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IEEE TRANSACTIONS ON POWER SYSTEMS 1
Optimal Charging of Electric Vehicles Taking
Distribution Network Constraints Into Account
Julian de Hoog, Member, IEEE, Tansu Alpcan, Senior Member, IEEE,MarcusBraz
il,
Doreen Anne Thomas, Senior Member, IEEE, and Iven Mareels, Fellow, IEEE
Abstract—The increasing uptake of electric vehicles suggests
that vehicle charging will have a significant impact on the elec-
tricity grid. Finding ways to shift this charging to off-pe
ak periods
has been recognized as a key challenge for integration of electric
vehicles into the electricity grid on a large scale. In this paper,
electric vehicle charging is formulated as a rece
ding horizon
optimization problem that takes into account the present and
anticipated constraints of the distribution network over a finite
charging horizon. The constraint set includes
transformer and
line limitations, phase unbalance, and voltage stability within the
network. By using a linear approximation of voltage drop within
the network, the problem solution may be
computed repeatedly in
near real time, and thereby take into account the dynamic nature
of changing demand and vehicle arrival and departure. It is shown
that this linear approximation of the
network constraints is quick
to compute, while still ensuring that network constraints are
respected. The approach is demonstrated on a validated model of
a real network via simulations that
use real vehicle travel profiles
and real demand data. Using the optimal charging method, high
percentages of vehicle uptake can be sustained in existing net-
works without requiring any fu
rther network upgrades, leading to
more efficient use of existing assets and savings for the consumer.
Index Terms—Distribution networks, electric vehicles, grid im-
pacts, optimization, receding horizon, smart charging.
NOMENCLATURE
Set of houses in the network .
Set of electric vehicles, each associated with a
house,
.
Set of discrete time intervals in charging horizon
.
Current (A) drawn by vehicle at time .
Current (A) drawn by household at time .
T
otal current (A) drawn at household
(from both
household and vehicle).
Manuscript received November 04, 2013; revised March 06, 2014; accepted
April 05, 2014. This work was supported by a Linkage Grant supported by
the Australian Research Council, Better Place Australia, and Senergy Australia.
Paper no. TPWRS-01408-2013.
J. de Hoog and D. A. Thomas are with the Department of Mechan-
ical Engineering, The University of Melbourne, Australia (e-mail: julian.
dehoog@unimelb.edu.au).
T. Alpcan, M. Brazil, and I. Mareels are with the Department of Electrical
and Electronic Engineering, The University of Melbourne, Australia.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2014.2318293
Total current (A) drawn by all single-phase loads
on phase
.
Stored energy (kWh) of vehicle at time .
Maximumallowedstoredenergy(kWh).
Charging efficiency factor.
Source voltage (V) at distribution transformer,
phase to neutral.
Nominal power rating (kVA) of distribution
transformer.
Current rating (A) of feeder backbone cable, phase
.
Current rating (A) of service line connecting house
to feeder.
Difference between voltage at transformer and
voltage at house
at time (V), phase to neutral.
I. INTRODUCTION
E
LECTRIC vehicles (EVs) are being pushed by govern-
mentsaroundtheworldasanalternativetofossilfuel
based transport. As a result, EV market share is starting to in-
crease: a recent report by the U.S. Department of Energy shows
that plug-in vehicle sales are more than double those of hybrids
when comparing the same stages of the technology life cycle
[1].
The charging of so many electric vehicles puts an additional
strain on the existing electricity grid. Since people are likely to
plug in when they arrive at home, there is a risk that vehicle
charging will coincide with peak demand. If vehicle charging is
not controlled, adverse impacts on the distribution network are
expected: power demand may exceed distribution transformer
ratings; line current may exceed line ratings; phase unbalance
may lead to excessive current in the neutral line; and voltages
at customers’ points of connection may fall outside required
levels [2]–[5]. However, these impacts can be alleviated if ve-
hicle charging is shifted to a time when there is more capacity
in the network, such as overnight. An effective shifting of EV
charging load to off-peak periods means that existing networks
can be used considerably more efficiently, reducing the need for
network upgrades and ultimately benefitting the consumer.
