A stochastic model for estimating electric vehicle arrival at multi-charger forecourts

open-source data, and is structured flexibly, making it adaptable to new sites as part of the technical and business planning process.


Introduction
The aim of electrifying transport is to reduce greenhouse gas (GHG) emissions.In the UK the anticipated reduction in the carbon dioxide emissions is between 50% and 87% by 2050 compared to 2014 due to EV uptake, improved efficiency of internal combustion engine vehicles and decarbonization of the electricity supply (Hill et al., 2019;Bharathidasan et al., 2022).Interest in trialing EVs and charging infrastructure is worldwide, for example: China (Zheng et al., 2020), Finland (Uimonen and Lehtonen, 2020), Ireland (Morrissey et al., 2016), New Zealand (Su et al., 2019), South Korea (Moon et al., 2018), UK (Robinson et al., 2013), and USA (Harris and Webber, 2014;Tehrani and Wang, 2015;Almutairi and Alyami, 2021).Barriers to the spread of EVs include limited availability of charging infrastructure, low charging power (long EV recharge time) and limited electrical range (Elghitani and El-Saadany, 2021;Goel et al., 2021).Most frequently charging infrastructure has been limited to individual home charging, adhoc on-street charging, and small collections of chargers at workplaces.For the normalization of EV use, charging needs to be more convenient, and it is proposed that an experience similar to current refuelling stations will be required i.e. public sites with many chargers available.However, this has attendant issues such as estimating the likely vehicle throughput and the impact on the local distribution grid (estimating the maximum instantaneous power demand).Neither of these issues are deterministic, but must be simulated stochastically.
Although EVs reduce tailpipe emissions, they introduce a new type of electric load (Birk Jones et al., 2022;Ahmad et al., 2022).It is widely recognized that grid constraints are an obstacle to greater EV uptake (Qian et al., 2011;Steen et al., 2012;Robinson et al., 2013;Harris and Webber, 2014;Morrissey et al., 2016;Su et al., 2019;Cheng et al., 2022).The majority of studies consider the effects of home charging (Qian et al., 2011;Harris and Webber, 2014;Tehrani and Wang, 2015;Brady and O'Mahony, 2016;Morrissey et al., 2016;Ul-Haq et al., 2018;Su et al., 2019;Almutairi and Alyami, 2021).Work place (Brady and O'Mahony, 2016;Uimonen and Lehtonen, 2020) and public (Qian et al., 2011;Morrissey et al., 2016) charging have also been considered.Most EV charging uses uncoordinated low to medium power charging points at home, work, and public places.Currently 81.4% of UK public installed charging points have power ratings ≤22 kW and only 3.9% are ≥100 kW (Zap-MAP, 2021).Home charging is limited to 3-5 kW requiring off-street parking often unavailable to those living in apartments or other high-density housing.Dedicated high-speed charging stations will allow drivers to charge without needing to arrive at their destination (Dixon et al., 2018).
The load profile of an EV is determined by several stochastic factors such as the daily travelled distance and the battery stateof-charge (SoC) at start of the day (Humayd and Bhattacharya, 2018).Deterministic modelling is not a suitable mean to estimate the EV load profile as it does not reflect the stochastic nature of the important variables, but non-deterministic approaches have not received sufficient attention (Buzna et al., 2021).Zhang et al. (2020) divide EV infrastructure modelling into computational geometry and origin-destination flow-based approaches.Different models have been reported to forecast/estimate the aggregated EVs load profile (Shepero et al., 2018;Daina et al., 2017).Modelling methods use a variety of statistical and probabilistic techniques (Tehrani and Wang, 2015;Brady and O'Mahony, 2016;Uimonen and Lehtonen, 2020;Zheng et al., 2020), agent-based models (Zhang et al., 2020), Gaussian mixture models (Powell et al., 2022), Monte Carlo simulation (Harris and Webber, 2014;Almutairi and Alyami, 2021;Sadhukhan et al., 2021), Markov chain (Ul-Haq et al., 2018;Yan et al., 2022), ensemble methods (Buzna et al., 2021), and deep learning (Zhu et al., 2019a,b).Most studies use travel data and vehicle specifications, but demographic data (Steen et al., 2012), building occupancy (Uimonen and Lehtonen, 2020), and user preference (Robinson et al., 2013;Moon et al., 2018) have also been used.Probability distribution functions have been used (Almutairi and Alyami, 2021) to represent variables such as the EV home arrival time, home departure time and daily mileage.The daily load profile was estimated for each EV and accumulated for all vehicles resulting in the total fleet load profile.The copula method has been used (Tehrani and Wang, 2015) to develop set of random variables with joint probability distributions.
In assuming that EVs charge at end of each trip to generate probability distributions throughout the day and giving different probabilities of charging to plug-in hybrid electric vehicles (PHEVs) and the battery electric vehicles (BEVs), Harris and Webber (Harris and Webber, 2014) used Monte Carlo simulations to estimate the load demand.In Brady and O'Mahony (2016), the dependence structure between the first departure time in the day, number of journeys per day and total distance travelled was modelled using a non-parametric copula function.Then, a travel pattern was generated considering uncertainty of the inputs.Upon arrival at a destination, a probabilistic charging decision model calculates the probability of charging given the EV battery state of charge, the available parking time and the current journey number.The model depends on using slow home/work chargers and assumed that it is more likely an EV charges after the first or second journey due to arriving at work.
