Implementation of an Optimal Control
Energy Management Strategy in a
Hybrid Truck ?
Dominique van Mullem ∗ , Thijs van Keulen ∗ , John Kessels ∗∗ ,
Bram de Jager ∗ , Maarten Steinbuch ∗ .
∗
Dept. of Mechanical Engineering, Control Systems Technology Group,
Eindhoven University of Technology, Eindhoven, The Netherlands
(e-mail: [email protected])
∗∗
TNO Science & Industry, Business Unit Automotive, Helmond,
The Netherlands (e-mail: [email protected])
Abstract: Energy Management Strategies for hybrid powertrains control the power split,
between the engine and electric motor, of a hybrid vehicle, with fuel consumption or emission
minimization as objective. Optimal control theory can be applied to rewrite the optimization
problem to an optimization independent of time. Estimation of the Lagrange parameter, e.g.,
by feedback on the battery State-Of-Charge (SOC), can be used to arrive at a real-time implementable strategy. Nevertheless, it is still required to solve a nonconvex optimization problem
with limited onboard computational power. This paper suggests to solve this optimization
problem, offline, for different values of the Lagrange parameter, crankshaft rotational speed, and
torque request. The resulting strategy is evaluated with simulations of a hybrid distribution truck
on two different velocity trajectories. The influence of several control parameters is investigated
also.
Keywords: Energy Management, Drive Train, Hybrid Configuration
1. INTRODUCTION
Hybrid Electric Vehicles (HEV’s) have, at least, two power
converters: usually an Internal Combustion Engine (ICE)
as prime mover, and an Electric Machine (EM) as secondary power converter. The EM enables energy recovery
during braking or driving downhill, this energy can be
used at a latter, more convenient, time to propel the
vehicle. The supervisory control algorithm, dealing with
the balanced generation and re-use of the stored energy, is
called Energy Management Strategy (EMS).
Several contributions have been made regarding the EMS
design for HEVs, see, e.g., Pisu and Rizzoni (2007), Sciarretta and Guzzella (2007) for an overview. EMS methods
can be divided in two classes. Firstly, noncausal methods
Lin et al. (2003) that require exact knowledge of the
power and velocity trajectories and secondly, causal or
real-time implementable methods Musardo et al. (2005)
that try to minimize fuel consumption without knowledge
of the upcoming trajectories. In general, the noncausal
strategies are used to benchmark and to design real-time
implementable strategies.
? The research leading to these results has received funding from the
ENIAC Joint Undertaking and from Senter-Novem in the Netherlands under Grant Agreement number 120009, and is part of a more
extensive project in the development of advanced energy management control for urban distribution trucks which has been made
possible by TNO Business Unit Automotive in cooperation with DAF
Trucks NV.
A solution can be derived using optimal control theory:
a Lagrange multiplier adjoins the system dynamics to
the fuel cost function. Applying the Pontryagin Maximum
Principle (PMP) simplifies the fuel minimization problem
to an optimization independent of time. Furthermore, from
the necessary conditions of optimality it follows that the
optimum is described with a multiplier which is constant.
This is exploited by real-time strategies that estimate the
Lagrange parameter based upon real-time available vehicle information, e.g., the battery State-Of-Charge (SOC)
Koot et al. (2005). Nevertheless, assuming that the Lagrange parameter could be predicted or estimated in a
suitable way, the resulting optimization is nonconvex. Furthermore, for a real-time EMS, additional demands on the
EMS are imposed by limited onboard computational power
and battery SOC estimation accuracy.
This paper adapts the real-time EMS presented in van
Keulen et al. (2008) and Koot et al. (2005), and presents
a fast method to effectively solve and implement the remaining nonconvex optimization problem in an onboard
vehicle Electronic Computation Unit (ECU). The optimization problem is solved offline for different values of
the Lagrange parameter, crankshaft rotational speed, and
torque request. The resulting optimal torque split is stored
in look-up tables that can be implemented in the vehicle.
Moreover, tuning rules for the SOC feedback gain and
initial guesses of the Lagrange parameter are presented.
