yiyuo

1. Optimal Control for Plug-in Hybrid Electric Vehicle Applications
Stephanie Stockar, Vincenzo Marano, Giorgio Rizzoni and Lino Guzzella
Abstract— Plug-In Hybrid Electric Vehicles (PHEVs) are a
promising solution to reduce fuel consumption and emissions,
due to their capability of storing energy in the battery through
direct connection to the grid. In order to achieve the highest
benefits from this technology, a suitable energy management
strategy that optimizes the vehicle energy efficiency must be
defined.
The present work proposes a supervisory controller for
PHEVs, which explicitly accounts for the on-board electricity
consumption during vehicle operations. The approach is based
on the formulation of an optimal control problem that is
solved by the Pontryagin’s minimum principle to produce a
solution that can be implemented on-line. Simulation results
are presented to illustrate the developed energy management
strategy.
I. INTRODUCTION
Recent improvements in lithium-ion battery technology
are making PHEVs a viable solution to reduce petroleum
consumption and emissions in the transportation sector. For
this reason, they are currently receiving great interest in the
United States.
Similar to the conventional Hybrid Electric Vehicles
(HEVs), PHEVs include two different sources of power
on-board that can be controlled to achieve higher energy
efficiency, lower pollutant emissions and lower operation
cost than a conventional vehicle. Furthermore, PHEVs have
the capability of storing energy in the battery through direct
connection to the energy grid. This poses further challenges
for the energy management strategy, such as accounting for
the cost of the recharging operations or allowing for battery
depletion during the vehicle operation.
Compared to conventional HEVs, where the supervisory
energy management strategy maintains the battery around
a nominally constant State of Charge (SOC), the ability of
PHEVs to recharge directly from the power grid provides the
opportunity to deplete the battery, with further improvements
of the vehicle fuel economy and emissions. On the other
hand, this feature poses complex challenges for the definition
of an energy management strategy. In fact, most of the
available electrical energy is supplied from the grid, hence
introducing the vehicle to grid interactions in the energy
optimization problem. This implies that the performance of
PHEVs are influenced by additional variables, such as the
generation mix, fuel and electricity cost.
S. Stockar and L. Guzzella are with the Department of Mechanical
and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland. Email:
stockas@student.ethz.ch, lguzzella@ethz.ch
V. Marano (corresponding author) and G. Rizzoni are with the Ohio State
University Center for Automotive Research, Columbus, OH 43212, USA.
Email: marano.8@osu.edu, rizzoni.1@osu.edu
Several control approaches have been proposed in the
literature on this topic [1], [2], [3], [4], [5], mostly based on
heuristic methods. Although relatively simple to implement,
rule-based control schemes do not guarantee the optimality
of the results, as the overall performances rely on the
particular vehicle structure and driving conditions. On the
other side, Stochastic Dynamic Programming (SDP) shows
improvements in fuel economy, operation costs and emission
but it requires a significant amount of statistical driving data
for the model validation. In addition, the policy evaluation
need huge computational time. This is usually overcome
by the implementation of look-up tables where the off-line
estimated control policy data are stored.
The Equivalent Consumption Minimization Strategy
(ECMS) is a well-known energy management algorithm for
HEVs that can be easily implemented on-line and presents
good results in term of fuel consumption reduction [6], [7],
[8]. In addition, the calibration effort is very small compared
to heuristic controllers, and can be easily adapted to different
architectures without changing the algorithm structure [9].
Recent studies have attempted at adapting the ECMS to
the supervisory control of PHEVs [10], [11], [12]. From the
results obtained, it appears that the best results in terms of
fuel economy can be achieved if the energy management
strategy is able to track a desired profile for the battery
SOC. In case of charge-sustaining HEVs, this profile is
nominally constant, hence relatively independent from the
driving conditions. In case of PHEVs, however, the primary
power source is electric, implying that the battery may be
considerably depleted at the end of a driving schedule.
Consequently, in order to specify a desired SOC profile
that depletes the battery and minimizes the vehicle fuel
consumption, the duration of the vehicle duty cycle must
be known a priori.
