#  Mind The Gap: Modelling Difference Between Censored and Uncensored Electric
Vehicle Charging Demand

Frederik Boe Hüttel, Filipe Rodrigues and Francisco Câmara Pereira

###### Abstract

Electric vehicle charging demand models, with charging records as input, will
inherently be biased toward the supply of available chargers. These models
often fail to account for demand lost from occupied charging stations and
competitors. The lost demand suggests that the actual demand is likely higher
than the charging records reflect, i.e., the true demand is latent
(unobserved), and the observations are censored. As a result, machine learning
models that rely on these observed records for forecasting charging demand may
be limited in their application in future infrastructure expansion and supply
management, as they do not estimate the true demand for charging. We propose
using censorship-aware models to model charging demand to address this
limitation. These models incorporate censorship in their loss functions and
learn the true latent demand distribution from observed charging records. We
study how occupied charging stations and competing services censor demand
using GPS trajectories from cars in Copenhagen, Denmark. We find that
censorship occurs up to $61\%$ of the time in some areas of the city. We use
the observed charging demand from our study to estimate the true demand and
find that censorship-aware models provide better prediction and uncertainty
estimation of actual demand than censorship-unaware models. We suggest that
future charging models based on charging records should account for censoring
to expand the application areas of machine learning models in supply
management and infrastructure expansion.

_K_ eywords Electric mobility, Electric Vehicle charging demand, Latent
mobility demand, Bayesian modelling.

##  1 Introduction

Electric vehicles (EV) and charging stations have become increasingly
prominent in energy and transportation infrastructure planning [Jakobsen et
al.,, 2020, Bauer et al.,, 2021]. Adequate infrastructure is a crucial driving
factor for large-scale EV adoption [Yi and Shirk,, 2018, Murugan and
Marisamynathan,, 2022]. However, the planning and expansion of charging
infrastructure is a complex task that requires consideration of various
factors such as charging station placement, demand forecasts, and integration
with existing power systems and road infrastructure [Deb,, 2021]. Machine
learning has emerged as a promising tool to address some of the challenges
associated with EV charging infrastructure planning, particularly charging
demand prediction, which is well-suited for machine learning approaches [Buzna
et al.,, 2019]. Forecasts of the charging demand can provide energy providers
and charging station operators with insights into the energy requirements of
the current stations, aiding in operations and maintenance [Yi et al., 2022b,
].

However, one distinguishing feature of charging stations is their shared
nature, which affects how the charging demand should be modelled. Shared
mobility services, like charging stations, often experience censorship, where
demand is lost due to limited supply or competing services [Hüttel et al.,,
2022]. Consequently, observations from charging stations may not accurately
reflect the behaviour of service users, resulting in biased data and the
subsequent models [Gammelli et al.,, 2020]. Specifically, for a charging
station with two plugs, the station can never record a demand above two cars
charging, even though more cars may need to recharge. The capacity of a
charging station acts as an upper limit for the demand each station can
observe, resulting in data that only contains a fraction of the true
unobserved (latent) demand distribution. Figure 1 illustrates these scenarios.

Machine learning models trained with maximum likelihood, such as Mean Squared
Error or Mean Absolute Error, on the observed data of EV charging demand do
not account for the unobserved demand, which limits their ability to predict
future demand for additional charging stations. This can lead to
underestimating the true demand and, in turn, restrict operators’ ability for
strategic expansion and operational decisions. In contrast, censorship-aware
models can provide estimates of actual charging demand by incorporating
censorship into their loss functions and learning the true latent demand
distribution from observed charging records [Hüttel et al.,, 2022]. By doing
so, these models are better equipped to model frustrated demand from occupied
charging stations and lost demand to competing service providers, resulting in
more accurate demand forecasts for infrastructure planning.

Therefore, we propose to extend current models to be censorship-aware to
estimate the true latent charging demand. By incorporating censorship-aware
techniques into the development of EV charging demand models, we can improve
the accuracy and reliability of these models for infrastructure planning and
policy making. Our study concentrates on the two scenarios where censorship of
demand may occur: firstly, when demand is lost due to occupied charging
stations (frustrated demand) [Larsen,, 2016], and secondly, when competing
services diminish the demand for a given provider [Li et al.,, 2021]. Through
a counterfactual analysis of demand in Copenhagen, Denmark, we provide
insights into the challenges and limitations of traditional machine learning
models when censorship is ignored and demonstrate how censorship-aware models
can mitigate these issues. In summary, the main contributions of the work are:

  * •

Previous research on electric vehicle demand modelling has mostly relied on
observational records that do not account for frustrated demand and demand
lost to competing services. In this study, we extend the models to be
censorship-aware, which has the advantage that it can be applied to observed
charging records to estimate the true latent demand.

  * •

To analyse and validate the performance of our censorship-aware model, we
conduct a counterfactual study on the charging demand in Copenhagen, Denmark,
by modelling the censorship of various queue models and competing services.

  * •

We demonstrate the advantages of incorporating censorship into the EV charging
demand models, particularly in scenarios where competing services and queue
models are present. Specifically, our findings highlight the potential
benefits of using these models in expanding and operating charging
infrastructure.

Figure 1: The figure shows the two instances of censoring of electric vehicle
(EV) charging demand Left: The demand is censored due to lost opportunities,
where EV drivers can not charge due to occupied charging stations. Right: The
demand is censored due to competing services, where EV drivers can choose
between competing services (Blue and Red chargers).

We organise the rest of the work as follows. Section 2 reviews related work on
modelling EV charging demand and censored modelling in the transport and
machine learning domain. After the review, section 3 introduces two
censorship-aware models with different distributional assumptions of the
charging demand. Firstly, the Tobit model assumes a Gaussian demand
distribution, and secondly, a quantile regression approach offers a semi-non-
parametric distribution fit of the demand. It also covers how to model the
spatial and temporal correlations between stations with graph neural networks.
Section 4 introduces the data we base our models on, and section 5 describes
our experimental setup, where we compare censorship-aware models with unaware
models. In section 6, we discuss some of the advantages and limitations of our
modelling approach. Lastly, section 7 concludes our main findings and outlines
future research directions.

##  2 Related Work

Over the last few years, there has been a growing interest in developing
accurate models for predicting EV charging demand. Researchers have studied
this problem from various perspectives, including simulation-based modelling
[Jin et al.,, 2023], queuing theory [Rich et al.,, 2022], and statistical
models [Amara-Ouali et al.,, 2022]. In this literature review, we focus on
machine learning and deep-learning models for modelling the EV charging demand
and highlight how observational data is used to model the demand.
Additionally, we delve into censored modelling techniques and how they are
applied to problems in transportation.

###  2.1 Electric Vehicle Charging Demand.

Generally, there is a large variation in the models used to model and forecast
the EV charging demand. Traditional statistical models, such as the
Autoregressive Integrated Moving Average (ARIMA) and its variations, have been
widely employed to forecast the EV charging demand, as they can model the
temporal correlations of the demand [Louie,, 2017, Amini et al.,, 2016, Kim
and Kim,, 2021]. These models offer interpretable parameters but may lack
flexibility and non-linear fitting capabilities. As a result, various machine
learning models, such as random forest [Almaghrebi et al.,, 2020, Lu et al.,,
2018, Ullah et al.,, 2021], support vector machines [Xydas et al.,, 2013,
Majidpour et al.,, 2016, Sun et al.,, 2016, Almaghrebi et al.,, 2020], and
gradient boosting [Almaghrebi et al.,, 2020, Buzna et al.,, 2019], have been
used to forecast EV charging demand. These models often incorporate temporal
and external features, such as weather, to enhance their predictive
performance.

In recent years, recurrent neural networks (RNN) have become popular in demand
forecasting due to their ability to handle large data sets and learn a
signal’s temporal features [Yi et al., 2022b, ]. One widely used RNN variant
for forecasting EV demand is the long short-term memory (LSTM) [Zhu et al.,,
2019, Kim and Kim,, 2021, Van Kriekinge et al.,, 2021, Ma and Faye,, 2022,
Boulakhbar et al.,, 2022]. The LSTM is known for its non-linear fitting
capabilities. However, its application is limited to the temporal aspect of
the charging demand, as it fails to capture the complex spatial correlations
between individual charging stations. Yi et al., 2022b  propose to mitigate
this limitation by clustered nearby charging stations and forecast demand at
each cluster with a sequence-to-sequence LSTM architecture. Another approach
is to apply temporal graph neural networks, which combine recurrent neural
networks and graph convolutions to incorporate spatial and temporal features
into the forecast [Hüttel et al.,, 2021]. A particular advantage of the graph
neural network is that it can capture demand patterns at different spatial
resolutions instead of at an aggregated level [Li et al.,, 2021], and it
learns the complex correlation between spatial locations [Tygesen et al.,,
2023]. However, the application of graph neural networks in EV charging demand
modelling is still sparse.

