Vehicle Trajectory Optimization for Application in Eco-Driving Felicitas Mensing1,2 , Rochdi Trigui1 , Eric Bideaux2 1 IFSTTAR Bron - LTE 2 AMPERE - INSA Lyon Abstract—To reduce fuel consumption in the transportation sector research focuses mainly on the development of more efficient drive train technologies and alternative drive train designs. Another and immidiately applicable way found to reduce fuel consumption in road vehicles is to change vehicle operation such that system efficiency is maximized. The concept of Ecodriving refers to the change of driver behavior in a fuel saving way or more generally in an energy saving way. In this paper system efficiency of a vehicle is optimized using a dynamic programming optimization approach. Given a drive cycle a so called ’eco-drive cycle’ is identified in which a vehicle performs the same distance with the same stops in equivalent time, while consuming less fuel. I. I NTRODUCTION The transportation sector is a major contributor to air pollution and consumer of scarce non-renewable fossil fuels [1]. With the rising fuel prizes and environmental concerns researchers are currently investigating new technologies to increase fuel efficiency and reduce emissions for road vehicles. Alternative drive train designs like the electric and hybrid vehicles are expected to have high potentials to decrease fuel consumption due to their ability to regenerate kinetic energy in the braking phase. In addition, in electric vehicles the electricity used can be generated from alternative, renewable energy sources. Another approach taken to increase system efficiency in vehicles is to improve current drive train technologies. By increasing the efficiency of individual components in the drive train overall system efficiency is maximized. A third way to reduce fuel consumption of road vehicles is to adapt driver behavior such that vehicle operation is optimized. Here an already existing vehicle configuration is used such that, for a desired mission, the fuel consumed is minimized. This concept of adapting driver behavior in a fuel saving way is often referred to as ’Eco-Driving’. It is well known that fuel economy in road vehicles does not only depend on the drive train but also on its operation. This has first been demonstrated in 1977 by Schwarzkopf [2]. Since then several works have shown that optimizing vehicle operation for a given mission can result in significant gains in system efficiency [3], [4], [5], [6]. Another study done by Ericsson [7] shows that certain driving patterns have an important effect on emissions and fuel consumption. Implementing and applying their results in real time driving has been difficult since an interface between the optimization and the driver has to be designed. Today several approaches to develop eco-driving strategies can be found. In most countries 978-1-61284-247-9/11/$26.00 ©2011 IEEE eco-driving courses are offered. But these have shown to reduce fuel consumption only for a short time period [8]. Other eco-driving tools consist of a display, which gives the driver advice on the optimal vehicle operation [9], or an active accelerator pedal, which influences the rate of acceleration of the driver by applying a resistance on the gas pedal [10]. In this work we suggest a strategy to compute a so called eco-drive cycle, given a general drive cycle. Using numerical optimization tools the best operation, which results in the same trip distance, with equivalent stops, in the same final time, will be computed for a given drive train configuration. For verification the vehicle will be simulated in a forward facing manner over the original drive cycle as well as over the determined eco-drive cycle. It is expected that the fuel consumption will be reduced from the original drive cycle to the eco-drive cycle. In analyzing the results the potentials in applying the suggested optimization algorithm to eco-driving will be investigated. In the following the system considered for optimization is presented. In Section III the approach suggested in this work is introduced. The method of optimization is discussed and the generation of an eco-drive cycle, given a general drive cycle, is explained. Results of application of the proposed strategy in simulation will be shown in Section IV. II. S YSTEM M ODEL The concept suggested in this paper is applicable to any vehicle architecture: conventional, electric or