Probabilistic forecasts of time and energy flexibility in battery electric vehicle charging

Abstract

Users charging the batteries of their electric vehicles in an uncoordinated manner can present energy systems with a challenge. One possible solution, smart charging, relies on the flexibility within each charging process and controls the charging process to optimize different objectives. Effective smart charging requires forecasts of energy requirements and parking duration at the charging station for each individual charging process. We use data from travel logs to create quantile forecasts of parking duration and energy requirements, approximated by upcoming trip distance. For this task, we apply quantile regression, multi-layer perceptrons with tilted loss function, and multivariate conditional kernel density estimators. The out-of-sample evaluation shows that the use of local information from the vehicle's travel data improves the forecasting accuracy by 13.7% for parking duration and 0.56% for trip distance compared to the data generated at the charging stations. In addition, the analysis of a case study shows that using probabilistic forecasts can control the interruption of charging processes more efficiently compared to point forecasts. Probabilistic forecasting leads up to 7.0% less interruptions, which can cause a restriction in drivers' mobility demand. The results show that charging station operators benefit from leveraging the driving patterns of electric vehicles. Thereby, smart charging and the application of the proposed models as benchmarks models for the related forecasting tasks is an improvement for the operators.

Full text

Probabilistic forecasts of time and energy flexibility in battery electric vehicle charging
Julian Hubera,b,∗∗, David Dannb,∗, Christof Weinhardtb,,
aFZI Research Center for Information Technology, Haid-und-Neu-Str. 10-14, 76131 Karlsruhe
bKarlsruhe Institute for Technology (KIT), Fritz-Erler-Str. 23, 76133 Karlsruhe
Abstract
Users charging the batteries of their electric vehicles in an uncoordinated manner can present energy systems with a challenge. One
possible solution, smart charging, relies on the flexibility within each charging process and controls the charging process to optimize
different objectives. Effective smart charging requires forecasts of energy requirements and parking duration at the charging station
for each individual charging process. We use data from travel logs to create quantile forecasts of parking duration and energy
requirements, approximated by upcoming trip distance. For this task, we apply quantile regression, multi-layer perceptrons with
tilted loss function, and multivariate conditional kernel density estimators. The out-of-sample evaluation shows that the use of
local information from the vehicle’s travel data improves the forecasting accuracy by 13.7 % for parking duration and 0.56 % for
trip distance compared to the data generated at the charging stations. In addition, the analysis of a case study shows that using
probabilistic forecasts can control the interruption of charging processes more efficiently compared to point forecasts. Probabilistic
forecasting leads up to 7.0 % less interruptions, which can cause a restriction in drivers’ mobility demand. The results show
that charging station operators benefit from leveraging the driving patterns of electric vehicles. Thereby, smart charging and the
application of the proposed models as benchmarks models for the related forecasting tasks is an improvement for the operators.
Keywords: charging coordination, demand-side flexibility, electric vehicles, probabilistic forecasts, smart charging
1. Introduction
The charging of battery electric vehicles (BEVs) challenges
the electricity grid for several reasons. First, the maximum
charging power is very high compared to other household ap-
pliances. Second, charging multiple BEVs often has a high si-
multaneity. To overcome these challenges, charging station op-
erators can apply smart charging. Smart charging adjusts the
charging power of individual charging processes by reducing
or postponing them. In this way, charging station operators can
exploit low prices on electricity markets (Huber and Weinhardt,
2018), provide system services (Staudt et al., 2018), or integrate
more renewable energy (Schuller et al., 2015).
Smart charging relies on flexibility in the charging demand.
Charging demand is flexible if the BEV’s parking duration at
the charging station is longer than the time needed to fulfil the
BEV’s energy demand. Ludwig et al. (2017) introduce the dif-
ference between two types of flexibility for industrial demand-
side management, i. e., energy and time flexibility. We transfer
this idea to the charging process of BEVs: Energy flexibility
is the difference between the minimal required state of charge
(SoC) and a fully charged battery. For instance, if the BEV
drivers are confident that their next trip’s energy demand does
not require a full SoC, they can provide energy flexibility by
∗Corresponding author
∗∗Principal corresponding author
Email addresses: julian.huber@fzi.de (Julian Huber),
david.dann@kit.edu (David Dann )
articulating that they do not require a full SoC at the end of the
charging process. Time flexibility results from the difference in
time required to reach the final SoC and the parking duration
between arrival and departure at the station. Higher flexibility
sets broader constraints to the optimization of the charging pro-
cess. For instance, reducing charging power to reach the same
final SoC over a more extended period allows flattening load
peaks.
To apply smart charging, the charging station operator re-
quires a definite valuation of the time and energy flexibility of
each charging process. Otherwise, the operator who uses smart
charging runs the risk to constrain BEV drivers’ mobility by
the time of departure because the SoC is not sufficient for the
BEV drivers to reach their intended destination. To mitigate the
risk of insufficient SoC, smart charging system operators mostly
rely on the users’ agreement to charge flexibly. Here, the users
accept that they can not charge the BEV immediately and con-
tinuously at full capacity until the BEV reaches its full SoC.
To do so, the users can communicate their flexibility by ex-
plicitly entering the desired SoC and planned time of departure
for each charging process. Alternatively, the users can define
profiles that fit their driving habits and can be adapted if nec-
essary (Huber et al., 2019b). Setting these defaults is not a
trivial task as drivers have to consider multiple requirements
and objectives, that could be aided by decision support systems
that provide forecast on the required SoC and parking duration
(Flath et al., 2012).
To assist charging station operators and drivers in this task,
forecasts could help to predict the charging flexibility and en-
Preprint submitted to Applied Energy January 9, 2020

