# Adaptive Second Order Sliding Mode Control of a Fuel Cell Hybrid System for Electric Vehicle Applications

Authors: Jianxing Liu; Yue Zhao; Bo Geng; Bing Xiao
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

We present an adaptive-gain second order sliding mode (SOSM) control applied to a hybrid power system for electric vehicle applications. The main advantage of the adaptive SOSM is that it does not require the upper bound of the uncertainty. The proposed hybrid system consists of a polymer electrolyte membrane fuel cell (PEMFC) with a unidirectional DC/DC converter and a Li-ion battery stack with a bidirectional DC/DC converter, where the PEMFC is employed as the primary energy source and the battery is employed as the second energy source. One of the main limitations of the FC is its slow dynamics mainly due to the air-feed system and fuel-delivery system. Fuel starvation phenomenon will occur during fast load demand. Therefore, the second energy source is required to assist the main source to improve system perofrmance. The proposed energy management system contains two cascade control structures, which are used to regulate the fuel cell and battery currents to track the given reference currents and stabilize the DC bus voltage while satisfying the physical limitations. The proposed control strategy is evaluated for two real driving cycles, that is, Urban Dynamometer Driving Schedule (UDDS) and Highway Fuel Economy Driving Schedule (HWFET).


---

## Body

## 1. Introduction

Polymer electrolyte membrane fuel cells (PEMFCs) have emerged as the most prominent technology for energizing future’s automotive world. They are clean, quiet, and efficient and have been widely studied in automotive applications over the past two decades due to their relatively small size, light weight, and easy manufacturing [1–3]. Despite these advantages, there are still major issues concerning cost, liability, and durability to be addressed before they become a widely used alternative to internal combustion engines.One of the main drawbacks of the FCs is slow dynamics dominated by the air-supply system and fuel-delivery system. As a result, the FCs cannot properly respond to rapid load changes in vehicle power demand which will cause a high voltage drop in a short time recognized as a fuel-starvation phenomenon [4]. Therefore, an additional energy storage system, such as Lithium ion (Li-ion) batteries, is usually used to improve the slow response dynamics of the PEMFC. The PEMFC hybrid electric vehicle (HEV) with Li-ion battery storage possesses the advantages of the PEMFC and Li-ion battery stack (large energy and power capacities). The PEMFC provides the main power requirements during normal driving condition, while the Li-ion battery stack discharges or stores energy to assist the PEMFC to meet peak power demand. Previous research works have shown that hybridization of FC vehicles with batteries provides the benefits of cost and performance as well as fuel economy [5, 6].In order to achieve the optimal operation performance of the hybrid system, suitable model and controller development efforts are needed [7–13]. In the hybrid configuration, the DC/DC converters adapt the voltage of the fuel cell and the battery to the DC bus voltage. This allows an active control of the power sharing between each source. Therefore, a power management strategy is needed to determine this power sharing while respecting system constraints (e.g., battery state of charge (SOC) limits, power limits, etc.) [14–16]. In [17], a load sharing algorithm using fuzzy logic control was studied for a fuel cell/supercapacitor HEV. In [18], a fuel cell/battery/supercapacitor hybrid bus was studied by a fuzzy power management controller, with the effectiveness of the fuzzy controller demonstrated by some designed tests. Recently, energy management based on proportional-integral (PI) controllers have been proposed [6, 19]. Although PI controllers are easy to be tuned on-line which provide a practical solution to many applications, they can not be robust and reliable in the presence of disturbances or uncertainties.Sliding mode technique is known for its insensitivity to external disturbance due to operating conditions, high accuracy, and finite time convergence [20–24]. Sliding mode control (SMC) has found wide applications in various dynamical systems, such as stochastic systems [25], Markovian jumping systems [26, 27], and fuzzy systems [28, 29]. However, the main drawback of SMC is the so-calledchattering phenomenon [30]. Adaptive sliding mode has been proven to be an efficient solution which does not require the bound of uncertainties [31–33]. The control gains are adapted dynamically to counteract the uncertainties/perturbations, which ensure the sliding mode. To the best of the authors’ knowledge, it is the first attempt to apply the adaptive SOSM control to the application of HEV. In this paper, we have proposed a cascade control strategy for the FC hybrid system, using an adaptive-gain super twisting algorithm (ASTW) [31]. The gains of this algorithm are adapted dynamically which ensures the convergence of the tracking error in finite time, withouta priori knowledge of the upper bound of uncertainties and perturbations. A state based control algorithm is proposed to generate the power reference for the fuel cell stack. In addition, controllers are also designed for the fuel cell and battery converters to track the given current references and stabilize the DC bus voltage.The rest of the paper is divided as follows: the modelling of the FC hybrid system is given in Section2. Section 3 discusses the designs of the power management control strategy, which generates the reference power for the FC. Then, the control strategies for the FC and battery converters using the proposed algorithm are provided. In Section 4, simulation results are given for two real driving cycles, UDDS and HWFET. Finally, the major conclusions are presented in Section 5.


## 2. Model of the Hybrid Fuel Cell Power System

Figure1 shows the schema of the studied hybrid system for the vehicle applications. The hybrid system is composed of the PEMFC, the Li-ion battery, FC unidirectional DC-DC converter, battery bidirectional DC/DC converter, and tracking motor.Figure 1
Schematic of the fuel cell/battery hybrid power system.


### 2.1. Fuel Cell System Modelling

In this paper, a 30 kW fuel cell composed of 90 cells in series is modeled considering its dynamics [34]. The fuel cell behavior is highly nonlinear and is dependent on several variables such as current density, stack temperature, membrane humidity, and reactant partial pressures. Several assumptions are considered: (1) the stack temperature and humidity in the fuel cell cathode are well controlled; (2) the water inside the cathode is only in vapor phase and any extra water in liquid phase is removed from the channels; (3) the input reactant flows are humidified in a consistent and rapid way and the high pressure compressed hydrogen is available; (4) the anode pressure is well controlled to follow the cathode pressure; and (5) the current dynamics of the motor which directly drives the compressor is negligible due to its small time constant as compared to the mechanical dynamics [35].


#### 2.1.1. Fuel Cell Stack Voltage

A single fuel cell operating voltage can be modeled as(1)
V
f
c
=
E
-
v
a
c
t
-
v
o
h
m
-
v
c
o
n
c
,where E is the open circuit voltage and v
a
c
t, v
o
h
m, and v
c
o
n
c present the activation loss, ohmic loss, and concentration loss, respectively.The open circuit voltageE is expressed as(2)
E
=
1.229
-
0.85
·
1
0
-
3
T
f
c
-
298.15
+
4.3085
·
1
0
-
5
T
f
c
l
n
p
H
2
+
1
2
l
n
p
O
2
,where T
f
c is the fuel cell stack temperature and p
H
2, p
O
2 are the partial pressures of hydrogen and oxygen, respectively [34].The activation lossv
a
c
t, ohmic loss v
o
h
m, and concentration loss v
c
o
n
c are expressed as follows.(1)
v
a
c
t
=
v
0
+
v
a
1
-
e
-
b
1
i is due to the difference between the velocity of the reactions in the anode and cathode [36], where i
=
I
s
t
/
A
f
c is the current density, I
s
t (A) is the stack current, and A
f
c (cm2) is the active area. v
0 (volts) is the voltage drop at zero current density, and v
a (volts) and b
1 are constants that depend on the temperature and the oxygen partial pressure [37, 38]. The values of v
0, v
a, and b
1 can be determined from a nonlinear regression of experimental data.
(2)
v
o
h
m
=
i
R
o
h
m is due to the electrical resistance of the electrodes, and the resistance to the flow of ions through the electrolyte [36]. R
o
h
m (Ω
·
c
m
2) represents the fuel cell internal electrical resistance.
(3)
v
c
o
n
c
=
i
b
3
i
/
i
m
a
x
b
4 results from the drop in concentration of the reactants due to the consumption in the reaction. b
3, b
4, and i
m
a
x are constants that depend on the temperature and the reactant partial pressures. i
m
a
x is the current density that generates the abrupt voltage drop.


