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1
A Scalable Stochastic Model for the Electricity Demand of
Electric and Plug-in Hybrid Vehicles
Mahnoosh Alizadeh∗, Anna Scaglione∗, Jamie Davies†, and Kenneth S. Kurani†
Abstract—In this paper we propose a stochastic model, based
on queueing theory, for Electric Vehicle (EV) and Plug-in
Hybrid Electric Vehicle (PHEV) charging demand. Compared
to previous studies, our model can provide 1) more accurate
forecasts of the load using real-time sub-metering data, along
with the level of uncertainty that accompanies these forecasts;
2) a mathematical description of load, along with the level of
demand flexibility that accompanies this load, at the wholesale
level. This can be useful when designing Demand Response and
Dynamic Pricing schemes. Our numerical experiments tune the
proposed statistics on real PHEV charging data and demonstrate
that the forecasting method we propose is more accurate than
standard load prediction techniques.
I. INTRODUCTION
Due to environmental and economic factors, the use of
electric vehicles (EV) and plug-in hybrid EVs (PHEV) is
expected to rise considerably in the near future [1]. Hence, in
recent years, several researchers have examined the potential
impacts of a large-scale integration of EVs on the power grid
(for a comprehensive review see [2]). For example, in [3], PJM
ran market model simulations to determine the effects of the
addition of one million EVs to its load under 3 different load
management scenarios. In [4], the electricity demand of the
CAISO CAMX (California Independent System Operator for
the California-Mexico Power Area) region, which accounts for
80% of the load of California, is examined under several EV
adoption rates. In [5], [6], the effects of EVs on distribution
network losses and voltage levels are examined.
Mapping a given level of EV penetration into a diurnal load
pattern requires a model for customer charging behavior. The
first step in providing such a model is to answer the following
essential questions: 1) when do vehicles arrive where a charger
is available? 2) how often do customers request battery charge
when their vehicle is parked? 3) how much energy is required
per each charge event? 4) how much flexibility accompanies
each charge request?
Due to the scarcity of real-world data, and pressed by the
need to assess the effects of EV load on the grid, most of the
present literature hypothesizes simple models of aggregate ar-
rival rates of EVs and their requests to charge, sometimes using
internal combustion engine vehicles (ICEVs) travel patterns.
Historical ICEV travel patterns are used in [7], [8], which
propose a set of rules to map ICEV data into synthetic EV
load traces that can then be used for Mote Carlo simulations.
Attempts at mathematically modeling the stochastic process of
battery charging vary in sophistication. For example, in [4], EV
∗Department of Electrical and Computer Engineering, University of Cal-
ifornia Davis, email: malizadeh@ucdavis.edu , †Institute of Transportation
Studies, University of California Davis. This work has been funded by DOE
under CERTS. Parts of this work was presented at Allerton 2010.
arrival times are modeled using a normal distribution with a
mean of 6 p.m. and a standard deviation of 30 minutes. Both
[9] and [10] develop a complete stochastic model for the load,
by modeling arrivals of charge requests as a Poisson process
and using queueing theory to derive the statistics of the load.
However, neither of these models are based on real data. In
fact, [9] simulates the arrival of vehicles as a homogeneous
Poisson process and proposes to model the length of charge
requests as an exponentially distributed. These, as well as
conjectures in [10], are not supported neither by the real
charging data, nor by the synthetic traces generated in [7], [8].
We showcase these discrepancies in our numerical results.
Our contribution: In this paper, we answer the essen-
tial questions mentioned above, observing the characteristic
features of real PHEV charging data [11], and provide a
stochastic mathematical model for EV/PHEV aggregate load.
Having access to such a model has key benefits that go beyond
assessment of impacts of future EV load on the power grid
via dynamic simulations. The model is useful for providing
more accurate short-term load forecasting, especially when
real-time sub-metering data is available. Crisper short-term
forecasts of the volatile charging load of EVs in real-time can
help the system operator in dispatching generation with the
lowest possible costs in the electricity market and becoming
less dependent on ancillary services. Furthermore, modeling
the statistics of the load can facilitate planning and operation
for a demand response (DR) aggregator that manages battery
charging of vehicles. There is a growing literature on EV de-
mand scheduling, e.g. [12], [13], [14], [15], that relies on such
statistics. Last but not least, thanks to its scalability, our model
helps describe the flexibility of EV/PHEV electricity demand
to the system operator (see [16] for a detailed explanation).
Note that our model presumes that we are looking at a large
population of vehicles (theoretically infinite), with probability
of a charging in the interval [t, t +dt)that is in the order
of λ(t)dt. The rate function λ(t)captures aggregate customer
behavioral characteristics, and can vary widely if one considers
a small population. Hence, this model is not suitable to capture
the PHEV/EV load at a feeder for a few houses, but can
be used for substations and charging stations. Also, we leave
for future work the inclusion of geographical information that
would allow us to understand the fraction of demand supported
by a certain portion of the grid, and therefore ensure that
physical power system constraints are met (e.g., transformer
capacity limits). Either through the car Global Positioning
System receiver, or through smart-phones applications, the
vehicle location can be made available, and it will enrich
the modeling and prediction engine. Location data would also
allow us to estimate carbon dioxide emissions associated with
the increase in electricity generation to charge the vehicles.
2
The paper is organized as follows. In Section II, we provide
a queuing model to capture the load due to EV/PHEV charging
that is not centrally controlled. We showcase how this model
can be used for load forecasting in Sections II-A and II-C.
Next, in Section III, we describe how the statistical information
required to use our model can be estimated from sub-metering
data. In Section IV-A, we introduce a scalable architecture for
telemetry, monitoring, and management of charging requests.
We argue that our model can fit arrival and charging data that
are influenced by DR strategies, most notably pricing and di-
rect load management (respectively in Sections II-B and IV-B).
Next, we proceed to give preliminary numerical estimates of
the model parameters using real-world PHEV home charging
data in Section V. Lastly, in Section VI, we numerically assess
the performance of our real-time forecasting algorithm.
II. UN CO NT ROL LE D EV L OAD MODELING
Queuing theory is often used to mathematically analyze
the effects of customers randomly arriving and being served
by a system. The process of EVs being randomly plugged
in and charged by the grid fits this paradigm. To model
the charging load imposed by EVs on the grid, we assume
that upon plugging in their vehicle, customers pick from a
set of possible charging levels. The instantaneous charging
power (rate) then remains constant throughout the charge. For
example, standard charging rates for electric vehicles include
level 1 (home) charging with a 1.1 kW rate, and 3.3 kWs
or 6.6 kWs for level 2 (home and workplace) charging. To
capture the temporal variations of demand due to EV charging
that is not centrally controlled, we utilize a set of Mt/GIt/∞
service systems, each representing a different charging rate.
For example, with the three mentioned charging rates, the
aggregate demand can be modeled as the workload of three
service systems. Scenarios that fall into this category include
unmanaged EV charging load, and EV load under rate-based
load shifting incentives such as dynamic pricing.
