an external input. We consider the latter case, assuming that an independent transmis-
sion controller exists to select the gear based mainly on drivability considerations.
The vehicle speed is considered as an external input for the energy management
strategy, as is the total torque request
T
pwt
(or power request
P
req
), which is gener-
ated by the driver via the accelerator pedal or by the speed controller in simulation,
according to the general control architecture shown in Fig.
3.2
. From (
8.1
), if
T
pwt
and the gear index
i
tr
are imposed, the gearbox input torque is determined as
T
gb
=
T
eng
+
T
mot
=
T
pwt
g
tr
(
i
tr
)
·
g
fd
.
(8.4)

8.2
Parallel Architecture
93
As discussed in Sect.
3.4
, the standard degree of freedom for the optimization
problem is the battery power
P
batt
. It is directly and bi-univocally related to the
motor torque,
T
mot
, because the motor speed is imposed by external inputs and the
motor efficiency only depends on torque and speed (neglecting temperature effects).
This allows using the motor torque as the control variable, which is more immediate
for this powertrain architecture: in fact, given the torque of the electric motor, the
engine torque is found by difference from
T
gb
:
T
eng
=
T
gb
−
T
mot
.
(8.5)
The fuel power (proportional to the fuel consumption) is computed using the map
in Fig.
8.2
, so it is a function of engine speed and torque. Through (
8.2
) and (
8.5
), it
is computed as a function of vehicle speed, total torque request, and motor torque as
P
fuel
=
Q
lhv
˙
m
f
(
T
eng
, ω
eng
)
=
P
fuel
(
T
gb
,
T
mot
,
v
veh
).
(8.6)
System dynamics
. The system dynamic equation represents the evolution of the
battery state of charge as a function of the state itself and of the control input
T
mot
(or
P
batt
, for what said earlier). According to (
2.37
), the state of charge variation is
˙
SOC
= −
1
η
sign
(
I
(
t
))
coul
Q
nom
⎡
⎣
V
oc
(
SOC
)
2
R
0
(
SOC
)
−
V
oc
(
SOC
)
2
R
0
(
SOC
)
2
−
P
batt
R
0
(
SOC
)
⎤
⎦
(8.7)
with the battery parameters
V
oc
(
SOC
)
and
R
0
(
SOC
)
shown in Fig.
8.4
.
It turns out it is more practical to use the electrochemical energy variation in place
of state of charge as state in this case. According to the formulation introduced in
Sect.
5.3.1
:
E
ech
=
E
batt
·
(
SOC
(
t
0
)
−
SOC
(
t
)
)
(8.8)
with
E
batt
=
V
oc
,
nom
Q
nom
η
coul
; the system dynamic equation is
˙
E
ech
=
P
ech
= −
E
batt
˙
SOC
.
(8.9)
The battery power is
P
batt
=
P
em
,
e
(
T
mot
, ω
mot
),
(8.10)
where
P
em
,
e
(
T
mot
, ω
mot
)
is the electrical power required by the electric machine to
produce the torque
T
mot
, at the speed
ω
mot
(which is a function of the vehicle speed
via the gearbox ratio
i
tr
). Therefore the state equation is a function only of
SOC
and
P
batt
, which in turn depends on the control input
T
mot
, and the external inputs
v
veh
,
i
tr
.

94
8
Case Studies
Control constraints
. The value of engine and motor torque must remain within
their respective limitations:
T
mot
,
min
(ω
mot
)
≤
T
mot
≤
T
mot
,
max
(ω
mot
),
(8.11)
T
ice
,
min
(ω
eng
)
≤
T
eng
≤
T
ice
,
max
(ω
eng
),
(8.12)
and, in addition, the motor power is saturated according to the minimum and maxi-
mum available electric power
2
:
P
batt
,
min
(
SOC
)
≤
P
mot
,
e
≤
P
batt
,
max
(
SOC
),
(8.13)

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