This paper is organized as follows: in Section 2 a vehicle
model is presented, Section 3 gives an overview of EMS
methods, Section 4 describes the implementation of a realtime EMS, simulation results are presented in Section 5.
Finally, conclusions and recommendations are summarized
in Section 6.
ICE efficiency
800
700
2. VEHICLE MODEL
Tice [Nm]
600
The vehicle considered, for implementation and testing, is
a medium duty, hybrid electric truck. The primary power
source is an ICE which has a peak power of 185 kW,
the EM has a peak power of 44 kW. The battery has a
rated capacity of 2 kWh. The EM is placed between the
gearbox and ICE, in a parallel configuration. Since not all
the engine auxiliaries (e.g., steering pump, brake assist) are
electrified, engine stop-start is not implemented in this test
vehicle. An overview of the drivetrain topology is given in
Fig. 1. It is assumed that the force at the wheel Fw , and
the vehicle forward velocity v follow from the desired route
profile. Besides, it is assumed that the gearshift strategy
is prescribed such that the torque request Treq and crank
shaft rotational velocity ω are input to the torque split
problem described in this paper.
500
high η
400
decreasing η
300
200
100
0
800
1000
1200
1400
1600
ω [RPM]
1800
2000
Fig. 2. Engine static efficiency.
Electric Machine Efficiency
decreasing η
Pf
Tice
Engine
Clutch
+
high efficiency
Fw , v
Treq , ω
Gearbox
SOC
Battery
Ps
Pb
Electric
M achine
Tem [Nm]
Ef
Tice,max
900
Tem
Fig. 1. Drive train topology.
2.1 Engine
The fuel consumption of the ICE is modeled by a nonlinear static map, relating the engine torque Tice and
rotational speed ω towards fuel rate ṁf . The engine map is
measured on a chassis dynamometer. The resulting engine
efficiency map is given in Fig. 2. The model does not take
thermal and transient effects into account. The fuel power
Pf is defined with:
Pf = ṁf (Tice , ω) HLV
(1)
here, HLV is the lower heating value of the fuel, for diesel
equal to 42.5 MJ/kg.
2.2 Electric machine
The EM is a liquid cooled permanent magnet type motor,
with a peak power of 44 kW. The dynamic efficiency, ηem
of the EM and inverter as function of rotational speed ω
and torque Tem , is visualized in Fig. 3. When providing
traction towards the wheels, Tem is defined positive and
vice versa. The required electric power Pb is given by:
Pb = f (Tem , ω)
(2)
2.3 Battery
The SOC of the battery depends on the battery current Is
according to:
˙
SOC(t)
= Is (t)
(3)
0
500
1000
ω [RPM]
1500
2000
2500
Fig. 3. Efficiency contour lines of the electric machine.
For the losses in the battery, an internal resistance model
is used, see Pop et al. (2008, p. 103):
2
Ps
R
(4)
Ps = P b +
V
in which, R is the battery SOC internal resistance, V is
the battery nominal voltage, and Ps is defined as:
Ps = Is V
(5)
where SOC dependency of V and R, transient effects and
temperature influences are neglected.
3. ENERGY MANAGEMENT STRATEGIES
3.1 Problem definition
The goal of the EMS is to find the control input u(t) which
minimizes the cost function J(t, u), defined as:
Z tf
J=
ṁf (t, u) HLV dt
(6)
0
where the powersplit u for the vehicle is defined by:
Tice = u Treq
Tem = (1 − u) Treq
(7)
in which, u ≥ 0, and Treq is the torque request from the
driver. Note that for u = 0, the vehicle is fully driven by
the EM, for u = 1 the traction is provided solely by the
ICE and for u > 1 the battery is recharging. Furthermore,
ω is prescribed. In order to have a charge sustaining EMS,
it is required that the initial SOC is equal to the end SOC
level. This is often referred to as the endpoint constraint.
