Smart Grid and Renewable Energy, 2013, 4, 398-408
http://dx.doi.org/10.4236/sgre.2013.45046 Published Online August 2013 (http://www.scirp.org/journal/sgre)
Model-Based Quantification of Load Shift Potentials and
Optimized Charging of Electric Vehicles
Tobias Hahn1, Martin Schönfelder2, Patrick Jochem2, Vincent Heuveline3, Wolf Fichtner2
1Institute of Process Engineering in Life Sciences, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany; 2Chair of Energy
Economics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany; 3University of Heidelberg, Interdisciplinary Center for
Scientific Computing, Heidelberg, Germany.
Received June 4th, 2013; revised July 4th, 2013; accepted July 11th, 2013
Copyright © 2013 Tobias Hahn et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Managing the charging process of a large number of electric vehicles to decrease the pressure on the local electricity
grid is of high interest to the utilities. Using efficient mathematical optimization techniques, the charging behavior of
electric vehicles shall be optimally controlled taking into account network, vehicle, and customer requirements. We
developed an efficient algorithm for calculating load shift potentials d efined as the range of all charg ing curves meeting
the customer’s requirements and respecting all individual charging and discharging constraints over time. In addition,
we formulated a mixed integer linear program (MIP) applying semi-continuous variables to find cost-optimal load
curves for every v ehicle participating in a load shift. This prob lem can be solved b y e.g. branch-and-b ound algorithms.
Results of two scenarios of Germany in 2015 and 2030 based on mobility studies sho w that the load shifting potential of
EV is significant and contribute to a necessary relaxation of the future grid. The maximum charging and discharging
power and the average battery capacity are crucial to the overall load shift potential.
Keywords: Electric Mobility; Load Shifting; Charging Management; Optimization
1. Introduction
Tomorrow’s energy systems will exhibit a tremendous
complexity reflected mainly by their dynamic behavior
and heterogeneity. This will apply to Germany in par-
ticular because of the constantly increasing share of po-
wer generation from renewable energy sources (RES) as
well as the ongoing decentralization of the po wer system.
The regionally strongly differing potentials of RES will
lead to an increasing spatial imbalance o f electricity gen-
eration and demand. Moreover, their fluctuating charac-
ter will require various measures, such as more flexible
power plants, a grow ing capacity of effi cient ener gy sto r-
age systems, and the application of Demand Side Man-
agement (DSM) methods.
At the same time, ongoing and extensive research and
development activities regarding electric vehicles (EV)
will facilitate the latters’ market penetration. A signifi-
cant share of EV in the passenger car fleet will lead to an
increasing demand for electric energy especially during
peak hours, if no controlled charging strategy is applied
[1]. This could jeopardize the local power grid [2].
However, due to the normally long parking times of ve-
hicles used for individual mobility purposes, the degrees
of freedom in charging EV are comparably high [3].
Therefore, controlled (e.g. delayed) charging of EV as
one DSM approach can help to avoid load peaks. More-
over, controlled charging strategies could support the
integration of fluctuating electricity su pply based on RES
[4,5]. Even more services can be provided, if the possi-
bility of feeding electricity back into the grid is consid-
ered (e.g. auxiliary services [6]).
To determine the technical potential of load shifting by
EV, the complex interplay of millions of consumers and
stationary and mobile energy producers has to be evalu-
ated with many specific restrictions and different levels
of aggregation and control points. This task is highly
complex in view of the heterogeneity and variety of indi-
vidual objects, which additionally differ in their temporal
availability.
Using efficient mathematical optimization techniques,
the charging behavior of electric vehicles shall be opti-
mally controlled taking into account network, vehicle,