Problem definition Gathering
information
Preliminary
(exploratory)
analysis
Choosing and
fitting models
Using and
evaluating a
forecasting model
Objective
Implementation Feature reduction
and model
development
Model selection
on via cross-
valdiation
Evaluation of
forecast results on
test-set and in
case study
Scheduling in
smart charging Feature
generation
Figure 1: Implementation of the basic steps of forecasting (Hyndman and Athanasopoulos, 2018) in this paper.
sure that the flexibility allowed by the system does not interfere
with the mobility needs. In particular, flexibility can be pre-
dicted by using forecasts for the expected parking duration and
energy requirement of the next trip.
Alternatively to the users setting their preferences, the charg-
ing station operator can use such forecasts to identify charg-
ing events with high flexibility in order to intervene without the
users noticing. The system must be designed conservatively, so
that the times it overestimates the parking duration or underes-
timates the energy requirement for the next trip are kept at a
minimum. To make conservative estimates, the forecasts have
to consider their uncertainty so that the charging system opera-
tor is confident not to control the inflexible charging events.
Meanwhile, various parties are positioning themselves as
charging station operators to take financial advantage of the
flexibility in electric mobility. Power grid operators want to use
the flexibility for system services and congestion management,
home-owners aim to maximize their self-consumption, whilst
utilities and public charging infrastructure operators want to
benefit from temporal changes in electricity prices. Also, man-
ufacturers of electric vehicles are pushing into the market of
charging infrastructure and energy trading, e. g., by offering
vehicle-to-home solutions (Weiller and Neely, 2014). Com-
pared to the other players, the vehicle manufacturers have an
important asset for determining charging flexibility: Global Po-
sitioning System (GPS) data of the vehicle provides them with
location data. This location data may enable a better prediction
of parking duration and energy requirements. At business level,
the question arises how big the advantage of this data is and
whether charging station operators should invest in the acquisi-
tion of this data or leave the field to the vehicle manufacturers.
Whilst previous scholars analyze the mobility patterns of car
users (Zumkeller et al., 2011) and flexibility potentials of differ-
ent BEV user groups (Schuller et al., 2015), to the best of our
knowledge there is no established forecaster that allows predict-
ing the flexibility of individual BEV charging events. Further-
more, it is still unclear what kind of data is required to predict
the time and energy flexibility of BEVs.
To address this striking research gap, this paper describes
the development of a quantile forecast for parking duration and
upcoming trip distance. We base our analysis on a German
data set of travel logs and follow the framework for forecast
development provided by Hyndman and Athanasopoulos
(2018) shown in Figure 1. First, we outline the requirement
for forecasts as an input for smart charging systems. Next,
we generate features from the travel logs to resemble the
information that is available to charging stations from historical
parking events or tracing users’ travel data with smartphone
applications. Next, we discuss feature selection reduction
and propose forecasters. Following, we select a model based
on cross-validation performance and discuss the forecast-
ing results on the test set. Finally, we show how charging
station operators can profit from using quantile forecasts to
solve congestion or provide flexibility an auxiliary market.
In doing this, we answer the following research questions (RQ):
RQ1: To what extent does location data improve quantile
predictions of parking durations?
RQ2: To what extent does location data improve quantile
predictions of trip distances?
RQ3: Can probabilistic forecasts help to reduce the number
of mobility impairment in BEV charging coordination?
We answer RQ1 and RQ2 comparing the cross-validated per-
formance of forecasters with and without location data. RQ3 is
answered by using the best forecast from the model selection
step to schedule BEV charging loads from the test set using a
greedy heuristic.
With this paper, we contribute by deriving interpretable fea-
tures that can assist charging station operators to forecast park-
ing duration and trip distance to predict BEV charging flexibil-
ity. We use these features to propose four forecasters that are
evaluated on an open data set and can serve as a benchmark
for further model development. The evaluation of the forecast
results shows that charging station operators should acquire lo-
cation data to improve forecast accuracy and use probabilis-
tic forecasters instead of point forecast when using real-world
scheduling problems.
The structure of the paper bases on the framework mentioned
above: After providing the problem definition in the first sec-
tion, we describe the related work on forecasting in BEV charg-
ing in Section 2. Next, we introduce the data and derive fea-
tures used for the evaluation in Section 3. Section 4 describes
the methodology, including the choosing and fitting of the fore-
2

casters. The forecast results are presented in Section 5. In Sec-
tion 6, we present a case study to evaluate the improvements
obtained by the forecasts. The paper closes with Section 7, dis-
cussing the results and a conclusion and practical implications
in Section 8.
2. Related Work
The increased number of BEVs is being accompanied by sci-
entists working on how to meet the challenges of increasing en-
ergy demand and the high simultaneity of charging processes.
In Xin et al. (2010), the authors discuss what kind of forecasts
are valuable to foster the integration of BEVs into the energy
system and charging station planning and operation. They men-
tion the forecast of electricity cost (i. e., electricity price and
charging demand), occupation of charging stations, and the de-
velopment of the electric vehicle population. Indeed, several
authors (e. g., Schellenberg and Sullivan 2011, Pl¨
otz et al. 2014,
and Gnann et al. 2015) try to predict the long-term develop-
ment of BEV diffusion for different countries. Given that these
are, by comparison, long-term forecast, covering years, such
forecasts are difficult to evaluate with real-world observations.
While these forecasts provide a valuable foundation for strate-
gic planning, their contribution to the day-to-day operation of
smart charging systems is limited.
On the contrary, Wi et al. (2013) describe a smart charg-
ing method for smart homes with photo-voltaic systems. Their
method relies on a time-series model for the feed-in of the
photo-voltaic systems and they assume perfect foresight on the
BEV charging. Similarly, Schuller et al. (2015) assume perfect
foresight on the mobility data provided by the German Mobility
Panel (Zumkeller et al., 2011) for scheduling charging to inte-
grate fluctuating renewable energies using a linear optimization
model. Their analyses show that different user types, e. g., re-
tirees and full-time workers, show different charging flexibil-
ity. Likewise, Sadeghianpourhamami et al. (2018) find three
distinct behavioural clusters in 387,524 BEV charging sessions
from the Netherlands. The behavioural clusters differ in the lo-
cation and the flexibility of the charging situation. They identify
parking to charge, e. g., on a long journey, charging near home,
e. g., overnight, and charging near work.
Table 1 lists work concerned with forecasting (upper half)
and simulating (lower half) of short-term charging demand.
While simulations do not try to predict real-world data, they
aim to provide realistic scenarios for BEV charging behaviour.
In contrast, the forecasts in the upper half are evaluated against
out-of-sample data.
While most papers forecast charging demand of an aggre-
gated BEV fleet, only a few (Bikcora et al. 2016, Ai et al.
2018) forecast demand or occupation of single charging sta-
tions or households. Most forecasts and simulations rely on
conventional trip data (i.e., from cars with internal combus-
tion engine), assuming that they share BEV users’ mobility pat-
terns. Only a few studies rely on rather small samples of BEV
charging (Bikcora et al. 2016, Ai et al. 2018). Forecasters are
different machine learning, time-series (Amini et al., 2015) or
explicit models of the charging behaviour (Xing et al., 2019).
Both the input features of simulations and forecasts of BEV
charging mainly rely on historic charging and parking patterns.
Driving patterns and SoC patterns are most often used in sim-
ulations, but not in forecasts. In contrast, some forecaster use
weather (Arias and Bae, 2016) or calendar data (Xydas et al.,
2013) as external variables.
As most forecasters predict the aggregated charging load of
BEV fleets and simulation results are not evaluated compared to
ground-truth data, the predictability of a single BEVs charging
behaviour is still unclear. The analyses mostly build on con-
ventional trip data or historic charging patterns, which provide
only limited information about the flexibility. While many sim-
ulations consider spatial mobility patterns as an essential input,
they seldom considered in the short-term forecast.
While there is no research on forecasting a single BEV’s
parking duration or trip distance and no insights on consider-
ing the uncertainty in such forecasts, there is vast literature on
probabilistic energy load and price forecasting (Hong and Fan,
2016). As the problem is relatively similar in terms of domain
and time frame, in the following, we rely on findings and meth-
ods from this domain.
The review of existing literature shows that forecasting en-
ergy demands and parking duration of BEVs has not received
enough attention. In particular, to our knowledge, there is no
research on forecasting a single BEV’s parking duration or trip
distance as a proxy for its flexibility. Besides, there are no in-
sights on considering the uncertainty in such forecasts as most
prior work focuses on aggregated charging loads using point
forecasts. Therefore, this paper provides charging station op-
erators with novel insights regarding three aspects. First, we
provide forecasts that allow assessing the flexibility of charging
events compared to mere load forecasts. Second, probabilis-
tic forecasts for individual parking events allow the account for
their uncertainty. Third, we evaluate the importance of histori-
cal driving patterns as an input feature for such forecasters.
3. Data
Evaluating the impact of location features requires a data set
of mobility patterns. We found only one publicly available data
set containing trip trajectories in Xing et al. (2019). As this data
stems from a ride-hailing company in China, it does not com-
pare to residential BEV users charging at home. To the best of
our knowledge, there are no public data sets of BEV charging
combined with location data of BEVs. For instance, Lee et al.
(2019) recently published an extensive data set of charging pro-
cesses of public charging stations in the US. However, this data
set does not provide information about the BEVs location data
and, consequently, is not suited to answer the posed research
questions. Hence, we use the data set provided by Zumkeller
et al. (2011)1that is also used in the analysis of Schuller et al.
(2015). The data set covers travel logs of 6.465 German car
users.
1available at at https://daten.clearingstelle-verkehr.de/192/
3