Hence the boundary conditions are:
SOC(0) = SOC0 , SOC(tf ) = SOC0
(8)
The system dynamics (3) can be adjoined to the integrant
of the cost function (6), using the multiplier function λ(t),
leading to the Hamiltonian:
H = Pf (t, u) + λ(t)Ps (t, u)
(9)
To minimize H subject to the boundary conditions (8), the
PMP can be applied, see Guzzella and Sciarretta (2005),
which states that if the control is optimal, then there
exists a nontrivial λ(t), such that the following necessary
conditions are satisfied:
• the differential equation on the Lagrange multiplier
holds:
−∂H
λ̇(t) =
= 0 ⇒ λ = const
(10)
∂SOC
• the Hamiltonian has a global minimum with respect
to u:
u∗ = arg min H(λ∗ , u)
(11)
u∈U
From the necessary conditions it can be concluded that, i)
the optimal Lagrange multiplier λ is constant, ii) given the
optimal Lagrange multiplier λ∗ , the optimization problem
reduces to an optimization only depending on vehicular
parameters at the current time. Furthermore, λ has a
physical meaning, it represents the relative incremental
cost of the primary and secondary power converter. For a
known velocity and power trajectory, numerical methods
can be used, see, e.g., Delprat et al. (2004), to find an
optimal value for λ∗ , and an optimal powersplit trajectory
u∗ (t).
3.2 Real-time implementation
Calculation of u∗ and λ∗ , requires that the exact future
driving cycle is known. In practice, this a-priori knowledge
is not available. Besides, when driving conditions differ
slightly from the driving cycle used for optimization, overor undercharging of the battery is likely to occur. Nevertheless, the PMP necessary conditions of optimality, can
be used to arrive at a real-time implementable strategy.
These strategies are often referred to as Equivalent
Consumption Minimization Strategy (ECMS), see, e.g.,
Guzzella and Sciarretta (2005), Paganelli et al. (2000),
here the Lagrange multiplier is approximated. In Koot
et al. (2005) the approximation results from a feedback
on the SOC:
λ(t) = λ0 + K(SOCdes − SOC(t))
(12)
here, λ0 is an initial value, K is a feedback gain parameter.
SOCdes is the setpoint for the SOC. Furthermore, when
taking upcoming route information into account, this
strategy can be adapted to follow a time-dependent SOC
trajectory, see Kessels and van den Bosch (2008). Note
that even for large changes in vehicle parameters and drive
cycles, see van Keulen et al. (2008), deviations in λ∗ , are
smaller than 20 %. Therefore, the feedback gain K can
be kept small, to ensure that changes around SOCdes are
permitted.
Assuming that (12) provides an acceptable estimation of
λ, due to the kind of engine and electric machine nonlinearities, determining the cost function (9), the resulting
optimization still remains nonconvex. The next section
describes a numerical approach to solve (11), subject to
(3), (8) and (10), offline and to implement the solution via
look-up tables. The real-time EMS uses interpolation to
obtain a real-time powersplit.
4. IMPLEMENTATION
In the previous section it was shown that optimal control
can be applied to arrive at an optimization independent
of time. Besides, it was noticed that the optimal value of
λ depends on the actual vehicle and drive cycle characteristics. These important observations are used to design a
real-time EMS.
Besides, processor load plays an important role in realtime implementation, efforts have to be made in order to
reduce computational burden: the EMS is implemented on
a standard ECU, which runs at 100 Hz. Therefore, a realtime optimization of u∗ , taking into account the nonconvex
cost function (9), takes too much effort.
To obtain acceptable computation times, the introduction
of look-up tables is proposed. For Treq > 0, a calculation
based on (11) is used in Section 4.1, while for Treq < 0,
implementation is done on a heuristic base in Section 4.2.
To find an instantaneous optimal split ratio, (7), the
drivetrain model of Section 2 is used. The aim is to find
a solution for u∗ , while still fulfilling the driver torque
demand Treq which follows from the desired trajectory.
