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Energy-Optimal Adaptive Cruise Control based on Model Predictive Control
Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World The Federation of Automatic Control Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 12563–12568
Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control based on Model Predictive Control based on Model Predictive Control based on Model Predictive Control based on Model Predictive Control
Andreas Weißmann, Daniel G¨ orges and Xiaohai Lin Andreas o Andreas Weißmann, Weißmann, Daniel Daniel G¨ G¨ orges rges and and Xiaohai Xiaohai Lin Lin Andreas Weißmann, Daniel G¨ orges and Xiaohai Lin Juniorprofessorship for Electromobility, University of Kaiserslautern, Juniorprofessorship for of Juniorprofessorship for Electromobility, Electromobility, University of Kaiserslautern, Kaiserslautern, Erwin-Schr¨ odinger-Straße 12, 67663 University Kaiserslautern, Germany Juniorprofessorship for Electromobility, University of Kaiserslautern, Erwin-Schr¨ o dinger-Straße 12, 67663 Kaiserslautern, Germany Erwin-Schr¨ o dinger-Straße 12, 67663 Kaiserslautern, Germany E-mail: weissmann|goerges|[email protected] Erwin-Schr¨ odinger-Straße 12, 67663 Kaiserslautern, Germany E-mail: weissmann|goerges|[email protected] E-mail: weissmann|goerges|[email protected] E-mail: weissmann|goerges|[email protected] Abstract: In this paper an approach for an energy-optimal adaptive cruise control based Abstract: In an approach for adaptive cruise control based Abstract: In this this paper paper an (MPC) approach for an an energy-optimal energy-optimal adaptive cruise control of based on model predictive control is presented. The approach uses the knowledge the Abstract: In this paper an (MPC) approach for an energy-optimal adaptive cruise control of based on model predictive control is presented. The approach uses the knowledge on model predictive control (MPC) is presented. The approach uses the knowledge of the the given route to precalculate a position-dependent energy-optimal speed trajectory using dynamic on model predictive control (MPC) is presented. The approach uses the knowledge of the given precalculate aaadditional position-dependent energy-optimal speed trajectory using dynamic given route route to to while precalculate position-dependent energy-optimal speedroad trajectory using dynamic programming taking information like speed limits, slope and travel time given route to while precalculate aadditional position-dependent energy-optimal speedroad trajectoryand using dynamic programming speed travel programming while taking taking additional information information like speed limits, limits, roadisslope slope travel time time into account during the optimization. The model like predictive controller usedand to control the programming while taking additional information like speed limits, roadisslope and travel time into account during the optimization. The model predictive controller used to control the into account during the vehicle optimization. The model predictive controller is usedspeed to control the traction force of the host such that the vehicle speed follows the optimal trajectory into account during the optimization. The model predictive controller is used to control the traction of vehicle such that follows optimal speed traction force of the the host host vehicle such constraints that the the vehicle vehicle speed follows the optimal vehicle speed trajectory trajectory as good force as possible while ensuring like speed distance to athe preceding or speed traction force of the host vehicle such constraints that the vehicle speed follows the optimal vehicle speed trajectory as good as possible while ensuring like distance to a preceding or speed as good as possible while ensuring constraints like distance to a preceding vehicle or speed limits. To show the benefits of the approach, a comparison of the energy consumption between as goodToasshow possible while ensuring constraints like distance to a preceding vehicle or speed limits. the of the aa comparison of the consumption between limits. To showvehicle the benefits benefits of preceding the approach, approach, comparison ofroute the energy energy consumption between the controlled and the vehicle on the same is performed. For the speed limits. To showvehicle the benefits of preceding the approach, a comparison ofroute the energy consumption between the controlled and the vehicle on the same is performed. For the speed the controlled vehicle andvehicle, the preceding on the same performed.show For the speed profile of the preceding data ofvehicle real test drives is route used. is Simulations that the the controlled vehicle andvehicle, the preceding vehicle on the same route is performed.show For the speed profile the data test drives is that profile of of leads the preceding preceding vehicle, data of ofofreal real test drives is used. used. Simulations Simulations show that the the approach to a significant reduction the energy consumption compared to the preceding profile of leads the preceding vehicle, data ofofreal test drives is used. Simulations showpreceding that the approach to significant reduction the compared to approach leads to aa route. significant reductionthe of simulations the energy energy consumption consumption compared to the the preceding vehicle on the same Furthermore indicate that the approach achieves high approach leadssame to a significant reductionthe of the energy consumption compared to the preceding vehicle on indicate that the vehicle savings on the the same route. Furthermore the simulations simulations indicate that car. the approach approach achieves achieves high high energy evenroute. with aFurthermore poor prediction model for the preceding vehicle savings on the same route. Furthermore the simulations indicate that car. the approach achieves high energy even with a poor prediction model for the preceding energy savings even with a poor prediction model for the preceding car. energy even with aFederation poor prediction model for the preceding car. Ltd. All rights reserved. © 2017, savings IFAC (International of Automatic Control) Hosting by Elsevier Keywords: Adaptive cruise control, Model predictive control, dynamic programming, cloud, Keywords: Keywords: Adaptive Adaptive cruise cruise control, control, Model Model predictive predictive control, control, dynamic dynamic programming, programming, cloud, cloud, prediction Keywords: Adaptive cruise control, Model predictive control, dynamic programming, cloud, prediction prediction prediction 1. INTRODUCTION interface for control of the vehicle speed and acceleration 1. INTRODUCTION INTRODUCTION interface for control control oftothe the vehicleACC speedfunctionality and acceleration acceleration 1. interface for vehicle speed and has inspired researchof combine with 1. INTRODUCTION interface for control oftothe vehicleACC speedfunctionality and acceleration has inspired research combine with In recent years the reduction of the energy consumption in has inspired research to combine ACC functionality with different control strategies to improve energy efficiency and has inspired research to combine ACC functionality In recenthas years the reduction reduction of the energy energy consumption in different different control strategies to improve improve energy efficiencywith and In recent years the of the consumption in vehicles become a major research topic. Reducing the control strategies to energy efficiency and safety alike. In recenthas years the reduction of the energy consumption in safety different control strategies to improve energy efficiency and vehicles become major research topic. Reducing and the alike. vehicles has become aais major research Reducing the energy consumption appealing fromtopic. an economical alike. vehiclesconsumption has become ais major research topic. Reducing and the safety safety alike. energy appealing from an economical In (Turri et al., 2016) an approach was proposed that energy consumption is appealing an economical and In (Turri et environmental perspective due tofrom increasing energy costs al., 2016) an approach wasspeed proposed that energy consumption is appealing from an economical and In (Turri et 2016) approach was proposed that environmental perspective due to increasing energy costs uses MPC to al., follow an an energy optimal trajectory environmental perspective due increasing energy point costs uses and emission requirements andto from a functional In (Turri etto al., 2016) an approach wasspeed proposed that MPC follow an energy optimal trajectory environmental perspective due tofrom increasing energy point costs uses MPC to follow an energy optimal speed trajectory and emission requirements and a functional derived by to dynamic programming. For both, MPC and and emission requirements andrange from of a electric functional point derived of view, e.g. for increasing the vehicles. uses MPC follow an energy optimal speed trajectory by dynamic programming. For both, MPC and andview, emission requirements andrange from of a electric functional point derived byprogramming, dynamic programming. both, MPClimits and of e.g. for for increasing the vehicles. dynamic informationFor about speed of view, e.g. increasing the range of electric vehicles. Considerable research has been devoted to constructive derived by dynamic programming. For both, MPC and dynamic programming, information about speed limits limits of view, e.g. for increasing the range of electric vehicles. dynamic information speed Considerable research has been been devoted to(‘downsizing’) constructive and road programming, slope are regarded during about optimization. The Considerable has devoted constructive approaches likeresearch more efficient engine designsto dynamic programming, information about speed limits and road slope are regarded during optimization. The Considerable research has been devoted to constructive and road slope are regarded during optimization. The approaches like more efficient engine designs (‘downsizing’) considered scenario is platooning of heavy-duty vehicles. approaches like more efficient engine more designsrecent (‘downsizing’) or lightweight materials. Another research considered and road slope are isregarded during optimization. The scenario platooning of heavy-duty heavy-duty vehicles. approaches like materials. more efficient engine more designsrecent (‘downsizing’) is platooning of vehicles. or lightweight Another research considered The resultsscenario show that this approach leads to good fuel or lightweight materials. Another more recent research direction is focused on the benefits that can be obtained considered scenario is platooning of heavy-duty vehicles. The results show that this approach leads to good fuel or lightweight materials. more recent obtained research The results show Turri that this leads good fuel direction is focused focused oncomplete theAnother benefits that can savings. However, et al.approach (2016) do not to consider the direction is on the benefits that One can be be obtained by the control of the vehicle. widely used The results show that this approach leads to good fuel savings. However, Turri traffic et al. al. (2016) (2016) doapplication not consider consider the direction is focused oncomplete the benefits that One can be obtained savings. However, Turri et do not the by the control of the vehicle. widely used influence of interfering and the of by the control of the vehicle. Oneiswidely used influence approach to reduce thecomplete energy consumption the adjustsavings. However, Turri traffic et al. (2016) not consider of interfering and the thedoapplication application of the by the control of the complete vehicle. Oneiswidely used influence of traffic and of the approach to reduce reduce the energy consumption consumption the adjustadjustapproach to interfering other vehicle classes. approach to the energy is way, the ment of the driving behavior in an efficient which influence of interfering traffic and the application of the approach to other vehicle classes. approach to reduce the energy consumption is way, the adjustapproach to other vehicle classes. ment of the the driving behavior in an an efficient efficient which ment of driving behavior in way, which has been shown to be very promising in reducing the approach to other vehicle approach classes. based on MPC and In this paper a similar mentbeen of the driving behavior in an efficient way, which has shown to be very promising promising in et reducing the In this paper paper similar approach The basedMPC on is MPC and has been shown to very in reducing the fuel consumption andbe emissions in (Mierlo al., 2004). In this aa similar approach based on MPC dynamic programming is proposed. usedand for has been shown to be very promising in reducing the fuel consumption andcan emissions in (Mierlo et eco-driving al., 2004). 