Examining the case for long-range battery electric vehicles with a generalized description of driving patterns

Transportation Research Part C 108 (2019) 1–11

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Examining the case for long-range battery electric vehicles with a generalized description of driving patterns

T

Michael A. Tamor Arizona State University, School for the Future of Innovation in Society, PO Box 875603, Tempe, AZ 85287-5603, United States

ARTICLE INFO

ABSTRACT

Keywords: Electric vehicle Electric range Charging Acceptance

Usage data from fleets of instrumented vehicles in several US cities have been used in numerous studies of the prospective costs, benefits and customer acceptance of battery-electric vehicles (BEV). In turn, the results of these studies can be used to design policies and strategies that promote electrification of personal transportation. Any broader application of the results of these regional studies carries the implicit assumption that vehicle usage in the affected population is similar to that of population selected for the underlying usage study. Given this projection of behavior from one population onto others, replacement of the raw usage data with a statistical representation of the heterogeneity of vehicle usage should be equally valid while reducing elaborate studies of large data sets to just a few equations and spreadsheet calculations. We demonstrate this analytic approach in a study of the trade-off between increasing cost and convenience of incremental all-electric range (AER), and the cost of fast chargers needed whenever range is insufficient for a given day of travel. Three scenarios are considered: (1) A single range was assigned to all vehicles regardless of individual usage, (2) a common tolerance for the frequency of charging away from home was ascribed to all users and the range computed accordingly, and (3) the range that minimized the combined cost of batteries and electricity was computed for each vehicle. All three methods suggest optimal fleet-average range of approximately 175 km. However, both the frequency of visits to fast-charging stations and the energy drawn from those chargers is reduced by allowing users to choose the range that best matches their needs. For example, allowing a choice of range that results in a fleet average range of 200 km in place of a common fixed range of 200 km, the frequency of visits to fast chargers is reduced by 25% and the energy drawn from those chargers is reduced by over 50%. These results suggest that the total cost of universal BEV deployment can be reduced by allowing users to choose the BEV range that matches their needs.

1. Introduction Most descriptions of a ‘low-carbon’ economy that meets the goals of the Paris Climate Agreement (COP21) call for electrification of nearly all light-duty road transportation (IEA, 2017; Williams et al., 2012; Chu and Majumdar, 2012). The pace and ultimate success of this transition will depend on minimizing the total cost, including infrastructure, of electrified transportation while meeting customer needs. In principle the low cost of renewable electricity relative to fossil fuel (lower still relative to substitute renewable fuels) should make battery-electric vehicles (BEV) economically attractive in general. However, for any plausible cost of storage batteries, the considerations of cost and utility are in conflict: greater range meets more customer needs for occasional long

E-mail address: [email protected]. https://doi.org/10.1016/j.trc.2019.09.003 Received 20 December 2018; Received in revised form 5 September 2019; Accepted 5 September 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.