The upper and lower limits on Treq are given by:
(Tdrag (ω) + Tem,min (ω)) ≤ Treq ≤ Tice,max (ω)
(13)
the subscripts ‘min’ and ‘max’, indicate the lower and
upper limits of the EM and ICE, which are defined as
function of rotational speed, ω. Notice that the maximum
drivetrain torque is limited by Tice,max and, therefore, only
depends on the ICE torque.
4.1 Offline optimization for positive torque request
To obtain a real-time optimal split ratio, which is able to
run on the ECU, an offline numerical optimization is performed. The resulting optimal solution for all ‘operating
points’ is stored in look-up tables. The schematic overview
is presented in Fig. 4. For every point with Treq > 0 the
following calculation is performed:
• step 1: a vector of operating points is defined for Treq ,
ω and λ. Vectors are indicated in boldface:
Treq = 0 + mi ∆T, mi ∈ [0 . . . ni ]
with ∆T =
Tice,max
, (14)
ni
ω = ωmin + mj ∆ω, mj ∈ [0 . . . nj ]
ωmax − ωmin
with ∆ω =
, (15)
nj
4.2 Negative torque request
For Treq < 0, two different situations can occur. Firstly,
the area which is indicated gray in Fig 4, the drive train is
commanded ‘engine only’ mode. Secondly, when (Treq <
Tdrag ) the EM recuperates the remaining kinetic energy.
Since fuel consumption data for negative torque output is
not available, these cases are handled in a heuristic way.
It is assumed that engine braking is more efficient than
idling the engine. This situation changes when stop-start
is enabled.
 +
for Treq > 0
u (eq. 20)
u∗ (ω, Treq ) = 1
for Tdrag ≤ Treq ≤ 0

Tdrag /Treq
for Treq < Tdrag
(21)
Now, a complete solution for the split-ratio u under all
circumstances is implemented.
5. SIMULATION RESULTS
λ = λmin + mk ∆λ, mk ∈ [0 . . . nk ]
λmax − λmin
with ∆λ =
, (16)
nk
where ni , nj and nk represent the number of points
for which the optimal split ratio will be calculated.
Due to memory limitations of the ECU, the number
of points has to be limited.
• step 2: a feasible EM torque region for every (ni × nj )
point given by (14) and (15) is defined by:
Tem = Tem,min + ml ∆Tem , ml ∈ [0 . . . nl ]
Tem,max (ω) − Trgn
(17)
with ∆Tem =
nl
here, the maximum amount of regeneration torque
Treg is defined by:
Treg = max (Treq − Ticemax (ω)), Temmin (ω) (18)
The implemented EMS is evaluated under different driving conditions with the drivetrain model as presented in
Section 2, where the simulation model contains also the
vehicle longitudinal dynamics and a driver model. Two different routes are simulated, see SAE J2711 (2002). Firstly,
the Manhattan cycle, representing low speed operation,
see Fig. 5. Secondly, is the Urban Dynanometer Driving
Schedule (UDDS), which mimics higher speed operation,
see Fig. 6. Simulation is performed on a distance vs. speed
base to ensure that the total driven distance is constant.
Since for both the hybrid and conventional drivetrain
the ICE does not shutdown, stop times in the cycle are
neglected. For a range of input parameters λ0 and K, the
sensitivity of the fuel consumption is investigated.
50
Velocity [km/h]
Fig. 4. Schematic overview of implementation.
30
20
10
Immediately from (7), Tice follows:
0
(19)
Since this calculation is performed offline and so not
limited by the ECU memory, ∆Tem can be kept
sufficiently small, to ensure accurate calculation of
the optimal split ratio.
• step 3: the equivalent costs can be calculated for
the whole set (ni × nj × nk ) of operating points by
substituting (19) into (1), (2) and taking into account
battery losses by (4). The optimal split ratio for
positive torque requests can be calculated by:
arg min(Pf + λ(k) Ps )
(20)
u+
i,j,k =
Treq (i) ω(j)
The result of this calculation for different values of λ is
given in Fig. 7. The split-ratio u+ is given as function
of Treq and ω. A low speed and torque region is present,
where full electric drive is preferred. The recharging area
‘grows’ for a larger value of λ, and vice versa for the
assist area. Between grid points, u is determined by linear
interpolation.