2004). dynamic In this paper a similar approach The basedMPC on is MPC and programming is proposed. proposed. used for fuel consumption and emissions in (Mierlo et al., Such an adjustment be done by using an dynamic programming is The MPC is used for following an energy-optimal speed trajectory while addifuel consumption and emissions in (Mierlo et al., 2004). Such an adjustment adjustment can be be done by usingefficient an eco-driving eco-driving dynamic programming is proposed. The MPCwhile is used for following an energy-optimal energy-optimal speed trajectory addiSuch an can using an assistance, which is used to done make by energy driving following an whilevarious additionally regarding preceding speed traffic trajectory and ensuring Such an adjustment can be usingefficient an eco-driving assistance, which is used used to done make by energy driving followingregarding an energy-optimal speed trajectory whilevarious additionally preceding traffic and ensuring assistance, which is to make energy efficient driving strategies, e.g. shifting strategies or speed proposals for tionally regarding preceding traffic and ensuring various constraints. This optimal speed trajectory is calculated assistance, e.g. which is usedstrategies to make energy efficient driving strategies, shifting ordriver speed proposals for constraints. tionally regarding preceding traffic and ensuring various This optimal speed trajectory is calculated calculated strategies, e.g. shifting strategies speed proposals for astrategies, certain route, accessible to theor via a humanconstraints. This optimal speed trajectory is offline for the given route before the trip using DP and e.g. shifting strategies or speed proposals for a certain route, accessible to the driver via a humanconstraints. This optimal speed trajectory is calculated offline for the given route before the trip using DP and amachine certaininterface route, accessible to the driver via a systems human- offline (HMI). Eco-driving assistance for theonline given for route before the trip using DP and is then used the MPC. The precalculation ala certain route, accessible to the driver via a humanmachine interface (HMI). Eco-driving Eco-driving assistance systems offline for theonline given for route before the trip using DP and is then used the MPC. The precalculation almachine interface (HMI). assistance systems using dynamic programming (DP) to derive an energyis then used online for the MPC. The precalculation allows a higher complexity of the DP optimization problem machine interface (HMI). Eco-driving assistance systems using dynamic programming (DP) to to vehicles derive an an energyis then used online for the MPC. precalculation allows higher complexity of the the DPThe optimization problem using dynamic programming (DP) derive energyoptimal speed profile for conventional have been lows aa higher complexity of DP optimization problem (i.e. high accuracy of energy consumption representation) using dynamic programming (DP) to vehicles derive an energyoptimal speed profile for conventional have been lows a higher complexity of the DP optimization problem (i.e. high accuracy of energy energy consumption representation) optimal speede.g. profile for conventional vehicles have been (i.e. investigated in (Luu et al., 2010) and for electric high accuracy of consumption representation) while keeping the online optimization problem of the MPC optimal speede.g. profile for conventional vehicles have been while investigated in et (Luu et al., al.,2012; 2010) and for electric (i.e. high accuracy of energy consumption representation) keeping the online optimization problem of the the speed MPC investigated e.g. in (Luu et 2010) and for electric vehicles e.g. in (Dib al., 2011, Lin et al., 2014). while keeping the online optimization problem of MPC very simple. A method to derive the energy-optimal investigated e.g. in et (Luu et al.,2012; 2010) and for 2014). electric very vehicles e.g. in (Dib al., 2011, Lin et al., whilesimple. keepingAthe onlinetooptimization problem of the speed MPC method derive the energy-optimal vehicles e.g. in (Dib et al., 2011, 2012; Lin et al., 2014). simple. method programming to derive the energy-optimal speed trajectory viaA dynamic for an electric vehicle vehiclesadvanced e.g. in (Dib et al., 2011, 2012; Lin et(ADAS) al., 2014). With driver assistance systems like very very simple. Adynamic method programming to derive the energy-optimal speed trajectory via for an electric vehicle dynamic programming foral., an electric vehicle With advanced driver (ACC) assistance systems (ADAS) like trajectory has alreadyvia been introduced in (Lin et 2014) and will With advanced driver assistance systems like adaptive cruise control many vehicles(ADAS) are by now trajectory via dynamic programming foral., an electric vehicle has already been introduced in (Lin et 2014) and and will With advanced driver (ACC) assistance systems (ADAS) like has already been introduced in (Lin et al., 2014) will adaptive cruise control many vehicles are by now be used in this work as well. It should be mentioned that adaptive cruise control (ACC) many vehicles are by now endowed with the technical equipment for implementing hasused already been introduced et be al.,mentioned 2014) and that will be in this this work as trajectory well. in It (Lin should adaptive cruise control (ACC) many vehicles are by now be used in work as well. It should be mentioned that endowed with strategies the technical equipment for implementing for this optimal speed a further constraintthat on endowed with the technical equipment for implementing these driving without direct interaction with be used in this work as well. It should be mentioned for this optimal speed trajectory a further constraint on endowed with strategies the technical equipment for implementing for this optimal speed trajectory a further constraint on these driving without direct interaction with the trip time is considered. The results in this paper are these driving strategies direct interaction the driver. ACC provideswithout additional safety benefits with due the for this optimal speed trajectory a further constraint on tripto time is considered. considered. Thecan results in this this paper are are thesedriver. driving strategies without direct interaction with the trip time is The results in paper the ACC provides additional safety benefitsof due related electric vehicles, but be easily extended for the driver. ACC provides additional safety benefits due to the distance control functionality. The availability an the trip time is considered. The results in this paper are related to electric vehicles, but can be easily extended for thethe driver. ACC provides additionalThe safety benefitsofdue to distance control functionality. availability an related to electric vehicles, but can be easily extended for to the distance control functionality. The availability of an to the distance control functionality. The availability of an related to electric vehicles, but can be easily extended for
Copyright © 2017, 2017 IFAC 13074 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 13074 Copyright ©under 2017 responsibility IFAC 13074 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 13074 10.1016/j.ifacol.2017.08.2196
Proceedings of the 20th IFAC World Congress 12564 Andreas Weißmann et al. / IFAC PapersOnLine 50-1 (2017) 12563–12568 Toulouse, France, July 9-14, 2017
Offline Cloud
Dynamic programming Optimal speed trajectory vref Communication
vref,α,vmax,vmin vp,d
Measurement e.g. Radar,Lidar
Geographical information - Speed limits vmax,vmin - Altitude profile α
vref,α,vmax,vmin Host vehicle Ft Vehicle MPC Controller Position x
Route
v
vp,d Preceding vehicle
Online
Fig. 1. Considered scenario with host and preceding vehicle use in conventional vehicles by additionally considering the shifting strategy. For the online part, namely the MPC, the main goal is to control the traction force of the vehicle in a way that the vehicle speed follows the energy-optimal speed trajectory if there is no preceding traffic or keep the distance in a safe range if there is any interfering traffic. During the optimization further constraints, e.g. maximum traction and brake force, are used to ensure a feasible vehicle behavior. To evaluate the MPC, several simulations are performed where the host vehicle follows a preceding vehicle on the same fixed route. The considered host vehicle in this work is a Nissan Leaf. The speed profile of the preceding vehicle is based on real test drives on the considered route. The energy consumption and trip time for the controlled vehicle and the preceding vehicle are compared to each other. The main contribution of this work consist in uniquely combining an offline optimization using dynamic programming with a precise model to derive an optimal speed profile and an online optimization implementing ACC functionality using MPC together with a simple model which provides a compromise between accuracy and complexity. The resulting new ADAS framework provides a combination of ACC functionality with the capability of following an energy optimal reference speed profile. This framework is evaluated by comprehensive simulations based on real world data. The paper is organized as follows. In Section II the optimization problem is formulated and an outline of the optimization problem used for dynamic programming is given. In Section III the test scenario and the simulation results of the proposed MPC are presented. Conclusions are finally provided in Section IV. 2. PROBLEM FORMULATION In Fig. 1 the considered scenario with the host vehicle and a preceding vehicle is illustrated. The proposed approach can be divided into an offline and an online part. The offline part, as shown in the upper dotted box of Fig. 1, includes the communication of data between the host and a cloud. Specifically the host sends his desired route to
the cloud or alternatively a desired destination if routing should be done in the cloud. The cloud then derives the speed limits and altitude profile along the route for use in dynamic programming and the MPC later on. After that the optimal speed trajectory is determined via dynamic programming. The geographical information as well as the optimal speed trajectory is sent to the host vehicle for use in the online part, namely the MPC. The online part includes following the optimal speed trajectory while ensuring a safe distance to a potential preceding vehicle. The online part is illustrated in the lower dotted box in Fig. 1. It is worth mentioning that solving a DP problem for a certain route can take very long due to the high complexity that increases with the length of the route. To solve this issue, the calculated optimal speed trajectories are stored in the cloud to avoid a recalculation upon the next request, which is especially beneficial when considering commuters. If a calculation cannot be avoided, it is possible to simply use the conventional ACC until the optimal speed trajectory is received. 2.1 System dynamics The system dynamics for the considered scenario can be expressed as dk+1 = dk − vk Ts + vp,k Ts vk+1 = vk + ak Ts
(1) (2)
where v and a are the velocity and acceleration of the controlled vehicle and Ts is the sampling time. d describes the distance between the host and the preceding vehicle which depends on the prediction of the velocity of the preceding vehicle vp . To include the traction force Ft as the controlled variable and the driving resistances, the longitudinal dynamics are used to replace a in (2). The longitudinal dynamics are given by meq a = Ft − Fr − Fa − Fg (3) with meq = mv + mr being the equivalent mass including the vehicle mass mv and the influence of the rotating masses mr . The considered driving resistance forces are: aerodynamic resistance force Fa = 21 cf Af ρa v 2 with the air density ρa , the air drag coefficient cf and the projected frontal area Af , neglecting the influence of the wind speed, rolling resistance force Fr = mv gcr with the gravitational acceleration g and the rolling resistance coefficient cr , and grading resistance force Fg = mv gα with the road angle α. It should be mentioned that in Fg and Fr the small-angle approximations sin(α) = α and cos(α) = 1 are used instead of the trigonometric functions. By substituting (3) into (2) and rearranging, the equation describing the velocity can be written as 1 Ts Ft − mv gcr − ρa Af ca vk2 − mv gαk . vk+1 = vk + meq 2 (4) To handle the nonlinear term vk2 resulting from the aerodynamic resistance force in (4), the predicted state sequence resulting from the last optimization is used instead of vk2 , assuming that the deviations of the predicted state sequence between two subsequent optimization steps are very small.