The “smart charging” problem is well studied, and many ap-
proaches have been proposed to achieve this behaviour. Several
studies pursue distributed methods, in which charge points or
vehicles make individual decisions on whether to charge or not
using local information [5]–[7]. The advantage of such methods
is that they usually do not require any particular metering or
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2 IEEE TRANSACTIONS ON POWER SYSTEMS
communication infrastructure. The disadvantages are that they
may not be able to fully make use of available system capacity
due to limited knowledge of the network’s current state, and that
they may be difficult to regulate.
Other studies use full network state information and con-
trol vehicle charge rates centrally [5], [8]–[14]. One such ap-
proach uses both quadratic and dynamic programming to mini-
mize network losses and voltage deviations [8]. Across a variety
of case studies, it is shown that this approach reduces losses,
improves voltage stability, and decreases peak load when com-
pared with uncontrolled charging. The relationships between
losses, load factor, and load variance are further explored in [9].
Reference [10] proposes a way of expressing centralized EV
charging using results from recent optimal power flow studies.
An optimal problem formulation aims to minimize generation
and charging costs while satisfying all the constraints posed by
the network, and the optimal power flow problem takes into ac-
count both elastic and inelastic loads. Reference [12] similarly
tries to maximize the revenue that a distribution network op-
erator might have in response to network operating costs and
fluctuating wholesale prices. Network voltages are taken into
account by iteratively solving a linear program until all voltages
fall within required levels. Reference [13] uses a two-stage opti-
mization approach, in which the minimum required peak is de-
termined in the first stage, and load fluctuation is minimized in
the second stage by choosing controllable vehicle charge rates
within the allowed peak load. The focus is on cost-benefitanal-
ysis and distribution network constraints are not considered. A
model predictive control framework is proposed in [14] that
minimizes the cost of energy consumption. Network demand
is limited via tracking of a reference load profile definedbythe
grid operator.
A further centralized approach uses linear programming to
maximize the total vehicle charging power that the distribution
network allows, while formulating network limitations as con-
straints [11]. Sensitivity analysis is applied to determine how
sensitive the voltage of each node in the network is to the addi-
tion of EV load. The optimal solution allows for the modelled
network to sustain an EV penetration of 50%, as compared with
only 16% in the uncontrolled case.
However, many existing methods suffer from one or more of
the following drawbacks: 1) The charging period is often mod-
elled as a static time interval (typically overnight), in which
vehicles do not arrive or depart; 2) distribution network con-
straints, in particular voltage drop, are often estimated, or only
included in an indirect way; and 3) phase unbalance is often ig-
nored, and limitations are expressed for the network as a whole
when instead they should be expressed on a phase-by-phase
basis.
In reality, vehicles may arrive and depart unexpectedly. Dis-
tribution network constraints can already be breached at very
low vehicle uptake rates: in previous work we have found that
with uncontrolled charging there is a risk of voltage dropping
below distribution code limitations at EV penetrations of only
10% [5] (a finding that is in line with several studies elsewhere,
e.g., [3], [11]). In the networks we have studied (where each
house is connected single-phase), phase unbalance has been a
major factor, with the most heavily loaded phase sometimes
having close to double the total load of the least loaded phase.
In this paper we build on existing work by addressing
these concerns, and provide a novel optimal smart charging
algorithm that allows large numbers of vehicles to be charged
without adverse effects on the network. We express smart
charging as an efficient linear optimization problem that takes
into account both the present and anticipated constraints of
the distribution network over the full finite charging horizon.
Recalculating our solution in discrete intervals allows for the
dynamic nature of vehicle arrival/departure to be accommo-
dated. We model voltage drop explicitly as a linear constraint,
on a phase-by-phase basis, in every lateral of the network. We
maintain separate loading constraints for each phase.
Using this framework, we examine two possible objectives:
in the first, we aim to provide as much charging power to the
vehicles as the network will allow. In the second, we take fluc-
tuations in the electricity price into account and aim to charge
all vehicles within a specified time period at minimum cost.
Our methods are implemented and simulated on a validated
model of a real distribution network, using real travel profiles to
simulate EV behavior and real demand profi
les obtained in the
network we are modelling. While we use linear approximations
when solving our optimization problem, the simulations are run
in a fully complex, unbalanced, three-phase load flow scenario.