#### 2.1.2. Gas Dynamics ofp
H
2 and p
O
2

The dynamics ofp
H
2, p
O
2 are expressed as follows:(3)
d
p
H
2
d
t
=
R
H
2
T
f
c
V
a
n
W
H
2
,
i
n
-
W
H
2
,
o
u
t
-
W
H
2
,
r
e
a
c
t
,
d
p
O
2
d
t
=
R
O
2
T
f
c
V
c
a
W
O
2
,
i
n
-
W
O
2
,
o
u
t
-
W
O
2
,
r
e
a
c
t
,where W
H
2
,
i
n is the mass flow rate of hydrogen gas entering the anode, W
H
2
,
o
u
t is the mass flow rate of hydrogen gas leaving the anode, W
H
2
,
r
e
a
c
t is the rate of hydrogen reacted, W
O
2
,
i
n is the mass flow rate of oxygen entering the cathode, W
O
2
,
o
u
t is the mass flow rate of oxygen leaving the cathode, W
O
2
,
r
e
a
c
t is the rate of oxygen reacted, R
H
2 is the hydrogen gas constant, R
O
2 is the oxygen gas constant, and V
a
n and V
c
a are the anode and cathode volumes, respectively. More details are provided in [34].


#### 2.1.3. Thermal Dynamics

The dynamic temperature model is described by a lumped thermal model [39]:(4)
m
s
t
C
p
s
t
d
T
f
c
d
t
=
Q
˙
s
o
u
-
W
c
C
p
c
T
f
c
-
T
c
,
i
n
,
Q
˙
s
o
u
=
I
s
t
-
T
f
c
Δ
s
4
F
+
v
a
c
t
+
I
s
t
R
o
h
m
,where m
s
t is the heat mass of the stack, C
p
s
t and C
p
c are the specific heat, W
c is the coolant flow rate considered as a control variable, T
c
,
i
n is the coolant temperature at the stack inlet, and Q
˙
s
o
u is the internal energy source. The latter is calculated as a function of the stack current, temperature, electrical resistance of stack layers R
o
h
m, Faraday’s number F, and the entropy change Δ
s. The physical parameters were obtained through extensive experimentation.


#### 2.1.4. Auxiliary Components

The main power consumer of the fuel cell system is the air compressor, which can consume up to 30% of the fuel cell power under high load conditions [40]. Therefore, the net power from the fuel cell system P
f
c
,
n
e
t can be calculated as(5)
P
f
c
,
n
e
t
=
n
V
f
c
I
f
c
-
P
c
p
-
P
a
u
x
,where n is the number of cells and P
c
p and P
a
u
x are the power consumed by the compressor and auxiliary components (such as the cooling pump, radiator fan), respectively. In this study, P
a
u
x is assumed to be constant and P
c
p is calculated as(6)
P
c
p
=
C
p
T
a
t
m
η
c
m
η
c
p
p
s
m
p
a
t
m
γ
-
1
/
γ
-
1
W
c
p
,where p
s
m is the supply manifold pressure which is equal to the pressure at the compressor outlet, p
a
t
m is the atmospheric pressure, C
p is the specific heat capacity of air, γ is the ratio of the specific hears of air, T
a
t
m is the ambient temperature, W
c
p is the compressor flow rate, and η
c
m and η
c
p are the compressor motor efficiency and compressor efficiency, respectively. Figure 2 shows the efficiency curve of the fuel cell system; it can be found that the optimal operating point is near 8.8 kW.Figure 2
PEMFC efficiency curve.Remark 1.
As discussed in [41], the time constants of the gas dynamics and the stack temperature dynamics are estimated to be the order of magnitude of 1
0
-
1 and 1
0
2, respectively. Therefore, the mass storage effects in the channels are not considered.


### 2.2. Fuel Cell Converter Model

A classical boost-type unidirectional DC/DC converter is selected as FC power converter because it can be operated in the current control mode in a continuous condition mode [42]. It adapts the DC voltage supplied by the fc to the desired DC bus voltage (420 V). By acting on the switching signal u
f
c, it is possible to determine the load power distribution between FC and battery. The average model of the FC converter is expressed as follows:(7)
L
f
c
d
i
f
c
d
t
=
V
f
c
-
r
f
c
i
f
c
-
u
f
c
V
b
u
s
,where r
f
c is the total equivalent series resistance in the FC converter, u
f
c
=
1
-
D
f
c, and D
f
c is the duty ratio of the switch S
1.


### 2.3. Battery Model

The Li-ion battery with peak power 40 kW and capacity 20 Ah is modeled as a voltage source in series with a resistance [14, 43], which is given as(8)
V
b
a
t
t
=
V
o
c
-
r
b
a
t
t
i
b
a
t
t
,where the open circuit voltage (V
o
c) and the internal resistance (r
b
a
t
t) are both functions of the battery SOC. The SOC is defined as the ratio of charge stored in the battery to the maximum charge capacity Q
b
a
t
t:(9)
d
S
O
C
d
t
=
-
i
b
a
t
t
Q
b
a
t
t
,where i
b
a
t
t is the battery current.In order to expressi
b
a
t
t it is noted that the instantaneous power delivered by the battery to the load equals P
b
a
t
t
=
V
b
a
t
t
i
b
a
t
t. Then, the rate of change of S
O
C gives(10)
d
S
O
C
d
t
=
-
V
o
c
-
V
o
c
2
-
4
r
b
a
t
t
P
b
a
t
t
2
r
b
a
t
t
Q
b
a
t
t
.The solution (10) is feasible for negative power demands which means the battery is in charge mode and maximizes efficiency for positive power demands (discharge mode).


### 2.4. Battery Converter Model

A simple boost-type bidirectional DC/DC converter is used to connect the battery to the high voltage bus, enabling the charge and discharge of the battery. The cascade control structure of the bidirectional converter is realized through the outer control loop (the DC bus energy loop) and the inner control loop (the battery current loop). The dynamic of the current control loop are designed to be much faster than that of the DC bus energy regulation loop.The average model of the battery converter is expressed as follows:(11)
L
b
a
t
t
d
i
b
a
t
t
d
t
=
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
u
b
a
t
t
V
b
u
s
,where r
b
a
t
t is the total equivalent series resistance in the battery converter, u
b
a
t
t
=
1
-
D
b
a
t
t, and D
b
a
t
t is the duty ratio of the switch S
2.The DC link capacitive energyy
b
u
s is given versus FC power P
f
c, battery power P
b
a
t
t, and load power P
l
o
a
d by the following equation:(12)
y
˙
b
u
s
=
P
f
c
,
0
+
P
b
a
t
t
,
0
-
P
l
o
a
d
,where y
b
u
s
=
1
/
2
C
b
u
s
V
b
u
s
2, P
f
c
,
0
=
P
f
c
,
n
e
t
-
r
f
c
i
f
c
2, and P
b
a
t
t
,
0
=
P
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
2.


### 2.5. Vehicle Dynamics Modelling

The power train is modeled as a point-mass [14](13)
M
d
v
d
t
=
F
t
r
a
c
+
F
b
r
a
k
-
C
r
M
g
c
o
s
α
-
1
2
ρ
a
A
f
C
D
A
f
v
2
-
M
g
s
i
n
α
,where F
t
r
a
c is the vehicle traction force, F
b
r
a
k is the friction brake force, M is the vehicle mass, v is the vehicle velocity, and other parameters are defined in Table 1.Table 1
Parameters for the vehicle.
Parameters
Value
Vehicle massM
1500 kg
Rolling resistanceC
r
0.01
Drag coefficientC
D
0.3
Frontal areaA
f
2.4 m2
Air densityρ
a
1.2 kg/m3
Differential ratio
5.0
Differential efficiency
97%
Wheel radius
0.29 mThen, the power required to drive the tracking motor is calculated as(14)
P
l
o
a
d
=
F
t
r
a
c
v
η
m
η
d
i
f
f
-
s
i
g
n
F
t
r
a
c
,where η
d
i
f
f is the efficiency of the differential and η
m is the efficiency of the tracking motor which is a function of motor torque τ
m and its rotation speed ω
m, that is, η
m
=
η
(
τ
m
,
ω
m
).In the next section, the control strategy of FC and battery will be presented using super-twisting sliding mode.


