On the use of Dynamic Programming in eco-driving cycle computation for electric vehicles

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1 On the se of Dynamic Programming in eco-driving cycle comptation for electric vehicles D Maamria, Kristan Gillet, Gillame Colin, C Noillant, Yann Chamaillard To cite this version: D Maamria, Kristan Gillet, Gillame Colin, C Noillant, Yann Chamaillard. On the se of Dynamic Programming in eco-driving cycle comptation for electric vehicles. 216 IEEE Conference on Control Applications (CCA), Sep 216, Benos Aires, Argentina. pp , 216, Proceedings of the 216 IEEE Conference on Control Applications (CCA). < <1.119/CCA >. <hal > HAL Id: hal Sbmitted on 21 Jl 217 HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 On the se of Dynamic Programming in eco-driving cycle comptation for electric vehicles D. Maamria a, K. Gillet a, G. Colin a, Y. Chamaillard a and C. Noillant b Abstract This paper considers the problem of eco-driving for electric cars. This problem is formlated as an Optimal Control Problem (OCP) aiming at minimizing the vehicle s energy consmption over fixed time and distance horizons. The impact of battery parameter variations and axiliary power demands on the optimal vehicle velocity comptation are stdied from a model complexity viewpoint. Simlation reslts are presented and discssed to illstrate the sggested simplifications. I. INTRODUCTION The expected depletion of fossil fel sorces, climate change de to polltion and an increase in overall energy demands are major challenges for the atomotive indstry. More generally, energy efficiency is increasingly becoming a major concern. For this prpose, highly efficient powertrain and lightweight atomobiles are being developed. Frthermore, eco-driving is now considered as a major soltion to redce the energy consmption linked to transportation system. In this context, several stdies investigating the problem of the vehicle speed trajectory optimization have been reported [1] [4]. In these stdies, an Optimal Control Problem (OCP) to determine the velocity trajectory that minimizes the energy consmption nder final time and distance constraints was formlated and solved. In particlar, the eco-driving problem for electric vehicles was addressed in [1] [3]. Usally, two state variables are considered in the OCP: the position and the speed of the vehicle. In [1], the eco-driving problem was stdied and solved for an electric car powered by a DC-type motor. The model sed was based on an analytical expression of the electric power demanded by the electric machine. In [2], the same problem sing similar modeling assmptions as [1] was addressed for an electric car powered by a permanent-magnet synchronos machine by taking into accont physical limitations on the control actions. The advantage of the proposed soltion is its relatively low comptational time compared to Dynamic Programming (DP). Later, in [3], a more representative (realistic) model of the electric machine was sed while neglecting the dependance of the internal battery resistance and the open circit voltage on the battery State Of Charge (SOC) [5] as commonly assmed in the literatre dealing with energy management system design for hybrid electric vehicles [6] [9]. a D. Maamria, K. Gillet, G. Colin and Y. Chamaillard are with Laboratoire PRISME, Université d Orléans 4572 Orléans, France djamaleddine.maamria@gmail.com b C. Noillant is with PSA Pegeot Citroën, Direction Recherche Innovation & Technologies Avancées (DRIA), France In this paper, a similar eco-driving problem to [3] is considered. The dependance of battery parameters on the SOC is taken into accont in the optimization. This increases the nmber of state variables from 2 (vehicle position and speed) to 3 (SOC, vehicle position and speed). The objective is to select the right level of modeling to optimize the accracy/complexity trade-off. A main motivation is that the nmber of state variables greatly impacts the sed nmerical methods. Considering additional state variables increases the level of complexity and the comptational brden. This observation holds for DP, and for methods sing Pontryagin Minimm Principle (PMP) or direct formlations (e.g. collocation methods) [1], [11]. A special focs, in this paper, is on the qantification of the gain in energy consmption by inclding the SOC dynamics in the optimization. The paper is organized as follows. In Section 2, the vehicle model is described. The calclation of eco-driving cycles and the qestions addressed in this paper are detailed in Section 3. Section 4 discsses nmerical and simlation reslts. In light of the reslts, some conclsions on the trade-off between the complexity of the models sed to calclate ecodriving cycles and the optimality of the associated soltions are drawn. A. Motion eqations II. VEHICLE MODELING The vehicle is modeled on the longitdinal axis. The motion of the vehicle is the reslt of the forces that are applied on its body. According to Newton s law of motion, the vehicle speed v satisfies the differential eqation: (m + m rot ) dv(t) = F t (t) F r (t), (1) where F t is the traction force to be provided by the electric machine, F r is the sm of resistance forces and m is the total vehicle mass. The term m rot is an eqivalent mass of the rotating parts. It acconts for the overall inertia of the wheels (n tire j tire ) and for that of the electric machine (j rot ): m rot = n tire j tire + j rot rtire 2, where r tire is the wheel radis. The force F r comprises the rolling resistance force, the aerodynamic drag force and a force de to the road grade. Its expression is given by: F r (t) = c + c 1 v(t) + c 2 v(t) 2, (2) where c i, i = {, 1, 2} are the coefficients of the road load eqation (this expression of F r was employed in [6], [12]).