Table 1: Related work on forecasting (upper part) and simulation (below mid-rule) of battery electric vehicle charging.
Predicted Vari-
able
Entity Input Features Data Models
Arias and Bae (2016) Charging load BEV fleet Weather, Historical
load
Conventional trip data Decision Trees
Xydas et al. (2013) Charging load BEV fleet Calendar variables,
Historical load
Conventional trip data Support Vector
Machines
Amini et al. (2015) Charging load Parking lot Historical load Power system load
data, Assumptions on
BEVs
ARIMA
Bikcora et al. (2016) Charging load,
Occupation
Charging station Historical load of
close charging sta-
tions
Charging station data Generalized linear
models
Ai et al. (2018) Charging time Household Historical load 20 days of charging
data from one house-
hold
Random Forrest,
Naive Bayes, Ad-
aBoost, GBoost,
ANN
Xing et al. (2019) Charging load
(spatial)
BEV fleet Trip trajectories,
departure time,
assumptions
Conventional trip data Explicit model
Kisacikoglu et al. (2018) Charging load Charging station Historical load 20 BEVs charging
loads
Fitted distribu-
tions
Zhang et al. (2014) Charging load
(spatial)
BEV fleet Land Use, BEV
population, parking
behaviour
Conventional trip data Monte-Carlo Sim-
ulation
Wang et al. (2019) Charging load
(location type)
BEV fleet SoC, first travel,
driving time, driving
distance
Conventional trip data Monte-Carlo Sim-
ulation
Yi and Bauer (2015) Charging load
(spatial)
BEV Parking duration,
SoC, time of arrival
and departure
Conventional trip data Explicit model
Omran and Filizadeh (2013) Charging load
(spatial), park-
ing events
BEV fleet Drivers decisions,
SoC, parking du-
ration, driving
distance
Conventional trip
data, assumptions on
drivers’ decisions
Location-specific
fuzzy decision
making system
4

Table 2: Example of travel log data for two drivers.
Driver Mo 00:00 Mo 00:15 ... Su 23:30 Su 23:45
distance 1 0 0 ... 20 7
location 1 home home ... driving leisure
distance 2 5 0 ... 0 0
location 2 shopping shopping ... home home
(a) Parking duration in hours (b) Trip distance in km
Figure 2: Scatter plots for parking duration and trip distance at home in the data-set plotted by the time of arrival.
The car users recorded their mobility behaviour for a whole
week. The participants started logging their mobility patterns
on different days. In the end, the data was aggregated starting
on Monday. The time of data collection was selected so that
there were no holidays during the recording period. During the
week, the participants noted where they went, e. g., work, shop-
ping, home, the mode of transportation, and the distance trav-
elled by car. This data results in mobility diaries in 15 minutes
resolution. Table 2 shows an extract of a travel log. User 1, for
instance, is at home on Monday from 00:00 to 00:30. On Sun-
day 23:30 to 23:45, she is driving 20 km, in the following 15
minutes, she drives for seven more km and arrives at a leisure
activity.
As an input, we use these travel log as time series data. We
generate a new time series of parking events with entries at
each arrival of a car at home. This time series contains dif-
ferent users. As the time series for each user is rather short (one
week of data), learning weekly patterns to accurately forecast
the next week is unlikely. However, we assume that users’ mo-
bility behaviour follows a weekly pattern. Therefore, we use
the whole data set of all users to train the model. The result-
ing time series contains all parking events in the data set with
the ID of the user and lagged features of the specific user (e. g.,
previous parking location).
While we cannot learn and forecast the weekly behaviour for
a single user, we assume that there are similarities within the
user types (as found by Schuller et al. (2015)) from which the
forecasters may learn. Consequently, we decide to derive ag-
gregated features for single parking events, as described in the
following. Another merit of this procedure is that the data sam-
pling for training, validation, and testing is independent of the
overall temporal order of the data. This allows us to use cross-
validation and an evaluation of the models’ performance over
the whole week (and not only the end of each week). For the
cross-validation, we use data from the end of the week to train
models that predict events that precede the training data. While
some researchers propose rolling-origin evaluations, to simu-
late a realistic situation, where all the training data precedes the
validation and test data (Tashman, 2000), Bergmeir and Ben´
ıtez
(2012) show that this has no adverse effect on the out-of-sample
performance.
For further analyses, we make the following assumptions.
We assume that the mobility patterns of conventional car users
are the same for BEV users. Pasaoglu et al. (2014) analyse the
potential of mobility surveys of conventional car owners to sup-
port studies on the impact of electric vehicles. They do not men-
tion any differences in conventional car owners or BEV drivers.
With insufficient BEV data available, most studies in Table 1
use trip data from conventional cars. As most trips are rather
short, 150 km range could cover 95 % of trips of German resi-
5

dential car users (Lunz and Sauer, 2015). Therefore, we see no
reason why BEV drivers would (have to) behave differently.
However, we cut offthe longest trips at 350 km, which is
the range of a current Tesla Model 3 (Fiori et al., 2016). Apart
from that, we apply no further cleansing as we see no unre-
alistic outliers in Figure 2. We next restructured the data, so
that one observation represents a parking event, i. e., when a
car is not used for driving. Note that a parking event is usu-
ally the moment when a BEV is parked and connected to the
charging station. As most private BEV users charge their car at
home (Hardman et al., 2018), we reduced the data to the park-
ing events at home for the following evaluation. This resulted
in a total of 38.086 parking events at home. We further derived
the set of features listed in Table 3 for all parking events.
For each parking event of index i, we make a forecast based
on the data available on the start of the parking event at the time
of arrival tabeing the forecast origin. Note that there are two
variables of interest when scheduling the charging of a BEV
by predicting its flexibility. First, the parking duration dideter-
mines the time flexibility by giving the time frame that is avail-
able for charging the BEV. The longer the BEV is parking at
the charging station, the higher the time flexibility and opportu-
nity for load shifting. Next, the user requires enough charge to
reach the next charging station. The survey conducted by Hu-
ber et al. (2019b) shows that BEV drivers do not always require
a full SoC after each charging process but are willing to pro-
vide flexibility in general. Therefore, the charging process also
has certain energy flexibility, i. e., the difference between 100%
SoC and the SoC required for the next trip, which is highly cor-
related with the trip distance liof the next trip. Next, we aim to
forecast trip distance and parking duration to improve the inputs
for smart charging systems.
As mentioned above, different forecasters might have ac-
cess to different amounts of data. The charging station oper-
ator knows the parking duration and energy requirements of
the past charging processes at the station for billing. For ac-
counting, the operator will usually also have an identification
of the BEV driver, which enables the operator to learn from the
driver’s behaviour patterns. With one week of data, we only ob-
serve around five parking events per driver. As this is too short
to learn distance patterns of a single user, we use the user type
as an input feature. This feature set τ, available to the charging
station operator, is denoted as τstation in the following.
We define location features as information that requires
knowledge on the BEVs position over time, e. g., retrieved by
GPS data. This information is available for vehicle manufac-
turer by on-board GPS or can be obtained by the charging sta-
tion operator by tracking the BEV driver via a smartphone ap-
plication. As many people already share GPS data with applica-
tion providers via their smartphone with little privacy concerns,
we assume a charging station operator could obtain such data
by its applications.
From the travel logs, we obtain the following location fea-
tures: the last destination before arriving at the station (imple-
mented as a category e. g., work, shopping, leisure), duration
and distance of the last drive to the parking position, and the
number, distance, and duration of previous trips of the same
calendar day. The complete feature set, including location data,
is denoted as τall.
In the way we derived features from the travel log, an arbi-
trary number of further features can be derived. We tested fur-
ther features in pretests and evaluated them in terms of out-of-
sample forecasting accuracy, but they did not improve models’
performance significantly. Consequently, we remained with the
features described in Table 3 as further features render under-
standing and interpreting the results more difficult.
Figure 2 presents the parking duration and trip distance for
all parking events in the data set over a day. The histogram
on the x-axis shows that most parking events at home occur
either around noon or in the evening. Most parking durations
(left graph) are either rather short (below one day). Only a few
parking events cluster around multiples of 24 hours, represented
as the horizontal bands in the graph. These bands drop to the
right, as cars that arrive later often still depart at similar times
in the next morning. The trip distance in the right graph does
not show such a distinctive characteristic. Most trips starting
at home are rather short (below 50 km). With the same data
set, Pl¨
otz et al. (2017) show that this distribution of trip length
in Germany corresponds to a Weibull distribution. In so far,
charging processes with high flexibility (promising for smart
charging) are the ones with high parking duration and short trip
distances. The first group is in the upper segment of the left
graph, where the time flexibility is high as only a portion of the
parking duration is needed to charge the BEVs. The latter are
the processes in the lower section of the right graph that have
high energy flexibility as only little SoC is needed to fulfil the
energy requirements of the upcoming trip.
4. Methodology
Next, we develop promising forecasters to identity the most
flexible charging processes. As we find no benchmark models
for this problem in literature, we decide to use forecasters used
in probabilistic (electricity) load forecasting, which is, besides
price forecasting, the most common forecasting task in the en-
ergy domain (Hong et al., 2016).
In probabilistic forecasting, a forecaster (model) Fdoes not
aim to predict a single value ˆy, but the distribution of the fore-
casted value ˆ
P(Y<y). The forecaster, hereby, often relies on
a set of features τthat have some predictive power on the fore-
casted variable y.
F:τ→ˆ
P(Y<y).
In many cases, the forecasters do not predict a complete prob-
ability distribution P(Y<y). Instead, they predict a selected set
of quantiles Q. A real number qais an a-quantile of Pif
P((−∞,qa]) ≥aand P([qa,+∞]) ≥1−a.
In this way, many forecasters predict a set of quantiles Qto
approximate the probability distribution ˆ
P(Y<y). The quantile
prediction allows accounting for the uncertainty in the forecast.
Having information on the expected probability distribution is
6

