ${\mathsf{\mu }}_{EVSE}—\mathrm{the} \mathrm{charging} \mathrm{station} \mathrm{restoration} \mathrm{intensity} \left(\mathrm{repair} \mathrm{rate}\right);$
• ${\lambda }_{EVSE}—\mathrm{the} \mathrm{charging} \mathrm{station} \mathrm{failure} \mathrm{intensity} \left(\mathrm{failure} \mathrm{rate}\right);$
• $q_{EVSE}—\mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{the} \mathrm{charging} \mathrm{station} \mathrm{being} \mathrm{out} \mathrm{of} \mathrm{service}.$

Thus, it can be concluded that in order to calculate the probability of V2G service from the EndUs perspective, the opposite value will be needed, i.e., the probability of the charging station being in the operating state p_EVSE. It is determined by Equation (16).

$$p_{EVSE}=1-q_{EVSE}=\frac{{\mu }_{EVSE}}{{\lambda }_{EVSE}+{\mu }_{EVSE}}$$

(16)

The second event relates to the availability of the V2G charging station at the service location. Availability of a V2G charging station is understood as unrestricted access to this station provided to the particular uEV by the V2Gsp or EndUs. In other words, no other vehicle that is not a member of the V2G Program may be parked at the station or no maintenance work may be taking place at the station. Regardless of the reason, such an event should be treated as the absence of an electric vehicle at the service location. Accordingly, the methodology for calculating the values of restoration and failure stream parameters can be adopted as in [47]. Analogously to Equation (16), the probability value of charging station availability was determined, which is presented in Equation (17).

$$p_{DPL}=\frac{{\mu }_{DPL}}{{\lambda }_{DPL}+{\mu }_{DPL}}$$

(17)

where:

• ${\mu }_{DPL}—\mathrm{the} \mathrm{intensity} \mathrm{of} V2G \mathrm{charging} \mathrm{station} \mathrm{availability} \mathrm{comeback};$
• ${\lambda }_{DPL}—\mathrm{the} \mathrm{intensity} \mathrm{of} \mathrm{the} V2G \mathrm{charging} \mathrm{station} \mathrm{transition} \mathrm{into} \mathrm{an} \mathrm{occupied} \mathrm{state};$
• $p_{DPL}—\mathrm{the} \mathrm{probability} \mathrm{of} V2G \mathrm{charging} \mathrm{station} \mathrm{availability}.$

By introducing Equations (16) and (17) into Equation (12), a formula for calculating the probability of service from the EndUs side is obtained, which is shown in Equation (18).

$$P\left(EndUs\right)=p_{EVSE} ·p_{DPL}=\frac{{\mu }_{EVSE}}{{\lambda }_{EVSE}+{\mu }_{EVSE}}·\frac{{\mu }_{DPL}}{{\lambda }_{DPL}+{\mu }_{DPL}}$$

(18)

The second probability value (P(V2Gsp)) that needs to be calculated relates to the V2G Program operator, i.e., the V2Gsp. In this case, the research reported in [47] was also referenced. The authors in [47] presented the role of the V2Gsp as an aggregator and created a two-state model for such an actor. Thus, it is possible to bring the unavailability of the aggregator to the unavailability of the V2Gsp, which will occur if the supervisory control system fails. Then, the V2Gsp will not be able to execute its tasks. In further research, the reliability model of the V2Gsp can be extended to consider the reliability of individual components of the system architecture. Using Equation (19), a relation that allows to calculate the probability of V2G service from V2Gsp perspective is presented.

$$P\left(V2Gsp\right)=1-P\left(F_S\right)=1-q_S=1-\frac{{\lambda }_S}{{\lambda }_S+{\mu }_S}$$

(19)

where:

• ${\lambda }_S—\mathrm{the} \mathrm{intensity} \mathrm{of} \mathrm{failures} \mathrm{in} \mathrm{the} V2Gsp \mathrm{supervisory} \mathrm{control} \mathrm{system};$
• ${\mu }_S—\mathrm{the} \mathrm{intensity} \mathrm{of} \mathrm{repairs} \mathrm{of} \mathrm{the} V2Gsp \mathrm{supervisory} \mathrm{control} \mathrm{system}$;
• $q_S—\mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{the} V2Gsp \mathrm{supervisory} \mathrm{control} \mathrm{system} \mathrm{failure}.$

The third probability value needed to calculate the total service probability relates to the V2G Program participant (uEV). As shown in Equation (14), it is determined by three components:

• P(EC)—the probability of the user’s reaction to providing the service at a given time, e.g., receiving an economic incentive or the willingness to perform a service due to the lifestyle of the uEV;
• P(US_EV)—the probability of electric vehicle availability because of its lack of usage;
• 1-P(INT)—the probability of service not being interrupted.

Of these three probabilities, only one can be determined easily by using a two-state model, and that is the probability 1-P(INT). The probability of service interruption can be calculated similarly to the probability of V2G charging station availability. The reason for that is that in [47] the authors do not specify what EV absence is. They even mentioned that it can refer to hardware failure or human action. Due to the lack of other available sources that could confirm other values than assumed, it was decided to use the aforementioned approximation. The calculation of the 1-P(INT) component is shown by Formula (20).

$$1-P\left(INT\right)=1-q_{INT}=1-\frac{{\lambda }_{INT}}{{\lambda }_{INT}+{\mu }_{INT}}$$

(20)

where:

• ${\lambda }_{INT}—\mathrm{the} \mathrm{intensity} \mathrm{of} \mathrm{transition} \mathrm{to} \mathrm{the} \mathrm{state} \mathrm{of} \mathrm{interruption} \mathrm{of} V2G \mathrm{service};$
• ${\mu }_{INT}—\mathrm{the} \mathrm{intensity} \mathrm{of} \mathrm{restoration} \mathrm{from} \mathrm{interruption} \mathrm{of} V2G \mathrm{service}$;
• $q_{INT}—\mathrm{the} \mathrm{probability} \mathrm{of} V2G \mathrm{service} \mathrm{interruption}.$

4.2. Method for Determining Initial Probability Values of V2G Service Provision Based on Survey Data

The other two event types related to V2G Program participants are strongly dependent on the behavior of potential users. The novelty of this paper is to try to use the collected survey data to estimate “initial” values for these events’ probabilities. The first of these events is the reaction of a V2G Program participant to a proposed V2G service offer. The probability of this event will vary depending on the time of day, as the economic conditions of such a service are different at night and during the day. Additionally, during the course of the research, the opinions of users who expressed a high willingness to provide EV discharging services at a V2G home charging station were taken into account (V2GH). However, due to the different types of service provided (as described in Section 2), probability values for this case will be calculated in future studies. Therefore, the probability values of P(EC) for providing the service at the location of another end user secured by the V2G Program $P{\left(EC\right)}_t{}^{V2GL}$ were distinguished. In order to determine the parameter $P{\left(EC\right)}_t{}^{V2GL}$, the V2G Program operator must first conduct surveys, which require the questions described in Appendix A, and in particular those shown in Figure A9, Figure A10 and Figure A11. Once the questions are answered, the indicator can be determined from Equation (21). It is recommended to obtain a sample of at least 100 respondents, but it should be emphasized that this does not have to be a representative sample as mentioned in Section 3.

