Vehicle Fuel Consumption Optimization using Model Predictive Control based on V2V communication

Transcription

1 Vehicle Fuel Consumption Optimization using Model Predictive Control based on V2V communication A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Junbo Jing, B.S. Graduate Program in Electrical and Computer Science The Ohio State University 2014 Master s Examination Committee: Ümit Özgüner, Advisor Giorgio Rizzoni

2 c Copyright by Junbo Jing 2014

3 Abstract As people are working hard on improving vehicle s fuel economy, a large portion of fuel consumption in everyday driving is wasted by vehicle driver s inexperienced operations and inefficient judgments. This thesis proposes a system that optimizes the vehicle s fuel consumption in automated car-following scenarios. The system is designed able to work in the initial stage of implementing Vehicle-to-Vehicle (V2V) communications. The system is developed based on Model Predictive Control (MPC). With a given prediction of the preceding vehicle s speed, the system controls the vehicle s throttle and brake to follow the preceding vehicle with an optimal velocity profile. The control problem is formed into a quadratic programming optimization problem using real vehicle parameters. Active-set algorithm is adopted for optimization, and the computation speed can satisfy real-time computations. The control results show a significant fuel saving benefit of up to 15%, with car-following safety ensured and ride comfort cared. To provide the prediction horizon for the MPC based system, a preceding vehicle speed prediction algorithm and a leading vehicle speed prediction algorithm are developed in this thesis. The preceding vehicle s speed is predicted by analyzing the transmission of speed disturbances along the convoy using Intelligent Driver Model (IDM). The information needed is obtained through V2V communication, and the ii

4 algorithm does not require a high V2V penetration rate. The estimated car-following behavioral parameters are clustered online for improved prediction accuracy. The algorithm can provide a prediction horizon of seconds depending on the convoy length. The leading vehicle speed prediction algorithm is developed to extend the prediction horizon. The algorithm predicts the leading vehicle s free road driving and approaching speed when a rather large gap to the next vehicle appears. The leading vehicle s historical speed profile is decoded into a driver operation state sequence and forms a Markov chain. Markov Model is used for speed prediction. The information required by the algorithm is simply speed profiles and car-following distance profiles, which can be easily obtained by cooperating with the already existing Adaptive Cruise Control (ACC) systems. iii

5 Dedication I dedicate my research work to my advisor Professor Ümit Özgüner. Thank you for admitting me as your student in Control and Intelligent Transportation Research Lab. I really appreciate those opportunities to work on autonomous vehicles and intelligent transportation. It is in this lab I can really pursue my childhood dream of becoming an automobile engineer. And thank you for all your encouragement. There are harsh days I can t solve a thing and make mistakes, but you don t blame me. There are small brilliant days I make tiny little creations, and you tell me you like it. Your simple thank you s really make my day and make those harsh days worthwhile. I truly enjoy working in this lab. iv

6 Acknowledgments This research is partially supported by Ford Motor Company. v

7 Vita November 14, Born - Zhengzhou, China B.S. Electrical Engineering, Shanghai Jiao Tong University 2012-present Electrical and Computer Engineering, The Ohio State University. Fields of Study Major Field: Electrical and Computer Engineering vi

8 Table of Contents Page Abstract Dedication Acknowledgments Vita List of Tables ii iv v vi ix List of Figures x 1. Introduction Research Background V2V & V2I Communication Recent Research on Transportation Efficiency Research in This Thesis Organization of this Thesis Optimizing Vehicle Fuel Consumption using Model Predictive Control Introduction and Literature Review Vehicle Model Construction Longitudinal Dynamic Vehicle Model Fuel Consumption Estimation Model MPC Controller MPC Controller Design Optimization Constraints Problem Convexity Quadratic Programming Optimization vii

9 2.6 Algorithm Tests and Improvements Basic Scenario Tests Eliminating Unnecessary Brakes Complex Speed Profile Test Multiple MPC Controlled Vehicle Convoy Four Vehicle Convoy Simulation Ten Vehicle Convoy Simulation Speed Prediction of the Preceding Vehicle Introduction and Literature Review Scenario description and modeling Scenario Description Traffic Modeling Preceding vehicle speed prediction General Description of the Algorithm Car-following Parameter Estimation Initialization Car-following Parameter Estimation Online Parameter Clustering Speed Prediction Speed Prediction of the Leading Vehicle Scenario Description System Requirement Leading Vehicle Speed Prediction Scenario Modeling Finite State Machine of Driver s Operations Speed-to-Action Decoder Speed Prediction using Markov Model Conclusions and Future Work Conclusions Future Work Bibliography viii