Queueing service models are traditionally represented using
Kendall’s shorthand A/B/C notation, in which the first term
(A) denotes the interarrival time distribution, which is a
time-dependent Markovian (Poisson) distribution in our case,
represented by the standard notation Mt. The second term
(B) denotes the service time (charging duration) distribution,
which is a general time-dependent distribution here, repre-
sented by GIt. Note that the service times are assumed to be
independent and identically distributed. Finally, the third term
(C) is the number of servers. The number of servers in the
model Mt/GIt/∞is assumed to be infinite. This is because,
when no central demand control technique is exercised, each
EV is provided with energy immediately as it arrives in the
system and no queue is actually ever formed. The number of
customers receiving service from each queue, multiplied by the
associated charging rate of that queue, provides us with the
total power consumption of vehicles served under that queue.
Our model is based on the following elements:
1) Assignment to queues: we assume that customers may
use one of Γdistinct charging rates, which, as mentioned
above, translates into being assigned to one of Γservice
Fig. 1. Γparallel Mt/GIt/∞service systems for vehicle battery charge
systems. We denote the charging power of the vehicles
using charging rate γas ¯
Rγ, and assume that this power
is constant throughout the charge cycle. Thus, if we
denote the total number of vehicles receiving service
from the γ-th queue at time tas Nγ(t)(see Fig. 1), we
can write the aggregate charging load at time tas
L(t) =
Γ
X
γ=1
¯
RγNγ(t),(1)
2) Poisson arrivals: we model the events of vehicles
arriving in queues to request charge as a point process
(a series of random arrival times ta
j, with separate events
indexed by j). Thus, for arrivals in each service queue,
we consider a non-homogeneous Poisson process with
a random arrival rate λγ(t). We show in Section V-E
why we think this is an appropriate model for charging
events. In a Poisson process, the arrival of one customer
carries no information about the arrival of others;
3) Service times: each plugged-in vehicle has an associated
duration of charge S. We denote by Ft,γ
S(s)the Cu-
mulative Distribution Function (CDF) of the duration of
charge Srequired by each arriving vehicle using the γ-th
charging rate at time t, considered to be independent and
identically distributed for customers in the same queue.
Next, using assumptions 2 and 3, we find the statistics of
Nγ(t)and thus, that of the charging load in (1). As mentioned,
these statistics could be beneficial for load forecasting and
demand response scheduling.
A. Forecasting Load on the Previous Day or Earlier
Here we forecast the load by calculating its statistics,
including its expected value, which we denote by E[L(t)].
Note from (1) that we will have a full statistical description
of L(t)if we can learn the probability distribution of Nγ(t).
Since the arrival rate of customers in queues and their charge
durations is non-homogeneous, the statistics of Nγ(t)are time
varying. We begin by assuming that the system started at
t=−∞. Then, we can use results in [17] to find the statistics
of Nγ(t). This assumption is usually made for initializing
non-stationary models and is appropriate for forecasting EV
charging load on the day-ahead, since EV charging cycles are
limited to a maximum of 10-12 hours. The most interesting
result for a Mt/GI/∞system, proven by [18],[19], is that
3
Fig. 2. Arrival rates are modified by setting load shifting rate incentives
Nγ(t)is a Poisson random variable with mean mγ(t)given
by:
mγ(t) = E[Nγ(t)] = Zt
−∞
(1 −Ft,γ
S(t−u))λγ(u)du, (2)
which we can calculate if we are provided with estimates of
λγ(t)and Ft,γ
S(s). The integral is basically summing up the
expected number of arrivals at all times u<twhich are
expected to still be receiving service at time t, i.e., their service
time is expected to be longer than t−u.
We can now combine (1) and (2) to say that the aggregate
load L(t)of the vehicles on the next day is the sum of Γ
scaled Poisson random variables, with time varying mean
E[L(t)] =
Γ
X
γ=1
¯
Rγmγ(t).(3)
B. The Effect of Demand Response Techniques: Rate-based
Load Shifting Incentives
It is widely believed that without proper management, the
load due to electric vehicles can cause price spikes and de-
crease safety margins in the power grid [20]. In order to study
the effects of rate-based charge incentives at specific times or
locations on the aggregate load, we need to further expand
our model to capture how these incentives can influence the
arrival rates λγ(t), i.e., how customers make a decision to plug
in their EV in response to dynamic tariffs. Thus, we define a
new mapping that models these influences:
I:{λγ(t), γ = 1,...,Γ} 7→ {µγ(t), γ = 1,...,Γ},
where µγ(t)denotes the modified customer arrival rates due
to the incentives (see Fig. 2). The mapping Ican in general
be non-linear, have memory, and be dependent on various
parameters such as location, grid conditions, time of day,
etc. Customers may be incentivized to change their charging
behavior through dynamic pricing or cheaper charging rates
available at work or public charging stations.
After learning the mapping I, potentially from empirical
econometric studies, one can simply use (2) and (3) to map
these modified arrival rates to aggregate load:
E[L(t)] =
Γ
X
γ=1
¯
RγZt
−∞
(1 −Ft,γ
S(t−u))µγ(u)du. (4)
Note that there exists another category of demand response
techniques where vehicle charging is directly controlled by
a central control unit. The effects of these techniques on our
model and on the forecast will be later studied in Section IV-B.
Next, we use this same model to forecast the load accurately
in real time.
C. Real-time Load Forecasting
During real-time operations, the forecast unit can use a
metering infrastructure to harness side information about the
exact duration and rate of charge requested by vehicles once
they are plugged in. Thus, to provide an accurate load forecast,
the distribution of Nγ(t)shall no longer be initialized at
t=−∞. Rather, the forecast unit can use the latest informa-
tion available on Nγ(t). Here we assume that the information
on the queue index γand the charge length Sfor each charge
event is revealed through sub-metering when the car is plugged
in, since the initial battery level allows to establish beforehand
how much time is needed for a full charge. When a full charge
is not possible, we assume that the amount of charge that can
be delivered before the customers depart is explicitly revealed
using sub-metering data.1This is important because there is
no uncertainty about the service time of each vehicle. Hence,
through telemetry, it is possible to determine exactly how many
cars, among the ones that have previously started charging,
will remain active at a future time t. Thus, the only uncertainty
in load forecasting will be represented by the incoming new
requests between the current time t◦and future time t.
This approach is what we call the Observable Arrival
Information (OAI) predictor. Specifically, let the present time
be t◦, and Nγ(t◦)be the customers present in the γ-th queue.
Then the OAI predictor models the number of cars present in
the system at time t>t◦as:
Nγ(t) = Nγ
new(t) + νγ(t),(5)
where Nγ
new(t)is the number of vehicles that arrived in the
queue after t◦and are actively charging at time t.Nγ
new(t)
is a random variable and the problem of determining its
distribution is similar to the one we looked at in Section
II-A. Thus, Nγ
new(t)is a Poisson random variable with the
mean given in 2, with λγ(t)set to zero for t≤t◦. The term
νγ(t), on the other hand, is deterministic and equal to Nγ(t◦)
minus the number of cars that departed (i.e. completed their
charge) before time t. We need to calculate this term using sub-
metering information. Let us denote the set of cars that arrived
in the γ-th queue before time t◦by Pγ
t◦, and the arrival time
and charge duration associated with the j-th charge request as
(ta
j, Sj). Ignoring the communication network delay, we can
write νγ(t)as,
νγ(t) = X
j∈Pγ
t◦
u(Sj−(t−ta
j)), t ≥t◦(6)
with u(.)as the unit step function. Given this, the real-time
forecast of load due to battery charging is given by
E[L(t)] = PΓ
γ=1 ¯
RγE[Nγ(t)] =
PΓ
γ=1 ¯
Rγ(E[Nγ
new(t)] + νγ(t)) (7)
1If the departure time of the customer is considered random, this technique
will still work but yet another level of uncertainty should be modeled (jobs
dropping out of the queue, before they are fully served).