$$P{\left(EC\right)}_t{}^{V2GL}=\frac{N_{uEV}^{V2GL}\left(t\right)}{R_{YES}^{V2GL}+R_{DK}^{V2GL}}$$

(21)

where:

• $P{\left(EC\right)}_t{}^{V2GL}—$ the probability of V2G Program participant reaction on the provision of the V2G service at the EndUs location at time t;
• $N_{uEV}^{V2GL}\left(t\right)—$ the number of respondents willing to provide V2G service at the EndUs location at time t;
• $R_{YES}^{V2GL}—$ the total number of “Yes” responses in the survey question regarding the willingness to provide V2G service at the EndUs location;
• $R_{DK}^{V2GL}—$ the total number of “Don’t know” responses in the survey question regarding the willingness to provide V2G service at the EndUs location.

Based on the collected survey data, it can be assumed that the condition of providing the service at time t would be to find out how many respondents would like to provide the service during that period. In addition, those respondents who did not know whether such services would be provided were also included. This is an intentional measure that can help in the risk analysis of the uncertainty of V2G service delivery.

The second indicator needed to calculate the probability P(uEV) is the probability of electric vehicle availability related to the lack of use at a given point in time P(US_EV). In order to quantify this value, it is first necessary to refer to the survey data. The vehicle usage profile shown in Figure A4 will be helpful. The proposed survey examined 3-h intervals due to a willingness toward simplifying the questionnaire. However, the target version of the survey prepared by the V2Gsp should reduce the time frame to 1-h intervals. This will allow for a more accurate estimation of initial probability values of P(US_EV). However, the methodology for calculating the specific probability value will also be broadened to include other components in relation to the EV usage profile. Namely, the calculation method also includes respondents who answered that they own an ICE vehicle and are considering purchasing an electric vehicle within 5 years. This procedure is intended to increase the accuracy of calculating the probability of P(US_EV), assuming that current users of ICE vehicles will maintain their travel patterns, i.e., travel at similar times. The methodology for calculating P(US_EV) is presented by Equations (22) and (23).

$$P_1{\left(U{S}_{EV}\right)}_t=\frac{N_{US}^{EV}\left(t\right)+N_{US}^{ICE}\left(t\right)}{R_{EV}+R_{ICE}}$$

(22)

$$P{\left(U{S}_{EV}\right)}_t=1-P_1{\left(U{S}_{EV}\right)}_t$$

(23)

where:

• $N_{US}^{EV}\left(t\right)—\mathrm{the} \mathrm{number} \mathrm{of} \mathrm{respondents} \mathrm{that} \mathrm{indicated} \mathrm{they} \mathrm{are} \mathrm{using} \mathrm{an} EV \mathrm{in} \mathrm{time} \left(t\right);$
• $N_{US}^{ICE}\left(t\right)—\mathrm{the} \mathrm{number} \mathrm{of} \mathrm{respondents} \mathrm{who} \mathrm{intend} \mathrm{to} \mathrm{purchase} \mathrm{an} EV \mathrm{in} 5 \mathrm{years} \mathrm{of} \mathrm{having} \mathrm{used} \mathrm{an} ICE \mathrm{vehicle} \mathrm{in} \mathrm{time} \left(t\right);$
• $R_{EV}—\mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} EV \mathrm{users};$
• $R_{ICE}—\mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} ICE \mathrm{users} \mathrm{who} \mathrm{intend} \mathrm{to} \mathrm{purchase} \mathrm{an} EV \mathrm{in} 5 \mathrm{years} \mathrm{of} \mathrm{having} \mathrm{used} \mathrm{an} ICE \mathrm{vehicle} \mathrm{in} \mathrm{time} \left(t\right);$
• $P_1{\left(U{S}_{EV}\right)}_t—\mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{vehicle} \mathrm{usage} \mathrm{in} \mathrm{time} \left(t\right);$
• $P{\left(U{S}_{EV}\right)}_t—\mathrm{the} \mathrm{probability} \mathrm{of} \mathrm{electric} \mathrm{vehicle} \mathrm{availability} \mathrm{because} \mathrm{of} \mathrm{its} \mathrm{lack} \mathrm{of} \mathrm{usage} \mathrm{in} \mathrm{time} \left(t\right).$

It is assumed that the final solution should include the ability for V2Gsp to collect location data from the on-board computers of individual uEVs, and then this data would be used to build short-term prediction models. Unfortunately, this solution requires a large invasion of user privacy, which may be a problem for V2G Program implementation [51].

The last value that will be determined using the survey data is a factor that determines the ratio of mandatory to optional mode vehicles—the MOD coefficient. This ratio will be based solely on survey responses in the first phase and will be updated on the last workday of the month in a later phase of the V2G Program implementation. Ultimately, it should count customers who are assigned to mandatory or optional mode, according to Formulas (24) and (25).

$$MO{D}_{MAN}^{V2GL}=\frac{N_{uE{V}_{V2GL}}^{MAN}}{N_{uE{V}_{V2GL}}}$$

(24)

$$MO{D}_{OPT}^{V2GL}=\frac{N_{uE{V}_{V2GL}}^{OPT}}{N_{uE{V}_{V2GL}}}$$

(25)

where:

• $N_{uE{V}_{V2GL}}^{MAN}—\mathrm{the} \mathrm{number} \mathrm{of} V2G \mathrm{Progam} \mathrm{participants} \mathrm{in} \mathrm{mandatory} \mathrm{mode} \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{type} \mathrm{of} V2G \mathrm{service};$
• $N_{uE{V}_{V2GL}}^{OPT}—\mathrm{the} \mathrm{number} \mathrm{of} V2G \mathrm{Progam} \mathrm{participants} \mathrm{in} \mathrm{optional} \mathrm{mode} \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{type} \mathrm{of} V2G \mathrm{service};$
• $N_{uE{V}_{V2GL}}—\mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} V2G \mathrm{Program} \mathrm{participants} \mathrm{for} \mathrm{a} \mathrm{given} \mathrm{type} \mathrm{of} V2G \mathrm{service}.$

The MOD coefficient should be based on the results of social surveys to determine the initial values. However, in order to accurately determine the values of the coefficients for the aforementioned conditions, it is necessary to analyze more closely the results of the questionnaires for the participants who would be interested in a given mode of service provision—the SUR coefficient. This is deduced from answering the questions described in

Table A10. Detailed responses to question (17).