It is shown that linear approximation in this manner is an
effective, fast way to find a charge scheduling solution. It is also
shown that existing networks can sustain high penetration rates
of electric vehicles, without significant additional investment
into network assets required.
II. U
NDERLYING MODEL
A. Preliminaries
We aim to determine the charging rates of vehicles in a radial
distribution network served by one transformer. We assume that
vehicles’ charging rates may be controlled centrally, and may be
set to any value within a given continuous range. We consider
this a realistic assumption since this was recently demonstrated
in the Australian Victoria Electric Vehicle Trial as part of a pilot
load control project [15]. We further assume that the network
operator has access to the following information:
• network size (number of houses);
• network structure (including line segment lengths and in-
dividual phase connections);
• line specifications (impedance per km, nominal current
ratings);
• transformer specifications (nominal power rating).
This information may be used by a central controller to de-
termine the best set of charging rates for all vehicles currently
connected, so that they may charge in a way that maximizes a
given objective while not violating any constraints in the net-
work. In making this decision, we are looking not just at the
current point in time, but at the best possible solution for a finite
future charging horizon in discrete intervals. Since underlying
conditions may change unexpectedly (such as vehicles arriving
or departing), the solution for the full charging horizon is re-
computed after each discrete time interval.
The distribution networks we have studied have had a high
power factor. Fig. 1 shows data obtained in the network pre-
sented in this paper; more than 99% of data points have power
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DE HOOG et al.: OPTIMAL CHARGING OF ELECTRIC VEHICLES TAKING DISTRIBUTION NETWORK CONSTRAINTS INTO
ACCOUNT 3
Fig. 1. Cumulative frequency histogram of power factor measurements in the
data set provided by our utility partner. This histogram reflects 43 317 data
points (14 439 for each phase) gathered in the period 17–27 August 2012 in
thenetworkdescribedinSectionVII-A.
factor 0.95 or higher. We therefore consider it a reasonable sim-
plification to use a DC-equivalent model of our distribution
network when formulating our optimization problem. This is
common practice when the angle between source voltage and
load voltage is very small [16]. By doing this we keep our con-
straint set linear, and in most cases it leads to conservative con-
straints. However, when we run our simulations to examine our
solutions (Section VII), we conduct unbalanced, three-phase
AC load flow analyses in fully realistic scenarios that have been
validated to have a high correlation to reality.
B. Notation
Our notation is as outlined before Section I. Let
be the set
of
houses in the network. (In Australia, houses are typically
connected single-phase.) Let
beasetof households
owning electric vehicles. We assume charging decisions can be
made in discrete time intervals; let
be the discretized charging
horizon having
intervals.
We denote current at point of connection
at time as
(current due to household load) and (current due to vehicle
load). Total current at household
at time is
We model the network as a three-phase wye-connected
system. Total current on a given phase is the sum of all currents
at any points of connection on that phase:
C. Vehicle Batteries
We use stored energy (kWh) as a measure of how charged the
vehicle batteries are. The storedenergyofthebatteryofthe
th
vehicle at time must satisfy .Weestimate
the future stored energy of a battery using the following:
(1)
where
is the nominal voltage of the grid, is current
supplied to the vehicle, is the size of our discretized time
interval and
is an efficiency factor (we use 0.9) that takes into
account energy lost due to AC/DC conversion and cooling. The
voltage at point of connection will usually not vary beyond 5%
(limits are imposed by the Electricity Distribution Code [17]),
so we consider this a sufficiently accurate way of estimating a
battery’s stored energy over future time intervals. If necessary, a
conservative nominal voltage may be chosen so that the stored
energy of a battery at a future time interval is not overestimated.
We assume that batteries can be charged at variable rates,
provided minimum and maximum rates are not exceeded (as
demonstrated in a pilot load control trial using commercially
available electric vehicles [15]).
Vehicles arrive and depart independently of one another.
Each vehicle has a charging horizon, with a target of being
fully charged at time
.
III. DECISION VARIABLES
In principle, the purpose of our optimization problem is to
determine the amount of power delivered to the grid-connected
electric vehicles. However, because EV charge points follow the
voltage from the grid, which is highly regulated, a charge cur-
rent decision is essentially equivalent (in fact, the J1772 stan-
dard for car battery charging is specified in terms of charging
current, not power [18]). In view of this, we choose the currents
supplied to each grid-connected vehicle as our decision vari-
ables. This further allows us to keep our problem formulation
linear in our decision variables.