4
Notice that in order to use (3) and (7) to provide day-
ahead and real-time load forecasts, the following information
is required:
1) The PDF of charge lengths for each charging rate
(queue), Ft,γ
S(s);
2) Forecasts of future arrival rates of vehicles to request
charge at different charging rates (queues), λγ(t);
3) The triplet (γ , Sj, ta
j)for each vehicle already plugged
in, acquired from sub-metering. This information is
specifically required to calculate νγ(t).
In the next section, we address how a load forecast unit
can acquire the required statistics mentioned above, namely
Ft,γ
S(s)and λγ(t), from real-world data. These estimates
should be updated adaptively to account for temporal and
geographical variations in customer behavior. Besides Ft,γ
S(s)
and λγ(t), one can also look at the statistics of laxity (flex-
ibility) of charge requests, which is not directly used by the
forecast unit but is relevant for demand response purposes.
After addressing these in Section , in Section IV-A, we propose
scalable model to capture and aggregate the triplets (γ, Sj, ta
j)
with low communication and storage requirements.
III. LEA RN IN G TH E STATISTICS
Given a large sample set of submetered data on vehicle
charge requests, estimating Ft,γ
S(s)is a straightforward task.
We can either provide rescaled histograms, or find analytical
distributions that fit the data well enough. In order to pro-
vide non-homogeneous distributions, we can assign different
distributions to different hours of the day or week.
However, the real challenge lies in estimating future values
of the arrival rates λγ(t). The complexity is twofold: 1) current
values of λγ(t)are not observable. The information we can
gather from the metering infrastructure provides the number
of vehicles that actually plugged in and requested charge
from a certain queue γ, which is a specific realization of the
random Poisson arrival process with mean λγ(t); 2) future
values of λγ(t)are random and correlated with its historical
values. Fortunately, the same problem is faced in researching
techniques to forecast future call volumes for call centers with
stochastic inhomogeneous demand [21], [22], [23], [24]. The
most popular model for the call arrivals is an inhomogeneous
(or piece-wise constant) Poisson process with a random arrival
rate [23]. We show in Section V-E why we think this model
is also appropriate for EV charging events.
Thus, next, given historical information about the number of
vehicles plugged in at each charging rate, we wish to estimate
the arrival rate λγ(t)for a future time t. For brevity and ease
of notation, in the rest of this section, we drop the superscript
γand estimate λ(t)for only one charging rate.
A. Estimating and Forecating λ(t)from Historical Data
We wish to estimate the inhomogeneous rate function λ(t)
that fully specifies the statistics of the EV Poisson arrival pro-
cess. Since the customer behavior depends on many underlying
factors, surely this rate is not best modeled as a deterministic
function of the day of the week. It will have a random
evolution that we need to predict on a daily basis, similar
to classical load forecasting techniques. The assumption that
this rate is random, on top of the random nature of Poisson
arrivals, leads to a doubly stochastic model for the arrivals.
Here, we approximate the arrival rate as being piecewise
constant for intervals lasting (24 hours)/K (e.g. quarter or
half hours) and we divide the day into Kdiscrete epochs.
Consequently, we denote the arrival rates on day jby a vector
λj= [λ((j−1)K+`)]`=0,...,K−1.
The goal of this section is to provide a method to fore-
cast the vector λjfor a future day. Successive values of
the arrival rate are not independent and give valuable in-
formation for predicting future values of the series. One
important tool to predict future values of such sequences is
time series analysis [25]. However, since the rate profiles
are unobservable, we need to build our forecasting model
on the corresponding count profiles, which for day jare in
our model given by the vector cj= [c((j−1)K+`)∼
Poiss(λ((j−1)K+`))]`=0,...,K−1. Note that our ultimate
goal in observing the count profiles cjis to estimate the rate
vectors λjof our statistical model.
Since the counts c(t)are realizations of Poisson random
variables with different means λ(t), their associated variances,
which is identical to the mean values, is time-varying as
well. In order to solve numerical issues that arise from this,
we can use a simple variance-stabilizing transformation. For
example, we can follow the suggestion in [23] and use a
slightly modified version of the Anscombe square-root trans-
form [26], which transforms Poisson distributed data like c(t)
to approximately normally distributed data with a constant
variance of 1/4. The transformation is as follows:
c(t)∼Poiss(λ(t)) approx.
−−−−−→ x(t) = rc(t) + 1
4∼ N(θ(t),1
4),
(8)
where θ(t) = pλ(t). Following our previous notation, we
also denote the vectors containing transformed counts x(t)
and their means θ(t)for day jas xjand θj.
Now, there are two types of forecasts that have to be
performed: 1) predict the arrival rates for the next day from
historical data. This is referred to as inter-day forecasting; 2)
dynamically update the forecasts during the day as new data
becomes available in real-time. We call this intra-day forecast
updating. Next, we address how we can provide such forecasts
from historical and real-time data.
1) Interday Forecasting: Transformed count vectors xi
and xjrepresenting different days are statistically dependent
random variables. However, we consider them conditionally
independent given the latent transformed rate vectors θiand
θj. As we saw, θj’s are unobservable (latent) variables and
need to be estimated. The first step in addressing this problem
is to adopt a model that can describe the correlation between
successive values of θj’s. Time series models are a popular
choice here. However, due to the large dimension of the
vectors θj, multivariate time series models may not be the best
way forward. Thus, to model inter-day dependencies among
the θj’s, we adopt a model similar to what is proposed in [24]
and we first reduce their dimension, using a low dimensional
factor model to describe the correlation between the arrival
5
rates on different days. Specifically, we assume:
θj=
κ
X
i=1
αj(i)ui,(9)
with κ < K and u0
iuj= 0 for i6=i0. To minimize the residual
mean squared error associated with a given κ,ui’s can be
chosen to be the principal components of the variables θj. The
ui’s are assumed time-invariant in (9) and have a much slower
dynamics than αi(j). Alternatively, if we define the vector
aj= [αj(1), . . . , αj(κ)]Tand the matrix U= [u1,...,uκ],
(9) can be written as
θj=Uaj.(10)
This approach maps the latent vector variable θjinto κ
scalar latent variables αj(i). The correlation between suc-
cessive values of θjis modeled through the αj(i). Due to
the characteristics of singular value decompositions, a vector
containing consecutive values of αj(i)’s for different days j
would be orthogonal to that of αj(m)for m6=i. Thus, it is
reasonable to assume that αj(i)and αj(m)are uncorrelated
and the series αj(i)can be modeled using separate univariate
time series models. Denote by the index tj= 1,...,7the day
of the week corresponding to day j. In our model we assume
that αj(i)’s will have a periodic mean of ¯αtj(i), corresponding
to the day of the week. Then, if we adopt an AR(1) model for
each αj(i), we can write
αj(i)−¯αtj(i) = βi(αj−1(i)−¯αtj−1(i)) + wj(i),(11)
where wj(i)∼ N(0, σ2
i)is white noise and is uncorrelated for
different i’s. Higher degree models can be adopted if proven
to be necessary by real-world data. In an AR(1) model, the
transformed daily count vector xjis conditionally independent
of all other count vectors in the historical dataset given θjfor
day jand the previous day’s transformed count vector xj−1.
