Real-time energy management of theVolvo V60 PHEV based on a closed-form

minimization of the HamiltonianViktor Larsson1, Lars Johannesson1, Bo Egardt1

Andreas Karlsson2, Anders Lasson2

1Department of Signals and Systems, Chalmers University of Technology2Volvo Car Corporation

v

Sou ce: Vol o Cars

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Background� 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

� Some trips will exceed the electric range of the PHEV

- Gradual discharge can reduce fuel consumption

� Objective is to implement a strategy with controllable discharge rate

distance

So

C

Battery State of Charge vs Distance

CSCD

1

0

start end

electric driving range

distance

So

C

Battery State of Charge vs Distance

Gradual discharge

1

0

start end

High resistive losses

Electric conversion losses Lower electric losses

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Outline

� Energy management system

� Simpli�ed powertrain model

� Minimizing the Hamiltonian

� Implementation in Simulink

� Simulations & Vehicle tests

� Conclusions

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The energy management system� 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

Real-time controllerInstantaneous power request

Optimal control problem

Feedforward information

Setpoints - engine - motor - etc.

Energy management system

Vehicle states

Real-time level

Predictive level

Predicted driving

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The energy management system� The energy management problem is to minimize overall energy cost

J∗ = minu(·)

G(x(tf))︸ ︷︷ ︸cost to recharge

+

∫ tf

t0

g(u(t), t) dt︸ ︷︷ ︸cost for fuel

s.t. x(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� The real-time controller is based on ECMS1

- derived from the Pontryagin principle

- control at each sample is obtained by minimizing the Hamiltonian

u∗ = argminu∈U

H(x, u, s) = argminu∈U

{g(u)︸︷︷︸fuel rate

+s · f (x, u)︸ ︷︷ ︸dSoCdt

}

- s is the equivalence factor which depend on future driving conditions

� The ECMS-strategy is implemented in an ECU

- important with low computational and memory demand

⇒ minimize the Hamiltonian analytically

1Equivalent Consumption Minimization Strategy

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Simpli�ed powertrain model

� Equivalent circuit battery model, x = dSoCdt

= −Voc−√

V 2oc−4RinPb

2RinQ

� Transmission ratios r with e�ciency η (no dynamics)

� Engine fuel rate a�ne in torque, g = c0(ωe)Te + c1(ωe)

� Electrical power of the motor quadratic in torque

Pm = d0(ωm)T2m + d1(ωm)Tm + d2(ωm)

� Electrical power of the generator a�ne in torque

Pg = e0(ωg)Tg + e1(ωg), Tg ≤ 0

96 rad/s(ICE speed)

289rad/s(ICE speed)84 rad/s

293 rad/s26 rad/s

1152 rad/s

+ -

electricmotor

batteryengine

clutch

trans-mission

integrated starter generator

clutch

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Minimizing the Hamiltonian� With the simple powertrain model the Hamiltonian is given by

H(x, u, s) = g(u)︸︷︷︸fuel rate

+s · f (x, u)︸ ︷︷ ︸dSoCdt

= c0(ωe)Te + c1(ωe)− sVoc −

√V 2oc − 4RinPb

2RinQ

where the battery power is: Pb = d0T2m + d1Tm + d2 + e0Tg + e1 + Pa

� The torque balance equation is

Td︸︷︷︸traction request

= ηrrrTm︸ ︷︷ ︸motor torque

+ ηfrfrgb(Te +rgηgTg)︸ ︷︷ ︸

input torque to gearbox

� Assume engine is on with a �xed gear rgbcontrol 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� Solve torque balance equation for engine torque

Te(Tm, Tg) =Td − rrηrTm − η−1g ηfrfrgrgbTg

ηfrfrk

two independent control variables ⇒ u = [Tm Tg]

� Substitute Te(Tm, Tg) into the Hamiltonian

H(Tm, Tg) = c0Td − rrηrTm − η−1g ηfrfrgrgbTg

ηfrfrk+ c1

− sVoc −√V 2oc − 4Rin(d0T 2

m + d1Tm + e0Tg + d2 + e1 + Pa)

2RinQ

which is convex in Tg and Tm!

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Minimizing the Hamiltonian� The minimizing generator torque becomes

T ∗g (Tm) = argminTg

H(Tm, Tg)

=V 2oc −

( e0ηgsQc0rg

)2 − 4Rin(d0T2m + d1Tm + d2 + e1 + Pa)

4Rine0

� Substitute T ∗g (Tm) into H and minimize with respect to motor torque

T ∗m = argminTm

H(Tm, T∗g (Tm)) =

e0ηrrrηg − d1ηfrfrgrgb2d0ηfrfrgrgb

minimizing T ∗m independent of equivalence factor and traction request!

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Minimizing the Hamiltonian� 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

� 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

0 50 100 150

−100

−50

0

Tm

[Nm

]

Speed [km/h]

Optimal Traction Motor Torque vs. Speed and Gear

0 25 50 75 100 125 1501

3

6

Gea

r [−

]

Motor torquegear number

Feasible set ofcontrol signals

Generator Torque

Motor Torque

Level curve of analytic solution

Unconstrained optimum

0

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ECMS ImplementationVehicle data- gear ratios- battery data- efficiencies- etc...