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 399
and customer requirements. For individual vehicles con-
nected to a charging station and given electricity price
levels, a cost minimizing charge function is determined
with the customer requirements being taken into consid-
eration. The local load shifting potential is defin ed as the
range of alternative charging functions that meet the
customer requirements. However, not all of these charg-
ing functions may be economically optimal for the cus-
tomer. The electricity supplier therefore needs to pay for
the customer’s permission to choose a different load
curve that better fulfills the global conditions (e.g. plan-
ed generating capacity, network utilization, etc.). In this
way, system efficiency can be increased considerably by
appropriate Demand Side Management.
This paper presents a model to compute the range of
actual load shift potentials under the constraints of indi-
vidual preferences, technical restrictions, and the elec-
tricity demand of EV. The model will be used as a basis
for further investigations, e.g. to optimize the network
utilization under macro-economic and energy-economic
criteria by adjusting the charging functions of each in-
volved EV with respect to individual technical restric-
tions. Recently, numerous works were published in the
research field addressed by this paper. Especially the op-
timization of EV charging subject to network constraints
is addressed with several models seeking to determine
grid-compatible charging strategies [7-10] or simulating
the grid effects of different charging strategies [1,11,12].
Other works focus on methods to influence consumer
behavior to achieve grid-optimal EV charging [13,14].
Some Multi-Agent Systems (MAS) have also been pro-
posed [15-17] to achieve a decentralized optimal behav-
ior. In contrast to the works mentioned, our focus is on
the online determination of load shift potentials for the
next hours. This is the relevant information for the local
distribution system operator (DSO). Furthermore, our mo-
del allows for the use of actual load shifts within the si-
mulated potentials (e.g. by a central instance). The soft-
ware updates the aggregated potentials of the fu ture with-
out neglecting the underlying constraints. This is achieved
by constantly re-optimizing the distributed charging stra-
tegies subject to updated constraints.
The paper is divided into seven sections. Section 2 in-
troduces the mathematical model to calculate the aggre-
gated load shift potential. Besides the formal definition
of load curves, the mathematical description of the indi-
vidual as well as of the aggregated load shift potential is
given. In the third section the optimization problem for
the generation of optimal load curves is presented. Fur-
thermore, the optimization problem is adapted to deter-
mine the upper and lower limits of the charging functions
used to compute th e load shift potential o f all vehicles. In
Section 4, DSM operation and processing of load shifts
are described. Section 5 covers the application of the
model to two realistic scenarios while Section 6 gives an
outloo k b efore coming to t he co nc lusions in Section 7.
2. Mathematical Model
This section describes all quantities and mathematical
definitions involved in the optimization problem to be
sol v e d i n Section 3. The load curve of a vehicle can be any
function in time, which fulfills the customer requirements
and follows the technical restrictions. The main customer
requirement is a full battery at the end. The technical restric-
tions include upp er and lower limits for th e charge and feed
power. The upper limits are usually imposed by the b attery
or charging station, while the lower limits are due to a cer-
tain threshold which ensures a flow of energy.
Furthermore, the available amount of energy shall not
drop below the minimum reserve and the battery’s ca-
pacity must not be exceeded. For the grid, the quantities
of interest are the amounts of energy required and avail-
able ra ther than the indi vidual states of charge or reser ve
levels. The constraints are formulated with the help of
these quantities in the sections below.
2.1. Load Curve
For a vehicle V that is to be charged with a certain
amount of energy 0
V
E in a time interval V T



,,
be
tt the load curve

V
Lt to reach the desired
amount of energy during charging time is required, i.e.


d,
VVV
TLttE


 (1)
where we assume, that this is technically feasible. In case
the charging time is not sufficient to fill the battery, the
desired amount of energy shall be set to the amount that
can be reached.
The maximum capacity constraint can be expressed by
staying below the required amount of energy at any point
of the charging process,

d<.
b
t
VVe
tLttE tt


 (2)
The same applies to the amount of available regenera-
tive energy 0
V
E,

d<.
b
t
VVe
tLttE tt


 (3)
The technical restrictions include upper and lower lim-
its for th e charge and feed power.




,min ,max
0<: >0,
VV
PtLtPttLt

  (4)




,min ,max
0>: <0.
VV
PtLtPttLt

  (5)
Figure 1 shows an example of a load curve within the
admissible ranges. The limits as well as the load curve
may be discontinuous. In this case, the load curve is
piecewise constant.