To train and fit charging demand models, researchers often rely on _observed_
records from charging stations [Flammini et al.,, 2019, Kim and Kim,, 2021,
Van Kriekinge et al.,, 2021, Buzna et al.,, 2021, Ma and Faye,, 2022,
Boulakhbar et al.,, 2022]. For example, Amara-Ouali et al., [2022] used public
charging records from multiple cities to estimate demand, while Yi et al.,
2022a  leveraged charging records to validate their PageRank-based demand
predictions. While charging records are an efficient way to estimate energy
and power demand for charging stations, these models and data have
limitations. Specifically, observed charging records may not reflect the true
demand due to limited supply and competition from alternative charging
services. As a result, these models may not be suitable for infrastructure
expansion planning as they cannot accurately extrapolate beyond the observed
supply and estimate the latent demand. The issue of censoring is often
neglected in the research on EV demand forecasting. Therefore, in contrast to
previous research, we propose to extend the model to be censorship-aware to
make more accurate forecasts of the true latent demand for charging.

###  2.2 Censored Modelling of Mobility demand

The modelling of censored observations is a well-established field, which
includes models such as the Tobit model [Tobin,, 1958] and the censored
quantile regression [Powell,, 1986, Yu and Stander,, 2007]. As with the EV
charging demand models, these traditional models are typically linear but
recently have been extended to non-linear variations such as censored SVM
[Shim and Hwang,, 2009], Random Forests [Li and Bradic,, 2020], and lately
neural networks [Jia and Jeong,, 2022].

In mobility research, Gaussian processes (GP) have been the go-to model for
censored distribution modelling [Gammelli et al.,, 2020, 2022]. The GP assumes
that the latent demand follows a Gaussian distribution and uses a Tobit
likelihood function to provide a probabilistic fit. Gammelli et al., [2020]
used the GP to model the latent demand for bike-sharing systems and showed
that censorship-aware models better fit the demand for shared bikes. Xie et
al., [2023] uses the censored GP to model the latent demand for bike-sharing
systems and use the predictions for efficient resource allocation in bike-
sharing systems. However, the GP faces scalability limitations when the data
set size becomes too large [Liu et al.,, 2020] and the Gaussian assumption
might be too simplistic for dynamic demand profiles [Huber et al.,, 2020].

As an alternative, neural networks can model the censored demand distribution,
particularly with censored quantile regression models [Hüttel et al.,, 2022].
The censored quantile regression is not limited to the estimation of a few
moments, such as the mean and standard deviation, allowing it to fit non-
symmetrical distributions [Peled et al.,, 2019]. For instance, Hüttel et al.,
[2022] proposed using neural networks and censored quantile regression models
to model shared mobility demand, including shared electric vehicle use. They
proposed a censored quantile regression model, which provided a better
uncertainty estimation than a Tobit model. Moreover, censored quantile
regression neural networks have been applied to biomedical data analysis [Jia
and Jeong,, 2022] and in Pearce et al., [2022], they studied the effect of
censorship in survival analysis.

Although censorship is a common issue in transportation research, there is
currently a lack of machine-learning techniques for modelling and analysing
censored EV charging demand in the literature. To address this gap and build
upon previous works, we propose to model the charging demand with censorship-
aware models, allowing a more accurate estimation of the true latent demand
for charging. Specifically, we extend censorship-aware neural networks to deep
architectures, including graph neural networks, to better capture the complex
relationships and dependencies among charging stations.

##  3 Methodology

To extend models to be censorship-aware in both the temporal and spatial
dimensions, we derive the loss functions for the Tobit and censored quantile
regression models. Since EV charging demand is influenced by various spatial
factors, such as the location and availability of nearby charging stations and
temporal factors, such as daily and seasonal patterns, we extend the loss
functions to temporal graph neural networks. We start by introducing the
notations for censored modelling.

###  3.1 Censored modelling

A data set $\mathcal{D}=\\{(x_{i},y_{i})\\}_{i=1}^{n}$ is said to be
_censored_ if the target values ($y_{i}$) are clipped observations of a
corresponding latent (true) value $y^{*}_{i}$. For each observation, there
exists a threshold value $\tau_{i}$, which makes observations either left-
censored or right-censored (or both), such that:

$y_{i}=\begin{cases}y^{*}\,,&y^{*}_{i}>\tau_{i}\\\
\tau_{i}\,,&y^{*}_{i}\leq\tau_{i}\end{cases}\text{ in left-censorship,}$ (1)

$y_{i}=\begin{cases}y^{*}\,,&y^{*}_{i}<\tau_{i}\\\
\tau_{i}\,,&y^{*}_{i}\geq\tau_{i}\end{cases}\text{ in right-censorship.}$ (2)

The threshold value $\tau_{i}$ in censored modelling can be observed,
unobserved, fixed, or stochastic [Pearce et al.,, 2022, Hüttel et al.,, 2022].
In right censoring, $\tau_{i}$ acts as an upper limit for the observations,
while in left censoring, it acts as a lower limit. The primary objective of
censored modelling is to approximate the latent value $y^{*}$ by using the
data set $\mathcal{D}=\\{(x_{i},y_{i},\tau_{i})\\}_{i=1}^{n}$ and a model with
parameters $\theta$. The model is denoted as $f_{\theta}$, where $\theta$ is a
set of parameters learned through stochastic gradient descent on a loss
function.

In the context of EV charging demand modelling, the variable $y^{*}_{i}$
denotes the true demand for charging, whereas $y_{i}$ corresponds to the
observed demand, and $\tau_{i}$ is the demand that the charging infrastructure
can serve. If the demand exceeds this threshold $\tau_{i}$, it will be
truncated at the value of $\tau_{i}$, resulting in right-censored EV charging
demand. It is important to note that one can switch between right and left
censoring by negating $y_{i}$ and $\tau_{i}$. In the following section, we
focus on the right-censoring scenario.

###  3.2 The Tobit model

The _Tobit_ model is commonly used in censored modelling [Tobin,, 1958]. It
modifies $\tau_{i}$ to an indicator variable $l_{i}$ set to 1 if $y_{i}$ is
censored and 0 otherwise. The model assumes that the latent value, $y^{*}$,
follows a Gaussian distribution given by,

$y^{*}_{i}=f_{\theta}(x_{i})+\varepsilon_{i},\quad\varepsilon_{i}\sim\mathcal{N}(0,\sigma^{2})\,.$
(3)

The likelihood function for the Tobit model can be derived as follows:

  * •

If $l_{i}=0$, then the observation is not censored, and the parameter can be
estimated with normal maximum likelihood. The likelihood function is given by

$\mathcal{L}(\theta,x_{i})=\varphi(y_{i}|f_{\theta}(x_{i}))$ (4)

where $\varphi$ is the Probability Density Function of a Gaussian
distribution.

  * •

if $l_{i}=1$, then $y_{i}$ is censored, and the likelihood function is given
by

$\mathcal{L}(\theta,x_{i})=1-\Phi(y_{i}|f_{\theta}(x_{i}))\,,$ (5)

where $\Phi$ is the Cumulative Density Function of a normal distribution
[Gammelli et al.,, 2020].

Assuming that all observations are independent and we have observed $l_{i}$,
combining the two censorship cases provides the following likelihood for a
censored data set $\mathcal{D}$,

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\prod_{i\in\mathcal{D}}\left(\underbrace{\left(\varphi(y_{i}|f_{\theta}(x_{i})\right)^{(1-l_{i})}}_{\text{Uncensored
part}}\underbrace{\left(1-\Phi(y_{i}|f_{\theta}(x_{i}))\,\right)^{l_{i}}}_{\text{Censored
part}}\right)\,.$ (6)

The likelihood function is split into two parts: the censored case and the
uncensored case. In many cases, it is common to fit models with the log-
likelihood instead of the likelihood itself,

$\log\mathcal{L}\left(\theta,\mathcal{D}\right)=\sum_{i\in\mathcal{D}}\Big{(}(1-l_{i})\log\left(\varphi\left(y_{i}|f_{\theta}(x_{i})\right)\right)+l_{i}\log\left(1-\Phi\left(y_{i}|f_{\theta}(x_{i})\right)\right)\Big{)}\,.$
(7)

In the context of neural networks and deep learning, the models are extended
to have two outputs. One output represents the mean $\mu_{\theta}(x_{i})$,
while the other represents the standard deviation $\sigma_{\theta}(x_{i})$.
The output for the standard deviation is passed through a SoftPlus function to
ensure it stays positive and differentiable. This makes the modelling
assumption that
$y_{i}^{*}\sim\mathcal{N}(\mu_{\theta}(x_{i}),\sigma_{\theta}(x_{i}))$. The
models’ outputs are used as the parameters for $\varphi$ and $\Phi$

###  3.3 Censored quantile regression

An alternative to the Tobit model for estimating censored data is censored
quantile regression [Powell,, 1986]. Rather than estimating the mean and
standard deviation of the latent distribution, quantile regression estimates a
set of quantiles
$\hat{Q}_{\theta}(x_{i})=\\{\hat{q}_{\theta,k}(x_{i})\\}_{k=1}^{K}$, where
$\hat{q}_{\theta,k}(x_{i})$ is the estimated $k$-th quantile of $y^{*}_{i}$.
Multiple quantiles can be combined to model the target distribution of
$y_{i}^{*}$. It is important to note that $k$ must be constrained to the
$\left[0,1\right]$ range.

