Eco-Platooning of Autonomous Electrical Vehicles Using Distributed Model Predictive Control

Transcription

1 Eco-Platooning of Autonomous Electrical Vehicles Using Distributed Model Predictive Control Aaron Lelouvier, Jacopo Guanetti and Francesco Borrelli Abstract We present a distributed predictive controller architecture for the coordination of a group of electrical connected autonomous vehicles CAVs. The objective is the energy optimal operation of the group for a given desired distance and arrival time of each CAV. In the distributed implementation, the agents exchange the predicted trajectories and iteratively improve the local solutions. The proposed approach computes the speed trajectories of each CAV to minimize the energy consumption of the entire group: this can involve travelling in sub-groups micro-platoons and performing the necessary formation joining and dismantling manoeuvres. We use simulation to show the effectiveness of the proposed approach and compare it to a centralized optimal control strategy. I. INTRODUCTION The transportation sector is currently undergoing important changes related to vehicle connectivity, automation, and electrification. Connected and automated vehicles CAVs are the focus of much academic research and industrial development efforts. CAVs can enable the reduction of road accidents and the improvement of traffic flow and road throughput [1] [4]. On another front, electric vehicles EVs enable operation without local pollutant emissions [5]. Their main drawbac is often identified in the limited electric driving range; this is particularly inconvenient in long trips, due to the relatively long charging times. CAVs have the potential to actively control their speed profile and thus their energy consumption [6] [8]. This feature is particularly desirable in EVs, where energy efficiency affects not only the cost of operation, but also the driving range and the overall user satisfaction. Connectivity and driving automation for energy efficiency have been thoroughly studied for heavy-duty vehicles performing freight transportation [9]. In heavy duty vehicles, energy savings from reduced inter-vehicular distance can be substantial. However, energy savings are not trivially achieved tracing a small inter-vehicular distance, because aggressive throttling is generally detrimental for fuel economy [9]. When shifting the focus on supervisory control, travel plans and constraints are usually nown in advance, at least for vehicles belonging to the same operator. A centralized planner is therefore realistic and desirable [10]. In CAVs for personal transportation, using a centralized planning approach is less realistic. Travel plans and constraints may not be completely nown in advance, and may change over time. Moreover, travelled distances tend to be This wor was not supported by any organization Department of Mechanical Engineering, University of California at Bereley, CA, USA. {jacopoguanetti,fborrelli}@bereley.edu shorter, while the flexibility on the arrival time is lower than in freight transportation. In this paper, we propose a distributed MPC approach for energy-saving platoon planning. Our controller optimizes the vehicles speed profiles, accounting for the trip constraints and seeing opportunities for energy savings. The spontaneous formation of small platoons can result from the optimization process. Clearly, platoons are created when the pah in common to the vehicles is long enough to justify possible additional energy expense to reach other vehicles. The trip constraints are set independently for each agent, but they all contribute to the common goal of energy minimization, therefore an iterative approach is proposed to achieve a consensus on the trajectories. Performance of the proposed approach is assessed with a set of simulations. Our simulation setup includes lower control layers for longitudinal speed and vehicle overtaing, as well as a centralized counterpart of our proposed controller. Our distributed implementation attains good performance when compared to the centralized benchmar, and can result in significant improvements when compared to vehicles driving alone. The paper is organized as follows. Section II describes the vehicle model. Section III discusses the control hierarchy and the lower control layers. Section IV formulates the eco-platooning problem and solves it via centralised MPC. Section V, proposes the distributed MPC approach to ecoplatooning. Section VI shows our simulation results. The paper ends with some concluding remars. II. SYSTEM MODEL In this section, we introduce our models for the vehicles longitudinal and lateral dynamics, for the battery charge dynamics, and for the aerodynamic drag. These models account for a single vehicle i, belonging to the group G of vehicles considered in the platoon planning problem. A. Vehicle model We use the following inematic bicycle model [11] s i +1 = si + t sv i cos ψ i + βi, 1a y i +1 = yi + t sv i sin ψ i + βi, 1b F i t, F i r, F i a,, 1c v i +1 = vi + t s m ψ i +1 = ψi + t s v i l sin r β i, 1d β i = tan 1 lr l r + l f tanδ i, 1e