Question:	In Exchange for a Fixed Monthly Remuneration, would you be Willing to Stand by to Discharge an Electric Vehicle during the Hours Set by the V2G Program Operator, being Aware of Potential Fines for not Providing the Service?
Yes	29
No	68
Yes, but with other additional benefits	32
No answer	1
Total number of respondents	130

and

Table A11. Detailed responses to question (18).

Question:	In Exchange for a Financial Benefit for a Single Discharging Action, Would You Be Able to Provide Discharging Services without Incurring Fines for not Providing the Service?
Yes	103
No	25
No answer	2
Total number of respondents	130

. Then, one must subtract those who answered “No” to both of the questions in Table A10 and Table A11, so that the result obtained will be the denominator of the SUR coefficient. Thus, the following methodology is proposed:

• Examine the number of respondents who declared “Yes” to the question presented in

  Table A9. Detailed responses to question (13).

  Question:	If You Owned an Electric Car, Would You Join a V2G Program Which Involves Traveling to a Point of Service (e.g., an Industrial Facility), Plugging Your Car into a Charging Station at the Order of a Third-Party Operator, and then Discharging the Battery to a Specified Level in Exchange for Financial Benefits?
  Yes	30
  No	74
  Don’t know	26
  Total number of respondents	130

  and then answered "Yes” and “Yes but with additional benefits” to the question: “In exchange for a fixed monthly remuneration, would you be willing to stand by to discharge an electric vehicle during the hours set by the V2G Program operator, being aware of potential fines for not providing the service?” and answered “No” in the question: “In exchange for a financial benefit for a single discharging action, would you be able to provide discharging services without incurring fines for not providing the service?”. Thus, the number of V2G Program participants for a given service provision location in mandatory only mode can be calculated.
• Examine the number of respondents who declared “Yes” to the question presented in Table A9 and then answered “No” to the question: “In exchange for a fixed monthly remuneration, would you be willing to standby to discharge an electric vehicle during the hours set by the V2G Program operator, being aware of potential fines for not providing the service?” and answered “Yes” to the question: “In exchange for a financial benefit for a single discharging action, would you be able to provide discharging services without incurring fines for not providing the service?”. Thus, the number of V2G Program participants for a given service location in optional mode only can be calculated.
• Examine the number of participants who declared “Yes” to the question presented on Table A9 and then answer “Yes” and “Yes but with additional benefits” to the question: “In exchange for a fixed monthly remuneration, would you be willing to standby to discharge an electric vehicle during the hours set by the V2G Program operator, being aware of potential fines for not providing the service?” and answered “Yes” to the question: “In exchange for a financial benefit for a single discharging action, would you be able to provide discharging services without incurring fines for not providing the service?”. Thus, the number of V2G Program participants for a given service location who said they were willing to participate in both modes can be calculated. Due to the aforementioned fact, these V2G Program participants are eligible for the mandatory mode first.

The mathematical notation of the aforementioned methodology is shown by Equations (26)–(28).

$$SU{R}_{MAN}^{V2GL}=\frac{N_{MA{N}_{ONLY}}^{V2GL}+N_{MA{N}_{OPT}}^{V2GL}}{N_{uEV}^{V2GL}-N_{NOTINT}{}_L}$$

(26)

$$SU{R}_{OPT}^{V2GL}=\frac{N_{OP{T}_{ONLY}}^{V2GL}}{N_{uEV}^{V2GL}-N_{NOTIN{T}_L}}$$

(27)

where:

• $SU{R}_{MAN}^{V2GL}—\mathrm{the}$ coefficient of respondents willing to participate in the V2G Program at the EndUs location in mandatory mode;
• $SU{R}_{OPT}^{V2GL}—\mathrm{the}$ coefficient of respondents willing to participate in the V2G Program at the EndUs location in optional mode;
• $N_{MA{N}_{ONLY}}^{V2GL}—\mathrm{the}$ number of respondents interested in the V2G Program at the EndUs location only in mandatory mode;
• $N_{OP{T}_{ONLY}}^{V2GL}—\mathrm{the}$ number of respondents interested in the V2G Program at the EndUs location only in optional mode;
• $N_{MA{N}_{OPT}}^{V2GL}—\mathrm{the}$ number of respondents interested in the V2G Program at the EndUs location in mandatory and optional mode;
• $N_{NOTIN{T}_L}—\mathrm{the}$ number of respondents, who would like to discharge their vehicle at an EndUs location but are not interested in the V2G Program in mandatory and optional modes.

Thus, it can be concluded that

$$SU{R}_{MAN}^{V2GL}+SU{R}_{OPT}^{V2GL}=1$$

(28)

Considering all the coefficients, the initial parameters of the V2G Program model can be calculated from the perspective of a V2Gsp. A graphical summary of the aforementioned deliberations is shown in Figure 3.

The last phase of the methodology, the validation phase, deserves special emphasis. In the uEV path, it was pointed out that parameters need to be recalculated after the first month of V2G Program operation. This is crucial to maintain the relative accuracy of the model calculations. A detailed description of this step is described in Section 4.3.

4.3. Method for Calculating the Probability Value P(EC) after the First Month of V2G Program Settlement

The user reaction probability P(EC) is extremely difficult to estimate due to the stochastic decision-making process of users. It is also one of the more sensitive factors affecting the V2G Program model operation. In order to calculate the probability $P{\left(EC\right)}_t{}^{V2GL}$ after the first settlement month of the V2G Program, it is proposed to use the author’s algorithm, which in its assumptions will take into account the mobility of uEVs in the area of operation of a V2Gsp. Therefore, the description of the area of operation of the V2Gsp should be introduced first. It will be defined similarly to the one in [34]. Therefore, let G denote the area of operation of the V2Gsp in which each vehicle can be located at a point g described by the following relation [34]:

$$g\left(x_G, y_G\right)\in G$$

(29)

where x_G and y_G are the geometric coordinates of the selected point g.

Subsequently, a square grid with a side length of 1 km must be created in area G. Therefore, the grid will consist of NG squares, which can be described as follows:

$$G_{\mathrm{i}}=\left[\left(x_G,y_G\right);(x_G, y_{G^{\prime }});\left(x_{G^{\prime }},y_G\right);(x_{G^{\prime }}, y_{G^{\prime }})\right]$$

(30)

Knowing that

$$x_{G^{\prime }}-x_G=1 \mathrm{km} \wedge y_{G^{\prime }}-y_G=1 \mathrm{km}$$

(31)

and

$$x_G\le x_{G_{max} }\wedge y_G\le y_{G_{max} }$$

(32)

where x_Gmax and y_Gmax are the boundary values of the area.