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 400
2.2. Individual Load Shift Potential
Usually, there is a large number of load curves

i
V
L
which satisfy the conditions in 2.1 . The range of all these
load curves then defines the load shift potential


St:



.
i
V
i
StL t (6)
This means that for every point in S, we can find at
least one load curve passing this point and thus allowing
for shifting this point in time. As the load shift potential
is a subset of the technically admissible range of load
curves, S is a union of three disjoint sets:

 
,max,min,min,max
:,0 ,.
VV VV
St
StSt StSt
 

  (7)
The colored areas in Figure 1 show the subsets. If the
EV arrives with a low battery level, it cannot discharge in
the beginning, which is depicted by the growing lower
area. In order to fulfill constraint (1), the EV cannot dis-
charge in the en d, which is why the lower area is shrink-
ing again.
2.3. Accumulated Load Shift Potential of
Vehicles
For the DSM, it is reasonable to divide the load shift po-
tentials into three sub-potentials for the producible, un-
avoidable, and compensable load:
 Negative load shift potential (load generation): The
area between maximum load generation and the cur-
rent load curve
 

,max
:,
VV
V
RtSt Lt


 (8)
 Positive load shift p o ten tia l (load reduction): The area
between the current load curve and the minimum load
generation
 

,min
:max0, ,
VV
V
RtLtSt


 (9)
 Super-positive load shift potential (power genera-
tion):
The area between 0 (or the current load curve, if power
is currently generated) and the maximum power genera-
tion
 



,max
min 0,.
VV
V
RtLtSt


 (10)
Figure 2 shows the load shift potential for a vehicle,
which is unable to discharge in the b eginning. This is the
result of a common requirement: The battery is first
charged to a specified minimum with full power in order
to be able to reach important places, e.g. a hospital. In the
end, the upper level might be reduced due to reservation
of power for preconditioning of the vehicle, see Section 4.1.
Figure 1. Example load curve and range of values.
Figure 2. Exemplary load shift potentials for one vehicle.
Dividing the load shift potential into three sub-poten-
tials allows to draw them separately versus the non-
manageable grid load, as shown in Figure 3. The lower
edge of R+ may provide additional information on the
unavoidable load when it deviates from the grid load.
The impact of vehicle-to-grid functionality can be di-
rectly read from R*, as only participating vehicles add t o it.
3. Optimization Problem
DSM starts with offering a time-flexible tariff taking into
account all expected fluctuations in electricity generation
and demand which is supposed to make the customer
schedule highly energy-consuming activities accordingly.
Other cost functions are also possible, e.g. based on the
CO2 emissions. In reality, however, the reaction to dy-
namic tariffs depends on the customer’s acceptance [18],
which is why we assume only reasonably behaving cus-
tomers. Furthermore, there should be a tariff for feeding
energy from the battery into the grid, which should not
correspond to the charging tariff in the same interval.

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 401
Figure 3. Accumulated load shift potential for all vehicles
shown with non-manageable (static) load.
Otherwise, a cost-optimal load cu rve might be oscillatin g
instead of just staying at zero.
3.1. Generation of Optimal Load Curves
Finding a cost-optimal load curve for a complex tariff
with case differentiation or a direct algorithm will be
difficult in the future when taking into account more
complex constraints like battery wear and other physical
effects. Even now, the absolute tariffs are not the only
decisive factor. Their ratios have to be considered as well.
It is thus reasonable to calculate the solution by formu-
lating an optimization problem. Non-linearities are to be
avoided in order to apply faster linear programming tech-
niques.
An optimal load curve ,V opt
L is defined as an admis-
sible load curve that minimizes the cost for a given tar-
iff/weigh t functi on

Wt:


 
,d.
min
V optV
T
Lt V
Lt LtWtt
 (11)
As there are two intervals enclosing zero, where charg-
ing or discharging is not allowed, the domain becomes
non-convex. Without simplification, solution of the op-
timization problem will require a large expenditure [19].
Hence, we discretize the problem and construct con-
straints in order to select values from the admissible
range only.
We can expect the cost function to be piecewise con-
stant in the future, as all other forms of tariff will most
probably over-strain the customer. Without loss of gen-
erality, we expect the intervals where the tariff/weight
function

Wt is constant to be of constant length (e.g.
one hour), such that we can use a vector form


i
Ww.
As we are only interested in non-oscillating load curves
which are constant over the time intervals, we use the
same vectorized form to define load curve representa-
tions L.
We now consider different tariffs for charging and
feeding

,
,,
for 0.
for <0
iVi
iViiVi
wl
wl wl






 (12)
To avoid case differentiation, V L is decomposed into
charging and discharging curves, as shown in Fi gur e 4,
.
VVV
LZZ

