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Optimal battery electric vehicles range: A study considering heterogeneous travel patterns, charging behaviors, and access to charging infrastructure
Journal Pre-proof Optimal battery electric vehicles range: A study considering heterogeneous travel patterns, charging behaviors, and access to charging infrastructure Yue Zhou, Ruoxi Wen, Hewu Wang, Hua Cai PII:
S0360-5442(20)30052-9
DOI:
https://doi.org/10.1016/j.energy.2020.116945
Reference:
EGY 116945
To appear in:
Energy
Received Date: 2 August 2019 Revised Date:
27 November 2019
Accepted Date: 9 January 2020
Please cite this article as: Zhou Y, Wen R, Wang H, Cai H, Optimal battery electric vehicles range: A study considering heterogeneous travel patterns, charging behaviors, and access to charging infrastructure, Energy (2020), doi: https://doi.org/10.1016/j.energy.2020.116945. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Optimal Battery Electric Vehicles Range: A Study Considering Heterogeneous Travel Patterns, Charging Behaviors, and Access to Charging Infrastructure Yue Zhoua, Ruoxi Wena, Hewu Wangb, and Hua Caia,c* a. School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, United States b. State Key laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, 100084, China c. Environmental and Ecological Engineering, Purdue University, West Lafayette, IN 47907, United States
∗Corresponding author E-mail: [email protected] (H. Cai).
Tel: 765-494-7701
Address: 315 N. Grant Street, West Lafayette, IN 47907, United States
Abstract The choice of battery range (all-electric driving range) for battery electric vehicles (BEVs) is an important issue for both BEV adopters and BEV makers. This paper proposes a model to identify the minimum BEV battery range that can satisfy given travel demands, considering the opportunities to charge at existing public charging stations and the uncertainties in charging decision making. We conducted a Stated Preference survey to study the charging decision making and analyzed the data using the Latent Class model to generate the model coefficients for charging decisions making. The proposed approach can better identify the needed battery range than the often-used simplified charging rules. We applied the model to a case study of Beijing to evaluate the needed battery range for taxis and private vehicles. For taxis, BEVs with 220-miles battery range are able to satisfy the travel demands for about 90% of the drivers. For private vehicles, a 300-miles range is needed to cover the travel demands of 90% of the drivers, while a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Simplified charging rules tend to underestimate the range needs for taxis but overestimate the range needs for private vehicles.
Keywords: Battery Range; Electric Vehicles; Mixed Integer Nonlinear Programming (MINLP); Charging Behaviors; Charging Infrastructure 1
1
Introduction
Growing concerns over the environmental impacts of fossil fuels have led to the development of various alternative vehicle and fuel technologies. Battery electric vehicles (BEVs) offer great opportunities to reduce the transportation greenhouse gas (GHG) emissions when they are powered by clean electricity (e.g., generated from solar or wind) [1]. Considering the potential environmental benefits, the government and industries are making great efforts to support the BEV adoption through policy incentives and the development of charging infrastructure and battery technology. However, only 4% of the potential buyers have purchased BEVs or plug-in hybrid electric vehicles (PHEVs) in the U.S. [2] There are mainly three concerns that prevent the potential consumers to adopt BEVs. The first one is the limited all-electric driving range, which refers to the total distance the BEV can travel with a fully charged battery (in this study, we refer to the all-electric driving range as battery range). Drivers are concerned about finding public charging stations when they need to charge, because charging stations are not as ubiquitous as the gas stations. Second, long charging time and the associated inconvenience is another concern. Unlike refilling the traditional internal combustion engine vehicles (ICEVs) which can be done within a few minutes, fully recharging a BEV may take hours, depending on the battery range and the charging power [3]. The third concern is BEVs’ relatively high price. BEVs are more expensive compared to their ICEV counterparts. While BEVs with larger batteries (longer range) can help alleviate range anxiety, they also cost even more than BEVs with smaller battery ranges. Therefore, for potential BEV adopters, knowing the minimum required battery ranges that can satisfy their travel demands will help address these concerns.
Given the vast differences in battery ranges among the available BEV models, choosing a BEV model bounds the driver to the BEV model’s battery range. BEVs with smaller batteries will require more frequent recharging and have the risks of not being able to complete long trips. Charging in the middle of a trip will cause significant inconvenience due to the long charging time, let along the potential risk of not being able to find charging stations when needed. On the other hand, however, BEVs with larger batteries require higher costs and also reduce the BEV’s fuel economy due to the weight of the battery. To determine the optimal (minimum) required battery range, the following key factors need to be considered: 1) the users’ travel demands (e.g., the trip distance and locations of the trip origins and destinations), 2) charging infrastructures 2
(e.g., the locations of home, work, and public charging stations), and 3) individual charging behaviors (e.g., under which conditions the drivers will charge the BEVs).
Most of the existing literature that studies the optimal BEV battery range has two major simplifications: 1) charging will only happen at home and/or work place and 2) everyone will follow the same simple rule to make charging decisions. The first simplification ignores the opportunities to recharge at public charging stations. Pearre et al. determined the minimum battery range by simplifying the problem with the assumption that the BEV drivers would fully charge the vehicles at home every night [4]. Weiss et al. also assumed that drivers could only charge at home to analyze the battery range needed to satisfy the travel demands of 1.3 million private vehicles in the Greater Stuttgart Area in Germany with an agent-based simulation model [5]. Stark et al. further considered work place charging by assuming that drivers recharged every time when they reach home and work locations [6]. Smith et al. found that the minimum battery range needed for the BEVs can be reduced by 40% under the scenarios that people recharge both at work place and at home, compared to home-charge only scenarios [7].
The earlier studies only considered home or work place charging because the public charging infrastructure has not been developed at that time [4, 7, 8]. However, the number of public charging stations have been increasing rapidly in recent years [9]. Public charging is also becoming more convenient with more fast chargers introduced into the market [10]. These changes enable the BEV drivers to recharge the vehicles during the time when they are shopping, having dinner, waiting for the next passenger (for taxis) etc. Ignoring these public recharging opportunities in the model may overestimate the battery range needed to meet the daily travel demands. According to Neubauer et al., the availability of the public charging infrastructure strongly affects the range anxiety of potential BEV adopters [11]. Overestimating the required battery range would mislead the consumers to choose a more expensive BEV model, reducing the affordability of BEV adoption. However, even with its fast development, the public charging stations are still far from ubiquitous coverage. It is important to evaluate the locations of the trip destinations and the nearby charging stations to determine whether public charging is feasible at each stop. Therefore, incorporating the public charging into the model can help better estimate the needed battery range and the cost of BEV adoption. 3
The second simplification neglects the complexity and heterogeneity of individual recharging behaviors. Björnsson and Karlsson estimated the minimum battery range needs assuming that the drivers would recharge the vehicles when parking time exceeds 10 hours, 4 hours, and 0.5 hour [12]. Shi et al. evaluated the needed battery range for private vehicles and taxis in Beijing, China, assuming that the BEVs will charge using the dwell time when they are within a predefined service range of charging stations [13]. Dong et al. assumed that the drivers would only charge at home except when the state of charge (SOC) exceeds a certain comfort level [14]. Greaves et al. assumed that the drivers would only recharge at home if parking time at home exceeds 60 minutes [8]. However, the recharging behavior of different people is different and can be affected by a variety of factors [14, 15]. Through analyzing the real-word charging data and the stated preference survey data, researchers have found that the battery state of charge (SOC), dwell time, cost to recharge, excess range to home, and the distance of the upcoming trip have significant impact on the charging decision of the drivers [1, 16-18]. Researchers have also shown that the recharging behavior of different drivers can be assigned into different categories [1, 17-25]. Ge et al. clustered the PHEV drivers into two classes [16]. In one class, the drivers made recharging decisions mainly according to the recharge cost. In the other class, the drivers avoided using gasoline and recharged more frequently [16]. Recharging decisions based on only one simple rule will not be able to reflect the complexity of charging decisions and may lead to underestimated or overestimated value of the required battery range, if the drivers charge less or more frequently than the charging scenarios based on the simple rules. Therefore, considering the different charging behaviors also plays a critical role in better analyzing the required battery range.