10 List of Tables Table Page 2.1 Specifications for Lincoln MKS Vehicle Fuel Consumption of Different Prediction Horizons Average Calculation Time of Different Prediction Horizons Each Vehicle s Fuel Consumption of a 4 MPC Vehicle Convoy Each Vehicle s Fuel Consumption of a 10 MPC Vehicle Convoy Driver Parameters in Different Traffic Conditions Parameters of the Simulated Three Kinds of Drivers The Transition Matrix of the Driver Operation Markov Model ix

11 List of Figures Figure Page 2.1 An Example of the Position Constraints Test Results with 15 Seconds Horizon in the Deceleration-to-Acceleration Scenario Test Results with 15 Seconds Horizon in the Acceleration-to-Deceleration Scenario The Optimized Control within the Prediction Horizon Test Results with Unnecessary Brake Prevented in the Decelerationto-Acceleration Scenario Test Results with Unnecessary Brake Prevented in the Accelerationto-Deceleration Scenario Speed Profiles with Different Prediction Horizons Car-following Distance Profiles with Different Prediction Horizons Engine Torque Profiles with Different Prediction Horizons Brake Force Profiles with Different Prediction Horizons Fuel Saving Advantage and Average Calculation Time Speed Profiles of 4 MPC Controlled Vehicles Convoy Car-following Distance Profiles of 4 MPC Controlled Vehicles Convoy 39 x

12 2.14 Engine Torque Profiles of 4 MPC Controlled Vehicles Convoy Brake Force Profiles of 4 MPC Controlled Vehicles Convoy Speed Profiles of a 10 MPC Controlled Vehicles Convoy Car-following Distance Profiles of a 10 MPC Controlled Vehicles Convoy Speed Profile Details of the 10 Vehicles Convoy The Traffic Scenario Traffic Condition from the View of the Ego Vehicle Speed Profiles of the Simulated Vehicle in the Convoy The Created Imaginary Convoy The Convergence of IDM with Well Estimated Parameters Solitary Parameter Estimation Results Parameter Estimation Combined with Online Clustering The Preceding Vehicle s Speed Prediction using IDM Preceding Vehicle Speed Prediction Result Preceding Vehicle Speed Prediction Result Scenario Definition of the Relative Leading Vehicle Scenario Definition of the Absolute Leading Vehicle The Simulated Scenario of the Leading Vehicle s Driving The Simulated Aggressive Driver s Driving The Simulated Normal Driver s Driving The Simulated Conservative Driver s Driving xi

13 4.7 A FSM Summarizes the Vehicle s Speed Changes The FSM Summarizes the Vehicle Driver s Operations The Calculated Vehicle Torque and Brake Profile for a Given Speed Profile Demonstration of Each Period of the Throttle Phase The Fitted Engine Torque Profile and the Calculated Engine Torque Profile (Throttle Phase 1) The Speed Profile Calculated from the Fitted Torque Profile and the Actual Speed Profile (Throttle Phase 1) Demonstration of Each Period of the Brake Phase The Fitted Brake Force Profile and the Calculated Brake Force Profile (Brake Phase) The Speed Profile Calculated from the Fitted Brake Profile and the Actual Speed Profile (Brake Phase) Demonstration of Each Period of the Throttle Phase The Fitted Engine Torque Profile and the Calculated Engine Torque Profile (Throttle Phase 3) The Speed Profile Calculated from the Fitted Torque Profile and the Actual Speed Profile (Throttle Phase 3) The Markov Chain of the Leading Vehicle Driver s operations The Predicted Speed Profile of an Aggressive Driver The Aggressive Driver s Actual Speed Profile The Predicted Speed Profile of a Normal Driver The Normal Driver s Actual Speed Profile xii

14 4.24 The Predicted Speed Profile of a Conservative Driver The Conservative Driver s Actual Speed Profile xiii

15 Chapter 1: Introduction 1.1 Research Background Driving economically is becoming a more and more important issue for today s traffic. It s not only related with the driver s every day expenses on fuel, but also has a significant influence on world environment. Data collected by United States Environmental Protection Agency shows that during the year of , about 32% of total CO 2 emissions and 27% of total green house emissions in US came from transportation sector [1], of which 60% were from cars and light trucks [2]. Apart from improving the fuel efficiency of the current generation vehicles and developing new energy vehicles, guiding the drivers to drive in a more economical way can also achieve markable reduction on both fuel consumption and emissions. In a series of driving tests carried out by the authors of [3], the results show that a reduction of 5-25% in fuel consumption can be achieved by training the drivers to drive in an economical style, and NO x emission is also reduced by 4% on average. On the contrast, aggressive driving behaviors containing many unnecessarily heavy accelerations and decelerations can increase up to 33% more fuel consumption on highways and 5% more in suburbs [4]. 1