Our decision variables are therefore the currents supplied to
all charging vehicles over all intervals in the charging horizon,
whichmaybedenotedbythematrix
.
.
.
.
.
.
where and . We can rewrite matrix as a vector
by using its column vectors.
The full solution to our problem is recomputed at each in-
terval (e.g., every 5 min). The number of decision variables may
change from one interval to the next, depending on whether ve-
hicles have arrived or departed.
IV. S
YSTEM CONSTRAINTS
Our full set of system constraints at any time interval may
be written in the standard format
, where the matrix
and the vector result from the grid and battery conditions at
each interval in our horizon. At any point in time there may be
many vehicles in the network, and for each of these vehicles
charge rates must be chosen for all intervals within a given fu-
ture horizon. As a result, the number of decision variables and
constraints can grow quickly (both often in the thousands for
our case study). We therefore make a series of approximations
that allow us to express the constraints in the distribution net-
work in a linear form. This ensures the optimization problem
can be solved very quickly, and makes our method suitable for
near-real-time decisions. Our simulation results (Section VII),
which are conducted in a fully complex, unbalanced, 3-phase,
validated system, suggest that our approximations are justified
since this way of formulating the problem successfully avoids
any network constraints being violated. The effects of lineariza-
tion are also further discussed in Section VII-E.
A. Nominal Power Rating of Transformer
Transformers have a nominal power rating
that should
not be significantly exceeded—if it is, transformer lifetime is
reduced. However, transformers often run at 130% of
or
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4 IEEE TRANSACTIONS ON POWER SYSTEMS
more, which is sustainable as long as there is an accompanying
cooling period. We express this constraint on a phase-by-phase
basis:
%
where is the phase-to-neutral voltage at transformer. Since
we are capping total power for each phase instead of for the
system as a whole, these are conservative constraints.
B. Nominal Current Ratings of Lines
Power lines have a variety of limitations, the most important
of which is current rating. If this is exceeded, cables can be dam-
aged. Let cable
have current rating . In the networks we
have studied, backbone and service lines typically have different
specifications. Thus we introduce separate constraints for each
phase of the backbone, and further add individual constraints
for every service line in the network:
C. Voltage Drop
Voltage must be maintained within lower and upper limits
at every point of connection.
1
If these limits are not respected,
household loads can be adversely affected.
Consider the simplified network in Fig. 2, where each load
is a combination of household and electric vehicle loads, and
all are on the same phase. The total voltage drop from
transformer to house 4 can be approximated by considering a
DC-equivalent circuit:
and in general at house
Since are a combination of the existing household and elec-
tric vehicle load, and since we are modelling the network as
mainly resistive (using only real power), this becomes a linear
expression. We can now formulate a constraint at every house
at every future time interval to ensure that voltage is high
enough:
Since networks may have multiple laterals, we must consider
what happens when there is a split. Consider the simplified net-
work in Fig. 3. The voltages in each lateral, at
and ,arenot
independent since they share an impedance . To ensure that
1
In Australia, voltages at point of connection must be maintained at 230 V,
% %, i.e., in the range 216 V–253 V [17].
Fig. 2. Simple network.
Fig. 3. Simple network with two branches.
voltage remains within limits at all locations in the network, we
consider the worst case when expressing this as a constraint:
where represents the voltage drop across . This can be
generalized to
where are all piecewise voltage drops from source to
house at time . In other words, at each interval, each house
will have its own unique constraint that is still linear in the cur-
rents through all other houses and vehicles in the network. The
performance of this linearization is explored as part of our sim-
ulation case study in Section VII-E.
D. Phase Unbalance
Phase unbalance can lead to overheating of motors, may have
negative effects on electrical equipment, and leads to higher cur-
rent in the neutral (which in turns contributes to voltage drop).
Phase unbalance is typically expressed in terms of percent
negative sequence voltage. However, to keep our constraint set
linear, we express phase unbalance in terms of
, the percent
deviation from average phase load:
Note that these constraints are linear when we multiply both
sides of the equation by the denominator.