This study aims to fill these gaps by proposing an optimization model to identify the required minimum battery range for BEVs, considering individual travel demands, locations of public charging infrastructure, and individual charging decision making. The model extends the work by Shi et al. [13] by including charging behaviors captured from a stated preference survey. To the best of our knowledge, this is the first study that considers stochastic charging decision making in the battery range optimization model. In the model proposed in this study, the BEV drivers can choose to recharge at public recharging stations. The availability of the public 4
charging infrastructures is based on real-world data. To determine under which scenarios the drivers will recharge, we conducted a Stated Preference survey. Latent Class model is used to analyze the survey data to generate the coefficients for charging decision making. This model is implemented using the trajectory data of taxis and private vehicles in Beijing, China as a case study. We chose Beijing as the case study city because the BEV sales in China has been rapidly increasing and Beijing is ranked as the best-selling city for EVs in China [26]. The BEV sales is expected to continually grow due to the policy support (e.g., purchase subsidies and relaxed regulation for license plate limit for BEV adopters) [27]. The results of this study can help inform policy and decision making for potential BEV adopters, government policy makers, and BEV manufacturers.
2
Method
2.1 General Model This section proposes a model to minimize the battery range (all-electric range) while satisfying given travel demands for BEV adopters, considering their access to public charging infrastructure, and accounting for the heterogeneous charging behaviors. The model assumes that daily trip information (i.e. trip start/end time, location, trip distance, and the dwell time at each destination) is known. At each trip destination, the driver can make a decision on whether or not to recharge the BEV based on a variety of factors such as the time available to recharge and the total cost of recharge (more details in Section 2.2). The travel demands for each driver in each
day can be described as a trip chain that contains a total of trips and + 1 stops (i.e. the
trip occurs between the stop and the + 1 stop). To minimize the inconvenience of
BEV adoption, the selected battery range should be sufficient to complete the trip chain as originally scheduled in the day as using a gasoline-fueled vehicle.
We propose a Mixed Integer Nonlinear Programming (MINLP) model to determine the minimum required battery range to satisfy the known travel demands. The model contains two sets of decision variables. The first set is the remaining battery energy at the stop (right
before taking the trip, . The other set is the decision on whether or not to recharge the
BEV at the stop, . We assume that the BEV starts the day with a fully charged battery. So 5
minimizing the required battery range is equivalent to minimizing the battery energy before taking the first trip (Eq. 1).
Objective Function
Subject to:
Min (1)
= − + , = 1, … , − 1; (2)
= min − + ,
, = 1, … , − 1; (3)
− ≥ 0, = 1, … , ; (4)
− ( − ) ≥ 0, = 1, … , − 1; (5)
= 0 or 1; = 2, … , ; (6)
≥ 0; = 1, … , ; (7)
The variables are defined as below:
The remaining battery energy before taking the trip (after charging, if recharged, at the stop). For the modeling purpose, the battery energy is measured in miles (e.g., the
amount of miles that the BEV can travel before running out of electricity). is the
remaining battery energy when fully charged before starting the trip chain, which is the BEV’s battery range.
The amount of battery energy that the BEV recharges at the stop. Similar to , is also measured in miles.
Binary variable indicating whether the BEV driver chooses to recharge at the stop or not.
= 1 represents that the driver recharges at the stop, while = 0 means
otherwise.
Distance of the trip (mile).
Charging rate of the nearest charging station at the stop (kW).
Time available for recharge at the stop (hours). Assumed to be equal to the dwell time at
the stop.
Fuel economy (electricity consumption rate) of the BEV (kWh/mile).
6
The first constraint (Eq. 2) ensures the conservation of energy. The remaining battery energy decreases as the BEV travels and increases with recharging at the stops. Eq. 3 determines the amount of energy that can be recharged at each stop. No energy can be charged if the BEV driver decides not to charge, or in other words, when = 0. When the driver chooses to recharge, the
amount of energy that is recharged is a function of the available charging time ( ), the charging rate at this charging station ( ), and the battery capacity (the battery cannot be over charged).
To linearize Eq.3, we combined Eq.2 and Eq. 3 and transformed them to the following inequalities (Eq. 8 and Eq. 9).
( − + ) − ( − + ) ≥ 0 ,
− ( − + ) ≥ 0,
= 1, … , − 1; (8)
= 1, … , − 1; (9)
Eq. 4 stipulates that the remaining battery energy before the trip ( ) must be sufficient to
complete the trip. A larger battery would be required if Eq. 4 cannot be satisfied. Eq. 5
ensures that the battery energy cannot decrease when the BEV is parked at a stop. Eqs. 6 and 7 defines and as binary and positive variables, respectively.
The model is built based on the following key assumptions: •
The trip chain will not change when the drivers switch from a gasoline vehicle to a BEV (this simplification has been made in many studies, such as in [17, 28-30]).
•
The driver starts the day with the battery fully charged, assuming that the vehicle’s last trip of the day ends at home with overnight charging. This assumption is made in many other studies, such as in [5, 8, 12].
•
The decision on whether or not to recharge at a certain stop is only determined by the factors listed in Section 2.2. In reality, additional factors may affect the charging decisions.
2.2
Determining the Recharge Probability
The probability that the drivers choose to recharge at the stop ( ) can be affected by many factors. Based on the literature review, the following five factors are included in this study to determine the likelihood that BEV drivers recharge at a stop:
7
1)
The time available to recharge the vehicle at the stop ( ). Longer parking time
provides the BEV drivers higher incentive to charge during the dwell time [16]. Jabeen et al. used a mixed logit model to study the recharging behavior of BEV drivers through a
Stated Preference survey. They found that was most sensitive to and the total cost of the recharging [22]. Similar results have also been shown in the studies [18, 23, 31].
2)
The distance to the nearest charging station from the stop. Currently, finding charging
stations for BEVs is not as convenient as finding gas stations. Given the long charging time, it is unrealistic for the drivers to wait at the charging stations. The distance from the trip destination to the charging station represents the detour the driver will have to make. Existing studies have shown that distance to the nearest charging station (detour distance [32], distance from origin to charging station [33]) is a key factor for making charging decisions. Models for optimal charging station siting also consider as being heavily influenced by the closeness of the nearest charging station [34, 35].
3)
The total cost to recharge at the stop (including both the charging cost and additional
parking cost, if any). If a farther charging station is much cheaper than the closer one, the
driver may trade the longer detour for the reduced cost. Wen et al. studied the recharging behavior of plug-in electric vehicles (PEVs) using a Mixed Logit model and a Latent Class (LC) model. They concluded that the total cost has the most significant impact on the decision on whether or not to recharge [18]. Ge et al. and Daina et al. also considered the cost as a very important factor [16, 31].
4)
The remaining battery energy at the stop before recharging ( , − , ). Lower battery energy may cause the drivers to be more anxious and prompt them to recharge.
Additionally, if the remaining battery energy is not sufficient to complete the next trip, the drivers will have a very strong incentive to recharge. On the other hand, if the battery energy is almost full, the motivation to recharge would be low [18, 24]. Daina et al. also
revealed a strong connection between and the remaining battery energy with a Hazard model [36].
8
5) The maximum amount of battery energy that can be recharged at the stop. If the battery is almost full or if the time available for charging is short, the amount of energy that can be charged into the battery will be limited. As a result, charging at this stop may not justify the hassles that the drivers need to go through to detour to visit the charging station and set up the charger and payment process. Studies of recharging behaviors have shown that the percentage of range obtained is a significant factor for the recharging decisions [16, 18].
We
represent
the
aforementioned
- = ./, , /0, , /1, , /2, , /3, 4, where:
factors
as
a
normalized
vector
x1,k = the time available to recharge the vehicle at the stop;
x2,k = the distance to the nearest charging station from the location of the stop;
x3,k = the total cost to recharge at the stop;
x4,k = the remaining battery energy at the stop before recharging;
x5,k = the maximum amount of battery energy that can be recharged at the stop.
Because different factors could have values in varied ranges, we normalized the factors into the range of [0, 1] by using the maximum values for each factor. The recharging decision is based on the total utility of the driver at the kth stop, 5 , where 5 = 67 + 6- . The driver will recharge
when the utility is greater than 0 and not otherwise. Therefore, we define the following relationship between - and :
;< (67 + 6- ) > 0, = 1 ,
= 2, … , ; (10)
;< (67 + 6- ) ≤ 0, = 0, = 2, … , ; (11)
Eqs.10 and 11 can be rewritten as Eqs.12 and 13 to be incorporated into the general model presented in Section 2.1 as constraints.
(67 + 6- ) ≥ 0, = 2, … , ; (12)
(67 + 6- )( − 1) ≥ 0 , = 2, … , ; (13)
To quantify the coefficients of 67 and 6, we conducted a survey to evaluate how the factors
affect the recharging behaviors (discussed in Section 2.5).