16 The effort of introducing eco-driving strategies has been long existing since 1983 [5], which was first given in the form of driving courses. Many automobile companies nowadays are also giving tutorial activities to their customers, so as to promote techniques to maximum their products fuel saving potential. For the future years, with the development in sensors, communication technology, and computation speed, human s driving can be more assisted and improved by electronics. Even if there are some complicated computations that are not likely to be accomplished on board, cloud computing technology still can give real time solutions [6]. The Intelligent Transportation System (ITS), which has been promoted by the US Department of Transportation (DOT), is being developed to improve traffic mobility and safety, while reducing the fuel consumption and emissions of ground transportation [7]. With this system, human drivers will get a lot more traffic information to help make more reasonable decisions. One of the main assistances from ITS will be exploiting a vehicle s maximum fuel saving potential by suggesting a proper speed profile, or even regulating the vehicle s speed profile. 1.2 V2V & V2I Communication The development of ITS is strongly related with information sharing among vehicles and infrastructures, which are named as Vehicle-to-Vehicle (V2V) communication and Vehicle-to-Infrastructure (V2I) communication. V2V communication enables the vehicles nearby to exchange information no less than vehicle position, speed, and location in real time [8], and V2I communication shares the operational information between vehicles on road and the road infrastructure [9]. Both V2V and V2I communications are designed to provide drivers of early warnings of the existing hazards, 2

17 help analyze and prevent potential risks, increase road capacitance and traffic flow speed, and decrease overall fuel consumptions and emissions. Those objectives are usually interconnected with each other. More synchronized and smoother flow of traffic is more likely to lead to safer driving, higher road capacitance, flow rate, and energy efficiency. 1.3 Recent Research on Transportation Efficiency The recent research of ITS on the efficiency aspect are mainly around the topics of efficient traffic light passing, traffic platoon formation/ speed synchronization, and cooperating with Adaptive Cruise Control (ACC) for economical speed profiles. For urban scenarios, one of the most dominant factors on vehicles fuel consumption is the stop-and-goes caused by traffic lights. Researchers are suggesting the method of having traffic lights broadcast duty cycle information to the nearby vehicles through V2I communications. The authors in [10] minimize vehicles fuel consumption by optimizing the vehicle s acceleration and deceleration process when passing the traffic light. Their method is based on the assumption that the traffic light s duty cycle is obtained via V2I, and is focused on determining the best acceleration and deceleration profile to pass the traffic light with minimum engine torque requirement and idling time. For the similar scenario, the authors in [11] give an analytical solution using a linearized vehicle longitudinal model and a numerical solution for comparison purpose. The authors conclude that the analytical method gives almost as accurate solutions as numerical solution, but with significant lower computation complexity which makes it possible for onboard computations. The researchers in [12] develop a green light optimized speed advisory application, and implement some simulations of 3

18 different penetration rates to evaluate the effectiveness. One interesting result is that as the penetration rate increases, not only the vehicles equipped with such system have higher benefit in fuel consumption, but the vehicles not equipped also benefits. And the overall traffic flow speed is improved. In [13] a more advanced solution for future intersection management system is proposed. The proposed system divides the road into grid cells, and reserves each grid for each vehicle for a specific time window, so as to increase traffic capacity and reduce fuel consumption and emissions by eliminating traffic lights. The system is based on the assumption that all the vehicles on road are fully autonomous, being able to V2V and V2I with reliable communication connections, and cooperating as a complete system. The research is more about the final stage of ITS in the future. Some research is more focused on utilizing traffic information from different sources cooperatively, and provide more globally oriented economical driving assistances. The authors in [14] introduce an algorithm to generate a TrafficMap via vehicular ad-hoc communication, so as to provide the drivers with traffic speed condition profiles far ahead of each lane. The TrafficMap is composed by a series of continuously generated and renewed entries from different V2V vehicles, and the speed information is compressed for higher broadcasting efficiency. Reference [15] describes a cooperative traffic situation analyzing system, which predicts the traffic speed by observing nearby vehicles behaviors through V2V and V2I communications along with static road information. Based on real time observation and prediction of traffic speed changes of each intersections, the system can therefore assist ego vehicle driver to drive e- conomically in urban traffic. The authors in [16] develop an eco-routing navigation system to find the route which requires minimum fuel consumptions and emissions. 4

