2014 Electrical Power and Energy Conference
An Optimal Control Solved by Pontryagin’s
Minimum Principle Approach for a Fuel
Cell/Supercapacitor Vehicle
Hanane HEMI
Jamel GHOUILI
Ahmed CHERITI
Université de Moncton
Moncton, NB, Canada
Université du Québec à Trois-Rivières
Trois-Rivières, QC, Canada
Email: hanane.hemi@umoncton.ca
Université de Moncton
Moncton, NB, Canada
Email: jamel.ghouili@umoncton.ca
Université du Québec à Trois-Rivières
Trois-Rivières, QC, Canada
Email: ahmed.cheriti@uqtr.ca
Abstract—A new real time optimal control based on Pontryagin’s minimum principle approach is proposed in this article.
The optimal control problem is formulated as an equivalent
consumption minimization strategy (ECMS), which must be
solved using the Pontryagin minimum principle (PMP). The
proposed approach manages the power required and sources,
depending on the unknown driving cycle. It is implemented
by using the Matlab/Simulink software and its development
tools in real time without any study in the off time or drive
cycle and driving conditions. This approach is simplified on two
major equations, the first calculates the costate variable in real
time and the second deduces the optimal fuel cell power. Also,
this approach has to satisfy the power requirement, reduce the
hydrogen consumption, and maintain the supercapacitor state
of charge (SOC) bounded for the unknown driving cycles. The
simulation results obtained show that these objectives are satisfied
using this approach, even though these results are suboptimal in
the global drive cycle due at the unknown drive final time.
allowing for the control problem to be solved and implemented
in an algorithm.
Reference [9] details two optimal control formulations
solved via dynamic programming. These formulations were
used for short trips. The inconvenient of this strategy is that the
dynamic programming is not used online; thus, the problem
solution search must be offline.
In reference [10] an optimal control problem based on
Pontryagin’s Minimum Principle is developed. The advantage
of this strategy is that it is implemented in real time without
knowing future driving conditions. It only requires knowledge
of the cruise time and the available negative energy during
braking.
Reference [11] presents a comparison of the constant and
variable Costate of an optimal control scheme based on the
Minimum Principle.
I. I NTRODUCTION
The need to minimize noxious CO2 and greenhouse gas
emissions has led to an increase in the use of hybrid vehicles
in recent years. These vehicles include thermal hybrid vehicles,
electrical vehicles equipped with a battery, and fuel cell
vehicles.
Recently several energy management strategies have been
suggested to control the distribution of power between the
two sources and the load. Reference [1] presents the energy
management system based on an equivalent consumption
minimization strategy (ECMS). Reference [2] develops a real
time optimal energy management strategy based on the determined dynamic programming (DDP) strategy. References
[3]-[6] propose a fuzzy logic control system.
Reference [7] proposes an optimal control based on Pontryagin’s minimum principle approach. The simulation results
are the same as the global optimal control based on dynamic
programming results.
Reference [8] formulates a cost-functional optimal control
problem based on the cumulative CO2 produced by the
vehicle. Pontryagins minimum principle was then applied to
reduce a global optimization problem to a local minimization,
978-1-4799-6038-5/14 $31.00 © 2014 IEEE
DOI 10.1109/EPEC.2014.15
Reference [12] sets an optimal control problem solved
using Pontryagin’s minimum principle approach. The optimal
Costate function has to be approximated via optimal simulations of the normalized city cycles. This parameter is tested
in forward simulations.
Several studies in the literature indicate that the resolution
of an optimal control problem depends on a few parameters
that must be tuned according to future driving conditions;
knowing that the future conditions are important yields a
global optimal solution. In this paper, a real time optimal
control problem based on Pontryagin’s minimum principle
power management strategy is implemented in a hybrid vehicle
with multiple power sources. The fuel cell is the primary
power source, and the secondary power sources are the battery
and supercapacitor.
This paper is structured in four sections as follows. Section
II describes the proposed power management strategy. Section
III presents a simulation results and section IV presents the
conclusions.