E. Battery
It is important not to exceed a battery’s maximum capacity.
This can be expressed by the following constraint:
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DE HOOG et al.: OPTIMAL CHARGING OF ELECTRIC VEHICLES TAKING DISTRIBUTION NETWORK CONSTRAINTS INTO
ACCOUNT 5
Batteries further have minimum and maximum possible
charging currents that should be respected to protect the battery
and ensure efficient charging:
F. Charging Targets
Each vehicle has a target of reaching at least 95% of its max-
imum stored energy within a finite, specific time
from the
moment it starts charging:
(2)
V. O BJECTIVE FUNCTIONS
We now examine two possible objective functions:
A. Greedy Charging (GC)
Our first objective is greedy: we want to only maximize the
stored energy of all the vehicles without considering pricing or
fairness issues:
(3)
Using the battery state evolution (1), the problem may be equiv-
alently expressed as
(4)
In other words, we want to provide as much possible charging
current as the network will allow.
B. Greedy Charging With Pricing (GCP)
Our second objective is to minimize the cost of charging (a
problem of interest, for example, to a charging provider or ag-
gregator). We use the dynamically changing spot price of elec-
tricity,
,asaparameter:
(5)
Since we have charging targets as a constraint (2), our solu-
tion must contain non-zero values for
. However, this ob-
jective means that higher charge rates are chosen when the spot
price is low.
VI. PROBLEM COMPLEXITY
In this section we explore the complexity of formulating the
problem using decision variables as described in Section III and
constraints as described in Section IV. We assume as before
that there are
houses having charging vehicles, making
decisions over a horizon having intervals.
Using our problem formulation, there will be
decision
variables. The number of constraints is summarized in Table I;
the only constraint that may not be immediately clear is phase
unbalance. For this, we need to ensure that each phase does not
deviate from average phase load, and since there are absolute
TABLE I
NUMBER OF CONSTRAINTS FOR
HOUSES HAVI N G CHARGING VEHICLES
OVER A CHARGING HORIZON OF INTERVALS
values involved we have two constraints for each phase for each
time interval. Summing the rows presented in Table I leads to a
total of constraints.
As an example, for the case study explored in Section VII
we will have 114 houses with 57 vehicles, making decisions in
15-min intervals over an 8-h horizon (for a total of 32 intervals).
In the worst case (when all vehicles are charging), there will
thus be 1824 decision variables and 11 385 constraints. By lin-
earizing the constraints, this problem can be solved in a matter
of seconds on a standard desktop PC using the MATLAB Op-
timization Toolbox. This speed of computation becomes all the
more important if the method is to be applied to multiple net-
works, or at a higher level such as the substation.
VII. I
MPLEMENTATION AND RESULTS
In the preceding sections, several simplifications were made
to allow us to keep our problem formulation and constraints
linear. In this section we implement and test our solution on a
validated model of a real network, and run fully complex, unbal-
anced, three-phase load flow analyses in each interval. Testing
our solution under fully realistic conditions allows us to confirm
whether our simplifications were justified or not.
A. Simulator, Data, and Case Study
Our simulations were conducted in POSSIM,
2
a tool devel-
oped at The University of Melbourne for analysis of distribu-
tion networks. POSSIM is a C++ based simulator that uses a
MATLAB SimPowerSystems backend for model building and
load flow analyses.
To conduct a realistic case study, we developed a model of a
real suburban distribution network in Melbourne containing 114
customers [Fig. 4(a)]. Detailed demand data was provided by
the network operator for each phase [Fig. 4(b)]. Individual phase
connections were not provided, but could be estimated using
aggregated load data. As can be seen, this is a fairly unbalanced
network with 50 houses connected to phase A, 43 houses to
phase B, and 21 houses to phase C. Each house was assigned an
average load profile on a per-phase basis using the data obtained
in the network, having both active and reactive power demand.
The network is served by a 300 kVA transformer. Line im-
pedances for both backbone and service lines were provided
by the network operator, as were distances between poles and
lengths of individual service lines. A validation cycle in which
we compared our simulated voltages and currents to those mea-
sured in the real network concluded that on average, voltages
2
POSSIM: POwer Systems SIMulator. Available at http://www.possim.org.