Real-time energy management of the Volvo V6 PHEV based on a closed-form minimization of the Hamiltonian Viktor Larsson 1, Lars Johannesson 1, Bo Egardt 1 Andreas Karlsson 2, Anders Lasson 2 1 Department of Signals and Systems, Chalmers University of Technology 2 Volvo Car Corporation Source: Volvo Cars
Background ˆ The nominal strategy in the V6 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 1 Battery State of Charge vs Distance 1 Battery State of Charge vs Distance High resistive losses Electric conversion losses Lower electric losses SoC CD CS SoC Gradual discharge electric driving range start distance end start distance end
Outline ˆ Energy management system ˆ Simplied powertrain model ˆ Minimizing the Hamiltonian ˆ Implementation in Simulink ˆ Simulations & Vehicle tests ˆ Conclusions
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.
The energy management system ˆ The energy management problem is to minimize overall energy cost J = min u( ) G(x(t f )) + }{{} cost to recharge tf g(u(t), t) dt t } {{ } cost for fuel s.t. ẋ(t) = f(x(t), u(t), t) x(t ) = x 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,...)
The energy management system ˆ The real-time controller is based on ECMS 1 - 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) }{{} fuel rate +s f(x, u) }{{} 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
Simplied powertrain model ˆ Equivalent circuit battery model, ẋ = dsoc dt = V oc ˆ Transmission ratios r with eciency η (no dynamics) ˆ Engine fuel rate ane in torque, g = c (ω e )T e + c 1 (ω e ) ˆ Electrical power of the motor quadratic in torque P m = d (ω m )T 2 m + d 1 (ω m )T m + d 2 (ω m ) ˆ Electrical power of the generator ane in torque P g = e (ω g )T g + e 1 (ω g ), T g V 2 oc 4R in P b 2RinQ + - 293 rad/s 26 rad/s 96 rad/s (ICE speed) battery engine transmission 84 rad/s 1152 rad/s 289rad/s (ICE speed) clutch electric motor integrated starter generator clutch
Minimizing the Hamiltonian ˆ With the simple powertrain model the Hamiltonian is given by H(x, u, s) = g(u) }{{} fuel rate +s f(x, u) }{{} dsoc dt = c (ω e )T e + c 1 (ω e ) s V oc V 2 oc 4R in P b 2R in Q where the battery power is: P b = d T 2 m + d 1 T m + d 2 + e T g + e 1 + P a ˆ The torque balance equation is T d }{{} traction request = η r r r T m }{{} motor torque + η f r f r gb (T e + r g η g T g ) }{{} input torque to gearbox ˆ Assume engine is on with a xed gear r gb control variables: engine/motor/generator torque { T e T m T g } two degrees of freedom in meeting the traction request T d
Minimizing the Hamiltonian ˆ Solve torque balance equation for engine torque T e (T m, T g ) = T d r r η r T m ηg 1 η f r f r g r gb T g η f r f r k two independent control variables u = [T m T g ] ˆ Substitute T e (T m, T g ) into the Hamiltonian T d r r η r T m ηg 1 η f r f r g r gb T g H(T m, T g ) = c + c 1 η f r f r k s V oc V 2 oc 4R in (d T 2 m + d 1 T m + e T g + d 2 + e 1 + P a ) 2R in Q which is convex in T g and T m!
Minimizing the Hamiltonian ˆ The minimizing generator torque becomes T g (T m ) = arg min T g H(T m, T g ) = V 2 oc ( e η g s Qc r g ) 2 4R in (d T 2 m + d 1 T m + d 2 + e 1 + P a ) 4R in e ˆ Substitute T g (T m ) into H and minimize with respect to motor torque T m = arg min T m H(T m, T g (T m )) = e η r r r η g d 1 η f r f r g r gb 2d η f r f r g r gb minimizing T m independent of equivalence factor and traction request!