3 This model considers only the forces in the longitdinal direction. Variations of friction parameters dring crves, wind forces, and other distrbances are neglected. B. Transmission model The driver s torqe demand and the vehicle speed are directly calclated from the wheel speed profile, elevation profiles and the transmission ratio. The reslting torqe vale T wh can be positive (traction) or negative (braking). The electric machine torqe T e is related to the torqe reqested at the wheel T wh by: T wh (t) = r tire F t (t) = η t R t T e (t), (3) where η t is the transmission efficiency and R t is the constant motor-to-wheel transmission ratio. Similarly, the rotational speed ω e of the electric machine is related to the vehicle speed v by: ω e (t) = R t v(t) r tire. C. Electric machine model The electric machine is modeled by a qasi-static map describing either the electric power or its efficiency. The electric power P m consmed (in traction mode) or spplied to the battery (in recperation mode) is of the form: P m = P m (T e, ω e ), where P m is the electric power map of the electric machine. Normalized Electric Torqe [N.m] Electric Power [kw] Normalized Electric Motor Speed [rpm] Fig. 1. Electric Machine Power [kw] Electric Power Max Elec Torqe Min Elec Torqe This map incldes the losses in the electric machine and the power electronic devices. The electric machine torqe is limited by speed-dependent pper and lower bonds of the form (bold red and black lines in Figre 1): T emin (ω e ) T e T emax (ω e ). D. Axiliary power demand model A constant or a piecewise constant power demanded by axiliaries (radio/tape player, lights, air conditioning and heating systems) is considered. This power is provided by the battery. E. Battery model The battery is sally represented by an eqivalent circit model comprising a voltage sorce U ocv in series with an electric resistance R b, both of which vary with ξ, the battery state of charge (SOC) [6], [13]. The expression of the battery crrent I b is given by [6]: I b = 1 2R b (ξ) (U ocv (ξ) U 2 ocv(ξ) 4R b (ξ) P b ), where P b is the power reqested from the battery given by: P b = P m +. The dynamics of ξ is given by: dξ(t) = I b(t) Q, where Q is the nominal battery capacity. In order to simplify the notation, the dynamics of ξ considering a given initial condition ξ is written as: dξ(t) = g(v(t), ξ(t), T e (t)), ξ() = ξ. The inner (electrochemical) battery power is defined by: P ech (v, ξ, T e ) = I b (v, ξ, T e ) U ocv (ξ). The model parameters are smmarized in Table I. TABLE I VEHICLE MODEL PARAMETERS Description Vale Unit m Vehicle mass 13 kg r tire Wheel radis.34 m n tire Wheel nmber 4 j tire Wheel inertia 4.28 kg m 2 c Constant coefficient of road load N c 1 Linear coefficient of road load.3 N/(m/s) c 2 Qadratic coefficient of road load.377 N/(m/s) 2 η t Transmission efficiency.925 R t Motor-to-wheel transmission ratio Q Battery nominal capacity 288 C R b Internal battery resistance mω U ocv Open circit voltage V III. ECO-DRIVING An eco-driving methodology consists in finding the optimal way to redce the overall energy consmption [2], [14]. For a given road, the objective is to find the best speed profile minimizing the vehicle power consmption knowing that the vehicle starts from a point A at rest and mst reach a destination point B in a dration t f, with a zero velocity. This kind of qestion can be formlated as an OCP [4], [14]. A. Problem nder consideration The objective in this stdy is to investigate the impact of the battery parameters SOC dependance and the axiliaries power demand on the optimal speed trajectory. The battery parameters SOC dependance is stdied here from a model complexity viewpoint to find a trade-off between the comptation time (indced by the model complexity) and