87
II. F ORMULATION OF O PTIMAL C ONTROL PROBLEM
The Hamiltonian of the problem can be defined as equation
(10) [9], [7], [10]:
The optimal control problem is formulated as equivalent
fuel consumption. Also, this approach has to protect the
supercapacitor from overcharging during the repetitive braking
energy accumulation. The equivalent fuel consumption is
composed by two equation. The first converts supercapacitor
power consumption to an equivalent amount of fuel, and
the second presents the hydrogen consumption by fuel cell.
The objective function J is given in equation (1) [1], [11].
Minimizing the objective function lead up to find the optimal
power split between the fuel cell and supercapacitors. The
control action u(t) = Pf c is considered as fuel cell power. The
supercapacitor power is given in equation (2). The equation
(3) is a weighting factor to achieve supercapacitor state of
charge (SOC) regulation. The equation (4) gives the relation
between the supercapacitor state of charge and the state of
energy (SOE).
tf
(ṁf (Pf c (t)) + s(t)P sc(t)) dt
(1)
J=
˙
H = ṁf (Pf c (t)) + s(t)P sc(t) + λ(t)SOE(t)
where λ(t) is the Costate function. The state and the adjoint
equations are defined as equations (11) and (12) [7], [10].
∂H(t, SOE(t), Pf c (t), λ(t))
˙
SOE(t)
=
∂λ
(2)
SOE(t) = SOC 2 (t)
(4)
∂H(t, SOE(t), Pf c (t), λ(t))
(12)
∂SOE
According to Pontryagin’s minimum principle, the optimal
control variable, Pf∗c , is obtained using equation (13).
∂H(t, SOE(t), Pf c (t), λ(t))
=0
∂Pf c
Vsc(t)
Vsc.max
Isc (t) = C V̇sc (t)
(3)
III. S IMULATION RESULTS AND DISCUSSION
The MATLAB/Simulink suite and SimPowerSystems and
SimDriveline library are used to model the electrical and
mechanical elements of hybrid vehicles. The fuel cell is a
400 cell, 288 Vdc , 100 kW Proton Exchange Membrane Fuel
Cell(PEMFC). The fuel cell parameters are shown in Tab.II.
The supercapacitor is a 288 V , 27.78 F , and the supercapacitor
used is a BMOD0500 P016. The electric motor is a 288 Vdc ,
100 kW permanent magnet synchronous machine (PMSM).
The mechanical model of the vehicle is modeled by using the
SimDriveline library [15]. The vehicle parameters are given
in table Tab.I. They are obtained from Honda Clarity vehicles
[16].
(5)
(6)
where ṁf is instantaneous fuel cell consumption (kg/s), Pr
and Pf c and Puc are required, fuel cell and supercapacitor
power (W ), LHVH2 is low heat value of hydrogen (M j/kg).
The dynamic system is presented in equation (7).
˙
SOE(t)
=
2Psc
2
CVsc.max
TABLE I: Vehicle parameters
(7)
2
where C is the supercapacitor capacity (F ) and Vsc.max
is the
supercapacitor maximum voltage (V ).
Equation (7) is determined using equation (5) and (6)
˙
SOE(t)
˙ 2 (t)
=
SOC
˙
= 2SOC(t)SOC(t)
Vsc
V̇sc
= 2 Vsc.max Vsc.max
Vuc
Isc
= 2 Vsc.max
C.Vsc.max
2Psc
=
CV 2
Dimensions
weight
Occupancy
Overall length (inches)
Overall width (inches)
Overall height (inches)
Tread (front/rear,inches)
Wheelbase (inches)
Vehicle weight (kg)
Number of occupants
Maximum speed (mph)
190,3
72,7
57,8
62,2/62,8
110,2
1625
4
100
(8)
TABLE II: Fuel cell parameters
Open circuit voltage
Nominal stack efficiency
Operating temperature
Nominal Air flow rate
Nominal fuel supply pressure
Nominal air supply pressure
Nominal composition H2 (air)
Nominal composition O2 /H2 O(air)
sc.max
Equation (9) presents the system local constraints.
⎧
⎪
⎨Pf c.min ≤ Pf c ≤ Pf c.max
Psc.min ≤ Psc ≤ Psc.max
⎪
⎩
SOCsc.min ≤ SOCsc ≤ SOCsc.max
(13)
The resolution of the necessary conditions of equations (11),
(12) and (13) leads to the optimal solution of Pontryagin’s
minimum principle. The boundary condition must be satisfied
(equation (14)) [18]. In this paper, the future conditions of
the driving cycle are unknown; in fact, the final time, tf , is
unknown.