With the V2Gsp operating area divided into squares, one needs to assign the number of V2G Program participants to each square at a given hour τ on day D_d. It is important to keep in mind that particular uEVs are constantly travelling, so it is necessary to average this number to indicate a single value. Therefore, it is proposed to use a solution borrowed from the analysis of electricity demand, i.e., to use an hourly averaging based on 15-min measurements. This is represented by Formula (33):

$$N_{G_i, \tau , D_d}^{uEV}=\frac{{\sum }_{t_{15}=1}^4{N}_{G_i, t_{15, D_d}}^{uEV}}{4}$$

(33)

where:

• $N_{G_i, \tau , D_d}^{uEV}—$ the number of uEVs in a given square G_i of the V2Gsp operation area on day D_d at hour τ;
• $N_{G_i, t_{15}, D_d}^{uEV}—$ the number of uEVs in a given square G_i of the V2Gsp operation area on day D_d at each quarter of hour τ;

It is worth noting that, by including a given uEV in the number of vehicles, $N_{G_i, t_{15, D_d}}^{uEV}$ is understood as finding this uEV in the area of the square G_i in the selected quarter of the hour τ on day D_d, i.e., the vehicle monitoring system should record the x_g and y_g coordinates of the given uEV defined for the given square G_i. Thus, the state of the number of vehicles at hour τ on day D_d can be presented by the following matrix:

$${\mathit{N}}_{{\mathit{G}}_{\mathit{i}}, \mathit{\tau }, {\mathit{D}}_{\mathit{d}}}^{\mathit{u}\mathit{E}\mathit{V}}=\left[\begin{array}{ccccc}{N}_{G_1, \tau , D_d}^{uEV}& N_{G_2, \tau , D_d}^{uEV} & N_{G_3, \tau , D_d}^{uEV}& \cdots & N_{G_n, \tau , D_d}^{uEV}\\ N_{G_{n+1}, \tau , D_d}^{uEV}& \cdots & \cdots & \cdots & \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \cdots & \cdots & \cdots & \cdots & N_{G_{NG}, \tau , D_d}^{uEV}\end{array}\right]$$

(34)

where:

• i$—$ the successive squares of area G;
• NG$—$ the maximum number of squares in area G.

Then, knowing the number of vehicles in the given squares, one can proceed to calculate the probability P(EC), which relates to the area of the operation of the V2Gsp. The first step in this process is to collect data on the number of reactions in hour τ on day D_d, resulting in acceptance of V2G service provision $ACP{T}_{\tau , D_d}$ and data on all correctly received reactions on that day $N{S}_{ \tau , D_d}$, i.e., those that were delivered to the uEV and either accepted or rejected. Due to the necessity of predicting the probability value P(EC), it is then necessary to group the data by each day of the week. Thus, for a selected day of the week (e.g., Monday), this can be expressed in the form of a matrix:

$$\mathit{A}\mathit{C}\mathit{P}{\mathit{T}}_{\mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{ccccc}ACP{T}_{1, D_1}& ACP{T}_{2, D_1} & ACP{T}_{3, D_1}& \cdots & ACP{T}_{24, D_1}\\ ACP{T}_{1, D_2}& ACP{T}_{2, D_2}& ACP{T}_{3, D_2}& \cdots & ACP{T}_{24, D_2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ ACP{T}_{1, D_d}& ACP{T}_{2, D_d}& ACP{T}_{3, D_d}& \cdots & ACP{T}_{24, D_d}\end{array}\right]$$

(35)

where:

• $ACP{T}_{\tau , D_d}—\mathrm{the}$ number of reactions in hour τ on day D_d, ending with acceptance of V2G service provision;
• τ$—$ each hour of the day D_d;
• d$—$ the same days of the week consecutively in a month (e.g., Mondays).

Similarly, for other days of the week, matrices would be determined as in Equation (35). In a similar manner, the matrix for the number of all calls and requests for V2G service provided to users should be determined:

$$\mathit{N}{\mathit{S}}_{ \mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{ccccc}N{S}_{1, D_1}& N{S}_{2, D_1} & N{S}_{3, D_1}& \cdots & N{S}_{24, D_1}\\ N{S}_{1, D_2}& N{S}_{2, D_2}& N{S}_{3, D_2}& \cdots & N{S}_{24, D_2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ N{S}_{1, D_d}& N{S}_{2, D_d}& N{S}_{3, D_d}& \cdots & N{S}_{24, D_d}\end{array}\right]$$

(36)

Then, the probability of user reaction can be calculated for the whole V2Gsp operating area—hereafter called the system probability $P^{SYS}{\left(EC\right)}_{\tau , D_d}$. This process is presented in Equations (37) and (38).

$${\mathit{P}}^{\mathit{S}\mathit{Y}\mathit{S}}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\frac{ACP{T}_{\tau , D_d}}{N{S}_{ \tau , D_d}}$$

(37)

$$P^{SYS}{\left(EC\right)}_{ \tau , D_d}=\left[\begin{array}{ccccc}{P}^{SYS}{\left(EC\right)}_{1, D_1}& P^{SYS}{\left(EC\right)}_{2, D_1} & P^{SYS}{\left(EC\right)}_{3, D_1}& \cdots & P^{SYS}{\left(EC\right)}_{24, D_1}\\ P^{SYS}{\left(EC\right)}_{1, D_2}& P^{SYS}{\left(EC\right)}_{2, D_2}& P^{SYS}{\left(EC\right)}_{3, D_2}& \cdots & P^{SYS}{\left(EC\right)}_{24, D_2}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ P^{SYS}{\left(EC\right)}_{1, D_d}& P^{SYS}{\left(EC\right)}_{2, D_d}& P^{SYS}{\left(EC\right)}_{3, D_d}& \cdots & P^{SYS}{\left(EC\right)}_{24, D_d}\end{array}\right]$$

(38)

where:

• $N{S}_{ \tau , D_d}—$ all correctly received reactions at each hour τ of the day Dd;
• $ACP{T}_{\tau , D_d}—$ the number of reactions in hour τ on day D_d, ending with acceptance of V2G service provision.

Then, over the settlement month for a given τ hour, the average value of the system probabilities for successive groups of days of the week should be determined:

$$P^{SYS}{\left(EC\right)}_{\tau , avg}=\frac{{\sum }_{d=1}^{N_D}{P}^{SYS}{\left(EC\right)}_{\tau , D_d}}{N_D}$$

(39)

where:

• d—the same days of the week consecutively in a month (e.g., Mondays);
• ND—the number of the same days of the week in a month (e.g., the number of all Mondays in a given month).