2 where the suffix is the discrete time index and s i denotes vehicle i s longitudinal position, y i its lateral position, v i the velocity, ψ i the inertial heading angle, β i the sideslip angle, and δ i the steering angle. The parameters are the vehicle mass m and the distances from the center of mass to the front wheels l f and the rear wheels l r. F i t is the traction force at the wheels, F r i is the rolling friction force, F a i is the aerodynamic drag force. When travelling on flat roads, they are respectively defined as F i t, = g r r w T i m,, F i r, = m c r g, F i a, = ρ c d A f 2 v i 2a 2b 2, 2c where g r is the gear ratio, r w the wheel radius, c r is the rolling friction factor, g the gravitational constant, ρ the air density, c d the air-drag coefficient and A f the frontal surface of the vehicle. We assume that g r is fixed [5], while T m and δ are controllable inputs. In the interest of simplified control design, a model of the longitudinal dynamics only is also introduced v i +1 = vi + t s m s i +1 = si B. Battery dynamics F i t, F i r, F i a,, 3a + t sv i. 3b In [12], the electric power drained from the battery by the motor is modelled as P i m, = b g 2 r 1 T i m, r vi + b 2 T m, i, 4 w The parameters are set to b 1 = 1 and b 2 = 0.18 to fit the efficiency map of a 70 W motor taen from [13]. The dynamics for the battery state of charge ξ are defined as ξ i +1 = ξi t s P i m, η ɛ, 5 b where η is the battery efficiency and ɛ b its total energy capacity. C. Aerodynamic drag model For a vehicle i following another vehicle j, the aerodynamic drag coefficient c i d generally depends on the intervehicular distance d = s j s i. For d > 0, an experimental characterization 1 is found in [14]; we approximate that characteristic as c appr d = c nom d, 6 where c nom = For d < 0, c d d = c nom holds. The switching behaviour is approximated introducing the smooth function Sd Sd = tanh100d d s, 7 1 The characterization in [14] considers heavy duty vehicles. To the best of the authors nowledge, no similar data for personal vehicles are publicly available. Air drag reduction may be less relevant in cars than in trucs, resuting into platoon formation less convenient. Fig. 1: Aerodynamic drag coefficient c d as a function of longitudinal and lateral distances d and d y. c cont d = c nom 1 Sd + c appr d Sd, 8 where d s is the distance where the switch occurs. We set d s = 5 m to ensure that the maximum reduction of aerodynamic drag occurs at a safe inter-vehicular distance. Finally, as shown in Figure 1, we also model the effect of the lateral deviation y = y j y i as ccont d c d d, y = c nom 1 + e 2 y 1. 9 c nom III. CONTROL SYSTEM ARCHITECTURE Our objective is to minimize the total energy that the vehicles belonging to group G drain from their batteries, given constraints on the destination and the arrival time. The individual speed profiles are optimized to minimize the energy consumption. Forming platoons that minimize the relative distance is desirable, unless the energy saved by doing so is compensated by the energy spent to form the platoons for instance by speeding up to reach other vehicles. We approach the above problem with a trajectory planning controller, that produces reference signals for lower level tracing controllers. The control hierarchy, inspired by [10], includes the following three levels from high to low. Eco-platooning controller - solves the eco-platooning planning problem and outputs a prediction for the velocity trajectory of the vehicle. Overtaing controller - receives the longitudinal prediction from the eco-platooning controller and determines the optimal lateral trajectory, avoiding collisions and performing overtaing as needed. Trajectory tracing controller - receives the longitudinal and lateral trajectory predictions from the upper levels and computes the corresponding optimal torque and steering angle of the vehicle. In the following subsections, we briefly introduce the two lower layers of the hierarchy.