hybrid drivetrain. For simplicity it will here be demonstrated on the example of a conventional drive-train. For optimization purposes a backward (inverse) vehicle model, where the engine operation is computed given the operation of the vehicle, is needed. The vehicle considered is a compact passenger car with a mass (M) of 1020kg. The vehicle can be modeled as a standard drive train as seen in the schematic in Figure 1 including the final drive reduction, a 5-speed gear box, a clutch, auxiliary losses and the engine. The engine used in this work represents a 1.5L diesel engine with common rail direct fuel injection. The vehicle model was constructed using the VEHLIB library, which was previously developed at IFSTTAR [11]. In this inverse modeling approach the losses and operations of each component is calculated in the direction opposite to the power flow. In general the vehicle is propelled by the engine, the power source is therefore the internal combustion engine. Given the output of the engine the acceleration of as an input to the simulation. In order to follow the desired speed profile the driver input was implemented using a PID controller. III. M ETHODOLOGY In the following the velocity trajectory optimization problem will be discussed and a method to generate an eco-drive cycle is presented. A. Optimization Figure 1. Clio 1.5L Drive Train Schematic the vehicle can be computed. In this approach, however, the required operation of the engine is back-calculated assuming a certain speed and acceleration of the vehicle. The force required at the wheels to satisfy a certain velocity and acceleration of the vehicle can be calculated in a quasistatic approach. The force, often referred to as resistance force, is computed as a sum of the rolling resistance (Froll ), the aerodynamic drag (Fdrag ), the road grade resistance (Fgrade ) and the acceleration force (Fa = Ma): Optimization of a velocity profile is a well known problem in literature. Several approaches can be found where researchers optimize the energy utilization of road vehicles for a given trip [3], [4], [5], [6]. While most works consider energy or fuel as the cost function to be optimized, other studies have been done where time consumed over a trip is minimized [12]. In this study the velocity profile of a vehicle is optimized such that a given mission is achieved in a fixed time considering overall fuel consumption as the cost. The system to be optimized can then be described in a discrete way by 1 xi+1 = xi + vi Δt + ai Δt 2 2 (III.1) vi+1 = vi + ai Δt (III.2) Fres = Froll + Fdrag + Fgrade + Fa . With this and the vehicle speed (v) the wheel operation can be expressed as: Tw = Fres ∗ Rtire ωw = v/Rtire where Rtire represents the radius of the tire. where x is the distance, v is the longitudinal vehicle velocity, a is the longitudinal vehicle acceleration and t the time. The system consists of two states, the distance driven (x) and the velocity of the vehicle (v). The control variable is represented by the acceleration (a). The objective function is defined by t=t Given the wheel operation the engine operation can be found by modeling the losses of the final drive, the gear box, the clutch and the auxiliaries. In this paper the efficiency of the final drive reduction and the gears are assumed to be constant with respect to operating torque and speed. The gear selection is optimized such that the instantaneous fuel rate is minimized. Losses in the clutch exist if the engine speed is bellow idle and the clutch is slipping. The auxiliary components are assumed to absorb constant 300W. The diesel engine is modeled using an engine efficiency map with maximum efficiency of 40%. Knowing the engine speed and torque the instantaneous fuel consumption can then be calculated using a look up table. Using this model the instantaneous fuel consumption can be computed as a function of vehicle velocity (v) and vehicle acceleration (a): m f˙uel = f (v, a) This backward facing model can now be utilized in the optimization process to find the fuel consumption for a chosen vehicle speed and acceleration. Once optimal operation was determined as a velocity trajectory over time the gains of this eco-driving algorithm was verified with a forward facing model. Rather than defining the vehicle speed and acceleration the driver operation of the gas and break pedal is here used t=t J = ∑t=0f J f uel (t) = ∑t=0f m f˙uel (t) with the following constraints applied to the final and initial states: x(0) = x0 x(t f ) = x f v(0) = v0 v(t f ) = v f tf = T Commonly used optimization methods to solve the energy utilization problem for road vehicles are the Pontryagin’s Maximum Principle [13] and the Dynamic Programming Optimization Method [13]. In this work a 3-dimensional dynamic programming approach, after the example of Hooker [14], was chosen. In the dynamic programming optimization the search for the optimal trajectory is simplified using the Bellman principle while searching from the final state backward in time. The Bellman Principle of Optimality states the following [13]: ’An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.’ In general the dynamic programming method has two major parts, as shown in Figure 2. These are the calculation of optimal cost and indexes to the final state X[N−a+1,i2 , j2 ] and the optimal cost from X[N−a+1,i2 , j2 ] to X f for all possible [i2 , j2 ]. The cost of the optimal trajectory at N − a is then given by: ∗ ∗ = min(J[N−a,i1 , j1 −>N−a+1,i2 , j2 ] + J[N−a+1,i ) J[N−a,i 1 , j1 ] 2 , j2 ] i2 , j2 velocity v ∗ Storing the costs J[N−a,i, j] and indexes I[N−a,i, j] for all a, i, and j the optimal cost from Xo to X f is found after N iterative steps. In order to find the sequence of states used to result in this cost the indexes are used to retrace the trajectory forward in time. Xo Dynamic Programming Optimization Flowchart J*[N-1,i,j->N,if,jf] Initially the possible range of time, distance and speed are discretized and a state Xk,i, j refers to the state [ ]T t(k) x(i) v( j) . The initial and final distance and speed at t(0) and t(N) are fixed and denoted ⎤ ⎤ ⎡ ⎡ to tf Xo = ⎣ xo ⎦ and X f = ⎣ x f ⎦. vo vf Beginning the search for the optimal trajectory from the ∗ final state the optimal costs J[N−1,i, j] = J[N−1,i, j−>N,i f , j f ] and indexes I[N−1,i, j] of the respective trajectories are stored for all possible states at the last time step N − 1. The state transition diagram to visualize the process can be seen in Figure 3. With this we can assume that the optimal trajectories for all possible values of i2 and j2 at some point in time N − a + 1 are known and their costs to reach the desired final state are ∗ . The optimal trajectory from some state given by J[N−a+1,i 2 , j2 ] X[N−a,i1 , j1 ] to the final state can then be found by comparing the sum of costs of the state transitions between X[N−a,i1 , j1 ] to dis backwards in time under utilization of the Bellmann Principle and the computation of the optimal trajectories retracing the stored indexes forward in time. In the following the method of dynamic programming optimization in the three dimensional approach will be described in more detail: In a 3-dimensional approach three variables are to be defined at each state, here time (t), distance (x) and velocity (v): ⎤ ⎡ ⎤ ⎡ t x1 ⎣ X = x2 ⎦ = ⎣ x ⎦ x3 v tan ce x Figure 2. time t tf xf Xf Figure 3. Dynamic programming optimization method Using dynamic programming the optimal solution is found by searching all possible proceeding state values at each step in time. The computational cost of this 3-dimensional method is very high in comparison with a two dimensional approach. For example, given a problem with two dimensions and K possible values at each dimension the method searches at each of the K possible time steps the K possible state values. The computational cost becomes K ∗ KC = K 2C, where C is the cost to calculate one edge. However, when the problem consists of three dimensions (as here), and again K possible values are given for each dimension, the method has to search each of the K time steps at all possible combinations (K 2 ) of the other two variables. The resulting computational cost becomes K ∗ K 2C=K 3C. Hence the computation time of a dynamic programming optimization grows exponentially with the dimension size. A three dimensional approach with the axis time, distance and speed is used here because it is necessary in order to fix and satisfy initial and final conditions in all of these dimensions. In future work, and in order to implement the developed concept in real time more rapid optimization methods should be explored. As previously stated the system can be described with two states, therefore using a 3 dimensional approach for optimization results in a dependency of the three axis. While 1 Δx = ai Δt 2 2 (III.3) Δv = ai Δt (III.4) speed [km/h] 100 50 0 0 200 0 2000 400 600 800 1000 time Maximum speed as function of distance 1200 150 100 50 0 Figure 4. 