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journeys but the cost of that infrequently-used range undermines the economics of BEV ownership. This trade-off can be assessed only in the context of long-baseline (at least months) vehicle usage information that captures both the day-to-day and vehicle-to-vehicle variations in travel behavior. Usage data at any higher level of aggregation, such as ensemble daily, weekly or annual travel distributions gathered from surveys impute a common travel ‘pattern’ to all prospective BEV users, which is certainly not the case. Anonymized data from fleets of instrumented vehicles in several cities are available in the National Renewable Energy Laboratory Transportation Secure Data Center (NREL) and are used in numerous studies of electrification (e.g. Pearre et al., 2011; Khan and Kockelman, 2012; Kontou et al., 2019). However, any generalization of the findings of those studies, such as a technology or policy recommendation, carries with it the assumption that the affected population is very similar to that in the underlying usage study. Given this inevitable projection of behavior from one population onto others – a universal log of the usage of all vehicles being unavailable – a statistically similar representation of vehicle usage should serve just as well and would be computationally much more convenient than repeatedly processing the same large data sets. Furthermore, if that representation is well-behaved mathematically, many analyses might be reduced to just a few equations. Tamor et al. (2015) proposed one such representation based on the population-wide distributions of the parameters that describe individual vehicle usage, specifically, the distribution of daily travel distances. In this paper we demonstrate the use of this representation in a simplified assessment of the trade-off between all electric range (AER), frequency of charging away from home and the corresponding costs of incremental range and public charging infrastructure. Most studies of acceptance and benefits of vehicle electrification are concerned with the early stages of deployment when strategic placement of charging infrastructure is essential to promoting the acceptance and use of BEV. The objective is geographic coverage adequate to serve a modest, but growing population of BEV. In this context, the consequences of insufficient range or inability to charge along the desired route have been represented by combinations of a cost penalties for lost time and alternative transportation on days when AER is insufficient (Kang and Recker, 2014; Barter et al., 2015), and an inconvenience threshold where the BEV is rejected entirely if range is insufficient too often (Khan and Kockelman, 2012; Pearre et al., 2011; Tamor et al., 2015). Nicholas et al. (2019) conducted a detailed study of charging needs to support a mixed fleet of 3.6 million long- and short-range battery-electric (BEV) and plug-in hybrid electric (PHEV) vehicles in 2025. The results suggest the need for roughly 10,000 DC fast-chargers (DCFC), but also reveal a high sensitivity to the mix of vehicle types; a modest increase in the fraction of long-range BEV resulted in a significant decrease in the number of fast-chargers needed. However, infrastructure investment decisions that we make today should consider the goal of virtually complete electrification of personal transportation by midcentury, only decades from today. To capture the scale of that infrastructure, we consider the end-state where all 240 million personal vehicles are all-electric, the territorial coverage of ultra-fast public charging is complete, and station capacity (the number of plugs per location) is scaled to energy needs. In other words, a case where the inconvenience of time lost for finding alternative transportation is replaced by the more familiar inconvenience of an occasional mid-day recharge hardly different from stopping for fuel today. This conversion to BEV as personal vehicles will require some combination of economic forces such as very high fuel prices and strong government policy that make BEV ownership favorable relative to conventional vehicles. Therefore, we consider cost of ownership only in terms of BEV range, i.e. the cost of batteries, and the cost of electricity at home and at fast-charging facilities. Implicit in this analysis is the assumption that usage of this future fleet of electric vehicles is the same as that of conventional vehicles today. As suitable data sets or plausible models for daily usage become available, other scenarios such as a general shortening of trips due to urban densification, shifting to other modes, household deployment of a very short range ‘commuter’ BEV, ride-sharing or autonomous vehicles could be analyzed with the same methodology. 2. Statistical description of travel For simplicity, this treatment assumes that every vehicle starts its travel day from home with a full charge regardless of AER, and the probability that vehicle i will travel a distance x on a given day is

fi (x ) =

i

×

wi e ki

x / ki

+ (1

wi ) (ui

x)

(1)

where λi is the probability that the vehicle is driven at all and δ(ui − x) is the delta function with value 1 where x = ui, and zero when x ≠ ui. For convenience, this one-day travel distance is referred to as a ‘trip’ even though it is likely to consist of multiple shorter legs with stops between. The first term in Eq. (1) represents a broad ‘random’ distribution of short and long daily trips selected with probability wi and the second term is a repeated ‘habitual’ daily trip of distance ui selected with probability 1-wi. This is a simplification of the distribution proposed by Tamor et al. (2015) that included a Gaussian distribution of habitual trip distances. That work showed that the five (now four with the elimination of the width of the Gaussian) parameters describing each vehicle are not correlated, the distributions of the key parameters ki and ui are themselves well represented by the log-logistic distribution,

f (k i ) =

ki Zk

1

1+

ki Zk

2

and g (ui ) =

ui Zu

1

1+

ui Zu

2

,

(2)

and a single value can be assumed for λ and w. The effectiveness of this statistical distribution as a substitute for large longitudinal data sets has been confirmed in multiple studies. Tamor et al. (2015) showed that Eq. (1) provided a good representation of usage for at least 95% of the vehicles in each of four studies: Mileage-Based User Fee Demonstration Project (Minnesota, 2006), 133 vehicles in greater Minneapolis-St. Paul; Puget 2

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Table 1 Fit parameters for Eq. (2) as applied to the observed distributions of daily travel distance in studies of three US cities, Germany (Tamor et al., 2015) and Beijing China (He et al., 2016). The Typical US values are simply the average of the parameters for the three US cities and are not populationweighted averages for the nation. Survey values of annual VKT for the US are from the National Household Travel Survey while those for Germany and China are from the International Road Federation World Road Statistics (IRF, 2009). ki (km)

Puget Sound Minneapolis-St. Paul Atlanta Germany Beijing Typical US

µi (km)