 (13)
The objective is then reformulated to
,
.
min VV
ZZ
VV
WZ WZ




 (14)
It has to be ensured that both curves lie within the al-
lowed ranges and charging and discharging do not hap-
pen at the same time. The admissible ranges are
,,,min,,max,
0,,
ViViViVi
zzPP


 

(15)
,,,max,,min,
0,.
ViViViVi
zzPP
 

 

(16)
Many modern mixed integer programming (MIP)
solvers directly support this kind of semi-continuous
variables, such that no additional integers have to be
manually introduced to fulfill the either-or condition
above.
On the other hand, without adding further variables,
the constraint that only one of the functions V
Z

and
V
Z

can be non-zero at a time, can only be written in a
bilinear form,
0,
VV
ZZ


 (17)
which is not allowed in most conventional MIP solvers.
Specialized algorithms exist [20], but usually take more
computing time. We hence introduce binary indicator
functions

, 0,1n
VV
CF (18)
as also shown in Figure 4. The additional constraints
,,max, ,
,
Vi ViVi
cP zi


 (19)
,,max, ,
,
Vi ViVi
f
Pzi


 (20)
ensure the correct setting of V C and V
F
. For example,
,>0 Vi
z implies ,1
Vi
c

.
Figure 4. Decomposition into charging and discharging
curves and introduction of binary indicator s.

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 402
The substitution of 0
VV
ZZ

 is given by
,,
1.
Vi Vi
cf i (21)
Finally, the constraints formulated in (1)-(3) can be
rewritten for a charging time of n intervals of length t

:
 Full charging

,,
1
,
n
Vi ViV
i
tz zE
 


 (22)
 Upper capacity limit

,, <
1
<
k
ViViVk n
i
tz zE
 

 
 (23)
 Lower capacity limit

,, <
1
<.
k
ViViVk n
i
tz zE
 

 
 (24)
This final formulation is completely linear, it consists
of the objective (14), constraints (19)-(24), and admissi-
ble ranges (15), (16) and (18).
The problem can be solved with established methods
like branch- and-bound [21 ] or branch- and-cut, which ex-
tends the former algorithm by cutting planes. The im-
plementation chosen here is lp_solve [22], a free branch-
and-bound solv er supporting semi-continuous variables.
In case semi-continuous variables are not available,
two additional constraints are needed:
,,min, ,
,
Vi ViVi
cP zi

 (25)
,,min, ,
.
Vi ViVi
f
Pzi

 (26)
In this way, the problem can be solved with conven-
tional MIP routines as offered by computing environ-
ments like Maple. As the integer variables are binary, it
is possible to consider all configurations and solve a set
of linear problems only. This is no practical method, but
punctuates the global optimality of the solution.
From the variety of heuristic methods, simulated an-
nealing would be suited well for mixed integer problems.
However, as the computation times with lp_solve are in
the range of a few seconds ev en for large lo ad shifts on a
standard dual-core office PC, they were not considered.
The degrees of freedom of the optimization problem can
also be reduced by joining intervals with equal cost val-
ues.
3.2. Load Shift Potential
The method described above can also be used to compute
the load shift potential by changing the objective func-
tion to finding the charge curve reaching a maximum or
minimum value at a certain point in time. Four optimiza-
tion problems would have to be solved for every vehicle
and point in time, which would be associated with too
high a computational expenditure. As the objective func-
tions are simpler in this case, a direct algorithm can be
formulated for determining the four boundaries:
,minV
S The algorithm has to deliver the requested amount
of energy without exceeding the upper and lower
limits of the battery capacity. The maximum
amount of energy within the charging window is
given by ,max,Vj
j
tP



. Subtracting V E

results
in the amount of energy which is not needed for
charging.
The lower power limit of each interval can be
found by shifting this area within the charging
window, such that we can calculate

,minVi
S as
the minimum power level necessary by
,min,
1
,min, ,max,,max,
min ,.
Vi
ViViV jV t
j
S
PPP E

 








 