Quantile regression models estimate the conditional quantile of the target
distribution $y$, and for a single quantile $q_{\theta,k}$, the model can be
expressed as:

$y^{*}_{i}=f_{\theta}(x_{i})+\varepsilon_{q,i}$ (8)

where $\varepsilon_{q,i}$ is the $q$-th conditional quantile of
$\varepsilon_{i}$ [Yu and Stander,, 2007].

For an uncensored quantile regression model, fitting is done by minimising the
tilted loss function,

$\rho_{q}(e)=\max\\{qe\,,(q-1)e\\}\,.$ (9)

with a targeted quantile $q$ and the residual $e$. The tilted loss is
essentially a modified version of the $\ell_{1}$ loss, where a tilting
parameter $q$ is introduced. When estimating a given quantile (e.g., $q=0.05$
quantile), the tilted loss $q$ penalises overestimation more than
underestimation by scaling the absolute value of the residual with its
probability $q$ [Rodrigues and Pereira,, 2020]. The parameters of a quantile
regression model $\theta$ are estimated by minimising the tilted loss over the
entire data set. The loss function becomes:

$\mathcal{L}(\theta,\mathcal{D})=\sum_{i\in\mathcal{D}}\rho_{q}\left(y_{i}-f_{\theta}(x_{i})\right).$
(10)

For a censored quantile model with left censorship, we observe $y_{i}$ instead
of $y_{i}^{*}$, such that $y_{i}=\max\left\\{\tau_{i},y^{*}_{i}\right\\}$. If
we want $f_{\theta}(x)$ to estimate the quantile of $y^{*}$, the loss function
becomes [Yu and Stander,, 2007]:

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\exp\Bigg{\\{}-\sum_{i\in\mathcal{D}}\rho_{q}(y_{i}-\max\\{\underbrace{\tau_{i}}_{\text{Censored
part}},\underbrace{f_{\theta}(x_{i})}_{\text{Uncensored
part}}\\})\Bigg{\\}}\,,$ (11)

It is important to note that since the likelihood functions above are based on
left censoring, the target values should be negated to convert the left
censoring scheme to a right censoring one, as follows:

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\exp\Bigg{\\{}-\sum_{i\in\mathcal{D}}\rho_{q}(-y_{i}-\max\\{-\tau_{i},-f_{\theta}(x_{i})\\})\Bigg{\\}}\,,$
(12)

To estimate the entire set of quantiles $Q_{\theta}$, one can extend a neural
network or deep learning model to have $K$ outputs corresponding to the number
of desired quantiles [Hüttel et al.,, 2022]. The likelihood function for this
case is given by:

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\sum_{q\in
Q}\exp\Bigg{\\{}-\sum_{i\in\mathcal{D}}\rho_{q}\left(-y_{i}-\max\left\\{-\tau_{i},-f_{\theta}(x_{i})\right\\}\right)\Bigg{\\}}\,,$
(13)

where $Q$ is the set of quantiles to estimate.

###  3.4 Censored Temporal Graph Convolution Neural Networks

Temporal Graph Convolution Neural Networks (T-GCNs) are deep learning models
that can effectively model non-linear relationships in both spatial and
temporal dimensions of time series data. By combining graph convolution and
recurrent layers, T-GCNs are highly effective in extracting spatial and
temporal features from data [Zhao et al.,, 2020].

The spatial structure in T-GCN is modelled with a graph $G=(N,E)$ with $m$
nodes. The set of nodes is represented by $V=\\{v_{1},v_{2},\dots,v_{m}\\}$,
and the edges between the nodes are represented by the set $E$. In a graph
neural network, the dimensions of $x_{i}$ are equivalent to the number of
nodes in the graph ($m$). Hence, $x_{i}$ can be expressed as a vector of
values $\mathbf{x}_{i}\in\mathcal{R}^{m\times 1}$. This can include external
features such as weather or time of day. In temporal modelling, the target
variable $\mathbf{y}_{i}$ and the input variable $\mathbf{x}_{i}$ are often
derived from the same signal $\mathbf{x}$. In such cases, lagged values of the
target are used to forecast future values, e.g., we have
$\mathbf{x}_{i}=\mathbf{x}_{t-l:t}$ and $\mathbf{y}_{i}=\mathbf{x}_{t+1}$,
where $l$ lagged values are used to forecast the target. In the case of
censored modelling, the forecasting process can be expressed as follows

$\mathbf{x}^{*}_{t+1}=f_{\theta}(G,\mathbf{x}_{t-l:t})+\bm{\varepsilon}\,,$
(14)

Where $\mathbf{x}^{*}_{t+1}$ is the future true latent demand for each node in
the graph, while $\bm{\varepsilon}$ depends on the choice of the loss function
(as discussed in the previous section). The model
$f_{\theta}(G,\mathbf{x}_{t-l:t})$ takes as input the graph structure $G$ and
a window of past observations $\mathbf{x}_{t-l:t}$ and produces a forecast for
the next time step.

##### Spatial Modelling

T-GCNs leverage the topology of the charging stations to define the edge
weights of the graph $G$. A common method to weight an edge between nodes $i$
and $j$ is via a Gaussian Kernel weighting function, defined as [Shuman et
al.,, 2013],

$e_{ij}=\exp(-h(z_{i},z_{j}))\,.$ (15)

Here, $z_{i}$ and $z_{j}$ denote the latitude and longitude of the nodes,
respectively, and $h$ represents the Haversine Distance between the nodes
measured in kilometres [Hüttel et al.,, 2021]. This makes nodes close to each
other have a large weight, i.e. a strong connection in the graph. Using the
set of edges $E$, we can construct the adjacency matrix $A\in R^{m\times m}$,
representing the network’s topology.

To capture features from the topological structure, a graph convolutional
network (GCN) uses graph convolutions to model the relationship between nodes
in a graph [Kipf and Welling,, 2016]. In the case of T-GCNs, a two-layered GCN
model is often employed. The model is formulated as follows

$f(A,\mathbf{x}_{t})=\sigma\left(\widehat{A}\,\operatorname{Relu}\left(\widehat{A}\mathbf{x}W_{0}\right)W_{1}\right)\,,$
(16)

Here,
$\widehat{A}=\widetilde{D}^{-\frac{1}{2}}\widetilde{A}\widetilde{D}^{-\frac{1}{2}}$
is the adjacency matrix with a self-connection structure (i.e. it has non-zero
entries in the diagonal), and $\widetilde{D}$ is the degree matrix with
$\widetilde{D}_{ii}=\sum_{j}\widetilde{A}_{ij}$. The weight matrices in the
first and second layers are denoted by $W_{0}$ and $W_{1}$, respectively. The
activation function $\sigma(\cdot)$ is applied element-wise, and
$\operatorname{Relu}()$ is the rectified linear unit activation function [Zhao
et al.,, 2020].

##### Temporal Modeling

The GCN extends to a temporal signal by combining the GCN with long short-term
memory layers (LSTM) [Hochreiter and Schmidhuber,, 1997, Zhao et al.,, 2020].
The key equations of the T-GCN with an LSTM cell can be summarised as follows,
where $f(A,\mathbf{x}_{t})$ is the graph convolution from Equation 16:

$\displaystyle i_{t}$
$\displaystyle=\sigma_{g}\left(W_{i}f(A,\mathbf{x}_{t})+U_{i}h_{t-1}+b_{i}\right)\,,$
(17) $\displaystyle f_{t}$
$\displaystyle=\sigma_{g}\left(W_{f}f(A,\mathbf{x}_{t})+U_{f}h_{t-1}+b_{f}\right)\,,$
(18) $\displaystyle o_{t}$
$\displaystyle=\sigma_{g}\left(W_{o}f(A,\mathbf{x}_{t})+U_{o}h_{t-1}+b_{o}\right)\,,$
(19) $\displaystyle\tilde{c}_{t}$
$\displaystyle=\sigma_{c}\left(W_{c}f(A,\mathbf{x}_{t})+U_{c}h_{t-1}+b_{c}\right)\,,$
(20) $\displaystyle c_{t}$ $\displaystyle=f_{t}\odot
c_{t-1}+i_{t}\odot\tilde{c}_{t}\,,$ (21) $\displaystyle h_{t}$
$\displaystyle=o_{t}\odot\sigma_{h}\left(c_{t}\right)\,.$ (22)

The trainable weights are contained in the matrices $W_{i-c}$ and $U_{i-c}$
and the vectors $b_{i-c}$. The initial values of the cell and hidden states
are set to $c_{0}=0$ and $h_{0}=0$, respectively. The Hadamard product is
denoted by $\odot$, the forget gate activation is represented by $f_{t}$, the
input/update gate activation is represented by $i_{t}$, and the output gate
hidden activation is represented by $o_{t}$. The Sigmoid function is denoted
by $\sigma_{g}$, and the hyperbolic tangent activation function is represented
by $\sigma_{c}$ and $\sigma_{h}$ [Hochreiter and Schmidhuber,, 1997].
Therefore, the set of parameters $\theta$ comprises the matrices of the
weights matrices from Equation 16, and all the weights in the LSTM ($W_{i-c}$
and $U_{i-c}$ and the vectors $b_{i-c}$)

In summary, the T-GCN models capture the complex topological structure using
the GCN and the temporal structure of the data using the LSTM layers [Zhao et
al.,, 2020]. A single forward pass produces a forecast for each node in the
graph.