aVKT

Zk

α

Zu

β

〈λ〉

〈w〉

Model

Survey

47 78 59 54 33 60

3.0 4.2 3.4 3.1 2.0 3.5

36 62 53 23 38 50

2.9 3.5 2.4 1.8 2.2 3.0

0.80 0.66 0.89 0.69 0.80 0.80

0.62 0.62 0.61 0.65 0.44 0.63

15,287 19,448 22,687 14,115 15,466 19,323

15,918 22,279 19,777 12,416 14,125 18,200

Sound Regional Council Traffic Choices Study (PSRC, 2008), 446 vehicles in the greater Seattle area; Commute Atlanta Value Pricing Program (Guensler et al., 2002; Ogle et al., 2005), 651 vehicles in greater Atlanta, and the Europe Field Operations Test (euroFOT, 2012), 100 midsized Ford vehicles in several German cities. Using the frequency of ‘inconvenience’ days (those requiring mid-day charging or use of alternative transportation) as the criterion for BEV acceptance as suggested by Pearre et al. (2011), BEV acceptance for a range of acceptance thresholds (1, 3, 9 and 27 days per year)was computed for all four studies by both brute force trip counting from the raw data and using the statistical distribution (Eq. (2)) with virtually indistinguishable results. These results are also essentially identical to those of Pearre et al. (2011) for Atlanta and of Khan and Kockelman (2012) for Puget Sound (though each chose slightly different values for the acceptance thresholds). The electrification potential of substituting one BEV in multi-vehicle households (MVHH) in Puget Sound was estimated assuming that the BEV was used for the longer daily trip whenever possible and inconvenience was incurred only when both vehicles exceed the AER on a given day (Tamor and Milacic, 2015). Again, the results were very similar to those of Khan and Kockelman (2012), with minor differences stemming from their choice to always substitute the BEV for the conventional vehicle that travels less on average or less on any given day. More recently, He et al. (2016) found similar effectiveness for characterizing usage of 459 private vehicles in Beijing. The resulting utility factor (UF, the fraction of ensemble travel that can be electrified with BEV of a given range without mid-day charging) is virtually identical to that found by Wang et al. (2015) in their study of 112 private vehicles in Beijing. Table 1 shows the values of the distribution parameters for three US cities and Germany (Tamor et al., 2015) and Beijing, China (He et al., 2016). The Typical US values are simply the average of those for the US cities and should not be interpreted as a population-weighted average for the nation. Table 1 also compares the annual vehicle-kilometers traveled (VKT) computed from Eq. (5a) (below) for each region to regional or national surveys. The resulting distributions of ki and ui are shown in Fig. 1. For readability the all-electric range (AER) will be represented as R in all equations. The total distance, di, traveled by vehicle i in one year is obtained by integration of Eq. (1) over the domain of daily trip distance, xi, using the factors in Table 1:

di = 365 ×

i

( (1

wi ) ui + wi

xe

0

x ki dx

)

(3)

which simplifies to (4)

di = 365(0.3ui + 0.5ki ). The average annual distance, D for the ensemble of vehicles is found by summing over all vehicles,

(

D = 365 0.3

0

ug (u) du + 0.5

0

)

kf (k ) dk ,

(5a)

Fig. 1. The log-logistic distribution of daily distance distribution parameters generated by inserting Typical US parameters from Table 1 into Eq. (2). 3

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D = 365 0.3Zu

/sin

+ 0.5Zk

/sin

,

(5b)