Rewriting (11) for all i= 1, . . . , κ in vector form gives
aj−¯
atj=D(aj−1−¯
atj−1) + wj,(12)
where ¯
atj= [¯αtj(1),..., ¯αtj(κ)],D= diag[β1, . . . , βκ], and
wj= [wj(1), . . . , wj(κ)].
With (11), our model is complete. For given values of the
parameters (σi,¯αtj(i), βi,ui), we can estimate the θj’s from
the transformed count vectors xjand use these estimates to
forecast future arrival rates. By assigning normal diffuse priors
to α1(i), i = 1, . . . , κ, this estimation would be straightfor-
ward using classical Bayesian techniques such as maximum
a posteriori (MAP) or minimum mean square error (MMSE)
estimators [27]. In fact, with an AR(1) model, this can be
easily done adaptively on a day-to-day basis, using standard
Kalman filtering iterations [28]. Define yj=xj−U¯
atjas
the transformed count vector with the average weekly trend
removed. Then, (8),(10) and (12) can be combined to write
the dynamics of yjas
a0
j=Da0
j−1+wj,(13)
yj=Ua0
j+vj,(14)
where yj’s are the observations, vjis zero mean Gaussian
white noise ∼ N(0,1
4I)and is independent from wj, and
a0
j=aj−¯
atj. The dynamical model in (13) is in accordance
with the framework of the Kalman filter.
In summary, in order to forecast arrival days for the next
day (or later) on day j, one must perform the following steps:
1) Transform the observed count vector cjto xjvia (8);
2) Remove the weekly trend U¯
atjfrom xjto get yj;
3) Apply the Kalman filter update step to estimate a0
j;
4) Apply the Kalman filter predict step the predict a0
j+1;
5) Add the weekly component ¯
atjto forecast aj+1
6) Map this forecast of aj+1 into a forecast of the trans-
formed rate θj+1 using (10);
7) Apply the inverse Anscombe transform to the forecast
of θj+1 to get a forecast of the actual rate vector λj+1
However, note that we are not provided with the model
parameters (σi,¯αtj(i), βi,ui). We have to learn them from
historical data. Due to the presence of latent variables λj, and
because of the non-linear nature of the model as a function of
the unknown parameters, this is not a straightforward problem
to solve analytically. Several approaches exist for calculat-
ing Maximum Likelihood (ML) estimates of the parameters
(equivalent to a MAP estimate with a uniform prior). We
initialize the model by assigning the first κsingular vectors
of the historical transformed count matrix to the vectors
ui. The rest of the parameters can then be estimated using
iterative parameter estimation techniques with latent variables
(an Expectation Maximization framework [29]). Note that the
eigen-structure of the rate profiles can be better estimated as
well once our estimates of the model parameters improve.
Thus, the vectors u1,...,uκare also updated dynamically
using recursive subspace tracking methods [30].
2) Intraday Updates: Now we have a forecast of the arrival
rates for the next day (indexed by j+ 1). During real-
time operations, the aggregator in charge of the vehicles will
gradually observe the realized values of the random daily
arrival counts, transformed to yj+1, through the metering
infrastructure. At hour mof day j+ 1, the first melements of
yj+1 is available, which we denote by y(m)
j+1. As these entries
become available, the load forecast unit can update its forecast
of λj+1. The update can be carried out in two steps:
1) Direct estimation: if we denote by U(m)the first mrows
of Uand by v(m)
j+1 the first melements of vj+1, then
y(m)
j+1 =U(m)a0
j+1 +v(m)
j+1 (15)
can be used to directly estimate estimates of a0
j+1
from y(m)
j+1 and thus, λj+1 using steps (5), (6) and (7)
mentioned above. Denote this estimate by ˆ
λintra
j+1 .
2) Penalized updating: With low amounts of arrival count
data, the direct estimation step is not accurate. Thus, we
can use a weighted sum of ˆ
λintra
j+1 and the day ahead
forecast as our estimate of λj+1.
IV. DIG ITA L (QUAN TI ZE D) SUBMETERING AND DIS PATCH
A. Classification of Charge Requests
In Section II-C, we proposed a real-time forecasting tech-
nique in which forecast errors were reduced by incorporating
submetering data. We divided the charging load at the future
6
Fig. 3. Quantized sub-metering can help facilitate real-time forecasting and
demand response. Each charging rate γis mapped into Qγqueues.
time tinto two parts: load due to vehicles that arrived before
the current time t◦(denoted by set Pγ
t◦), and load due to
vehicles that will arrive between t◦and t(denoted by set
Pγ
t− Pγ
t◦). Thus, the load at a future time is given by
L(t) =
Γ
X
γ=1
¯
RγNγ(t)(16)
=
Γ
X
γ=1
¯
RγNγ
new(t) + νγ(t)
=
Γ
X
γ=1
¯
RγX
j∈Pγ
t◦
u(Sj−(t−ta
j)) +
X
j∈Pγ
t−Pγ
t◦
u(Sj−(t−ta
j)),
where νγ(t)is deterministic and completely observable from
submetering data. To track its value and improve the forecasts
of L(t)as described in Section II-C, we need the analog
triplets (γ, Sj, ta
j)for each charge request.
But, high telemetry and storage costs and modeling com-
plexity are two faces of the same coin. Thus, it is desirable
to reduce the burden of communicating and storing the triplet
(γ, Sj, ta
j)for every single vehicle. Similar to the basic princi-
ples of all digital communication systems, in [31] we proposed
to quantize the tuple (γ, Sj). This is carried out through a set
of classifiers Ψγ(Sj)that quantize the charging durations Sof
vehicles using the γ-th charging rate onto a set of Qγpossible
values Cγ
q, q = 1, . . . , Qγ. The effect of this clustering is
twofold: 1) it obviously provides a digital representation of the
charging load that can be communicated; 2) it separates the
charge requests into Q=PγQγclasses of service. We denote
the class of all vehicles that use a charging rate γand charge
duration qby a tuple (γ, q ). This classification of demand
highly reduces out storage costs, as well as computational
costs when using demand response techniques later in Section
IV-B. We also choose to discretize time since forecast updates
are not carried out continuously and, thus, we do not need to
store the state of the queues at every time instant. Hence, from
this point on, tis used to denote the index of discrete time
epochs equally distanced by 4.
With this quantized charge description, at each epoch t, the
new information that needs to be stored is the number of new
arrivals in each class (γ, q)since the previous epoch, which
we denote by aγ
q(t). Consequently, νγ(t)can now be written
as,
νγ(t) =
Qγ
X
q=1
t◦
X
k=t◦−Cγ
q
aγ
q(k)Π t−k
Cγ
q,(17)
where Π(.)denotes a unit pulse function between [0,1].
Similarly, if we expand the summation over time beyond the
current time t◦, we can model Nγ(t) = Nγ
new(t) + νγ(t)as,
Nγ(t) =
Qγ
X
q=1
t
X
k=t−Cγ
q
aγ
q(k)Π t−k
Cγ
q,(18)
where the arrival counts aγ
q(t)beyond t◦are not known yet
but can be forecasted.
There are many benefits in using (18) over (16), the most
notable of which is scalability. No matter how large the vehicle
population is, we only need to store Qnumbers per time index
tto keep track of the sub-metered load. Also, when simulating
EV load, we only need to generate QPoisson random variables
per time index tto generate random vehicle arrivals, which is
extremely simple.
