Time-optimal Energy Management of the Formula Power Unit Mauro Salazar Dr. Philipp Elbert Zürich, 5.2.27 Prof. Dr. Chris Onder
The 24-22 Power Unit The Power Unit in the Chassis 2
The 24-22 Power Unit Comparison 3
The 24-22 Power Unit 5
Minimise Lap Time Before KERS Min Lap Time Max Power Hybrid Formula Min Lap Time? Optimal Power Every tenth of a second counts! 6
The 24-22 Power Unit Comparison 23-24: downsizing, turbocharging and hybridization Engine: 29 23 24 Displacement 2.4 L.6 L Air intake Nat. asp. Forced Limit rpm 8 3 Cylinders V8 V6 Fuel kg Max. flow rate kg /h Energy recovery system: Power 6 kw 2 kw Energy storage.4 MJ 4. MJ 7
The 23 Power Unit Energy storage system:.4 MJ Motor-generator unit kinetic (MGU-K): ±6 kw ES K Wheel B Hydraulic brakes E Nat. aspirated 2.4 L V8 engine 8
The 24-22 Power Unit Energy storage system: 4 MJ ES to K: 4 MJ/lap Motor-generator unit kinetic (MGU-K): ±2 kw ES K K to ES: -2 MJ/lap Wheel H Motor-generator unit heat (MGU-H): ±6 kw B Hydraulic brakes E Fuel consumption Max: kg/race Turbocharged.6 L V6 engine 9
Time-optimal Control of the Formula Power Unit Battery + - E-Motor Limited H C Turbo K Gearbox www.formula.com < % Power Split = % Thrust and Power Split E-Motor MGU-K T Tank 5kg Engine ICE Optimal Control Policy? Preq Speed Driver 5 5 2 25 s [m] 3 35 4 45
Constrained Optimization Problem Optimal Control Problem: Minimise Lap Time 3 Velocity (km/h) 25 2 5 T lap = 39.2s Max. velocity velocity 5 2 4 6 8 2 4 6 8 2 Is there a better MGU-K actuation using the same amount of electric energy? Power (kw) 6 4 2 2 4 Boost Engine Brake Motor min T lap =min Z S v ds 6 2 4 6 8 2 4 6 8 2 Distance (m) Engine back-off and motor braking Distribute energy over straight
Constrained Optimization Problem Optimal Control Problem: Minimise Lap Time 3 Velocity (km/h) 25 2 5 T lap = 38.9s Max. velocity Velocity 5 2 4 6 8 2 4 6 8 2 Power (kw) 6 4 2 2 4 Engine Brake Motor 6 2 4 6 8 2 4 6 8 2 Distance (m) Thermal recharge Boost Motor back-off Engine back-off and motor braking Deploy all energy as soon as possible at highest power 2
The 24-22 Power Unit Powertrain Model Energy storage system: 4 MJ ES to K: 4 MJ/lap Motor-generator unit kinetic (MGU-K): ±2 kw ES K K to ES: -2 MJ/lap Wheel H Motor-generator unit heat (MGU-H): ±6 kw B Hydraulic brakes E Fuel consumption Max: kg/race Turbocharged.6 L V6 engine 3
Pi/Pi, Motor Generator Unit - Kinetic.5 DC Motor Model U dc dc = R I + U k P k,dc dc k,dc = U dc I P k,loss k,loss = R I 2 P k =! k T k = U k I P k,dc k,loss k,dc = P k,loss + P k P k,dc k,dc = R (apple! k ) 2 Pk 2 + P k {z } k Approximation P k,dc k,dc = k Pk 2 + P k U a U dc L = L a I a R a U k U i T k T a! k ω m er armature field shaft Pk,dc/Pk,dc,.5.5.5 Measured Model.5.5.5.5.5 P k /P k, Fig. 9: A quadratic model of the losses in the MGU-K was identified. The normalized mean error is 2.4%, whereas the T m 4.5 P b /P b, Fig. : A quadratic model of the losse was identified. The normalized mean the outliers reflect the neglected ope Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal measurement Control Strategies noise. for a Hybrid Electric Race Car, IEEE TCST, 27 The internal battery power P i (s), resp change in the state of energy E b (s), is P i (s)=a b P b (s) 2 + P Figure shows the quadratic fit to mea outliers stem from the dynamics of the for by the model. Equation (4) can be expressed in turn by means of a second-order conic dt ds (s)+f i(s) F b (s) dt ds (s) 2 pa As for (22) and (37), the equality in inequality, which will hold with equalit is required to minimize the lap time. battery state of energy E b (s) are mode in the energy level DE b (s)=e b (s) E dde b ds (s)= F i F i(s)
Battery Open Circuit Voltage Model U b = R I + U oc Measured Model P b = U b I P b,loss = R I 2 P i = U oc I P b = P b,loss + P i Pi/Pi,.5.5 P b = R U 2 oc P 2 i + P i.5.5 P b /P b, Approximation P i = b P 2 b + P b Fig. : A quadratic model of the losses in the energy storage was identified. The normalized mean error is 7.4%, whereas the outliers reflect the neglected open circuit voltage and measurement noise. Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 The internal battery power P i (s), responsible for the actual change in the state of energy E b (s), is approximated by 5