where the integrals in Eq. (5a) are easily evaluated as the first moment of the log-logistic function (Eq. (5b)). [Note that the subscript i referring to individual vehicles has been dropped in integrals over the ensemble distribution of vehicles.] For the Typical US parameters in Table 1, D = 19,300 km (11,900 miles), nicely in line with the 2009 US National Household Travel Survey (11,284 miles for all personal vehicles). The 34% of ‘habitual’ travel distance inferred from Eq. (5b) is consistent with the NHTS finding that 29% of vehicle travel is ‘to or from work’ (NHTS, 2017). The distributions of ki and ui play very different roles. The value of ui is an actual travel distance, making it trivial to determine whether a given habitual trip is within the AER of the vehicle. For the large ranges in this study, habitual travel is not an issue; only 11% of this synthetic population have habitual daily travel distance more than 100 km and only 1.5% have habitual distance over 200 km. While it is possible that some buyers will choose a very short range that does require at-work charging, such a vehicle would have very limited use while the present trend is toward longer ranges. This suggests that work-place charging will have limited impact on the utility of BEV as commuting vehicles, and we need only focus on the frequency of longer daily travel distances captured in the ‘random’ distribution. All the metrics of BEV usage – fraction of travel electrified, frequency and amount of charging, etc. - can be evaluated by simple integration of the domain of k from 0 to infinity. A possible criticism of the daily distance distribution in Eq. (1) is the large number of parameters. Several alternative statistical distributions have been considered in the literature. With a stated desire for computational simplicity, the options have been limited to right-skewed, non-negative two-parameter distributions: log-normal, Gamma, and Weibull (Lin et al., 2012; Plötz et al., 2017). [Note here that the exponential distribution is a special case of the Weibull distribution.] While none of these has been proven clearly superior to the others, they fall short of the fidelity and utility of the distribution of Eq. (1) in several respects. (1) None can simultaneously exhibit a finite, decreasing probability of short-distance travel days and a peak at a non-zero distance, whereas as nearly every individual daily distance distribution exhibits this characteristic. In other words, Eq. (1) more closely resembles the actual data. (2) By using the ensemble averages of λ and w, Eq. (1) is also reduced to two parameters that describe each individual distance distribution. (3) By identifying and separating a repeated daily trip distance (the ‘habitual’ trip) the fit to the long tail of occasional long trips is improved. Also, with the assumption that at least one destination is common to all habitual travel days for a given vehicle, the benefits of ‘at-work’ charging can be evaluated separately from that of public charging. (4) The distributions of the parameters describing the individual distance distributions (the ‘metadistributions’, Eq. (2)) have physical meaning that can be compared to survey data. Fig. 2 shows the ensemble cumulative fraction of travel as function of daily travel distance evaluated using Eqs. (6) and (9) below. The figure shows that the characteristic log-normal-like shape is not reflective of a single underlying daily distance distribution but is the sum of two distributions with quite different shapes. The electrification of habitual travel can be nearly complete (94% electrified without daytime charging) with AER of 200 km simply because longer repeated daily travel distances are so rare. In contrast, electrification of the same fraction of ‘random’ travel requires 500 km range if daytime charging is not available. Finally, although several researchers have parameterized individual daily-distance distributions, this work is the first to take advantage of the distribution of those parameters to create an entirely analytic representation of personal vehicle usage. Note that the simplified version of Eq. (1) does not associate values of ui and ki for each vehicle. For simplicity here, it is assumed that each vehicle travels the ensemble average of 6502 km in habitual distance each year using home-charging only, and only ‘random’ travel days are considered in the assessment of charging needs away from home. This is correct for the ensemble when charging for habitual trips is not needed (or is simply ignored) and is also correct for energy used in random trips. However, it does conceal the variation in total energy needs among individuals. Two vehicles driven the same annual distance may have very different balances of random and habitual driving distances and therefore very different need for charging away from home. Analyses that must reflect the paired values of ki and ui entail convolution integrals. For example, to evaluate a metric as a function of the total distance traveled by each vehicle the convolution would be over f(k) * g(k′) where k′ = (d − 0.5 k)/0.3 from Eq. (4). For some purposes, it may be necessary to retain the individual values of all parameters. This makes the integrals considerably more complex but does not negate the advantages of the analytic approach.

Fig. 2. The cumulative fraction of travel for the entire population as a function of daily travel distance (upper curve). This is equivalent to the fraction of travel that would be electrified if all daily travel with total distance less than the AER was accomplished in a BEV and all longer trips in a conventional vehicle. The middle curve is the cumulative fraction of random travel only and the lower curve is that for habitual travel only. 4

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3. Technology assumptions 3.1. Vehicles Vehicle assumptions in this analysis are based on the recent Argonne National Laboratory (ANL) benchmark study of the ownership cost and GHG reduction potential of BEV and PHEV (Elgowainy et al., 2016). The entire fleet of light-duty vehicles is represented by a single mid-size sedan with an estimated ‘real-world’ energy consumption of 0.19 kWh/km (0.31 kWh/mile) independent of AER. (With this simplification, energy and travel distance can be treated interchangeably.) As in the ANL report, incremental battery capacity for range above a 50 km minimum is assumed to cost $125/kWh in the near term (2025) with the potential to achieve $85/kWh in the long term (2045). The assumption of a single all-electric range for all vehicles – analogous to the ‘standard’ 500–700 km range of gasoline vehicles - serves the purpose of demonstrating the usefulness of an analytic representation of travel behavior but is unrealistic in several respects. First, one vehicle type cannot represent the energy consumption of a real fleet of comprising a variety of types, but so long as the range of all vehicles is the same, it does properly reflect the frequency of visits to public charging facilities. Similarly, BEV energy consumption will increase somewhat with range due to increased battery weight, but the frequency of charging is unaffected. Second, customers in very hot or very cold climates might select a larger label range in order to assure sufficient range under extreme conditions. This could drive oversizing of the battery by as much as 30% (e.g. purchasing a vehicle with 285 km range in good conditions in order to assure 200 km range in extreme conditions) (Yuksel and Michalek, 2015). Third, it is possible that a ‘standard’ range will not emerge, and customers will be able to select the AER that best suits their individual needs. This possibility is explored through two scenarios: (1) where a ‘universal tolerance’ for visiting fast-charging facilities is imputed to all users and the range selected to exactly meet that tolerance, and (2) where the electric range is selected to minimize the combined cost of incremental battery capacity and the incremental cost of electricity from DCFC. 3.2. Charging Each vehicle is assumed to be fully charged at home before each travel day. The costs of construction and operation of DCFC locations are based on the ‘Ultimate/Corridor’ scenario described in the Idaho National Laboratories report on DC fast charging system design (Francfort et al., 2017). The $1.7 M capital cost for a 1.06 MW unit servicing three charging plugs translates to $660 k per plug. This is somewhat higher than the $270 k for each 125 kW Tesla charger estimated by Keeney (2016) and is likely to decline with experience and scale. In the ‘Ultimate/Corridor’ scenario, each plug will be used for an average of 14.8 charges per day with an effective electricity cost of 77¢ per kWh delivered, nearly ten times the cost of home charging. If compared to an HEV rated at 72 km/ gallon (45 mpg), this is equivalent to purchasing fuel for $3.50 per gallon. While certainly more expensive than home charging or petroleum fuel today, this is in-line with the estimated cost of renewable transportation fuel and so is unlikely to be a deterrent to BEV adoption (Connolly et al., 2014; Grahn, et al., 2014). The usage rate defined in the INL report is in good agreement with a simple estimate based on conventional fuel stations. At present, there are roughly 110,000 retail gasoline stations in the US (US Census, 2012), serving 240 million vehicles (Davis et al., 2018). Assuming an average of 16 ‘pumps’ at each station (based on personal observation, a reliable statistic being unavailable) and that each conventional vehicle is refueled every 500 km, this equates to an average of 13 visits to each pump each day. For convenience, the above assumptions and their sources are summarized in Table 2. 4. Example applications The degree to which the travel of a given vehicle can be electrified is assessed by considering three subsets of travel: (1) the distance, d1, accumulated on days of travel that can be completed without additional charging (x < R), (2) the distance, d2, accumulated on days with trip distance x > R up to the point where the vehicle must be re-charged in order to complete that day’s travel, (3) the distance, d3 accumulated when completing those longer distances (i.e. covering the remaining x − R) using energy from public charging. For a single vehicle i, the annual distance accumulated on days with d < R is: Table 2 BEV vehicle and charger cost assumptions and sources. Note that the estimated number of plugs per location impacts the number of locations, but not the total number and cost of individual DCFC plugs. Parameter