To this extent, the present work aims at revising the super-
visory control problem for PHEV applications, proposing a
more general formulation for the ECMS for charge depleting
operations. Starting from an optimal control problem formu-
lation and by applying the Pontryagin’s minimum principle,
it is possible to reduce a global optimization problem into a
local (instantaneous) minimization problem [13], which can
be implemented on-line with low computational effort.
The study presented in this paper is conducted in simula-
tion, using validated energy-based models of hybrid power-
trains.
II. DESCRIPTION OF THE VEHICLE SIMULATOR
The simulator used for this study was developed at
The Ohio State University Center for Automotive Research
2010 American Control Conference
Marriott Waterfront, Baltimore, MD, USA
June 30-July 02, 2010
FrA10.1
978-1-4244-7425-7/10/$26.00 ©2010 AACC 5024

2. TABLE I
DESCRIPTION OF THE MAIN VEHICLE COMPONENTS.
Component Type Specifications
Chassis Mid-size
SUV
2005 Chevrolet Equinox
Engine 1.9l Diesel 4 Cylinder, 16v, Euro 4, 103 kW @
4000 rpm, 305 Nm @ 2000 rpm
Belted Starter/
Alternator
Permanent
Magnet
Kollmorgen Servomotor, 10.6 kW
Nominal Power, 80 Nm Peak
Torque, 4150 r/min Max Speed
Energy Storage NiMH
Batteries
38 Panasonic Prismatic
Modules,7.2V, 6.5 Ah
Transmission 6 speed auto-
matic
450 Nm torque capacity
Electric Motor AC induction 32 kW , 185 Nm of peak torque
(OSU-CAR). The simulator builds upon the energy-based
model of a conventional hybrid electric vehicle [14], [15],
[16], designed for the Challenge-X student competition
project and implemented in MATLAB - Simulink environ-
ment. The vehicle model was validated on driving tests data
collected during over three years of Challenge-X competition
and the vehicle components validation was performed on
laboratory tests data, [14], [15], [17].
A. The Vehicle Platform
The HEV model used in the study is based on a se-
ries/parallel power-split hybrid architecture. Fig. 1 outlines
the vehicle drivetrain, while Table I describes main vehicle
components.
Fig. 1. Challenge X vehicle platform.
The proposed configuration includes a Diesel engine cou-
pled to a Belted Starter Alternator (BSA) on the front axle
and an Electric Motor (EM) on the rear axle [18].
The above configuration allows for a variety of modes such
as pure electric drive, electric launch, engine load shifting,
motor torque assist, and regenerative braking. A simple block
diagram of the power flows on the vehicle is shown in Fig.
2.
B. Control Oriented Model of the Battery
The vehicle simulator was converted to PHEV by replac-
ing the model of the existing energy storage system with a
Fig. 2. Block diagram of the drivetrain power flows.
TABLE II
LI-ION BATTERY DATA.
Total Energy 10kWh
Nominal Voltage 3.3 V
Number of cells in series 90
Number of cells in parallel 15
Nominal Capacity per cell 2.3 Ah
10kWh Li-Ion battery, which enables for an all electric range
of approximately 17 miles [19]. The battery data are reported
in Table II.
A simple energy-based dynamic model for the battery was
built and implemented in the presented simulator. According
to the equivalent circuit analogy, the battery system dynamics
is described by the following equation:
Vbatt(t) = Voc − R · Ibatt(t) (1)
where the open-circuit voltage Voc and internal resistance R
are functions of the battery SOC, which is defined as:
SOC(t) =
Q (t)
Qmax
(2)
where Q (t) is the capacity of the battery at time t and Qmax
represents the maximum battery capacity. The model has
been validated on a set of laboratory test data [17].
It is worth observing that the above model represents a
strong approximation of the real behavior of a battery, for
example neglecting the dependence of the model parameters
on depth of discharge and temperature. Such assumptions
have been formulated exclusively with the objective of
developing a simple dynamic model that is practical for real-
time applications and control design.
The above model structure is in fact considered in the
formulation of the optimal control problem for the energy
management strategy. In this case, the battery system dy-
namics is described by the state equation:
d
dt
SOC(t) = η
Ibatt(t)
Qmax
(3)
where
η =
ηbatt if Pbatt ≤ 0;
1
ηbatt
if Pbatt > 0.
with ηbatt representing the efficiency of the battery and
power electronics during charging and discharging opera-
tions.