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2.2 Cost function
12565
7000
For the desired objective of following an energy-optimal speed trajectory vref it is a reasonable approach to use a cost function that punishes the deviation of the vehicle speed from its energy-optimal reference speed. A quadratic cost function N −1 2 q(vk − vref,k )2 + pFbrake,k , (5) J=
F
Traction force Ft (N)
t,max
y1=c 1 v + c 2
6000 5000 4000 3000 2000
k=0
is chosen which represents the weighted deviations to the reference speed vref over the control horizon N . The 2 additional term Fbrake is used to punish the usage of hydraulic brake force Fbrake as the regenerative braking of the electric vehicle is favorable over hydraulic braking. In (5) q > 0 and p > 0 denote weighting factors. 2.3 Constraints
To ensure realizable results of the optimization, several constraints need to be considered. From the physical limitations of the electric motor a maximum traction force arises. For the considered electric vehicle this constraint is derived by transforming the maximum torque over angular velocity diagram Tmax (ωmotor ) into maximum traction force over velocity diagram Ft,max (v). The transformation )γη Rw follows from Ft,max (v) = Tmax (ωRmotor and v = ωmotor γ w with the wheel radius Rw , the gearbox efficiency η, and the gear ratio γ. The resulting diagram is shown in Fig. 2. To obtain a convex constraint the maximum traction force is approximated as indicated by the red dotted line in Fig. 2. From this Ft,k,max = c1 vk + c2 (6) follows with c1 and c2 being identified constants.
1000 0
0
18
36
54
72
90
108
126
144
162
v (km/h)
Fig. 2. Maximum traction force over velocity diagram Another important constraint follows from the ACC functionality, which provides a safe distance to the preceding vehicle. In this approach a constant headway h and a fixed minimum distance dmin is used, leading to dmin + hvk ≤ dk . (10) In addition, speed limits can be addressed by vmin,k ≤ vk ≤ vmax,k ,
(11)
with 0 ≤ vmin,k mainly used to prevent driving backwards if the distance gets too small. The upper speed limit vmax,k is based on the admitted speed along the route and is derived during the offline part as indicated in Fig. 1. However, it also can be easily derived online, using GPS and map data. 2.4 MPC optimization problem
The braking system of the car has limitations in terms of the maximum deceleration. The limitations can be divided into the regenerative braking capability of the electric motor and the braking capability of the hydraulic brake for modeling. As a permanent magnet synchronous motor is used in the particular electric vehicle, it is reasonable to describe the regenerative braking limits in the generator mode, which are expressed by a negative traction force Ft , in the same way as in the motor mode by mirroring the diagram in Fig. 2 to the generator mode. The hydraulic brake force Fbrake is only included as a soft constraint, allowing the use of additional braking force if the regenerative braking limits are exceeded to ensure safety constraints. The usage of the hydraulic brake force is unfavorable considering the energy efficiency and its usage therefore punished in the cost function (equation (12)).
With the system dynamics, the cost function, and the constraints the online optimization problem for MPC, which is solved at every time instance can be summarized as N −1 2 min (12) q(vk − vref,k )2 + pFbrake,k
The limits for regenerative braking (−Ft,k,max ) and the hydraulic brake force Fbrake result from Ft,k,brake,max = −c1 vk − c2 − Fbrake,k . (7)
2.5 Optimal speed trajectory
The hydraulic brake force also has limits which simply follow from the technical specification as 0 ≤ Fbrake ≤ Fbrake,max . (8) Summarizing (6) to (8) the overall constraint resulting from the traction force and the braking system limitations can be formulated as −c1 vk − c2 − Fbrake,k ≤ Ft,k ≤ c1 vk + c2 . (9)
Ft,k ,Fbrake,k
k=0
subject to
Ts 1 (Ft− mv gcr− ρa Af ca vk2− mv gαk ) meq 2 dk+1 = dk − vk Ts + vp,k Ts − c1 vk − c2 − Fbrake,k ≤ Ft,k ≤ c1 vk + c2 0 ≤ Fbrake,k ≤ Fbrake,max dmin,k + hvk ≤ dk vmin,k ≤ vk ≤ vmax,k vk+1= vk+
(13a) (13b) (13c) (13d) (13e) (13f)
In transportation it is in general a problem to find a suitable compromise between the energy consumption and trip time. Dynamic programming (DP) is used to address this problem in the proposed approach. The used problem formulation for DP and a precise model of the electric vehicle has been introduced in (Lin et al., 2014) and will be briefly reviewed in the following. Due to the position-dependent nature of the road angle α and speed limits vmin/max , it is convenient to use
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a position-dependent representation of the system. This requires a transformation of the velocity v(t) and position x(t) into v(x) with a discretization step of ∆x instead of Ts . This discretization leads to a partition of the route with the overall length xmax into M segments of length ∆x for which the optimal speed trajectory is calculated. The desired optimal speed trajectory results from the optimization problem M −1 tk (vk )Pbattery,total,k (vk )+βtk (vk )+λ(Tmotor,k (vk ))2 min vk
Stored Data
Equations (15a) and (15b) represent the initial state for the trip. The constraints (15c)-(15e) are used to keep the SOD of the battery in safe boundaries during the trip and to set defined final values for the SOD and the final velocity at the end of the trip. The constraints (15f) and (15g) are the same as in (9) and (11), addressing the system limitations on maximum traction force and maximum braking force as well as the speed limits. The result of the optimization is the optimal speed trajectory v ∗ (x) as well as the overall trip time Tduration which is calculated M −1 from Tduration = k=0 tk . It is worth mentioning that dynamic programming already considers the maximum speed limit vmax and passes this to the MPC in form of v ∗ (x) and therefore does not need to be considered in the optimization problem of the MPC. 3. SIMULATION AND RESULTS 3.1 Simulation setup The simulation setup is shown in Fig. 3. In the figure a hat on a variable ˆ ∗ indicates that the predicted sequence over the horizon N is passed on to the next block. The prediction block contains the prediction of vp over the horizon N which will be described later. The conversion from
MPC
^ x(t)
Urban
140
^
Ft(t) System Model
v(t),d(t)
Interurban
Highway
120
Velocity v (km/h)
subject to
where SOD denotes the state of discharge of the battery of the electric vehicle. In (14) the term tk (vk )Pbattery,total,k (vk ) is the representation of the energy consumption for a single segment, with tk (vk ) being the trip time for ∆x in segment k. Pbattery,total,k (vk ) represents the power drawn from the battery including electric motor, electrified auxiliary units and so on. The terms βtk (vk ) and λ(Tmotor,k (vk ))2 are used to tune the trip time and the ride comfort respectively by weighting the trip time and motor torque Tmotor . An increase of β leads to a reduction in the trip time and an increase of λ improves the ride comfort by punishing the usage of high torque for accelerating and braking. However, in the following only the trip time and therefore β is considered and tuned.
Prediction
^ vp(t)
Fig. 3. Block diagram of the simulation structure
(14)
(15a) (15b) (15c) (15d) (15e) (15f) (15g)
v^max(t),v^min(t)
vref(x),α(x) vmax(x),vmin(x) vp(t)
k=0
v0 = vinit SOD0 = SODinit vM ∈ [vmin,M , vmax,M ] SODM ∈ [SODmin,M , SODmax,M ] SODk ∈ [SODmin,k , SODmax,k ] vk ∈ [vmin,k , vmax,k ] Tmotor,k ∈ [Tmin,k , Tmax,k ]
^ v(t),d(t)
^ ^ vref(t),α(t)
100 80 60 40 Optimal speed trajectory vref
20
Upper speed limit vmax 0
0
2000
4000
6000
8000
10000 12000 14000 16000 18000
Position x (m)
Fig. 4. Optimal speed trajectory vref discrete-position data to discrete-time data is performed by means of the predicted position sequence x ˆ which is derived from the system model and the solution of the optimization problem (12). For the speed profile vp data acquired from real trips is used. The data was gathered from a commuter on a route from Kaiserslautern to Landstuhl in Germany with a length of 17.8 km. This route contains urban and interurban segments as well as a highway. The data amounts to 85 trips in total and therefore contains a wide variety of possible traffic conditions. To obtain a realistic trip time for the optimal speed trajectory vref , the mean value over all 85 trips is used which results in a desired trip time of 868 s. The dynamic programming is implemented with the opensource software DPM which was developed at the ETH Z¨ urich (Elbert et al., 2013). Solving the optimization problem (14) with DPM takes 38.5 s per run. It must be mentioned that several runs are necessary to reach the desired trip time of 868 s by tuning the parameter β. However, in practice this procedure needs to be done only once and the information can then be stored in the cloud and sent to the host upon the next request for that route. The resulting position-dependent optimal speed trajectory vref is shown in Fig. 4 together with the road type and the upper speed for each segment. It can easily be seen that most of the time the speed is close to the speed limit vmax so that the desired trip time of 868 s can be achieved. The final speed boundaries defined by (15c) range from 60 km/h to 120 km/h for the different trips and leads to the braking maneuver which can be observed near the end position at 17.8 km. This braking maneuver indicates the behavior to drive as slow as possible for minimum energy consumption. This also shows the necessity to punish the trip time. The speed trajectory vref is stored for use in the MPC. In the simulated scenario the host vehicle should follow the reference trajectory vref while ensuring a safe distance to
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the preceding vehicle with the speed profile vp starting at a distance of d = 150 m ahead. A sampling time of Ts = 0.2 s and a horizon of N = 100 is used for the simulation. The initial speed of the host vehicle is chosen to be the same as the initial speed of the preceding vehicle vp,init . The simulation is performed for all 85 profiles of vp . To predict vp over the horizon N , a prescient MPC (PMPC) and a frozen-time MPC (FTMPC) approach is used for the simulations. PMPC assumes full knowledge of the preceding car’s velocity over the horizon N and represents the best achievable prediction. FTMPC uses only knowledge from the present measurements of vp (known e.g. from a recent measurement with radar/lidar/camera) and assumes this to be constant over the horizon N . It represents therefore a very simple to implement and rather poor prediction. To show the energy saving potential of this approach, the difference in energy consumption between the controlled and the preceding vehicle for every profile vp is calculated using the electric vehicle model introduced in (Lin et al., 2014).