Value

Source

BEV Energy Consumption Battery Cost (2025) Battery Cost (2045) DCFC Capital Cost DCFC Electricity Cost Home Electricity Cost DCFC Plug Usage DCFC Plugs per Location

0.19 kWh/km $125/kWh $85/kWh $660 k/plug $0.77/kWh $0.10/kWh 14.8 charges/day 16

Elgowainy et al. (2016) “ “ Francfort et al. (2017) “ EIA (2018) Francfort et al. (2017) Similar to Retail Gasoline

5

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Fig. 3. The fraction of travel distance associated with travel days that are completed within the AER (D1, top curve), distance accumulated on travel days with distance greater the AER (D2, middle curve) and distance covered using energy drawn from public facilities on travel days with distance exceeding the AER (D3, bottom curve). It is assumed that every trip begins at home with a full charge.

d1 (ki ) = 365

1 ki

i wi

R 0

x ki dx

xe

= 182 ki

(R + k i ) e

R ki

.

(6)

Similarly, the annual travel distance accumulated on travel days with distance exceeding R up to the point where the battery is recharged after driving distance R is

d2 (ki ) = 365

1 ki

i wi

Re

R

x ki dx

= 182Re

R ki

(7)

and the annual distance accumulated in the remainder of those long trips after on-roach charging is

d3 (ki ) = 365

1 ki

i wi

ki e

R

x ki dx

= 182ki e

R ki .

(8)

The sum of these three becomes Eq. (1) as R goes to infinity. The ensemble average for each of travel subset is given by integrating dj(k) weighted by the f(k), the frequency distribution for values of k (Eq. (2)):

Dj =

0

f (k ) dj (k ) dk.

(9)

Fig. 3 shows the fraction of each of the three travel subsets as a function of AER. As expected, an extremely long-range BEV will need supplementary charging only very rarely. Conversely, attempting to fulfill all travel needs with a BEV150 similar to many on the road today would entail filling roughly 10% of energy needs away from home. From the customer point of view, the amount of energy drawn from public charging may not be as significant as the frequency of the need for a mid-day charge. The number of days per year that a given vehicle, i, with range R must visit a public charger at least once, ni, is easily computed as

ni = 365

i wi

e

R / ki

= 182e

(10)

R / ki .

When multiple visits on the same day are included, the number of charges over the course of a year is significantly larger:

ni = 182e

R / ki /

(1

R / ki

e

).

(11)

The ensemble average number of visits to chargers away from home is each year, N, is then

N=

0

f (k ) n (k ) dk .