5025

3. As a final remark, the modular structure of the simulator
allows for more detailed battery models to be used in
replacement of the one developed. To this extent, physically-
based models could be considered in order to study the
battery pack behavior and possibly aging in relation with the
vehicle duty cycles and the supervisory energy management
strategy.
III. FORMULATION OF THE OPTIMAL CONTROL
PROBLEM FOR PHEV ENERGY MANAGEMENT
The objective of this study is to apply the optimal control
theory to define the supervisory energy management strat-
egy for PHEV applications, starting from the Pontryagin’s
minimum principle [13], [20].
Compared to the corresponding formulation for charge-
sustaining HEVs presented in [7], [21], the constraint on the
final SOC: SOC(ta) = SOC(tb) is here removed in order
to allow for charge depleting operation.
Furthermore, the usage of the battery is no longer related
to an equivalent fuel mass flow rate, as this formulation
prevents from accounting for the power received from the
grid. For this reason, the function to be optimized must be
redefined, for instance by including the total operating costs
of the vehicle as proposed in [3], or the total CO2 emissions.
In this paper, the latter approach is considered, namely
defining the cost function based on the cumulative CO2
emissions produced by the vehicle during a driving path:
JP HEV (u(t)) =
tb
ta
˙mCO2,f (t) + ˙mCO2,e(t)dt (4)
where mCO2,f (t) is the mass of the CO2 produced by the
engine and mCO2,e(t) is the mass of CO2 produced as result
of the electric energy on-board consumed. If the former term
is directly related to the engine fuel consumption, the latter
can be estimated based on the battery energy consumed at
the end of the driving path and the average CO2 content due
to the electricity generation mix [22].
In order to later apply the optimal control theory to the
PHEV system, it is necessary to associate the CO2 mass
flow rates produced during the vehicle operation to system
variables. For this study, the corresponding CO2 masses are
calculated as follows:
˙mCO2,f = κ1 · Pf (t)
˙mCO2,e = κ2 ·
Pbatt(t)
ηch
(5)
where κ1 and κ2 represent the specific CO2 content in the
fuel and in the electricity (consumed) per kWh. For instance,
κ1 = 0.294kg/kWh is assumed for the Diesel engine and
κ2 = 0.567kg/kWh for the USA electricity production
scenario. These parameters were estimated using the GREET
software [22]. The term ηch = 0.86 represents the battery
charging efficiency when the vehicle is connected to the grid.
According to the form of the cost function, the bigger
the ratio between κ1 and κ2, the more the controller will
privilege the electric energy over the fuel.
In order to account for the energy stored in the battery, an
additional variable is defined in this study, namely the State
of Energy (SOE):
SOE =
Ebatt(t)
Emax
(6)
where Emax = Qmax ·VOC is the maximum energy that can
be stored in the battery.
Considering the SOE as the new state variable instead of
the SOC, it is possible to rewrite the system state equation
as follows:
d
dt
SOE(t) = −η
Pbatt(t)
Emax
(7)
where Pbatt is the battery power, which is defined positive
if charging the battery: Pbatt = −I(t) · Vbatt(t). Note that,
if Vbatt(t) = VOC, then SOE = SOC for any time t.
The control variable u(t) for the considered problem is a
two-dimensional vector defined as:
u(t) = [Pbatt; PEM,el/Pbatt] (8)
where the first element is the total battery power and the latter
represents the power split between the rear electric motor
and the belted starter/alternator. According to the power
flow diagram in Fig. 2, it is possible to state the following
balances:
Ptot(t) =PICE(t) + PBSA,el · ηBSA + PEM,el · ηEM
Pbatt(t) =PBSA,el + PEM,el
(9)
where Ptot(t) is the total power request to the hybrid
driveline. According to Fig. 2, Ptot = Pwh,front +Pwh,rear.