Then, the probability value $P^{SYS}{\left(EC\right)}_{\tau , avg}$ should be dependent on the number of uEVs that are located in each square of the operation area of the V2Gsp. It is assumed that the probability of service for the EndUs located in a square with a higher number of uEVs will be higher than in the case of an area less densely covered with uEVs. Therefore, for each square of the area G, determine the ratio of the average number of uEVs located in the area of that square in each hour τ for the same weekdays in a month, to the average of the sums of all uEVs in the area G for each hour τ for the same weekdays in a month. This is represented by Equations (40)–(44):

$$N_{G_i, \tau , D_d}^{avg}=\frac{{\sum }_{d=1}^{N_D}{N}_{G_i, \tau , D_d}^{uEV}}{N_D}$$

(40)

$$N_{G_i, \tau , D_d}^{SUM}={\displaystyle \sum }_{i=1}^{N_G}{N}_{G_i, \tau , D_d}^{uEV}$$

(41)

$$N_{G_i, \tau , D_d}^{SUMavg}=\frac{{\sum }_{d=1}^{N_D}{N}_{G_i, \tau , D_d}^{SUM}}{N_D}$$

(42)

$$W_{G_i, \tau , D_d}=\frac{N_{G_i, \tau , D_d}^{avg}}{N_{G_i, \tau , D_d}^{SUMavg}}$$

(43)

$${\mathit{W}}_{\mathit{G}\mathit{i} \mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{ccccc}{W}_{G_1, \tau , D_d}& W_{G_2, \tau , D_d} & W_{G_3, \tau , D_d}& \cdots & W_{G_n, \tau , D_d}\\ W_{G_{n+1}, \tau , D_d}& \cdots & \cdots & \cdots & \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \cdots & \cdots & \cdots & \cdots & W_{G_{NG}, \tau , D_d}\end{array}\right]$$

(44)

where:

• $N_{G_i, \tau , D_d}^{avg}$—the average number of uEVs located in the area of that square in each hour τ for the same weekdays in a month;
• $N_{G_i, \tau , D_d}^{SUM}$—the sum of all uEVs in the area G for each hour τ for the same weekdays in a month;
• $N_{G_i, \tau , D_d}^{SUMavg}$—the average value of the $N_{G_i, \tau , D_d}^{SUM}$ in the area G for each τ hour for the same weekdays in a month;
• ND—the number of the same days of the week in a month (e.g., the number of all Mondays in a given month);
• NG—the maximum number of squares in area G;
• $W_{G_i, \tau , D_d}$—the ratio between $N_{G_i, \tau , D_d}^{avg}$ to $N_{G_i, \tau , D_d}^{SUMavg}$.

Then, it is necessary to determine the average value from every value of matrix (44)—$W_{G_i, \tau , D_d}^{avg}$:

$$W_{G_i, \tau , D_d}^{avg}=\frac{{\sum }_{i=1}^{NG}{W}_{G_i, \tau , D_d}}{NG}$$

(45)

Based on the obtained calculations, the system probability value can be decomposed as a function of the number of uEVs in a given square at a given hour τ for selected days of the week in the settlement month. This process is represented by Equations (46) and (47):

$$P^{G_i}{\left(EC\right)}_{\tau ,D_d}=P^{SYS}{\left(EC\right)}_{\tau , avg}+\left(W_{G_i, \tau , D_d}-W_{G_i, \tau , D_d}^{avg}\right)$$

(46)

$${\mathit{P}}^{{\mathit{G}}_{\mathit{i}}}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{\tau },{\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{ccccc}{P}^{G_1}{\left(EC\right)}_{\tau ,D_d}& P^{G_2}{\left(EC\right)}_{\tau ,D_d} & P^{G_3}{\left(EC\right)}_{\tau ,D_d}& \cdots & P^{G_n}{\left(EC\right)}_{\tau ,D_d}\\ P^{G_{n+1}}{\left(EC\right)}_{\tau ,D_d}& \cdots & \cdots & \cdots & \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \cdots & \cdots & \cdots & \cdots & P^{G_{NG}}{\left(EC\right)}_{\tau ,D_d}\end{array}\right]$$

(47)

where $P^{G_i}{\left(EC\right)}_{\tau ,D_d}$ is the probability of the user’s reaction of V2G service provision in the square Gi at a given hour τ for selected days of the week D_d in the settlement month.

Obtained from Equation (46), the probability of the user’s reaction of V2G service provision in the square Gi $P^{G_i}{\left(EC\right)}_{\tau ,D_d}$ refers to the settlement month M. However, for the purpose of forecasting the probability value P(EC) in month M+1, the trend of changes from month M and M−1 should also be considered. Hence,

$$P_{M+1}^{G_i}{\left(EC\right)}_{\tau ,D_d}=P_M^{G_i}{\left(EC\right)}_{\tau ,D_d}+\Delta P$$

(48)

$$\Delta P=P_M^{G_i}{\left(EC\right)}_{\tau ,D_d}-P_{M-1}^{G_i}{\left(EC\right)}_{\tau ,D_d}$$

(49)

5. Results and Discussion

In order to validate the methodology presented in Section 4.1 and Section 4.2, exemplary calculations were performed that will determine the initial parameters for calculating the total probability of V2G service, as well as simulation on targeted P(EC) values. At the end of this section, the results from the calculations and the values assumed in [34] will be compared.

The first probability that will be evaluated concerns EndUs. It should be calculated according to Formula (18). As mentioned in the previous section, the methodology and values needed to calculate it will be taken from the literature. First, the probability that the charging station will operate normally should be calculated using Equation (16). The values presented in [50] were used. It was assumed that ${\mu }_{EVSE}=\mathrm{20,000}$ per 1 million hours and ${\lambda }_{EVSE}=11.3266$ per 1 million hours [50]. The value of ${\lambda }_{EVSE}$ was determined for the non-repairable two-phase interleaved topology. Thus, having the aforementioned data, the probability of operation of the charging station can be calculated according to Equation (16):

$$p_{EVSE}=\frac{{\mu }_{EVSE}}{{\lambda }_{EVSE}+{\mu }_{EVSE}}=\frac{20,000}{20,000+11.3266}=0.99943$$

(50)

Then, according to Equation (11), the probability of V2G charging station availability for a V2G Program participant should be calculated. In [47], an approach was presented in which, regardless of the factor (hardware or human activity) affecting the absence of an EV at a charging location, such an event should be treated equally. Therefore, the values from [47] were adopted and are as follows: ${\mu }_{DPL}=1\times {10}^{-2}$ per hour and ${\lambda }_{DPL}=5\times {10}^{-4}$ per hour. The probability of charging station availability is thus equal to

$$p_{DPL}=\frac{{\mu }_{DPL}}{{\lambda }_{DPL}+{\mu }_{DPL}}=\frac{1\times {10}^{-2}}{1\times {10}^{-2}+5\times {10}^{-4}}=0.9524$$

(51)

Therefore, the probability P(EndUs) is equal to

$$P\left(EndUs\right)=p_{EVSE} ·p_{DPL}=0.99943 ·0.9524=0.9518$$

(52)

Another probability to be evaluated concerns the V2Gsp. Once again, the data presented in [47] was used, and P(V2Gsp) was calculated assuming that

$$\begin{array}{l}{\lambda }_S=1 \mathrm{per} \mathrm{year}\\ {\mu }_S=99 \mathrm{per} \mathrm{year}\\ P\left(V2Gsp\right)=1-P\left(F_S\right)=1-q_S=1-\frac{{\lambda }_S}{{\lambda }_S+{\mu }_S}=1-\frac{1}{99+1}=0.99\end{array}$$