2.3
Battery Range Optimization Model (BROM)
9
The coefficients 67 and 6 play a critical role in the decision on whether or not to recharge at a
certain stop. The model presented in Section 2.1 would be suitable if the individual coefficients can be obtained for different drivers or if the coefficients were the same for all BEV drivers. However, in the real world, different drivers may have difference preferences (e.g., some drivers might be more sensitive to the cost of recharge while some may weigh more on the distance to the charging station). Furthermore, for the purpose of system analysis and policy planning for a
large system (e.g., at the city scale), it is impossible to have the individual coefficients for everyone. Therefore, to account for this stochasticity, we adopted Kim et al.’s approach and grouped the drivers into clusters to derive the coefficients for each cluster [25]. Kim et al. found that EV drivers could be assigned into two clusters: regular users and random users. Drivers in the regular user cluster are BEVs owners who drive the BEV regularly. Drivers in the random cluster are those who rent BEV for travel or other temporary needs. Thus, different coefficients should be applied to each cluster [25]. Therefore, in our model, we assumed that drivers belong to cluster ? with a probability of @A . Within the cluster ?, the drivers share the same charging
preferences (i.e. the same coefficients of 67,A and 6A ). The model can then be modified accordingly as follows.
Objective Function
C
Min B @A A, (14) AD
Subject to:
(67,A + 6A - ) A, ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (15)
(67,A + 6A - )( A, − 1) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (16)
A, .A − A, + 4 − ( A, − A, + ) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (17)
A,
− (A, − A, + ) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (18)
A, − ≥ 0, ? = 1, . . . , ;, = 1, … , (19)
A, − .A, − 4 ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (20)
A, = 0 or 1, ? = 1, . . . , ;, = 2, … , (21) A, ≥ 0, ? = 1, . . . , ;, (22)
where the variables are defined as below:
10
@A
The probability that the BEV driver belongs to cluster ?, where ?=1 or 2 with two
A,
Remaining battery energy before taking the trip if the driver belongs to cluster ?.
A,
Binary variable indicating whether the BEV driver chooses to recharge at the stop
clusters.
A, is the battery range for the BEV if the driver belongs to cluster ?.
or not when he/she belongs to cluster ?. A, = 1 represents that the driver recharges at
67,A , 6A
the stop while A, = 0 means otherwise.
Coefficients of charging probability if the driver belongs to cluster ?.
Distance of the trip (mile).
Charging rate of the nearest charging station at the stop (kW).
Time available for recharge at the stop (hour). Assumed to be equal to dwell time at
the stop.
Electricity consumption rate of BEV (kWh/mile).
The BROM model is similar to the general model presented in Section 2.1, except that: •
The driver of a BEV may belong to cluster ? with a probability of @A . Each cluster has its
•
Solving the battery range minimization model is actually solving two parallel general
own set of 67,A and 6A .
models. The objective function is the expected value of the minimum battery size when the drivers belong to different clusters
2.4
Case Study
We implemented the battery range minimization model on a case study of the taxis and private vehicles in Beijing, China because this city’s large and continuously growing BEV sales [26]. Assuming that the BEV adoption will not modify the current trip chain of gasoline vehicle drivers, we used the trip chain data of 11,881 gasoline taxis and 104 private vehicles over one week as the model inputs for the travel demands. For each day, the data includes the distance travelled for each trip, the location and parking time at each stop, and trip start and end time. More detailed information of the data can be found in our previous work [13, 28]. The availability of charging infrastructures at each trip destination is evaluated based on real world 11
data. A total of 522 public recharging stations exist in Beijing at the time of this study. In order to implement the model, we have to further determine the charging decision coefficients (67,A and 6A ). Therefore, we conducted a stated preference survey as discussed in Section 2.5. 2.5
Stated Preference Survey on the Recharging Behaviors
To collect information on the heterogeneous individual recharging preferences and determine the
appropriate coefficients (67,A and 6A ) for the battery range minimization model, we conducted a
web-based Stated Preference survey. Stated Preference (SP) survey is a method to study consumers’ evaluation of multi-attribute products and services [37]. Our survey asks the participants whether they would like to recharge under different scenarios characterized by the five factors discussed in Section 2.2. This method has been used to study recharging behavior in the existing literature, such as in [16, 18, 22].
The survey presents a series of scenarios to the participants and collects their stated charging preferences. Every participant needs to answer three sets of questions, assuming to be the driver of three types of BEVs with battery ranges of 70, 100, and 200 miles (the amount of miles that the BEV can travel when fully-charged). Each set of questions assumes a unique battery range and three stops that the driver will visit after three trips. The participants are asked whether they will recharge the BEV at each stop or defer charging to later stops. The scenarios are linked and the decisions made in earlier questions will change the scenarios presented in later questions.
For each set of the questions, all participants start with the same initial question and then branch out to different scenarios depending on their answers. There are a total of seven possible questions. Each participant will only answer three of them, following one of the paths in Figure 1.
12
Figure 1: The design of a question set for the SP survey. Green diamonds represent logic test while the rectangular boxes represent all the possible questions. Each question describes a scenario characterized by the five factors affecting as defined in Section 2.2. The scenarios were designed in a way that the participants have to assess the
trade-offs between different factors. For example, the participants are asked to decide on whether or not to charge at the current stop where charging is more expensive (than recharging at the next stop) but the charging station is much closer. An example of the questions is presented in the Supplementary Information. A tutorial video is provided to help the participants better understand the survey. The survey was distributed online from September 25th, 2017 to January 8th, 2018. There are a total of 334 respondents, out of which 132 are valid and complete.
Using the data collected by the SP survey, we can determine the parameters needed for the model presented in Section 2.3: 1) 2)
The probability that a BEV driver belongs to cluster ?, @A
The coefficients 67,A and 6A to determine the recharge probability A,
Latent Class (LC) model is used to determine these parameters because its demonstrated effectiveness in relevant literature. Ge et al. used the LC model to analyze the factors that make the PHEV drivers charge more frequently than the battery electric vehicle (BEV) drivers [16]. Yu et al. and Wen et al. also applied LC model to analyze to what degree the different factors (e.g., cost, charging power, and dwell time) affect the recharging behavior [17, 18]. The LC model assumes that all individuals can be separated into a finite set of clusters (; clusters). Here, 13
the coefficient 6A is captured by allocating respondents to different clusters in a probabilistic manner. Each cluster has different coefficients, but within each cluster, the coefficients stay the same [16].
The implementation of the LC model requires two steps in R: (1) process the survey data using mlogit.data and (2) implement the LC model with the gmnl package. The BEV drivers (the participants) make a decision on whether to recharge based on the total utility (5E,F, ) which is defined as:
5E,F, = -E,F, 6F , G = 1,2,3 … H; I = 1,2; = 1,2,3 … 21 (25) #
where -E,F, is a normalized 5 × 1 vector of the observed situation, containing the distance to
the nearest charging station, total cost, etc.; n is the index of the participants; m is the index for alternatives to recharge now (m = 1) or defer recharging (m = 2); and k is the index of different
stops. 6F is the coefficients for driver G and assumed to vary depending on to which cluster
the driver belongs.
Driver G belongs to class ? with the probability @F,A . Therefore,
6F = 6A w. p. @F,A ? = 1.2 (26)
where ∑CAD @F,A = 1 and @FA > 0. Here, @F,A is unknown. In the LC model, we first
generated a random OA and improved @F,A through iterations. @F,A =
exp(OA ) ; ? = 1,2 (27) ∑CAD exp(OA )
For the given data, binary index RE,F, = 1 if the driver G choses alternative I on the
stop. Otherwise, this index is 0. The unconditional probability of the driver G for the sequence of
choices is
^
X
E
exp.-E,F, 6A 4 Y
F (S ) = T UV V W X ∑ED exp(-EF 6A ) = B
C
^
X
E
Z[,\,]
exp.-E,F, 6A 4
_ <(6A )6A
Y @F,A UV V W X ∑ED exp.-E,F, 6A 4
AD
Z[,\,]
_ (28)
The LC model optimizes @A,` and 6A using the expectation–maximization (EM) algorithm.
14
2.6
Comparing the proposed model to recharging with simplified charging rules
To evaluate how the identified minimum battery range using the proposed model is different from those using simplified rules, we have also conducted a simulation using the following simplified charging rules to determine the minimum battery range: 1)
The time available to recharge exceeds 150 minutes. The current 7kW charger is able to recharge 30 miles of range per hour [38]. Drivers who possess a BEV with battery range around 100 miles is able to recharge around 80% of the battery in 150 minutes.
2)
Distance to the nearest charging station is less than one mile. Drivers might not be willing to recharge if the nearest charging station is too far. It takes around 20 minutes to walk one mile [13].
3)
The unit cost to recharge is less than ¥0.2/mile, considering both charging and parking. Based on the electricity cost of ￥0.5/kWh, BEV fuel economy of 0.35kWh/mile, and a charging efficiency of 88%, the unit cost of charging at home is about ¥0.2/mile in Beijing. So this rule represents the scenario that the drivers take advantage of cheaper charging when it is available.