3 A. Overtaing controller Consider a vehicle i approaching a vehicle j. We define an ellipsoidal safe region around vehicle j by setting [15] s i s j 2 + yi y j 2 } 2l {{ 2w } c cs i,y i,s j,y j 1, where l and w are respectively the length and width of vehicle j. A MPC controller with horizon H o is set up to compute the optimal lateral trajectories satisfying the constraints of the safety zones. The overtaing MPC controller for vehicle i solves the following optimization problem. Problem 1: min y i t t,...,yi t+ho 1 t s.t. t+h o y i t yi ref 2 =t s i +1 t = si y i +1 t yi t s t + t sv i t, t < ẏ max, = t,..., t + H o 1, c c s i t, yi t, sj t, yj t 1, = t,..., t + H o, j N i, s i t t = si t. In Problem 1, ẏ max is the maximum rate of change of the lateral position and N i is the set of neighbouring cars of vehicle i, N i = G\i. v i is the velocity trajectory received from the eco-platooning controller. y ref is the reference lateral position the centre of the left lane. B. Trajectory tracing controller The tracing MPC controller has horizon H t and uses the bicycle model 1. For vehicle i, we define the cost function t+h t J i t = c y y i t y opt i 2 +c v v i t v opt i 2 =t and solve, at each time step, the following problem. Problem 2: min T i i,...,t m,t t m,t+h t 1 t δ i t t,...,δi t+h t 1 t J i t s.t. dynamics in 1, T min T i m, t T max, v min v i t vmax, T i m,+1 t T i m, t t s < T max, ψ i +1 t ψi t t s < ψ max, = t,..., t + H t 1, y i t t = yi t, ψ i t t = ψi t. In Problem 2, c y and c v are tuning coefficients, y i opt are the lateral trajectories received from the overtaing controller, T max, T min, v max, v min are upper and lower bounds on T m and v, T max and ψ max are the maximum rates of change of T m and ψ. IV. PERFORMANCE BENCHMARKING VIA CENTRALIZED CONTROL In this section, we formalize the eco-platooning planning problem, and we propose a centralized receding horizon solution, that is used as a performance benchmar in our simulations. A. Terminal set and cost design We constrain the vehicle position at the end of the prediction horizon such that the desired arrival time and destination remain feasible driving at maximum speed after the end of the MPC horizon s lb = st 0 + s arr t arr T v max, 10 where t 0 is the current time, H the controller horizon, t arr the maximum remaining time to arrival and s arr the remaining distance to arrival. A terminal cost penalizes the deviation from desired values of position and velocity J vel = c vel v des vt f 2, J pos = c pos s des st f 2, 11a 11b where c vel and c pos are tuning parameters, v des is the constant velocity that allows to cover the remaining distance s arr in time t arr, and the s des is the corresponding position at the end of the horizon v des = s arr st 0, t arr t 0 12a s des = st 0 + v des H. 12b To ensure that the remaining battery charge is sufficient to reach destination, a prediction for the battery charge is also included in the receding horizon. Driving alone at velocity v min requires the traction torque T vmin = r wm g r g c r + ρ A f c nom 2 m v2 min, 13 which results in a constant discharge rate of the battery ξ vmin = 1 g r b 1 T vmin v min + b 2 Tv 2 ηɛ b r min. 14 w The terminal lower bound on the state of charge is then set as ξ lb = ξ min t arr t 0 t horizon ξ vmin, 15 where ξ min is the minimal charge at arrival.

4 B. Proximity penalisation As some preliminary simulations confirmed, the formulation introduced so far may bring the vehicles in a platoon to frequently switch positions in general, there may be multiple locally optimal formations. In practice, frequent switching of positions in the platoon is undesirable. To mitigate this issue, we introduce a soft constraint penalizing vehicles driving closer than d s = 5 m on the longitudinal axis, s i s j 2 25 λ i,j, where λ i,j 0 is the associated slac variable. C. Centralised optimisation problem For each vehicle, the stage cost includes the drained battery power 4 and a penalty on the slac λ i,j l i g r = b 1 T m i v i + b 2 T m i 2 + c λ λ i,j 2. r w j N i The optimization problem solved by the centralized controller is: Problem 3: min T i i,...,t m,t t m,t+h 1 t i G G i=1 J i pos + J i s.t. dynamics in 3, 5, vel + t+h 1 =t T min T i m, t T max, v min v i t vmax, s i t sj t 2 25 λ i,j, λ i,j 0, = t,..., t + H 1, s i t+h t si lb, ξi t+h t ξi lb, i G, j N i. V. DISTRIBUTED AVERAGING CONTROLLER l i We now introduce the proposed distributed controller. After discussing agents coupling, we define the optimization problem solved by each vehicle, and we describe the iterative procedure used to reach a consensus between the vehicles. A. Agent coupling term For a fixed velocity trajectory v i it holds that v i +1 vi t s = c r g ρci d A f v i 2 2m + g r T i m, m r w. If the vehicle is travelling alone then the aerodynamic coefficient is c i d = c nom, otherwise it is reduced by a quantity that depends on the inter-vehicular distance, c i d = c nom c i d di,j. Correspondingly, the motor torque is also i reduced by a quantity T m, that is easily found to be T i m = r w 2 g r ρ A f v i t 2 c i d di,j. T i m Plugging into the battery power equation 4, we estimate the energy saved by that vehicle j due to the proximity to vehicle i as v j 3 j c d dj,i l i j red = b 1ρ A f 2 + b 2 rw ρ A f 2 g r v j 2 c j d dj,i 2. B. Distributed optimization problem The distributed controller computes the optimal speed trajectory for vehicle i, that minimizes its driving cost and maximizes the energy savings of neighbouring cars N i, while satisfying the constraints on the position, time and state of charge at the arrival. The cost function of the distributed controller for vehicle i sums J pos, i J i J i d = t+h =t l i vel and l j i red j N i. 16 At each time step, the controller of vehicle i solves the following optimization problem. Problem 4: min T i i,...,t m,t t m,t+h 1 t J i dist + J i pos + J i vel s.t. dynamics in 3, 5, C. Iterative averaging T min T i m, t T max, v min v i t vmax, s i t si t 2 25 λ i,j, λ i,j 0, s i t+h si lb, ξi t+h ξi lb, j N i. The previously derived optimization problem finds the optimal speed trajectory of the each vehicle i, given the trajectories of the neighbouring vehicles N i. Each vehicle i can then transmit the computed trajectory, and the others can recompute a new optimal trajectory. By iterating this procedure, a consensus between the vehicles is achieved, and the energetic performance of the group of vehicles is expected to improve. The iterative procedure performed at each run of the distributed controller is defined in Algorithm 1, where ζ i n,t collects the trajectories of s i, vi, and ξi, for = t,..., t + H, at time t and iteration n. The output of Algorithm 1 at time t is denoted as ζ,t. i The iterative averaging scheme above is inspired by the iterative DMPC described in Section of [16], which reviews recent results in distributed MPC for non-linear systems. For this iterative DMPC scheme it is possible to prove stability, but not convergence to the corresponding centralised solution. A formal analysis of the stability and convergence for our distributed averaging controller is left to future research.

5 Algorithm 1 Iterative averaging procedure for vehicle i if t = 0 then Initialize ζ i 1,0 assuming constant velocity to destination. else Initialize ζ i 1,t trajectories with ζi,t t s. end if Transmit ζ i 1,t to each vehicle j N i. Receive ζ j 1,t from each vehicle j N i. while condition do ζ i n+1,t Obtain Compute ζ i n+1,t solving Problem 4. i as the average of ζ n+1,t and ζi n,t. Transmit ζ i n+1,t to each vehicle j N i. Receive ζ j n+1,t from each vehicle j N i. end while Output ζ i,t. TABLE I: Physical constants and system parameters. Gravity g 9.81 m/s 2 Air density ρ 1.22 g/m 3 Air-drag coefficient c nom Friction coefficient c r Car mass m 1200 g Energy capacity of the battery ɛ b 70 Wh Motor efficiency η Motor parameter 1 b Motor parameter 2 b Fixed motor gear g r 3 - Wheel radius r w 0.3 m Frontal surface of the cars A f 3 m 2 Distance to rear axles l r 2 m Distance to front axles l f 2 m Vehicle length l 4 m Vehicle width w 2 m Minimum velocity on highway v min 100/3.6 m/s Maximum velocity on highway v max 130/3.6 m/s Minimum torque T min -100 N m Maximum torque T max 100 N m Norm of the difference Iteration Fig. 2: Norm of the difference between the trajectories at the new and previous iteration, using the distributed controller. Figure 2 shows the absolute difference