 (27)
,maxV
S To find the upper limit, the amount of energy not
needed for charging has to be shifted more spe-
cifically: When discharging at a high state of
charge, the amount of chargeable energy increases.
This means that charging should start with full
power from interval i until the end or until V
E

is
reached.
If we do not have ,max, ,max,ViVi SP

 yet, we try
discharging from interval 1 to 1i, such that we
can increase the power level at i in a stepwise
manner.
,maxV
S The discharging power limit can be computed
similarly: The higher the state of charge, the more
energy can be discharged. This means that charg-
ing starts with full power, leaving out the interval i
and bearing the upper limit V E in mind. When
the battery capacity is reached, ,max,Vi S is low-
ered in steps to ,max,Vi P as long as there are inter-
vals left for charging.
,minV
S This boundary is equal to ,minV S, if ,max,Vi S

,min,Vi
P, otherwise it is set to zero.
4. DSM Operation
The accumulated load shift potential is highly time-de-
pendent, as vehicles are constantly arriving and departing.
At the same time, the set of alternative load curves for
each connected EV is constantly reduced when values
are fixed over time.
The operation of the DSM, including this update of
potentials, is given in Listing 1. In a time step, arrival
events have to be processed first (Line 2), then all re-
maining vehicles are updated with the amount of energy
they received during the last interval. Departing vehicles
are automatically removed in this step.
At this point, all information items have been updated
to compute the load shift potential (Line 3) and process

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 403
1 for step=1 to noSteps
2 —Process mobility events
3 Computeshiftpotentials// acc .to3.2.
4 for request=1to noShiftRequests
5 —Save targetloads (request)
6 for interval=1 to duration(request)
7 —Sort vehicles
8 —Determine participating vehicles
9 for vehicle=1to noInvolvedVehicles
10 —Shiftloadcurve// acc .to3.1.
11 —Fixthisintervalin load curve
12 —Updateshiftpotential(vehicle )
—
Listing 1. Overview of the DSM cycle.
possible load shift requests (Line 4). Load shift requests
are issued by the supplier in reaction to unforeseen de-
viations from the planned load curve or because of eco-
nomic considerations. The direct algorithm for computa-
tion of potentials is cheap and can be parallelized. Proc-
essing of load shift requests can be distributed as well,
but requires more communication to exactly achieve the
requested shift.
Load shifts over several intervals are processed in
steps in temporal order (Line 6). The average goal values
are saved, as the cumulative load changes while shifting
(Line 5). A request is first compared to the potentials,
thereafter the satisfiable part of it is assigned to the par-
ticipating vehicles. Normally, the vehicles with the long-
est remaining charging time offer the highest flexibility
and load shift potential. Hence, they are sorted in de-
scending order according to their departure times (Line 7).
The partial shifts are performed with a greedy algo-
rithm. For one vehicle after another (Line 9), the maxi-
mum possible shift is performed by generating a new
curve (Line 10) which attains a minimum or maximum.
When processing the next partial load shift, the previ-
ously treated intervals are fixed (Line 11), such that the
achieved shift cannot be reverted accidentally.
Temporal resolution influences the runtime and accu-
racy of the simulation. On the one hand, it seems rea-
sonable to choose a small time step for more precise load
curves, as vehicles are constantly arriving and departing.
On the other hand, a longer interval gives more reliable
information on the shift potential. When processing shifts
over several intervals, the potential changes after the first
step, such that the request might not be fulfilled in the
second. In this sense, a load shift is understood to be an
averaged shift referred to the medium load value in the
interval and longer time steps of 5 to 10 minutes are pre-
ferred. Figure 5 shows the load curve and potentials of
one vehicle in a demonstrator application.
4.1. Preconditioning
Preconditioning demonstrates exemplary the integration
of further constraints into the model. Constant energy
Figure 5. Exemplary vehicle with load curve (black), upper
and lower power limits (red/green) and shift potentials
(areal). For a sample tariff, discharging between 17:45 and
18:45 as well as between 20:30 and 21:30 is beneficial and
full charging postponed accordingly .
supply during parking and the strictly limited amount of
energy while moving make it reasonable to start air con-
ditioning before leaving. Depending on the difference to
the preferred temperature, preconditioning is started 10
to 20 minutes before departure. The necessary amount of
energy for heating or cooling is calculated from the ther-
mal energy
.
heat
Ec mT

 (28)
The outside temperature curve is specified in the XML
data and the vehicle is assumed to arrive with the goal
temperature.
The product of mass and heat capacity is set to a typi-
cal value of 100 kJ/kg and thermal transfer through the
vehicle body to 75 W/K [23].
To calculate the amount of energy for preconditioning,
we first determine the interior temperature at the begin-
ning of preconditioning via
   