Attention has been paid to characterizing charging demand using surveys and by analysing recorded charging events.A Korean survey (Moon et al., 2018) estimated the increase in the electrical load due to the expansion of EVs and its affect on the electric grid.The survey included questions about EV users' preferences such as time of the day, location and type of charging point used to recharge the EVs.This showed that the users prefer to use fast public charging points during the peak hours of the day while charging at home during the night.Fast charging points are preferred by users of the public chargers which will be commercially available in the short to medium term (Morrissey et al., 2016).A study of charging habits in the north east of the UK combined both private and organizational users (Robinson et al., 2013).
Analysis of 7704 charging events showed that private users peak demand was at the evening when the drivers arrive home (home charging) while the organizational drivers primarily charge their EVs at the work.Charging events at a wide variety of locations and types of chargers in the San Francisco area have been analysed and modelled to estimate charging demand (Powell et al., 2022).
Unscheduled EV charging places stress on the electric grid (Harris and Webber, 2014).In Zhu et al. (2019b,a), different deep learning approaches have been used and compared for short-term load forecasting of EV charging.Real charging events recorded in Shenzhen, China, have been used for the training and testing of the models.By segmenting the fleet (Zheng et al., 2020) studied mid-and-long term load forecasting using a probabilistic approach incorporating market evolution, giving projections of future EV ownership.To enhance the power system operation and to help setting the network reinforcement or upgrade plans, stochastic modelling for EV load demand was proposed (Qian et al., 2011).Comparison between four charging scenarios (Qian et al., 2011): (1) uncontrolled domestic charging, (2) uncontrolled off-peak domestic charging, (3) smart domestic charging and (4) uncontrolled public charging, showed that the uncontrolled domestic charging causes the highest increase in the system peak load.The model assumed that private EVs charge once every two days while company EVs charge once per day.
Two models estimate loads for fast-charging stations in a stochastic manner: a station with six 50 kW chargers (Deng et al., 2018) and 50 stations each with one 50 kW charging point (Korolko et al., 2016) (defaults to home-charging with a 10% station-use probability).The fleet size was defined according to the national average size of EVs per installed charger (with uniform distribution).Moradipari et al. (2021) studied how to influence where EV users charge to reduce both the aggregate congestion at fast-charging points installed at different locations and impact on the electric grid.An energy management approach (Huo et al., 2017) forecasts charging demand assuming a Poisson process where EVs start with 10%-40% SoC, terminating at 80%.For a 3-charger station 100 EVs were assumed to arrive every day.Google Maps Popular Times data for all petrol stations in the UK was used in a Monte Carlo model to estimate the EV arrival at a fast-charging station (Dixon et al., 2018).Probability distributions were assigned to EV initial and final SoC while converting the arrival rate to a demand profile.Most models do not consider location-dependent factors/parameters, instead using an average station occupancy.
Most previous models rely on finding the periods of time EVs are at home/work/public place, then checking the charging probability during these intervals.Others concerned about fast charging estimate an average occupancy of the charging connection or study the aggregate influence of dispersed charging points on the electric grid.Typically, studies have been concerned with power demand from EVs at a highly aggregated system level e.g.Tang and Wang (2016), Yi and Bauer (2016), Zhang et al. (2020), Kavianipour et al. (2021) and Pan et al. (2020) and are focused on infrastructure planning for a geographic area or region.The charging requirements of EVs has been incorporated into a stochastic transport model as part of a smart city simulation (Duan et al., 2021).
In this paper, EV arrival at a multi-charger electric forecourt is treated like conventional (liquid) refuelling stations, which is expected to be the preferred method in a mature EV market.To the authors best knowledge, work reported in the literature regarding EV charging demand allocation/estimation lacks a detailed study concerning a dedicated multi-charger forecourt with EVs charging during a journey.The aim of this paper is to model EV arrival rates and power demand for a dedicated charging station.The model helps optimize the forecourt operation by using parameters specific to the forecourt location, such as registered BEV/PHEVs by zipcode and real-world traffic flow on nearby roads, to forecast the EVs arrival rate and the corresponding power/energy demand.Moreover, the model has the following features which are not fully incorporated in a single model in the literature: 1.A stochastic approach incorporating uncertainty in the parameters controlling the charging process.2. Considers probabilistic EV charging from both residents and travelling EVs (different populations).3. PHEVs and BEVs are treated independently with different probabilities.4. Real-world sale statistics are used to derive information about the EV market share and fleet composition in different populations instead of assuming a national average EV distribution.5. Accounts for seasonality in the parameters.6. Considers charging at other places other than the electric forecourt (competitive charging).
The approach is to use real-world open-source data to ensure that the method is widely applicable.For validation, a convenient case-study is the UK's first dedicated multi-charger station, located at Braintree in Essex County.It is an integrated, utility-scale site consisting of 24 ultra-rapid DC chargers (of various power ratings from 90 kW to 350 kW) with charging times of less than 30 min, plus 6 AC chargers (up to 22 kW) and 6 T superchargers.This facility may mitigate the burden on the local distribution system the as the site is connected to the 33 kV grid.The model will help predict likely near-term site use, understand patterns of use to assist in developing similar sites, optimize revenue, and maintain assets.