Internal Combustion Engine with Turbocharger Willans Approximation and Algebraic TC Operation P e = e P f P e, P h = h P f.2 2.8 Measured Model Ph/Ph, Measured Model Pe/Pe,.6.4 2.2.2.4.6.8.2 P f /P f,.2 Likelihood.5..5.2.2.4.6.8.2.2.4.6.8.2 P f /P f, P f /P f, Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 6
The 24-22 Power Unit Powertrain Model Energy storage system: 4 MJ ES to K: 4 MJ/lap Motor-generator unit kinetic (MGU-K): ±2 kw ES K K to ES: -2 MJ/lap Wheel H Motor-generator unit heat (MGU-H): ±6 kw B Hydraulic brakes E Fuel consumption Max: kg/race Turbocharged.6 L V6 engine 7
Optimal Control Problem Longitudinal Dynamics Minimise Lap-time Constraints x = B @ ẋ = f(x, u) Speed Fuel SOC C u = ES2K A K2ES ICE MGU-K min u Z T Subject to dt Stage Max Speed Max Power Terminal Fuel Load SOC ES2K apple 4MJ K2ES 2MJ Vehicle Dynamics Speed 5 5 2 25 3 35 4 45 s [m] 8
Optimal Control Problem Maximum Optimum v [-].5 5 5 2 25 3 35 4 45 s [m] Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 Pe, Pk [-].5 ICE MGU-K mf,eb [kg, MJ].5 3 2 5 5 2 25 3 35 4 45 s [m] s [m] Fuel Battery 5 5 2 25 3 35 4 45 9
Optimal Control Problem Eb,min (MJ) 4 3 2 2 3 4 2 3 4 Distance (km).8.6.4.2 -.2 -.4 -.6 -.8 - Normalized MGU-K Power Fuel (%) Fuel (%) 5 95 2 3 4 5 95.5.6.7.8.9 Distance (km).8.6.4.2 Normalized Fuel Power.8.6.4.2 Normalized Fuel Power D Fig. 9: The optimal fuel power P s (normalized) over a Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 2
Optimal Control Problem.8.8.6.7.4.6.2.5 T (s).8.6 T (s).4.3.4.2.2..2 4 3 2 2 3 4 E b,min (MJ). 95 5 Fuel (%) D Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 2
Time-optimal Control of the Formula Power Unit Battery + - Limited Flows E-Motor H C Turbo K Gearbox www.formula.com < % Power Split = % Thrust and Power Split E-Motor MGU-K T Tank 5kg Engine ICE Optimal Control Policy? Preq Speed Driver 5 5 2 25 s [m] 3 35 4 45 22
Optimal Control Problem Longitudinal Dynamics Minimise Lap-time Constraints x = B @ ẋ = f(x, u) Speed Fuel SOC C u = ES2K A K2ES ICE MGU-K min u Z T Subject to dt Stage Max Speed Max Power Terminal Fuel Load SOC ES2K apple 4MJ K2ES 2MJ Numerical Convex Optimization Vehicle Dynamics Non-smooth Pontryagin Minimum Principle v [-] Pe, Pk [-] mf,eb [kg, MJ].5 Maximum Optimum 5 5 2 25 3 35 4 45 s [m].5 ICE MGU-K.5 5 5 2 25 3 35 4 45 s [m] 3 2 Implementation? Fuel Battery 5 5 2 25 3 35 4 45 s [m] J (,x)=min u2u Z T g(x, u)dt + h(x(t )) s.t. ẋ(t) =f(x, u) 8t 2 [,T] x() = x x 2 X =[a, b]! g(x, u)+ [a,b](x) H(x, u, )=g(x, u)+ T f(x, u) Ψ [a,b] (x) a b x 2 @ u H(x,u, )=Q(u, ) Ebbesen, Salazar, Elbert, Bussi and Onder: Time-optimal Control Strategies for a Hybrid Electric Race Car, IEEE TCST, 27 Combine Salazar, Elbert, Ebbesen, Bussi and Onder: Time-optimal Control Policy for a Hybrid Electric Race Car, IEEE TCST, 27 23
Implementation and Results.8 Implement the optimal control policy with simple look-up tables: interpolations!.2 = % P req < % Lap Time VS Consumption Thrust and Power Split Fuel VS Battery v [-] Total Power Kinetic Costate λ [-] Pu [-].6.4 2 3 4 s [m].5.5.2 2 3 4 s [m].8.6.4.2.2.4.2.2.4.6.8.2.4 λ Kinetic Costate Variable Speed 5 5 2 25 3 35 4 45 s [m] Power Split.8 ICE MGU-K.6 Salazar, Elbert, Ebbesen, Bussi and Onder: Time-optimal Control Policy for a Hybrid Electric Race Car, IEEE TCST, 27 Power Split Pe, Pk [-].4.2.2.4.2.2.4.6.8.2 P u [-] Total Power P? e P? k 24
Results One Race Lap in Barcelona t [ms] Power [-] Speed [kmph] 4 2.5 5 5 ICE MGU-K Position [m] Feedforward Optimal 5 5 2 25 3 35 4 45 Salazar, Bussi, Grando and Onder: Optimal Control Policy Tuning and Implementation for a Hybrid Electric Race Car, IFAC AAC 26 25
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