## 2.1. Fuel Cell System Modelling

In this paper, a 30 kW fuel cell composed of 90 cells in series is modeled considering its dynamics [34]. The fuel cell behavior is highly nonlinear and is dependent on several variables such as current density, stack temperature, membrane humidity, and reactant partial pressures. Several assumptions are considered: (1) the stack temperature and humidity in the fuel cell cathode are well controlled; (2) the water inside the cathode is only in vapor phase and any extra water in liquid phase is removed from the channels; (3) the input reactant flows are humidified in a consistent and rapid way and the high pressure compressed hydrogen is available; (4) the anode pressure is well controlled to follow the cathode pressure; and (5) the current dynamics of the motor which directly drives the compressor is negligible due to its small time constant as compared to the mechanical dynamics [35].


### 2.1.1. Fuel Cell Stack Voltage

A single fuel cell operating voltage can be modeled as(1)
V
f
c
=
E
-
v
a
c
t
-
v
o
h
m
-
v
c
o
n
c
,where E is the open circuit voltage and v
a
c
t, v
o
h
m, and v
c
o
n
c present the activation loss, ohmic loss, and concentration loss, respectively.The open circuit voltageE is expressed as(2)
E
=
1.229
-
0.85
·
1
0
-
3
T
f
c
-
298.15
+
4.3085
·
1
0
-
5
T
f
c
l
n
p
H
2
+
1
2
l
n
p
O
2
,where T
f
c is the fuel cell stack temperature and p
H
2, p
O
2 are the partial pressures of hydrogen and oxygen, respectively [34].The activation lossv
a
c
t, ohmic loss v
o
h
m, and concentration loss v
c
o
n
c are expressed as follows.(1)
v
a
c
t
=
v
0
+
v
a
1
-
e
-
b
1
i is due to the difference between the velocity of the reactions in the anode and cathode [36], where i
=
I
s
t
/
A
f
c is the current density, I
s
t (A) is the stack current, and A
f
c (cm2) is the active area. v
0 (volts) is the voltage drop at zero current density, and v
a (volts) and b
1 are constants that depend on the temperature and the oxygen partial pressure [37, 38]. The values of v
0, v
a, and b
1 can be determined from a nonlinear regression of experimental data.
(2)
v
o
h
m
=
i
R
o
h
m is due to the electrical resistance of the electrodes, and the resistance to the flow of ions through the electrolyte [36]. R
o
h
m (Ω
·
c
m
2) represents the fuel cell internal electrical resistance.
(3)
v
c
o
n
c
=
i
b
3
i
/
i
m
a
x
b
4 results from the drop in concentration of the reactants due to the consumption in the reaction. b
3, b
4, and i
m
a
x are constants that depend on the temperature and the reactant partial pressures. i
m
a
x is the current density that generates the abrupt voltage drop.


### 2.1.2. Gas Dynamics ofp
H
2 and p
O
2

The dynamics ofp
H
2, p
O
2 are expressed as follows:(3)
d
p
H
2
d
t
=
R
H
2
T
f
c
V
a
n
W
H
2
,
i
n
-
W
H
2
,
o
u
t
-
W
H
2
,
r
e
a
c
t
,
d
p
O
2
d
t
=
R
O
2
T
f
c
V
c
a
W
O
2
,
i
n
-
W
O
2
,
o
u
t
-
W
O
2
,
r
e
a
c
t
,where W
H
2
,
i
n is the mass flow rate of hydrogen gas entering the anode, W
H
2
,
o
u
t is the mass flow rate of hydrogen gas leaving the anode, W
H
2
,
r
e
a
c
t is the rate of hydrogen reacted, W
O
2
,
i
n is the mass flow rate of oxygen entering the cathode, W
O
2
,
o
u
t is the mass flow rate of oxygen leaving the cathode, W
O
2
,
r
e
a
c
t is the rate of oxygen reacted, R
H
2 is the hydrogen gas constant, R
O
2 is the oxygen gas constant, and V
a
n and V
c
a are the anode and cathode volumes, respectively. More details are provided in [34].


### 2.1.3. Thermal Dynamics

The dynamic temperature model is described by a lumped thermal model [39]:(4)
m
s
t
C
p
s
t
d
T
f
c
d
t
=
Q
˙
s
o
u
-
W
c
C
p
c
T
f
c
-
T
c
,
i
n
,
Q
˙
s
o
u
=
I
s
t
-
T
f
c
Δ
s
4
F
+
v
a
c
t
+
I
s
t
R
o
h
m
,where m
s
t is the heat mass of the stack, C
p
s
t and C
p
c are the specific heat, W
c is the coolant flow rate considered as a control variable, T
c
,
i
n is the coolant temperature at the stack inlet, and Q
˙
s
o
u is the internal energy source. The latter is calculated as a function of the stack current, temperature, electrical resistance of stack layers R
o
h
m, Faraday’s number F, and the entropy change Δ
s. The physical parameters were obtained through extensive experimentation.


### 2.1.4. Auxiliary Components

The main power consumer of the fuel cell system is the air compressor, which can consume up to 30% of the fuel cell power under high load conditions [40]. Therefore, the net power from the fuel cell system P
f
c
,
n
e
t can be calculated as(5)
P
f
c
,
n
e
t
=
n
V
f
c
I
f
c
-
P
c
p
-
P
a
u
x
,where n is the number of cells and P
c
p and P
a
u
x are the power consumed by the compressor and auxiliary components (such as the cooling pump, radiator fan), respectively. In this study, P
a
u
x is assumed to be constant and P
c
p is calculated as(6)
P
c
p
=
C
p
T
a
t
m
η
c
m
η
c
p
p
s
m
p
a
t
m
γ
-
1
/
γ
-
1
W
c
p
,where p
s
m is the supply manifold pressure which is equal to the pressure at the compressor outlet, p
a
t
m is the atmospheric pressure, C
p is the specific heat capacity of air, γ is the ratio of the specific hears of air, T
a
t
m is the ambient temperature, W
c
p is the compressor flow rate, and η
c
m and η
c
p are the compressor motor efficiency and compressor efficiency, respectively. Figure 2 shows the efficiency curve of the fuel cell system; it can be found that the optimal operating point is near 8.8 kW.Figure 2
PEMFC efficiency curve.Remark 1.
As discussed in [41], the time constants of the gas dynamics and the stack temperature dynamics are estimated to be the order of magnitude of 1
0
-
1 and 1
0
2, respectively. Therefore, the mass storage effects in the channels are not considered.


## 2.1.1. Fuel Cell Stack Voltage

A single fuel cell operating voltage can be modeled as(1)
V
f
c
=
E
-
v
a
c
t
-
v
o
h
m
-
v
c
o
n
c
,where E is the open circuit voltage and v
a
c
t, v
o
h
m, and v
c
o
n
c present the activation loss, ohmic loss, and concentration loss, respectively.The open circuit voltageE is expressed as(2)
E
=
1.229
-
0.85
·
1
0
-
3
T
f
c
-
298.15
+
4.3085
·
1
0
-
5
T
f
c
l
n
p
H
2
+
1
2
l
n
p
O
2
,where T
f
c is the fuel cell stack temperature and p
H
2, p
O
2 are the partial pressures of hydrogen and oxygen, respectively [34].The activation lossv
a
c
t, ohmic loss v
o
h
m, and concentration loss v
c
o
n
c are expressed as follows.(1)
v
a
c
t
=
v
0
+
v
a
1
-
e
-
b
1
i is due to the difference between the velocity of the reactions in the anode and cathode [36], where i
=
I
s
t
/
A
f
c is the current density, I
s
t (A) is the stack current, and A
f
c (cm2) is the active area. v
0 (volts) is the voltage drop at zero current density, and v
a (volts) and b
1 are constants that depend on the temperature and the oxygen partial pressure [37, 38]. The values of v
0, v
a, and b
1 can be determined from a nonlinear regression of experimental data.
(2)
v
o
h
m
=
i
R
o
h
m is due to the electrical resistance of the electrodes, and the resistance to the flow of ions through the electrolyte [36]. R
o
h
m (Ω
·
c
m
2) represents the fuel cell internal electrical resistance.
(3)
v
c
o
n
c
=
i
b
3
i
/
i
m
a
x
b
4 results from the drop in concentration of the reactants due to the consumption in the reaction. b
3, b
4, and i
m
a
x are constants that depend on the temperature and the reactant partial pressures. i
m
a
x is the current density that generates the abrupt voltage drop.