Table 3: List of predicted variables and location and input features.
Feature Unit Description τall τstation
i1 index of the parking event
ta
itime start of the parking event
Trip distance km length of the next trip following the starting event ˆ
l|i, τall ˆ
l|i, τstation
Parking duration min duration of the parking event ˆ
d|i, τall ˆ
d|i, τstation
Hour of arrival [1, 24] at start of the parking event
Weekday of arrival [1, 7] at start of the parking event
User [1, 6.465] id of the user
User type [1, 6] occupation of the user: full-time, half-time, retiree,
education, homekeeper, unemployed
Previous parking location [1,7] last destination before arriving at home (work,
home, shopping, service, leisure, vacation, business
trip)
Previous trip duration min duration of the last drive to the parking position
Previous trip distance km distance covered by the last drive to the parking po-
sition
# Previous trips 1 number of previous drives of the calendar day
PPrevious trips duration min duration of all previous drives of the calendaric day
PPrevious trips distance km distance covered by all previous drives of the calen-
daric day
a valuable property in various use cases. For instance, they pro-
vide more information in decisions under uncertainty, as they
provide information on the interval in which the realized out-
come lies within a given probability. Especially for scheduling
BEV charging, it is not only essential to know whether the start-
ing time will be short or long, but also whether the forecaster
is confident in its forecast. To derive a forecast for parking du-
ration and trip distance, we follow the framework outlined in
Figure 1:
After prepossessing (described in previous chapter), we shuf-
fle and split the data randomly in two parts. 85 % for train-
ing and model selection (training and validation set) and a
15 % hold-out sample for final evaluation (test set). For model,
parameter, and feature selection, we perform five-fold cross-
validation by splitting the 85 % training and validation set in
five different folds. By choosing five-fold cross-validation, the
validation sets have a similar proportion of the total data as the
test set (17 % compared to 15 %).
To answer RQ1 and RQ2 regarding the value of location in-
formation, we differentiate two feature sets noted in Table 3: A
set τall that contains all features including location information
and set τstation that only includes data available at the charging
station. We compare the cross-validation results for the fore-
caster with the smallest average difference in performance be-
tween two feature sets τall and τstation.
To substantiate the results of the model selection, we then
evaluate the results on the hold-out test set. In contrast to
measures for point forecasts e. g., Root Mean Squared Error
(RMSE) or Mean Absolute Percentage Error (MAPE), pure ac-
curacy measures for probabilistic forecasts are usually hard to
interpret, as they do not provide an average or percentage devia-
tion. Moreover, the framework of Hyndman and Athanasopou-
los (2018) prescribes to use and test the forecasts in a use case.
Following this framework, we apply the trained forecasters in a
case study scheduling the interruption of BEVs’ charging pro-
cesses. We use these results to answer RQ3 on the value of
probabilistic forecast in BEV scheduling problems. In the fol-
lowing section, we first describe error measures used as criteria
for model selection. We then present the selected feature sets
and forecasters.
4.1. Selected Features
Some of the features listed in Table 3 contain similar infor-
mation. For instance, the previous trip duration and distance
should show a high correlation. As some forecasters might
perform better, when the set of input features is reduced, e. g.,
quantile regression (Guyon and Elisseeff, 2003), we consider
performing a feature selection step for both forecasted vari-
ables. For both forecasted variables and feature sets (τstation and
τall), we perform a lasso regression (Ludwig et al., 2015) on the
training and validation set. As the results of the lasso regres-
sion show non-zero coefficients for all features and forecasted
variables, we decide to omit the feature reduction. The decision
to do so is also motivated by the fact that the feature selection
is based on a point forecast, while the final models aim at prob-
ability density forecast. A feature may have little information
on the average parking duration or trip distance, i. e., not im-
proving the point forecasts accuracy, but high information for
predicting other segments of the probability density.
4.2. Error Measures
Point forecasts in energy forecasting are mostly evaluated in
terms of RSME or MAPE (Hippert et al., 2001). While the
7