B. The Effect of Demand Response Techniques: Controlled
Activations
The proposed load classification can also be used to charac-
terize the dispatchability of the EV/PHEV load by an aggrega-
tor. An aggregator that wants to apply a demand management
technique needs to model how its control signals will affect
the load. In Section II-B, we saw that the effects of dynamic
pricing techniques can be captured through modeling their
effects on the aggregate arrival rates λγ(t). Another scenario
is that vehicle charge requests can be managed by a central
unit that directly controls the exact times at which vehicles
charge their batteries.
Under a direct demand management program, the number
of requests that start being served by each queue at every time
instant is controlled and customers that arrive in the grid to
receive charge need to wait to receive an authentication from
the control unit. This can be captured by denoting the number
of vehicles that start charging at time tfrom the (γ, q)-th
service class by dγ
q(t), while a number of vehicles are held
in the queue (see Fig. 3). With this new definition, we can
rewrite the aggregate charging load in (16), under a central
demand control program, as
L(t) =
Γ
X
γ=1
¯
Rγ
t
X
k=t−Cγ
Q
Qγ
X
q=1
dγ
q(k)Π t−k
Cγ
q.(19)
Consequently, studying the effects of different demand man-
agement techniques on the load would translate into under-
standing how the customer’s initial arrival pattern aq(t)is
affected by the control signal and yields dq(t). We denote the
function that describes this relationship as F(.). The arguments
of this function can be as simple as just aq(t), in the case of,
7
Fig. 4. Smart load forecasting with DR
for example, a heuristic threshold based demand management
program holding back demand if it exceeds a certain value. At
the other extreme, the control algorithm, described by F(.),
can have many input arguments and may not be an analytical
function, but an optimization problem that should be solved
numerically (see e.g. [31]).
The description in (19) can be generalized to also account
for Vehicle-to-Grid (V2G) scenarios. To do so, we can assume
that appliances waiting to receive service from each queue
can choose between various service options. For example, a
vehicle requiring charge for one hour can be charged and leave,
or it can discharge the battery now and receive a longer charge
afterwards. The details of how this can be achieved are beyond
the scope of this paper and will be addressed in future work.
Note that, in the case where a DR aggregator is manipulating
the access rate of vehicles to receive charge, the OAI predictor,
which merely used estimates of the arrival rates aq(t)to
predict the future load, now needs to forecast the values of
dq(t)instead. There are two possible cases: 1) the load forecast
unit has an analytical description of how the dq(t)’s will
be determined in response to an arrival rate of aq(t), due
to F(.); 2) the forecast unit needs to learn the behavior of
each aggregator’s demand control scheme from historical data
available on the aq(t)and dq(t)’s, i.e., estimate the function
F(.). The required interactions to forecast load in these cases
is shown in Fig. 4. Again, this can be easily extended to
include multiple charging rates or general charging pulses g(.).
V. NUMERICAL EX PE RI ME NT S - PART I:
TUNING MOD EL PARAMETERS TO HISTORICAL DATA
Now that we have described how the system operator or
an aggregator of electric vehicles can use sub-metering data
to model the full statistics of charging demand for load fore-
casting or demand response purposes, we wish to numerically
calibrate the parameters of our model to historical data. We
provide these initial estimates with the caveat that they can be
further improved when more data is available, considering that
they might prove useful as a reference for research purposes.
Due to the limitations of the real-world data we have access
to, here we only look at level-1 home charging of PHEVs (Γis
equal to 1 since we only have one charging level). Hence, for
brevity, we eliminate the subscript γfrom our notation from
this point on.
A. Data set description
We will compute the parameters of our model as follows.
The probability density function (PDF) of charge durations
and laxity (slack time) of PHEV charging is based on data
from PHEVs driven and charged by households [11], provided
by the UC Davis PH&EV center. These data confirm that
the random number of vehicles being plugged in per unit of
time, which we refer to as vehicle arrival process, has time-
varying statistics. However, these statistics cannot be accu-
rately learned from the PHEV database, due to its small size.
Thus, we propose a new methodology to derive the parameters
of the arrival model using ICEV data. More specifically, we
infer a second distribution of charging events using the 2009
National Household Travel Survey (NHTS) [32], emulating
the example of [7] in building synthetic traces.
The NHTS gathers information about daily travel patterns
of different types of households. The available data include
mode of transportation, duration, distance, and purpose of
travel for ICEV owners. The specific parameters found for this
portion of the model may be unrealistic for PHEVs and EVs
in three ways. First, as [11] and [33] discuss, the probability
that a PHEV or EV is charged upon its arrival at a charging
location (taken to be home in this instance) 1) decreases with
a longer driving range per battery charge; 2) increases with the
distance the vehicle has been driven since its last charge; 3)
increases with the time duration the vehicle is parked before
it is driven again; and 4) increases as a function of how far
the vehicle is expected to be driven subsequent to the charge.
Because the entries in the NHTS travel data each cover only
one day, the last point in particular cannot be incorporated
into any charging probability distribution derived from [32],
if charging is based on an expectation of the next day’s
travel. Second, we are not as sanguine as [7] that deviations
between the observed travel in [32] and travel by PHEVs and
EVs will decline in the future. Observations from [11] and
[33] indicate that households change driver assignments to
vehicles, trips, and activities, choose new locations, and buy
different vehicles in response to the new driving and charging
characteristics of PHEVs and EVs. That is, we view it as
more likely that any future NHTS containing large numbers of
PHEVs and EVs will be different from, not similar to, any data
consisting primarily of ICEVs. Third, the amount of charged
required by the vehicle is a function of several parameters
such as driving/road conditions, driving habits, vehicle model,
battery condition, dynamic prices, climate control settings,
and user selected modes (charge depleting, blended, or charge
sustaining). We cannot account of all of these parameters
when mapping NHTS to PHEV charge requests. Although all
of these issues will affect the statistics for PHEV charging
events derived here, this does not limit our model from being
re-parameterized when new larger data sets of PHEV or EV
charging become available.
B. Statistics of Charge Duration S
Remember that in this work, the charging period of each
vehicle is modeled as an independent identiacally distributed
random variable Sfor vehicles plugged in at the same charging
8
TABLE I
ENE RGY P ER T RAVEL ED MI LE
Vehicle Type kWh per mile
Sedans 0.18 - 0.3
Vans 0.3 - 0.4
SUVs 0.4- 0.5
Trucks 0.5 - 0.7
level and time. In this section we provide an appropriate PDF
for Sfor the case of PHEV level-1 home charging. This
answers the third essential question posed in the introduction:
how much energy is required per each charge event?
Using level 1 charging, the full charging time of our studied
PHEVs would be between four and five hours (four to six
kWh of battery capacity). The actual power to charge typically
starts at higher than the average rate and tapers off as the
battery approaches a full charge. We ignore this change (as it
has little impact on the duration of charging) and assume the
PHEV charging power is equal to an average of ¯
R= 1.1kW,
consistent with our real-world data.