## 2.1.2. Gas Dynamics ofp
H
2 and p
O
2

The dynamics ofp
H
2, p
O
2 are expressed as follows:(3)
d
p
H
2
d
t
=
R
H
2
T
f
c
V
a
n
W
H
2
,
i
n
-
W
H
2
,
o
u
t
-
W
H
2
,
r
e
a
c
t
,
d
p
O
2
d
t
=
R
O
2
T
f
c
V
c
a
W
O
2
,
i
n
-
W
O
2
,
o
u
t
-
W
O
2
,
r
e
a
c
t
,where W
H
2
,
i
n is the mass flow rate of hydrogen gas entering the anode, W
H
2
,
o
u
t is the mass flow rate of hydrogen gas leaving the anode, W
H
2
,
r
e
a
c
t is the rate of hydrogen reacted, W
O
2
,
i
n is the mass flow rate of oxygen entering the cathode, W
O
2
,
o
u
t is the mass flow rate of oxygen leaving the cathode, W
O
2
,
r
e
a
c
t is the rate of oxygen reacted, R
H
2 is the hydrogen gas constant, R
O
2 is the oxygen gas constant, and V
a
n and V
c
a are the anode and cathode volumes, respectively. More details are provided in [34].


## 2.1.3. Thermal Dynamics

The dynamic temperature model is described by a lumped thermal model [39]:(4)
m
s
t
C
p
s
t
d
T
f
c
d
t
=
Q
˙
s
o
u
-
W
c
C
p
c
T
f
c
-
T
c
,
i
n
,
Q
˙
s
o
u
=
I
s
t
-
T
f
c
Δ
s
4
F
+
v
a
c
t
+
I
s
t
R
o
h
m
,where m
s
t is the heat mass of the stack, C
p
s
t and C
p
c are the specific heat, W
c is the coolant flow rate considered as a control variable, T
c
,
i
n is the coolant temperature at the stack inlet, and Q
˙
s
o
u is the internal energy source. The latter is calculated as a function of the stack current, temperature, electrical resistance of stack layers R
o
h
m, Faraday’s number F, and the entropy change Δ
s. The physical parameters were obtained through extensive experimentation.


## 2.1.4. Auxiliary Components

The main power consumer of the fuel cell system is the air compressor, which can consume up to 30% of the fuel cell power under high load conditions [40]. Therefore, the net power from the fuel cell system P
f
c
,
n
e
t can be calculated as(5)
P
f
c
,
n
e
t
=
n
V
f
c
I
f
c
-
P
c
p
-
P
a
u
x
,where n is the number of cells and P
c
p and P
a
u
x are the power consumed by the compressor and auxiliary components (such as the cooling pump, radiator fan), respectively. In this study, P
a
u
x is assumed to be constant and P
c
p is calculated as(6)
P
c
p
=
C
p
T
a
t
m
η
c
m
η
c
p
p
s
m
p
a
t
m
γ
-
1
/
γ
-
1
W
c
p
,where p
s
m is the supply manifold pressure which is equal to the pressure at the compressor outlet, p
a
t
m is the atmospheric pressure, C
p is the specific heat capacity of air, γ is the ratio of the specific hears of air, T
a
t
m is the ambient temperature, W
c
p is the compressor flow rate, and η
c
m and η
c
p are the compressor motor efficiency and compressor efficiency, respectively. Figure 2 shows the efficiency curve of the fuel cell system; it can be found that the optimal operating point is near 8.8 kW.Figure 2
PEMFC efficiency curve.Remark 1.
As discussed in [41], the time constants of the gas dynamics and the stack temperature dynamics are estimated to be the order of magnitude of 1
0
-
1 and 1
0
2, respectively. Therefore, the mass storage effects in the channels are not considered.


## 2.2. Fuel Cell Converter Model

A classical boost-type unidirectional DC/DC converter is selected as FC power converter because it can be operated in the current control mode in a continuous condition mode [42]. It adapts the DC voltage supplied by the fc to the desired DC bus voltage (420 V). By acting on the switching signal u
f
c, it is possible to determine the load power distribution between FC and battery. The average model of the FC converter is expressed as follows:(7)
L
f
c
d
i
f
c
d
t
=
V
f
c
-
r
f
c
i
f
c
-
u
f
c
V
b
u
s
,where r
f
c is the total equivalent series resistance in the FC converter, u
f
c
=
1
-
D
f
c, and D
f
c is the duty ratio of the switch S
1.


## 2.3. Battery Model

The Li-ion battery with peak power 40 kW and capacity 20 Ah is modeled as a voltage source in series with a resistance [14, 43], which is given as(8)
V
b
a
t
t
=
V
o
c
-
r
b
a
t
t
i
b
a
t
t
,where the open circuit voltage (V
o
c) and the internal resistance (r
b
a
t
t) are both functions of the battery SOC. The SOC is defined as the ratio of charge stored in the battery to the maximum charge capacity Q
b
a
t
t:(9)
d
S
O
C
d
t
=
-
i
b
a
t
t
Q
b
a
t
t
,where i
b
a
t
t is the battery current.In order to expressi
b
a
t
t it is noted that the instantaneous power delivered by the battery to the load equals P
b
a
t
t
=
V
b
a
t
t
i
b
a
t
t. Then, the rate of change of S
O
C gives(10)
d
S
O
C
d
t
=
-
V
o
c
-
V
o
c
2
-
4
r
b
a
t
t
P
b
a
t
t
2
r
b
a
t
t
Q
b
a
t
t
.The solution (10) is feasible for negative power demands which means the battery is in charge mode and maximizes efficiency for positive power demands (discharge mode).


## 2.4. Battery Converter Model

A simple boost-type bidirectional DC/DC converter is used to connect the battery to the high voltage bus, enabling the charge and discharge of the battery. The cascade control structure of the bidirectional converter is realized through the outer control loop (the DC bus energy loop) and the inner control loop (the battery current loop). The dynamic of the current control loop are designed to be much faster than that of the DC bus energy regulation loop.The average model of the battery converter is expressed as follows:(11)
L
b
a
t
t
d
i
b
a
t
t
d
t
=
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
u
b
a
t
t
V
b
u
s
,where r
b
a
t
t is the total equivalent series resistance in the battery converter, u
b
a
t
t
=
1
-
D
b
a
t
t, and D
b
a
t
t is the duty ratio of the switch S
2.The DC link capacitive energyy
b
u
s is given versus FC power P
f
c, battery power P
b
a
t
t, and load power P
l
o
a
d by the following equation:(12)
y
˙
b
u
s
=
P
f
c
,
0
+
P
b
a
t
t
,
0
-
P
l
o
a
d
,where y
b
u
s
=
1
/
2
C
b
u
s
V
b
u
s
2, P
f
c
,
0
=
P
f
c
,
n
e
t
-
r
f
c
i
f
c
2, and P
b
a
t
t
,
0
=
P
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
2.