the velocity trajectories at two consecutive iterations, in a case where G includes 5 vehicles. The trajectories computed by the distributed controller appear to converge after 10 iterations. To limit the computational time, the termination condition of the distributed controller is chosen so that the iterations stop after 5 iterations. A. Simulation setup VI. SIMULATION RESULTS Our simulation setup, including vehicles models and the MPC controllers described above, is implemented in Julia [17]. Optimization problems were modeled using the JuMP [18] modelling language and the IpOpt [19] solver. Because the Problems 3 and 4 are non-linear and non-convex, we only expect to achieve local optimality. The assumed model parameters are summarized in Table I. The horizon for the eco-platooning controllers is H = 120 s. To compare the energy consumption, a solo controller is introduced as a baseline: it is a fully decentralised MPC controller that performs the trip planning for each vehicle, without considering information from other vehicles into account. TABLE II: Two-vehicles: simulation conditions. Vehicle 1 2 Initial Position m Initial Velocity m/s Initial Charge Final Position m Desired Arrival Time s Minimum Charge B. Two-vehicles scenario In this simulation, we consider two vehicles that have the same target distance and arrival time, but are separated by an initial gap see Table II. Figure 3a shows that, with the solo controller, the two vehicles follow the same velocity trajectory. The initial position gap is maintained throughout the trip. Figure 3b shows that, with the centralized controller, vehicles 1 and 2 respectively accelerate and slow down to form a platoon at t = 25 s. Afterwards, vehicles drive at identical velocity, until they disband by respectively slowing down and accelerating, to reach their destinations. The distributed controller computes similar trajectories to the centralised controller, as seen in Figure 3c. Table III compares the energy savings achieved by the centralized and the distributed controllers, when compared to the solo controller. The distributed controller achieves a small performance loss compared to the centralised controller. Figure 4 shows the trajectories of the state of charge of vehicle 1 with each high-level controller. TABLE III: Two vehicles scenario: total energy consumption. Solo Centralised Distributed Energy MJ Savings %

6 Velociy in m/s Velociy in m/s Velociy in m/s Vehicle 1 Vehicle a Solo Controller b Centralised Controller. 28 c Distributed Controller. Fig. 3: Closed-loop velocity trajectories in the 2 vehicles scenario. C. Five-vehicles scenario This scenario describes a more complex scenario involving five vehicles. The selected initial and final conditions, summarized in Table IV, mae overtaing manoeuvres necessary for each vehicle to successfully reach its destination. Figures 5a and 5b show the position trajectories for all vehicles, using the centralised and the distributed controllers, respectively. The trajectories and platoon formations attained by the two controllers differ, however the overall trip planning is similar. The energetic performance is summarized in table V. The State of charge Solo Centralised Distributed Fig. 4: State of charge of vehicle 1 in the 2 vehicles scenario. TABLE IV: Five-vehicles: simulation conditions. Vehicle Initial Position m Initial Velocity m/s Initial Charge Final Position m Desired Arrival Time s Minimum Charge savings achieved by both eco-platooning controllers compared to the solo controller are in the order of 10 %, slightly smaller than in the two-vehicles scenario. Clearly, every specific scenario has different potential for improvement, regardless of the control approach used. In this case, the distributed controller achieves a slightly better performance than the centralised one. This is not surprising because, as mentioned above, our solutions to Problems 3 and 4 are only locally optimal. D. Open loop comparison: Merging trajectories To better understand the different performance of the centralized and distributed controllers, we now compare their open-loop trajectories. Five vehicles are equally separated by 50 m and travel at a nominal velocity of 32 m/s. Figures 6a and 6b show the open-loop planned position trajectories computed by the two controllers. In both cases, all vehicles form a platoon at t = 40 s. However, the merging manoeuvres to form the platoon appear better coordinated when using the centralised controller. VII. CONCLUSIONS In this paper we proposed a distributed MPC approach for a platoon planning problem. Our setup assumes electrical vehicles for personal transportation, and the goal is to minimize the overall energy consumption while meeting constraints on time, distance and battery state of charge at the TABLE V: Five vehicles scenario: total energy consumption. Solo Centralised Distributed Energy MJ Savings %

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8 [17] The julia language homepage. [18] Jump - julia for mathematical optimization. readthedocs.io/en/latest/. [19] Coin-or website for ipopt. Ipopt.