Fig. 5 shows the probability distribution function of charge
durations Sjderived from the PH&EV center data (E[Sj] =
189 minutes). We tested numerous distribution functions that
seem to fit the charge duration data. We found that a lognormal
distribution ln N(5.03,0.782)best fits the data. Thus, we
conclude that a suitable PDF to model the charge duration
of PHEVs under level-1 home charging is
fPHEV
S(s)≈1
0.78√2πs exp−(ln s−5.03)2
2∗0.782(20)
where the unit of sis minutes. The expected value of this
random variable is 207 minutes. The q-q plot in Fig. 6
compares the distribution to the samples. Another step to
further improve this fit is to clip the above PDF at a maximum
of 300-350 minutes. To see how much this improves the fit,
we cut the PDF at 320 minutes and distributed the samples
that were above this value uniformly in [280,320] min (see
Fig. 6).
Here, we cut off the distribution at a specific charge dura-
tion, which translates into a specific battery size. In the case
of different battery sizes we expect that either a mixture of
lognormal RVs or a single distribution with a tail that combines
the cutoff effects of different battery capacities will fit the data.
One important issue that we wish to bring forward is that
most of the home charging events in this dataset happen after
4 pm and continue throughout the evening and night hours.
Thus, we do not have a large enough number of samples
from charging events that happened during morning hours
and the ones we have are shorter than the charges at night
hours. Hence, the PDF in (20) is most appropriate to fit night
charging events and the distribution fPHEV
S(s)may not be
stationary and could be a function of time. However, when
we studied the distribution of mileage driven before returning
home from NHTS entries, and divided the night and day hours
data, we observe very similar statistics. This suggests that any
method converting travel miles into home charges will lead to
a stationary distribution for the charge PDF, which at moment
is neither confirmed nor denied by the real data. For the sake
0 60 120 180 240 300 350
0
0.005
0.01
0.015
0.02
0.025
0.03
Duration (minutes)
Probability
PDF of recharge durations from
UC Davis data
Fig. 5. The PDF of charging durations obtained from PH&EV center data.
The noisiness is due to the rather small size of the dataset from UC.
0 50 100 150 200 250 300
0
50
100
150
200
250
300
350
Quantiles of distribution
Quantiles of input sample
Fig. 6. q-q plot of PHEV charge estimates versus the traditional and the
clipped lognormal distributions
of illustrating our model, we proceed to use a stationary PDF
of the charge requests for home charging, even though our
model in Section II is compatible with a non-stationary charge
request distribution.
C. Statistics of Laxity of Charge Requests
Next, we look at one parameter that may be interesting to
DR aggregators in charge of managing EV charging load.
We answer the fourth essential question mentioned in the
introduction about the amount of flexibility accompanying
each charge request. Fig. 7(a) shows the PDF of the laxity
(slack time) of the charging requests gathered by UC Davis
[11]. The PDF has two distinct peaks, one at 1-2 hours and
the next at 8-10 hours. As seen in Fig. 7(b), we can confirm
from the data that daytime requests contribute mostly to the
first peak, while nighttime requests mostly belong to the
second peak. Consequently, we fit two different probability
density functions to represent the laxity offered by daytime
and nighttime charge requests. We found that an exponential
distribution with a mean of 1.089 best fits the daytime data,
9
(a)
0 5 10 15 20 25 30
0
0.05
0.1
0.15
0.2
0.25
Laxity (hours)
Probability
PDF of laxity from PH&EV center data
(b)
0 5 10 15 20 25 30
0
0.2
0.4
0.6
0.8
1
1.2
Laxity (Hours)
Probability
PDF of laxity of daytime charge requests
Exponential fit − daytime
PDF of laxity of nighttime charge requests
Lognormal fit − nighttime
Fig. 7. (a) PDF of laxity of charge requests (b) PDF of laxity separated for
nighttime and daytime requests. Two distinct peak are observed.
while the nighttime laxity is best represented by a lognormal
distribution ln N(2.25,0.42). Due to a lack of enough samples,
we refrain from finding a joint density for the charge laxity
and length. However, we acknowledge that these two variables
are most likely correlated.
D. Mapping NHTS Entries into Charge Durations
As mentioned, the size of the real-world data set available
to us is rather small (620 samples). While this is a reasonable
amount of data to derive the charge duration and laxity PDF,
one cannot learn vehicle arrival statistics (the rate λ(t)) from
this data set. One possible way to have more statistics on
PHEV travel patterns, and thus arrivals, is to expand our
sample set by using a methodology similar to [7], [8] and
convert the NHTS mileage records traveled by ICEVs into
PHEV arrival and charge durations [32]. Since NHTS entries
only record one day worth of travel patterns per household for
ICEVs, a common assumption made to convert miles to kWhs
is that vehicles are charged at least once a day. If we adopt
this assumption, we end up mapping at least one arrival of the
customer at home per day into a battery charge request. To
showcase why we think this is possibly inaccurate, we will
introduce a simple miles to kWhs conversion next.
We assume that, on average, the energy required per each
traveled mile is uniformly distributed as in Table I [34]. Also,
we assume that the PHEVs are in a charge depleting mode
until they run out of charge, when they switch to consuming
gas. Therefore, if we denote the mileage traveled by vehicle i
as Mi, the average energy required per mile by i, the charge
rate by Ri, the battery capacity by Υi, and the amount of time
the vehicle is parked at home as ρi, we can write the mapping
from NHTS entries into charge durations Sias
Si= min Υi
Ri
,i
Ri
Mi, ρi.(21)
Fig. 8 displays the probability density function of the home
charging durations Siin minutes derived from the mileage
entries based on (21). We can see that a large percentage of
charge requests have a rather short duration, which is neither
consistent with the real-world data (c.f. Fig. 4) nor intuitive
(E[Si] = 102 minutes). We envision that this problem will
be eliminated if we account for the random nature of plug-in
events. In fact, drivers are unlikely to charge at the end of every
single day if they have only traveled a rather short distance
and their battery is not depleted. The data from the UC Davis
PH&EV center confirm that charging usually happens when
the battery is nearly empty. Consequently, lacking sufficient
arrival data in the PH&EV center dataset, we propose to
account for this random plug-in behavior so as to transform
NHTS entries to realistic PHEV arrival time data. The NHTS
entries, unlike the PHEV data set, are abundant and expand
our sample set sufficiently to denoise the estimate.
If customers do not plug in their vehicles every night, the
total mileage driven by vehicle iin (21), Mi, will be the sum
of daily traveled mileages over the days following the previous
charge event. Thus, denoting the miles traveled by vehicle i
on the j-th day since its last battery charge by Mi,j, we have
Mi=
Ki
X
j=1
Mi,j ,(22)
where Kiis the number of days since the customer has last
charged the battery. Note that Kiis a random number. We
assume a simple Bernoulli model to capture the customers
decisions to charge their vehicles, with a few possible options
on how to model the success probability. Picking different
options would affect the statistics of Ki. Since the NHTS only
gives us information on a single day of travel per vehicle i,
i.e., one Mi,j per household i, we propose two strategies to
model Ki.
The first strategy is to assume that the plug-in probability
is only a function of the last day’s mileage, Mi,Ki. We found
that by choosing
Pplug(Mi,Ki) =
0.05, Mi,Ki<1.5
0.1,1.5< Mi,Ki<8
0.5,8< Mi,Ki<16
1,16 < Mi,Ki
,(23)
a two-sample Kolmogorov-Smirnov test does not reject the
hypothesis that the 620 real-world samples and the 150,000
charging amounts derived from the NHTS database come
from the same probability distribution (at a 5% significance
level). However, note that this may not be the optimal way of
capturing the customer’s decision, which will most probably
depend on Miinstead of Mi,j .