## 2.5. Vehicle Dynamics Modelling

The power train is modeled as a point-mass [14](13)
M
d
v
d
t
=
F
t
r
a
c
+
F
b
r
a
k
-
C
r
M
g
c
o
s
α
-
1
2
ρ
a
A
f
C
D
A
f
v
2
-
M
g
s
i
n
α
,where F
t
r
a
c is the vehicle traction force, F
b
r
a
k is the friction brake force, M is the vehicle mass, v is the vehicle velocity, and other parameters are defined in Table 1.Table 1
Parameters for the vehicle.
Parameters
Value
Vehicle massM
1500 kg
Rolling resistanceC
r
0.01
Drag coefficientC
D
0.3
Frontal areaA
f
2.4 m2
Air densityρ
a
1.2 kg/m3
Differential ratio
5.0
Differential efficiency
97%
Wheel radius
0.29 mThen, the power required to drive the tracking motor is calculated as(14)
P
l
o
a
d
=
F
t
r
a
c
v
η
m
η
d
i
f
f
-
s
i
g
n
F
t
r
a
c
,where η
d
i
f
f is the efficiency of the differential and η
m is the efficiency of the tracking motor which is a function of motor torque τ
m and its rotation speed ω
m, that is, η
m
=
η
(
τ
m
,
ω
m
).In the next section, the control strategy of FC and battery will be presented using super-twisting sliding mode.


## 3. Energy Management System

The energy management system implemented is based on the states based control algorithm and a SOSM control which generate the FC and battery reference currents, respectively. Then, the FC and battery current controls are realized with SOSM controllers due to its insensitivity to external disturbance, high accuracy, and finite time convergence. In this paper, the EMS are considered and designed based on the requirements given in Table2.Table 2
EMS design requirements.
Fuel cell parameters
P
fc
min
⁡
2 kW
P
fc
opt
13 kW
P
fc
max
⁡
30 kW
i
fc
min
⁡
2 A
i
fc
max
⁡
300 A
τ
fc
0.25 s
L
fc
2
·
1
0
-
3 H
r
fc
0.1Ω
FC current dynamic limitation
40 A/s
Battery parameters
P
batt
min
⁡
−40 kW
P
batt
max
⁡
40 kW
SOC
min
⁡
30%
SOC
max
⁡
90%
L
batt
2
·
1
0
-
3 H
r
batt
0.1Ω
V
bus
*
420 VNote that FC has a slow dynamic response, so that the battery functions to compensate the FC dynamic performance to avoid the FC starvation problem [44], supply the overpower demand, and absorb the regenerative braking power. The FC functions to supply the power to both the DC bus capacitor C
bus and the battery to be charged when its SOC is low. The control of the whole system is based on the SOC of the battery [6]:(i)
when SOC is lower than SOC reference, FC is necessary to charge the battery through the bidirectional DC/DC converter;
(ii)
when SOC is higher than its reference value, then the battery charging current reference is set to zero and the FC current reference is reduced to zero. For transient conditions, the battery supplies all load variations.The main objective of the control is to regulate DC bus voltagev
bus and prevent the FC suffering from fast current demand. Two SOSM controllers have been proposed for the unidirectional boost FC converter and the bidirectional battery converter. The bidirectional property allows the management of charge/discharge cycles of the battery tank.


### 3.1. Adaptive-Gain Supertwisting Sliding Mode Control

Consider a single-input uncertain nonlinear system(15)
x
˙
=
f
x
+
g
x
u
,
y
=
s
x
,
t
,where x
∈
X
⊂
R
n is a state vector, u
∈
U
⊂
R is a bounded control input, and f
(
x
), g
(
x
) are sufficiently smooth uncertain functions. The control objective is to force the sliding variable s
(
x
,
t
) and its time derivative and s
˙
(
x
,
t
) to zero in finite time, that is, s
(
x
,
t
)
=
s
˙
(
x
,
t
)
=
0.Assumption 2 (See [31]).
The relative degree of the system (15) equals one with respect to u and the internal dynamics are stable.
Therefore, under the Assumption2, the first time derivative of s
(
x
,
t
) can be presented in the following form:(16)
s
˙
=
∂
s
∂
t
+
∂
s
∂
x
f
x
︸
φ
x
,
t
+
∂
s
∂
x
g
x
︸
γ
x
,
t
u
,
=
φ
x
,
t
+
γ
x
,
t
u
.Assumption 3.
The time derivative of the functionφ
(
x
,
t
) is bounded, |
φ
˙
(
x
,
t
)
|
≤
δ with an unknown value δ, ∀
x
∈
X.Assumption 4.
There exist positive butunknown constants γ
m and γ
M, such that ∀
x
∈
X, 0
<
b
m
<
|
γ
(
x
,
t
)
|
<
b
M and |
γ
˙
(
x
,
t
)
|
≤
B.
The following ASTW algorithm is considered [31] (17)
u
=
-
α
s
1
/
2
s
i
g
n
s
+
ν
,
ν
˙
=
-
β
s
i
g
n
s
,where α
=
α
(
s
,
t
) and β
=
β
(
s
,
t
) are the adaptive gains to be determined.
The system (16) can be rewritten as(18)
s
˙
=
-
α
γ
x
,
t
s
1
/
2
s
i
g
n
s
+
ω
,
ω
˙
=
-
β
γ
x
,
t
s
i
g
n
s
+
φ
˙
x
,
t
+
γ
˙
x
,
t
ν
,where ω
(
0
)
=
0.
Given thatβ
≤
β
∗
>
0 [31] and Assumption 4, it follows that(19)
φ
˙
x
,
t
+
γ
˙
x
,
t
ν
≤
δ
+
B
∫
0
t
‍
β
d
τ
,
≤
δ
+
B
β
∗
t
=
δ
1
,for someunknown positive constant δ
1.
The control gainsα and β are adapted dynamically [31](20)
α
˙
=
ω
1
γ
1
2
s
i
g
n
s
-
μ
,
i
f
α
>
α
m
,
η
1
,
i
f
α
≤
α
m
,
β
=
ε
1
α
,where ω
1, γ
1, μ, η
1, ε
1, and α
m are arbitrary positive constants.Remark 5.
The practical sliding mode is established in finite timet
F, that is, |
s
|
≤
μ
1, |
s
˙
|
≤
μ
2. At the beginning, |
s
(
0
)
|
>
μ and α
(
0
)
>
α
m, and the adaptive gains α and β dynamically increase in order to stabilize the system. Once the practical sliding mode is achieved, the gains start to decrease.


### 3.2. FC Reference Power Calculation

In this study, state based control algorithm proposed in the work [45] is used to calculate the FC reference power P
f
c
∗. The FC reference power is determined based on the load power and battery SOC. Three levels for the SOC range are considered: high (S
O
C
>
90
%), normal (60
%
≤
S
O
C
≤
85
%), and low (S
O
C
<
30
%). Eight states derived from the works of [45–47] are given as follows.Case 1.
SOC is high andP
l
o
a
d
<
P
f
c
m
i
n. In this case, the FC operates at its minimum power P
f
c
∗
=
P
f
c
m
i
n, only supplies auxiliary components.Case 2.
SOC is high andP
f
c
m
i
n
≤
P
l
o
a
d
≤
P
f
c
m
a
x. In this case, the FC operates with a load following strategy P
f
c
∗
=
P
l
o
a
d and the battery will supply the fast load demand when necessary.Case 3.
SOC is high andP
l
o
a
d
≥
P
f
c
m
i
n. In this case, the FC is demanded to generate its maximum power P
f
c
∗
=
P
f
c
m
a
x and the battery will provide additional power to assist the FC.Case 4.
SOC is normal andP
l
o
a
d
<
P
f
c
o
p
t. In this case, the FC operates at its optimum power P
f
c
∗
=
P
f
c
o
p
t.Case 5.
SOC is normal andP
f
c
o
p
t
≤
P
l
o
a
d
≤
P
f
c
m
a
x. In this case, the FC operates with a load following strategy, which works the same as Case 2.Case 6.
SOC is normal andP
l
o
a
d
≥
P
f
c
m
a
x. In this case, the FC is demanded to generate its maximum power P
f
c
∗
=
P
f
c
m
a
x; it works the same as Case 3.Case 7.
SOC is low andP
l
o
a
d
<
P
f
c
m
a
x. In this case, the FC must generate the load power plus the battery charging power, P
f
c
∗
=
P
f
c
m
a
x
+
P
b
a
t
t
c
h
a
r.Case 8.
SOC is low andP
l
o
a
d
≥
P
f
c
m
a
x. In this case, the FC is required to generate the maximum power, P
f
c
∗
=
P
f
c
m
a
x.
The hysteresis control of the battery SOC is shown in Figure3.Figure 3
State based control hysteresis.