RSME expresses the error in the unit of the forecasted variable,
MAPE is a relative error measure that somewhat allows com-
paring the forecasters’ performance on different data sets. The
MAPE is calculated as:
MAPE =1
n
n
X
i=1
|ˆy−y
y| · 100%.
As Figure 2 shows, the distribution of data is very skewed
in both parking duration and trip distance. To provide a error
measure insensitive against very high observations, i. e., out-
liers being parking events with very the long parking duration
and trip distance, we follow the recommendation by Armstrong
and Collopy (1992) and also use the median of the Absolute
Percentage Errors (MdAPE):
MdAPE =median|ˆy−y
y| · 100%.
On the contrary, the evaluation of probabilistic forecast is
less straight forward. Gneiting and Katzfuss (2014) and Gneit-
ing and Raftery (2007) discuss prediction spaces, calibration,
and sharpness as possible evaluation criteria and provide guide-
lines for proper scoring rules. Such scoring rules assign a nu-
merical score S(F,y) to the probabilistic forecast Fand the ob-
served value y. They propose the Continuous Ranked Proba-
bility Score and Dawid–Sebastiani Score as a more practical
alternative.
Meanwhile, the energy load forecasting community estab-
lishes the Pinball Score a quasi-standard by using it in the eval-
uation of the global energy forecasting competition (Hong et al.
2016, Hong and Fan 2016). Accordingly, we decide to use the
Pinball Score instead of the other available measures mentioned
before. The Pinball loss for each quantile qais calculated by
La(qa,y)=((1 −a
100 )·(qa−y),if y<qa
a
100 ·(y−qa),if y≥qa.
This definition results in a tilted loss function resembling the
trajectory of a pinball hitting the barrier having the following
properties. If the observed value matches the predicted quan-
tile, the loss function is zero for this quantile and non zero for all
other quantiles. For the median (50 % quantile), the loss func-
tion becomes symmetric, returning half the absolute error. In
case the observed value is higher than predicted quantile value,
the Pinball loss Lareturns a high penalty, especially for high
quantiles.
We use the MAPE and MdAPE to evaluate the point fore-
casts being the prediction of the 50 % quantile. The Pinball
Score is the average Pinball loss Lafor all considered quan-
tiles in Q. For the probabilistic forecasts, we use the Pinball
Score for two reasons. First, it resembles the loss function in
quantile regressing, which is a forecaster often used to generate
quantile predictions. Second, Pinball Score is well established
and well understood in the energy forecasting domain, which
makes it easier for fellow researchers to interpret and compare
the results.
4.3. Selected Forecasters
In the following subsection, we describe the selected fore-
casters. These forecasters are based on models found in the
energy forecasting literature, e. g., wind, solar, load, price. For
each forecaster, we briefly describe the general idea and ap-
plications in energy forecasting. We further describe how we
implement the forecaster for our evaluation.
4.3.1. Naive Benchmark
Forecasting often uses a simple (naive) model to evaluate the
improvements achieved by the introduction of the more sophis-
ticated models developed. For point forecasts of time series, a
standard naive benchmark is to use the last observed value as a
forecast for the next point in time (Hyndman and Athanasopou-
los, 2018). Similarly, the mean observation in the training set
can be used as a forecast. Analogously, we use the historical
distribution, i. e., quantiles, in the training set as a forecast for
the distribution for each observation in the test set. The his-
tograms on the right in Figure 2 resemble these naive forecast-
ers.
4.3.2. Quantile Regression
Koenker and Bassett (1978) introduced quantile regression
in 1978. Quantile regression extends the ordinary least-squares
estimation of conditional mean models to allow for the esti-
mation of an ensemble of models for several conditional quan-
tile functions. Since then it has become a standard tool for
forecasting uncertainty in the energy domain (e. g., in Kaza
2010, Hammoudeh et al. 2014, Hong et al. 2016, or Liu et al.
2017). For our analysis, we use the implementation provided
in Python package statmodels (Seabold and Perktold, 2010).
In the following, we name the forecasters based on the model
and the set of input features QRτ(Quantile Regression), M LPτ
(Multi-Layer Perceptron), K DEτ(Kernel Density Estimator),
and Naive.
4.3.3. Quantile Regression with Multi-Layer Perceptron
The approach of quantile regression has been adapted by us-
ing similar loss functions within artificial neural networks to
model non-linear relationships (Taylor 2000, Cannon 2011) and
quickly spread into the load forecasting community (Hippert
et al. 2001, He et al. 2016). In our analysis, we rely on the
implementation provided by Abeywardana (2018). To answer
RQ1 and RQ2 on feature importance, we do not require the best
forecast possible. Instead, we aim to find whether the addi-
tional features improve the forecast independent from the fore-
caster used. Consequently, we keep the neural network reason-
ably simple. We do not put much effort into the optimization
of hyper-parameters, i. e., number of hidden nodes and layer,
selection of activation function, and feature representation. As
we use separate models for each quantile, the artificial neural
network has a single output neuron predicting the quantile. We
implement the neural network using Tensorflow (Abadi et al.,
2016) and Keras (Chollet et al., 2015) by defining a tilted loss
function that is the Pinball loss L(qa,y) (defined above).
8

Based on a grid search in the cross-validation, the neural net-
works consist of two hidden layers with a number of units re-
sembling the minimum of the number of input features or at
least twenty. Such shallow networks are usually sufficient for
comparable forecasting tasks (Park et al., 1991). We use a linear
rectifier unit as an activation function (Li and Yuan, 2017):
f(x)=max(0,x).
The network is trained using the Adam algorithm (Kingma
and Ba, 2014). To prevent over-fitting, training stops if the val-
idation error does not improve for 10 epochs.
4.3.4. Multivariate Conditional Kernel Density Estimator
On the contrary to quantile regression, Kernel Density Es-
timators (KDE) provide a full probability density (John and
Langley, 1995). The estimated density function bases on the
aggregation of kernels (often Gaussian distributions) based on
historical data. An essential parameter in fitting such models is
the bandwidth of the kernels influencing the smoothness of the
fitted distribution. Conditional KDEs expand this concept to
picture the conditional distribution given the realization of other
continuous or discrete input variables. They have been used as
a forecaster for smart meter data (Arora and Taylor, 2016) and
wind power (Jeon and Taylor, 2012). We use the implementa-
tion provided by the Python package statsmodels (Seabold and
Perktold, 2010). This package also includes bandwidth selec-
tion heuristic (Bashtannyk and Hyndman, 2001) that we use in
training the forecasters.
5. Forecasting Results
In this section, we first describe the results of the model se-
lection step in the cross-validation and evaluate the value of
local information answering RQ1 and RQ2. We further check
the results of the model selection step by presenting the results
on the hold-out test set.
5.1. Cross-Validation
Figure 3 and Table 4 show the results of the five folds in
the cross-validation. The dots in Figure 3 indicate the out-of-
sample performance of the forecasters on different data splits.
The out-of-sample performance results from the average Pin-
ball Score of all predictions in the concerning hold-out sam-
ple. The hold-out sample is either one of the five validation sets
from the cross-validation or the test set (see Section 4). For the
parking duration, the M LP shows the best results on average.
The QR also outperforms the naive benchmark for both feature
sets. The K DE, however, does not achieve good results and per-
forms worse than the naive benchmark. The variance in the Pin-
ball Score of the KDEτstat ion with the smaller feature set is fairly
high compared to the quantile regression models. For every
single forecaster (QR,MLP,K DE), the forecast with the local
information τall outperforms the forecast using only station data
τstation. Table 4 shows that Pinball Score is a good indicator of
the MdAPE. Forecasters with a low Pinball Score also show a
low MdAPE, around 30 % for the best forecasters. The MAPE,
on the other hand, is very high (around 300 %). The high values
are likely caused by the occurrence of high values in the distri-
bution, i. e., the occurrence of very high parking durations, and
cannot be clearly associated with the Pinball Score.
There is an underlying similarity in the pattern for the pre-
dictions of trip distance. On average, QR and K DE outperform
the naive benchmark. However, the improvements of both QR
forecasters and M LPτstation using only station data are very small.
The MLPτall with the full data set appears to be better on aver-
age than the other models. The model fitted and predicted the
data very well on four of the five cross-validation folds (low
bias). In contrast, it does not manage to predict the data well
in the fifth fold (high variance). As the trip distance is limited
to 350 km (maximum range and therefore, maximum charg-
ing requirement BEVs) in preprocessing, the distribution is less
skewed (see Figure 2). As a result, MAPE (around 100 %) and
MdAPE (around 60 %) are lower than those for trip distance.
5.2. Value of Location Information
To answer RQ1 and RQ2 on the value of location informa-
tion, we investigate the forecaster (QR) with the smallest differ-
ence in average Pinball Score between the full feature set τall
containing local information and the reduced feature set τstation.
We test the null hypothesis that there is no difference in fore-
casting accuracy, i. e., Pinball Score, between the same fore-
caster using the different feature sets:
H0:LQRτstation =LQRτall .
As the Pinball Scores are not normally distributed, see Fig-
ure 3, and the samples are not independent, the data does not
fulfil the assumptions for t-test or ANOVA. Consequently, we
test the hypothesis using a Wilcoxon signed-rank test. This test
requires paired samples, chosen at random, and a variable on
an interval or ordinal scale. Wilcoxon test’s requirements are
met, as the cross-validation sets are created by shuffling the
data. We compare the results of the same folds (one to five)
and the Pinball Score provides an ordinal scale. For both, park-
ing duration and trip distance, the test results in a test statistic
of V=15, rejecting the null hypothesis at a confidence level of
p>0.03125. Consequently, we answer RQ1 and RQ2: The use
of location data improves quantile predictions of both parking
durations and trip distances.
5.3. Model Selection
On a practical level, the user of the forecast must decide what
forecast to rely on an unseen data set. Typical criteria for model
selection are the complexity of the learned classifier, general-
ization accuracy on new examples, and the amount of training
data available and needed to achieve high accuracy (Engle and
Brown, 1986). The amount of training data available is usually
given ex-ante for each forecasting task and cannot be changed.
Less complex models are often easier to interpret and less likely
to overfit the training data (compare QR and MLP results for
trip distance). Meanwhile, high accuracy on an out-of-sample
fit, e. g., using cross-validation, is usually the most important
criterion in established literature (Hong et al. 2015, Hippert
9