25.1 s. To further evaluate the controller performance, it is beneficial to split the results into two datasets. The first dataset contains the results for which the host vehicle can completely follow the reference trajectory vref . This dataset contains 45 of the simulation results. This is the case if the preceding vehicle has a higher speed than the host vehicle and hence there is never the necessity to leave the reference trajectory due to the distance constraint (13e). The second dataset contains the data in which a deviation from the reference trajectory vref is necessary to fulfill the distance constraint (13e). This dataset contains 40 of the simulation results. The results for the first dataset are shown in Table 2 and those for the second dataset in Table 3. Table 2. Results first dataset PMPC FTMPC
Table 1. Overall results δ¯ -7.5% -7.2%
δmin -18.4% -17.9%
δmax 3.7% 3.6%
∆T¯ 25.15s 25.17s
∆Tmin 0.8s 0.8s
∆Tmax 84.2s 84.2s
PMPC FTMPC
δmin -16.9% -16.9%
δmax 0.4% 0.4%
∆T¯ 39.9s 39.9s
∆Tmin 5.2s 5.2s
∆Tmax 84.2s 84.2s
δ¯ -7.8% -7.2%
δmin -18.4% -17.9%
δmax 3.7% 3.6%
∆T¯ 8.5s 8.5s
∆Tmin 0.8s 0.8s
∆Tmax 55.8s 55.8s
From Table 2 it can be seen that the results are the same for PMPC and FTMPC. This is not unexpected as in both cases the same speed trajectory is followed and therefore the trip time and energy consumption are the same. For this dataset the preceding vehicle is faster and therefore has a shorter trip time than the controlled vehicle. However, as the speed trajectory is followed the desired trip time of 868 s (vref ) can nearly be reached with an offset of approximately +2.4 s that results from the fact that the controlled vehicle starts at a different speed than vref,init . Table 3 shows the results of the second dataset. The mean value of the trip time difference (8.5 s) is now fairly small as we partially have to follow the slower speed of vp in certain sections due to the distance constraint. The mean values of the energy consumption reduction are with 7.8% for PMPC and 7.2% for FTMPC very close and indicates that the proposed approach is only influenced slightly by the used prediction model for vp . 130
From Table 1 it can be seen that a good overall energy reduction of 7.5% for PMPC and 7.2% for FTMPC can be achieved considering all datasets with the best achievable reduction of 18.4% and 17.9%, respectively. It is to be mentioned that only four profiles of vp led to an increased energy consumption for the host vehicle in comparison to the preceding vehicle, with three of them being < 0.5% and one of them being 3.7%. This results from the fact that a human driver and therefore vp does not always fulfill the speed limits, allowing to take a more beneficial working point than the constrained reference trajectory vref and thus resulting in an overall lower energy consumption for the preceding vehicle. The difference of the trip time for PMPC and FTMPC is for both cases almost identical with a mean value of
Velocity v (km/h)
PMPC FTMPC
δ¯ -7.2% -7.2%
Table 3. Results second dataset
3.2 Results The proposed control structure and control strategy as shown in Fig. 3 is implemented and simulated in MATLAB and evaluated with an electric vehicle model in Simulink. To evaluate the performance of the proposed approach, the relative energy consumption difference δ of the host (Ehost ) and the preceding vehicle (Eprec ) is computed from Ehost − Eprec δ= . (16) Eprec The overall results for the simulations with PMPC and FTMPC are shown in Table 1. In the table δ¯ is the mean value over the simulation results for the 85 vp profiles. The minimum and maximum of the achieved relative energy consumption difference for all results is denoted by δmin and δmax , respectively. In addition the difference in trip time ∆T = thost − tprec is considered. The mean values over all datasets as well as the minimum and maximum are presented. As a following scenario is considered, the host vehicle is always slower than the preceding vehicle.
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Fig. 5. Simulation result for bad traffic conditions To highlight the advantages and drawbacks of the approach as well as the influence of the prediction model, the results of a simulation with very bad traffic conditions, namely stop and go traffic in the urban section of the
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However, the deviation from the reference trajectory, which is necessary to fulfill the distance constraint, leads to an increase in the trip time. The reason is straightforward as vref is precalculated for a trip time of 868 s and a deviation at any position will lead to an increased trip time > 868 s. A possibility to deal with this could be a mechanism which is trying to catch up the ‘lost’ time by increasing the remaining reference trajectory such that the desired trip time can still be fulfilled at the cost of reducing the achievable energy savings. Another drawback is resulting from the fact that the cost function does not include any representation of the energy consumption in order to keep the online optimization problem simple. All energy-related information is included in the optimal speed trajectory vref derived by DP. Therefore after deviations from vref no aspects regarding the energy consumption are considered during the online optimization. 4. CONCLUSIONS In this paper an energy-optimal adaptive cruise control based on MPC has been proposed. Dynamic programming is used to precalculate and store an energy-optimal speed trajectory for a given route, considering geographical information and constraints like speed limits, which then is submitted to a host vehicle upon request. The precalculation is done offline in a cloud before the trip,
Distance d (m)
In this case we have δ = −11.2% for PMPC and δ = −10% for FTMPC. Furthermore ∆T = 55.8 s for both prediction models. The reason for the difference in δ can be seen in Fig. 5 between 200 s and 500 s. It is obvious that PMPC (blue line) with full knowledge over the horizon N recognizes braking and acceleration maneuvers of vp (red line) beforehand and leads to a smoother behavior. FTMPC (black line) has only knowledge of the current velocity vp and therefore has no smoothing behavior but rather follows the velocity of the preceding vehicle, resulting in a higher energy consumption than PMPC due to more and stronger acceleration maneuvers. From the figure it can also be seen that in a major part of the trip the host vehicle can follow the reference trajectory vref (green line) very well for both prediction models, which finally leads to a high energy saving. Deviations to slower speeds only appear when the distance constraint becomes active at some point. This is shown in Fig. 6 (PMPC). The figure shows the result of the optimization applied to the system model over the horizon N with the velocity in the top subplot and the distance to the preceding car in the bottom subplot. From this it can be seen that the host velocity v (blue line) stays as close as possible to the reference trajectory vref (green dotted line), which follows from the problem formulation of the MPC (12) with the goal of minimizing the deviation to vref . It follows that a deviation from vref only occurs when the distance constraint becomes active during the optimization. This is the main reason why FTMPC is fairly close to the performance of PMPC as the prediction model only is taken into account in the explained scenario.
Velocity v (km/h)
route, is presented. In Fig. 5 the implemented velocity over the trip time is shown for the preceding vehicle vp , the host vehicle with PMPC and FTMPC, and the reference trajectory vref for PMPC.
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Fig. 6. Optimization result at t=210 s which allows the usage of a complex and very accurate representation of the energy consumption in DP. A simple MPC framework is then used online in the host vehicle to follow the optimal speed trajectory while ensuring a safe distance to the preceding vehicle. To ensure realizable inputs, limitations on the traction and braking force are included in the optimization as linear constraints. Simulations in a following scenario with speed profiles of real test drives for the preceding vehicle have shown a significant reduction of the energy consumption, using either FTMPC or PMPC as prediction model for the preceding vehicle. REFERENCES Dib, W., Chasse, A., Di Domenico, D., Moulin, P., and Sciarretta, A. (2012). Evaluation of the energy efficiency of a fleet of electric vehicle for eco-driving application. Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, 67(4), 589–599. Dib, W., Serrao, L., and Sciarretta, A. (2011). Optimal control to minimize trip time and energy consumption in electric vehicles. In Proceedings of the 2011 IEEE Vehicle Power and Propulsion Conference. Institute of Electrical and Electronics Engineers (IEEE). Elbert, P., Ebbesen, S., and Guzzella, L. (2013). Implementation of dynamic programming for n-dimensional optimal control problems with final state constraints. IEEE Transactions on Control Systems Technology, 21(3), 924–931. Lin, X., G¨orges, D., and Liu, S. (2014). Eco-driving assistance system for electric vehicles based on speed profile optimization. In 2014 IEEE Conference on Control Applications (CCA). Institute of Electrical & Electronics Engineers (IEEE). Luu, H.T., Nouveli`ere, L., and Mammar, S. (2010). Dynamic programming for fuel consumption optimization on light vehicle. IFAC Proceedings Volumes, 43(7), 372– 377. Mierlo, J.V., Maggetto, G., de Burgwal, E.V., and Gense, R. (2004). Driving style and traffic measures-influence on vehicle emissions and fuel consumption. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 218(1), 43–50. Turri, V., Besselink, B., and Johansson, K.H. (2016). Cooperative look-ahead control for fuel-efficient and safe heavy-duty vehicle platooning. Accepted for IEEE Transactions on Control Systems Technology.