The control and state variables are subject to constraints in
order to respect limitations from the drivetrain components
and for safe vehicle operations. In particular, the battery SOE
(in principle defined between zero and one) is usually limited
in order to avoid operating conditions that may result in
battery abuse and premature aging [23]:
SOEmin ≤ SOE(t) ≤ SOEmax (10)
Further constraints may posed by the components of the
drivetrain, which are typically subject to power limitations:
Pbatt,min ≤Pbatt(t) ≤ Pbatt,max
PEM,min ≤PEM (t) ≤ PEM,max
PBSA,min ≤PBSA(t) ≤ PBSA,max
(11)
IV. PONTRYAGIN MINIMUM PRINCIPLE
The above optimization problem may be solved through
numerical approaches (such as dynamic programming [24]),
or by applying analytical methods. In particular, the Pon-
tryagin’s minimum principle [20] is here adopted to solve
the optimal control problem, whose state dynamics can be
described by the equation:
˙x(t) = f(x(t), u(t), t) (12)
5026

4. and with the cost functional defined as:
J(u) =
tb
ta
L(x(t), u(t), t)dt + K(xb, tb) (13)
The Pontryagin’s minimum principle converts a global
optimal control problem into a local minimization problem,
thereby reducing the computational requirements and allow-
ing one to solve the problem in continuous time domain.
The theorem introduces the Hamiltonian function:
H(t, u(t), x(t), λ(t)) = L(t, u(t), x(t))+λ(t)·f(t, u(t), x(t))
(14)
which has to be minimized at each time t to provide the
optimal control policy uo
(t):
uo
(t) = arg min
u
H(t, u, λ(t)) (15)
A. Necessary Condition for Optimality
If uo
(t) is the optimal control policy, then the following
conditions are satisfied:
i) ˙xo
(t) = ∇λH|o = f(xo
(t), uo
(t), t)
ii) ˙λo
(t) = −∇xH|o
iii) xo
(ta) = xa
iv) xo
(tb) ∈ S
v) H(xo
(t), uo
(t), λo
(t), t) ≤ H(xo
(t), u(t), λo
(t), t)
If the state x(t) is bounded, an additional term is intro-
duced in the Hamiltonian function in order to account for
this limitation. The corresponding Lagrange multiplier is a
scalar denoted by μl and subject to the following necessary
condition:
vi) μo
l (t) ≥ 0
B. Application to PHEV Energy Management
For the PHEV control problem, the Hamiltonian function
is calculated according to the mass flow rates defined above
in Eq. (5). In this case, the Hamiltonian function becomes:
H(x(t), u(t), λ(t), t) =
= κ1 · Pf (t) + {
κ2
ηch
−
λ(t) · η
Emax
+
μ · η
Emax
} · Pbatt(t)
(16)
with
μ =
⎧
⎨
⎩
−μl if SOE(t) ≥ 0.95;
μl if SOE(t) ≤ 0.25;
0 else.
where μl is the scalar Lagrange multiplier for the inequality
constraints on the SOE and is typically determined itera-
tively.
With the substitution of μ and the notation introduced, the
usage of the battery will be penalized when the SOE is at
its lower bound, while it will be facilitated if the battery is
fully charged.
The necessary condition for the co-state λo
(t) is:
ii) ˙λo
(t) = −∇xH|o = −
∂
∂x
κ1Pf (t)−
−
∂
∂x
κ2 · Pbatt(t)
ηch
+
∂
∂x
Pbatt(t) · η
Emax
· (λ(t) + μ(t))
(17)
with μo
l (t) ≥ 0. The ODE for the co-state λo
(t) can be
furthermore simplified because neither Pf (t) nor Pbatt(t) are
explicit functions of the SOE (or SOC).
This assumption is not valid for the battery efficiency. In
fact, according to the battery model presented above, the
battery power is Pbatt(t) = Ibatt(t) · Vbatt(t), while the
maximum battery power during discharging is Pmax(t) =
Ibatt(t)·VOC(SOC). This will further penalize any operation
at low SOE, when the battery efficiency is lower.
The battery efficiency for the discharging phase is defined
as: ηbatt = Vbatt(t)/VOC. Inserting this expression in Eq.
(17), the co-state ODE can be rewritten as follows:
˙λo
(t) =
⎧
⎪⎨
⎪⎩
Pbatt(t)
Emax
· ∂
∂x ηbatt · (λ(t) + μ(t)) Ibatt < 0
−Pbatt(t)
Emax·η2
batt
· ∂
∂x ηbatt · (λ(t) + μ(t)) Ibatt ≥ 0
where ηbatt is a function of the SOC.