(53)

Then, the probability of service provision by the V2G Program participant uEV was calculated. The first event concerns the operation of the V2G charging station due to the uninterrupted service by the uEV—component $1-P\left(INT\right)$. Analogous considerations for the probability of charging station availability from Equation (51) should be used [47]. Given these considerations, it can be assumed that

$$\begin{array}{l}{\mu }_{INT}=1\times {10}^{-2} \mathrm{per} \mathrm{hour} \\ {\lambda }_{INT}=5\times {10}^{-4} \mathrm{per} \mathrm{hour} \\ 1-P\left(INT\right)=1-\frac{{\lambda }_{INT}}{{\lambda }_{INT}+{\mu }_{INT}}=1-\frac{5\times {10}^{-4}}{1\times {10}^{-2}+5\times {10}^{-4}}=0.95\end{array}$$

(54)

The next calculation will be based on the survey results presented in Appendix A. The first probability to be calculated will be the reaction probability of the V2G Program participant. As mentioned in Section 4, the first calculated parameter will be the probability of user reaction P(EC). It is worth noting that in the conducted survey time frames were used. Ultimately, the values of the aforementioned probabilities should be variable for each hour of the day, but in order to simplify the survey form, time frames were used. In the process of calculating the probability $P{\left(EC\right)}_t{}^{V2GL}$, the distribution of answers to question (13) was used. The crucial values are presented in Table A9. In

Table 2. Distribution of responses to the question regarding the hours of potential V2G service provision location of EndUs and calculated probability $P{\left(EC\right)}_t{}^{V2GL}$ values.

Question:	(14) During What Hours Would You Be Able to Provide an Electric Vehicle Discharging Service at the Point of Service (e.g., Industrial Facility)?	$\mathit{P}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{t}}{}^{\mathit{V}2\mathit{G}\mathit{L}}$
00:00–06:00	12	0.21
06:01–09:00	10	0.18
09:01–12:00	21	0.38
12:01–15:00	23	0.41
15:01–18:00	9	0.16
18:01–21:00	17	0.30
21:01–23:59	23	0.41

, the answers from Figure A10 are extracted and according to Equation (21) the probability $P{\left(EC\right)}_t{}^{V2GL}$ is calculated.

Exemplary calculations are shown by Equation (55).

$$P{\left(EC\right)}_{12:01–15:00}{}^{V2GL}=\frac{N_{uEV}^{V2GL}\left(t\right)}{{\sum }_{k=1}^{R_{YES}^{V2GL}}{N}_k+{\sum }_{l=1}^{R_{DK}^{V2GL}}{N}_l}=\frac{23}{30+26}=0.41$$

(55)

The final component needed to calculate the probability P(A) is the probability of electric vehicle availability related to lack of use at a given point in time P(US_EV). As with P(EC), the calculation will be done based on the surveys and within the defined time frames. During the surveys, 51 respondents indicated that they own a BEV or PHEV, and after a further analysis of the responses, 22 respondents indicated that they currently own an ICE vehicle and plan to purchase an EV within the next 5 years. It should be noted that the number of respondents who plan to purchase a vehicle within 5 years and are users of an ICE vehicle is from post-processing of the survey data. According to Equation (22), the sum of these indices will be the set of all users to be considered when calculating the initial probability value P₁(US_EV). Each respondent could select multiple responses, i.e., multiple hourly intervals.

Table 3. Calculation of probability values P(US_EV).

Question:	What Hours Do You Use Your Car Most Often?		Probability of Using Vehicle P1(USEV)	Probability of Electric Vehicle Availability Related to the Lack of Use P(USEV)
Hours:	BEV and PHEV Owners	ICE Owners, Who Intend to Buy an EV within 5 Years
00:00–06:00	2	1	0.04	0.96
06:01–09:00	30	12	0.58	0.42
09:01–12:00	28	4	0.44	0.56
12:01–15:00	29	4	0.45	0.55
15:01–18:00	36	15	0.70	0.30
18:01–21:00	22	6	0.38	0.62
21:01–23:59	5	4	0.12	0.88

shows the distribution of responses to the question regarding the hours of movement of users and calculates the probabilities P(US_EV). It is also worth mentioning that in order to calculate the probability P(US_EV), one must first calculate the value P₁(US_EV), i.e., the actual value indicating the share of active vehicles in traffic, according to Formula (23).

Exemplary calculations are shown by Equations (56) and (57).

$$P_1{\left(U{S}_{EV}\right)}_{12:01–15:00}=\frac{N_{US}^{EV}\left(t\right)+N_{US}^{ICE}\left(t\right)}{{\sum }_{m=1}^{R_{EV}}{N}_m+{\sum }_{n=1}^{R_{ICE}}{N}_n}=\frac{29+4}{51+22}=0.45$$

(56)

$$P{\left(U{S}_{EV}\right)}_{12:01–15:00}=1-P_1{\left(U{S}_{EV}\right)}_t=1-0.4521=0.55$$

(57)

Knowing that P(uEV) is defined by Equation (14), its value for each time frame can be calculated.

Table 4. Calculation of probability values P(uEV).

Hours:	Probability of Service Not being Interrupted 1-P(INT)	Probability of Electric Vehicle Availability Related to the Lack of Use P(USEV)	Probability of uEV Reaction on the Provision of the V2G Service at the EndUs Location $\mathit{P}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{t}}{}^{\mathit{V}2\mathit{G}\mathit{L}}$	Probability Relates to the V2G Service Provision at an EndUs Location from a uEV Perspective P(uEV)V2GL
00:00–06:00	0.95	0.96	0.21	0.20
06:01–09:00		0.42	0.18	0.07
09:01–12:00		0.56	0.38	0.20
12:01–15:00		0.55	0.41	0.21
15:01–18:00		0.30	0.16	0.05
18:01–21:00		0.62	0.30	0.18
21:01–23:59		0.88	0.41	0.34

shows the calculated uEV probability values, whereas the total probability of V2G service provision is the product of three separable events: P(EndUs), P(V2Gsp), and P(uEV), defined by Equation (5).

Table 5. Calculation of probability values P(A).

Hours:	Probability Relates to the V2G Service Provision at an EndUs Location from a uEV Perspective P(uEV)V2GL	Probability Relates to the V2G Service Provision from a V2Gsp Perspective P(V2Gsp)	Probability Relates to the V2G Service Provision from an EndUs 1 Perspective P(EndUs)	Total Probability of V2G Service Provision at an EndUs Location P(A) V2GL
00:00–06:00	0.20	0.99	0.9518	0.18
06:01–09:00	0.07			0.07
09:01–12:00	0.20			0.19
12:01–15:00	0.21			0.20
15:01–18:00	0.05			0.04
18:01–21:00	0.18			0.17
21:01–23:59	0.34			0.32

^1. In this case, EndUs is the secured customer of the V2G Program. This could be a household or other industrial facility.

shows the calculated probability values P(A) for the selected time frames.