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Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control Energy-Optimal Adaptive Cruise Control based on Model Predictive Control based on Model Predictive Control based on Model Predictive Control based on Model Predictive Control
Andreas Weißmann, Daniel G¨ orges and Xiaohai Lin Andreas o Andreas Weißmann, Weißmann, Daniel Daniel G¨ G¨ orges rges and and Xiaohai Xiaohai Lin Lin Andreas Weißmann, Daniel G¨ orges and Xiaohai Lin Juniorprofessorship for Electromobility, University of Kaiserslautern, Juniorprofessorship for of Juniorprofessorship for Electromobility, Electromobility, University of Kaiserslautern, Kaiserslautern, Erwin-Schr¨ odinger-Straße 12, 67663 University Kaiserslautern, Germany Juniorprofessorship for Electromobility, University of Kaiserslautern, Erwin-Schr¨ o dinger-Straße 12, 67663 Kaiserslautern, Germany Erwin-Schr¨ o dinger-Straße 12, 67663 Kaiserslautern, Germany E-mail: weissmann|goerges|[email protected] Erwin-Schr¨ odinger-Straße 12, 67663 Kaiserslautern, Germany E-mail: weissmann|goerges|[email protected] E-mail: weissmann|goerges|[email protected] E-mail: weissmann|goerges|[email protected] Abstract: In this paper an approach for an energy-optimal adaptive cruise control based Abstract: In an approach for adaptive cruise control based Abstract: In this this paper paper an (MPC) approach for an an energy-optimal energy-optimal adaptive cruise control of based on model predictive control is presented. The approach uses the knowledge the Abstract: In this paper an (MPC) approach for an energy-optimal adaptive cruise control of based on model predictive control is presented. The approach uses the knowledge on model predictive control (MPC) is presented. The approach uses the knowledge of the the given route to precalculate a position-dependent energy-optimal speed trajectory using dynamic on model predictive control (MPC) is presented. The approach uses the knowledge of the given precalculate aaadditional position-dependent energy-optimal speed trajectory using dynamic given route route to to while precalculate position-dependent energy-optimal speedroad trajectory using dynamic programming taking information like speed limits, slope and travel time given route to while precalculate aadditional position-dependent energy-optimal speedroad trajectoryand using dynamic programming speed travel programming while taking taking additional information information like speed limits, limits, roadisslope slope travel time time into account during the optimization. The model like predictive controller usedand to control the programming while taking additional information like speed limits, roadisslope and travel time into account during the optimization. The model predictive controller used to control the into account during the vehicle optimization. The model predictive controller is usedspeed to control the traction force of the host such that the vehicle speed follows the optimal trajectory into account during the optimization. The model predictive controller is used to control the traction of vehicle such that follows optimal speed traction force of the the host host vehicle such constraints that the the vehicle vehicle speed follows the optimal vehicle speed trajectory trajectory as good force as possible while ensuring like speed distance to athe preceding or speed traction force of the host vehicle such constraints that the vehicle speed follows the optimal vehicle speed trajectory as good as possible while ensuring like distance to a preceding or speed as good as possible while ensuring constraints like distance to a preceding vehicle or speed limits. To show the benefits of the approach, a comparison of the energy consumption between as goodToasshow possible while ensuring constraints like distance to a preceding vehicle or speed limits. the of the aa comparison of the consumption between limits. To showvehicle the benefits benefits of preceding the approach, approach, comparison ofroute the energy energy consumption between the controlled and the vehicle on the same is performed. For the speed limits. To showvehicle the benefits of preceding the approach, a comparison ofroute the energy consumption between the controlled and the vehicle on the same is performed. For the speed the controlled vehicle andvehicle, the preceding on the same performed.show For the speed profile of the preceding data ofvehicle real test drives is route used. is Simulations that the the controlled vehicle andvehicle, the preceding vehicle on the same route is performed.show For the speed profile the data test drives is that profile of of leads the preceding preceding vehicle, data of ofofreal real test drives is used. used. Simulations Simulations show that the the approach to a significant reduction the energy consumption compared to the preceding profile of leads the preceding vehicle, data ofofreal test drives is used. Simulations showpreceding that the approach to significant reduction the compared to approach leads to aa route. significant reductionthe of simulations the energy energy consumption consumption compared to the the preceding vehicle on the same Furthermore indicate that the approach achieves high approach leadssame to a significant reductionthe of the energy consumption compared to the preceding vehicle on indicate that the vehicle savings on the the same route. Furthermore the simulations simulations indicate that car. the approach approach achieves achieves high high energy evenroute. with aFurthermore poor prediction model for the preceding vehicle savings on the same route. Furthermore the simulations indicate that car. the approach achieves high energy even with a poor prediction model for the preceding energy savings even with a poor prediction model for the preceding car. energy even with aFederation poor prediction model for the preceding car. Ltd. All rights reserved. © 2017, savings IFAC (International of Automatic Control) Hosting by Elsevier Keywords: Adaptive cruise control, Model predictive control, dynamic programming, cloud, Keywords: Keywords: Adaptive Adaptive cruise cruise control, control, Model Model predictive predictive control, control, dynamic dynamic programming, programming, cloud, cloud, prediction Keywords: Adaptive cruise control, Model predictive control, dynamic programming, cloud, prediction prediction prediction 1. INTRODUCTION interface for control of the vehicle speed and acceleration 1. INTRODUCTION INTRODUCTION interface for control control oftothe the vehicleACC speedfunctionality and acceleration acceleration 1. interface for vehicle speed and has inspired researchof combine with 1. INTRODUCTION interface for control oftothe vehicleACC speedfunctionality and acceleration has inspired research combine with In recent years the reduction of the energy consumption in has inspired research to combine ACC functionality with different control strategies to improve energy efficiency and has inspired research to combine ACC functionality In recenthas years the reduction reduction of the energy energy consumption in different different control strategies to improve improve energy efficiencywith and In recent years the of the consumption in vehicles become a major research topic. Reducing the control strategies to energy efficiency and safety alike. In recenthas years the reduction of the energy consumption in safety different control strategies to improve energy efficiency and vehicles become major research topic. Reducing and the alike. vehicles has become aais major research Reducing the energy consumption appealing fromtopic. an economical alike. vehiclesconsumption has become ais major research topic. Reducing and the safety safety alike. energy appealing from an economical In (Turri et al., 2016) an approach was proposed that energy consumption is appealing an economical and In (Turri et environmental perspective due tofrom increasing energy costs al., 2016) an approach wasspeed proposed that energy consumption is appealing from an economical and In (Turri et 2016) approach was proposed that environmental perspective due to increasing energy costs uses MPC to al., follow an an energy optimal trajectory environmental perspective due increasing energy point costs uses and emission requirements andto from a functional In (Turri etto al., 2016) an approach wasspeed proposed that MPC follow an energy optimal trajectory environmental perspective due tofrom increasing energy point costs uses MPC to follow an energy optimal speed trajectory and emission requirements and a functional derived by to dynamic programming. For both, MPC and and emission requirements andrange from of a electric functional point derived of view, e.g. for increasing the vehicles. uses MPC follow an energy optimal speed trajectory by dynamic programming. For both, MPC and andview, emission requirements andrange from of a electric functional point derived byprogramming, dynamic programming. both, MPClimits and of e.g. for for increasing the vehicles. dynamic informationFor about speed of view, e.g. increasing the range of electric vehicles. Considerable research has been devoted to constructive derived by dynamic programming. For both, MPC and dynamic programming, information about speed limits limits of view, e.g. for increasing the range of electric vehicles. dynamic information speed Considerable research has been been devoted to(‘downsizing’) constructive and road programming, slope are regarded during about optimization. The Considerable has devoted constructive approaches likeresearch more efficient engine designsto dynamic programming, information about speed limits and road slope are regarded during optimization. The Considerable research has been devoted to constructive and road slope are regarded during optimization. The approaches like more efficient engine designs (‘downsizing’) considered scenario is platooning of heavy-duty vehicles. approaches like more efficient engine more designsrecent (‘downsizing’) or lightweight materials. Another research considered and road slope are isregarded during optimization. The scenario platooning of heavy-duty heavy-duty vehicles. approaches like materials. more efficient engine more designsrecent (‘downsizing’) is platooning of vehicles. or lightweight Another research considered The resultsscenario show that this approach leads to good fuel or lightweight materials. Another more recent research direction is focused on the benefits that can be obtained considered scenario is platooning of heavy-duty vehicles. The results show that this approach leads to good fuel or lightweight materials. more recent obtained research The results show Turri that this leads good fuel direction is focused focused oncomplete theAnother benefits that can savings. However, et al.approach (2016) do not to consider the direction is on the benefits that One can be be obtained by the control of the vehicle. widely used The results show that this approach leads to good fuel savings. However, Turri traffic et al. al. (2016) (2016) doapplication not consider consider the direction is focused oncomplete the benefits that One can be obtained savings. However, Turri et do not the by the control of the vehicle. widely used influence of interfering and the of by the control of the vehicle. Oneiswidely used influence approach to reduce thecomplete energy consumption the adjustsavings. However, Turri traffic et al. (2016) not consider of interfering and the thedoapplication application of the by the control of the complete vehicle. Oneiswidely used influence of traffic and of the approach to reduce reduce the energy consumption consumption the adjustadjustapproach to interfering other vehicle classes. approach to the energy is way, the ment of the driving behavior in an efficient which influence of interfering traffic and the application of the approach to other vehicle classes. approach to reduce the energy consumption is way, the adjustapproach to other vehicle classes. ment of the the driving behavior in an an efficient efficient which ment of driving behavior in way, which has been shown to be very promising in reducing the approach to other vehicle approach classes. based on MPC and In this paper a similar mentbeen of the driving behavior in an efficient way, which has shown to be very promising promising in et reducing the In this paper paper similar approach The basedMPC on is MPC and has been shown to very in reducing the fuel consumption andbe emissions in (Mierlo al., 2004). In this aa similar approach based on MPC dynamic programming is proposed. usedand for has been shown to be very promising in reducing the fuel consumption andcan emissions in (Mierlo et eco-driving al., 2004). 