4 the optimality of the soltion (energy consmption). Two qestions are addressed: 1) What is the maximm benefit of considering the dependance of battery parameters in the optimal speed trajectory calclation? 2) What is the impact of the axiliaries power demand on the optimal speed trajectory? B. OCP formlation The cost fnction (4) to be minimized is the electrochemical battery energy in traction over a fixed time window of dration t f : J() = P ech (v(t), ξ(t), (t)), (4) where the control variable is the electric machine torqe: (t) = T e (t). This optimization is carried ot nder the following differential eqations: dv(t) dx(t) dξ(t) = f(v(t), (t)), v() =, (5) = v(t), x() =, (6) = g(v(t), ξ(t), (t)), ξ() = ξ, (7) where x is the position of the vehicle and the fnction f is calclated by combining (1, 2, 3): f(v, ) = 1 ( c c 1 v c 2 v 2 + η t R t ). m + m rot r tire Since the speed and the electric machine torqe are limited and the final position and speed are set, the optimization mst be performed nder the following state and inpt constraints: v(t) [, v max (x(t))], (8) (t) [T emin (ω e (t)), T emax (ω e (t))], (9) x(t f ) = D, (1) v(t f ) =, (11) where D is the total traveled distance. The speed limits are given as a fnction of the vehicle position and not of time [15]. On the other hand, the final vale of ξ is free as the traction of the vehicle is ensred only by the electric energy (no additional energy sorce is available on board as for hybrid electric cars). To smmarize, the following OCP can be defined: (OCP ) : min J() (12) nder the dynamics (5, 6, 7), state and inpt constraints (8, 9) and the final constraints (1, 11). C. Simplified OCP The dynamics of ξ is taken into accont in the (OCP) described in (12) becase of the internal battery resistance R b and the open circit voltage U ocv dependance on ξ. To redce the calclation time and the model complexity, mean constant vales of R b and U ocv can be considered in the model sed to calclate the optimal soltion. In this case, the cost fnction (4) becomes of the form: J s () = P ech (v(t), ξ, (t)), where ξ is a fixed vale sed to calclate mean vales of R b and U ocv. As the cost fnction J s and the dynamics of v and x are independent of ξ, the nmber of the state variables is ths redced from 3 (v, x, ξ) to 2 (v, x). The following simplified OCP can be defined: (OCP s ) : min J s () (13) nder the dynamics (5, 6), state and inpt constraints (8, 9) and the final constraints (1, 11). D. Nmerical solving method The OCPs defined in (12) and (13) can be solved by nmeros methods. The soltion considered here is based on Dynamic Programming (DP) [16]. Considering the cost fnction J to be minimized: J = L(X(t), (t), t), from a mathematical viewpoint, Bellman s principle can be formlated as follows: let t [, t f [ and X(t) R n be given, then for all real r [t, t f ], the cost-to-go fnction V satisfies: { r V (X(t), t) = min L ( X(τ), (τ), τ ) dτ + V ( X(r), r )} U t This eqation is solved recrsively and backward. In order to redce the calclation time, the method sggested in [3], [4], [17] is sed, where an additional tnable term β t f is added to the cost fnction as a terminal cost: J() = J s () = P ech (v(t), ξ(t), (t)) + β t f, P ech (v(t), ξ, (t)) + β t f. The constant tnable parameter β penalizes the final time to obtain almost the same time dration as the initial driving cycle. Ths, two new OCPs can be defined as follows: (OCP ) : min J(), (14) (OCP s ) : min Js () (15) To calclate the right vale of the tnable parameter β, a root-finding method can be sed to drive the final time error to zero as done in [15], [18]. It was shown in [3], [4] that the soltions of the problems (OCP ) and (OCP s ) converge to the soltions of (OCP ) and (OCP s ) respectively when the obtained final time is almost the same as the initial driving cycle.