##### Censored Graphs

We extend the model’s architecture to integrate the censored likelihood
functions to make the T-GCN censorship-aware. Specifically, we extend the
output to provide either the mean and standard deviation of a Gaussian
distribution ($\mathcal{N}(\mu_{\theta}(x_{i})),\sigma_{\theta}(x_{i})$) or
the set of quantiles ($\hat{Q}_{\theta}(x_{i})$) for each node in the graph.
For simplicity in notation, the predicted values for time $t+1$ are denoted as
$y_{i}$. Each graph node’s threshold value, $\tau_{i,v}$, depends on its
corresponding supply and can vary across the nodes. The loss function is
computed individually for each node, $v$, and then summed. The loss function
for the Tobit T-GCN becomes

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\sum_{i\in\mathcal{D}}\sum_{v\in
V}\Big{(}(1-l_{i,v})\log\left(\varphi\left(y_{i,v}|f_{\theta}(x_{i,v}\right)\right)+l_{i,v}\log\left(1-\Phi\left(y_{i,v}|f_{\theta}(x_{i,v})\right)\right)\Big{)}\,,$
(23)

And the loss function for the censored quantile regression T-GCN becomes

$\mathcal{L}\left(\theta,\mathcal{D}\right)=\sum_{q\in
Q}\exp\Bigg{\\{}-\sum_{i\in\mathcal{D}}\sum_{v\in
V}\rho_{q}\left(y_{i,v}-\max\left\\{\tau_{i,v},f_{\theta}(x_{i,v})\right\\}\right)\Bigg{\\}}\,.$
(24)

It is worth noting that the likelihood functions for both censored and
censored quantile regression models are not specific to a particular neural
network architecture and can also be utilised with other types of neural
networks. One notable advantage of censored models is their ability to handle
observed charging records, given that the information regarding which
observations are censored is available. This property enables the
approximation of the true charging demand, even when the observed demand is
incomplete due to censorship.

##  4 Counterfactual study

To test the hypothesis that observed EV charging demand is censored, we
conduct a _counterfactual_ study using GPS trajectories of internal combustion
engine cars (ICE) in the metropolitan area of Copenhagen, Denmark. The ICE
trajectories estimate both observed and unobserved charging demand in the
city, assuming that the ICE cars are replaced with EVs. We model the state-of-
charge (SoC) of the EV fleet based on the GPS trajectories and charging
dynamics from public charging stations, similar to the approach proposed by Tu
et al., [2016]. This study provides us with the _true_ unobserved charging
demand, allowing us to assess the censorship levels that occur throughout
space and time as we model each charging station with a queuing model. We call
it a counterfactual study because we ask, ”What would have happened with
charging demand if the ICE cars from the data set had been EVs instead?” The
overall procedure is illustrated in Figure 2, while Algorithm 1 provides a
summary of the study’s steps, which are further explained in Appendix A. The
study aims to obtain charging records for both charged and uncharged vehicles,
enabling us to assess the censorship in EV charging demand and examine its
spatial and temporal characteristics.

Figure 2: Illustration of the experimental procedure. Firstly, the GPS
trajectories of Internal Combustion Engine (ICE) vehicles are modelled as EVs
using Algorithm 1 (for detailed information, refer to Appendix A).
Subsequently, the observed charging demand and the true demand are derived
from the study. The observed demand is utilised for training a censorship-
aware T-GCN model, which is then employed to predict the true demand. In
practice, the true demand is not directly observable because of censoring.
Figure 3: Overview of the total daily trips in the data set. From the 14th to
the 20th of October, there is a drop in daily trips due to a holiday week in
Denmark. Therefore, there are fewer trips due to a reduced traffic flow from
commuting.

###  4.1 Data

The data consist of GPS trajectories from ICE cars in Denmark, where locations
are saved every 20 seconds. Each GPS location contains an ID of the car and
can be stitched together into a complete GPS trajectory, which forms a trip
for the car. Each trip contains a start coordinate, an end coordinate, and the
distance driven by a car. Cars’ parking time can be inferred as the time
between trips. To ensure the users’ privacy, the trajectories are randomised
by adding a noise distance between 50 and 500 meters to each trip’s endpoints.
The GPS trajectories of the cars were collected over three months, from
September to November 2019 and are uniformly distributed across the country
(Denmark) and vehicle segments. The GPS trajectories do not suffer from the
biases identified in other EV studies, namely, having early adopters bias the
behaviour in the data sets. In total, 32664 cars are observed in the capital
region (Copenhagen), which accounts for $5.71\%$ of the cars in the city (A
total of 571627 cars for the period [Statistik-Banken,, 2022]). In the
observations period, the penetration rate of EVs in Copenhagen was $1\%$, and
for the fall of 2022, the penetration rate has increased to $2.5\%$
[Statistik-Banken,, 2022]. We visualise the number of daily trips for the
entire period in Figure 3. In total, we have 1.5 million trips with a median
of 10 trips pr. Car.

##### Charging Stations in Copenhagen

In this study, the charging infrastructure in Copenhagen was modelled based on
the charging infrastructure from 2021. To obtain the locations of EV charging
stations, we scraped the open-source map provided by uppladdning, [2021]. The
scraped chargers have varying charging power ranging from slow chargers with
3.7 kW to fast chargers with 150 kW. The extracted information provides us
with the locations and charging power for each charging station in the city.
This information was then used to simulate the locations where ICE cars would
have charged had they been EVs.

###  4.2 Queuing Models

We use three different queuing methods to handle multiple cars wanting to
charge. We theorise that the observed demand is censored, as opportunities are
lost when charging stations are occupied. We model three different queues,
which all have real-world applications. The three queues are as follows:

  * •

Gas station: Cars queue up like an ordinary gas station and wait for their
turn to charge. The EVs charge to 80% SoC before driving away. This behaviour
is primarily associated with fast chargers and is often called a smart queue
[Monta,, 2021].

  * •

3-Hour: In this queue, EVs are only allowed to charge for 3 hours before
driving away, regardless of the SoC of the car. Cars that arrive at an
occupied station will not be charged. This behaviour reflects policies (e.g.
Denmark [FDM,, 2022]) where there is a (typically 3 hrs) time limit for
parking and charging.

  * •

First comes - First serve queue: The last queue follows a first come, first
serve approach, where the EV that arrives first charges for the entire
duration until the next trip. Cars that arrive at an occupied station will not
be charged. This corresponds to opportunistic behaviour.

The different queuing methods proposed in our study result in varying levels
of censoring. For example, the gas station queue can accommodate all users as
long as they are willing to wait. In contrast, the first come - first serve
queue can only serve users who find an available charger. We maintain
consistency in our queuing approach across all charging stations for one
study.

###  4.3 Outline of the Counterfactual Study

To study the charging behaviour of ICE cars as if they were EVs, we randomly
sample different numbers of cars from the GPS trajectories, corresponding to
different EV penetration rates. The GPS trajectories originally have a
penetration rate of $5\%$ for all cars. However, since the penetration rate of
EVs was roughly $1\%$ in 2019 and $3\%$ in 2022, we vary the penetration rate
from $1\%$ to $5\%$ and test the three different queuing models. We model the
cars’ charging behaviour using the sampled cars and the queuing model,
assuming they were EVs. We randomly sample the battery sizes according to the
market distribution of different EV types as described in subsection A.1. We
also calculate their willingness to charge based on the state of charge
consumption of a trip. The charging station is probabilistically selected from
each end trip’s five closest charging stations. Each car follows the procedure
in Algorithm 1.

Algorithm 1 Overview of the counterfactual study.

Generate a fleet of electric vehicles.

Set queuing model.

Set SoC for the fleet of cars $SoC\sim\mathcal{N}(0.6,0.2),0.20\leq SoC\leq
1$.

for each GPS trajectory do

Find the car id of the GPS trajectory.

Compute energy consumption for the car.

Update SoC-level for the car.

Compute willingness to charge for the car.

Compute were to charge based on the end location of the trip.

Add car to queue.

Compute energy demand based on the time spent charging.