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 T m = if engine is on decision is how much to charge with generator T m [Nm] 5 1 Optimal Traction Motor Torque vs. Speed and Gear Motor torque gear number 25 5 75 1 125 15 1 Speed [km/h] 6 3 Gear [ ] Level curve of analytic solution Unconstrained optimum Generator Torque Feasible set of control signals Motor Torque
ECMS Implementation Equivalence factor - s = s - 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 - Tm given implicitly - Tg = - Te = - Check constraints - Compute Joff Engine On Case - Tm = - Tg given by Eq. - Te given implicitly - Check constraints - Compute Jon Compare the values of Jon and Joff Engine on/off Generator torque reference Velocity reference + - Driver model Torque demand ECMS Vehicle states Engine on/off Torque reference Vehicle velocity Vehicle plant
Implementation in Simulink (VSim) <Tem> 3 Optimal Torques ICE On <Tice> <Tisg> <Tem> 1 Coefficients ICE, ISG, ERAD <c> <c1> <d> <d1> <d2> <e> Pbat <Tice> 1 ICE On Data <e1> <Tisg> ############ Battery Power Eq. (44) ############# <Paux> <c_em> fuel cost fuel cost ############# J_on computation Eq. (43) ############# 2 Other Parameters <c_f> <Voc> <Rin> <Q_bat> <lambda> 4 u n-d T(k k f Prelookup Map1Dnp1 ################### dsocdt Computation Eq. (12) in document ################### x eps dsoc/dt -1 eq battey cost J_on 2 1^-6 equivalence factor lambda 1 <ICE_state> Penalty to turn on the ICE <State_sw_co>
Simulations in VSim ˆ Equivalence factor s adapted to track a linearly decreasing SoC-reference ˆ Left gure, ECMS reduces fuel consumption with about 1% ˆ Right gure, ECMS does not decrease fuel consumption 15 Hyzem Highway + FTP75 15 FTP75 + Hyzem Highway Speed [km/h] 1 5 Speed [km/h] 1 5 Discharge Trajectories Discharge Trajectories SoC Nominal strategy ECMS SoC Nominal strategy ECMS 2 4 6 Distance [km] 2 4 6 Distance [km]
E TR LE CEN 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 Speed profile of test drive 1 5 Logged SoC estimate SoC reference SoC estimate SoC [ ] Speed [km/h] 15 2 4 Distance [km] 6 B D VEH E SW H Y IC ISH RI D
Vehicle tests ˆ The ane generator model and the quadratic motor model gives good approximations of the battery power Estimated battery power engine on Power [kw] 5 5 1 P b measured P b estimate (ECMS) 275 277 279 281 Time [s] Estimated battery power engine off 3 Power [kw] 2 1 1625 17 1775 185 Time [s]
Conclusion ˆ An optimized discharge can decrease fuel consumption with up to 1% - reduction depends very much on the driving pattern ˆ Analytic solutions can decrease computational demand signicantly - code increases ECU RAM usage with.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
D E W S H S I R B Y H C E E N R T Acknowledgments
Speed [km/h] 15 1 5 Hyzem Highway + FTP75 Speed [km/h] 15 1 5 FTP75 + Hyzem Highway SoC SoC SoC [ ] SoC [ ] Engine On/Off State Engine On/Off State On Off On Off Relative Torque 1 Normalized Generator Torque Nominal strategy CDCS Blended ECMS 2 4 6 Distance [km] Relative Torque 1 Normalized Generator Torque Nominal strategy CDCS Blended ECMS 2 4 6 Distance [km]
ECMS Implementation ˆ Engine o case Engine and generator torque zero, Te = Tg = - motor torque given by traction demand, Tm = g (T d ) - J o = s V oc V 2 oc 4R in P b (T m) 2RinQ ˆ Engine on case Motor torque zero, T m = - generator torque by derived equation, T g = g 1 (s) - engine torque def. by traction dem. and generator, T e = g 2 (T d, T g ) - J on = c T e + c 1 s V oc V 2 oc 4R in P b (T g ) 2RinQ ˆ Engine on/o is decided by comparing J on and J o state = min{j on, J o } ˆ Equivalence factor is adapted to track a linearly decreasing SoC-ref. s = s + F (x ref x)