In the second strategy, instead, we assume that there is a
constant probability of plugging in, no matter the miles, equal
to Pplug. Hence, Kiwill be a geometric random variable with
a success probability of Pplug, i.e.,
P(Ki=n) = (1 −Pplug)n−1Pplug (24)
10
0 50 100 150 200 250 300 350
0
0.002
0.004
0.006
0.008
0.01
0.012
Duration (minutes)
Probability
PDF of daily recharge durations
estimated from NHTS data
Fig. 8. The PDF of home charging durations obtained from NHTS is clearly
not similar to that of the real-world data. We envision that this is because
NHTS alone cannot account for the random nature of plug-in events.
0 5 10 15 20 25 30 35 40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Miles driven before returning home
Probability
PDF of miles driven − returning home before 5 pm
PDF of miles driven − returning home after 5 pm
Fig. 9. Distribution of miles driven before returning home, derived from
NHTS
Next, we need to estimate the parameter Pplug based on the
real-world data. We can learn the PDF of the miles driven
on a single day, Mi,j, from the NHTS data (see Fig. 9). If
we assume that the mileage driven on consecutive days are
independent random variables2, we can numerically calculate
the PDF of Mifor different values of Pplug. Next, we record
the log-likelihood that the 620 samples available from the
PH&EV center data originated from each of these PDFs. The
results indicate that our optimal estimate is P∗
plug = 0.4.
A further improvement would be to make the probability
of plug-in a function of Mi, the number of miles driven since
the last charge. This would mean that, if the battery is nearly
empty, the customer will charge it with a higher probability.
A simple model for this case would be
Pplug(Mi) = P1, Mi<Threshold
P2,else (25)
If we choose the threshold in (25) to be equal to 5 miles,
we can estimate P∗
1= 0.15 and P∗
2= 0.65 by performing a
likelihood ratio test on the real-data.
With this, we have answered the second essential question
posed in the introduction: how often do customers request
2It is likely this is not the case, and that one would have to account for serial
correlation in travel across days for individual drivers, but we are avoiding
the complexities associated with making these distinctions.
battery charge when their vehicle is parked? Here, we choose
to proceed with the plug-in probability (23) to map the event
of an ICEV arriving at home in the NHTS database to a PHEV
plug-in event for our numerical experiments. Now we assume
that we have an abundant amount of historical data on PHEV
arrivals. Next, we show why we think a non-homogeneous
Poisson process is an appropriate choice for modeling arrivals,
as promised previously.
E. Uniform Conditional Test for a Poisson process
In order to show that an inhomogeneous Poisson arrival
model is a reasonable model for PHEVs, we construct a
test of null hypothesis based on vehicle travel patterns from
the NHTS data. The null hypothesis is that vehicle arrivals
for charging follow a constant rate Poisson process in short
intervals of time (the length of this interval is chosen to be
30 minutes in our case). To be consistent with our charging
model, we assumed that vehicles request charge only when
they arrive at home. The decision of customers to plug-in was
assumed to be a Bernoulli random variable with a success
probability given by (23). It is worth noting that, no matter
the choice of plug-in probability, random selections of Poisson
processes are still Poisson processes.
Before explaining how we carry out this test, we need to
address an issue emerging from the NHTS trip start and end
times. These times are almost always rounded to the nearest
5 minutes. Also, we observed that 1/2hour intervals appear
much more often than expected. Therefore, following [23],
we added a random dithering to the data by adding normally
distributed noise with a variance of 2 minutes to trip end and
start times that were multiples of 5 and, with a variance of 15
to recorded times that were multiples of 30 minutes.
The idea behind the test of the Poisson hypothesis was
explained first in [35]. Assume that there are a total of
Narrival events in an interval (0, t), occurring at times
t1, . . . , tN. Then, if we condition on N, the variables ti/t are
independent and uniformly distributed between (0,1). Thus,
to test the conditional uniformity of the arrival times, standard
tests of uniformity can be applied to this data (for example,
a Chi-squared test of uniformity) [36]. We used a one-sample
Kolmogorov-Smirnov test to compare the distributions of the
normalized arrival times with the U(0,1) distribution. The
test was repeated for different hours of the day and the null
hypothesis was never rejected. Fig. 10 shows a q-q plot for
arrivals between 13:30 to 14:00 on Mondays. Quantile-quantile
(q-q) plots are used to check, in a non-parametric fashion,
whether two sample set originated from the same probability
distribution. If the two sets come from similar distributions, the
points in the q-q plot will approximately lie on the line y=x.
These tests have been run with standard Matlab functions.
Obviously none of these tests can prove that the model is
truthful, but they can at least not reject it.
F. Principal Components of Vehicle Arrival Counts
The NHTS gives us an ample amount of arrival count data
for vehicle plug-ins, sorted by the day of the week. Unfor-
tunately, since we do not know the exact date of each event,
11
Fig. 10. q-q plot comparing conditionally normalized EV arrival times
between 13:30-14:00 on a Monday to a uniform distribution.
(a)
1
2
3
4
5
6
0
5
10
15
20
25
−0.4
−0.2
0
0.2
0.4
(b)
Fig. 11. (a) The first sixth principal components (index 1 on the x axis is the
principal component with the highest variance). (b) Periodic mean of the six
principal component coefficients (the x axis represents day of the week and
the y axis is the principal component index)
we cannot numerically demonstrate the principles described
in Section III-A to calculate ML estimates of the coefficients
of a time series model describing the αj(i)’s, e.g., the AR
coefficient βiand the noise variance σ2
iin (11). However,
due the large size of the dataset, the principal components
of the weekly arrival data give us a reliable estimate of the
factors ui, i = 1, . . . , κ in (9). We perform an SVD on the
transformed cumulative arrival count matrix [x1,x2,...,x7],
with each count vector xtdescribing the arrival counts at
different hours of a specific the week of the day t. It is
interesting to mention that the six most important ui’s, shown
in Fig. 11.a, account for 96% of the variance. Thus, the model
in (9) reduces the size of the data (in our case from 96 to
6 elements per day) and make the next steps of predicting
future arrival rates more tractable. One noteworthy observation
is that u1is very similar to the weekday travel patterns while
u2compensates for the different trends observed in weekend
travel patterns. The next step is to estimate the periodic mean
of the principal component coefficients, i.e, ¯αt(1),...,¯αt(κ)
for t= 1,...,7and κ= 6. Fig. 11.b displays this periodic
mean for the first sixth principal component coefficients.
VI. NUMERICAL EX PE RI ME NT S PART II:
REA L-TIME LOA D FORECASTING
Our experiments in this Section will showcase the advan-
tages of sub-metering PHEVs separately through real-time
smart meter data, just for forecasting purposes. We refer to
the technique, presented in Sections II-C and IV-A, as the OAI
predictor (prediction that uses Observable Arrival Information)
and compare it to what we refer to as the classical prediction,
which models the load as an ARMA process [37]. Univariate
methods are frequently used for short-term load forecasting
because exogenous variables such as the weather are assumed
to change very smoothly over shorter time scales [38].