100 kJ11.
75 W
inout in
TttT tTt  (29)


0.
in goal
TT (30)
We now assume that the target temperature is ap-
proached linearly and, hence, compute thermal transfer
during preconditioning as
1
75 .
2
transgoal in
W
PTT
K
  (31)
The heating or cooling power is then
/1
100 .
a cgoalin
p
rec
kJ
PTT
Kt
 (32)
In order to obtain the final value, which is subtracted
from the upper power limit ,maxV
P, we assume a typical
coefficient of performance of 1.52 and a ventilation con-
sumption of 5 00W [23].
/
.500
1.52
a ctrans
prec
PP
PW


(33)

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 404
5. Exemplary Application
To quantify the potentials, two scenarios are constructed
as outlined in Table 1. The results given in the following
sections have to be interpreted as an upper limit for the
load shift potential, since it is assumed that all vehicles
are connected to the grid during parking time.
Scenario 1 (year 2015) is based on the goals of the
German Federal Government, which were developed in
the so-called National Electric Mobility Platform (NPE)
and published in its second report [24]. Scenario 2 (year
2030) is based on a forward projection of these goals, as
shown in Figure 6.
The data on the behavior of the vehicles are required
from a representative German mobility study which is
based on realistic driving profiles for more than 32,000
distinct routes [25]. Due to the limited size of the data set,
only some of the EV assumed to exist in total are explic-
itly represented in the simulation. As a simple scaling of
the data set would not have increased the significance,
each scenario is limited to 10,000 vehicles and approx.
16,000 distinct routes. This number of vehicles corre-
sponds to the size of the original data set and the as-
sumptions regarding the participation in DSM measures.
However, the calculated load shift potentials are scaled
up to the total number of EV in each scenario.
Table 1. Basic information on applied scenarios.
Parameter Scenario 1 Scenario 2
Year 2015 2030
Duration (starting at 12 noon) 24 hours 24 hours
Total number of EV o n the road 200,000 6,000,000
Vehicles participating in DSM 50% 90%
Maximum power at home 3.68 kW 10 kW
Maximum power at public places 10 kW 22 kW
Number of selec t ed vehicles 10,000 10,000
Approx. number of distinct
routes 16,000 16,000
Scale-up factor 20 600
Share of hybrid EV 60% 40%
Average battery capacity 9635 Wh 18,239 Wh
Vehicles participating in V2G 30% 50%
Figure 6. Predicted number of EV in Germany.
The simulation uses a predefined static load curve
which is independent of the power demand of the EV.
This exogenous load cu rve correlates with the p redefined
tariffs shown in Figure 7 and illustrates a realistic load
situation. Preconditioning was not considered in these
examples to produce more general results without any
dependencies on a given whether situation.
5.1. Scenario 1 (Year 2015)
The NPE goals [24] include one million EV on German
roads in the year 2020. A non-linear extrapolation using
the current number of vehicles and the NPE schedule
leads to an approximate number of 200,000 vehicles in
the year 2015 as shown in Figure 6. In this scenario, the
average battery capacity is 9635 Wh because of the
comparably high share of hybrid EV which generally
have a significantly lower battery capacity than pure EV.
Furthermore the share of vehicles which participate in
V2G is only 30% due to an assumed lack of acceptance.
After initialization, the DSM computes the aggregated
load shift potential of the currently connected vehicles.
Perfect foresight is assumed regarding the departure of
connected vehicles, but not for new arrivals. This leads to
a realistic representation of the degrees of freedom for
the simulated DSM to respond to load shift requests in
each individual time step. Running the simulation with-
out any load shift requests leads to a bare visualization of
the varying potentials over the corresponding time hori-
zon. Table 2 shows selected results, the notations refer to
the definitions given in Section 2.3.
The results show that this setup already leads to sig-
nificant load shift potentials. Especially the constantly
high negative potential is remarkable. However, if the
execution of a load shift is requested, the corresponding
potentials in future time slots will decrease significantly.
Figures 8 and 9 show the aggregated load shift potential
over the considered time horizon, without taking the pos-
sible realization of load shifts into account. As the load
shift potential changes in each time step i t it is in fact
two-dimensional. The curves in Figures 8 and 9 show
the specific load shift potential available at i t when the
point in time i
tt

is reached.
Figure 7. Predefined grid load curve and tariffs for the
simulation.