4)
The maximum battery energy that can be recharged is greater than 15 miles. Drivers are able to recharge 15 miles in half an hour with a 7kW charger. The drivers potentially would be more likely to recharge the vehicles if a decent amount of energy can be recharged (when they have more than 30 minutes of parking time and when the battery is not almost full).
5)
The remaining battery at the next stop is less than 20 miles, if not recharging at the current stop. To avoid range anxiety after the upcoming trip, the drivers may be motivated to recharge before taking the trip.
3 3.1
Results and Discussions Charging Decision Coefficients Estimation using Latent Class Model
The LC model requires a pre-defined number of clusters. We adopted Kim et al.’s approach and grouped the drivers into two clusters [25]. According to the LC model, the probability of belonging to cluster 1 and cluster 2 is 0.49 and 0.51, respectively. In cluster 1, three of the factors, the time available for recharge, the distance to the nearest charging station, and the total 15
recharge cost, had significant influence on the recharging decision (Table 1). In cluster 2, drivers tend to make more random decisions.
Table 1: The LC model results
Estimate
67
6
0
-0.003
0.015
0.011
0.004
0.009
0.005
0.010
z value
-2.964
-2.929
-4.227
0.902
1.565
Pr(>z)
0.003
0.003
0.000
0.367
0.118
Significant
**
**
***
-0.1253
0.0022
-0.0039
-0.0126
0.0041
0.0079
0.2900
0.0065
0.0049
0.0060
0.0046
0.0076
z value
-0.4319
0.3446
-0.7901
-2.0953
0.9023
1.0326
Pr(>z)
0.6658
0.7304
0.4295
0.0361
0.3669
0.3018
Stand Error
Significant
Codes Notes:
3.2
63
-0.040
Estimate
Significant.
62
-0.013
Error
Cluster 2
61
0.033
Stand Cluster 1
60
* 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
6 : Time Available to recharge 60: Distance to nearest charging station 61: Total cost to recharge 62 : Remaining battery at current stop 63: Maximum battery can be recharged at current stop
Minimum Battery Range Required for Taxis
Applying the proposed model and charging decision coefficients to taxi trip chain data, we can identify the required battery range for taxis in Beijing based on each day’s travel demands. The minimum battery ranges for the taxi drivers vary on different days because of the variances in daily travel patterns. We calculated the minimum battery range for each single day between March 3rd and March 9th, 2009 selecting the highest minimum battery range among the seven days as the recommended range. With this battery range, the driver is able to finish all the travel demand within the week. The distribution of the suggested battery ranges is presented in Figure 2.
16
To satisfy the travel demands of 90% of taxi drivers, a BEV with a full battery range of 220 miles is required, while satisfying 99.5% of travel demand requires a battery range of 330 miles.
Figure 2 The cumulative distribution function (CDF)of the suggested battery ranges for Beijing taxis
To study the different recharging behaviors of the two clusters, we compared the drivers’ recharging frequency on March 3rd, 2009. The recharging frequency is calculated as the number of times the drivers chose to recharge in a trip chain divided by the total number of stops (excluding home) in the trip chain. The drivers in different clusters have significant different charging behaviors: about 90% of the drivers in cluster 2 chose not to recharge at all during the day, while only 20% of the drivers in cluster 1 does not recharge (Figure 3). This result shows that drivers in cluster 1 is more proactive in charging the BEVs.
17
Figure 3: The empirical CDF of the percentage of stops to recharge
3.3
Minimum Battery Range for Private Vehicles
We also implemented the battery range minimization model using the historical travel data of private vehicles. Same as Section 3.2, we selected 7 days as the study period. For each driver, we determined the minimum battery range needed for each day and then picked the highest battery range within the 7 days as the recommended range. The CDFs of the suggested battery range for the private vehicles are presented in Figure 4, in comparison to that of the taxis. For the private vehicles, a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Comparing the histogram of the suggested battery range of taxis and private vehicles, we can see that the majority of the private vehicles only need a small battery range (Figure 5). However, compared to the suggested battery range for the taxis, the distribution of the minimum battery range for private vehicles has a long tail. A battery range of approximately 300 miles is needed to satisfy the travel demand of 90% of the private vehicle drivers according to the model. The travel patterns of some of the private vehicles make them unsuitable for BEV adoption, such as having long inter-city trips or traveling in the regions with low charging infrastructure support).
18
Figure 4: The empirical CDF of the suggested battery range
Figure 5: The histogram of the suggested battery range 3.4
Comparing the minimum battery range identified using simplified rules
Comparing to the minimum battery ranges determined using simplified charging rules for taxi drivers, we found that all rules, except for Rule 1 (time available to recharge exceeds 150 minutes) and Rule 3 (unit cost of recharging is less than ¥0.2/mile), underestimate the required battery range compared to the proposed model to satisfied one-day travel demand (Figure 6a). This result shows that EV drivers may not necessarily use every charging opportunity to recharge 19
the vehicle and may not only wait for long dwell time or cheap charging opportunity to recharge. Unlike the taxis, all rules overestimate the required battery range compared to the proposed model for private vehicles (Figure 6b). This result indicates that private vehicle drivers utilize more diverse charging opportunities than those described by the simple rules. Considering the complexity and heterogeneity of charging decisions is important in determining the needed battery range for BEVs.
a)
b) Figure 6: The minimum battery range estimated using different rules for (a) taxis and (b) private vehicles (Rule 1: Time available to recharge exceeds 150 minutes; Rule 2: Distance to the nearest charging station is less than 1 mile; Rule 3: The unit cost of recharging is less than ¥0.2/mile, considering both charging cost and parking cost; Rule 4: The maximum battery energy that can be recharged is greater than 15 miles; and Rule 5: The remaining battery energy at the next stop, if not recharging at current stop, is less than 15 miles.)
4
CONCLUSION AND FUTURE WORK
This study proposes a battery range optimization model to identify the minimum BEV battery range that can satisfy given travel demands for BEV drivers, considering the opportunities to charge at existing public charging stations and the uncertainties in charging decision making. To assess the probability of whether a driver recharges at each stop, we conducted a Stated Preference survey. Based on the collected data from the survey, the LC model is used to analyze and predict the different recharging behaviors of the two clusters of BEV drivers. The models were implemented using trajectory data of Beijing taxi and private vehicles. For the taxis in Beijing, BEVs with a battery range of 220 miles are able to satisfy the travel demand for about 20
90% of the drivers. For private vehicle, a range of 300 miles is needed to cover the travel demands of 90% of the drivers, while a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Comparing the identified minimum required battery range from the proposed model and those using simplified recharging rules, our results show that most of the considered rules may cause underestimation of the minimum required battery range for taxis and overestimation of the range needs for private vehicles, showing the importance to consider the charging behavior in BEV models. These results also show that, with the existing charging infrastructure in Beijing, the benefits of providing policy incentives and technology development for BEVs with large battery sizes exceeding 220 miles may be limited. Simulation tools and models such as the one presented in this study can be helpful tools to enable individual consumer, industry experts, and policy makers evaluate the BEV range needs at the individual level, fleet level, and city scale, respectively.
While this study has the merit of being the first to consider stochastic charging behaviors in the BEV range optimization models, it has three major limitations that can be improved in future studies. First, we only conducted one set of survey and used the derived charging coefficients for both taxis and private drivers. Different driver groups may have different charging behaviors. Future studies obtaining the charging coefficients specifically for each driver group can make the model more accurate. Second, it is possible that the charging decision making derived from the stated preference survey is different from the actual charging decisions. Comparing the stated charging preferences to the revealed actual charging decisions can help better evaluate to what extent the survey results reflect the actual charging decision making. Lastly, because the problem is formulated as a MINLP, the model becomes hard to solve when the number of decision variables become too large (e.g., solving the model using one month’s trip data). Alternative ways to formulate the recharging probability may help improve the scalability of the model.
In summary, this study proposed a model and framework to integrate the charging behavior and real world public charging infrastructures into determining the minimum battery range needed to meet the given travel demands. The identified distribution of the minimum battery ranges to adopt BEVs can be used as inputs for BEV related studies (e.g., charging infrastructure
21
development, environmental impacts modeling, and vehicle design etc.) and inform decision making for policy makers, industry experts, and consumers.