(a) Parking Duration (b) Trip Distance
Figure 3: Cross-validation and test set performance of different forecasters evaluated using Pinball Score.
et al. 2001). This factor can be directly measured in the out-of-
sample performance of the model. Often, the model selection
is not based on statistical analysis but the rule of thumbs. For
instance, ridge regression chooses the model with the lowest
complexity within one standard error of the best model (Hastie
et al., 2009) while Hong et al. (2015) test for the right weather
station combinations to forecast electricity load and selects the
set with the highest out-of-sample accuracy. Given the box-
plots in Figure 3, M LPτall reliably outperforms all other fore-
casters and should be selected. For trip distance, MLPτall also
has the lowest average Pinball Score. However, it shows poor
performance on the fifth fold. Therefore, we decide to use the
second-best forecaster ORτall .
5.4. Model Performance on Test Set
The evaluation results on the test set in Figure 3 and Table
4 confirm the results of the cross-validation. For parking dura-
tion, the test set results are better than the average in the cross-
validation for all forecasters but KDEτall . The selected model
MLPτall remains to be the model with the lowest Pinball Score.
In the trip distance, there is a small worsening for the selected
model QRτall . However, it still remains the second-best per-
forming model in the test set. The best model MLPτall shows
a substantial improvement, as it is compared to the average of
cross-validations (containing the outlier in the fifth fold). The
MLP also seems to profit from the larger amount of training
data as the model is retrained on the set that was split in train-
ing and cross-validation before. On the test set, the relative
improvement in Pinball Score achieved using the location data
in MLPτall compared to M LPτstat ion is 0.56 % for trip distance
and 13.7 % for parking duration.
5.5. Error Analysis
Figures 4 a and b show the distribution of the Pinball Score
for different days of the week predicted with ORall. Each vi-
olin shows the kernel density estimation and box plot of the
errors (i. e., Pinball Score) in the test set for different weekdays.
For both parking duration and trip distance, the Pinball Score is
rather low for most predictions. However, there are a few obser-
vations, where the prediction is very far off. There is no clear
pattern for these outliers over the week. Predictions of park-
ing duration (Figure 4 a) are, on average, worse on Fridays and
Saturdays showing a substantial number of poor predictions on
Fridays. For trip distance (Figure 4 b), the highest variation in
forecasting errors is on Sundays with no evident patterns in the
average values over the week.
As we see no clear pattern for the outliers in the day of week
and time of day, we look into the relationship between forecast-
ing error and the observed values in Figures 4 c and d. The
figures show the Absolute Percentage Error (APE) in red and
Pinball Score (grey) for all the observations in the test set. The
APE compares the prediction of the 50 % quantile with the ob-
served value, while the Pinball Score compares all predicted
quantiles to the observation.
For both parking duration and trip distance, we see very high
APE values when the observed variable assumes small val-
ues. The high error values are an artefact of the error mea-
sure: For instance, if trip distance or parking duration is very
short (e. g., 15 min) and the prediction is close to the average
parking duration around 500 min, this results in high APEs (for
instance: (500 min-15 min)/15 min =3,233 %). For higher
observed values, the APE gets better for both trip distance and
parking duration.
Meanwhile, we see an increase in Pinball Score with higher
observations for both predicted variables. Here, the relative in-
crease in Pinball Score is higher for parking durations (where
the variance in the data is higher (see Figure 2).
6. Case Study
For the case study, we assume that a charging station oper-
ator can control the charging of all BEVs in the test set. Fig-
10

Table 4: Average cross-validation and test set performance of different forecasters.
Forecasted
Variable
Model Cross-Validation Test Set
Pinball Score MAPE [%] MdAPE [%] Pinball Score MAPE [%] MdAPE [%]
Parking Duration
Naive 202.10 381.05 40.15 198.15 394.22 40.54
QRτstation 180.20 302.85 32.14 177.52 302.36 32.29
QRτall 179.35 301.97 32.13 176.84 302.49 31.91
MLPτstation 176.89 307.03 28.18 173.96 304.20 28.41
MLPτall 174.77 295.18 30.55 171.91 299.37 29.83
KDEτstation 218.66 335.83 49.19 205.10 425.42 38.43
KDEτall 208.82 375.18 40.89 209.01 511.73 42.56
Trip Distance
Naive 4.97 110.15 60.76 4.95 108.17 60.00
QRτstation 4.89 111.54 63.84 4.86 108.25 62.50
QRτall 4.53 116.91 56.94 4.54 120.23 56.48
MLPτstation 4.88 110.37 62.54 4.85 110.29 63.10
MLPτall 4.92 99.83 57.33 4.19 103.35 45.42
KDEτstation 5.77 190.51 100.38 5.41 137.50 66.67
KDEτall 5.53 160.84 74.58 5.54 161.34 73.26
(a) Distribution of Pinball Score for parking duration for different days. (b) Distribution of Pinball Score for trip distance for different days.
(c) Pinball Score and APE for parking duration in relation to observed values. (d) Pinball Score and APE for trip distance in relation to observed values.
Figure 4: Forecasting errors in the test set predicted with QRall .
11