Knowing the values of the probabilities P(A) calculated according to the methodology presented in this publication, it is possible to compare them with the previously assumed values of the probabilities in [34]. It turns out that only for P(V2Gsp) was the correct value assumed. The biggest differences are in the results of P(uEV). Such low P(uEV) values imply that the search area mentioned in [34] should be much larger and may not meet the users’ expectations for the distances traveled between their current location and the location where the service is provided.

It should also be emphasized that using the survey data approach to calculate the initial probabilities of V2G service provision requires the aforementioned surveys to be carried out in the most accurate manner. It should be kept in mind that the accuracy of the calculations will be affected not only by the number of respondents, but also by the way the questions are asked. Therefore, the future V2Gsp should conduct these surveys with the help of specialists in social research. Considering the fact that initial values will be needed to establish boundary conditions in the first months of V2G Program implementation, it is necessary to use advanced forecasting tools in the future to verify the calculations performed.

The coefficients determining the proportion of users in mandatory and optional mode for both types of services, i.e., at home and at the EndUs location, were also calculated. The results are presented in Equations (58) and (59) and

Table 6. Parameters required to calculate SUR coefficients.

Parameter	Value
$N_{uEV}^{V2GL}$	30
$N_{MA{N}_{ONLY}}^{V2GL}$	1
$N_{MA{N}_{OPT}}^{V2GL}$	21
$N_{OP{T}_{ONLY}}^{V2GL}$	7
$N_{NOTINT}{}_L$	1

shows the parameters needed to calculate the aforementioned coefficients.

$$SU{R}_{MAN}^{V2GL}=\frac{N_{MA{N}_{ONLY}}^{V2GL}+N_{MA{N}_{OPT}}^{V2GL}}{N_{uEV}^{V2GL}-N_{NOTIN{T}_L}}=0.7586$$

(58)

$$SU{R}_{OPT}^{V2GL}=\frac{N_{OP{T}_{ONLY}}^{V2GL}}{N_{uEV}^{V2GL}-N_{NOTINT}{}_L}=0.2414$$

(59)

Analyzing the results presented in Equations (58) and (59) and comparing them to the assumed values in [34], it can be concluded that the willingness of declared mandatory users is well above expectations. Nevertheless, it should be mentioned that the SUR coefficients consider only committed participants, which can also be misleading in observing the assumptions of the V2G Program.

In order to perform calculations for the target solution while calculating the probability P(EC), a number of simulations must be performed on random data. It should be emphasized that the indicated values do not represent the real state of the uEV distribution in the real location. The following simulation conditions were established:

• The number of days for analysis: d = 4;
• Analysis for one-hour τ;
• The number of service acceptances and all successful reactions determined randomly;
• The operating area of a V2Gsp (G) is 6 km × 6 km;
• In each square G_i, a random number of vehicles was assigned according to $N_{G_i, \tau , D_d}^{uEV}\in <0, 3000)$

.

The matrices describing the number of vehicles at a given hour τ on days d = 1, d = 2, d = 3, and d = 4 are shown in Equations (60)–(63), according to Relation (34).

$${\mathit{N}}_{{\mathit{G}}_{\mathit{i}}, \mathit{\tau }, {\mathit{D}}_{\mathbf{1}}}^{uEV}=\left[\begin{array}{cccccc}625& 399& 64& 394& 747& 92\\ 628& 577& 513& 605& 27& 553\\ 618& 13& 406& 760& 557& 105\\ 307& 122& 566& 267& 405& 495\\ 662& 421& 622& 287& 448& 381\\ 473& 652& 228& 122& 295& 51\end{array}\right]$$

(60)

$${\mathit{N}}_{{\mathit{G}}_{\mathit{i}}, \mathit{\tau }, {\mathit{D}}_{\mathbf{2}}}^{uEV}=\left[\begin{array}{cccccc}245& 489& 550& 425& 550& 659\\ 410& 277& 520& 588& 506& 228\\ 403& 282& 199& 315& 349& 249\\ 553& 356& 340& 266& 498& 109\\ 457& 330& 263& 284& 5637& 545\\ 446& 313& 137& 335& 618& 455\end{array}\right]$$

(61)

$${\mathit{N}}_{{\mathit{G}}_{\mathit{i}}, \mathit{\tau }, {\mathit{D}}_{\mathbf{3}}}^{uEV}=\left[\begin{array}{cccccc}254& 515& 708& 883& 986& 389\\ 19& 153& 413& 766& 942& 746\\ 225& 582& 653& 258& 424& 241\\ 127& 1004& 316& 314& 651& 164\\ 674& 333& 938& 18& 8& 829\\ 679& 781& 820& 625& 1009& 406\end{array}\right]$$

(62)

$${\mathit{N}}_{{\mathit{G}}_{\mathit{i}}, \mathit{\tau }, {\mathit{D}}_{\mathbf{4}}}^{uEV}=\left[\begin{array}{cccccc}299& 1404& 424& 2200& 1210& 2520\\ 935& 810& 553& 1353& 1039& 1698\\ 977& 145& 920& 293& 1131& 2505\\ 351& 1354& 148& 718& 2438& 1971\\ 253& 424& 2147& 374& 1777& 2535\\ 779& 1423& 2179& 881& 997& 855\end{array}\right]$$

(63)

Then, it is necessary to note the matrices that represent the number of acceptances of the V2G service $ACP{T}_{\tau , D_d}$ and the number of all successful uEV reactions $N{S}_{ \tau , D_d}$, i.e., all accepted and rejected requests and demands. According to Formulas (35) and (36) for the selected simulation cases, these will be vectors with four rows and one column, as one hour of the day for the selected group of weekdays in a month was analyzed (no. of days in the analysis is d = 4). Thus,

$$\mathit{A}\mathit{C}\mathit{P}{\mathit{T}}_{\mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{c}644\\ 456\\ 583\\ 700\end{array}\right]$$

(64)

$$\mathit{N}{\mathit{S}}_{ \mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{c}1400\\ 901\\ 1800\\ 2421\end{array}\right]$$

(65)

Therefore, the system probability for each day can be determined according to Equation (37), and then the matrix for days d = 4 can be determined according to Equation (38):

$${\mathit{P}}^{\mathit{S}\mathit{Y}\mathit{S}}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{c}0.460\\ 0.506\\ 0.324\\ 0.289\end{array}\right]$$

(66)

The average system probability value $P^{SYS}{\left(EC\right)}_{\tau , avg}$ was then determined according to Equation (39):

$$P^{SYS}{\left(EC\right)}_{\tau , avg}=\frac{0.460+0.506+0.324+0.289}{4}=0.395$$

(67)