4000 6000 distance [m] 8000 10000 12000 NEDC Drive Cycle with maximum speed limitations B. Drive Cycle Generation Choosing Δx and Δt the grid size for Δv is found by 2Δx Δt (III.5) Δt 2 With this definition all resulting distances, xi , for a chosen speed, vi , at time ti fall on the defined grid. Due to the fact that the three dimensions, (x, v,t), in the dynamic programming computation are dependent the initial state might not be reachable by all paths within one final step in the calculation. This becomes obvious looking at the last step where xi , vi , xi+1 , vi+1 , and Δt are fixed by the computed optimal paths up to the (i+1)th step and the initial conditions (at time i). Given these values the resulting system of equations consists of two equations with only one unknown, which is not solvable. Hookers publication does not mention any adaptation of his calculation to this fact. In our calculation it was found that leaving the second to last iterative step (at t=1) free, and independent of the grid, can ensure that all possible trajectories computed can reach the initial state. This process is demonstrated in the following. Expanding Equations (III.1) and (III.2) for two time steps results in the following four equations: Δv = 1 xi+1 = xi + vi Δt + ai Δt 2 2 (III.6) vi+1 = vi + ai Δt (III.7) 1 xi+2 = xi+1 + vi+1 Δt + ai+1 Δt 2 2 (III.8) vi+2 = vi+1 + ai+1 Δt (III.9) Replacing xi+1 and vi+1 in Equations (III.5) and (III.6) with Equations (III.3) and (III.4) results in 1 1 xi+2 = (xi + vi Δt + ai Δt 2 ) + (vi + ai Δt)Δt + ai+1 Δt 2 2 2 (III.10) vi+2 = (vi + ai Δt) + ai+1 Δt drive cycle 150 speed [km/h] Hooker[14] used interpolation to fix the resulting distance, xi , for a given time, ti , and speed, vi , to the grid it was found in these studies that the dependency of the three dimensions allows to make an intelligent choice of grid size that makes interpolation unnecessary. The step size of each dimension is determined by the following procedure: Initially vi is assumed to be 0. Equations (III.1) and (III.2) can then be reduced to (III.11) With these 2 Equations, which represent the second to last and last time step in the dynamic programming process, the two unknowns ai and ai+1 can be computed with the fixed variables xi , vi , xi+2 , vi+2 , and Δt. Using this concept all possible trajectories can be explored for optimality. To quantify improvements due to eco-driving strategies a baseline has to be defined for comparison. Using standard and real-life drive cycles to represent average driving behavior the differences in fuel consumption and vehicle operation between general driving and eco-driving can be evaluated. A similar approach to compare gains of optimal to general vehicle operation was used by van Keulen in the development of optimal energy management for a hybrid electric truck [15]. Using a drive cycle, where the velocity is defined as a function of time, the constraints for an optimization are defined. A maximum speed profile corresponding to the speeds seen in the cycle is specified as a function of distance. Such a maximum speed function can be defined by fixing speed limits. Here the general limits of the roads in France were used and consist of 30km/h, 50km/h, 70km/h, 90km/h, 110km/h and 130km/h zones. Ensuring that the vehicle stops at the same distances as in the drive cycle an additional 0km/h limit is introduced. Such a maximum speed function dependent on distance can be seen in Figure 4. Here the velocity profile of the NEDC (New European Drive Cycle) is shown in the first plot. This speed profile is integrated over time to identify the distance traveled. In the second plot in blue the speed as a function of distance can be seen. In green the assigned maximum speed limits are shown. It can be seen that at several intermediate distances the vehicle is required to come to a full stop. The idling times at each of these stops is added after the optimization process. In this analysis it is assumed that the vehicle has to cover the same distance as specified by the drive cycle. Using the cycle length a final time constraint can be defined as well. This strategy should result in a fair comparison in fuel consumption and vehicle operation between general driving, represented by the drive cycle, and