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24
Highlights • • • • •
Proposed a Battery Range Optimization Model (BROM) Integrated heterogeneous charging behaviors into battery range optimization Evaluated the battery range needs for taxis and private vehicles in Beijing as case studies Simple charging rules may lead to overestimated battery needs for taxis Simple charging rules may lead to underestimated battery range needs for private vehicles
Declaration of interests ☒ 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0360-5442(20)30052-9
DOI:
https://doi.org/10.1016/j.energy.2020.116945
Reference:
EGY 116945
To appear in:
Energy
Received Date: 2 August 2019 Revised Date:
27 November 2019
Accepted Date: 9 January 2020
Please cite this article as: Zhou Y, Wen R, Wang H, Cai H, Optimal battery electric vehicles range: A study considering heterogeneous travel patterns, charging behaviors, and access to charging infrastructure, Energy (2020), doi: https://doi.org/10.1016/j.energy.2020.116945. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.
Optimal Battery Electric Vehicles Range: A Study Considering Heterogeneous Travel Patterns, Charging Behaviors, and Access to Charging Infrastructure Yue Zhoua, Ruoxi Wena, Hewu Wangb, and Hua Caia,c* a. School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, United States b. State Key laboratory of Automotive Safety and Energy, Tsinghua University, Beijing, 100084, China c. Environmental and Ecological Engineering, Purdue University, West Lafayette, IN 47907, United States
∗Corresponding author E-mail: [email protected] (H. Cai).
Tel: 765-494-7701
Address: 315 N. Grant Street, West Lafayette, IN 47907, United States
Abstract The choice of battery range (all-electric driving range) for battery electric vehicles (BEVs) is an important issue for both BEV adopters and BEV makers. This paper proposes a model to identify the minimum BEV battery range that can satisfy given travel demands, considering the opportunities to charge at existing public charging stations and the uncertainties in charging decision making. We conducted a Stated Preference survey to study the charging decision making and analyzed the data using the Latent Class model to generate the model coefficients for charging decisions making. The proposed approach can better identify the needed battery range than the often-used simplified charging rules. We applied the model to a case study of Beijing to evaluate the needed battery range for taxis and private vehicles. For taxis, BEVs with 220-miles battery range are able to satisfy the travel demands for about 90% of the drivers. For private vehicles, a 300-miles range is needed to cover the travel demands of 90% of the drivers, while a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Simplified charging rules tend to underestimate the range needs for taxis but overestimate the range needs for private vehicles.
Keywords: Battery Range; Electric Vehicles; Mixed Integer Nonlinear Programming (MINLP); Charging Behaviors; Charging Infrastructure 1
1
Introduction
Growing concerns over the environmental impacts of fossil fuels have led to the development of various alternative vehicle and fuel technologies. Battery electric vehicles (BEVs) offer great opportunities to reduce the transportation greenhouse gas (GHG) emissions when they are powered by clean electricity (e.g., generated from solar or wind) [1]. Considering the potential environmental benefits, the government and industries are making great efforts to support the BEV adoption through policy incentives and the development of charging infrastructure and battery technology. However, only 4% of the potential buyers have purchased BEVs or plug-in hybrid electric vehicles (PHEVs) in the U.S. [2] There are mainly three concerns that prevent the potential consumers to adopt BEVs. The first one is the limited all-electric driving range, which refers to the total distance the BEV can travel with a fully charged battery (in this study, we refer to the all-electric driving range as battery range). Drivers are concerned about finding public charging stations when they need to charge, because charging stations are not as ubiquitous as the gas stations. Second, long charging time and the associated inconvenience is another concern. Unlike refilling the traditional internal combustion engine vehicles (ICEVs) which can be done within a few minutes, fully recharging a BEV may take hours, depending on the battery range and the charging power [3]. The third concern is BEVs’ relatively high price. BEVs are more expensive compared to their ICEV counterparts. While BEVs with larger batteries (longer range) can help alleviate range anxiety, they also cost even more than BEVs with smaller battery ranges. Therefore, for potential BEV adopters, knowing the minimum required battery ranges that can satisfy their travel demands will help address these concerns.
Given the vast differences in battery ranges among the available BEV models, choosing a BEV model bounds the driver to the BEV model’s battery range. BEVs with smaller batteries will require more frequent recharging and have the risks of not being able to complete long trips. Charging in the middle of a trip will cause significant inconvenience due to the long charging time, let along the potential risk of not being able to find charging stations when needed. On the other hand, however, BEVs with larger batteries require higher costs and also reduce the BEV’s fuel economy due to the weight of the battery. To determine the optimal (minimum) required battery range, the following key factors need to be considered: 1) the users’ travel demands (e.g., the trip distance and locations of the trip origins and destinations), 2) charging infrastructures 2
(e.g., the locations of home, work, and public charging stations), and 3) individual charging behaviors (e.g., under which conditions the drivers will charge the BEVs).
Most of the existing literature that studies the optimal BEV battery range has two major simplifications: 1) charging will only happen at home and/or work place and 2) everyone will follow the same simple rule to make charging decisions. The first simplification ignores the opportunities to recharge at public charging stations. Pearre et al. determined the minimum battery range by simplifying the problem with the assumption that the BEV drivers would fully charge the vehicles at home every night [4]. Weiss et al. also assumed that drivers could only charge at home to analyze the battery range needed to satisfy the travel demands of 1.3 million private vehicles in the Greater Stuttgart Area in Germany with an agent-based simulation model [5]. Stark et al. further considered work place charging by assuming that drivers recharged every time when they reach home and work locations [6]. Smith et al. found that the minimum battery range needed for the BEVs can be reduced by 40% under the scenarios that people recharge both at work place and at home, compared to home-charge only scenarios [7].
The earlier studies only considered home or work place charging because the public charging infrastructure has not been developed at that time [4, 7, 8]. However, the number of public charging stations have been increasing rapidly in recent years [9]. Public charging is also becoming more convenient with more fast chargers introduced into the market [10]. These changes enable the BEV drivers to recharge the vehicles during the time when they are shopping, having dinner, waiting for the next passenger (for taxis) etc. Ignoring these public recharging opportunities in the model may overestimate the battery range needed to meet the daily travel demands. According to Neubauer et al., the availability of the public charging infrastructure strongly affects the range anxiety of potential BEV adopters [11]. Overestimating the required battery range would mislead the consumers to choose a more expensive BEV model, reducing the affordability of BEV adoption. However, even with its fast development, the public charging stations are still far from ubiquitous coverage. It is important to evaluate the locations of the trip destinations and the nearby charging stations to determine whether public charging is feasible at each stop. Therefore, incorporating the public charging into the model can help better estimate the needed battery range and the cost of BEV adoption. 3
The second simplification neglects the complexity and heterogeneity of individual recharging behaviors. Björnsson and Karlsson estimated the minimum battery range needs assuming that the drivers would recharge the vehicles when parking time exceeds 10 hours, 4 hours, and 0.5 hour [12]. Shi et al. evaluated the needed battery range for private vehicles and taxis in Beijing, China, assuming that the BEVs will charge using the dwell time when they are within a predefined service range of charging stations [13]. Dong et al. assumed that the drivers would only charge at home except when the state of charge (SOC) exceeds a certain comfort level [14]. Greaves et al. assumed that the drivers would only recharge at home if parking time at home exceeds 60 minutes [8]. However, the recharging behavior of different people is different and can be affected by a variety of factors [14, 15]. Through analyzing the real-word charging data and the stated preference survey data, researchers have found that the battery state of charge (SOC), dwell time, cost to recharge, excess range to home, and the distance of the upcoming trip have significant impact on the charging decision of the drivers [1, 16-18]. Researchers have also shown that the recharging behavior of different drivers can be assigned into different categories [1, 17-25]. Ge et al. clustered the PHEV drivers into two classes [16]. In one class, the drivers made recharging decisions mainly according to the recharge cost. In the other class, the drivers avoided using gasoline and recharged more frequently [16]. Recharging decisions based on only one simple rule will not be able to reflect the complexity of charging decisions and may lead to underestimated or overestimated value of the required battery range, if the drivers charge less or more frequently than the charging scenarios based on the simple rules. Therefore, considering the different charging behaviors also plays a critical role in better analyzing the required battery range.
This study aims to fill these gaps by proposing an optimization model to identify the required minimum battery range for BEVs, considering individual travel demands, locations of public charging infrastructure, and individual charging decision making. The model extends the work by Shi et al. [13] by including charging behaviors captured from a stated preference survey. To the best of our knowledge, this is the first study that considers stochastic charging decision making in the battery range optimization model. In the model proposed in this study, the BEV drivers can choose to recharge at public recharging stations. The availability of the public 4
charging infrastructures is based on real-world data. To determine under which scenarios the drivers will recharge, we conducted a Stated Preference survey. Latent Class model is used to analyze the survey data to generate the coefficients for charging decision making. This model is implemented using the trajectory data of taxis and private vehicles in Beijing, China as a case study. We chose Beijing as the case study city because the BEV sales in China has been rapidly increasing and Beijing is ranked as the best-selling city for EVs in China [26]. The BEV sales is expected to continually grow due to the policy support (e.g., purchase subsidies and relaxed regulation for license plate limit for BEV adopters) [27]. The results of this study can help inform policy and decision making for potential BEV adopters, government policy makers, and BEV manufacturers.