Model design
The model is developed from an outline given in Aboshady et al. (2021), to include seasonality, competitive charging with facilities beyond the site, and charging EVs from adjacent geographical districts.The structure of the proposed model is shown in Fig. 1 has three stages.Different input parameters are fed to the model in the first stage.The processing stage estimates the arrival rate and the corresponding load demand.Lastly, the model outputs the variables of interest.
For any day, the need to charge depends on parameters such as daily travelled distance and the battery state of charge (SoC) at the start of the day.Most of the parameters are probabilistic in nature.Therefore, a stochastic approach of EV arrival accounts for the uncertainty observable in real-world data.The arrival model first estimates the number of EVs arriving at the site at each of 48 half-hour slots (aligning with the National Grid settlement period).Then, we estimate the power and energy demand for these EVs.
The model uses two population-based estimations namely EV owners living or working nearby, and traffic passing along the main roads surrounding the site.For instance, an EV driver may plan to use the site for recharging during a long journey.Each population has different charging requirements and is modelled independently.For modelling different parameters, data from the literature is used.

The local population of EVs
The local fleet combines EVs in the adjacent areas plus 25% (assumed) of the next-nearest areas, concentric to the site.Regarding the Braintree site used as a case-study, this gives 406 BEVs and 303 PHEVs registered at the end of 2020 (Department for Transport, 2020b).Their behaviour is tied to the daily travelled distance.
Modelling EV arrival rate for the local population has two steps.Firstly, the number of EVs (N ev ) most likely needing to charge in the next 24 h is estimated.The second step uses the output of the first to estimate the number of EVs for each of the 48 half-hour periods.Each EV's SoC is related to the total distance travelled since the last charging event.Assuming the SoC decreases linearly with the distance travelled, the likely current SoC is calculated by where SoC c and SoC f are the current and final SoC respectively, and d and D are the total travelled distance (accumulated daily travelled distance since the last charging event) and total electrical range for the EV, respectively.The total electrical range, battery capacity, and charging rate depend on the EV model (EV Database, 2020).A database with the common EV models and their market share is constructed.From this database, an EV model is picked randomly, based on their relative market share, for each EV in the local fleet (Department for Transport, 2020b).The distribution modelling the uncertainty in the daily travelled distance is derived from the UK National Travel Survey (Department for Transport, 2017).The data (Fig. 2) is fitted to  a lognormal distribution (2).The mean and standard deviation coefficients for the distribution are 1.9 and 1.1, respectively.
where f (d|µ, σ ) is the probability density function at distance d, µ is the mean value and σ is the standard deviation.
If the likely SoC c for an EV is < 20% the EV needs charging in the next 24-h period, giving N ev for the local population.The sequence describing this first step is illustrated by Algorithm I (Appendix).Next, we estimate the EV arrival for the 48 halfhour slots using a plug-in time distribution.Real-world charging events recorded in Essex County in the UK. are used to build the seasonal plug-in time distribution (Department for Transport, 2020a).The data is fitted to a normal distribution (3) with mean and standard deviation given in Table 1, where f (t|µ, σ ) is the probability density function at t, µ is the mean value and σ is the standard deviation.
For each time slot, the likelihood calculated from the plug-in time distribution is used as a threshold value.A random integer between zero and an upper limit is generated.The upper limit for the first time slot is N ev and is recursively decreased for the remaining time slots to account for the EVs charged during the previous time slots.This integer (N) represents the maximum possible number of EVs arriving at this slot.To decide how many EVs arrive at the current time slot, N random values between 0 and 1 are generated.Each value is compared with the threshold value at this time slot.The EV is counted as a new arrival if the random value is less than or equal to the threshold value.The sequence describing this second step is illustrated by Algorithm II (Appendix).

Passing traffic population
Arrivals from this population depend on the daily flow on the nearby roads.For this case-study (Braintree site), three major roads are considered, A12, A120 and A131, using the flow on an average day of the year (Department for Transport, 2019).The variance in traffic flow between months of the year can showed that car trips longer than 25 miles are only 7% of the total trips.This is used to distinguish long journeys from the average EV daily traffic flow.Thus, the daily vehicle flow is scaled by two factors to reflect the EV flow travelling long distances.
The EV daily flow is distributed throughout the day using the normalized motor vehicle flow distribution by time of the day (Fig. 3) (Department for Transport, 2019).All weekdays show similar distributions with two peaks, and the weekends are similar with a single peak in the afternoon.A Gaussian mixture distribution, comprising multivariate Gaussian distribution components, is used for the weekdays (4).The mean, covariance, and proportion values for the distribution are given in Table 3.A generalized extreme value distribution (5) is used for the weekends with the distribution parameters (Table 3).These distributions are equivalent to the plug-in time distribution required with the local population modelling.Accordingly, each time slot has a maximum possible number of EVs (upper limit).For each time slot, several random values are generated between 0 and 1.The slot upper limit determines how many random values are generated.These values are compared with a threshold level (TF thr ).An EV is counted as a new arrival if the generated random value is less than or equal to the threshold level.The influence of the TF thr value is studied later in the competitive charging subsections.
where p (t) is the probability density function at t for a weekday, µ i , σ i and w i are the mean, the covariance, and the proportion value for the ith distribution and n is the number of distributions.
In this case, n = 2.
where f (t|k, µ, σ ) is the probability density function at t for a weekend, µ, σ and k are the location, the scale, and the shape parameters for the generalized extreme value distribution.

Competitive charging
As EVs may charge at other locations e.g.home, work, or public standalone charging points, the proposed model includes a competitive charging probability for each population.

Local population competitive charging factor
For each EV from the local population, N ev , a random value between 0 and 1 is generated and compared with the competition factor to decide if that EV is going to be a potential arrival from the local population or is going to charge elsewhere.If the random value is less than or equal to the competition factor, then this EV is counted as a possible new arrival.Otherwise, it is assumed that this EV is going to charge elsewhere.Accordingly, a modified value for N ev is calculated and used as an input to the second step of the local population algorithm.It is expected that the site is mostly used by BEVs rather than PHEVs because of their larger battery capacity, thus benefitting from higher charging rates.Hence, two competition factors are used, BEV com and PHEV com .

Passing traffic population competitive charging factor
The competitive charging is implemented by controlling the threshold level used to determine the new arrivals (TF thr ).PHEVs have the option of using the engine, so to account for the lower likelihood of charging, the PHEV market share is scaled down by a factor (PHEV rt ) when estimating the EV flow.Therefore, the BEV to the PHEV effective ratio participating in the passing traffic increases.The effective EV share in the passing traffic is therefore estimated by ( 6) with the second term representing the PHEV effective share in the passing traffic, EV share = BEV share + PHEV rt ×PHEV share (6) A study setting the competitive charging parameters is carried out in Section 3.

EV charging
Thus far, only the EV arrival rate is modelled.This section shows how to estimate the load demand corresponding to the estimated EV arrivals.As previously mentioned, a database for the common EV models is constructed and used to randomly pick the EV model for each new arrival based on the relative EVs share in the market (Department for Transport, 2020b).The EV model defines the rated charging power, battery capacity, and total electrical range.The BEVs and PHEVs in the local population is defined using the relative share of the common models.For the  passing traffic population, firstly the EV is chosen to be BEV or PHEV based on their relative share in the population.For example, if the BEV and PHEV relative share is 80% and 20% respectively.A random value is generated between 0 and 1.If the random value is less than or equal to 0.8 then the vehicle is considered a BEV, otherwise, a PHEV.Then a vehicle is selected using the relative share of the common BEVs and PHEVs, similar to the local population case.
The energy demand of an EV is drawn from the combination of initial SoC, final SoC and the EV battery capacity because the energy distribution may be biased based on the EV type.The likely initial SoC is estimated by (1) for the local population.However, it is not predicted for the passing traffic population.Common distributions for the initial SoC (Qian et al., 2011) and final SoC (Schäuble et al., 2017) are used (Fig. 4).Both initial and final SoC distributions are represented by generalized extreme value distribution with the parameters given in Table 4.
The EV charging status time step (∆t) is 2 min.The EV charging power is assumed to be the rated EV value.The charging session ends when the amount of energy charged is equal to the required energy.A 1% probability is assumed that the user terminates the charging session before its planned end.Therefore, the power delivered by each charger and the total power (from all chargers) is recorded with a resolution ∆t throughout the day.
The total power demand is calculated using ( 7), where N ch is the total number of chargers in the charging station.This power profile represents the expected load profile for the next day which can be integrated in the optimization process controlling the site operation.

Competitive parameters: setting and model sensitivity
A study is performed on the influence of the four parameters controlling the competitive charging namely BEV com , PHEV com , TF thr , and PHEV rt on the model output, and to guide setting the values.The total monthly energy required to charge the EVs and the percentage of charging sessions related to charging BEVs (%BEV ss ) are considered as the outputs of interest when setting the competitive charging parameters.As more charging events are for BEVs, the monthly energy demand is dominated by BEVs rather than PHEVs.So, the BEV com for the local population and the TF thr for the passing traffic population are the important parameters from an energy point of view.The other two parameters (PHEV com and PHEV rt ) control the %BEV ss level.
Deep testing is performed to (1) derive the relationship between the parameters and the model output, (2) define the feasible range for each parameter, and (3) provide a guide for setting the parameters.Testing is performed over a wide range of values for the parameters using the following tests.For all cases the model has been executed 15 times for each test point.

BEV com , TF thr and monthly energy
To study the relationship between the BEV com , the TF thr , and the energy demand, each of the BEV com and the TF thr were varied between 0.1 to 0.8 in 0.05 steps while keeping PHEV com and PHEV rt fixed at 0.1 and 0.25, respectively.Fig. 5 shows the average monthly energy.The standard deviation for the 15 model executions at each point has been calculated and the average standard deviation for all points is 0.57 MWh.The following observations can be made from Fig. 5: • the monthly energy increases with increasing any of the two parameters, and • both BEV com and TF thr have a close to linear relationship with total monthly energy.
The operational data suggests that the %BEV ss is > 70%.Over the whole feasible range, the change in the average %BEV ss was only 10% as shown in Fig. 6 meaning that the BEV com and TF thr have a limited effect on the %BEV ss .
Repeating this scenario, but with PHEV com and PHEV rt fixed at 0.25 and 0.05 showed a similar curve to that in Fig. 5.The average difference in the monthly energy is 0.8 MWh between the two cases.Comparing this difference with the standard deviation of the results (0.57 MWh), then the two curves are undistinguishable, and therefore, this difference can be ignored.In other words, the PHEV com and PHEV rt have a limited impact on the relationship between the BEV com and TF thr and the monthly energy.This is also true for the %BEV ss relationship with BEV com and TF thr .
From Fig. 5, different operating points can provide the same amount of energy.Assuming an energy level of 15 MWh, a possible setting from Fig. 5 for the TF thr and the BEV com is 0.55 and 0.65, respectively (exact corresponding energy is 14.75 MWh).PHEV com and PHEV rt were changed between 0 and 0.5 while keeping the TF thr and the BEV com fixed at 0.55 and 0.65, respectively.The average monthly energy and %BEV ss are shown in Figs.7  and 8, respectively.The average standard deviation is 0.64 MWh for the monthly energy and 1.6% for the %BEV ss .Lower PHEV rt values have less effect than higher PHEV rt values on the energy.Compared with the variation of BEV com and TF thr , it is clear that the BEV com and TF thr have the major impact on the monthly energy while the PHEV com and PHEV rt predominantly affects the %BEV ss .This conclusion does not eliminate any dependency between the parameters, though the result is stable with variation in BEV com and TF thr .Repeating this scenario, but with BEV com and TF thr fixed at 0.4 and 0.7 at the same energy level does not influence the relationship between the %BEV ss and the PHEV com and PHEV rt (Fig. 8).Furthermore, changing the monthly energy by 20% showed a <3% (%BEV ss > 70%) difference from that in Fig. 8.

%BEV ss level and monthly energy, BEV com and TF thr
This case tests the effect of changing the %BEV ss level, as defined by the PHEV com and PHEV rt , on the relationship between the monthly energy and the BEV com and TF thr .From Fig. 8, the %BEV ss at PHEV com of 0.1 and PHEV rt of 0.25, used in the first scenario, is equal to 77%.Both the PHEV com and the PHEV rt were set to 0.1 corresponding to a %BEV ss of 85% in Fig. 8.The average difference in the monthly energy between the first and third scenarios was only 0.33 MWh.The influence of changing the %BEV ss level (8% change between the two scenarios) is small and will be ignored.Therefore, the same curve points can be used for representing the relationship between the monthly energy and the BEV com and TF thr .

Summary
From the previous three scenarios, we conclude: • BEV com and TF thr are key in determining the monthly energy level (Figs. 5 and 7).Whilst PHEV com and PHEV rt affect more the %BEV ss (Figs. 6 and 8).
• The relationship between the monthly energy and the BEV com and TF thr is little affected by changing the %BEV ss level.
• A 20% change in the monthly energy causes a change of <3% in the relationship between %BEV ss and the PHEV com and PHEV rt for %BEV ss > 70%.
• For the same %BEV ss as defined by the two parameters PHEV com and PHEV rt , different settings for these two parameters do not affect the relationship between the %BEV ss and the BEV com and TF thr .
• For the same energy level, different combinations for the BEV com and TF thr can be used.Changing the setting point does not affect the relationship between the %BEV ss and the PHEV com and PHEV rt .
Accordingly, the following five steps are the proposed guide for setting the four parameters: 1. Save the data in Figs. 5, 6, and 8 with energy values rounded to the nearest quarter as lookup tables.2. For the required %BEV ss that matches the nominal site utilization, pick all possible PHEV com and PHEV rt combinations that satisfy this %BEV ss .3. Choose any setting for the PHEV com and PHEV rt .Noting that high PHEV rt values are not recommended.4. For the required energy demand, pick all possible BEV com and TF thr combinations that satisfy this demand.5.For the possible combinations in step 4, find the corresponding %BEV ss values (data saved from Fig. 6).Choose a setting for the BEV com and TF thr that provides a %BEV ss close to the required value.If different combinations satisfy this point, then choose any setting.
As the data in Fig. 6 was derived for a %BEV ss of 77% then the maximum value is 80%.Therefore, if the required %BEV ss is > 77%, step 5 in the previous procedure may be not achievable and it can be omitted.