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 405
Figure 8. Resulting limits of load shift potentials (10,000 EV) in Scenario 1 (Year 2015).
Figure 9. Resulting limits of load shift potentials (10,000 EV) in Scenario 2 (Year 2030).
5.2. Scenario 2 (Year 2030)
Again based on the NPE goals [24] and the German mo-
bility study [25], a non-linear extrapolation leads to an
estimated number of 6 million EV in Germany in the
year 2030 (Figure 6). Due to the assumed increasing
number of pure EV and the decreasing share of hybrid
EV as well as the expected development of battery tech-
nology, the average battery capacity is significantly high-
er than in the year 2015 (cf. Table 1). Furthermore, ris-
ing acceptance for V2G measures is expected, which is
why the share of participating vehicles is set to 50%.
The results given in Table 3 differ significantly from
those of Scenario 1. Due to the increased average battery
capacity, the aggregated load shift potential increases at
almost all points of the simulation. Consequently, the
DSM has higher degrees of freedom in utilizing load
shifting potentials. Especially the rise in the super-po-
sitive load shift potential is remarkable. This may be ex-
plained by the higher share of vehicles participating in
V2G measures.
To assess the results, a short example will be given
below. The released positive secondary power reserve in
Germany from October 24 to 30, 2011 reached a peak of
approx. 2 GW. This means that the providers have to
decrease their loads or to feed electricity into the grid.
This form of power reserve can only be provided by ve-
hicles which take part in V2G measures. In our nomen-
clature, the sum of positive and super-positive load shift
potential serves to fulfill requests of positive reserve
power. The results in Table 3 show that the peak demand
in the week considered could have been met by EV at
every time in the time horizon of the simulation. How-
ever, the duration of the request and the profitability for
the customer [26] have a significant influence on its fea-
sibility.

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 406
Table 2. Selected scaled-up simulation results of Scenario 1
(2015).
Load shift potential [kW]
EV load Negative Positive Super-p.
Number
of EV
1:00 pm 30,529 70,724 26,422 21,363 33,729
6:00 pm 26,614 90,552 21,739 50,603 63,870
3:00 am 1903 61,127 1718 57,434 76,281
8:00 am 22,019 109,945 15,758 66,233 79,145
Max. 36,951 115,365 29,497 82,894 90,994
Min. 1112 59,445 987 12,907 26,075
Table 3. Selected scaled-up simulation results of Scenario 2
(2030).
Load shift potential [MW]
EV load Negative Positive Super-p.
Number of
EV
1:00 pm 2067 14,122 1665 6555 1.824 M
6:00 pm 2241 20,078 1749 14,713 3. 445 M
3:00 am 150 17,675 143 17,055 4.098 M
8:00 am 1414 23,183 821 19,730 4.257 M
Max. 2347 25,596 1795 23,538 4.873 M
Min. 94 12,376 86 4330 1.408 M
5.3. Load Shifting
As explained in 4, the simulation result highly depends
on the temporal resolution. Choosing a larger time step
gives information on the attainable load shift over a
longer time period. Figure 10 shows the remaining av-
erage potential in percent in comparison to the 5-minute
time step used for the examples above. The difference
between 2015 and 2030 results from the higher participa-
tion in DSM measures, allowing higher short-term shifts.
Figure 11 shows the result of a load shift of +1500 kW
from 23:00 to 0:00 (approx. 25% of the 5 min. potential
at 23:00) for the 2015 scenario. Depending on the given
tariff, the effects of a shift might be visible for several
hours: The vehicles are forced to charge during 23:00
and 0:00, reduce their power or even discharge from 0:00
to 5:00, and fill up the battery between 5:00 and 6:00
because of a slightly cheaper tariff level (Figure 7).
Meanwhile, the negative potential is reduced to 2000
kW during the shift and considerably decreased after-
wards. This information could be used for drafting risk-
based DSM algorithms using load shift potentials as a
measure of remaining flexibility.
6. Outlook
The knowledge and software developed in the course of
this analysis is highly relevant for DSO and will be the
basis for various further developments, from grid devel-
opment planning to drafting business models.
Figure 10. Remaining negative shift potential in percent for
larger time steps.
Figure 11. Resulting absolute change in load and potentials
when performing a shift of 1500 kW from 23:00 to 0:00.
Real battery data or advanced battery models should
be used in further studies to assess the real wear of the
vehicle batteries. More reliable data for the future distri-
bution of EV and their corresponding user patterns would
also improve the results.
Stochastic influences should be included as well to
simulate customer reactions to unforeseen ev ents and the
partly unreliable communication between the agents. Not
only flexible and robust algorithms, but also communica-
tion protocols are needed for more realistic simulations.
Currently, no grid-imposed constraints are included.
Due to fast charging stations at home, a shortage in the
local grid might occur especially in the evening hours.
Besides the lowest-voltage grid, also the low- and me-
dium-voltage grid might be affected in some regions [26].
Furthermore, the time of charging has an impact on the
specific CO2 emissions (and other emissions) during EV
usage due to the time-dependent energy mix [27]. Both
issues could be incorporated in future studies.
In addition, simulations of the actual power flow will
have to be applied to include grid topology. This will be
achieved by coupling the algorithms for optimal power
flow computations presented in [28] with the existing
software. An adaptive algorithm for optimal control of
the distributed local grid as a function of the flexible de-
mand of other users or even flexible or fluctuating elec-
tricity supply by combined heat and power units or
photovoltaic systems might be an interesting extension of
this project. The variety of proposed DSM algorithms
(P2P, heuristic, deterministic) can be evaluated for dif-