2
Method
2.1 General Model This section proposes a model to minimize the battery range (all-electric range) while satisfying given travel demands for BEV adopters, considering their access to public charging infrastructure, and accounting for the heterogeneous charging behaviors. The model assumes that daily trip information (i.e. trip start/end time, location, trip distance, and the dwell time at each destination) is known. At each trip destination, the driver can make a decision on whether or not to recharge the BEV based on a variety of factors such as the time available to recharge and the total cost of recharge (more details in Section 2.2). The travel demands for each driver in each
day can be described as a trip chain that contains a total of trips and + 1 stops (i.e. the
trip occurs between the stop and the + 1 stop). To minimize the inconvenience of
BEV adoption, the selected battery range should be sufficient to complete the trip chain as originally scheduled in the day as using a gasoline-fueled vehicle.
We propose a Mixed Integer Nonlinear Programming (MINLP) model to determine the minimum required battery range to satisfy the known travel demands. The model contains two sets of decision variables. The first set is the remaining battery energy at the stop (right
before taking the trip, . The other set is the decision on whether or not to recharge the
BEV at the stop, . We assume that the BEV starts the day with a fully charged battery. So 5
minimizing the required battery range is equivalent to minimizing the battery energy before taking the first trip (Eq. 1).
Objective Function
Subject to:
Min (1)
= − + , = 1, … , − 1; (2)
= min − + ,
, = 1, … , − 1; (3)
− ≥ 0, = 1, … , ; (4)
− ( − ) ≥ 0, = 1, … , − 1; (5)
= 0 or 1; = 2, … , ; (6)
≥ 0; = 1, … , ; (7)
The variables are defined as below:
The remaining battery energy before taking the trip (after charging, if recharged, at the stop). For the modeling purpose, the battery energy is measured in miles (e.g., the
amount of miles that the BEV can travel before running out of electricity). is the
remaining battery energy when fully charged before starting the trip chain, which is the BEV’s battery range.
The amount of battery energy that the BEV recharges at the stop. Similar to , is also measured in miles.
Binary variable indicating whether the BEV driver chooses to recharge at the stop or not.
= 1 represents that the driver recharges at the stop, while = 0 means
otherwise.
Distance of the trip (mile).
Charging rate of the nearest charging station at the stop (kW).
Time available for recharge at the stop (hours). Assumed to be equal to the dwell time at
the stop.
Fuel economy (electricity consumption rate) of the BEV (kWh/mile).
6
The first constraint (Eq. 2) ensures the conservation of energy. The remaining battery energy decreases as the BEV travels and increases with recharging at the stops. Eq. 3 determines the amount of energy that can be recharged at each stop. No energy can be charged if the BEV driver decides not to charge, or in other words, when = 0. When the driver chooses to recharge, the
amount of energy that is recharged is a function of the available charging time ( ), the charging rate at this charging station ( ), and the battery capacity (the battery cannot be over charged).
To linearize Eq.3, we combined Eq.2 and Eq. 3 and transformed them to the following inequalities (Eq. 8 and Eq. 9).
( − + ) − ( − + ) ≥ 0 ,
− ( − + ) ≥ 0,
= 1, … , − 1; (8)
= 1, … , − 1; (9)
Eq. 4 stipulates that the remaining battery energy before the trip ( ) must be sufficient to
complete the trip. A larger battery would be required if Eq. 4 cannot be satisfied. Eq. 5
ensures that the battery energy cannot decrease when the BEV is parked at a stop. Eqs. 6 and 7 defines and as binary and positive variables, respectively.
The model is built based on the following key assumptions: •
The trip chain will not change when the drivers switch from a gasoline vehicle to a BEV (this simplification has been made in many studies, such as in [17, 28-30]).
•
The driver starts the day with the battery fully charged, assuming that the vehicle’s last trip of the day ends at home with overnight charging. This assumption is made in many other studies, such as in [5, 8, 12].
•
The decision on whether or not to recharge at a certain stop is only determined by the factors listed in Section 2.2. In reality, additional factors may affect the charging decisions.
2.2
Determining the Recharge Probability
The probability that the drivers choose to recharge at the stop ( ) can be affected by many factors. Based on the literature review, the following five factors are included in this study to determine the likelihood that BEV drivers recharge at a stop:
7
1)
The time available to recharge the vehicle at the stop ( ). Longer parking time
provides the BEV drivers higher incentive to charge during the dwell time [16]. Jabeen et al. used a mixed logit model to study the recharging behavior of BEV drivers through a
Stated Preference survey. They found that was most sensitive to and the total cost of the recharging [22]. Similar results have also been shown in the studies [18, 23, 31].
2)
The distance to the nearest charging station from the stop. Currently, finding charging
stations for BEVs is not as convenient as finding gas stations. Given the long charging time, it is unrealistic for the drivers to wait at the charging stations. The distance from the trip destination to the charging station represents the detour the driver will have to make. Existing studies have shown that distance to the nearest charging station (detour distance [32], distance from origin to charging station [33]) is a key factor for making charging decisions. Models for optimal charging station siting also consider as being heavily influenced by the closeness of the nearest charging station [34, 35].
3)
The total cost to recharge at the stop (including both the charging cost and additional
parking cost, if any). If a farther charging station is much cheaper than the closer one, the
driver may trade the longer detour for the reduced cost. Wen et al. studied the recharging behavior of plug-in electric vehicles (PEVs) using a Mixed Logit model and a Latent Class (LC) model. They concluded that the total cost has the most significant impact on the decision on whether or not to recharge [18]. Ge et al. and Daina et al. also considered the cost as a very important factor [16, 31].
4)
The remaining battery energy at the stop before recharging ( , − , ). Lower battery energy may cause the drivers to be more anxious and prompt them to recharge.
Additionally, if the remaining battery energy is not sufficient to complete the next trip, the drivers will have a very strong incentive to recharge. On the other hand, if the battery energy is almost full, the motivation to recharge would be low [18, 24]. Daina et al. also
revealed a strong connection between and the remaining battery energy with a Hazard model [36].
8
5) The maximum amount of battery energy that can be recharged at the stop. If the battery is almost full or if the time available for charging is short, the amount of energy that can be charged into the battery will be limited. As a result, charging at this stop may not justify the hassles that the drivers need to go through to detour to visit the charging station and set up the charger and payment process. Studies of recharging behaviors have shown that the percentage of range obtained is a significant factor for the recharging decisions [16, 18].
We
represent
the
aforementioned
- = ./, , /0, , /1, , /2, , /3, 4, where:
factors
as
a
normalized
vector
x1,k = the time available to recharge the vehicle at the stop;
x2,k = the distance to the nearest charging station from the location of the stop;
x3,k = the total cost to recharge at the stop;
x4,k = the remaining battery energy at the stop before recharging;
x5,k = the maximum amount of battery energy that can be recharged at the stop.
Because different factors could have values in varied ranges, we normalized the factors into the range of [0, 1] by using the maximum values for each factor. The recharging decision is based on the total utility of the driver at the kth stop, 5 , where 5 = 67 + 6- . The driver will recharge
when the utility is greater than 0 and not otherwise. Therefore, we define the following relationship between - and :
;< (67 + 6- ) > 0, = 1 ,
= 2, … , ; (10)
;< (67 + 6- ) ≤ 0, = 0, = 2, … , ; (11)
Eqs.10 and 11 can be rewritten as Eqs.12 and 13 to be incorporated into the general model presented in Section 2.1 as constraints.
(67 + 6- ) ≥ 0, = 2, … , ; (12)
(67 + 6- )( − 1) ≥ 0 , = 2, … , ; (13)
To quantify the coefficients of 67 and 6, we conducted a survey to evaluate how the factors
affect the recharging behaviors (discussed in Section 2.5).