Model output
The model has been executed using the values for different parameters e.g. the local fleet composition, distribution for plug-in time, and SoC distributions.For the competitive charging parameters, models will be unique if based on real operational data and specific local characteristics.The model has been tested at a monthly energy demand of 17 MWh, and a %BEV ss of 80% and 85%. 1 1 Monthly data obtained from Gridserve, priv.comm.For a %BEVss of 80% and following the proposed procedure to set the competitive charging parameters, three possible settings for the PHEVcom and PHEVrt and four different settings for the BEVcom and TFthr were found (Table 5).The model was run for the three possible settings for the PHEVcom and PHEVrt and the two extreme settings for the BEVcom and TFthr to show how changing the parameters affects the model output.For each setting group, the model has been executed for a month with 15 runs for each day.
Table 6 illustrates the average and standard deviation values for the energy and %BEV ss for the six test cases.Despite the data used to extract the parameters setting corresponds to %BEV ss of 77% (Fig. 5) and energy level of 15 MWh (Fig. 8), the model output for energy level of 17 MWh and %BEV ss of 80% are close to the desired values.
For a %BEV ss of 85% and demand of 17 MWh, the possible settings for BEV com and TF thr are given in Table 5.There are two possible combinations for (PHEV com , PHEV rt ) as (0, 0.25) and (0.1, 0.1).Setting the PHEV com and PHEV rt to 0.1 and running the code for the four possible BEV com and TF thr settings, the output is given in Table 7. Similar to the previous test, the obtained energy and %BEV ss are close to the desired values.It is worth noting that the maximum %BEV ss in Fig. 6 is 80%.Therefore, none of the BEV com and TF thr settings in Table 5 correspond to 85%.However, the %BEV ss obtained is close to the desired value ensuring that the BEV com and the TF thr do not have undue influence on the %BEV ss .
To illustrate the range of outputs, the EV arrival distribution for an iteration for one day is shown in Fig. 9 for the local and passing traffic populations.The corresponding charging power is shown in Fig. 10.The daily required charging energy over the whole month is given in Fig. 11.Finally, Figs. 12 and 13 show the distribution of the total monthly EV arrival from the two populations.The average arrival (15 iterations) at each time slot is shown together with the error bars representing the standard deviation.The monthly EV arrival matches the modelled plug-in time distribution and the traffic flow distribution for the local and passing traffic populations, respectively.

Conclusions and further work
The growth of EVs is slowed by a lack of easy to use and reliable charging infrastructure, especially rapid charging facilities with an experience similar to conventional refuelling stations.Such stations will likely be used mainly by BEVs rather than PHEVs because large capacity batteries require charging rates higher than can be accommodated by at-home on on-street facilities.
Designing and operating a large multi-charger station depends on an optimization process of parameters such as arrival rate and the technical specifications of different EV types.The parameters characterizing whether an EV needs to charge are shaped by the habits of drivers, thus in their nature are stochastic and not deterministic.A stochastic model for EV arrival has been developed to consider different uncertainties associated with daily travelled distance, vehicle battery SoC, start time of charging events, and vehicle parameters.In this way, the model can mimic observable power demand for a charging facility resulting from patterns in EV owner/driver behaviour and travel patterns.The model uniquely treats EVs as local and passing traffic populations, plus separating BEVs and PHEVs.The steps for estimating arrival rates from each population were discussed in detail, accounting for seasonality.The model incorporates competitive charging between the forecourt and other facilities.Sensitivity of the model to different parameters controlling the competitive charging was investigated, resulting in a procedure to better understand and set these parameters.The model addresses the gap in knowledge between a detailed physics-based power systems and power electronics model of the charging process, and a purely behavioural or economics view of EV fleet characteristics.
By using the UK's first dedicated large-scale multi-charger forecourt as a case study, the model was tested and validated with real-world data.The model was able to produce plausible EV arrival patterns for both of the population types, and reproduce the characteristic power demand at the site.All estimates incorporate uncertainty, reflecting the realistic variability of the parameters of interest.Together, these allow the site operator to plan daily activities, interact with the local grid operator, and to inform investment decisions.The model in independent of location, uses open-source data, and is structured flexibly, making it is relatively straightforward to adapt to any potential site as part of the technical and business planning process.
Future development will be directed to using real operational data for predicting daily demand, applying this model to other types of location e.g.city centre, and understanding how projected growth of EV use will affect the requirements for charging infrastructure and impact the electrical power distribution grid.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 4 .
Fig. 4. Initial and final state of charge distributions.

Table 1
Plug-in time distribution parameters.

Table 2
Average daily traffic flow by month, average = 100.Month Jan. Feb.Mar.Apr.May Jun.Jul.Aug.Sep.Oct.Nov.Dec.Table2).From the average daily vehicle flow, the expected EV flow is estimated using the EV market share.The licenced BEVs and PHEVs in the UK at the end of 2019 are 0.3% and 0.4%, respectively, of the total fleet (Department for Transport, 2020b).Therefore, the total EV share (0.7%) is used to scale the total vehicle flow to represent the passing EV flow.Not all vehicles on the roads are travelling long distances.The National Travel Survey dataset(Department for Transport, 2017)

Table 4
Parameters for the initial and final SoC distributions.