Save the energy demand from the charging event.

end for

###  4.4 Demand Profiles from the Counterfactual Study

We examine the resulting observed and unobserved energy demand from each GPS
trajectory. We find that censorship occurs and varies across the different
queues and penetration rates. The gas station queue experiences very little
censorship due to the nature of charging, where observations are only censored
if the EVs have a new trip while in the queue. However, the censoring
increases as we move to the different queue types, with the first-come queue
having the largest amount of censorship. We report the fraction of hours where
censorship occurs across the queue and the penetration rate in Table 1. For
the current penetration levels of EVs, we find that between $33\%$ and $53\%$
of hours are censored in the entire area. As the penetration rate increases,
censorship follows suit, which is a natural consequence, as the infrastructure
would need to expand to handle more demand from chargers.

| Penetration rate
---|---
Queue |  1% (Historical) |  2% |  3% (Current) |  4% |  5% (Data)
Gas station Queue |  $<$0.1% | 2.2% | 4.9% | 9.8% | 16.4%
3 Hour queue | 4.1% | 17.6% | 33.2% | 46.8% | 56.9%
First come - First serve Queue | 8.3% | 30.1% | 53.2% | 66.2% | 70.5%

Table 1: Fraction of hours where the observed demand is censored across the
different queuing models and penetration rate.

The study shows that most censoring in EV charging demand occurs during peak
demand hours in the temporal dimension, particularly during morning and
afternoon periods, as shown in Figure 4. These peak demand hours align with
the traffic patterns observed in the original GPS trajectories and are
consistent with findings from other studies on charging demand in urban areas.
Furthermore, the analysis revealed that charging demand varies across
different days, with lower demand observed on weekends than on typical
weekdays. Notably, despite the fluctuations in the total charging demand, the
amount of censored demand remains relatively stable, with no significant
variations in the proportion of censored demand observed across different
weekdays (Figure 4). These findings provide important insights into the
temporal patterns of EV charging demand and highlight the need for censorship-
aware models to accurately estimate the true latent demand distribution,
especially during peak demand periods.

Figure 4: Average charging demand, either grouped by the time of day (left) or day of the week (right). Rate: $5\%$ and queue: _First come - First serve_. Figure 5: Overview of the spatial structure in the counterfactual study. Small circles indicate the end of the GPS trajectories. The circle with a dark line is the charging station, and squares indicate the centroids of the clusters. Colours indicate which cluster they belong to. Cluster |  Location |  $\%$ of censored hours
---|---|---
0 | Western part of city | $61.03\%$
1 | South Easter part | $20.29\%$
2 | West of the city centre | $3.12\%$
3 | North Eastern part of the city | $16.31\%$
4 | Southern part of the city | $11.28\%$
5 | Western part of city | $24.33\%$
6 | North part of the city centre | $6.57\%$
7 | South Western part of the city | $20.16\%$
8 | North Western part of the city | $31.17\%$
9 | City center | $16.67\%$
Table 2: Table presenting the proportion of hours with censored observations
under a $5\%$ EV penetration rate and the First come - First serve Queue

We spatially aggregate the demand data with k-means clustering, which groups
charging stations into ten regions. This approach is necessary as demand at
individual charging stations can be highly unpredictable and sporadic, making
it difficult to model accurately. By clustering stations based on their
proximity, we account for regional variations in demand patterns. Each cluster
centre serves as a node in the temporal graph neural network To visually
represent the distribution of charging stations used in our study, we present
a map in Figure 5. We find that across the city, the censorship levels vary.
Table 2 shows the censorship levels for the $5\%$-penetration of EVs and the
_First come - first serve_ queue. In cluster 0, which covers the western part
of Copenhagen, $61\%$ of the hourly charging demand is censored. This high
level of censorship is attributed to only two regional charging stations
covering a large spatial area without having sufficient supply to meet the
demand. However, it is important to note that this area is primarily
residential, and most of the demand could be served by home charging, which is
not included in the study. On the other hand, in cluster 4, located in the
southern part of Copenhagen and an industrial area with low demand at night,
lower censorship levels and minimal demand at night are observed (Figure 6).
The study reports censorship levels across all the different queues,
penetration rates, and clusters in Appendix B.

Figure 6: This figure displays the charging demand obtained from the
Counterfactual study for cluster 0 and cluster 4 in the city. The observed
demand is input into the models to estimate the true latent demand.

To summarise, we conducted a counterfactual study to model the charging
behaviour of electric vehicles (EVs) based on GPS trajectories from internal
combustion engine (ICE) cars in the fall of 2019. This approach enabled us to
estimate the true unobserved charging demand that would have existed had these
cars been EVs. We find that the demand is censored and varies in space and
time. The charging demand for all the cars, including those who recharged and
those who did not, is obtained in the form of charging records similar to
those obtained by charging station operators. The observed charging records
are used as input to the Censored T-GCN models. The unobserved charging
records are utilised to validate the model’s accuracy in approximating the
true charging demand.

##  5 Experiments and Results

We conduct a series of experiments with the counterfactual study data. First,
we investigated the effect of various censorship schemes on modelling the
charging demand for the entire city while considering different penetration
rates and queuing models. Next, we compared and evaluated the performance of
censorship-aware and censorship-unaware models in a competitive setting where
only a fraction of the demand is observable 111Code is available at
https://github.com/fbohu/CensoredTGCN.

We label the models based on the likelihood functions they employ. We include
two censorship-unaware models: a _Gaussian_ model estimated using maximum
likelihood and a _quantile_ regression (QR) model trained with the uncensored
tilted loss function described in Equation 9. Additionally, we compare these
models with censorship-aware models: a _Tobit_ model (Equation 23) and a
_censored QR_ model (Equation 24). We adopt identical architectural choices
for all models and initialise their parameters with the same random seed. We
conduct ten runs for each experiment and report the average results across
these runs, along with their corresponding standard deviations.

###  5.1 Architectures

The most recent 168 hourly demand observations, representing a week’s worth of
data, are utilised as the input size for the temporal signal. To augment the
historical demand data, the type of day and hour are incorporated as external
input features encoded into cyclical features using sine and cosine. These
cyclical features are combined with the historical demand to generate time
series for each node in the graph. To prevent any one node from contributing
excessively to the loss function during training, each time series is scaled
between 0 and 1, as recommended by Zhao et al., [2020]. The model uses 16 and
8 channels for the graph convolutions and an LSTM with 32 hidden units. The
Adam optimiser [Kingma and Ba,, 2014] with a learning rate of 0.0003 and norm
clipping of 1.0 is used. During training, each model is trained for 1000
epochs with a batch size of 256, and an early stopping criterion of 0.001 on
the validation loss is applied. To evaluate the models, the data set is
divided into three sets: train, validation, and test, with a split of 80%,
10%, and 10%, respectively. The censored quantile regression models estimate
the 0.05, 0.5, and 0.95 quantiles. All models are implemented in Keras
[Chollet et al.,, 2015], and graph convolutions were performed using
StellarGraph [Data61,, 2018]

###  5.2 Evaluation Metrics

We assess the performance of our models by computing their tilted loss on the
test set ($\mathcal{D}_{\text{test}}$). We estimate their quantiles using the
fitted distributions for the Gaussian and Tobit models. Moreover, we use the
following metrics widely used to evaluate a probabilistic forecast: Interval
Coverage Percentage (ICP) and Mean Interval Length (MIL). To simplify
notation, we consider a single node in the graph.

ICP
$\displaystyle=\frac{1}{|\mathcal{D}_{\text{test}}|}\sum_{i\in\mathcal{D}_{\text{test}}}\begin{cases}1&\text{
if }\hat{q}_{\theta,0.05}(x_{i})\leq
y^{*}_{i}\leq\hat{q}_{\theta,0.95}(x_{i})\\\ 0&\text{otherwise}\end{cases}$
(25) MIL
$\displaystyle=\frac{1}{|\mathcal{D_{\text{test}}}|}\sum_{i\in\mathcal{D}_{\text{test}}}\left(|\hat{q}_{\theta,0.05}(x_{i})-\hat{q}_{\theta,0.95}(x_{i})|\right)$
(26)

The objective of a probabilistic model is to achieve a high Interval Coverage
Percentage (ICP) (Equation 25), close to 0.9, indicating that $90\%$ of the
observations fall within the prediction interval, while maintaining a low Mean
Interval Length (MIL) (Equation 26), which indicates that the model’s
predictions intervals are tight around the true value.

###  5.3 Total Demand Predictions

To assess the effectiveness of our models, we conducted initial experiments
using the censoring scheme outlined in Table 1, which was applied to all
charging stations and nodes. The penetration rate of electric vehicles (EVs)
was varied, with corresponding increases in censorship levels and
modifications to the queuing model. Our findings are presented in Table 3,
where we report the tilted loss on the true demand for the test set.”

Our findings suggest that the quantile regression-based models (QR and
Censored-QR) generally perform better than the censorship-unaware models. For
the Gas station queue, we did not observe any substantial distinction between
the censorship-aware and unaware models, primarily due to the low levels of
censorship in this queue type. However, as we varied the queuing model and
censorship levels, we found that the performance discrepancy between the
censorship-aware and unaware models increased. Specifically, the Censored QR
model is the best-performing model for the 3-hour and First come queue, with
high censorship levels.