2.3
Battery Range Optimization Model (BROM)
9
The coefficients 67 and 6 play a critical role in the decision on whether or not to recharge at a
certain stop. The model presented in Section 2.1 would be suitable if the individual coefficients can be obtained for different drivers or if the coefficients were the same for all BEV drivers. However, in the real world, different drivers may have difference preferences (e.g., some drivers might be more sensitive to the cost of recharge while some may weigh more on the distance to the charging station). Furthermore, for the purpose of system analysis and policy planning for a
large system (e.g., at the city scale), it is impossible to have the individual coefficients for everyone. Therefore, to account for this stochasticity, we adopted Kim et al.’s approach and grouped the drivers into clusters to derive the coefficients for each cluster [25]. Kim et al. found that EV drivers could be assigned into two clusters: regular users and random users. Drivers in the regular user cluster are BEVs owners who drive the BEV regularly. Drivers in the random cluster are those who rent BEV for travel or other temporary needs. Thus, different coefficients should be applied to each cluster [25]. Therefore, in our model, we assumed that drivers belong to cluster ? with a probability of @A . Within the cluster ?, the drivers share the same charging
preferences (i.e. the same coefficients of 67,A and 6A ). The model can then be modified accordingly as follows.
Objective Function
C
Min B @A A, (14) AD
Subject to:
(67,A + 6A - ) A, ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (15)
(67,A + 6A - )( A, − 1) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (16)
A, .A − A, + 4 − ( A, − A, + ) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (17)
A,
− (A, − A, + ) ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (18)
A, − ≥ 0, ? = 1, . . . , ;, = 1, … , (19)
A, − .A, − 4 ≥ 0, ? = 1, . . . , ;, = 1, … , − 1 (20)
A, = 0 or 1, ? = 1, . . . , ;, = 2, … , (21) A, ≥ 0, ? = 1, . . . , ;, (22)
where the variables are defined as below:
10
@A
The probability that the BEV driver belongs to cluster ?, where ?=1 or 2 with two
A,
Remaining battery energy before taking the trip if the driver belongs to cluster ?.
A,
Binary variable indicating whether the BEV driver chooses to recharge at the stop
clusters.
A, is the battery range for the BEV if the driver belongs to cluster ?.
or not when he/she belongs to cluster ?. A, = 1 represents that the driver recharges at
67,A , 6A
the stop while A, = 0 means otherwise.
Coefficients of charging probability if the driver belongs to cluster ?.
Distance of the trip (mile).
Charging rate of the nearest charging station at the stop (kW).
Time available for recharge at the stop (hour). Assumed to be equal to dwell time at
the stop.
Electricity consumption rate of BEV (kWh/mile).
The BROM model is similar to the general model presented in Section 2.1, except that: •
The driver of a BEV may belong to cluster ? with a probability of @A . Each cluster has its
•
Solving the battery range minimization model is actually solving two parallel general
own set of 67,A and 6A .
models. The objective function is the expected value of the minimum battery size when the drivers belong to different clusters
2.4
Case Study
We implemented the battery range minimization model on a case study of the taxis and private vehicles in Beijing, China because this city’s large and continuously growing BEV sales [26]. Assuming that the BEV adoption will not modify the current trip chain of gasoline vehicle drivers, we used the trip chain data of 11,881 gasoline taxis and 104 private vehicles over one week as the model inputs for the travel demands. For each day, the data includes the distance travelled for each trip, the location and parking time at each stop, and trip start and end time. More detailed information of the data can be found in our previous work [13, 28]. The availability of charging infrastructures at each trip destination is evaluated based on real world 11
data. A total of 522 public recharging stations exist in Beijing at the time of this study. In order to implement the model, we have to further determine the charging decision coefficients (67,A and 6A ). Therefore, we conducted a stated preference survey as discussed in Section 2.5. 2.5
Stated Preference Survey on the Recharging Behaviors
To collect information on the heterogeneous individual recharging preferences and determine the
appropriate coefficients (67,A and 6A ) for the battery range minimization model, we conducted a
web-based Stated Preference survey. Stated Preference (SP) survey is a method to study consumers’ evaluation of multi-attribute products and services [37]. Our survey asks the participants whether they would like to recharge under different scenarios characterized by the five factors discussed in Section 2.2. This method has been used to study recharging behavior in the existing literature, such as in [16, 18, 22].
The survey presents a series of scenarios to the participants and collects their stated charging preferences. Every participant needs to answer three sets of questions, assuming to be the driver of three types of BEVs with battery ranges of 70, 100, and 200 miles (the amount of miles that the BEV can travel when fully-charged). Each set of questions assumes a unique battery range and three stops that the driver will visit after three trips. The participants are asked whether they will recharge the BEV at each stop or defer charging to later stops. The scenarios are linked and the decisions made in earlier questions will change the scenarios presented in later questions.
For each set of the questions, all participants start with the same initial question and then branch out to different scenarios depending on their answers. There are a total of seven possible questions. Each participant will only answer three of them, following one of the paths in Figure 1.
12
Figure 1: The design of a question set for the SP survey. Green diamonds represent logic test while the rectangular boxes represent all the possible questions. Each question describes a scenario characterized by the five factors affecting as defined in Section 2.2. The scenarios were designed in a way that the participants have to assess the
trade-offs between different factors. For example, the participants are asked to decide on whether or not to charge at the current stop where charging is more expensive (than recharging at the next stop) but the charging station is much closer. An example of the questions is presented in the Supplementary Information. A tutorial video is provided to help the participants better understand the survey. The survey was distributed online from September 25th, 2017 to January 8th, 2018. There are a total of 334 respondents, out of which 132 are valid and complete.
Using the data collected by the SP survey, we can determine the parameters needed for the model presented in Section 2.3: 1) 2)
The probability that a BEV driver belongs to cluster ?, @A
The coefficients 67,A and 6A to determine the recharge probability A,
Latent Class (LC) model is used to determine these parameters because its demonstrated effectiveness in relevant literature. Ge et al. used the LC model to analyze the factors that make the PHEV drivers charge more frequently than the battery electric vehicle (BEV) drivers [16]. Yu et al. and Wen et al. also applied LC model to analyze to what degree the different factors (e.g., cost, charging power, and dwell time) affect the recharging behavior [17, 18]. The LC model assumes that all individuals can be separated into a finite set of clusters (; clusters). Here, 13
the coefficient 6A is captured by allocating respondents to different clusters in a probabilistic manner. Each cluster has different coefficients, but within each cluster, the coefficients stay the same [16].
The implementation of the LC model requires two steps in R: (1) process the survey data using mlogit.data and (2) implement the LC model with the gmnl package. The BEV drivers (the participants) make a decision on whether to recharge based on the total utility (5E,F, ) which is defined as:
5E,F, = -E,F, 6F , G = 1,2,3 … H; I = 1,2; = 1,2,3 … 21 (25) #
where -E,F, is a normalized 5 × 1 vector of the observed situation, containing the distance to
the nearest charging station, total cost, etc.; n is the index of the participants; m is the index for alternatives to recharge now (m = 1) or defer recharging (m = 2); and k is the index of different
stops. 6F is the coefficients for driver G and assumed to vary depending on to which cluster
the driver belongs.
Driver G belongs to class ? with the probability @F,A . Therefore,
6F = 6A w. p. @F,A ? = 1.2 (26)
where ∑CAD @F,A = 1 and @FA > 0. Here, @F,A is unknown. In the LC model, we first
generated a random OA and improved @F,A through iterations. @F,A =
exp(OA ) ; ? = 1,2 (27) ∑CAD exp(OA )
For the given data, binary index RE,F, = 1 if the driver G choses alternative I on the
stop. Otherwise, this index is 0. The unconditional probability of the driver G for the sequence of
choices is
^
X
E
exp.-E,F, 6A 4 Y
F (S ) = T UV V W X ∑ED exp(-EF 6A ) = B
C
^
X
E
Z[,\,]
exp.-E,F, 6A 4
_ <(6A )6A
Y @F,A UV V W X ∑ED exp.-E,F, 6A 4
AD
Z[,\,]
_ (28)
The LC model optimizes @A,` and 6A using the expectation–maximization (EM) algorithm.
14
2.6
Comparing the proposed model to recharging with simplified charging rules
To evaluate how the identified minimum battery range using the proposed model is different from those using simplified rules, we have also conducted a simulation using the following simplified charging rules to determine the minimum battery range: 1)
The time available to recharge exceeds 150 minutes. The current 7kW charger is able to recharge 30 miles of range per hour [38]. Drivers who possess a BEV with battery range around 100 miles is able to recharge around 80% of the battery in 150 minutes.
2)
Distance to the nearest charging station is less than one mile. Drivers might not be willing to recharge if the nearest charging station is too far. It takes around 20 minutes to walk one mile [13].
3)
The unit cost to recharge is less than ¥0.2/mile, considering both charging and parking. Based on the electricity cost of ￥0.5/kWh, BEV fuel economy of 0.35kWh/mile, and a charging efficiency of 88%, the unit cost of charging at home is about ¥0.2/mile in Beijing. So this rule represents the scenario that the drivers take advantage of cheaper charging when it is available.