Finally, according to the Pontryagin’s minimum prin-
ciple, the control policy denoted by uo
(t) is optimal if
H(xo
(t), u(t), λo
(t), t) presents a global minimum with re-
spect to uo
(t).
C. Algorithm Implementation
In order to implement the above algorithm, the torque
split factors fICE and fBSA are introduced. These variables
determine the ratio of the torque request that will be satisfied
by the engine and the BSA, respectively.
By conducting a simple energy balance to the drivetrain
shown in Fig. 2, it is possible to generate three matrices
containing all the possible torque combinations that satisfy
the power balance of Eq. (9), namely:
TICE(t) = fICE · Treq(t) ∈ Rnxm
TBSA(t) = fBSA · (1 − fICE) · Treq(t) ∈ Rnxm
TEM (t) = (1 − fBSA) · (1 − fICE) · Treq(t) ∈ Rnxm
(18)
Note that, the dimensions m and n are related to the chosen
resolution for the factors fICE and fBSA.
The battery and fuel power are then computed in order to
evaluate the Hamiltonian function as expressed in Eq. (16).
Specifically, the battery power is given by:
Pbatt(t) = PEM,el(t) + PBSA,el(t) (19)
where the power of the electric machines considers the
related efficiencies, computed from the rotational speeds and
the torque matrices previously generated.
The power associated to the fuel utilization is calculated
considering the lower heating value of the fuel, which is
assumed 43 MJ/kg.
The behavior of μ(t) is described by the following:
μ(t) = μ1(t) + μ2(t) (20)
μ1 =
μl if SOE ≥ 0.95
0 otherwise
and
μ2 =
−μl if SOE < 0.25
0 otherwise
5027

5. 0 200 400 600 800 1000 1200 1400 1600
0
20
40
60
80
100
120
Time[s]
Velocity[km/h]
Fig. 3. Vehicle velocity profile for the driving cycle considered in this
study
The differential equation (17) for the Lagrange multiplier
λo
(t) contains the derivative of the battery efficiency, which
is a function of the SOC, in x (the SOE). In order to evaluate
this expression, the following relationship is used:
∂
∂SOE
SOC =
VOC(SOC)
Vbatt(t)
(21)
This allows for the scheduling of the time-independents
terms of Eq. (17) in the parameter γ(SOC) and the ODE
can be finally rewritten as:
˙λo
(t) = Pbatt(t) · γ(SOC) · (λ(t) + μ(t)) (22)
At any time step, the combination of fICE and fBSA that
corresponds to the minimum of the Hamiltonian function
matrix is chosen as the solution of the problem. The algo-
rithm implementation above described allows for achieving
real-time operations.
V. SIMULATION RESULTS AND DISCUSSION
The developed control strategy was applied to the PHEV
simulator to evaluate the vehicle performance during driving
operations.
To this extent, a custom driving cycle was used for the
validation of the supervisory algorithm, extracted from a
database of real-world duty cycle data collected from a
PHEV fleet. The cycle considered is shown in Fig. 3
The results shown were obtained assuming the bat-
tery fully charged at the beginning of each test, namely
SOE(ta) = 0.95.
With reference to the driving profile shown in Fig. 3, the
value of the parameter λ0 and μ was chosen iteratively over a
large number of possible values in order to minimize the cost
functional JP HEV (the total CO2 emissions), which include
the on-board fuel and the energy stored from the grid.