SOE(ti ) = SOE(tf )
(14)
The supercapacitor state of charge is defined by equation (5),
and the supercapacitor current Isc is obtained by equation (6).
SOC(t) =
(11)
λ̇(t) = −
t0
P r(t) = Pf c (t) + Psc (t)
SOE(t) − 1
s(t) =
LHVH2 SOCmin SOCmax
(10)
(9)
88
V
%
0C
lpm
bar
bar
%
%
400
57
95
1698
3
3
99.95
21/1
Fig. 1: Fuel cell/Supercapacitor vehicle configuration
Fig. 1 shows Fuel cell/Supercapacitor vehicle configuration.
The power management strategy is detailed with green color.
The power required during a drive is calculated depending
the pedal position. The fuel cell power is determined in the
optimal control block.
The required fuel cell current is given by a lookup table
and compared the fuel cell measured current to control the
chopper by a PI controller. The reference current is also used
by the fuel cell system to feed the stack with Hydrogen and
Oxygen.
In the case of supercapacitor, a PI controller generates PWM
signals from the differences between the reference DC bus
voltage, which maintained at 288 V and the DC bus measured
voltage to control the bidirectional buck/boost chopper.
A braking chopper is added to protect the supercapacitor
from a supplement of energy during a regenerative braking.
In this simulation, the optimal control problem is solved
using Pontryagin’s minimum principle for the UDDS drive
cycle. This drive cycle was selected to analyze the performance
of the proposed power management strategies. Because this
cycle contains more accelerations and decelerations than other
drive cycles, the hybrid vehicle loses more energy and the
efficiency of the system decreases.
The simulation includes two parts to analyse the proposed
approach for different driving conditions. First, the drive cycle
is used, and the vehicle mass is 1625 kg. Second, the drive
cycle is multiplied by two Vref = 2×VU DDS , and the vehicle
mass is 2500 kg. The measured car speed in these two parts
is presented in Fig. 2 (a) and (b).
The optimal control block shown in figure 1 calculates
optimal fuel cell power during a drive cycle, and the supercapacitor power is deduced by the difference between the
required power and fuel cell power. Fig. 3 (a) and (b) illustrate
the power obtained from the two sources. The required power
(motor power) at the DC bus is allocated between the fuel cell
power and supercapacitor power. In the two speeds references,
the optimal fuel cell and supercapacitor power respect the
constrains imposed in the optimal control problem.
89
66
60
car speed
car speed reference
64
50
62
state of charge (%)
speed (km/h)
40
30
20
60
58
56
54
10
52
0
0
200
400
600
800
1000
1200
50
0
1400
time (s)
200
400
600
800
1000
1200
1400
800
1000
1200
1400
time (s)
(a)
(a)
120
100
car speed
car speed reference
100
90
state of charge (%)
speed (km/h)
80
60
40
20
0
0
200
400
600
800
1000
1200
80
70
60
50
1400
time (s)
(b)
40
0
200
400
600
time (s)
Fig. 2: Car speed with (a) mv = 1625 kg and Vref = VU DDS
(b) mv = 2500 kg and Vref = 2 × VU DDS
(b)
Fig. 4: Supercapacitor state of charge with (a) mv = 1625 kg
and Vref = VU DDS (b) mv = 2500 kg and Vref = 2×VU DDS
4
1.5
x 10
Motor power
Fuel cell power
Supercapacitor power
1
0.2
0
Power (w)
0.5
−0.2
−0.4
0
()
−0.6
−0.5
−0.8
−1
−1
0
200
400
600
800
1000
1200
−1.2
1400
time (s)
−1.4
(a)
−1.6
0
200
400
600
800
1000
1200
1400
800
1000
1200
1400
time (s)
4
8
x 10
Motor power
Fuel cell power
Supercapacitor power
6
(a)
1
4
−1
0
()
Power (w)
0
2
−2
−3
−4
−6
0
−2
−4
200
400
600
800
1000
1200
1400
time (s)
−5
0
(b)
200
400
600
time (s)
(b)
Fig. 3: Motor, fuel cell, and supercapacitor power with (a)
mv = 1625 kg and Vref = VU DDS (b) mv = 2500 kg and
Vref = 2 × VU DDS
Fig. 5: Costate variable with with (a) mv = 1625 kg and
Vref = VU DDS (b) mv = 2500 kg and Vref = 2 × VU DDS
90
To calculate this optimal fuel cell power, the knowledge of
the supercapacitor state of energy is required during a drive
cycle. This state of energy represents the optimal control block
input. This state of energy SOE(t) = SOC 2 (t) must be
maintained bounded to improve the supercapacitor life cycle
and protect it from surcharge.
