Real-time energy management of the Volvo V60 PHEV based on a closed-form minimization of the Hamiltonian

Transcription

1 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

2 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

3 Outline ˆ Energy management system ˆ Simplied powertrain model ˆ Minimizing the Hamiltonian ˆ Implementation in Simulink ˆ Simulations & Vehicle tests ˆ Conclusions

4 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.

5 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,...)

6 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

7 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 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

8 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

9 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!

10 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!

11 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 Speed [km/h] 6 3 Gear [ ] Level curve of analytic solution Unconstrained optimum Generator Torque Feasible set of control signals Motor Torque

12 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

13 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>

14 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 Distance [km] Distance [km]

15 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] Distance [km] 6 B D VEH E SW H Y IC ISH RI D

16 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] P b measured P b estimate (ECMS) Time [s] Estimated battery power engine off 3 Power [kw] Time [s]

17 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

18 D E W S H S I R B Y H C E E N R T Acknowledgments

19 Speed [km/h] Hyzem Highway + FTP75 Speed [km/h] 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 Distance [km] Relative Torque 1 Normalized Generator Torque Nominal strategy CDCS Blended ECMS Distance [km]

20 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)