### 3.3. Control of FC Converter

The detailed FC current regulation loop is portrayed in Figure4. From the FC reference power generated by the states control algorithm and the FC voltage, the FC reference current is determined which is limited to a range i
f
c
m
i
n
,
i
f
c
m
a
x. In the case of within the interval, the control of FC boost converter ensures current regulation (with respect to its reference value). In the case of outside the interval, the FC current will be saturated and the surge current is then provided or absorbed by the battery. Moreover, the FC dynamic limitation is also considered using a low pass filter, such that fast current demand is avoided. The error between the FC current and its reference is used in a SOSM controller in order to determine the duty cycle of the FC unidirectional converter [48].Figure 4
Control law of FC current.The sliding manifold of the FC current control loop is defined as(21)
s
1
=
i
f
c
,
r
e
f
-
i
f
c
.In view of (7), the first time derivative of s
1 is calculated as follows:(22)
s
˙
1
=
i
˙
f
c
,
r
e
f
-
V
f
c
-
r
f
c
i
f
c
-
u
f
c
V
b
u
s
L
f
c
.According to the adaptive SOSM algorithm [31], the FC current controller can be designed as follows:(23)
u
f
c
=
V
f
c
-
r
f
c
i
f
c
-
L
f
c
υ
1
s
1
V
b
u
s
,where υ
1
(
s
1
) takes the same structure as (17): (24)
υ
1
s
1
=
υ
11
s
1
+
υ
12
s
1
,
υ
˙
11
s
1
=
-
β
1
s
i
g
n
s
1
,
υ
12
s
1
=
-
α
1
s
1
1
/
2
s
i
g
n
s
1
,and the adaptive gains α
1 and β
1 are designed according to (20). In order to ensure the fast convergence of s
1, the parameters have been tuned as follows: ω
1
=
150, γ
1
=
2.5, μ
1
=
0.1, η
1
=
25, α
1
m
=
0.01, and ε
1
=
2.


### 3.4. Battery Control

A cascade control structure is proposed, which consists of the outer control loop (DC bus energy control) and the inner control loop (battery current control), as shown in Figure5. The DC bus energy regulation law generates the battery reference power P
b
a
t
t
∗. This reference value is then divided by the measured battery voltage V
b
a
t
t in order to obtain the battery reference current i
b
a
t
t
∗, which is limited by its maximum and minimum current (i
b
a
t
t
,
m
a
x and i
b
a
t
t
,
m
i
n).Figure 5
Control law of battery current.The error between the battery current and its reference is used in another SOSM controller in order to determine the duty cycle of the battery bidirectional converter. On the one hand, the battery is in the discharge mode when the system demands power from the battery, and the duty cycle of the battery converter increases in order to supply power to maintain the DC bus voltage. On the other hand, the battery is in the charge mode when there is excess power in the system, and the duty cycle of the battery converter decreases and, thus, the battery voltage increases. Therefore, the battery converter is controlled for DC bus energy regulation and is switching between the charge and discharge modes [46].


#### 3.4.1. External Control Loop

For the DC bus energy regulation, a SOSMC is employed, with the error between the bus energy and its reference as input. The sliding manifold of the outer loop is defined as(25)
s
2
=
y
b
u
s
∗
-
y
b
u
s
,where y
b
u
s
∗
=
0.5
C
b
u
s
V
b
u
s
∗
2 and y
b
u
s
=
0.5
C
b
u
s
V
b
u
s
2.In view of (12), the first time derivative of s
2 is calculated as follows:(26)
s
˙
2
=
-
P
f
c
,
0
-
P
b
a
t
t
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
.Then, the battery power reference is obtained as(27)
P
b
a
t
t
∗
=
υ
2
s
2
-
P
f
c
,
0
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
,where υ
2
(
s
2
) is the adaptive SOSM controller with the same structure of (20) and (24).The battery reference current is obtained as follows:(28)
i
b
a
t
t
∗
=
P
b
a
t
t
∗
V
b
a
t
t
.


#### 3.4.2. Inner Control Loop

The inner loop contains a SOSM current controller which generates the duty cycle of the battery bidirectional converter. The sliding manifold of the inner loop is defined as(29)
s
3
=
i
b
a
t
t
∗
-
i
b
a
t
t
.In view of (11), the first time derivative of s
3 is calculated as follows:(30)
s
˙
3
=
i
˙
b
a
t
t
∗
-
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
u
b
a
t
t
V
b
u
s
L
b
a
t
t
.The battery current controller u
b
a
t
t is designed using the adaptive SOSM algorithm(31)
u
b
a
t
t
=
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
L
b
a
t
t
υ
3
s
3
V
b
u
s
,where υ
3
(
s
3
) is the adaptive SOSM controller with the same structure of (20) and (24).Remark 6.
Due to the cascade control structure and the constant switching frequencyω
S of the power electronic, the cutoff frequency of the inner loop ω
I is ten times higher than that of the outer loop ω
O and less than ω
S, that is, ω
O
≪
ω
I
≪
ω
S. In order to have an effective cascade control system, it is essential that the inner loop responds much faster than the outer loop.


## 3.1. Adaptive-Gain Supertwisting Sliding Mode Control

Consider a single-input uncertain nonlinear system(15)
x
˙
=
f
x
+
g
x
u
,
y
=
s
x
,
t
,where x
∈
X
⊂
R
n is a state vector, u
∈
U
⊂
R is a bounded control input, and f
(
x
), g
(
x
) are sufficiently smooth uncertain functions. The control objective is to force the sliding variable s
(
x
,
t
) and its time derivative and s
˙
(
x
,
t
) to zero in finite time, that is, s
(
x
,
t
)
=
s
˙
(
x
,
t
)
=
0.Assumption 2 (See [31]).
The relative degree of the system (15) equals one with respect to u and the internal dynamics are stable.
Therefore, under the Assumption2, the first time derivative of s
(
x
,
t
) can be presented in the following form:(16)
s
˙
=
∂
s
∂
t
+
∂
s
∂
x
f
x
︸
φ
x
,
t
+
∂
s
∂
x
g
x
︸
γ
x
,
t
u
,
=
φ
x
,
t
+
γ
x
,
t
u
.Assumption 3.
The time derivative of the functionφ
(
x
,
t
) is bounded, |
φ
˙
(
x
,
t
)
|
≤
δ with an unknown value δ, ∀
x
∈
X.Assumption 4.
There exist positive butunknown constants γ
m and γ
M, such that ∀
x
∈
X, 0
<
b
m
<
|
γ
(
x
,
t
)
|
<
b
M and |
γ
˙
(
x
,
t
)
|
≤
B.
The following ASTW algorithm is considered [31] (17)
u
=
-
α
s
1
/
2
s
i
g
n
s
+
ν
,
ν
˙
=
-
β
s
i
g
n
s
,where α
=
α
(
s
,
t
) and β
=
β
(
s
,
t
) are the adaptive gains to be determined.
The system (16) can be rewritten as(18)
s
˙
=
-
α
γ
x
,
t
s
1
/
2
s
i
g
n
s
+
ω
,
ω
˙
=
-
β
γ
x
,
t
s
i
g
n
s
+
φ
˙
x
,
t
+
γ
˙
x
,
t
ν
,where ω
(
0
)
=
0.
Given thatβ
≤
β
∗
>
0 [31] and Assumption 4, it follows that(19)
φ
˙
x
,
t
+
γ
˙
x
,
t
ν
≤
δ
+
B
∫
0
t
‍
β
d
τ
,
≤
δ
+
B
β
∗
t
=
δ
1
,for someunknown positive constant δ
1.
The control gainsα and β are adapted dynamically [31](20)
α
˙
=
ω
1
γ
1
2
s
i
g
n
s
-
μ
,
i
f
α
>
α
m
,
η
1
,
i
f
α
≤
α
m
,
β
=
ε
1
α
,where ω
1, γ
1, μ, η
1, ε
1, and α
m are arbitrary positive constants.Remark 5.
The practical sliding mode is established in finite timet
F, that is, |
s
|
≤
μ
1, |
s
˙
|
≤
μ
2. At the beginning, |
s
(
0
)
|
>
μ and α
(
0
)
>
α
m, and the adaptive gains α and β dynamically increase in order to stabilize the system. Once the practical sliding mode is achieved, the gains start to decrease.