5 IV. NUMERICAL RESULTS To compte an eco-driving cycle, the following constraints [14] extracted from an initial driving cycle have to be considered: the same final distance x(t f ), the same nmber of stops and almost the same dration t f as the initial driving cycle. the vehicle speed limits depending on the position of the vehicle (x). To specify the speed limits, the following legal speed limits v lim were applied: v lim = [3, 5, 7, 9, 11, 13, 15] (km/h). The choice of this vector is not restrictive. Other speed limits can be sed. The process of identifying the appropriate speed limit can be described in two steps: 1) For each time t, find the maximm vale of j for which v lim (j) < v(x(t)) and v lim (j) v(x(t)). 2) v max (x(t)) = v lim (j). Two normalized driving cycles are considered: the EUDC cycle with a dration of 36s and traveled distance of 6.9km and the Worldwide harmonized Light vehicles Test Cycle (WLTC) with a dration of 1574s and traveled distance of 22.7km. The two OCPs defined in (14) and (15) are solved sing DP with the following grid parameters: δv =.1m/s for the vehicle speed and of δx = 2m for the distance. For the control inpt, a δ = 1N.m in the case of the EUDC Cycle and δ = 2N.m in the case of the WLTC cycle are sed. The notation δξ refers to the step of ξ (SOC) sed to solve the OCP (14). The initial vale of the SOC is ξ() = 9%. To make a fair comparison between the soltions of problems (14) and (15), the control trajectory in each case is applied to a (forward) simlation model considering the variation of R b and U ocv as fnctions of ξ. A. Impact of SOC dependent battery parameters The internal resistance R b and the open circit voltage U ocv depend on the SOC vale. The maximm variation of R b is abot 12% while the variation of the U ocv is 14% when the SOC changes between 9% and 2%. The impact of this variation on the optimal speed calclation is stdied below for two driving cycles. 1) EUDC cycle: The speed trajectories obtained from the DP are shown in Figre 2 verss distance and in Figre 3 verss time. The electric energy consmption in [%] and the time α needed to rn the DP are given in Table II. TABLE II TIME α NEEDED TO RUN THE DP AND ξ() ξ(t f ) FOR EUDC CYCLE. ξ() ξ(t f ) [%] α [s] EUDC: Initial Cycle Eco-EUDC: Simplified model Eco-EUDC: δξ = 1% Eco-EUDC: δξ =.5% Eco driving cycle: Fll model δξ=.5% Eco driving cycle: Fll model δξ=1% Fig Distance [m] Vehicle speed [km/h] vs distance [m] for EUDC cycle Fig. 3. Initial Cycle Eco driving cycle: Fll model δ ξ=.5% Eco driving cycle: Fll model δξ=1% Vehicle speed [km/h] vs EUDC cycle. Figres 2 and 3 show that the eco-speed trajectories satisfy the speed limits and that they are close. This can be confirmed from Table II where the difference in the final SOC between the cycles Eco-EUDC for δξ =.5% and Eco- EUDC with the simplified model is negligible (.1% on the energy consmption). On the other hand, the difference in the time needed to calclate these two cycles is significant (a ratio of 96). 2) WLTC cycle: For the previos case, the variation range of the SOC is not high. In order to ensre that the SOC sweeps a larger range, the WLTC cycle is sed. The speed profiles obtained are shown in Figre 4 verss distance and in Figre 5 verss time. The electric energy consmption in [%] and the time α needed to rn the DP are given in Table III. we highlight that the stops are indicated in distance and not in time as for the speed limits. Figres 4 and 5 show that the eco-speed trajectories satisfy the speed limits. From Table III, the difference in the final SOC between the cycles Eco-WLTC for δξ = 1% and

6 Eco driving cycle: Fll model δξ=1% Eco driving cycle: Fll model δξ=4% TABLE III TIME α NEEDED TO RUN THE DP AND ξ() ξ(t f ) FOR WLTC. ξ() ξ(t f ) [%] α [s] WLTC: Initial Cycle 8.23 Eco-WLTC: Simplified model Eco-WLTC: δξ = 4% Eco-WLTC: δξ = 1% = =.5kW =1kW =2kW Distance [m] Fig. 4. Vehicle speed [km/h] vs distance [m] for WLTC. x 1 4 Final time tf [s] Initial Cycle Eco driving cycle: Fll model δ ξ=1% Eco driving cycle: Fll model δξ=4% Fig. 5. Vehicle speed [km/h] vs for WLTC. Eco-WLTC with the simplified model is.1% (which is considered as negligible). On the other hand, the time needed to rn the DP increases by a ratio of 116 when considering ξ in the OCP. 3) Conclsion: Becase of the negligible sb-optimality indced and the redction in the calclation time, the nmerical reslts presented above sggest that it is sfficient to solve the OCP defined in (15) to calclate the optimal speed trajectory and to take the battery parameters dependance on the SOC into accont only in the (forward) simlation model. Note that similar analysis have been done for other normalized driving cycles: NEDC, Urban Artemis, Artemis Rral and Artemis highway cycles. B. Impact of the axiliaries power demand From the conclsion of the previos section, the OCP defined in (15) is considered in what follows. This optimization problem is solved in the case of the EUDC cycle (with a dration of 36s) for varios vales of β Fig. 6. β Relation between β and the final time t f [s]. 