For the two different driving conditions of the simulation,
the optimal SOE(t) = SOC 2 (t) trajectories are shown in
Fig. 4 (a) and (b). Those figures show that the the difference
between the supercapacitor initial time and final time is less
then 10%, knowing that this study is based on unknown drive
cycle and any informations about the drive final time.
The optimal control Costate function is presented in Fig. 5
(a) and (b) and calculated from two equations (11) and (12).
The supercapacitor state of energy for the different driving
conditions is important to calculate the costate variable. As
450
400
voltage (V)
300
250
200
150
100
0
200
400
800
1000
1200
1400
(a)
450
400
voltage (V)
350
Motor current
Fuel cell current
Supercapacitor current
40
Motor voltage
Fuel cell voltage
Supercapacitor voltage
300
250
200
20
150
0
100
0
200
400
600
800
1000
1200
1400
time (s)
−20
(b)
Fig. 7: Motor, fuel cell, and supercapacitor voltage with (a)
mv = 1625 kg and Vref = VU DDS (b) mv = 2500 kg and
Vref = 2 × VU DDS
−40
−60
0
600
time (s)
60
current (A)
Motor voltage
Fuel cell voltage
Supercapacitor voltage
350
200
400
600
800
1000
1200
1400
time (s)
(a)
Fig. 6 (a) and (b) present the current of these sources and the
required current (motor current) at the DC bus. Fig. 7 (a) and
(b) illustrate the fuel cell and supercapacitor voltages. The bus
voltage is maintained at 288 Vdc .
Tab.III gives the fuel cell consumption depending on the
vehicle conditions. The mass and speed are variable.
300
Motor current
Fuel cell current
Supercapacitor current
250
200
current (A)
150
100
50
0
−50
TABLE III: Hydrogen consumption
−100
−150
−200
0
Conditions
200
400
600
800
1000
1200
1400
Distance (km)
Duration (s)
Consumption (l)
time (s)
(b)
Fig. 6: Motor, fuel cell, and supercapacitor current with (a)
mv = 1625 kg and Vref = VU DDS (b) mv = 2500 kg and
Vref = 2 × VU DDS
VU DDS
mv = 1625 kg
7,29
1400
76,49
2 × VU DDS
mv = 2500 kg
14,568
1400
493,32
IV. C ONCLUSION
This paper presents an optimal control using Pontryagin’s
minimum principle approach for fuel cell/supercapacitor electrical vehicle.
The simulation results show that the power requirement
for the unknown driving cycles, and the power distribution
among various power source are satisfying. Also the results
indicate that the hydrogen consumption is lower during a
drive cycle, and the supercapacitor state of charge is bounded
at the interval desired. In addition, this approach is easy to
implement and hasn’t required any system expertise compared
with other strategies like Fuzzy logic method. This approach is
easy to be implemented in real time because it is based on the
shown in figures, this variable is not constant, it is calculated
in real time and not deduced from any off time study. This
method gives an independence of the system of knowledge
of future driving conditions to calculate this costate variable.
Equations (11 and 13) calculate the optimal fuel cell power
and a lookup table is used to deduce a fuel cell current.
The supercapacitor current is the deference between required
current and fuel cell, adding to that, the regenerative braking
current is absorbed by supercapacitor, so the braking chopper
absorbs the extra current that is not authorized by contractor.
91
instantaneous minimization of the Hamiltonian. Furthermore,
the future conditions of the driving cycle are unknown, the
final time, tf , is unknown. The Costate variable is calculated
in real time using equation (12) and it is not constant. This
approach is suboptimal due to the aforementioned reasons. By
adding an algorithm that predicts the future driving conditions,
this approach can yield the optimal solution.
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