## 3.2. FC Reference Power Calculation

In this study, state based control algorithm proposed in the work [45] is used to calculate the FC reference power P
f
c
∗. The FC reference power is determined based on the load power and battery SOC. Three levels for the SOC range are considered: high (S
O
C
>
90
%), normal (60
%
≤
S
O
C
≤
85
%), and low (S
O
C
<
30
%). Eight states derived from the works of [45–47] are given as follows.Case 1.
SOC is high andP
l
o
a
d
<
P
f
c
m
i
n. In this case, the FC operates at its minimum power P
f
c
∗
=
P
f
c
m
i
n, only supplies auxiliary components.Case 2.
SOC is high andP
f
c
m
i
n
≤
P
l
o
a
d
≤
P
f
c
m
a
x. In this case, the FC operates with a load following strategy P
f
c
∗
=
P
l
o
a
d and the battery will supply the fast load demand when necessary.Case 3.
SOC is high andP
l
o
a
d
≥
P
f
c
m
i
n. In this case, the FC is demanded to generate its maximum power P
f
c
∗
=
P
f
c
m
a
x and the battery will provide additional power to assist the FC.Case 4.
SOC is normal andP
l
o
a
d
<
P
f
c
o
p
t. In this case, the FC operates at its optimum power P
f
c
∗
=
P
f
c
o
p
t.Case 5.
SOC is normal andP
f
c
o
p
t
≤
P
l
o
a
d
≤
P
f
c
m
a
x. In this case, the FC operates with a load following strategy, which works the same as Case 2.Case 6.
SOC is normal andP
l
o
a
d
≥
P
f
c
m
a
x. In this case, the FC is demanded to generate its maximum power P
f
c
∗
=
P
f
c
m
a
x; it works the same as Case 3.Case 7.
SOC is low andP
l
o
a
d
<
P
f
c
m
a
x. In this case, the FC must generate the load power plus the battery charging power, P
f
c
∗
=
P
f
c
m
a
x
+
P
b
a
t
t
c
h
a
r.Case 8.
SOC is low andP
l
o
a
d
≥
P
f
c
m
a
x. In this case, the FC is required to generate the maximum power, P
f
c
∗
=
P
f
c
m
a
x.
The hysteresis control of the battery SOC is shown in Figure3.Figure 3
State based control hysteresis.


## 3.3. Control of FC Converter

The detailed FC current regulation loop is portrayed in Figure4. From the FC reference power generated by the states control algorithm and the FC voltage, the FC reference current is determined which is limited to a range i
f
c
m
i
n
,
i
f
c
m
a
x. In the case of within the interval, the control of FC boost converter ensures current regulation (with respect to its reference value). In the case of outside the interval, the FC current will be saturated and the surge current is then provided or absorbed by the battery. Moreover, the FC dynamic limitation is also considered using a low pass filter, such that fast current demand is avoided. The error between the FC current and its reference is used in a SOSM controller in order to determine the duty cycle of the FC unidirectional converter [48].Figure 4
Control law of FC current.The sliding manifold of the FC current control loop is defined as(21)
s
1
=
i
f
c
,
r
e
f
-
i
f
c
.In view of (7), the first time derivative of s
1 is calculated as follows:(22)
s
˙
1
=
i
˙
f
c
,
r
e
f
-
V
f
c
-
r
f
c
i
f
c
-
u
f
c
V
b
u
s
L
f
c
.According to the adaptive SOSM algorithm [31], the FC current controller can be designed as follows:(23)
u
f
c
=
V
f
c
-
r
f
c
i
f
c
-
L
f
c
υ
1
s
1
V
b
u
s
,where υ
1
(
s
1
) takes the same structure as (17): (24)
υ
1
s
1
=
υ
11
s
1
+
υ
12
s
1
,
υ
˙
11
s
1
=
-
β
1
s
i
g
n
s
1
,
υ
12
s
1
=
-
α
1
s
1
1
/
2
s
i
g
n
s
1
,and the adaptive gains α
1 and β
1 are designed according to (20). In order to ensure the fast convergence of s
1, the parameters have been tuned as follows: ω
1
=
150, γ
1
=
2.5, μ
1
=
0.1, η
1
=
25, α
1
m
=
0.01, and ε
1
=
2.


## 3.4. Battery Control

A cascade control structure is proposed, which consists of the outer control loop (DC bus energy control) and the inner control loop (battery current control), as shown in Figure5. The DC bus energy regulation law generates the battery reference power P
b
a
t
t
∗. This reference value is then divided by the measured battery voltage V
b
a
t
t in order to obtain the battery reference current i
b
a
t
t
∗, which is limited by its maximum and minimum current (i
b
a
t
t
,
m
a
x and i
b
a
t
t
,
m
i
n).Figure 5
Control law of battery current.The error between the battery current and its reference is used in another SOSM controller in order to determine the duty cycle of the battery bidirectional converter. On the one hand, the battery is in the discharge mode when the system demands power from the battery, and the duty cycle of the battery converter increases in order to supply power to maintain the DC bus voltage. On the other hand, the battery is in the charge mode when there is excess power in the system, and the duty cycle of the battery converter decreases and, thus, the battery voltage increases. Therefore, the battery converter is controlled for DC bus energy regulation and is switching between the charge and discharge modes [46].


### 3.4.1. External Control Loop

For the DC bus energy regulation, a SOSMC is employed, with the error between the bus energy and its reference as input. The sliding manifold of the outer loop is defined as(25)
s
2
=
y
b
u
s
∗
-
y
b
u
s
,where y
b
u
s
∗
=
0.5
C
b
u
s
V
b
u
s
∗
2 and y
b
u
s
=
0.5
C
b
u
s
V
b
u
s
2.In view of (12), the first time derivative of s
2 is calculated as follows:(26)
s
˙
2
=
-
P
f
c
,
0
-
P
b
a
t
t
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
.Then, the battery power reference is obtained as(27)
P
b
a
t
t
∗
=
υ
2
s
2
-
P
f
c
,
0
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
,where υ
2
(
s
2
) is the adaptive SOSM controller with the same structure of (20) and (24).The battery reference current is obtained as follows:(28)
i
b
a
t
t
∗
=
P
b
a
t
t
∗
V
b
a
t
t
.


### 3.4.2. Inner Control Loop

The inner loop contains a SOSM current controller which generates the duty cycle of the battery bidirectional converter. The sliding manifold of the inner loop is defined as(29)
s
3
=
i
b
a
t
t
∗
-
i
b
a
t
t
.In view of (11), the first time derivative of s
3 is calculated as follows:(30)
s
˙
3
=
i
˙
b
a
t
t
∗
-
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
u
b
a
t
t
V
b
u
s
L
b
a
t
t
.The battery current controller u
b
a
t
t is designed using the adaptive SOSM algorithm(31)
u
b
a
t
t
=
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
L
b
a
t
t
υ
3
s
3
V
b
u
s
,where υ
3
(
s
3
) is the adaptive SOSM controller with the same structure of (20) and (24).Remark 6.
Due to the cascade control structure and the constant switching frequencyω
S of the power electronic, the cutoff frequency of the inner loop ω
I is ten times higher than that of the outer loop ω
O and less than ω
S, that is, ω
O
≪
ω
I
≪
ω
S. In order to have an effective cascade control system, it is essential that the inner loop responds much faster than the outer loop.