2004). dynamic In this paper a similar approach The basedMPC on is MPC and programming is proposed. proposed. used for fuel consumption and emissions in (Mierlo et al., Such an adjustment be done by using an dynamic programming is The MPC is used for following an energy-optimal speed trajectory while addifuel consumption and emissions in (Mierlo et al., 2004). Such an adjustment adjustment can be be done by usingefficient an eco-driving eco-driving dynamic programming is proposed. The MPCwhile is used for following an energy-optimal energy-optimal speed trajectory addiSuch an can using an assistance, which is used to done make by energy driving following an whilevarious additionally regarding preceding speed traffic trajectory and ensuring Such an adjustment can be usingefficient an eco-driving assistance, which is used used to done make by energy driving followingregarding an energy-optimal speed trajectory whilevarious additionally preceding traffic and ensuring assistance, which is to make energy efficient driving strategies, e.g. shifting strategies or speed proposals for tionally regarding preceding traffic and ensuring various constraints. This optimal speed trajectory is calculated assistance, e.g. which is usedstrategies to make energy efficient driving strategies, shifting ordriver speed proposals for constraints. tionally regarding preceding traffic and ensuring various This optimal speed trajectory is calculated calculated strategies, e.g. shifting strategies speed proposals for astrategies, certain route, accessible to theor via a humanconstraints. This optimal speed trajectory is offline for the given route before the trip using DP and e.g. shifting strategies or speed proposals for a certain route, accessible to the driver via a humanconstraints. This optimal speed trajectory is calculated offline for the given route before the trip using DP and amachine certaininterface route, accessible to the driver via a systems human- offline (HMI). Eco-driving assistance for theonline given for route before the trip using DP and is then used the MPC. The precalculation ala certain route, accessible to the driver via a humanmachine interface (HMI). Eco-driving Eco-driving assistance systems offline for theonline given for route before the trip using DP and is then used the MPC. The precalculation almachine interface (HMI). assistance systems using dynamic programming (DP) to derive an energyis then used online for the MPC. The precalculation allows a higher complexity of the DP optimization problem machine interface (HMI). Eco-driving assistance systems using dynamic programming (DP) to to vehicles derive an an energyis then used online for the MPC. precalculation allows higher complexity of the the DPThe optimization problem using dynamic programming (DP) derive energyoptimal speed profile for conventional have been lows aa higher complexity of DP optimization problem (i.e. high accuracy of energy consumption representation) using dynamic programming (DP) to vehicles derive an energyoptimal speed profile for conventional have been lows a higher complexity of the DP optimization problem (i.e. high accuracy of energy energy consumption representation) optimal speede.g. profile for conventional vehicles have been (i.e. investigated in (Luu et al., 2010) and for electric high accuracy of consumption representation) while keeping the online optimization problem of the MPC optimal speede.g. profile for conventional vehicles have been while investigated in et (Luu et al., al.,2012; 2010) and for electric (i.e. high accuracy of energy consumption representation) keeping the online optimization problem of the the speed MPC investigated e.g. in (Luu et 2010) and for electric vehicles e.g. in (Dib al., 2011, Lin et al., 2014). while keeping the online optimization problem of MPC very simple. A method to derive the energy-optimal investigated e.g. in et (Luu et al.,2012; 2010) and for 2014). electric very vehicles e.g. in (Dib al., 2011, Lin et al., whilesimple. keepingAthe onlinetooptimization problem of the speed MPC method derive the energy-optimal vehicles e.g. in (Dib et al., 2011, 2012; Lin et al., 2014). simple. method programming to derive the energy-optimal speed trajectory viaA dynamic for an electric vehicle vehiclesadvanced e.g. in (Dib et al., 2011, 2012; Lin et(ADAS) al., 2014). With driver assistance systems like very very simple. Adynamic method programming to derive the energy-optimal speed trajectory via for an electric vehicle dynamic programming foral., an electric vehicle With advanced driver (ACC) assistance systems (ADAS) like trajectory has alreadyvia been introduced in (Lin et 2014) and will With advanced driver assistance systems like adaptive cruise control many vehicles(ADAS) are by now trajectory via dynamic programming foral., an electric vehicle has already been introduced in (Lin et 2014) and and will With advanced driver (ACC) assistance systems (ADAS) like has already been introduced in (Lin et al., 2014) will adaptive cruise control many vehicles are by now be used in this work as well. It should be mentioned that adaptive cruise control (ACC) many vehicles are by now endowed with the technical equipment for implementing hasused already been introduced et be al.,mentioned 2014) and that will be in this this work as trajectory well. in It (Lin should adaptive cruise control (ACC) many vehicles are by now be used in work as well. It should be mentioned that endowed with strategies the technical equipment for implementing for this optimal speed a further constraintthat on endowed with the technical equipment for implementing these driving without direct interaction with be used in this work as well. It should be mentioned for this optimal speed trajectory a further constraint on endowed with strategies the technical equipment for implementing for this optimal speed trajectory a further constraint on these driving without direct interaction with the trip time is considered. The results in this paper are these driving strategies direct interaction the driver. ACC provideswithout additional safety benefits with due the for this optimal speed trajectory a further constraint on tripto time is considered. considered. Thecan results in this this paper are are thesedriver. driving strategies without direct interaction with the trip time is The results in paper the ACC provides additional safety benefitsof due related electric vehicles, but be easily extended for the driver. ACC provides additional safety benefits due to the distance control functionality. The availability an the trip time is considered. The results in this paper are related to electric vehicles, but can be easily extended for thethe driver. ACC provides additionalThe safety benefitsofdue to distance control functionality. availability an related to electric vehicles, but can be easily extended for to the distance control functionality. The availability of an to the distance control functionality. The availability of an related to electric vehicles, but can be easily extended for
Copyright © 2017, 2017 IFAC 13074 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 13074 Copyright ©under 2017 responsibility IFAC 13074 Peer review of International Federation of Automatic Control. Copyright © 2017 IFAC 13074 10.1016/j.ifacol.2017.08.2196
Proceedings of the 20th IFAC World Congress 12564 Andreas Weißmann et al. / IFAC PapersOnLine 50-1 (2017) 12563–12568 Toulouse, France, July 9-14, 2017
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Fig. 1. Considered scenario with host and preceding vehicle use in conventional vehicles by additionally considering the shifting strategy. For the online part, namely the MPC, the main goal is to control the traction force of the vehicle in a way that the vehicle speed follows the energy-optimal speed trajectory if there is no preceding traffic or keep the distance in a safe range if there is any interfering traffic. During the optimization further constraints, e.g. maximum traction and brake force, are used to ensure a feasible vehicle behavior. To evaluate the MPC, several simulations are performed where the host vehicle follows a preceding vehicle on the same fixed route. The considered host vehicle in this work is a Nissan Leaf. The speed profile of the preceding vehicle is based on real test drives on the considered route. The energy consumption and trip time for the controlled vehicle and the preceding vehicle are compared to each other. The main contribution of this work consist in uniquely combining an offline optimization using dynamic programming with a precise model to derive an optimal speed profile and an online optimization implementing ACC functionality using MPC together with a simple model which provides a compromise between accuracy and complexity. The resulting new ADAS framework provides a combination of ACC functionality with the capability of following an energy optimal reference speed profile. This framework is evaluated by comprehensive simulations based on real world data. The paper is organized as follows. In Section II the optimization problem is formulated and an outline of the optimization problem used for dynamic programming is given. In Section III the test scenario and the simulation results of the proposed MPC are presented. Conclusions are finally provided in Section IV. 2. PROBLEM FORMULATION In Fig. 1 the considered scenario with the host vehicle and a preceding vehicle is illustrated. The proposed approach can be divided into an offline and an online part. The offline part, as shown in the upper dotted box of Fig. 1, includes the communication of data between the host and a cloud. Specifically the host sends his desired route to
the cloud or alternatively a desired destination if routing should be done in the cloud. The cloud then derives the speed limits and altitude profile along the route for use in dynamic programming and the MPC later on. After that the optimal speed trajectory is determined via dynamic programming. The geographical information as well as the optimal speed trajectory is sent to the host vehicle for use in the online part, namely the MPC. The online part includes following the optimal speed trajectory while ensuring a safe distance to a potential preceding vehicle. The online part is illustrated in the lower dotted box in Fig. 1. It is worth mentioning that solving a DP problem for a certain route can take very long due to the high complexity that increases with the length of the route. To solve this issue, the calculated optimal speed trajectories are stored in the cloud to avoid a recalculation upon the next request, which is especially beneficial when considering commuters. If a calculation cannot be avoided, it is possible to simply use the conventional ACC until the optimal speed trajectory is received. 2.1 System dynamics The system dynamics for the considered scenario can be expressed as dk+1 = dk − vk Ts + vp,k Ts vk+1 = vk + ak Ts
(1) (2)
where v and a are the velocity and acceleration of the controlled vehicle and Ts is the sampling time. d describes the distance between the host and the preceding vehicle which depends on the prediction of the velocity of the preceding vehicle vp . To include the traction force Ft as the controlled variable and the driving resistances, the longitudinal dynamics are used to replace a in (2). The longitudinal dynamics are given by meq a = Ft − Fr − Fa − Fg (3) with meq = mv + mr being the equivalent mass including the vehicle mass mv and the influence of the rotating masses mr . The considered driving resistance forces are: aerodynamic resistance force Fa = 21 cf Af ρa v 2 with the air density ρa , the air drag coefficient cf and the projected frontal area Af , neglecting the influence of the wind speed, rolling resistance force Fr = mv gcr with the gravitational acceleration g and the rolling resistance coefficient cr , and grading resistance force Fg = mv gα with the road angle α. It should be mentioned that in Fg and Fr the small-angle approximations sin(α) = α and cos(α) = 1 are used instead of the trigonometric functions. By substituting (3) into (2) and rearranging, the equation describing the velocity can be written as 1 Ts Ft − mv gcr − ρa Af ca vk2 − mv gαk . vk+1 = vk + meq 2 (4) To handle the nonlinear term vk2 resulting from the aerodynamic resistance force in (4), the predicted state sequence resulting from the last optimization is used instead of vk2 , assuming that the deviations of the predicted state sequence between two subsequent optimization steps are very small.