Queue |  Penetration rate |  1.0% |  2.0% |  3.0% |  4.0% |  5.0%
---|---|---|---|---|---|---
Gas station | Gaussian | $0.796\pm 0.001$ | $0.800\pm 0.001$ | $0.709\pm 0.001$ | $0.783\pm 0.001$ | $0.714\pm 0.001$
Tobit | $0.801\pm 0.009$ | $0.800\pm 0.007$ | $0.711\pm 0.004$ | $0.775\pm 0.003$ | $0.713\pm 0.011$
QR | $0.656\pm 0.003$ | $\mathbf{0.719\pm 0.002}$ | $\mathbf{0.668\pm 0.003}$ | $0.731\pm 0.006$ | $\mathbf{0.680\pm 0.004}$
Censored QR | $\mathbf{0.655\pm 0.006}$ | $0.720\pm 0.008$ | $0.670\pm 0.004$ | $\mathbf{0.731\pm 0.004}$ | $0.681\pm 0.005$
3-hour | Gaussian | $0.767\pm 0.002$ | $0.750\pm 0.002$ | $0.810\pm 0.002$ | $0.781\pm 0.002$ | $0.798\pm 0.002$
Tobit | $0.764\pm 0.006$ | $0.750\pm 0.002$ | $0.799\pm 0.007$ | $0.762\pm 0.005$ | $0.765\pm 0.006$
QR | $0.644\pm 0.004$ | $0.678\pm 0.005$ | $0.756\pm 0.003$ | $0.740\pm 0.003$ | $0.763\pm 0.002$
Censored QR | $\mathbf{0.641\pm 0.006}$ | $\mathbf{0.673\pm 0.004}$ | $\mathbf{0.748\pm 0.004}$ | $\mathbf{0.726\pm 0.006}$ | $\mathbf{0.736\pm 0.009}$
First come - First serve | Gaussian | $0.695\pm 0.001$ | $0.774\pm 0.001$ | $0.795\pm 0.001$ | $0.888\pm 0.001$ | $0.945\pm 0.001$
Tobit | $0.693\pm 0.004$ | $0.772\pm 0.010$ | $0.750\pm 0.003$ | $0.785\pm 0.010$ | $0.790\pm 0.016$
QR | $0.577\pm 0.007$ | $0.697\pm 0.005$ | $0.734\pm 0.003$ | $0.841\pm 0.011$ | $0.894\pm 0.007$
Censored QR | $\mathbf{0.574\pm 0.004}$ | $\mathbf{0.690\pm 0.005}$ | $\mathbf{0.709\pm 0.005}$ | $\mathbf{0.774\pm 0.005}$ | $\mathbf{0.765\pm 0.004}$

Table 3: Comparison of tilted loss summed for all nodes across different
models and EV penetration rates. Lower values indicate better performance, and
the best-performing model is highlighted in bold.

Figure 7: The ICP (left) and MIL (right) across the different penetration
rates and Queues. In general, the ICP should be close to 0.9 with a lower MIL
value, which means that the uncertainty can capture the target distribution
while allowing for uncertainty in the predictions.

In Figure 7, we present the results of the Interval Coverage Percentage (ICP)
and Mean Interval Length (MIL) metrics for the models on different queues and
penetration rates. The ICP represents the proportion of observations within
the prediction interval, while the MIL measures the average length of the
prediction intervals. Since the level of censoring varies between nodes, we
present the ICP for the cluster with the highest censorship level. The
Gaussian models, namely Gaussian and Tobit, generally show higher ICP and MIL
values, indicating larger prediction intervals than the non-parametric models.
As a result, they achieve a higher ICP than the QR models. On the other hand,
the QR models exhibit a narrower confidence interval, resulting in lower MIL
values, while their ICP is comparable to that of the Gaussian models.
Therefore, the best model is the one with an ICP close to $90\%$ and the
shortest possible MIL.

###  5.4 Application of Censored Demand Modelling

Figure 8: Demand predictions for cluster 0 in the first-come queue scenario,
with a 5% penetration rate. The black line represents the true demand, while
the observed demand is shown in grey. The blue and red lines correspond to the
predictions obtained using the quantile and censored quantile regression
models, respectively. The shaded red prediction interval is defined by the
quantiles $q_{\theta,0.05}(x_{i})$ and $q_{\theta,0.95}(x_{i})$

In this section, we discuss the advantages of using the censorship-aware model
over the censorship-unaware model and illustrate how it can influence future
expansion plans. While infrastructure expansion planning is a complex process,
we focus on the benefits of using censorship-aware models.

Cluster 0 is a cluster where a large fraction of observations is censored. The
censored QR model better fits the actual true demand than the censorship-
unaware QR model (Figure 8). The QR model provides a relatively accurate fit
of the observed demand and tends to forecast the demand sufficiently by
forecasting the general temporal patterns. However, the censored QR model
provides higher predictions of the charging demand than the system observes.
These elevated predictions suggest that potentially expanding the
infrastructure in cluster 0 is viable. One benefit of the censorship-aware
models is their estimation of the capacity increase required to meet the true
demand. The gap between the predicted demand and observed censored demand
could already serve as an indicator for expansion. However, the regressed
value and their corresponding uncertainty provide a size of the additions
needed to meet all demands. Therefore, censorship-aware models estimate the
capacity to meet demand, providing a reasonable basis for data-driven
decisions. Additionally, flexible supply could serve peak demand hours due to
the variation in demand throughout the day.

###  5.5 Competing Services

We now investigate a scenario where multiple EV charging providers operate in
a competitive environment. Our goal is to forecast the true demand for
operational expansions from the perspective of a single charging service
provider who only observes a fraction (market share) of the total charging
demand in the city. We vary the market share from $10\%$ to $95\%$, where a
$10\%$ market share means that $90\%$ of the true demand remains unobserved.
We do not assume any particular number of competitors in the market and only
use data one provider observes. To cluster the chargers, we follow the same
method as previously described. We sample the market share of chargers in the
entire city and censor the demand from all other charging stations.

We report our experiments’ results regarding tilted loss, ICP, and MIL in
Table 4. We find that the censorship-aware models outperform their unaware
counterparts. Specifically, the censored quantile regression model tends to
provide a better fit for the charging demand for low market shares while still
having a low performance of the ICP and MIL. As the market share increases and
the censorship decreases, the performance gap between the censorship-aware and
unaware models diminishes. However, for a market share of $95\%$, the
censorship-aware models still provide tighter uncertainty in their prediction
intervals while still having a high percentage of observations within their
intervals.

These results demonstrate the benefits of using censorship-aware models in a
competitive environment. The models provide a reliable basis for the charging
station’s operations and estimates of the demand that competitors can serve.
Estimating demand lost to competitors provides a competitive edge for future
expansions and the operation of the infrastructure from the perspective of a
charging station operator. This estimate can be used for strategic moves [Li
et al.,, 2021].

|  | Market share
---|---|---
Metric |  Model |  $10\%$ |  $25\%$ |  $50\%$ |  $75\%$ |  $95\%$
TL | Gaussian | $3.289\pm 0.005$ | $1.991\pm 0.007$ | $1.149\pm 0.016$ | $0.868\pm 0.006$ | $0.764\pm 0.007$
Tobit | $2.799\pm 0.053$ | $1.667\pm 0.030$ | $1.027\pm 0.010$ | $0.838\pm 0.009$ | $0.765\pm 0.006$
QR | $3.120\pm 0.023$ | $1.779\pm 0.013$ | $1.019\pm 0.009$ | $0.775\pm 0.004$ | $0.686\pm 0.004$
Censored QR | $\mathbf{2.636\pm 0.026}$ | $\mathbf{1.538\pm 0.009}$ | $\mathbf{0.940\pm 0.006}$ | $\mathbf{0.754\pm 0.007}$ | $\mathbf{0.684\pm 0.006}$
ICP | Gaussian | $0.566\pm 0.002$ | $0.660\pm 0.005$ | $0.766\pm 0.009$ | $0.820\pm 0.004$ | $0.867\pm 0.004$
Tobit | $0.567\pm 0.011$ | $0.713\pm 0.016$ | $0.807\pm 0.014$ | $0.839\pm 0.008$ | $0.866\pm 0.005$
QR | $0.398\pm 0.008$ | $0.557\pm 0.033$ | $0.725\pm 0.011$ | $0.772\pm 0.011$ | $0.812\pm 0.015$
Censored QR | $\mathbf{0.389\pm 0.016}$ | $\mathbf{0.642\pm 0.014}$ | $\mathbf{0.732\pm 0.023}$ | $\mathbf{0.768\pm 0.026}$ | $\mathbf{0.809\pm 0.008}$
MIL | Gaussian | $0.225\pm 0.006$ | $0.295\pm 0.003$ | $0.312\pm 0.006$ | $0.315\pm 0.005$ | $0.331\pm 0.004$
Tobit | $0.246\pm 0.009$ | $0.336\pm 0.011$ | $0.349\pm 0.005$ | $0.333\pm 0.007$ | $0.332\pm 0.006$
QR | $0.164\pm 0.005$ | $0.263\pm 0.003$ | $0.284\pm 0.006$ | $0.296\pm 0.006$ | $0.318\pm 0.005$
Censored QR | $\mathbf{0.256\pm 0.005}$ | $\mathbf{0.323\pm 0.005}$ | $\mathbf{0.319\pm 0.003}$ | $\mathbf{0.313\pm 0.004}$ | $\mathbf{0.323\pm 0.003}$

Table 4: Model performance for increasing market share with three different
evaluation metrics, Tilted loss (TL), Interval Coverage Percentage (IPC), and
Mean Interval Length (MIL). The bold values indicate the model with the lowest
Tilted loss

##  6 Discussion

Throughout our analysis, experiments and results, we have argued that the
observed EV charging demand from charging records is censored and, thus, does
not reflect the actual demand for charging. We conducted a counterfactual
study using ICE GPS trajectories and battery models to investigate this
hypothesis.