As shown in Fig. 4, the value of λ0 not only affects
the performance of the vehicle, but also defines the vehicle
operation mode. As λ0 decreases from its optimal value, the
electric energy usage is increasingly penalized. Therefore,
0
4
8
12
16
20
−20−15−10−505101520
0
0.2
0.4
0.6
0.8
1
μ
λ
0
FinalStateofEnergy[−]
Fig. 4. Final value of the SOE as function of the Lagrange multipliers for
the considered driving cycle
0
4
8
12
16
20
−20−15−10−505101520
250
260
270
280
290
300
310
λ
0
μ
CO
2
[g/km]
Fig. 5. CO2 emission due to vehicle operation for the considered driving
cycle
the controller will attempt at utilizing the engine rather than
discharging the battery. Conversely, if λ0 is greater than
λo
0, the controller will operate the vehicle in all-electric
mode, until the lower constraint on the battery SOE is
reached. Then, the controller will maintain charge-sustaining
operations around the predefined value SOEmin = 0.25. In
particular, with λ0 ≤ 0, the strategy does not deplete the
battery and the final SOE is of about the same value as the
initial one. For λ0 grater than 5, the vehicle operates mostly
in electric mode, while for values of λ0 between 0 and 5 the
battery is depleted but not until the lower SOE value.
The scalar Lagrange multiplier μ has no impacts on the
vehicle performance, but rather accounts for the boundary
conditions. As shown in Fig. 4, μ has to be chosen large
enough. In fact, if μ is too small, the SOE will exceed its
boundaries. Conversely, μ ≥ 10 leads to a SOE profile that
remains bounded for every value of λ0.
Fig. 5 shows the effect of different initial values for the
Lagrange multiplier λ and μ on the selected performance
index the cumulative CO2 emissions due to vehicle opera-
tion. The results indicate that, for the specific driving cycle
5028

6. TABLE III
FUEL ECONOMY AND CO2 EMISSION RESULTS
λ0 MPG l/100km gCO2/km
-10 23 10.3 304
4 38 6.2 296
6 55 4.3 278
10 52 4.5 289
0 200 400 600 800 1000 1200 1400 1600
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
StateofEnergy[−]
Time [s]
λ0
= −10
λ0
= +4
λ0
= +6
λ0
= +10
MPG = 23
MPG = 38
MPG = 55
MPG = 52
Fig. 6. Evolution of the SOE according to different λ0 and μ = 16 for
the considered driving cycle
considered, the benefits of electric drive are limited and the
resulting CO2 surface is almost flat. This is consequent to
the fact that the electricity used in the United States has a
large specific CO2 content since is produced from coal.
For the considered diving cycle, four different values of λ0
have been selected, with μ set to 16, to show the evolution
of the SOE over time. The main results are summarized in
Table III.
In particular, Fig. 6 shows that a negative value of λ0
results in charge sustaining operation at SOE = SOEmax =
0.95. By setting λ0 greater than 10, the vehicle is first
operated in electric drive and then, when the lower limit
of SOE is reached, the controller automatically switches to
charge sustaining mode.
In addition, two intermediate solutions are observed for
λ0 = 4 and λ0 = 6. With λ0 = 6 the battery is slowly
depleted during the cycle allowing the SOE for reaching
its lower bound only at the end of the driving pattern and
avoiding any charge sustaining operations. Same behavior
can be observed for λ0 = 6, but with the difference that,
at the end of the driving path, the battery has still energy
available.
Fig. 6 also reports the fuel economy corresponding to each
selected SOE profiles. The case with λ0 = −10 presents a
fuel economy of about 23 MPG (10.3 l/100 km), which is
consistent with the type of the cycle and vehicle. For λ0 = 4,
the energy is supplied by both energy sources, namely battery
and fuel. This results into an improved fuel economy of 38
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ICETorque[Nm]
0 200 400 600 800 1000 1200 1400 1600
−100
0
100
200
EMTorque[Nm]
0 200 400 600 800 1000 1200 1400 1600
−100
0
100
BSATorque[Nm]
Time [s]
λ
0
= −10
λ0
= +6
λ
0
= +10
Fig. 7. ICE, BSA and EM torques for different λ0 and μ = 16 for the
considered driving cycle
MPG (6.2 l/100 km). Obviously, the best results in term of
reduction of net fuel consumption are achieved when the
battery is completely depleted at the end of the cycle.
This condition can be achieved in two different ways,
namely EV Mode control and Blended Mode control [19].
With a proper calibration of the parameter λ0, the proposed
energy management strategy is able to reproduce both be-
haviors. The best value of the fuel economy is obtained for
λ0 = 6 and the SOE slowly decreases through the driving
path and reaches the lower bound only at the end of the
cycle. In this case the fuel resulting fuel economy is of
about 55 MPG (4.3 l/100 km). With the EV mode strategy,
switching from CD to CD operations, the estimated fuel
economy is 52 MPG (4.5 l/100 km). The strategy that avoids
CS operations also obtains better performance, but the overall
benefit remains marginal (within the 5.5%).