eco-driving, where the same distance with the same stops is driven and the target is reached in the same time. IV. R ESULTS To evaluate the gains of an eco-driving strategy and to understand the optimal operation of a conventional vehicle optimal operation of a Clio 1.5L vehicle was simulated using the NEDC as base cycle. Using the previously defined distance, time and maximum speed constraints the optimal speed profile was calculated and can be seen in Figure 5 together with the original cycle. Here the original NEDC cycle is displayed in green, the maximum speed constraints can be seen in blue and the computed optimal velocity for this trip is shown in Specific Engine Consumption [g/kWh] 140 cycle eco max speed NEDC 120 215 150 21 22 5 0 0 5 22 5 Torque [Nm] 100 80 22 velocity [km/h] 0 100 25 215 21 215 275 0 25 300 50 275 300 250 275 300 400 60 400 400 0 40 20 −50 0 0 0 2000 4000 6000 distance [m] 8000 10000 Figure 7. Figure 5. 500 1000 1500 2000 2500 Rotational Speed [rpm] 3000 3500 4000 4500 12000 Engine operation NEDC Preliminary Results: Drive cycle versus eco-drive cycle Specific Engine Consumption [g/kWh] 215 Speed profil 120 150 22 5 0 0 Torque [Nm] 0 200 400 600 time [s] 800 1000 5 20 22 40 0 5 22 100 60 0 80 25 21 215 215 velocity [km/h] 21 eco cycle NEDC cycle 100 300 275 300 250 275 300 400 1200 Gear selection 275 0 25 50 400 400 0 5 gear 4 −50 3 0 500 1000 1500 2000 2 2500 Rotational Speed [rpm] 3000 3500 4000 4500 1 0 Figure 6. 0 200 400 600 time [s] 800 1000 1200 Figure 8. Engine operation Eco-Drive Cycle Speed profile for NEDC cycle and eco-drive cycle red. In this graph, the cycles are shown as a speed dependent on distance whereas in Figure 6 speed is displayed with its dependency on time. Due to the difference in vehicle speed the eco-driver does not perform the same stops at the same time but rather at the same distance. Simulating the vehicle in a forward facing manner with the help of the VEHLIB [11] software the operation of the vehicle was compared for the original NEDC cycle and the eco-drive cycle. The two speed trajectories with its assigned gears can be seen in Figure 6, where the NEDC is plotted in green and the eco-drive cycle in red. The fuel consumption of the Clio over the NEDC is calculated to be 3.87L/100km while used in the most efficient way the same vehicle for the same trip in the same time can achieve a consumption of 3.26L/100km. It can be seen from the graph that the vehicle uses short but high acceleration phases to reach a rather low constant speed. In the NEDC the vehicle acceleration lasts longer and reaches a higher speed. Because of the fast acceleration of the ecodriver the two vehicle can still arrive at the final distance at the same time. In the given NEDC drive cycle the gear choice is given by the cycle. Utilizing this eco-driving strategy the gears are chosen to be optimal. That means for a given wheel speed and acceleration the gear that results in the least fuel consumption is chosen. The difference between the gear choice in the initial cycle and the eco-drive cycle is significant as demonstrated in Figure 6 plot 2. In Figures 7 and 8 the operation of the engine can be seen for the two cycles. While the engine is operated mostly in the low torque high speed region in the NEDC cycle in eco-driving the engine operation is pushed to very high or maximum torque operation. It should be noted here that maximum acceleration in this simulation is only limited by the maximum engine torque. The torque interrupts due to shifting are kept to a minimum such that the vehicle can follow the computed speed trajectory. Although the engine operation changes, average engine efficiency over the drive cycle was found to be almost constant. The engine operates at an average 28.5% over the NEDC cycle, while the eco-driver drives with an average engine efficiency of 29.0%. From these values it becomes obvious that the reduction in fuel consumption of almost 16% does not only come from an improvement in engine operation. This result shows that the losses in the engine contribute to the overall losses of the system but do not outweigh efficiency of the other components in a vehicle. In the concept of eco-driving the fuel consumption is reduced which is equivalent to maximizing overall efficiency of the vehicle. Contributing to this efficiency is the engine but also the drive train as well as the chassis of the vehicle. In Table I