One advantage of our counterfactual study is that it employs real-world
censoring instead of artificial censoring schemes commonly used in censored
modelling [Hüttel et al.,, 2022, Gammelli et al.,, 2020]. The censoring is
occurring due to the inclusion of queues and limited supply. Therefore, the
censoring scheme consistent with actual charging infrastructure provides a
more accurate representation of how the censoring occurs. Another advantage of
using the ICE GPS trajectories is that early EV adopters do not influence
them. Hence, they represent the desired behaviour of users when they are not
constrained by needing to recharge their cars. Our counterfactual study
provides an initial indication of the degree of underestimation of the
observed demand that is caused by censorship. Another advantage of using the
proposed method to estimate the censored demand is that it allows for the
inference of latent demand from observational data. In this work, the
counterfactual study is _just_ used to validate our censored models with a
realistic censorship scheme compared to an artificial one.

While ICE GPS trajectories were used in our study, naturally, EV trajectories
would provide a better proxy for the charging demand, as behavioural
differences exist between ICE and EV users. For example, the ICE cars do not
need to recharge; hence, they represent the desired behaviour of users as they
are not constrained by needing to recharge their cars. On the other hand, EV
users may be more conservative due to the slow charging speed and the shortage
of charging facilities, which could result in shorter trips per charge. This
aspect, however, is also likely to change with more efficient and widespread
technology.

The counterfactual study conducted in this work shares many similarities with
agent-based simulations but with the distinction that individual travel
behaviours, such as origins, destinations, and parking times, are determined
by GPS trajectories. The counterfactual nature of the study represents the
”what-if” scenario of running the same trips with different assumptions of the
EV. However, validating the simulation is challenging since data from the
period is not obtainable. Future studies could incorporate additional factors,
such as departure time and route, to improve the simulation’s accuracy.
Collecting more comprehensive data on EV charging behaviour would also be
beneficial to improve the results’ reliability and inform the development of
appropriate charging infrastructure and policies.

Our study acknowledges that our simplistic charging and battery models may
underestimate the charging demand, as we do not account for other key travel
metrics such as speed, acceleration, and ambient temperature [McNerney et
al.,, 2017]. These factors also affect battery consumption and result in
variations in the charging demand for each trip. Therefore, the censored
charging demand may be greater than we present here, highlighting the need for
censorship-aware models to model EV charging demand accurately. While a more
behaviourally sophisticated approach, such as a discrete choice model, would
be ideal, it requires data currently unavailable, such as the social
demographics and preferences of the users.

##  7 Conclusion and future research directions

In conclusion, we have addressed the issue of censorship in modelling electric
vehicle charging demand by extending machine learning models to be censorship-
aware. We argue that the demand observed by charging records is censored due
to lost opportunities to competition or supply limitations, which can
negatively impact censorship-unaware models’ ability to model the demand. It
is essential to address the gap between lost and observed demand to facilitate
efficient and cost-effective expansions and operations of charging stations.

We propose several methods to account for censorship in modelling EV charging
demand and extend them to spatiotemporal problems, namely the Tobit model and
the censored quantile regression. An advantage of the proposed method is that
it can be applied to any observed charging records. Specifically, we used a
set of GPS trajectories from ICE cars in Copenhagen, Denmark. We studied how
they would have behaved by modelling their behaviour and charging dynamics,
assuming the cars were EVs. This leads to various amounts of censorship across
space and time. Using the observed charging demand from this study, we
demonstrate the superiority of censorship-aware models over censorship-unaware
models in accurately modelling the latent distribution of charging demand. The
experiments show that the censored quantile regression model obtains a better
fit and uncertainty estimation of the demand than the Tobit model.
Furthermore, We compared censorship-aware and unaware models in a competitive
environment with different market share amounts. We found that censorship-
aware models are superior in modelling the demand, which could provide an edge
in a competitive environment.

Furthermore, while this study focuses on two specific scenarios of demand
censorship (frustrated demand due to occupied charging stations; diminished
demand due to competing services), other types of latent, lost, or unobserved
demand exist. This includes exploring additional factors such as charging
station unavailability, faulty equipment, pricing structures, and user
behaviour patterns that can contribute to censored demand. Future research
should also investigate the implications of incorporating mobile charging
stations into the existing infrastructure, as they can be used to uncover the
true demand.

##  8 Acknowledgements

The research leading to these results has received funding from the
Independent Research Fund Denmark (Danmarks Frie Forskningsfond) under grant
no. 0217-00065B.

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##  Appendix A Counterfactual study - appendix

###  A.1 Data

As mentioned, the data consist of GPS trajectories from ICE cars in Denmark,
where locations are saved every 20 seconds. Each GPS location contains an ID
of the car and can be stitched together into a complete GPS trajectory, which
forms a trip for the car. Each trip contains a start coordinate, an end
coordinate, and the distance driven by a car. Cars’ parking time can be
inferred as the time between trips. To ensure the users’ privacy, the
trajectories are randomised by adding a noise distance between 50 and 500
meters to each trip’s endpoints. The GPS trajectories of the cars were
collected over three months, from September to November 2019 and are uniformly
distributed across the country (Denmark) and vehicle segments. The GPS
trajectories do not suffer from the biases identified in other EV studies,
namely, having early adopters bias the behaviour in the data sets. In total,
32664 cars are observed in the capital region (Copenhagen), which accounts for
$5.71\%$ of the cars in the city (A total of 571627 cars for the period
[Statistik-Banken,, 2022]). In the observations period, the penetration rate
of EVs in Copenhagen is $1\%$, and for the fall of 2022, the penetration rate
has increased to $2.5\%$ [Statistik-Banken,, 2022]. We visualise the number of
daily trips for the entire period in Figure 9. In total, we have 1550509 trips
with a median of 10 trips pr. car.

Figure 9: Overview of the total daily trips in the data set. From the 14th to
the 20th of October, there is a drop in daily trips due to a holiday week in
Denmark. Therefore, there are fewer trips due to a reduced traffic flow from
commuting.

##### Charging Stations in Copenhagen

To model the charging infrastructure in Copenhagen, we use the charging
infrastructure from 2021. We scraped the infrastructure from uppladdning,
[2021], an open-source map containing locations of EV charging stations. The
scraped chargers have different charging power ranging from slow chargers with
3.7 kW charging to fast chargers with 150 kW. These provide us with the
locations and the power for each charging station in the city, which we use to
determine where the ICE cars would have charged, assuming they were EVs.

##### Market shares of Electric vehicles

Before we describe our battery and charging models, we generate the fleet of
EVs based on the distribution of EVs in Denmark. To generate the EV fleet, we
sample the EV characteristics for each car according to their market share
distribution. We report the ten most popular EV models in Denmark can be seen
in Table 5.

Ranking |  Model |  Count |  Range (km) |  Battery capacity (kWh)
---|---|---|---|---
1 | Tesla Model 3 SR | 8183 | 380.0 | 57.0
2 | Renault Zoe | 4050 | 315.0 | 52.0
3 | Tesla Model S | 3915 | 560.0 | 95.0
4 | Volkswagen ID.3 EV | 3353 | 350.0 | 58.0
5 | Nissan Leaf | 3033 | 225.0 | 37.0
6 | Hunydai Kona BEV | 2948 | 395.0 | 64.0
7 | Volkswagen ID.4 EV | 2473 | 400.0 | 77.0
8 | Kira Niro EV | 1890 | 370.0 | 64.0
9 | BMW i3 | 1642 | 235.0 | 37.9
10 | Volkswagen e-Up! | 1370 | 205.0 | 32.3
- | Others | 17399 | 313.0 | 60.0
Table 5: Top 10 most popular EV models in Denmark [Observatory,, 2021] with
manufacturing standards from [Database,, 2022].

##### Generation of EV fleet

We generate a fleet of EVs from the ICE using the market shares. Where each EV
corresponds to an ICE, with the same trips, for each EV we keep track of the
_SoC_ , _Range_ and _battery capacity_ for the vehicle.