(12)

For example, if the entire population were to attempt to use a BEV100 for all purposes, the ensemble would require charging away from home at least once on an average of 39 days per year. Because many vehicles will charge more than once on some days, this equates to an ensemble average of 59 charges away from home per year. The fraction of the population charging on fewer than n days each year, p(n), is estimated by first computing k(n), the value k at which at which ni = n,

k (n ) =

R/ln

n 365

i wi

,

(13)

and then integrating Eq. (2) (the cumulative of the log-logistic distribution) to compute p(n) as the probability of ki being less than k (n),

p (n ) = 1

1/ 1 +

k (n ) Zk

k

,

(14) 6

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Fig. 4. The fraction of total travel energy drawn from public chargers (upper curve) and the ensemble average probability of needing at least one charge on a ‘random’ driving day (lower curve) as a function of AER on a logarithmic scale. At very short range the probability of needing multiple charges in one day is significant.

By this calculation, 96% of a population equipped with BEV 100 would need public charging three or more times per year. In contrast, if equipped with BEV300, only 35% of the population would need public charging three or more times per year, and the average number of visits to chargers would fall to five visits per year with nearly zero days requiring multiple charges. While the impact of the need for public charging on BEV acceptance may be hard to quantify, its impact on the extent of public charging infrastructure can be estimated. Fig. 4 shows that the fraction of total travel energy that must be drawn from public charging facilities declines roughly exponentially with increasing AER. The frequency of visiting public chargers shows a similar but slightly steeper decline. [Although the shape might be expected consistent with the exponential tail in the individual daily travel distance distribution (Eq. (1)), the scale and shape of the of that near-exponential is a product of the distribution f(k) (Eq. (2)) and so is not at all obvious.] The steep decline suggests that a modest increase in range will translate to a significant reduction in the required public charging infrastructure. The number of charging facilities can be estimated by dividing the ensemble frequency of daily visits to charging stations (240 million times the one-day charging probability from Eq. (12) or Fig. 4) by the daily capacity of each station (16 plugs each serving 14.8 vehicles per day, on average). Although subject to very large error – strategic placement could reduce the number of stations, but longer charging times would force an increase – this suggests that only 13,000 fast charging locations could serve 240 million BEV300s as conveniently as conventional vehicles are served today. This is comparable to the 8000 locations necessary to place a charger within three miles of any point in the lower 48 US states (Wood et al., 2017). For comparison to studies that consider a more modest population of BEV, Fig. 5 shows the number of plugs needed to serve the energy needs of 1000 BEV as a function of electric range. Although based on different assumptions, these results are in good agreement with previous studies based on aggregated usage data. Rajagopalan et al. (2013) and Wood et al. (2017), both estimate 5 fast chargers for every 1000 BEV when conventional vehicles are simply replaced by BEV150, while the latter estimates 0.5 plugs are needed for each 1000 vehicles in a fleet of BEV400. It is important to note that although the requisite number of charging stations can be estimated from aggregated data, the frequency with which any one vehicle will visit a station cannot. As the typical BEV range gets longer, chargers are used by a shrinking fraction of the population. If all light-duty vehicles were BEV400, 80% of the energy from public charging would be drawn by only 7% of the vehicles and 70% of all vehicles would visit a charging station less than once per year. The rapid decrease in the number of fast chargers in response to a linear increase in AER suggests a crossover point where it is no longer cheaper to add range than to add chargers. Because the projected operating lives of vehicles (50% survival after 15 years, Davis et al., 2018, Table 3.12) and the assumed financial payback period for chargers (10 or 15 years, Francfort et al., 2017) are similar, their costs can be compared directly without annualization. The solid curves in Fig. 6 show the total cost per vehicle in

Fig. 5. The number of plugs required to serve each 1000 BEV as a function of their range. The symbols drawn are from Rajagopalan et al. (2013). 7

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Fig. 6. The total cost of incremental range above the very low value of 50 km reflecting the trade-off between the reduced number of chargers and the increased cost of batteries. The solid line is based on near-term cost projections (Elgowainy et al., 2016; Francfort et al., 2017). The dashed curve indicates the effect of future cost reductions combined with a 150 km minimum range.