## 3.4.1. External Control Loop

For the DC bus energy regulation, a SOSMC is employed, with the error between the bus energy and its reference as input. The sliding manifold of the outer loop is defined as(25)
s
2
=
y
b
u
s
∗
-
y
b
u
s
,where y
b
u
s
∗
=
0.5
C
b
u
s
V
b
u
s
∗
2 and y
b
u
s
=
0.5
C
b
u
s
V
b
u
s
2.In view of (12), the first time derivative of s
2 is calculated as follows:(26)
s
˙
2
=
-
P
f
c
,
0
-
P
b
a
t
t
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
.Then, the battery power reference is obtained as(27)
P
b
a
t
t
∗
=
υ
2
s
2
-
P
f
c
,
0
+
r
b
a
t
t
i
b
a
t
t
2
+
P
l
o
a
d
,where υ
2
(
s
2
) is the adaptive SOSM controller with the same structure of (20) and (24).The battery reference current is obtained as follows:(28)
i
b
a
t
t
∗
=
P
b
a
t
t
∗
V
b
a
t
t
.


## 3.4.2. Inner Control Loop

The inner loop contains a SOSM current controller which generates the duty cycle of the battery bidirectional converter. The sliding manifold of the inner loop is defined as(29)
s
3
=
i
b
a
t
t
∗
-
i
b
a
t
t
.In view of (11), the first time derivative of s
3 is calculated as follows:(30)
s
˙
3
=
i
˙
b
a
t
t
∗
-
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
u
b
a
t
t
V
b
u
s
L
b
a
t
t
.The battery current controller u
b
a
t
t is designed using the adaptive SOSM algorithm(31)
u
b
a
t
t
=
V
b
a
t
t
-
r
b
a
t
t
i
b
a
t
t
-
L
b
a
t
t
υ
3
s
3
V
b
u
s
,where υ
3
(
s
3
) is the adaptive SOSM controller with the same structure of (20) and (24).Remark 6.
Due to the cascade control structure and the constant switching frequencyω
S of the power electronic, the cutoff frequency of the inner loop ω
I is ten times higher than that of the outer loop ω
O and less than ω
S, that is, ω
O
≪
ω
I
≪
ω
S. In order to have an effective cascade control system, it is essential that the inner loop responds much faster than the outer loop.


## 4. Simulation Results and Discussion

The proposed hybrid system and control strategies have been implemented in MATLAB/Simulink and tested for the standard drive cycles, that is, the Urban Dynamometer Driving Schedule (UDDS) and Highway Fuel Economy Driving Schedule (HWFET). The UDDS and HWFET were used to represent typical driving conditions of light duty vehicles in the city and highway driving conditions, respectively. In this paper, the driving cycle implemented in the FC-Battery hybrid system is obtained from the simulation package Advanced Vehicle Vehicle Simulator (ADVISOR) [49]. Specific characteristics of the UDDS are shown in Table 3. The speed and mechanical power required by the electrical vehicle is shown in Figure 6. The implementation of the proposed EMS in the FC-battery hybrid system is shown in Figure 7.Table 3
Driving cycle specific characteristics [50].
Driving cycle
UDDS
HWFET
Duration
1369 s
765 s
Distance
12.07 km
16.45 km
Maximum speed
56.7 mph
59.90 mph
Average speed
19.58 mph
48.28 mph
Average accelerate
0.50 m/s2
0.19 m/s2
Average decelerate
−0.58 m/s2
−0.22 m/s2Figure 6
Standard driving cycles (UDDS and HWFET).
(a)
UDDS
(b)
HWFETFigure 7
Hybrid system configuration.The multirate simulation of the proposed FC-battery hybrid system has been carried out. Multirate approach realizes the achievement of realistic simulation results by taking into account some implementation issues:(1)
the control evaluation ratef
2 is less than the simulation rate f
1 (i.e., the integration was carried out according to the Euler method);
(2)
the power elements switch ratef
3 is less than the control evaluation rate f
2 due to switching loss.The parameters of the PEMFC system and multirate approach used in this study are shown in Tables4 and 5. Parameters associated with the FC converter control and battery converter control are given in Table 2.Table 4
PEMFC system parameters.
Symbol
Parameter
Value
n
Number of cells in stack
381
R
Universal gas constant
8.314 J/(mol K)
T
fc
Temperature of the fuel cell
353.15 K
F
Faraday constant
96485 C/mol
p
atm
Atmospheric pressure
1.01325
×
1
0
5 Pa
T
atm
Atmospheric temperature
298.15 K
V
ca
Cathode volume
0.01 m3
V
an
Anode volume
0.005 m3
x
O
2
,
atm
Oxygen mass fraction
0.23
M
a
Air molar mass
28.9644
×
1
0
-
3 kg/mol
M
O
2
Oxygen molar mass
32
×
1
0
-
3 kg/mol
M
N
2
Nitrogen molar mass
28
×
1
0
-
3 kg/mol
γ
Ratio of specific heats of air
1.4
C
p
Specific heat capacity of air
1004 J/(kg K)
η
cp
Compressor efficiency
80%
η
cm
Motor mechanical efficiency
98%Table 5
Parameters for the multirate simulation.
Symbol
Parameter
Value
f
1
Simulation rate
10 kHz
f
2
Sampling rate
2 kHz
f
3
Chopping frequency
1 kHzThe tests are performed under the same initial conditions: the initial batteryS
O
C
=
0.5, the initial FC voltage V
f
c
=
350
V, and the initial FC temperature T
f
c
=
353.15
K. The simulation results under UDDS and HWFET driving cycles are shown Figures 8–13.Figure 8
FC and battery powers under UDDS and HWFET driving cycles.
(a)
UDDS
(b)
HWFETFigure 9
FC power tracking and its error under UDDS and HWFET driving cycles.
(a)
UDDS
(b)
UDDS
(c)
HWFET
(d)
HWFETFigure 10
Battery SOC under UDDS and HWFET driving cycles.
(a)
UDDS
(b)
HWFETFigure 11
DC bus voltage performance under UDDS and HWFET driving cycles.
(a)
ASTW (UDDS)
(b)
PI (UDDS)
(c)
ASTW (HWFET)
(d)
PI (HWFET)Figure 12
Duty cycles of the FC and battery converters under UDDS and HWFET driving cycles.
(a)
UDDS
(b)
HWFETFigure 13
Hydrogen consumption under UDDS and HWFET driving cycles.
(a)
UDD
(b)
HWFETIt is shown from Figure8 that both the fuel cell and battery output powers are in their allowable operational ranges. In addition, the FC power works often at its optimal power which increases the the efficiency. The rate of change of the FC power is significantly alleviated using the state based control. Figure 9 presents the performance of the proposed ASTW controller for regulating the FC power to its reference value. Figures 9(b) and 9(d) prove the effectiveness of the controller. Figure 10 shows that the battery SOC are successfully sustained between S
O
C
m
i
n
=
0.3 and S
O
C
m
a
x
=
0.9 under both driving cycles.The DC bus voltage performance is compared with a well-tuned linear Proportional-Integral (PI) controller, as shown in Figure11. The proposed ASTW controller stabilized the DC bus voltage at the reference value V
b
u
s
∗
=
420
V which acts in the outer control loop, while PI control results in higher fluctuation around the desired value and higher voltage overshoot compared with the adaptive ASTW control. Faster convergence can also be observed from the enlarged view. Some overshoots of the DC bus voltage can be observed from Figure 11; this is because the power demands change rapidly at the DC bus during the driving process. Figure 12 gives the duty cycles of the FC unidirectional converter and battery bidirectional converter, which are generated from the FC current control loop and the battery current control loop, respectively. Both of the current control loop are based on the proposed ASTW algorithm, which act in the inner loop. Figure 13 shows the hydrogen consumption under UDDS and HWFET driving cycles.


## 5. Conclusions

This paper investigated the power management control for a FC hybrid system using a Li-ion battery stack as a secondary storage element. A hysteresis controller is proposed to determine the power reference for the fuel cell stack. The adaptive STW control is employed for the fuel cell converter to track the given power command while satisfying the power limitations. A cascade control structure is also designed for the battery converter, where the battery current control acts as the inner loop and the DC bus regulation acts as the outer loop which aims to stabilize the DC bus voltage. Both of the control loops are based on the adaptive STW algorithm. Finally, multirate simulations are carried out in the MatlabSimulink environment over two driving cycles, UDDS and HWFET. The simulations results have shown that the proposed approach is effective and feasible.



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*Source: 370424-2015-07-29.xml*