1) Constant axiliaries power demand: Varios constant axiliaries power demand are considered in the OCP (15). The final time t f and the electric energy consmption are given in Figres 6 and 7. As one can see from Figres 6 and 7, the axiliaries power demand shifts the final time t f for the same vale of β: the final time decreases when increases. The objective here is to stdy the impact of on the speed trajectories for the same final time t f. The speed trajectories having the same t f for varios vales of are given in Figre 8. These reslts indicate that the optimal speed profiles are the same for the vales of considered. The difference is only on the final SOC as illstrated in Figre 7 for a final time t f arond 36s. 2) Variable axiliaries power demand: A variable axiliaries power demand in [kw] defined by: 1, if x(t) D 3, =.5, if D 3 < x(t) 2D 3, (16), if x(t) > 2D 3, is considered. The optimal speed trajectory obtained for this variable is compared to the optimal speed calclated for = in Figre 9 (the two soltions mst have the same final time). The optimal speed profiles are the same for the two cases. 3) Conclsion: The nmerical reslts sggest that it is sfficient to set to zero in the model sed to calclate

7 = Variable 19 = 1 SOC()-SOC(tf) [%] =.5kW =1kW =2kW Final time t f [s] Fig. 7. Relation between t f [s] and energy consmption ξ() ξ(t f )[%]. Fig. 9. Speed trajectories for a variable defined in (16) = =.5kW =1kW =2kW Fig. 8. Speed trajectories for varios constant vales of. optimal speed trajectories and to take it into accont only in the forward simlation model. V. CONCLUSION The eco-driving problem for electric vehicles has been addressed. This problem is formlated as an OCP aiming at minimizing the electric energy consmption. The stdies condcted sggest that is not necessary to take into accont the battery parameters dependence on the SOC and the axiliaries power demand on the optimization problem. The impact of these simplifications on the optimal speed trajectory is negligible while ensring a reasonable time to rn dynamic programming. REFERENCES [1] N. Petit and A. Sciarretta, Optimal drive of electric vehicles sing an inversion-based trajectory generation approach, 18th IFAC World Congress, vol. 18, pp , 211. [2] W. Dib, A. Chasse, P. Molin, A. Sciarretta, and G. Corde, Optimal energy management for an electric vehicle in eco-driving applications, Control Engineering Practice, vol. 29, pp , 214. [3] F. Mensing, Optimal energy tilization in conventional, electric and hybrid vehicles and its application to eco-driving, Ph.D. dissertation, INSA Lyon, 213. [4] V. Monastyrsky and I. Golownykh, Rapid comptation of optimal control for vehicles, Transportation Research Part B, vol. 27, pp , [5] M. Miyatake, M. Kriyama, and Y. Takeda, Theoretical stdy on ecodriving techniqe for an electric vehicle considering traffic signals, in Proc. 9th IEEE Int. Conf. Power Electronics Drive Systems, pp , 211. [6] L. Gzzella and A. Sciarretta, Vehicle proplsion systems. Springer, 213. [7] A. Sciarretta, M. Back, and L. Gzzella, Optimal control of parallel hybrid electric vehicles, IEEE Transactions on Control Systems Technology, vol. 12, no. 3, pp , 24. [8] L. Serrao, S. Onori, A. Sciarretta, Y. Gezennec, and G. Rizzoni, Optimal energy management of hybrid electric vehicles inclding battery aging, American Control Conference, 211. [9] T. van Kelen, B. de Jager, D. Foster, and M. Steinbch, Velocity trajectory optimization in hybrid electric trcks, in Proc. American Control Conference, pp , 21. [1] M. Athans and P.-L. Falb, Optimal control: an introdction to the theory and its applications. Dover, 26. [11] C.-R. Hargraves and S.-W. Paris, Direct trajectory optimization sing nonlinear programming and collocation, Jornal of Gidance, Control and Dynamics, vol. 1, pp , [12] S-B. Ebbesen, Optimal sizing and control of hybrid electric vehicles, Ph.D. dissertation, ETH Zrich, 212. [13] F. Badin, Hybrid Vehicles. Editions TECHNIP, 213. [14] F. Mensing, E. Bideax, R. Trigi, J. Ribet, and B. Jeanneret, Ecodriving: an economic or ecologic driving style? Transportation Research Part C: Emerging Technologies, vol. 38, pp , 214. [15] F. Mensing, R. Trigi, and E. Bideax, Vehicle trajectory optimization for application in eco-driving, IEEE Vehicle Power and Proplsion Conference, pp. 1 6, 211. [16] D. Bertsekas, Dynamic programming and optimal control. Athena Scientific, 212. [17] H. Bovier, G. Colin, and Y. Chamaillard, Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles, Eropean Control Conference, pp , 215. [18] A. Sciarretta, G. D. Nnzio, and L. L. Ojeda, Optimal ecodriving control: Energy-efficient driving of road vehicles as an optimal control problem, IEEE Control Systems Magazine, 215.