Figure 7 compares the torques of electric motor, BSA and
ICE, according to the selected values for λ0. For λ0 = 10, the
vehicle is primarily driven by the EM with some contribution
from the BSA for regenerative braking and power boost.
When λ0 is equal to -10, the controller selects the ICE
over the electric drive and the BSA and EM are only used
for regenerative braking and to satisfy short-term power
demands. Finally, λ0 = 6 reproduces the behavior described
as Blended Mode Control in [19]. In fact, the SOE decreases
gradually during the driving path and reaches the SOE lower
bound only the end of the cycle, this allows for avoiding CS
operations.
Results show that the proposed on-line implementable
control strategy achieves good results in terms of fuel econ-
omy and CO2 emissions with a minimum calibration effort.
The proposed algorithm leads to vehicle performance within
5% of the optimal solution.
The presented energy management strategy is also able
to reproduce a series of vehicle behaviors, such as Charge
Sustaining (CS) at SOEmax, CS at SOEmin and Charge
Depleting operations, without the need of an additional
higher level rule-based controller responsible to determine
the vehicle operation.
5029

7. VI. CONCLUSION
This paper presents the definition of a supervisory con-
troller for Plug-in Hybrid Electric Vehicles that does not
require any a priori information and can be implemented
on-line. Starting from a general optimal control problem
formulation, a new cost functional is defined to account for
the electrical energy supplied from the grid, hence explicitly
considering the vehicle to grid interactions in the energy
optimization problem.
The Pontryagin’s minimum principle is then applied to re-
duce a global optimization problem to a local minimization.
This allows for the control problem to be solved for charge-
depleting operations and to be implemented on-line.
Following this approach, the controller calibration was
reduced to two parameters, namely the initial condition for
the co-state λ0 and the scalar Lagrange multiplier μ. The
calibration was done considering a real-world driving cycle.
Result shows that, the calibration of the parameter μ
determines the ability of the control strategy to avoid battery
operations at the SOE boundaries but does not impact the ve-
hicle performance. On the other hand, the initial condition of
the Lagrange multiplier λ0 determines the vehicle operations
hence directly impacts the vehicle performance.
A careful calibration of the parameter λ0 leads to the
solution of the optimal control problem and presents the
best fuel economy and lower CO2 emission. However, an
approximated value of λ0 leads to performance within 5%
of the optimal behavior noticed in the sweet spot.
REFERENCES
[1] Q. Gong, Y. Li, and Z. Peng, “Trip-based Optimal Power Management
of Plug-in Hybrid Electric Vehicles,” IEEE Transaction on Vehicular
Technology, vol. 57, 2008.
[2] D. Karbowski, A. Rousseau, S. Pagerit, and P. Sharer, “Plug-in Vehicle
Control Strategy: from global optimization to real-time application,”
22nd International Battery, Hybrid and Fuel Cell Electric Vehicle
Symposium, (EVS-22), 2006.
[3] S. Moura, H. Fathy, D.S. Callaway, and J. Stein, “Impact of Battery
Sizing on Stochastic Optimal Power Management in Plug-in Hybrid
Electric Vehicles,” IEEE International Conference on Vehicular Elec-
tronics and Safety, Columbus, OH, 2008.
[4] A. Rousseau, S. Pagerit, and D. Gao, “Plug-in Hybrid Electric Vehicle
Control Strategy Parameter Optimization,” 23nd International Battery,
Hybrid and Fuel Cell Electric Vehicle Symposium and Exhibition,
2007.
[5] P. Sharer, A. Rousseau, D. Karbowski, and S. Pagerit, “Plug-in
Hybrid Electric Vehicle Control Strategy: Comparison between EV
and Charge-Depleting Options,” SAE World Congress, SAE, 2008.
[6] G. Paganelli, S. Delprat, T. Guerra, J. Rimaux, and J. Santin,
“Equivalent consumption minimization strategy for parallel hybrid
powertrains,” Proceeding of Conference sponsored by Vehicular Trans-
portation Systems (VTS) and IEEE, 2002.
