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Real-time energy management of the
Volvo V60 PHEV based on a closed-form
minimization of the Hamiltonian
Viktor Larsson1, Lars Johannesson1, Bo Egardt1
Andreas Karlsson2, Anders Lasson2
1
Department of Signals and Systems, Chalmers University of Technology
2
Volvo Car Corporation
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Background
ˆ The nominal strategy in the V60 PHEV is rule-based
-Charge-Depletion followed by Charge-Sustaining mode
- based on precalibrated maps ⇒ not easy to change discharge rate
ˆ Some trips will exceed the electric range of the PHEV
- Gradual discharge can reduce fuel consumption
ˆ Objective is to implement a strategy with controllable discharge rate
Battery State of Charge vs Distance
Battery State of Charge vs Distance
1
1
Lower electric losses
SoC
SoC
High resistive Electric conversion
losses
losses
CD
Gradual discharge
CS
electric driving range
0
0
start
distance
end
start
distance
end
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Outline
ˆ Energy management system
ˆ Simplied powertrain model
ˆ Minimizing the Hamiltonian
ˆ Implementation in Simulink
ˆ Simulations & Vehicle tests
ˆ Conclusions
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The energy management system
ˆ Divided into a predictive level and a real-time level
- computations at predictive level using cloud computing or smartphone
- computations at real-time level in the vehicle Electronic Control Unit
Energy management system
Predictive level
Predicted driving
Optimal control problem
Feedforward
information
Real-time level
Instantaneous power request
Vehicle states
Real-time controller
Setpoints
- engine
- motor
- etc.
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The energy management system
ˆ The energy management problem is to minimize overall energy cost
Z tf
J ∗ = min
u(·)
G(x(tf )) +
g(u(t), t) dt
t0
| {z }
cost to recharge
|
{z
cost for fuel
}
s.t. ẋ(t) = f (x(t), u(t), t)
x(t0) = x0
x(t) ∈ X, u(t) ∈ U t)
- x = SoC is the state and f (x, u) the state dynamics
- u represents the control signal (torques, gear, engine state,...)
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The energy management system
ˆ The real-time controller is based on ECMS1
- derived from the Pontryagin principle
- control at each sample is obtained by minimizing the Hamiltonian
u∗ = arg min H(x, u, s) = arg min
u∈U
u∈U
g(u) +s · f (x, u)
|{z}
fuel rate
| {z }
dSoC
dt
- s is the equivalence factor which depend on future driving conditions
ˆ The ECMS-strategy is implemented in an ECU
- important with low computational and memory demand
⇒ minimize the Hamiltonian analytically
1
Equivalent Consumption Minimization Strategy
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Simplied powertrain model
ˆ Equivalent circuit battery model,
ẋ =
dSoC
dt
√
=−
Voc −
Voc2 −4Rin Pb
2RinQ
ˆ Transmission ratios r with eciency η (no dynamics)
ˆ Engine fuel rate ane in torque,
g = c0(ωe)Te + c1(ωe)
ˆ Electrical power of the motor quadratic in torque
Pm = d0(ωm)Tm2 + d1(ωm)Tm + d2(ωm)
ˆ Electrical power of the generator ane in torque
Pg = e0(ωg )Tg + e1(ωg ), Tg ≤ 0
+ 96 rad/s
293 rad/s
84 rad/s
battery
(ICE speed)
26 rad/s
1152 rad/s
289rad/s
(ICE speed)
clutch electric
motor
transmission
engine
clutch
integrated starter
generator
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Minimizing the Hamiltonian
ˆ With the simple powertrain model the Hamiltonian is given by
H(x, u, s) = g(u) +s · f (x, u)
|{z}
fuel rate
| {z }
dSoC
dt
= c0(ωe)Te + c1(ωe) − s
Voc −
p
Voc2 − 4RinPb
2RinQ
where the battery power is: Pb = d0Tm2 + d1Tm + d2 + e0Tg + e1 + Pa
ˆ The torque balance equation is
Td
|{z}
traction request
rg
Tg )
= η| r r{z
T
+
η
r
r
(T
+
r m
f f gb
e
}
ηg
motor torque
|
{z
input torque to gearbox
}
ˆ Assume engine is on with a xed gear rgb
control variables: engine/motor/generator torque Te Tm Tg
⇒ two degrees of freedom in meeting the traction request Td
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Minimizing the Hamiltonian
ˆ Solve torque balance equation for engine torque
Td − rr ηr Tm − ηg−1ηf rf rg rgbTg
Te(Tm, Tg ) =
ηf rf rk
two independent control variables ⇒ u = [Tm Tg ]
ˆ Substitute Te (Tm , Tg ) into the Hamiltonian
Td − rr ηr Tm − ηg−1ηf rf rg rgbTg
H(Tm, Tg ) = c0
+ c1
ηf rf rk
p
Voc − Voc2 − 4Rin(d0Tm2 + d1Tm + e0Tg + d2 + e1 + Pa)
−s
2RinQ
which is convex in Tg and Tm!
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Minimizing the Hamiltonian
ˆ The minimizing generator torque becomes
Tg∗(Tm) = arg min H(Tm, Tg )
Tg
=
Voc2 −
e0 ηg s 2
Qc0 rg
− 4Rin(d0Tm2 + d1Tm + d2 + e1 + Pa)
4Rine0
ˆ Substitute Tg∗ (Tm ) into H and minimize with respect to motor torque
Tm∗ = arg min H(Tm, Tg∗(Tm)) =
Tm
e0ηr rr ηg − d1ηf rf rg rgb
2d0ηf rf rg rgb
minimizing Tm∗ independent of equivalence factor and traction request!