Vehicle states- wheel speed- current gear- SoC - etc...

Interpolate param.- engine- generator - motor- etc...

- Tm given implicitly - Tg = 0- Te = 0

Engine Off Case

- Check constraints

- Compute Joff

- Tm = 0 - Tg given by Eq.- Te given implicitly

Engine On Case

- Check constraints

- Compute Jon

Torquedemand

Compare the values of Jon and Joff

Generator torque reference

+-

Velocityreference Driver

model

Torque demand ECMS

Vehicle plant

Vehicle states

Vehicle velocity

Engine on/off

Torquereference

Engine on/off

Data bus

Equivalence factor- s = s0 - tan(xref )-x

Compare the values of Jon and Joff

- including extra cost to change engine state

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Implementation in Simulink (VSim)

fuel cost

Pbat

dSoC/dt

Penalty to turn on the ICE

fuel cost

eq battey cost###################

dSoCdt ComputationEq. (12) in document

###################

############Battery Power

Eq. (44)#############

#############J_on computation

Eq. (43)#############

equivalence factor lambda

ICE OnData

1

x÷epsPrelookup

u k

f

Map1Dnp1

n-D T(k

-1

2

4

1

10^-6

Optimal TorquesICE On

3

Other Parameters2

Coefficients ICE, ISG, ERAD

1

<c0>

<c1>

<c_f>

<e0>

<d2>

<c_em>

<Voc>

<Rin>

<Q_bat>

<lambda>

<Tice>

<Tice>

<Tisg>

<Tisg>

<ICE_state>

<State_sw_co>

J_on

<Tem>

<Tem>

<d0>

<d1>

<Paux>

<e1>

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Simulations in VSim� Equivalence factor s adapted to track a linearly decreasing SoC-reference

� Left �gure, ECMS reduces fuel consumption with about 10%

� Right �gure, ECMS does not decrease fuel consumption

0

50

100

150Hyzem Highway + FTP75

Spe

ed [k

m/h

]

0 20 40 60

Discharge Trajectories

SoC

Distance [km]

Nominal strategyECMS

0

50

100

150FTP75 + Hyzem Highway

Spe

ed [k

m/h

]

0 20 40 60

Discharge Trajectories

SoC

Distance [km]

Nominal strategyECMS

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Vehicle tests� Controller code generated with TargetLink and tested in production PHEV

- test driving on public roads veri�es that the strategy works in practice

0

50

100

150Speed profile of test drive

Spe

ed [k

m/h

]

0 20 40 60Distance [km]

SoC

[−]

Logged SoC−estimate

SoC−referenceSoC−estimate

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Vehicle tests� The a�ne generator model and the quadratic motor model gives goodapproximations of the battery power

2750 2770 2790 2810

−10

−5

0

5

Pow

er [k

W]

Time [s]

Estimated battery power engine on

1625 1700 1775 1850

0

10

20

30

Pow

er [k

W]

Time [s]

Estimated battery power engine off

Pb measured

Pb estimate (ECMS)

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Conclusion� An optimized discharge can decrease fuel consumption with up to 10%

- reduction depends very much on the driving pattern

� Analytic solutions can decrease computational demand signi�cantly

- code increases ECU RAM usage with 0.17kB and ROM with 4.2kB

- same solution can be used in Approximate Dynamic Programming

� 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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AcknowledgmentsS

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0

50

100

150Hyzem Highway + FTP75

Spe

ed [k

m/h

]

SoC

SoC

[−]

OffOn

Engine On/Off State

0 20 40 60

01

Normalized Generator Torque

Rel

ativ

e T

orqu

e

Distance [km]

Nominal strategyCDCSBlended ECMS

0

50

100

150FTP75 + Hyzem Highway

Spe

ed [k

m/h

]

SoC

SoC

[−]

OffOn

Engine On/Off State

0 20 40 60

01

Normalized Generator Torque

Rel

ativ

e T

orqu

e

Distance [km]

Nominal strategyCDCSBlended ECMS

ECMS Implementation� Engine o� case ⇒ Engine and generator torque zero, T ∗e = T ∗g = 0

- motor torque given by traction demand, T ∗m = g0(Td)

- Jo� = −sVoc−√

V 2oc−4RinPb(T ∗

m)

2RinQ

� Engine on case ⇒ Motor torque zero, T ∗m = 0

- generator torque by derived equation, T ∗g = g1(s)

- engine torque def. by traction dem. and generator, T ∗e = g2(Td, T∗g )

- Jon = c0T∗e + c1 − s

Voc−√

V 2oc−4RinPb(T ∗

g )

2RinQ

� Engine on/o� is decided by comparing Jon and Jo�

state = min{Jon, Jo�}� Equivalence factor is adapted to track a linearly decreasing SoC-ref.

⇒ s = s0 + F (xref − x)

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