In the following steps, the four-day average of the number of uEVs in a given square was determined. In the exemplary results, the calculation for square G₁ is shown. Knowing that area G consists of a 6 × 6 km grid, it can be deduced that the number of NG squares is equal to 36, i.e., there are 36 G_i squares in area G. After calculating the sum of uEVs in the area for each day d according to Equation (41), the average of the sums for the four days was then obtained.

$$N_{G_i, \tau , D_d}^{avg}=\frac{625+245+254+299}{4}=356$$

(68)

$$N_{G_i, \tau , D_d}^{SUMavg}=\frac{14,487+14,116+18,853+42,020}{4}=22,369$$

(69)

By having the aforementioned values, the ratios $W_{Gi \tau , D_d}$ can be calculated according to Equation (43). Formula (70) shows the matrix of ratios $W_{Gi \tau , D_d}$. Then, the average value from the values of the matrix of ratios $W_{Gi \tau , D_d}$ was calculated according to Equation (45).

$${\mathit{W}}_{\mathit{G}\mathit{i} \mathit{\tau }, {\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{cccccc}0.016& 0.031& 0.020& 0.044& 0.039& 0.041\\ 0.022& 0.020& 0.022& 0.037& 0.028& 0.036\\ 0.025& 0.011& 0.024& 0.018& 0.028& 0.035\\ 0.015& 0.032& 0.015& 0.017& 0.045& 0.031\\ 0.023& 0.017& 0.044& 0.011& 0.031& 0.048\\ 0.027& 0.035& 0.038& 0.022& 0.033& 0.020\end{array}\right]$$

(70)

$$W_{G_i, \tau , D_d}^{avg}=\frac{1}{36}=0.0278$$

(71)

In the last step, it is necessary to calculate the modified system probability considering the uEV density in a given square $P^{G_i}{\left(EC\right)}_{\tau ,D_d}$. An example calculation for square G₁ is presented in Equation (72), based on Equation (46). The matrix of probabilities $P^{G_i}{\left(EC\right)}_{\tau ,D_d}$ is then presented in Equation (73). It should be emphasized that the trend of changes in the values of the probabilities was not evaluated in this case study, due to its purely illustrative nature.

$$P^{G_1}{\left(EC\right)}_{\tau ,D_d}=0.395+\left(0.016-0.0278\right)=0.383$$

(72)

$${\mathit{P}}^{{\mathit{G}}_{\mathit{i}}}{\left(\mathit{E}\mathit{C}\right)}_{\mathit{\tau },{\mathit{D}}_{\mathit{d}}}=\left[\begin{array}{cccccc}0.383& 0.398& 0.387& 0.411& 0.406& 0.408\\ 0.389& 0.387& 0.389& 0.404& 0.395& 0.403\\ 0.392& 0.378& 0.391& 0.385& 0.395& 0.402\\ 0.382& 0.399& 0.382& 0.384& 0.412& 0.398\\ 0.390& 0.384& 0.411& 0.378& 0.398& 0.415\\ 0.394& 0.402& 0.405& 0.389& 0.400& 0.387\end{array}\right]$$

(73)

Thus, it was verified that the methodology for calculating values of probability $P^{G_i}{\left(EC\right)}_{\tau ,D_d}$ allows us to determine them from the number of uEVs in a given grid square of the V2Gsp operating area.

6. Conclusions

This paper presented an original methodology for calculating the values of initial V2G service probabilities needed to estimate the number of electric vehicles (V2G Program participants) providing discharging services in a given area. In addition, the results of a survey on the willingness to participate in the V2G Program were presented.

The survey section of the article presented the results of this survey. The survey took place between 6 July 6 and 19 July 2021, during which 130 forms were collected. Additionally, five follow-up interviews were conducted in which current EV users evaluated the assumptions of the V2G Program based on their experience of driving an electric vehicle. From the survey results, it can be deduced that there is public interest in V2G technology. However, potential users are mainly interested in providing V2G services in their own households. At this point, it is worth noting that during the interviews and in the additional comments to the survey, a large number of respondents indicated that they would like to use the electric vehicle as an additional, backup energy source exclusively for their own household needs. This would be in contrast to the V2G Program’s intent, in which a regional operator would manage the energy in available EVs. Nevertheless, it does set some direction for potential operators to take while offering an interesting household business case. Furthermore, it can be read from the surveys that respondents are not very keen on moving between locations to provide V2G services. This highlights a problem that is mentioned in almost every publication on the impact of V2G on user behavior, which is the mobility behavior of V2G Program participants. There is another interesting finding from the surveys regarding electric vehicle charging locations. One of the questions asked current EV owners to indicate where they most often charge their vehicles, and respondents who do not have such a vehicle were asked to indicate the place where they would like to charge it. Interestingly, the responses for the two groups of respondents were consistent for charging at home, while there were big differences for charging in public places. It can be presumed that current EV owners have verified the possibility and chances of charging in public places.

The second part of the paper presented the original methodology for calculating the probability values based on the collected survey data. It was assumed that the survey results can be used as initial parameters in the business model of the V2G service provider. However, it should be emphasized that in the real conditions of V2G Program implementation, these parameters should be updated using advanced forecasting tools. This is due to the fact that, in the proposed V2G Program model, the service probability values are extremely important as they affect the number of vehicles that will be called to provide the service. There may be some doubt about the value of the calculated probabilities from the perspective of the V2G Program operation. During hours of peak power demand in the power system or in industrial plants, the availability of users providing the service is very low, as it does not exceed 20%. This result should encourage potential V2G Program operators to thoroughly analyze the business risks of this venture. In addition, the article presented a method for calculating the share of participants in mandatory and optional mode. in its assumptions, the mandatory mode involves a fixed revenue that would be received for the availability of V2G services, but also potential fines for not providing the service. During the simulation tests, there was a high percentage of participants willing to participate in the mandatory mode, with a significant number declaring their willingness to participate in both modes. This is important as it may guarantee a high-quality service, as perhaps a small proportion of all those willing would take part in active service provision, but those committed would be determined to provide the service reliably. The values of uEV participation in mandatory and optional modes as well as the values of probabilities should be updated on an ongoing basis during the operation of the V2G Program. A target methodology for calculating the probability of V2G Program participant reactions was also presented. From the perspective of V2G service provision, this is the most sensitive element that may affect the correct or incorrect estimation of the total probability of service provision, due to the impossibility of predicting human behavior. Therefore, it is necessary to predict the value of this probability based on historical data obtained during the operation of the V2G Program. This only underlines the fact that determining initial probability values based on survey data will be necessary to begin operation of the entire V2G Program.

In summary, the selection of initial parameters is crucial for the continued smooth operation of the V2G Program. An extremely important element is to conduct social research that will help to properly select V2G products and services in a given area of operation. Future research work in the area of the proposed V2G Program mechanism will focus on developing settlement models for individual parties and on enhancing survey questions that would even more adequately describe EV users’ behavior.