batteries for incremental range at $125/kWh (from Elgowainy et al., 2016), and for fast chargers at $660 k per plug (Francfort et al., 2017), assuming full usage of those chargers (14.8 visits per plug per day). The range at which the total cost is minimized is 175 km, above which the cost of additional batteries is greater than the savings from fewer chargers and reduced use of DCFC. Although, these costs are subject to very large errors as we project future technologies manufactured in unprecedented volumes, the steep decline in the number of chargers with electric range makes the result quite robust. In a hypothetical future scenario where the ‘baseline’ BEV has 150 km range, battery costs have declined to $85/kWh and the cost of DCFC equipment has declined by two-thirds, the range at which total cost is minimized is shifted to a slightly lower value. Because the cost of electricity from DCFC is dominated by depreciation of the electric equipment itself (two-thirds of the total station cost) the same result is found if the range-dependent mixture of expensive DCFC charging and cheaper home charging energy were used instead of the plug cost. Up to this point, our methodology has assumed a single AER for all vehicles and tested for the variation in usage of DCFC as a function of that range. However, the rather modest range that minimizes the combined investment in chargers and batteries would be unsatisfactory (in terms of inconvenience) and costly (in terms of the expensive energy from DCFC) to a significant subset of users whose travel needs require frequent use of DCFC. This calls into question the assumption that a long, ‘standard’ range for BEV will emerge as it has for conventional vehicles. We can use the analytic representation to examine the opposite extreme where users are able to select the exact range that best suits their needs. To do so, we must choose a metric of ‘suitability.’ As in earlier analyses of BEV acceptance, we can define a universal threshold for the frequency of the inconvenience of mid-day visits to fast-charging facilities. Because the inconvenience of stopping to charge is far less than that of finding alternative transportation, we expect that tolerance will be much higher, perhaps approaching the frequency with which conventional vehicles must be refueled (roughly 36 times per year assuming 500 km range and 18,000 km of driving). For an individual vehicle i, the range, Ri that results in n DCFC visits per year is given by

Ri (n) =

ki × ln

n 365

(15)

i wi

and the distribution of AER is obtained by simply scaling the distribution f(k) (Fig. 1). To ensure that the results are not distorted by implausibly long ranges for a few vehicles, the maximum allowed range is limited to 600 km. Fig. 7a shows the most ‘popular’ range (the peak of the distribution), the average range and the range that would be suitable to the 90th percentile of the user population as a function of the number of DCFC visits per year. Fig. 7b shows the fraction of energy drawn from fast-chargers and the (predetermined) one-day charging probability as a function of the average electric range. For comparison to the fixed-range case, the dashed curves are the same as in Fig. 4. At the low end of range (highest tolerance for charging, 27 days per year) the results are nearly identical. However, in the ‘custom-range’ case, the decrease in the need for charging with increasing average range is much more pronounced than in the ‘one-size-fits-all’ scenario where all vehicles have the same range. For an average range of 300 km, the frequency of charging at DCFC is reduced by nearly half and the energy from DCFC charging is reduced by 80%. This shows that redistributing the total investment in batteries to those vehicles that make most use of the added range can significantly reduce the requisite investment in fast-charging infrastructure. While the emergence of a single, universally accepted range or tolerance for inconvenience is uncertain at this early stage, it is reasonable to assume the future BEV users will prefer a range that minimizes their cost of ownership. In this simplified case we consider only the trade-off between the increasing fraction of more expensive DCFC charging as range is reduced and the cost of batteries as range is increased. From Eq. (8), the combined cost of incremental range above a minimum range, Rmin, and electricity for 15 years of driving by vehicle i is

Ci = 15 × 365

i wi

ki

f

e

R ki

h

1

e

R ki

+ (R

Rmin)

(16)

where ε is the energy consumption of the vehicle (0.19 kWh/km), βf is the cost of energy from DCFC ($0.77/kWh), βh is the cost of energy from the home charger ($0.10/kWh) and γ is the incremental cost of batteries ($135/kWh). Setting dCi/dRi = 0 and solving 8

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Fig. 7. (a) All-electric range that results in a given number of visits to fast chargers each year for each vehicle as a function of that number of visits. Lower curve: the peak of the distribution, thus the most popular range choice; Middle curve: the fleet-wide average range that measures the ensemble investment in batteries; Upper curve: the range that would assure as many or fewer visits for 90% of the user population. (b) The fraction of charging energy drawn from fast-chargers (upper solid curve) and the probability of visiting a fast charger on any given travel day (lower curve) as a function of the fleet-wide average range. For comparison, the dashed curves are drawn from the previous scenario where all vehicles are assigned the same range.

for Ri results in

Ri = ki × ln

15 × 365

i wi

(

f

h)

.

(17)