[7] A. Sciarretta, M. Back, and L. Guzzella, “Optimal Control of Par-
allel Hybrid Electric Vehicles,” IEEE Transaction on Control System
Technology, vol. 12, May 2004.
[8] A. Sciarretta and L. Guzzella, “Control of Hybrid Electric Vehicles
- a survey of optimal energy-management strategies,” IEEE Control
Systems Magazine, vol. 27, 2007.
[9] L. Guzzella and A. Sciarretta, Vehicle Propulsion System: Introduction
to Modeling and Optimization. Springer, second edition ed., 2007.
[10] P. Tulpule, V. Marano, and G. Rizzoni, “Effets of Different PHEV
Control Strategies on Vehicle Performance,” 2009 American Control
Conference, St. Louis, MO, 2009.
[11] P. Tulpule, S. Stockar, V. Marano, and G. Rizzoni, “Optimality
Assessment of Equivalent Consumption Minimization Strategy for
PHEV Applications,” in 2009 ASME Dynamic Systems and Control
Conference, Hollywood, CA, 2009.
[12] P. Tulpule, S. Stockar, V. Marano, and G. Rizzoni, “Comparative Study
of Different Control Strategies for Plug-in Hybrid Electric Vehicles,”
9th International Conference on Engines and Vehicles, Capri, NA,
Italy, SAE, September 2009.
[13] H. Geering, Optimal Control with Engineering Applications. Springer,
2007.
[14] K. Koprubasi, Modeling and Control of Hybrid-Electric Vehicle for
Drivability and Fuel Economy Improvements. PhD thesis, The Ohio
State University, 2008.
[15] The Ohio State University - Challenge-X. Team, “Final design and ve-
hicle technical specifications, Challenge-X 2006 fall technical report,”
tech. rep., The Ohio State University, 2006.
[16] M. Arnett, K. Bayar, C. Coburn, Y. Guezennec, K. Koprubasi,
S. Midlam-Mohler, K. Sevel, M. Shakiba-Herfeh, and G. Rizzoni,
“Cleaner Diesel Using Model-based Design and Advanced Aftertreat-
ment in a Student Competition Vehicle,” SAE World Congress, Detroit,
MI, 2008.
[17] Y. Hu, S. Yurkovich, Y. Guezennec, and R. Bornatico, “Model-
based Calibration for Battery Characterization in HEV Applications,”
American Control Conference, 2008.
[18] M. Canova, K. Sevel, Y. Guezennec, and S. Yurkovich, “Control of
the Start/Stop of a Diesel Engine in a Parallel HEV: Modeling and
Experiments,” ASME International Mechanical Engineering Congress
and Exposition, Chicago, IL, 2006.
[19] S.Stockar, P. Tulpule, V. Marano, and G. Rizzoni, “Energy, Economical
and Environmental Analysis of Plug-in Hybrid Electric Vehicles based
on Common Driving Cycles,” 9th International Conference on Engines
and Vehicles, Capri, NA, Italy, SAE, September 2009.
[20] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F.
Mishchenko, The Mathematical Theory of Optimal Processes. Inter-
science Publishers, 1962.
[21] L. Serrao, A Comparative Analysis of Energy Management Strategies
for Hybrid Electric Vehicles. PhD thesis, The Ohio State University,
2009.
[22] Argonne. National. Laboratory, “Greet 1.8b,” 2008.
[23] K. R. Genung, C. Lawrence, T. Markel, C. S. Hinds, R. McGill,
R. E. Ziegler, D. E. Smith. (Sentech, Inc.), D. Brooks. (EPRI),
S. W. Hadley, R. L. Smith, D.L. Greene. (ORNL), H. Wiegman.
(GE Global Research), and V. Marano. (OSU-CAR), “Plug-in Hybrid
Electric Vehicle, Value Proposition Study, phase 1, task 3: Technical
requirements and procedure for evaluation of one scenario,” tech.
rep., ORNL under contract No. DE-AC05-00OR22725,, January 2009.
Final Report.
[24] D. P. Bertsekas, Dynamic Programming and Optimal Control. Athena
Scientific, 1995.
5030