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Minimizing the Hamiltonian
ˆ Plot optimal motor torque vs. vehicle speed and gear shifting sequence
- negative motor torque implies charging through the road
⇒ Unconstrained optimum always outside of the feasible set U
ˆ Constrained optimum lies along the boundary of the feasible set
- in practice the optimal solution is along edge with Tm = 0
⇒ if engine is on decision is how much to charge with generator
Optimal Traction Motor Torque vs. Speed and Gear
0
Level curve of analytic solution
6
Generator
Torque
−50
3
Gear [−]
Tm [Nm]
Unconstrained
optimum
0
Motor torque
gear number
−100
0
25
50
75
Speed [km/h]
100
125
1
150
Feasible set of
control signals
Motor
Torque
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ECMS Implementation
Equivalence factor
- s = s0 - tan(xref-x)
Vehicle data
- gear ratios
- battery data
- efficiencies
- etc...
Vehicle states
- wheel speed
- current gear
- SoC
- etc...
Interpolate param.
- engine
- generator
- motor
- etc...
Torque
demand
Data bus
Engine Off Case
Engine On Case
- Tm given implicitly
- Tg = 0
- Te = 0
- Tm = 0
- Tg given by Eq.
- Te given implicitly
- Check constraints
- Check constraints
- Compute Joff
- Compute Jon
Compare the values of Jon and Joff
Engine on/off
Velocity
reference
+
Generator torque reference
Driver Torque
model demand
ECMS
Vehicle
states
Vehicle velocity
Engine
on/off
Vehicle
plant
Torque
reference
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Implementation in Simulink (VSim)
<Tem>
<c0>
<Tice>
<c1>
3
Optimal
Torques
ICE On
<Tisg>
<Tem>
<d0>
1
Coefficients
ICE, ISG, ERAD
Pbat
<d1>
<Tice>
1
ICE On
Data
<d2>
<e0>
<e1>
############
Battery Power
Eq. (44)
#############
<Tisg>
#############
J_on computation
Eq. (43)
#############
<c_em>
fuel cost
fuel cost
<Paux>
<c_f>
J_on
<Voc>
2
Other Parameters
<Rin>
###################
dSoCdt Computation
Eq. (12) in document
###################
<Q_bat>
<lambda>
u
k
n-D T(k
dSoC/dt
x
÷
eps
f
Prelookup Map1Dnp1
4
eq battey cost
-1
10^-6
2
equivalence factor lambda
1
<ICE_state>
Penalty to turn on the ICE
<State_sw_co>
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Simulations in VSim
ˆ Equivalence factor s adapted to track a linearly decreasing SoC-reference
ˆ Left gure, ECMS reduces fuel consumption with about 10%
ˆ Right gure, ECMS does not decrease fuel consumption
Hyzem Highway + FTP75
FTP75 + Hyzem Highway
Speed [km/h]
150
100
50
0
100
50
0
Discharge Trajectories
0
Discharge Trajectories
20
40
Distance [km]
60
Nominal strategy
ECMS
SoC
Nominal strategy
ECMS
SoC
Speed [km/h]
150
0
20
40
Distance [km]
60
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Vehicle tests
ˆ Controller code generated with TargetLink and tested in production PHEV
- test driving on public roads veries that the strategy works in practice
Speed profile of test drive
100
50
0
Logged SoC−estimate
SoC−reference
SoC−estimate
SoC [−]
Speed [km/h]
150
0
20
40
Distance [km]
60
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Vehicle tests
ˆ The ane generator model and the quadratic motor model gives good
approximations of the battery power
Estimated battery power engine on
Pb measured
Power [kW]
5
Pb estimate (ECMS)
0
−5
−10
2750
2770
2790
Time [s]
2810
Estimated battery power engine off
Power [kW]
30
20
10
0
1625
1700
1775
Time [s]
1850
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Conclusion
ˆ An optimized discharge can decrease fuel consumption with up to 10%
- reduction depends very much on the driving pattern
ˆ Analytic solutions can decrease computational demand signicantly
- code increases ECU RAM usage with 0.17kB and ROM with 4.2kB
- same solution can be used in Approximate Dynamic Programming
ˆ A route optimized system can be developed using existing technology
- precompution in smartphone app and/or using cloud computing
- no additional hardware required, low marginal cost to implement
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FTP75 + Hyzem Highway
Speed [km/h]
100
50
0
150
100
50
0
SoC [−]
SoC
SoC [−]
SoC
Engine On/Off State
Engine On/Off State
On
Off
Normalized Generator Torque
1
0
0
Nominal strategy
CDCS
Blended ECMS
20
40
Distance [km]
60
Relative Torque
On
Off
Relative Torque
Speed [km/h]
Hyzem Highway + FTP75
150
Normalized Generator Torque
1
0
0
Nominal strategy
CDCS
Blended ECMS
20
40
Distance [km]
60
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ECMS Implementation
ˆ Engine o case ⇒ Engine and generator torque zero, Te∗ = Tg∗ = 0
- motor torque given by traction demand, Tm∗ = g0(Td)
√
- Jo = −s
Voc −
∗)
Voc2 −4Rin Pb (Tm
2RinQ
ˆ Engine on case ⇒ Motor torque zero, Tm∗ = 0
- generator torque by derived equation, Tg∗ = g1(s)
- engine torque def. by traction dem. and generator, Te∗ = g2(Td, Tg∗)
√
- Jon =
c0Te∗
+ c1 − s
Voc −
Voc2 −4Rin Pb (Tg∗ )
2RinQ
ˆ Engine on/o is decided by comparing Jon and Jo
state = min{Jon, Jo}
ˆ Equivalence factor is adapted to track a linearly decreasing SoC-ref.
⇒ s = s0 + F (xref − x)
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