Figure 5: Aggregated number of charging processes (connected BEVs) at each
time-step in the test set.
ure 5 shows the aggregated number of assumed charging pro-
cesses in the test set. Most charging processes at home start in
the evening from Monday to Thursday. On Friday and Satur-
day, most charging processes start around noon. The test set
only contains a few charging processes starting on Sunday. If
no charging coordination is applied, the peak in numbers of
charging processes will likely lead to a peak in electricity con-
sumption, which can cause grid congestion on distribution level
(Salah et al., 2015). One way to avoid congestion is to inter-
rupt some of the charging processes in the BEV charging port-
folio. For example, the German legislator proposes to regis-
ter BEV charging stations as interruptible consumer facilities
(EnWG §19). For the distribution system operator (DSO) to
interrupt them if necessary, the station operator benefits from
reduced network charges. Other reasons for the operator to in-
terrupt some of the charging processes at point tcould be to
provide auxiliary services or the optimise against fluctuations
in the energy markets.
In the case study, we assume that a charging system operator
can control the charging of all the BEVs in the test set I. At each
point tin the test set, the charging station operators can decide
to interrupt an arbitrary number nof the running uncontrolled
charging processes It. For each tand n, the operators has to
decide for an order to interrupt the charging processes based
an a interruption heuristic Λ. The set of the first ninterrupted
charging processes at time tusing Λis It,n,Λ.
The BEVs are charged directly after arriving at home with
the maximum charging power of 2.3 kW, i. e., a CEE 7/4 plug
AC household socket charger outlet. To fulfil its driver’s mo-
bility needs, each BEV has to charge enough energy to last the
next trip assuming an energy consumption of 20 kWh/100 km
(United States Environmental Protection Agency and U.S. De-
partment of Energy, 2016). All assumptions are laid out in Ta-
ble 5.
This uncontrolled charging results in an uncontrolled charg-
ing period between ta
iand tu
i:
tu
i=ta
i+ηBEV ·li
Cmax .
Table 5: Assumptions for charging infrastructure, interruption duration, and
energy consumption in the case study.
Assumption Unit
Energy consumption ηBEV 20 kWh/100 km
Maximum Charging Power Cmax 2.3 kW
Duration of interruption ι60 min
6.1. Interruption Heuristic
Interruption strategies for BEV charging could follow on the
intuition of time and energy flexibility: As long as parking du-
ration is long enough to charge the BEV after the interruption,
the interruption has no negative influence on the mobility of
the BEV driver. Not impairing the mobility is very likely for
charging processes with the following characteristics. If the
next trip is rather short, the energy flexibility is high as only
a small amount of energy has to be procured during parking
duration. If the parking duration is long and time flexibility is
high, there is enough time to charge even a substantial amount
of energy. In contrast, interruptions leading to insufficient SoC
at the departure impair the mobility of the driver.
Interrupting the charging processes with the highest time or
energy flexibility can be implemented by different heuristics).
For instance, a first-come-first-out heuristic ΛFiFo could start
with interrupting the charging processes of the BEV that arrived
first.
In addition the charging station operator can use the forecast
for parking duration to estimate the quantile forecast of the time
of departure td
ifor each charging process:
ˆ
td
i,q=ta
i+ˆ
di,q.
We derive the forecast for the parking duration ˆ
difrom
MLPall. The charging station operator can now apply Λˆ
tby
interrupting the BEV with the highest remaining parking du-
ration first. To account for energy flexibility, we also predict
trip distance ˆ
li,qusing MLPall. Using Λˆ
l, the charging station
operator starts with interrupting the charging processes that are
likely to be followed by a short trip and therefore require little
time for charging. As benchmark, we also evaluate interrupting
the charging processes by drawing at random ΛRandom.
The charging station operator uses a heuristic Λ, described
in Algorithm 1 to interrupt a given number ntout of the cur-
rently running charging processes Itat point tin the test set.
Algorithm 1 results in a list of interrupted charging processes
IΛ
t,n.
Data: It,n,Λ;
Result: IΛ
t,nlist of interrupted charging processes;
initialization;
sort Itusing interruption heuristic Λ;
set IΛ
t,nthe first ncharging processes from It;
return IΛ
t,n;
Algorithm 1: Interruption heuristic Λfor interrupting n
charging processes in It
12

Each interruption in IΛ
t,ncan result in an insufficient charge
at departure if the duration of the interruption ιis so long that
not enough SoC can be charged at capacity Cmax to fulfil the
next driving event of trip distance liat td
i. Therefore, a mobility
behaviour µ(i, ι) of a charging process is impaired:
µ(i, ι)=




1 if (td
i−ta
i−ι)·Cmax <li·ηBE V
0 (td
i−ta
i−ι)·Cmax ≥li·ηBE V .
This assumes that BEVs arrive at the charging station with
a SoC of zero and that the BEV can be charged at their next
stop, so they do not get stranded there. While this assumption
might is not realistic, it allows for a fair comparison between
the heuristics as no other factors, i. e., SoC at arrival or location
of the next stop, biasing the results.
This allows to count the number impaired mobility events z
for interruption ncharging processes in tusing Λby:
z(t,n, ι, Λ)=X
i∈IΛ
t,n
µ(i, ι).
6.2. Evaluation Scenario
We apply the interruption heuristics to 5.709 parking events
in the test set. For each point 15 min time slot tin C, we de-
termine the number of running charging processes Itand count
the number of impaired mobility events z(t,n, ι, Λ) when inter-
rupting ncharging processes at t. For each time step, we apply
all interruption heuristics Λto interrupt one up to all running
charging processes. Note that for most time slots in the test set,
there are a number of charging processes that cannot be fully
charged even if uninterrupted.
To compare the performance of the different charging heuris-
tics, we calculate the quotient Zof impaired mobility events
and interrupted charging processes for interruption of one to all
charging processes for each point in time in the test set C:
Z(Λ)=1
672 X
t∈C
1
ntX
n∈It
z(t,n, ι, Λ)
nt
.
6.3. Case Study Results
This results in the share of mobility impairment Zplotted in
Figure 6. Drawing at random using ΛRandom impairs 25.1 % of
mobility events. The first-in-first-out ΛF iFo improves the share
of impaired mobility events down to 17.5 %. Using the pre-
dicted trip distance as a heuristic Λˆ
loutperforms the random
benchmark, but never outperforms ΛFiFo no matter which quan-
tile qis used.
There is a clear pattern in using Λˆ
td. The prediction of the
upper quantiles does not improve the scheduling of interrup-
tions. Using the median prediction (50 % quantile) result in a
higher interruption rate than the ΛFiF o and Λˆ
l. However, using
the lower quantiles (10-30 %) improves the scheduling down to
a 16.5 % interruption rate.
Using the 10 % quantile prediction ΛQ10 improves the rate of
impaired mobility events to 16.5 %. The relative improvement
of 7.0 % by using quantile predictions compared to point fore-
cast answers RQ3: Using probabilistic forecasts is beneficial in
BEV charging coordination.
Figure 6: Performance of interruption heuristics on the test set based on differ-
ent decision variables.
7. Discussion
This section starts with discussing the forecasts performance
and error analysis. We next address the findings from the case
study.
7.1. Forecasting Accuracy
All evaluated forecasters, except for conditional kernel den-
sity estimators, provide more accurate forecasting results com-
pared to the naive benchmark. In this way, even charging station
operators without location data may profit from the forecasters
developed in this paper. While the improvement of including
location features is statistically significant, it is quite low. How-
ever, the difference might further increase with better data (e.g.
trip trajectories).
However, even the limited location information in the data
set can improve such forecasts. We assume that forecasts can
be even more accurate if more data about the single users his
behaviour is available. In this case, other forecasters like neural
networks with long short-term memory could help to make use
of this additional temporal information.
As discussed before, MAPE and MdAPE are sensitive to-
wards low observation values in the data. The analysis of the
Pinball Score indicates that the accuracy drops with high obser-
vations of parking durations. Such observations are very sparse
in the data and probably not sufficiently considered in the model
as the models stop at predicting the 90 % quantile. Hence, there
are around 10 % of observations that are higher than the pre-
dicted 90 % quantile. Some of them can be much larger than
the predicted 90 % quantile resulting in a high Pinball loss. This
effect does not occur when predicting trip distance, where the
distribution is less skewed. However, this is not a problem for
most smart charging use cases where it is not crucial whether
the parking duration is long or very long (e. g., one compared
to several days). Therefore, in some use cases, it might be suffi-
cient to predict a flexibility ranking of charging processes or are
coarse classification different groups of short and long parking
duration. However, such classifications offer lower information
content compared to quantile predictions.
The forecasting errors for both parking durations and trip dis-
tance are not particularly small in terms of MAPE and MdAPE.
13