0
0.5
1
1.5
2
Distance [km]
2.5
3
3.5
Fig. 5. Manhattan drive cycle.
UDDS cycle
100
Velocity [km/h]
Tice = Treq − Tem
40
80
60
40
20
0
0
1
2
3
4
5
Distance [km]
6
7
8
9
Fig. 6. UDDS drive cycle.
5.1 Tuning rules for feedback parameters
The optimal control theory states that, for the given
configuration and drive cycle, a unique constant λ can be
λ = 2.4
900
800
800
700
700
Torque request [Nm]
Torque request [Nm]
λ = 2.1
900
600
500
400
300
600
500
300
200
200
100
100
0
1000
1500
2000
Rotational velocity ω [rpm]
Charge
400
0
2500
1000
1500
2000
Rotational velocity ω [rpm]
900
900
800
800
700
700
600
500
400
300
600
400
300
200
100
100
1000
1500
2000
Rotational velocity ω [rpm]
2500
Assist
500
200
0
no EM use
λ=3
Torque request [Nm]
Torque request [Nm]
λ = 2.7
2500
Full electric
0
1000
1500
2000
Rotational velocity ω [rpm]
2500
Fig. 7. Iso contour plot of power split for various values of λ, ω and Treq .
For the Manhattan and UDDS cycle these values are
λopt = 2.49 and λopt = 2.41, respectively. The influence
of the feedback parameters λ0 and K on the fuel consumption, compared to the optimum, is investigated. The
cycles are simulated with a SOC(0) of 0.5. After the cycle
is completed, the resulting SOC deviations are taken into
account and fuel usage is corrected using λopt to weight the
SOC difference in fuel, as was done in Delprat et al. (2004)
and Smokers et al. (2000). The results in terms of fuel
consumption for the two cycles are given in Fig. 8 and 9.
When K is set to 0, the choice of λ0 has a large influence on
fuel consumption. For the Manhattan cycle this behavior
is the most profound. For larger numbers of K, the fuel
consumption rises slightly and deviates from the optimal
solution, however, the λ0 setting has less influence. It can
be concluded that tuning parameters for this controller
should be with λ0 ≈ 2.55 and K ≥ 1.5 to ensure charge
sustaining behavior.
Since for real-time implementation an open-loop controller, (K = 0), is not preferable, some feedback should be
implemented. Fig. 10 and 11 shows the SOC trajectories
for different feedback gain K settings. For larger values
of K, the desired SOC is followed more strictly, still energy recuperation is done in a heuristic way, therefore the
differences in (equivalent) fuel economy remain small.
Fuel consumption for Manhattan cycle
0.37
Fuel consumption [L/km]
found, for which the result is optimal. For both cycles,
using the split maps as in Fig. 7, and K = 0, λ0 is varied
iteratively by bisection on SOC(tf ), until the boundary
conditions (8) hold. A similar approach can be found
in Delprat et al. (2004).
0.36
0.35
0.34
0.33
3
4
3
2.5
2
1
λ0 [−]
2
0
K [−]
Fig. 8. Fuel consumption for Manhattan cycle for various
feedback parameters.
6. CONCLUSION
Implementation of an optimal control energy management
strategy, in a standard onboard vehicle Electronic Computation Unit (ECU), is not trivial due to the nonconvex
properties of the optimization problem and the limited
available computation power. This paper suggests to handle energy recovery, during braking, in a heuristic way,
and to solve the optimization for positive power requests
offline. The optimal solution is stored in a look-up table,
and via interpolation used by the real-time strategy. Sim-
ulation results indicate that the proposed strategy obtains
fuel consumption results close to optimal, for a wide range
of feedback parameters, while keeping the battery stateof-charge within the bounds.
Fuel consumption for UDDS cycle
Experimental results with this strategy, implemented in a
hybrid electric truck, are foreseen.
Fuel consumption [L/km]
0.275
0.27
REFERENCES
0.265
0.26
0.255
3
3
2.5
2
1
2
λ0 [−]
0
K [−]
Fig. 9. Fuel consumption for UDDS cycle for various
feedback parameters.