Just as for the case of a fixed tolerance for inconvenience, the distribution of cost-minimizing range is simply scaled from f(k). For the Typical US parameters, the average range is 178 km, increasing to 197 km when the battery cost is reduced to $85/kWh. In short, individual cost minimization produces essentially the same result as a universal tolerance for visiting DCFC roughly 10 times per year. This result is generally similar to that found in a range-cost trade-off analysis using a much richer cost model combined with a much less representative usage model than those used here (Lin, 2014). Although the 140 km average range found in that work is somewhat shorter than that found here (170 km), the robustness of that estimate with variation of battery or charging cost is quite similar. 5. Discussion and conclusions It is important to note that the purpose of this work is to demonstrate the utility of an analytic representation of the heterogeneity of personal vehicle usage. The parameters describing that vehicle-to-vehicle and day-to-day variation were chosen to be typical of the US and are not ‘tuned’ to any specific city or region. However, the steep decline in need for charging away from home implies a high sensitivity to the value of Zk. Although the few available values for US cities fall in a quite narrow range (roughly 50–80 km) this modest variance has a disproportionate impact on the requisite number of fast-charging stations. From Fig. 4, a 10 km decrease (increase) in Zk will roughly halve (double) the number of chargers. Thus, at a given AER for all vehicles, Minneapolis-St. Paul would need roughly four times as many chargers per vehicle than the Puget Sound Region – a variance so large as to influence policy discussions even at this early stage. Table 1 shows good correspondence between the computed annual VKT and that reported in the NHTS. This indicates that the distribution parameters, most importantly Zk and Zμ, can be adjusted to match regional vehicle usage where detailed longitudinal data is unavailable. Other results such as an ‘optimal’ range respond linearly to this adjustment and so are much less sensitive to regional variations. Although, the representations of cost of ownership and other considerations in opting for a BEV are highly simplified the findings of this study do indicate some broad policy considerations as well as directions for more detailed or region-specific analysis. In the scenario where all BEV are assigned the same range, the results of this simple analysis of the role fast charging are in good agreement with those of previous, much more laborious studies. The important difference is the extreme ease with which the results were obtained. The balance between electric range and the capacity of the fast-charging network suggests an optimal range of only 170 km, 9

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above which the cost of fleet-wide deployment of additional range is greater than the cost saved by reducing the number of chargers. This range is greater than all but 3% of habitual trips, suggesting that installation of large numbers of slower workplace and public chargers will be of limited value. However, assigning 170 km range to all vehicles results in the need for approximately 90,000 fastcharging locations (of 16 plugs each), nearly the number of filling stations today, and imposes a great deal of inconvenience on the subset of users that drive the most (18% of vehicles would visit fast-chargers more than 27 times per year). Increasing the ‘standard’ BEV range to 300 km reduces the number of stations ten-fold with only 3% of vehicles needing such frequent fast charges but would result in higher total cost born by users who seldomly drive that far in one day. The analytic usage model was easily modified to test the case where electric range is matched to the usage of individual vehicles. The results of assigning the range that satisfies a common tolerance for the inconvenience (measured in DCFC visits per year) or the range that minimizes individual ownership cost are mathematically the same. The suggested fleet-average electric range is again approximately 180 km, yet by allowing deployment of greater range to the vehicles that need it most the investment in charging infrastructure is reduced without increasing the total investment in batteries. Regardless of regional variations, this shows that offering a choice of BEV range (and helping prospective buyers to choose appropriately) can be an important contributor to reducing the total cost of electrifying of personal transportation. While there are many reasons why a buyer would opt for greater range than indicated here - the comfort of reserve range and the anticipation of occasional long drives – there is little incentive to choose less range as inconvenience rises rapidly and habitual (commuting) travel needs might be compromised. With only a modest increase in average range to 240 km, well within the capabilities of several BEV on the market today - the requisite number of fast-charging locations falls in an attractive range. It is large enough (roughly 9000 with 16 plugs each) that geographic coverage can be as complete as that of conventional fuel retailers while size of facilities can increase continuously as the BEV population grows. With over a thousand BEVs sharing each ‘plug’, the cost of ultra-fast charging infrastructure should not be so large as to impede mass deployment of BEV. This result is consistent with the strategies of several BEV manufacturers that are building private networks of roughly 400 fast-charging stations along major transportation corridors to achieve national coverage (Wood et al., 2017). Although not the subject of this work, it is important to note that this modest fast-charging investment is possible only when all BEV have access to a ‘home’ charger. Estimates of the fraction of personal vehicles that do not have a dedicated home parking place that might be equipped with a Level-2 charger vary widely from 16% (UCS, 2013) to 44% (Traut et al., 2013). Again, using the analogy with gasoline stations, leaving only 10% of vehicles reliant on public charging would require tens of thousands of additional facilities. This high sensitivity to the population of ‘stranded’ BEV suggests that provision of an overnight charging location for every light-duty vehicle must be a long-term policy goal. It is hoped that this demonstration of a realistic, but tractable model of personal vehicle usage will encourage other researchers to use this computationally-efficient approach in more sophisticated studies of these issues. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.trc.2019.09.003. References Barter, G.E., Tamor, M.A., Manley, D.K., West, T.H., 2015. Implications of modeling range and infrastructure barriers to adoption of battery electric vehicles. Transp. Res. Rec. J. Transp. Res. Board, No. 2502 80–88. https://doi.org/10.3141/2502-10. Chu, S., Majumdar, A., 2012. Opportunities and challenges for a sustainable energy future. Nature 488 (7411), 294–303. Connolly, D., Mathiesen, B.V., Ridjan, L., 2014. A comparison between renewable transport fuels that can supplement or replace biofuels in a 100% renewable energy system. Energy 73, 110–125. 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