CE 191: Civil and Environmental EngineeringSystems Analysis
LEC 16 : Combined Design & Control ofa Fuel Cell Bus via Convex Programming
Work by Dr. Xiaosong Hu, Dr. Nikolce Murgovski, Prof. Bo EgardtSwedish Hybrid Vehicle Center
Chalmers University of Technology
Professor Scott MouraCivil & Environmental EngineeringUniversity of California, Berkeley
Fall 2014
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 1
Fuel Cell Hybrid Vehicles
AC Transit HyRoad Fuel Cell Bus
2016 Toyota Mirai
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 2
How Fuel Cells Work
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 3
Fuel Cell Hybrid Powertrain
Figure: Fuel cell hybrid powertrain. The vehicle is propelled by an electric machine(EM), which obtains energy from a fuel cell system (FCS), or an electric buffer(battery or supercapacitor). When EM operates as a generator, mechanical energyfrom the wheels is converted to (and stored as) electrical energy in the buffer.
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 4
Bus Line Velocity and Road Grade
Figure: Model of a bus line, expressed by demanded vehicle velocity and roadaltitude. The initial and final velocities and road altitudes, respectively, are equal,thus conserving kinetic and potential energy of the vehicle.
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 5
Research QuestionGiven a fixed bus line (i.e. velocity-road grade profile), optimize
Fuel cell & super capacitor component sizes
Energy management strategy for power-split
to minimize operating (hydrogen fuel) + component (FC + SC) costs
Unique FeaturesComponent sizes are static design variables (not time-varying)
Energy management strategy is a multi-stage control process(time-varying)
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 6
Optimization Framework
Figure: Optimization framework for simultaneous component sizing and energymanagement of a hybrid city bus. After user inputs are provided, the combinedoperational and components cost are minimized simultaneously, in order to obtainthe optimal power split control and sizes of powertrain components.
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 7
Optimization Problem
minimize Operation + Component Costsubject to: Driving cycle constraints,
Energy conversion and balance constraints,Buffer dynamics,Physical limits of components,...(For all time instances along the bus line).
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 8
Useful Properties of Convex Function
The function f is said to be concave if −f is convex.
An affine function h(x) = qx+ r is both concave and convex.
A quadratic function f(x) = qx2 + px+ r with domain f R is convex ifp ≥ 0.
A quadratic-over-linear function f(x, y) = x2/y withdom f =
{(x, y) ∈ R2 | y > 0
}is convex.
The geometric mean f(x, y) =√xy with dom f =
{(x, y) ∈ R2 | x, y ≥ 0
}is concave.
A nonnegative weighted sum f =∑
wifi, with wi ≥ 0, of convexfunctions fi, is a convex function.
A product f(x, y) = xy is generally not a convex function.
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 9
Longitudinal Vehicle Dynamics
Newton’s Second Law, Electric Machine Torque:
Tdem(sF, sB, t) =
(JV +m(sF, sB)
R2w
r2fg
)ωM(t) +
ρairAfcdR3w
2r3fg
ω2M(t)
+m(sF, sB)crRw
rfgg cosα(t) +m(sF, sB)
Rw
rfgg sinα(t)
Vehicle Mass:m(sF, sB) = m0 + sFmF + sBmB
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 10
Electric Motor
PM(TM, t) = a0(ωM) + a1(ωM) · TM(t) + a2(ωM) · T2M(t)
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 11
Fuel Cell
PFf (PFe, sF) = b0sF + b1 · PFe(t) + b2 ·P2Fe(t)
sF0 ≤ PFe(t) ≤ sFPFe,max
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 12
Supercapacitor
Energy Storage DynamicsEB(t) = −PB(t)
Resistive Losses
PB,loss(PB(t),EB(t)) =RC
2
P2B(t)
EB(t)
The SC energy level EB(t) is related to the number of SC cells sFn0 and
voltage V(t) according to EB(t) =CV2(t)
2 sBn0. Both pack energy, cell voltage,and electric current are limited according to
0 ≤ EB(t) ≤CV2
max
2sBn0
imin
√2n0
CEB(t)sB ≤ PB(t) ≤ imax
√2n0
CEB(t)sB
KEY STEP: Show these produce convex ineq. constraints, using convex fcnproperties
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 13
Objective Function
Minimize operational cost (consumed hydrogen) and component costs (FCand SC)
J(TM(t), PFe(t), sF, sB,EB(t), PB(t)) = wh
∫ tf
0PFf (PFe(t), sF)dt + wFsF + wBsB
subject to:
Longitudinal Vehicle Dynamics
Electric Motor Constraint Equations
Fuel Cell Constraint Equations
Supercapacity Constraint Equations
Must show:
Obj. Fcn. is convex w.r.t. design variables
Inequality Fcns are all convex w.r.t. design variables
Equality Fcns are all affine w.r.t. design variables
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 14
Objective Function
Minimize operational cost (consumed hydrogen) and component costs (FCand SC)
J(TM(t), PFe(t), sF, sB,EB(t), PB(t)) = wh
∫ tf
0PFf (PFe(t), sF)dt + wFsF + wBsB
subject to:
Longitudinal Vehicle Dynamics
Electric Motor Constraint Equations
Fuel Cell Constraint Equations
Supercapacity Constraint Equations
Must show:
Obj. Fcn. is convex w.r.t. design variables
Inequality Fcns are all convex w.r.t. design variables
Equality Fcns are all affine w.r.t. design variables
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 14
Objective Function
Minimize operational cost (consumed hydrogen) and component costs (FCand SC)
J(TM(t), PFe(t), sF, sB,EB(t), PB(t)) = wh
∫ tf
0PFf (PFe(t), sF)dt + wFsF + wBsB
subject to:
Longitudinal Vehicle Dynamics
Electric Motor Constraint Equations
Fuel Cell Constraint Equations
Supercapacity Constraint Equations
Must show:
Obj. Fcn. is convex w.r.t. design variables
Inequality Fcns are all convex w.r.t. design variables
Equality Fcns are all affine w.r.t. design variables
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 14
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 15
Optimal Results - I
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 16
Optimal Results - II
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 17
Summary
Optimize component sizes (design) and energy management (control)of fuel cell city bus
Enormously complicated problem
THE SECRET: convex model formulation
Solutions in less than 10sec→ rapid design iteration
This is just the beginning...
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 18
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 19
Additional Reading
[1]. X. Hu, L. Johannesson, N. Murgovski, B. Egardt. “Longevity-consciousdimensioning and power management of a hybrid energy storagesystem for a fuel cell hybrid electric bus”. Journal of Applied Energy,2014, doi:10.1016/j.apenergy.2014.05.013.
[2]. X. Hu, N. Murgovski, L. Johannesson, and B. Egardt, “Comparison ofthree electrochemical energy buffers applied to a hybrid bus powertrainwith simultaneous optimal sizing and energy management,” IEEETransactions on Intelligent Transportation Systems, vol. 15, no. 3, pp.1193Ð1205, June 2014. doi:10.1109/TITS.2013.2294675
[3]. B. Egardt, N. Murgovski, M. Pourabdollah, L. Johannesson.“Electromobility studies based on convex optimization: design andcontrol issues regarding vehicle electrification”. IEEE Control SystemsMagazine, vol. 34, no. 2, pp. 32-49, 2014,doi:10.1109/MCS.2013.2295709.
Prof. Moura | UC Berkeley CE 191 | LEC 16 - FCHV Design & Control Slide 20