SOC trajectory for Manhattan cycle
0.7
λopt
λ0 = 2.55, K = 1.5
0.65
λ0 = 2.55, K = 3.0
SOC [−]
0.6
0.55
0.5
0.45
0
0.5
1
1.5
2
distance [km]
2.5
3
Fig. 10. SOC trajectory for optimal control and ECMSopt .
SOC trajectory for UDDS cycle
0.6
λopt
λ0 = 2.55, K = 1.5
0.55
λ0 = 2.55, K = 3.0
SOC [−]
0.5
0.45
0.4
0.35
0.3
0
1
2
3
4
5
distance [km]
6
7
8
9
Fig. 11. SOC trajectory for optimal control and ECMSopt .
S. Delprat, J. Lauber, T.M. Guerra and J. Rimaux. Control of a Parallel Hybrid Powertrain: Optimal Control.
IEEE Trans. on Vehicular Tech., Vol. 53, No. 3, pp. 872881, May 2004.
L. Guzzella, A. Sciarretta. Vehicle Propulsion Systems.
ISBN 3-540-25195-2, Springer, 2005.
J. Kessels, P. van den Bosch. Electronic Horizon: Road
Information used by Energy management Strategies.
Int. J. of Intelligent Information and Database Systems,
Vol. 2, No. 2, pp. 187-203, 2008.
T. van Keulen, B. de Jager, M. Steinbuch. An Adaptive
Sub-Optimal Energy Management Strategy for Hybrid
Drive-Trains. Proc. of the 17th IFAC World Congress,
Seoul, Korea, July 6-11, 2008.
M. Koot, J. Kessels, B. de Jager, W. Heemels, P. van den
Bosch, M. Steinbuch. Energy Management Strategies
for Vehicular Power Systems. IEEE Trans. on Vehicular
Tech., Vol. 54, No. 3, pp. 1-11, May 2005.
C.C Lin, H. Peng, J. Grizzle, J. Kang. Power Management
Strategy for a Parallel Hybrid Electric Truck. IEEE
Trans. on Control Systems Tech., Vol. 11, No. 6, pp. 839349, November 2003.
C. Musardo, G. Rizzoni, B. Staccia. A-ECMS: An Adaptive Algorithm for Hybrid Electric Vehicle Energy Management. Proc. of the 44th IEEE Conf. on Decision and
Control, and the European Control Conf. 2005, Seville,
Spain, December 12-15, 2005.
G. Paganelli, T. Guerra, S. Delprat, J. Santin, M. Delhom, E. Combes. Simulation and Assessment of Power
Control Strategies for a Parallel Hybrid Car. Proc. of
the Institution of Mechanical Engineers, Part D: J. of
Automobile Engineering, Vol. 214, No. 7, pp. 705-717,
2000.
P. Pisu, G. Rizzoni. A Comparative Study of Supervisory
Control Strategies for Hybrid Vehicle Energy Managment. IEEE Trans. on Control Systems Tech., Vol. 15,
No. 3, pp. 506-518, May 2007.
V. Pop, H.J. Bergveld, D. Danilov, P.H.L. Notten. Battery Management Systems. ISBN 978-1-4020-6944-4,
Springer, 2008.
SAE J-2711. Recommended Practice for Measuring Fuel
Economy and Emissions of Hybrid-Electric and Conventional Heavy-Duty Vehicles. Issued 2002-09.
A. Sciarretta, L. Guzzella. Control of Hybrid Electric
Vehicles. IEEE Control Systems Magazine, Vol. 27,
No. 2, pp. 60-70, April 2007.
R.T.M Smokers, S. Ploumen, M. Conte, L. Buning, K.
Meier-Engel. Test Methods for Evaluating Energy
Consumption and Emission of Vehicles with Electric,
Hybrid and Fuel Cell Power Trains. Proc. of the EVS 17,
Driving new visions, Montreal, Canada, 1518 October
2000.