Model-Based Quantification of Load Shift Potentials and Optimized Charging of Electric Vehicles 407
ferent scenarios in the future, with CO2 minimizing stra-
tegies being taken into account.
7. Conclusions
A bottom-up simulation algorithm is presented, which
allows for determining the cumulative load curve and
load shifting potentials of EV in the charging process in a
generic energy grid. A direct algorithm was developed
for this purpose that can be efficiently applied to all ve-
hicles in parallel.
For a manual change of the grid load by requesting
load shifts, a cost-optimal shift is performed by altering
the original charging curve of some participating vehicles
within their individual constraints, e.g. departure with
desired state of charge of the battery. A mixed-integer
linear program (MIP) has been developed that can be
solved more efficiently than the ori ginal non-convex for-
mulation.
Results of two scenarios based on mobility studies
show that the load shifting potential of EV is significant
and contribute to a necessary relaxation of the future grid.
The maximum charging and discharging power and the
average battery capacity are crucial to the overall load
shift potential. However, the individual willingness to
participate will depend on the profitability for the cus-
tomer, which is why future projects should pay particular
attention to this issue.
8. Acknowledgements
This work was supported by the Energy Solution Center
(EnSoC), an association of major industrial corporations
and research institutions in Germany. EnSoC focuses on
projects using high-performance computing technology
in application-oriented energy research. The authors would
like to thank EnSoC for the continuous support and the
professional project management.
We acknowledge support by Deutsche Forschungsge-
meinschaft and Open Access Publishing Fund of Karl-
sruhe Institute of Technology.
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Nomenclature
Load Curve
V
L Continuous load curve of Vehicle V
V
L Vectorized, piece-wise constant load curve
V
Z
 Positive part of vectorized load curve
V
Z
 Negative part of vectorized load curve
Vehicle Properties
V
T Time interval of connection
V
E Amount of energy until full charge
V
E Amount of energy dischar gea ble
,minV
P Minimum charge power
,maxV
P Maximum charge power
,minV
P Minimum discharge power
,maxV
P Maximum discharge power
V
C Binary charge indicator
V
F
 Binary feed indicator
Load Shift Potential
V
S Continuous bounded load shift potential (LSP)
V
S Piece-wise constant bounded LSP
,minV
S Lower boundary of LSP through charging
,maxV
S Upper boundary of LSP through charging
,minV
S Upper boundary of LSP through feeding
,maxV
S Lower boundary of LSP through feeding
V
R

Negative LSP
V
R

Positive LSP
V
R

Super-positive LSP
System Properties
W Tariff/weight function
W

Piece-wise constant charge tariff
W

Piece-wise constant feed tariff