###  A.2 Battery Model

Firstly, we turn to our modelling of the Lithium ION batteries in the EV
fleet. Initially, we set the SoC levels of the fleet to follow a truncated
normal distribution as $SoC\sim\mathcal{N}(0.6,0.2),0.20\leq SoC\leq 1$. We
assume there is a linear relationship between the distance of a trip and State
of charge consumption for a trip and model the SoC consumption (or the
decrease in the SoC levels) as:

$\text{Consumption pr. trip}=\frac{\text{Distance of trip}}{\text{Range of car
type}}\,.$ (27)

After one trip, we update the EV with the SoC for the next trip as:

$\text{SoC}_{t+1}=\text{SoC}_{t}-\text{Consumption pr. trip}\,.$ (28)

####  A.2.1 Charging decision

We model the charging decision with a smooth beta function. We evaluate the
probability of charging after a trip as a CDF of a beta distribution with the
initial $x_{i}$ and the final SoC $x_{f}$ levels. [Hipolito et al.,, 2022] The
PDF of a beta distribution is

$f_{\beta}(a,b;x)=\frac{x^{a-1}(1-x)^{b-1}}{\Gamma(a)\Gamma(b)/\Gamma(a+b)}\,.$
(29)

We model the probability of charging with a $SoC\leq x$ as the CDF
($F_{\beta}(a,b;x)$) of the survival function

$S_{\beta}(a,b;x)=1-F_{\beta}(a,b;x)=1-\int_{0}^{x}f_{\beta}(a,b;t)dt$ (30)

Using this equation for the decision to charge can lead to counter intuitive
examples of charging, where cars charge even at high levels of $SoC$.
Therefore we make the vehicles context-aware by assessing the willingness to
charge based on the initial $SoC$ ($x_{i}$) and the final $SoC$ ($x_{f}$) of a
trip. We compute the willingness to charge based on the differences between
the CDFs for the initial SoC and the final SoC, normalised to the probability
of charging at the start÷

$W_{\beta}\left(a,b;x_{f},x_{i}\right)=\left[F_{\beta}\left(a,b;x_{i}\right)-F_{\beta}\left(a,b;x_{f}\right)\right]/F_{\beta}\left(a,b;x_{i}\right).$
(31)

Using the willingness to charge, we random draw if a car decides to charge.
The parameters $a$ and $b$ are estimated via non-linear regression We use
default parameters for the distributions, and the reader is referred to
[Hipolito et al.,, 2022] for a larger discussion on the willingness to charge
and the estimation of $a$ and $b$.

###  A.3 Where to charge

The willingness to charge gives us a willingness score between 0 and 1. We
randomly draw if you charge or not based on the willingness score. We employ a
simple discrete choice model to decide where to charge if the car needs to
charge We identify the five closest chargers based on the trip end point and
compute their utility as the distance between the endpoint of the trip and the
charger location. For each station of the five closest, we draw the
probability:

$P_{j}=\frac{\exp(D_{j})}{\sum\exp(D_{j})}\,,$ (32)

where $D_{j}$ is the distance between the charging station and the end trip
point the trip and $P_{j}$ is the probability of going to station $j$. We then
randomly select a charging station from the $P_{j}$ of the five closest
charging stations.

####  A.3.1 Charging model

We model the decision to charge as a probabilistic choice between the Once a
charger has been chosen, we approximate the charging of the EV with a piece-
wise linear model [Gjelaj et al.,, 2020]. The time spent charging will be
dependent on the queue for the study. Firstly, we compute the time it takes
for the car to have an SoC of more than $80\%$ as:

$T_{80}=0.8\left(\frac{\text{Battery capacity}-SoC\cdot\text{Battery
capacity}}{\text{Charger power}}\right)\,.$ (33)

We then compute the new SoC for the car based on the following. In Figure 10,
we show an example of the charging equation for three different charger types:

$\text{SoC}_{t+1}=\begin{cases}(\text{SoC}_{t}+\text{charging
time}*\frac{\text{Charger power}}{\text{Battery capacity}})&\text{if charging
time}<T_{80}\\\ 0.8+0.25((\text{charging time above
}T_{80})*\frac{\text{Charger power}}{\text{Battery
capacity}})&\text{else}\end{cases}\,.$ (34)

Figure 10: Charging curves for a _Tesla Model 3_ with a battery capacity of 57
kWh and three different charging capacities. A 7kW slow charger, a 22 kW
medium charger and a 50 kW fast charger.

Finally, we store the energy required for the charging. If a vehicle did not
charge due to occupied charging stations, we assume it would have charged
$80\%$ SoC and stored this as censored demand.

###  A.4 Summary of the counterfactual study

Using the points made above, we conduct the counterfactual study. We vary the
number of sample EVs to match the different penetration rates. In 2019, the
penetration rate of EVs was roughly 1% and in 2022, is roughly 3%. The GPS
trajectories themselves have a penetration rate of 5%. We run the study with
three different queuing models. We sample the car distribution according to
the count distribution in Table 5. Each car follows the procedure in Algorithm
1.

Algorithm 2 Overview of the counterfactual study.

Generate a fleet of electric vehicles.

Set queuing model.

Set SoC for the fleet of cars $SoC\sim\mathcal{N}(0.6,0.2),0.20\leq SoC\leq
1$.

for each GPS trajectory do

Find the car id of the GPS trajectory.

Compute energy consumption for the car (Equation 27).

Update SoC-level for the car (Equation 28).

Compute willingness to charge for the car (Equation 31).

Compute were to charge based on the end location of the trip (Equation 32).

Add the car to the queue.

Compute energy demand based on the charging time (Equation 34).

Save the energy demand from the charging event.

end for

##  Appendix B Censoring across the different clusters

|  | Penetration Rate
---|---|---
Queue |  Cluster |  1.0% |  2.0% |  3.0% |  4.0% |  5.0%
Gas station | 0 | $0.22\%$ | $1.26\%$ | $2.80\%$ | $5.08\%$ | $8.11\%$
1 | $0.04\%$ | $0.09\%$ | $0.22\%$ | $0.49\%$ | $1.09\%$
2 | $0.00\%$ | $0.00\%$ | $0.00\%$ | $0.18\%$ | $0.36\%$
3 | $0.00\%$ | $0.05\%$ | $0.27\%$ | $0.22\%$ | $0.63\%$
4 | $0.0v\%$ | $0.05\%$ | $0.14\%$ | $0.50\%$ | $1.40\%$
5 | $0.00\%$ | $0.05\%$ | $0.18\%$ | $1.13\%$ | $2.17\%$
6 | $0.00\%$ | $0.05\%$ | $0.09\%$ | $0.09\%$ | $0.36\%$
7 | $0.00\%$ | $0.40\%$ | $0.67\%$ | $1.91\%$ | $3.09\%$
8 | $0.00\%$ | $0.18\%$ | $0.31\%$ | $0.86\%$ | $2.22\%$
| 9 | $0.04\%$ | $0.10\%$ | $0.45\%$ | $0.49\%$ | $1.22\%$
3-hour | 0 | $2.08\%$ | $9.74\%$ | $20.80\%$ | $31.30\%$ | $41.96\%$
1 | $0.27\%$ | $1.00\%$ | $4.08\%$ | $6.25\%$ | $11.69\%$
2 | $0.00\%$ | $0.22\%$ | $0.18\%$ | $0.54\%$ | $1.45\%$
3 | $0.36\%$ | $1.63\%$ | $2.72\%$ | $4.80\%$ | $9.15\%$
4 | $0.09\%$ | $0.45\%$ | $1.72\%$ | $3.27\%$ | $5.66\%$
5 | $0.14\%$ | $0.50\%$ | $1.22\%$ | $3.53\%$ | $5.89\%$
6 | $0.04\%$ | $0.36\%$ | $0.58\%$ | $1.40\%$ | $2.62\%$
7 | $0.27\%$ | $1.54\%$ | $3.72\%$ | $7.52\%$ | $10.24\%$
8 | $0.63\%$ | $3.27\%$ | $7.11\%$ | $10.87\%$ | $13.18\%$
| 9 | $0.27\%$ | $1.04\%$ | $2.22\%$ | $5.21\%$ | $8.47\%$
First come - First serve | 0 | $4.17\%$ | $17.68\%$ | $38.70\%$ | $54.64\%$ | $61.03\%$
1 | $0.77\%$ | $2.36\%$ | $7.11\%$ | $12.00\%$ | $20.29\%$
2 | $0.04\%$ | $0.36\%$ | $0.58\%$ | $2.53\%$ | $3.12\%$
3 | $0.86\%$ | $3.49\%$ | $6.57\%$ | $10.74\%$ | $16.31\%$
4 | $0.13\%$ | $1.99\%$ | $4.17\%$ | $6.62\%$ | $11.28\%$
5 | $0.22\%$ | $1.04\%$ | $5.71\%$ | $13.68\%$ | $24.33\%$
6 | $0.22\%$ | $0.63\%$ | $1.31\%$ | $2.45\%$ | $6.57\%$
7 | $0.54\%$ | $2.62\%$ | $7.43\%$ | $12.96\%$ | $20.16\%$
8 | $1.27\%$ | $5.2\%$ | $12.96\%$ | $22.33\%$ | $31.17\%$
| 9 | $0.59\%$ | $2.22\%$ | $4.80\%$ | $11.69\%$ | $16.67\%$
Table 6: Percentage of hours where the observed demand is censored across the
different queues, penetration rates and clusters.