the efficiency values for the drive train can be found. As mentioned previously the engine efficiency has not changed by a large amount. Final Drive efficiency remains constant between the NEDC and the eco-drive cycle while gearbox efficiency has improved by 0.3%. It can be concluded from these values that the majority of losses reduced due to eco-driving are not found in the drive train nor in the engine. In Figure 9 the energy required by the vehicle to perform each of the two cycles is displayed. Cycle 1 here represents the NEDC cycle and cycle 2 stands for the derived eco-drive cycle. In this plot the energy required to move the inertia of the vehicle is shown in red, and the energy spent on overcoming the resistance forces can be seen in green. It is illustrated in Engine efficiency [%] Final Drive efficiency [%] Gearbox efficiency [%] NEDC 28.5 97.0 97.9 Table I E FFICIENCY OF DRIVE TRAIN FOR NEDC eco-drive cycle 29.0 97.0 97.6 AND ECO - DRIVE CYCLE 1200 Inertial Force Resistance Force 1000 Energy [Wh] 800 600 400 200 1 2 It can therefore be concluded that eco-driving is a good and immediately applicable way to reduce fuel consumption in road vehicle but in order to maximize gains in fuel economy overall system efficiency (engine-, drive train-, and chassis efficiency) has to be considered. This algorithm can easily be applied to other vehicle architectures. In order to compare optimal operation and potentials in eco-driving strategies it will be applied in the future to electric and hybrid vehicles. cycle Figure 9. Inertia Force and Resistance Force for NEDC and Eco-drive cycle this figure that the energy required to drive the computed ecodrive cycle is a lot less than the energy used to perform the original standard drive cycle. Energy needed to move the inertia of the vehicle is reduced by choosing short high acceleration rate but keeping the time spent on acceleration short. The velocity profile shows that the vehicle is accelerated with a rather high rate but for a very short time. In the standard drive cycle average acceleration rates are used but over a longer time window which then results in higher amount of energy consumed. The resistance forces are reduced by keeping the vehicle speed at a lower average speed. The losses on aerodynamic resistance are reduced by accelerating fast to a lower maximum speed. This driving behavior results then in the same trip time. It should be noted that this driving behavior might not be optimal with respect to the interactions with other vehicles on the road and driver comfort. In future studies constraints can be assigned on maximum acceleration and minimum velocity for certain parts of the trip, to represent for example a highway section. V. C ONCLUSION The work presented in this paper has shown that eco-driving strategies can improve fuel economy by a large amount. Using the dynamic programming optimization method to find optimal operation for a vehicle in off-line simulation has shown to be a good approach. However due to the high computational cost in calculation the method does not seem suitable for real-time implementation. For in-vehicle application in real-time faster optimization methods should be explored. Using GPS defined routes or predictive algorithms to define traffic simulations an application of numerical optimization methods on eco-driving seems to have high potentials. Under utilization of a standard drive cycle the potential gains due to eco-driving were investigated. By finding the optimal speed trajectory that results in the same distance driven over the same time using similar constraints on maximum velocity, stops and idle time vehicle performance was compared. It was found that improvements in fuel economy of up to 16% can be achieved by applying eco-driving strategies. In analyzing the resulting optimal vehicle operation it was found that losses are reduced only by a small amount in the drive train and the engine. However by making optimal choices on vehicle acceleration and vehicle speed the energy required to perform the mission was reduces by a large amount. ACKNOWLEDGMENT The French Environment and Energy Management Agency (Agence de l’Environnement et de la Maı̂trise de l’Energie, ADEME) is gratefully acknowledged for its support of this research. R EFERENCES [1] “Data and maps -EEA.” http://www.eea.europa.eu/data-and-maps. 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