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2.2 Cost function
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For the desired objective of following an energy-optimal speed trajectory vref it is a reasonable approach to use a cost function that punishes the deviation of the vehicle speed from its energy-optimal reference speed. A quadratic cost function N −1 2 q(vk − vref,k )2 + pFbrake,k , (5) J=
F
Traction force Ft (N)
t,max
y1=c 1 v + c 2
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k=0
is chosen which represents the weighted deviations to the reference speed vref over the control horizon N . The 2 additional term Fbrake is used to punish the usage of hydraulic brake force Fbrake as the regenerative braking of the electric vehicle is favorable over hydraulic braking. In (5) q > 0 and p > 0 denote weighting factors. 2.3 Constraints
To ensure realizable results of the optimization, several constraints need to be considered. From the physical limitations of the electric motor a maximum traction force arises. For the considered electric vehicle this constraint is derived by transforming the maximum torque over angular velocity diagram Tmax (ωmotor ) into maximum traction force over velocity diagram Ft,max (v). The transformation )γη Rw follows from Ft,max (v) = Tmax (ωRmotor and v = ωmotor γ w with the wheel radius Rw , the gearbox efficiency η, and the gear ratio γ. The resulting diagram is shown in Fig. 2. To obtain a convex constraint the maximum traction force is approximated as indicated by the red dotted line in Fig. 2. From this Ft,k,max = c1 vk + c2 (6) follows with c1 and c2 being identified constants.
1000 0
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Fig. 2. Maximum traction force over velocity diagram Another important constraint follows from the ACC functionality, which provides a safe distance to the preceding vehicle. In this approach a constant headway h and a fixed minimum distance dmin is used, leading to dmin + hvk ≤ dk . (10) In addition, speed limits can be addressed by vmin,k ≤ vk ≤ vmax,k ,
(11)
with 0 ≤ vmin,k mainly used to prevent driving backwards if the distance gets too small. The upper speed limit vmax,k is based on the admitted speed along the route and is derived during the offline part as indicated in Fig. 1. However, it also can be easily derived online, using GPS and map data. 2.4 MPC optimization problem
The braking system of the car has limitations in terms of the maximum deceleration. The limitations can be divided into the regenerative braking capability of the electric motor and the braking capability of the hydraulic brake for modeling. As a permanent magnet synchronous motor is used in the particular electric vehicle, it is reasonable to describe the regenerative braking limits in the generator mode, which are expressed by a negative traction force Ft , in the same way as in the motor mode by mirroring the diagram in Fig. 2 to the generator mode. The hydraulic brake force Fbrake is only included as a soft constraint, allowing the use of additional braking force if the regenerative braking limits are exceeded to ensure safety constraints. The usage of the hydraulic brake force is unfavorable considering the energy efficiency and its usage therefore punished in the cost function (equation (12)).
With the system dynamics, the cost function, and the constraints the online optimization problem for MPC, which is solved at every time instance can be summarized as N −1 2 min (12) q(vk − vref,k )2 + pFbrake,k
The limits for regenerative braking (−Ft,k,max ) and the hydraulic brake force Fbrake result from Ft,k,brake,max = −c1 vk − c2 − Fbrake,k . (7)
2.5 Optimal speed trajectory
The hydraulic brake force also has limits which simply follow from the technical specification as 0 ≤ Fbrake ≤ Fbrake,max . (8) Summarizing (6) to (8) the overall constraint resulting from the traction force and the braking system limitations can be formulated as −c1 vk − c2 − Fbrake,k ≤ Ft,k ≤ c1 vk + c2 . (9)
Ft,k ,Fbrake,k
k=0
subject to
Ts 1 (Ft− mv gcr− ρa Af ca vk2− mv gαk ) meq 2 dk+1 = dk − vk Ts + vp,k Ts − c1 vk − c2 − Fbrake,k ≤ Ft,k ≤ c1 vk + c2 0 ≤ Fbrake,k ≤ Fbrake,max dmin,k + hvk ≤ dk vmin,k ≤ vk ≤ vmax,k vk+1= vk+
(13a) (13b) (13c) (13d) (13e) (13f)
In transportation it is in general a problem to find a suitable compromise between the energy consumption and trip time. Dynamic programming (DP) is used to address this problem in the proposed approach. The used problem formulation for DP and a precise model of the electric vehicle has been introduced in (Lin et al., 2014) and will be briefly reviewed in the following. Due to the position-dependent nature of the road angle α and speed limits vmin/max , it is convenient to use
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a position-dependent representation of the system. This requires a transformation of the velocity v(t) and position x(t) into v(x) with a discretization step of ∆x instead of Ts . This discretization leads to a partition of the route with the overall length xmax into M segments of length ∆x for which the optimal speed trajectory is calculated. The desired optimal speed trajectory results from the optimization problem M −1 tk (vk )Pbattery,total,k (vk )+βtk (vk )+λ(Tmotor,k (vk ))2 min vk
Stored Data
Equations (15a) and (15b) represent the initial state for the trip. The constraints (15c)-(15e) are used to keep the SOD of the battery in safe boundaries during the trip and to set defined final values for the SOD and the final velocity at the end of the trip. The constraints (15f) and (15g) are the same as in (9) and (11), addressing the system limitations on maximum traction force and maximum braking force as well as the speed limits. The result of the optimization is the optimal speed trajectory v ∗ (x) as well as the overall trip time Tduration which is calculated M −1 from Tduration = k=0 tk . It is worth mentioning that dynamic programming already considers the maximum speed limit vmax and passes this to the MPC in form of v ∗ (x) and therefore does not need to be considered in the optimization problem of the MPC. 3. SIMULATION AND RESULTS 3.1 Simulation setup The simulation setup is shown in Fig. 3. In the figure a hat on a variable ˆ ∗ indicates that the predicted sequence over the horizon N is passed on to the next block. The prediction block contains the prediction of vp over the horizon N which will be described later. The conversion from
MPC
^ x(t)
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^
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v(t),d(t)
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where SOD denotes the state of discharge of the battery of the electric vehicle. In (14) the term tk (vk )Pbattery,total,k (vk ) is the representation of the energy consumption for a single segment, with tk (vk ) being the trip time for ∆x in segment k. Pbattery,total,k (vk ) represents the power drawn from the battery including electric motor, electrified auxiliary units and so on. The terms βtk (vk ) and λ(Tmotor,k (vk ))2 are used to tune the trip time and the ride comfort respectively by weighting the trip time and motor torque Tmotor . An increase of β leads to a reduction in the trip time and an increase of λ improves the ride comfort by punishing the usage of high torque for accelerating and braking. However, in the following only the trip time and therefore β is considered and tuned.
Prediction
^ vp(t)
Fig. 3. Block diagram of the simulation structure
(14)
(15a) (15b) (15c) (15d) (15e) (15f) (15g)
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v0 = vinit SOD0 = SODinit vM ∈ [vmin,M , vmax,M ] SODM ∈ [SODmin,M , SODmax,M ] SODk ∈ [SODmin,k , SODmax,k ] vk ∈ [vmin,k , vmax,k ] Tmotor,k ∈ [Tmin,k , Tmax,k ]
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Fig. 4. Optimal speed trajectory vref discrete-position data to discrete-time data is performed by means of the predicted position sequence x ˆ which is derived from the system model and the solution of the optimization problem (12). For the speed profile vp data acquired from real trips is used. The data was gathered from a commuter on a route from Kaiserslautern to Landstuhl in Germany with a length of 17.8 km. This route contains urban and interurban segments as well as a highway. The data amounts to 85 trips in total and therefore contains a wide variety of possible traffic conditions. To obtain a realistic trip time for the optimal speed trajectory vref , the mean value over all 85 trips is used which results in a desired trip time of 868 s. The dynamic programming is implemented with the opensource software DPM which was developed at the ETH Z¨ urich (Elbert et al., 2013). Solving the optimization problem (14) with DPM takes 38.5 s per run. It must be mentioned that several runs are necessary to reach the desired trip time of 868 s by tuning the parameter β. However, in practice this procedure needs to be done only once and the information can then be stored in the cloud and sent to the host upon the next request for that route. The resulting position-dependent optimal speed trajectory vref is shown in Fig. 4 together with the road type and the upper speed for each segment. It can easily be seen that most of the time the speed is close to the speed limit vmax so that the desired trip time of 868 s can be achieved. The final speed boundaries defined by (15c) range from 60 km/h to 120 km/h for the different trips and leads to the braking maneuver which can be observed near the end position at 17.8 km. This braking maneuver indicates the behavior to drive as slow as possible for minimum energy consumption. This also shows the necessity to punish the trip time. The speed trajectory vref is stored for use in the MPC. In the simulated scenario the host vehicle should follow the reference trajectory vref while ensuring a safe distance to
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the preceding vehicle with the speed profile vp starting at a distance of d = 150 m ahead. A sampling time of Ts = 0.2 s and a horizon of N = 100 is used for the simulation. The initial speed of the host vehicle is chosen to be the same as the initial speed of the preceding vehicle vp,init . The simulation is performed for all 85 profiles of vp . To predict vp over the horizon N , a prescient MPC (PMPC) and a frozen-time MPC (FTMPC) approach is used for the simulations. PMPC assumes full knowledge of the preceding car’s velocity over the horizon N and represents the best achievable prediction. FTMPC uses only knowledge from the present measurements of vp (known e.g. from a recent measurement with radar/lidar/camera) and assumes this to be constant over the horizon N . It represents therefore a very simple to implement and rather poor prediction. To show the energy saving potential of this approach, the difference in energy consumption between the controlled and the preceding vehicle for every profile vp is calculated using the electric vehicle model introduced in (Lin et al., 2014).