4)
The maximum battery energy that can be recharged is greater than 15 miles. Drivers are able to recharge 15 miles in half an hour with a 7kW charger. The drivers potentially would be more likely to recharge the vehicles if a decent amount of energy can be recharged (when they have more than 30 minutes of parking time and when the battery is not almost full).
5)
The remaining battery at the next stop is less than 20 miles, if not recharging at the current stop. To avoid range anxiety after the upcoming trip, the drivers may be motivated to recharge before taking the trip.
3 3.1
Results and Discussions Charging Decision Coefficients Estimation using Latent Class Model
The LC model requires a pre-defined number of clusters. We adopted Kim et al.’s approach and grouped the drivers into two clusters [25]. According to the LC model, the probability of belonging to cluster 1 and cluster 2 is 0.49 and 0.51, respectively. In cluster 1, three of the factors, the time available for recharge, the distance to the nearest charging station, and the total 15
recharge cost, had significant influence on the recharging decision (Table 1). In cluster 2, drivers tend to make more random decisions.
Table 1: The LC model results
Estimate
67
6
0
-0.003
0.015
0.011
0.004
0.009
0.005
0.010
z value
-2.964
-2.929
-4.227
0.902
1.565
Pr(>z)
0.003
0.003
0.000
0.367
0.118
Significant
**
**
***
-0.1253
0.0022
-0.0039
-0.0126
0.0041
0.0079
0.2900
0.0065
0.0049
0.0060
0.0046
0.0076
z value
-0.4319
0.3446
-0.7901
-2.0953
0.9023
1.0326
Pr(>z)
0.6658
0.7304
0.4295
0.0361
0.3669
0.3018
Stand Error
Significant
Codes Notes:
3.2
63
-0.040
Estimate
Significant.
62
-0.013
Error
Cluster 2
61
0.033
Stand Cluster 1
60
* 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
6 : Time Available to recharge 60: Distance to nearest charging station 61: Total cost to recharge 62 : Remaining battery at current stop 63: Maximum battery can be recharged at current stop
Minimum Battery Range Required for Taxis
Applying the proposed model and charging decision coefficients to taxi trip chain data, we can identify the required battery range for taxis in Beijing based on each day’s travel demands. The minimum battery ranges for the taxi drivers vary on different days because of the variances in daily travel patterns. We calculated the minimum battery range for each single day between March 3rd and March 9th, 2009 selecting the highest minimum battery range among the seven days as the recommended range. With this battery range, the driver is able to finish all the travel demand within the week. The distribution of the suggested battery ranges is presented in Figure 2.
16
To satisfy the travel demands of 90% of taxi drivers, a BEV with a full battery range of 220 miles is required, while satisfying 99.5% of travel demand requires a battery range of 330 miles.
Figure 2 The cumulative distribution function (CDF)of the suggested battery ranges for Beijing taxis
To study the different recharging behaviors of the two clusters, we compared the drivers’ recharging frequency on March 3rd, 2009. The recharging frequency is calculated as the number of times the drivers chose to recharge in a trip chain divided by the total number of stops (excluding home) in the trip chain. The drivers in different clusters have significant different charging behaviors: about 90% of the drivers in cluster 2 chose not to recharge at all during the day, while only 20% of the drivers in cluster 1 does not recharge (Figure 3). This result shows that drivers in cluster 1 is more proactive in charging the BEVs.
17
Figure 3: The empirical CDF of the percentage of stops to recharge
3.3
Minimum Battery Range for Private Vehicles
We also implemented the battery range minimization model using the historical travel data of private vehicles. Same as Section 3.2, we selected 7 days as the study period. For each driver, we determined the minimum battery range needed for each day and then picked the highest battery range within the 7 days as the recommended range. The CDFs of the suggested battery range for the private vehicles are presented in Figure 4, in comparison to that of the taxis. For the private vehicles, a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Comparing the histogram of the suggested battery range of taxis and private vehicles, we can see that the majority of the private vehicles only need a small battery range (Figure 5). However, compared to the suggested battery range for the taxis, the distribution of the minimum battery range for private vehicles has a long tail. A battery range of approximately 300 miles is needed to satisfy the travel demand of 90% of the private vehicle drivers according to the model. The travel patterns of some of the private vehicles make them unsuitable for BEV adoption, such as having long inter-city trips or traveling in the regions with low charging infrastructure support).
18
Figure 4: The empirical CDF of the suggested battery range
Figure 5: The histogram of the suggested battery range 3.4
Comparing the minimum battery range identified using simplified rules
Comparing to the minimum battery ranges determined using simplified charging rules for taxi drivers, we found that all rules, except for Rule 1 (time available to recharge exceeds 150 minutes) and Rule 3 (unit cost of recharging is less than ¥0.2/mile), underestimate the required battery range compared to the proposed model to satisfied one-day travel demand (Figure 6a). This result shows that EV drivers may not necessarily use every charging opportunity to recharge 19
the vehicle and may not only wait for long dwell time or cheap charging opportunity to recharge. Unlike the taxis, all rules overestimate the required battery range compared to the proposed model for private vehicles (Figure 6b). This result indicates that private vehicle drivers utilize more diverse charging opportunities than those described by the simple rules. Considering the complexity and heterogeneity of charging decisions is important in determining the needed battery range for BEVs.
a)
b) Figure 6: The minimum battery range estimated using different rules for (a) taxis and (b) private vehicles (Rule 1: Time available to recharge exceeds 150 minutes; Rule 2: Distance to the nearest charging station is less than 1 mile; Rule 3: The unit cost of recharging is less than ¥0.2/mile, considering both charging cost and parking cost; Rule 4: The maximum battery energy that can be recharged is greater than 15 miles; and Rule 5: The remaining battery energy at the next stop, if not recharging at current stop, is less than 15 miles.)
4
CONCLUSION AND FUTURE WORK
This study proposes a battery range optimization model to identify the minimum BEV battery range that can satisfy given travel demands for BEV drivers, considering the opportunities to charge at existing public charging stations and the uncertainties in charging decision making. To assess the probability of whether a driver recharges at each stop, we conducted a Stated Preference survey. Based on the collected data from the survey, the LC model is used to analyze and predict the different recharging behaviors of the two clusters of BEV drivers. The models were implemented using trajectory data of Beijing taxi and private vehicles. For the taxis in Beijing, BEVs with a battery range of 220 miles are able to satisfy the travel demand for about 20
90% of the drivers. For private vehicle, a range of 300 miles is needed to cover the travel demands of 90% of the drivers, while a 100-mile range battery is able to satisfy the need for 80% of the private drivers. Comparing the identified minimum required battery range from the proposed model and those using simplified recharging rules, our results show that most of the considered rules may cause underestimation of the minimum required battery range for taxis and overestimation of the range needs for private vehicles, showing the importance to consider the charging behavior in BEV models. These results also show that, with the existing charging infrastructure in Beijing, the benefits of providing policy incentives and technology development for BEVs with large battery sizes exceeding 220 miles may be limited. Simulation tools and models such as the one presented in this study can be helpful tools to enable individual consumer, industry experts, and policy makers evaluate the BEV range needs at the individual level, fleet level, and city scale, respectively.
While this study has the merit of being the first to consider stochastic charging behaviors in the BEV range optimization models, it has three major limitations that can be improved in future studies. First, we only conducted one set of survey and used the derived charging coefficients for both taxis and private drivers. Different driver groups may have different charging behaviors. Future studies obtaining the charging coefficients specifically for each driver group can make the model more accurate. Second, it is possible that the charging decision making derived from the stated preference survey is different from the actual charging decisions. Comparing the stated charging preferences to the revealed actual charging decisions can help better evaluate to what extent the survey results reflect the actual charging decision making. Lastly, because the problem is formulated as a MINLP, the model becomes hard to solve when the number of decision variables become too large (e.g., solving the model using one month’s trip data). Alternative ways to formulate the recharging probability may help improve the scalability of the model.
In summary, this study proposed a model and framework to integrate the charging behavior and real world public charging infrastructures into determining the minimum battery range needed to meet the given travel demands. The identified distribution of the minimum battery ranges to adopt BEVs can be used as inputs for BEV related studies (e.g., charging infrastructure
21
development, environmental impacts modeling, and vehicle design etc.) and inform decision making for policy makers, industry experts, and consumers.
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24
Highlights • • • • •
Proposed a Battery Range Optimization Model (BROM) Integrated heterogeneous charging behaviors into battery range optimization Evaluated the battery range needs for taxis and private vehicles in Beijing as case studies Simple charging rules may lead to overestimated battery needs for taxis Simple charging rules may lead to underestimated battery range needs for private vehicles
Declaration of interests ☒ 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: