Outline
Vehicle Propulsion Systems
Lecture 09
Repetition
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Case Study 6 Fuel Cell Vehicle and Optimal Control
Model simplification
Lars Eriksson
Associate Professor (Docent)
Formulating the optimal control problem
Vehicular Systems
Linköping University
Optimal Controllers Solutions
December 2, 2012
Some Additional Material – Fuel Consumption
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The Vehicle Motion Equation
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Gravitational Force
Newtons second law for a vehicle
mv
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d
v (t) = Ft (t) − (Fa (t) + Fr (t) + Fg (t) + Fd (t))
dt
Gravitational load force
–Not a loss, storage of potential energy
Fa
Fd
Fa
Ft
Fd
Fg
Ft
Fr
Fg
mv · g
Fr
α
mv · g
α
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Ft – tractive force
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Fa – aerodynamic drag force
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Fr – rolling resistance force
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Fg – gravitational force
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Fd – disturbance force
Up- and down-hill driving produces forces.
Fg = mv g sin(α)
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Flat road assumed α = 0 if nothing else is stated (In the
book).
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Deterministic Dynamic Programming – Basic algorithm
J(x0 ) = gN (xN ) +
N−1
X
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Examples of Short Term Storage Systems
gk (xk , uk )
k=0
xk+1 = fk (xk , uk )
Algorithm idea:
Start at the end and proceed backward in time to evaluate the
optimal cost-to-go and the corresponding control signal
x
k= 0
1
2
ta
N −1
N
t
tb
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Heuristic Control Approaches
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Fuel Cell Basic Principles
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Parallel hybrid vehicle (electric assist)
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Convert fuel directly to electrical energy
Let an ion pass from an anode to a cathode
Take out electrical work from the electrons
Fuel cells are stacked (Ucell ≤ 1V)
Determine control output as function of some selected
state variables:
vehicle speed, engine speed, state of charge, power
demand, motor speed, temperature, vehicle acceleration,
torque demand
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Overview of Different Fuel Cell Technologies
Hydrogen Fuel Storage
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Hydrogen storage is problematic – Challenging task.
Some examples of different options.
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High pressure bottles
Liquid phase – Cryogenic storage, -253◦ C.
Metal hydride
Sodium borohydride NaBH4
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Quasistatic Modeling of a Fuel Cell
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Fuel Cell Thermodynamics
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Starting point reaction equation
H2 +
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Causality diagram
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1
O2 ⇒ 2 H2 0
2
Open system energy – Enthalpy H
H = U + pV
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Reversible energy – Gibbs free energy G
G = H + TS
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Power amplifier (Current controller)
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Fuel amplifier (Fuel controller)
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Standard modeling approach
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Open circuit cell voltages
Urev = −
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∆G
,
ne F
Uid = −
∆H
,
ne F
Urev = ηid Uid
F – Faradays constant (F = q N0 )
Under load
Pl = Ifc (t) (Uid − Ufc (t))
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Fuel Cell Performance – Polarization curve
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Fuel Cell System Modeling
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Polarization curve of a fuel cell
Relating current density ifc (t) = Ifc (t)/Afc , and cell voltage
Ufc (t)
Describe all subsystems with models
P2 (t) = Pst (t) − Paux (t)
Paux = P0 + Pem (t) + Pahp (t) + Php (t) + Pcl (t) + Pcf (t)
em–electric motor, ahp – humidifier pump, hp – hydrogen
recirculation pump, cl – coolant pump, cf – cooling fan.
Curve for one operating condition
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Fundamentally different compared to combustion
engine/electrical motor
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Excellent part load behavior
–When considering only the cell
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Submodels for: Hydrogen circuit, air circuit, water circuit, and
coolant circuit
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Outline
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Problem Setup
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Run a fuel cell vehicle optimally on a racetrack
Repetition
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Model simplification
Formulating the optimal control problem
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Optimal Controllers Solutions
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Some Additional Material – Fuel Consumption
Path to the solution
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Start up lap
Repeated runs on the track
Measurements – Model
Simplified model
Optimal control solutions
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Problem Setup – Road Slope Given
Model Causality
Road slope γ = α(x)
Model causality – Dynamic model
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Model Component – Fuel Cell
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Fuel Cell – Polarization and Auxiliary Components
Current in the cell and losses
Ifc (t) = Ifc (t) + Iaux (t)
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Current and hydrogen flow
ṁH (t) = c9 Ifc (t)
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Next step: Polarization curve and auxiliary consumption
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Fuel Cell – Polarization and Auxiliary Components
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Model Component – DC Motor Controller
Polarization curve
Ust (t) = c0 + c1 · e−c2 ·Ifc (t) − c3 · Ifc (t)
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Auxiliary power
Paux (t) = c6 + c7 · Ifc (t) + c8 · Ifc (t)2
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DC motor voltage (from control signal u)
Uem (t) = κ ωem (t) + K Rem u(t)
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Current requirement at the stack
Ist =
Uem (t)Iem (t)
ηc Ust (t)
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Model Component – DC Motor
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DC motor current
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Iem (t) =
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Model Component – Transmission and Wheels
Tractive force
Uem (t) − κ ωem (t)
Rem
F (t) = ηt±1
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DC motor torque
Rotational speed
ωem (t) =
Tem (t) = κem Iem (t)
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γ Tem (t)
rw
γ v (t)
rw
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Model Compilation 1 – Vehicle
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Model Compilation 2 – Fuel Consumption
The vehicle tractive force can now be expressed as
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ηt γ
κem K u(t)
F (t) =
rw
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Fuel flow, ṁH (t) = c9 Ifc (t)
Ifc (t) =
K u(t)
Paux (Ist (t))
+
Ust (Ist (t)) ηc Ust (Ist (t))
K Rem u(t) + κem
γ
v (t)
rw
Dynamic vehicle velocity and position model
–Implicit nonlinear static function
d
v (t) =h1 u(t) − h2 v 2 (t) − g0 − g1 α(x(t))
dt
d
x(t) =v (t)
dt
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Simpler model
ṁH (t) = b0 + b1 v (t) u(t) + b2 u 2 (t)
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Outline
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Simplified Fuel Consumption – Validation
Repetition
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Model simplification
Formulating the optimal control problem
Optimal Controllers Solutions
Some Additional Material – Fuel Consumption
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Detour
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Outline
Repetition
Occam’s razor:
–The explanation of any phenomenon should make as few
assumptions as possible.
Shave of those who are unnecessary.
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Law of Parsimony: Among others a factor in statistics: In
general, mathematical models with the smallest number of
parameters are preferred as each parameter introduced
into the model adds some uncertainty to it.
Model simplification
Formulating the optimal control problem
Another viewpoint.
Try to simplify the problem you solve as much as possible.
–Neglect effects and be proud when you are successful!
Optimal Controllers Solutions
Some Additional Material – Fuel Consumption
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Optimal Control Problems
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PID Cruise Controller – Baseline for Comparison
Start of the cycle
v (0) = 0,
Simple controller for the start
x(0) = 0
Z
λ1 (tf ) = 0,
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u(t) = Kp (f vm − v (t)) + Ki
x(tf ) = xf = vm tf
Periodic route
x(tf ) = xf = vm tf ,
(f vm − vt (t))dt
f -tuning parameter ≈ 1 to allow for matching the average speed
x(0) = 0
λ1 (tf ) = λ1 (0),
t
0
v (tf ) = v (0)
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Outline
Problem Setup – Road Slope Given
Repetition
Road slope γ = α(x)
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Model simplification
Formulating the optimal control problem
Optimal Controllers Solutions
Some Additional Material – Fuel Consumption
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Fuel Optimal Trajectory – Start
Fuel Optimal Trajectory – Continuous Driving
Fuel optimal trajectory has 7% lower fuel consumption
Fuel optimal trajectory has 9% lower fuel consumption
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Outline
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Fuel Optimal Speed for Normal Driving
ICE vehicle (light weight 800 kg)
Repetition
Case study 6: Fuel Optimal Trajectories of a Racing FCEV
Model compilation
Model simplification
Formulating the optimal control problem
Optimal Controllers Solutions
Some Additional Material – Fuel Consumption
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Engine Map and Gearbox Layout
CI engine (light weight 800 kg)
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