However, we could not find any similar evaluations to bench-
mark the results. Nevertheless, even if the forecasting accuracy
seems low, the information derived from the probability fore-
cast still helps with ranking the flexibility of different charging
processes.
7.2. Forecast Application
The application of the quantile forecasts in the case study
shows that using forecasts for parking duration can improve
scheduling of BEV charging. In this case study, the forecasts of
the lower quantiles are more useful for this task than the point
forecasts of the median values. However, the improvements are
small compared to the first-in-first-out heuristic.
For interrupting the charging processes with high flexibility,
we use the low quantile value as a proxy for identifying charg-
ing events where we expect a long parking duration with high
certainty. Quantile predictions lower than 10 % other low quan-
tile might work as well.
We keep the heuristic rather simple and do not consider the
already achieved SoC at interruption time or a combination
of the estimated time of departure and estimated trip distance.
This simplification allows observing the benefits of probabilis-
tic forecast without any overlays. Using forecasts for trip dis-
tance do not show a distinct improvement for different quan-
tiles and are lower than the improvements for parking dura-
tion. Figure 2 shows that the distribution of trip distance is nar-
rower than the distribution for parking durations (i. e., observed
trip distances are closer together). We suspect that even with
lower forecast errors, in general, this makes it harder to rank
the charging processes regarding energy flexibility. Besides, in
the specific case study time flexibility is more relevant than en-
ergy flexibility. If we observe a very long trip, it is more likely
that its energy requirement cannot be fulfilled with or without
the interruption. On the contrary, it is much more useful to dif-
ferentiate the very short from the medium parking durations.
In case of congestion, a DSO might have to shed loads. In
this case, the DSO has to decide, which loads to shed. There are
some non-discriminating options (reducing maximum charg-
ing power for all BEVs affecting the congestion or interrup-
tion in order of connection time, e. g., ΛFi Fo in the case study.
However, the proposed forecasts for individual charging pro-
cesses provide additional information for this decision. In case
a charging station operator interrupts the drivers charging pro-
cess centrally, it seems unlikely that BEV drivers will accept
smart charging schemes that impair their mobility even if it
comes with a discount charging price. To overcome this chal-
lenge, the charging station operator might send out a notifica-
tion to the BEVs drivers to approve the interruption in case of
an congestion. In this case, the charging station operator profits
from an accurate forecast to decide which processes to interrupt
and which BEV drivers to notify and ask to approve the inter-
ruptions. In this case, a more accurate forecast can improve the
number of correct notifications, i. e., when the user accepts in-
terruption and can increase acceptance of such smart charging
schemes. Will and Schuller (2016) find that such personalized
smart charging can increase acceptance of smart charging.
8. Conclusion
In this paper, we describe forecasters for the time and en-
ergy flexibility of BEV charging that are needed for effective
scheduling of BEV charging. We focus on charging at home
and present how to derive a set of interpretable features from
travel logs, charging stations, or GPS data that can be used to
predict deciles of parking duration and trip distance as proxies
for time and energy flexibility.
Our evaluation is based on an open data set from German
drivers and shows that a multi-layer-perceptron with tilted loss
function achieves the most accurate results compared to quan-
tile regression, a conditional multivariate kernel density estima-
tor, and a naive benchmark. In particular, we find that using lo-
cation information of the BEV significantly increases the fore-
casting accuracy of decile forecasts evaluated in Pinball Score.
As the data set only provides limited data, i. e., one week, the
forecasters cannot rely on long-term usage patterns of single
BEV drivers. Given the limitations of the data set, we limited
our efforts in feature engineering and hyper-parameter tuning.
The proposed forecasters are rather simple to implement and
can serve as a benchmark for similar forecasting tasks.
While the findings demonstrate, how charging station opera-
tors can build powerful forecaster with simple models and lim-
ited data, it also shows that car manufacturers and other ac-
tors having access to location data have an competitive advan-
tage in forecasting BEVs charging flexibility and may allow to
build business cases around this information advantage. Conse-
quently, they will be capable of predicting charging flexibility
more accurate. Location data provides them with an advan-
tage in scheduling charging to optimize their energy procure-
ment (Kristoffersen et al., 2011) or even tab into new revenue
streams from balancing markets (Sarker et al., 2015). As us-
ing the location data has a significant benefit in predicting the
distributions of parking duration and trip distance, charging sta-
tion operators should aim to acquire such data, e. g., by using
smartphones apps providing driver data or cooperating with car
manufacturers. Otherwise, car manufacturers will have a com-
petitive advantage in operating smart charging systems.
The findings are backed by the evaluation of a case study us-
ing a greedy heuristic for scheduling the interruption of BEV
charging processes. The heuristic is simple and only uses one
out of nine deciles predicted in the forecast. The results show
that charging system operators should not stick to simple point
forecasts predicting the expected median or mean of the vari-
ables relevant for smart charging. Results show a 7.0 % relative
improvement using the 90 % decile forecast compared to using
conventional point (median) forecasts.
Besides peak shaving, other use cases could profit from such
forecasts. For instance, an aggregator using the flexibility from
BEV to provide system services must know about the parking
duration and energy demand of the BEVs to plan when he can
provide how much flexibility. Also, homeowners with PV gen-
eration and a home energy management system could profit
from such forecasts. If the homeowner connects her BEV to
the charging station, a local energy management system could
generate a probabilistic forecast for the flexibility of the BEV.
14

Based on the risk preference of the energy management sys-
tem could select a default charging settings that ensure a suffi-
ciently charged BEV in 99 % of the historical cases. If the BEV
driver knows that her mobility behaviour does not correspond
to this forecast, she could still overrule the forecast. Integrating
such a probability forecast into the strategies of energy manage-
ment systems or aggregators is another opportunity for further
research.
When predicting the distribution in a higher resolution (more
quantiles), the charging station operator could use this infor-
mation for more detailed evaluations (e. g., selecting charging
processes by their probability of parking longer than an arbi-
trary time). Based on such considerations, a probabilistic fore-
cast may constitute the basis for more sophisticated interruption
and scheduling algorithms, that also include the actual SoC and
expected trip distance.
Besides helping charging station operators in scheduling de-
cisions, the developed forecast can also aid the BEV drivers to
decide when to charge in a flexible charging mode, e. g., when
they have local generation powering their home charging sta-
tion. Such feedback can help at integrating more local, renew-
able energy in the mobility sector.
Further forecasts besides parking duration and energy de-
mand might be needed to improve BEV scheduling further. In
particular, the charging operator could profit from forecasts on
the occupancy of the charging stations before the BEV arrives
at the charging station.
Based on such forecasts, more sophisticated smart charging
methods, such as stochastic optimization, can be used to im-
prove the scheduling of BEV charging. In this way, proba-
bilistic forecasts of parking duration and trip distance will be
a valuable tool for integrating more BEVs in a grid friendly and
cost-effective manner.
Nomenclature
iIndex of parking event
τSet of input features
taTime of arrival at the parking station
tdTime of depature
ˆyPredicted variable
dParking duration
lTrip distance
qaa-quantile
QSet of quantiles
P(Y) Probability distribution of Y
LaPinball loss for quantile qa
LPinball Score is the average Pinball loss La
FτForecaster using τ
VTest statistic for Wilcoxon test
ΛInterruption heuristic
LtSet of running charging processes at t
ntNumber of running charging processes at t
It,n,ΛSet of first ninterrupted charging processes
ηBEV Energy efficiency of BEV
Cmax Maximum charging power
ιDuration of interruption
z(t,n, ι, Λ) number of impaired mobility events
Z(Λ) share of impaired mobility events
MAPE Mean Absolute Percentage Error
MdAPE Median Absolute Percentage Error
RMSE Root mean squared error
Acknowledgements
The authors want to thank the Germany Federal Ministry for
Economic Affairs and Energy for the funding and support. This
research was partly financed by the Smart Energy Showcases -
Digital Agenda for the Energy Transition (SINTEG) program.
We would also like to thank the reviewers for their feedback
and constructive comments
Data Availability
The data is available at Huber et al. (2019a)2.
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