As explained in Section II-C, an OAI predictor needs to
estimate the number of vehicles that will be receiving charge
from each queue at a future time t. This load is proportional
to the sum of a deterministic term ν(t)and a random term
Nnew(t)with mean m(t)in (2). To evaluate m(t), we use
two separate methods and compare their performances. In the
first method, we forecast the arrival rate of the cars using
the weekly component of the vehicle count vector and then
evaluate the integral (2) numerically. We refer to this method
as OAI1 predictor. In the second method, we use the results
presented in [9], which we refer to as OAI2 predictor. This
model assumes a homogeneous arrival process for the charging
requests. Accordingly, we picked the average arrival rate we
used for OAI1 as the arrival rate for OAI2, which was equal
to 12 vehicles per every half hour.
The methods were tested on the simulated load for a
substation serving a population of around 1000 PHEVs. The
simulation covers 60 days, 30 of which are used for training
the classical ARMA model and the forecast errors are averaged
over the remaining 30 days. For fairness, we set the base
load to be zero so that forecasting the base load does not
contribute to the error. We assume that the entire population of
PHEVs uses level 1 charging (1.1 kW). We assume that: 1) the
residents arrive to receive charge based on arrival rates derived
from the NHTS data; 2) the charging time Sof each car has
the same distribution (20). Our results, displayed in Fig. 11.a,
show the average absolute error (in kWs) of half-hour ahead
prediction using the OAI1, OAI2, and the classical methods. In
our simulations, the daily load due to battery charging of the
1000 PHEVs had a peak of around 250 kWs. One can clearly
observe that the classical prediction technique has an absolute
error that is up to 3-4 times worse than the OAI1 method.
Also, the results clearly showcase that a non-homogeneous
arrival model is necessary to capture EV charging load. Thus,
12
(a)
0 5 10 15 20
0
5
10
15
20
25
30
35
40
Time of day (hours)
Absolute error (kWs)
Mean absolute error of half hour ahead prediction for 58 days
Without PHEVs−
classic predictor
With PHEVs −
classic predictor
With PHEVs − OAI1
perfectly known λ(t)
With PHEVs − OAI1
forecasted λ(t)
With PHEVs − OAI2
(b)
0 5 10 15 20 25
0
20
40
60
80
100
120
140
160
Time (hours)
Absolute error (kWs)
Absolute error of half hour ahead prediction
Without PHEVs
With PHEVs − classic predictor
With PHEVs − OAI predictor
Fig. 12. Absolute error of half hour ahead prediction (a) arrivals based on
NHTS data (b) arrivals based on UC Davis PH&EV center data
a useful extension to the model proposed in [9] would be to
incorporate non-homogeneous arrivals.
Furthermore, to showcase the benefits of individual energy
requests being observable even if the statistics are incorrect,
we simulate the performance of both predictors when the
arrival events are taken from the PH&EV center data set, with
all events assumed to happen on the same day. Due to the
small size of this data, we refrain from training the ARMA
model or predicting the arrival rates for the OAI predictor.
Consequently, we used the same model developed on the
NHTS data for the classic predictor and used a persistence
predictor (i.e., ˆ
λ(`+ 1) = λ(`)) for the OAI predictor. This
clearly degrades the performance of both predictors. However,
as seen in Fig. 12, lowering the prediction accuracy of λ(t),
which is consistent with the scenario where the arrivals are
more volatile and unpredictable, will hurt the classic predictor
much more than the OAI predictor. This is simply due to
the fact that, even if prediction of λ(t)is inaccurate, the
OAI predictor still has zero forecast error on the load due
to previous charge requests.
VII. CONCLUSIONS
This paper provides a stochastic model useful to simulate
and predict the electricity demand resulting from a general
arrival and charging pattern for EVs/PHEVs. It also proposes a
methodology to gather and fuse information online to perform
future predictions that is useful both for the generation and
demand side management of EVs. The model is validated
using traces extracted from mapping NHTS data and via real
PHEV measurements from the UC Davis PH&EV center. We
would like to reiterate that the numerical values presented
for the parameters of our model can and should be re-
evaluated as new data (and in particular, multi-day charing
data) become available, charging infrastructure develops, and
business plans for demand side management/dynamic pricing
techniques emerge.
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[ ]Mahnoosh Alizadeh(S’ 08) is a PhD candi-
date at the University of California Davis. She received the B.Sc.
degree in Electrical Engineering from Sharif University of Technol-
ogy in 2009. Her research interests mainly lie in the area of real-time
control and optimization for cyber-physical systems, with a special
focus on designing scalable and economic demand side management
frameworks for the Smart Grid.
[ ]Anna Scaglione (S’97-M’99-SM’09-F’11) is
currently Professor in Electrical and Computer Engineering at the
University of California Davis. She joined UC Davis in 2008, after
leaving Cornell University, Ithaca, NY, where she started as Assistant
Professor in 2001 and became Associate Professor in 2006; prior to
joining Cornell she was Assistant Professor in the year 2000-2001,
at the University of New Mexico.
She has been a Fellow of the IEEE since 2011. She is the Editor in
Chief of the IEEE Signal Processing Letters, and served as Associate
Editor for the IEEE Transactions on Wireless Communications from
2002 to 2005, and from 2008 to 2011 in the Editorial Board of the
IEEE Transactions on Signal Processing, where she was Area Editor
in 2010-11. She has been in the Signal Processing for Communication
Committee from 2004 to 2009 and is in the steering committee
for the conference Smartgridcomm since 2010. She was general
chair of the workshop SPAWC 2005. Dr. Scaglione is the first
author of the paper that received the 2000 IEEE Signal Processing
Transactions Best Paper Award; she has also received the NSF
Career Award in 2002, and is a co-recipient of the Ellersick Best
Paper Award (MILCOM 2005), and the 2013 IEEE Donald G. Fink
Award. Her expertise is in the broad area of signal processing for
communication systems and networks. Her current research focuses
on signal processing algorithms for networks and for sensors systems,
with specific focus on Smart Grid, demand side management and
reliable energy delivery.
[ ]Jamie Davies received the B.S degree in
environmental science and policy in 2008, and the M.S. degree in
transportation in 2011, from the University of California at Davis.
From 2008 to 2011 he was a Graduate Student Researcher at the
Institute for Transportation Studies at UC Davis. Since 2011, he
has been Consumer Research Analyst at the Plug-in and Electric
Vehicle Research Center at UC Davis. His research emphasis is
on the design and practice of quantitative and qualitative research
to collect, measure and evaluate plug-in electric vehicle impacts,
policy goals and vehicle design potential based on consumer’s real
world travel and charging behavior. His broader research goals also
include understanding factors which affect PEV market and testing
the effectiveness of market development strategies on the sale of plug-
in electric vehicles to organizations and consumers.
Kenneth S. Kurani received the Ph.D. degree in civil and envi-
ronmental engineering from the University of California, Davis in
1992. He is presently an Associate Researcher at the Institute of
Transportation Studies at the University of California, Davis. He is
the co-author of nearly 100 publications including scholarly articles,
book chapters, and technical reports. His research focuses on the
intersection of lifestyle, automobility, energy, and the environment.
He is especially interested in enriching the behavioral approaches
to consumers responses to new transportation technologies such as
new propulsion systems and fuels for automobiles. Dr. Kurani was a
Federal Highway Administration Graduate Research Fellow in 1985
and a Chevron Fellow in 1991. He is a friend of the committee for the
Energy, Alternative Transportation Fuels, Travel Behavior and Values,
and User Information Systems Committees of the Transportation
Research Board.