25.1 s. To further evaluate the controller performance, it is beneficial to split the results into two datasets. The first dataset contains the results for which the host vehicle can completely follow the reference trajectory vref . This dataset contains 45 of the simulation results. This is the case if the preceding vehicle has a higher speed than the host vehicle and hence there is never the necessity to leave the reference trajectory due to the distance constraint (13e). The second dataset contains the data in which a deviation from the reference trajectory vref is necessary to fulfill the distance constraint (13e). This dataset contains 40 of the simulation results. The results for the first dataset are shown in Table 2 and those for the second dataset in Table 3. Table 2. Results first dataset PMPC FTMPC
Table 1. Overall results δ¯ -7.5% -7.2%
δmin -18.4% -17.9%
δmax 3.7% 3.6%
∆T¯ 25.15s 25.17s
∆Tmin 0.8s 0.8s
∆Tmax 84.2s 84.2s
PMPC FTMPC
δmin -16.9% -16.9%
δmax 0.4% 0.4%
∆T¯ 39.9s 39.9s
∆Tmin 5.2s 5.2s
∆Tmax 84.2s 84.2s
δ¯ -7.8% -7.2%
δmin -18.4% -17.9%
δmax 3.7% 3.6%
∆T¯ 8.5s 8.5s
∆Tmin 0.8s 0.8s
∆Tmax 55.8s 55.8s
From Table 2 it can be seen that the results are the same for PMPC and FTMPC. This is not unexpected as in both cases the same speed trajectory is followed and therefore the trip time and energy consumption are the same. For this dataset the preceding vehicle is faster and therefore has a shorter trip time than the controlled vehicle. However, as the speed trajectory is followed the desired trip time of 868 s (vref ) can nearly be reached with an offset of approximately +2.4 s that results from the fact that the controlled vehicle starts at a different speed than vref,init . Table 3 shows the results of the second dataset. The mean value of the trip time difference (8.5 s) is now fairly small as we partially have to follow the slower speed of vp in certain sections due to the distance constraint. The mean values of the energy consumption reduction are with 7.8% for PMPC and 7.2% for FTMPC very close and indicates that the proposed approach is only influenced slightly by the used prediction model for vp . 130
From Table 1 it can be seen that a good overall energy reduction of 7.5% for PMPC and 7.2% for FTMPC can be achieved considering all datasets with the best achievable reduction of 18.4% and 17.9%, respectively. It is to be mentioned that only four profiles of vp led to an increased energy consumption for the host vehicle in comparison to the preceding vehicle, with three of them being < 0.5% and one of them being 3.7%. This results from the fact that a human driver and therefore vp does not always fulfill the speed limits, allowing to take a more beneficial working point than the constrained reference trajectory vref and thus resulting in an overall lower energy consumption for the preceding vehicle. The difference of the trip time for PMPC and FTMPC is for both cases almost identical with a mean value of
Velocity v (km/h)
PMPC FTMPC
δ¯ -7.2% -7.2%
Table 3. Results second dataset
3.2 Results The proposed control structure and control strategy as shown in Fig. 3 is implemented and simulated in MATLAB and evaluated with an electric vehicle model in Simulink. To evaluate the performance of the proposed approach, the relative energy consumption difference δ of the host (Ehost ) and the preceding vehicle (Eprec ) is computed from Ehost − Eprec δ= . (16) Eprec The overall results for the simulations with PMPC and FTMPC are shown in Table 1. In the table δ¯ is the mean value over the simulation results for the 85 vp profiles. The minimum and maximum of the achieved relative energy consumption difference for all results is denoted by δmin and δmax , respectively. In addition the difference in trip time ∆T = thost − tprec is considered. The mean values over all datasets as well as the minimum and maximum are presented. As a following scenario is considered, the host vehicle is always slower than the preceding vehicle.
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Fig. 5. Simulation result for bad traffic conditions To highlight the advantages and drawbacks of the approach as well as the influence of the prediction model, the results of a simulation with very bad traffic conditions, namely stop and go traffic in the urban section of the
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However, the deviation from the reference trajectory, which is necessary to fulfill the distance constraint, leads to an increase in the trip time. The reason is straightforward as vref is precalculated for a trip time of 868 s and a deviation at any position will lead to an increased trip time > 868 s. A possibility to deal with this could be a mechanism which is trying to catch up the ‘lost’ time by increasing the remaining reference trajectory such that the desired trip time can still be fulfilled at the cost of reducing the achievable energy savings. Another drawback is resulting from the fact that the cost function does not include any representation of the energy consumption in order to keep the online optimization problem simple. All energy-related information is included in the optimal speed trajectory vref derived by DP. Therefore after deviations from vref no aspects regarding the energy consumption are considered during the online optimization. 4. CONCLUSIONS In this paper an energy-optimal adaptive cruise control based on MPC has been proposed. Dynamic programming is used to precalculate and store an energy-optimal speed trajectory for a given route, considering geographical information and constraints like speed limits, which then is submitted to a host vehicle upon request. The precalculation is done offline in a cloud before the trip,
Distance d (m)
In this case we have δ = −11.2% for PMPC and δ = −10% for FTMPC. Furthermore ∆T = 55.8 s for both prediction models. The reason for the difference in δ can be seen in Fig. 5 between 200 s and 500 s. It is obvious that PMPC (blue line) with full knowledge over the horizon N recognizes braking and acceleration maneuvers of vp (red line) beforehand and leads to a smoother behavior. FTMPC (black line) has only knowledge of the current velocity vp and therefore has no smoothing behavior but rather follows the velocity of the preceding vehicle, resulting in a higher energy consumption than PMPC due to more and stronger acceleration maneuvers. From the figure it can also be seen that in a major part of the trip the host vehicle can follow the reference trajectory vref (green line) very well for both prediction models, which finally leads to a high energy saving. Deviations to slower speeds only appear when the distance constraint becomes active at some point. This is shown in Fig. 6 (PMPC). The figure shows the result of the optimization applied to the system model over the horizon N with the velocity in the top subplot and the distance to the preceding car in the bottom subplot. From this it can be seen that the host velocity v (blue line) stays as close as possible to the reference trajectory vref (green dotted line), which follows from the problem formulation of the MPC (12) with the goal of minimizing the deviation to vref . It follows that a deviation from vref only occurs when the distance constraint becomes active during the optimization. This is the main reason why FTMPC is fairly close to the performance of PMPC as the prediction model only is taken into account in the explained scenario.
Velocity v (km/h)
route, is presented. In Fig. 5 the implemented velocity over the trip time is shown for the preceding vehicle vp , the host vehicle with PMPC and FTMPC, and the reference trajectory vref for PMPC.
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Fig. 6. Optimization result at t=210 s which allows the usage of a complex and very accurate representation of the energy consumption in DP. A simple MPC framework is then used online in the host vehicle to follow the optimal speed trajectory while ensuring a safe distance to the preceding vehicle. To ensure realizable inputs, limitations on the traction and braking force are included in the optimization as linear constraints. Simulations in a following scenario with speed profiles of real test drives for the preceding vehicle have shown a significant reduction of the energy consumption, using either FTMPC or PMPC as prediction model for the preceding vehicle. REFERENCES Dib, W., Chasse, A., Di Domenico, D., Moulin, P., and Sciarretta, A. (2012). Evaluation of the energy efficiency of a fleet of electric vehicle for eco-driving application. Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, 67(4), 589–599. Dib, W., Serrao, L., and Sciarretta, A. (2011). Optimal control to minimize trip time and energy consumption in electric vehicles. In Proceedings of the 2011 IEEE Vehicle Power and Propulsion Conference. Institute of Electrical and Electronics Engineers (IEEE). Elbert, P., Ebbesen, S., and Guzzella, L. (2013). Implementation of dynamic programming for n-dimensional optimal control problems with final state constraints. IEEE Transactions on Control Systems Technology, 21(3), 924–931. Lin, X., G¨orges, D., and Liu, S. (2014). Eco-driving assistance system for electric vehicles based on speed profile optimization. In 2014 IEEE Conference on Control Applications (CCA). Institute of Electrical & Electronics Engineers (IEEE). Luu, H.T., Nouveli`ere, L., and Mammar, S. (2010). Dynamic programming for fuel consumption optimization on light vehicle. IFAC Proceedings Volumes, 43(7), 372– 377. Mierlo, J.V., Maggetto, G., de Burgwal, E.V., and Gense, R. (2004). Driving style and traffic measures-influence on vehicle emissions and fuel consumption. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 218(1), 43–50. Turri, V., Besselink, B., and Johansson, K.H. (2016). Cooperative look-ahead control for fuel-efficient and safe heavy-duty vehicle platooning. Accepted for IEEE Transactions on Control Systems Technology.
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