19 The system evaluates a route s fuel requirement based on several data sources, such as travel demand, local congestion level, and numerous traffic trajectory snippets. The estimations are adjusted using an operational parameter set which quantifies the impacts of different factors on fuel consumption and emissions. The routing engine gives the most economical route based on the finalized estimation. For highway scenarios, speed disturbances in traffic flow are causing extra fuel consumption, and many phantom jams are reducing the traffic flow rate. Those phenomena are caused by inefficient human operations [17]. However, if the vehicles on highways can travel in a synchronized way, not only the traffic speed and highway capacity will be increased, but the energy consumption will also be decreased [18]. In [19], the authors formulate a traffic platoon for several vehicles based on V2V communication. The platoon operates by having one human driven vehicle as the leading vehicle which broadcasts its velocity information, and letting other autonomous vehicles receive the information and follow at the designated speed. In [20], the authors further exploit truck convoy s fuel saving potential on air-resistance and road terrains by forming platoons. They model the system into an optimization problem, and derive the optimal control for each truck within the platoon, so that the aerodynamic drag force can be largely reduced by small car-following distances. The authors in [17] propose a V2V based advisory system to synchronize each vehicle s speed, so as to eliminate the phantom jams on highways. They use the cellular automata (CA) model to estimate the changes in traffic density, and provide drivers operation suggestions based on logic judgments. They claim the traffic capacitance can be improved by 40% under full penetration rate, and average speed can be 30% faster for high traffic density conditions. 5

20 However, driving in platoons or synchronizing the vehicles are not so practical for a near term ITS plan. The formation of traffic platoons or synchronized traffic flow requires nearly full V2V penetration rate, and those vehicles involved should all cooperate and be engaged to a centralized control. Such cooperation is unlikely to be accomplished by human driver, because it lays too much operation burden on the drivers and highly depends on the drivers willingness. A stable and effective traffic synchronization system will require the vehicles engaged to be controlled automatically at least longitudinally. So it not only requires a high proportion of vehicles on road to communicate via V2V/V2I, but also requires those V2V/V2I vehicles to be at least partially drive-by-wire vehicles. A drive-by-wire vehicle is actuated using electronic control systems instead of the conventional mechanical parts. It has the major advantage that additional safety and comfort control systems can be easily implemented, since there is no direct driver control input into the vehicle [21]. Therefore, modifying a drive-by-wire vehicle will be mainly about software level and adding more sensors. Otherwise, automatically controlling a non-drive-by-wire vehicle will need the mechanical actuators that simulate human driver s operations on the vehicle s throttle, brake, steering wheel, etc, to be added inside the vehicle. Such actuators are not only usually expensive, but also occupy noticeable space, causing inconvenience for the driver. Therefore, it s not likely to massively modify those non-drive-by-wire vehicles on road to cooperate in an ITS which aims at synchronizing the vehicles in traffic. What s more, due to the cost and reliability issues, drive-by-wire technology is still not massively adopted among production vehicles. The first steer-by-wire production vehicle (Infinity Q50) was just announced by Nissan Motor Company in August, Throttle-by-wire is 6

21 a lot more popular among production cars, and there are also some models equipped with brake-by-wire system. But the current penetration rate is still rather low, and even massively modifying those throttle-by-wire & brake-by-wire vehicles will cost significant amount of money and time. As a result, travelling in platoons in everyday driving is not likely to happen within the decade. The more realistic solution for the near future is to optimize the efficiency of the driver s single vehicle, which is called the ego vehicle. 1.4 Research in This Thesis This thesis proposes a system to minimize the ego vehicle s fuel consumption in the car-following scenario on highways by providing optimal speed profiles. The optimal speed profiles are calculated using Model Predictive Control (MPC), and the information required is obtained from onboard sensors and V2V communication. The proposed system can be viewed as an upgrade to the current Adaptive Cruise Control (ACC) system. It enables the ego vehicle to minimize the amount of fuel consumed in following the vehicle immediately ahead of it, which is called the preceding vehicle in this paper. The longitudinal dynamic model and fuel consumption rate estimation model of the ego vehicle are required by MPC to generate effective control. Based on the predicted preceding vehicle s velocity within a horizon of seconds, the system calculates optimal engine torque and brake torque to control the vehicle. The controlled vehicle will drive at an optimal speed, ensuring minimum fuel consumption, safe car-following, and ride comfort. Speed prediction of the front vehicle is achieved by observing the traffic flow and estimating the driver behavioral parameters. As the disturbances of speed propagate 7

22 from upstream along the traffic flow with delays and oscillations in amplitude [22] [23], the preceding vehicle driver will operate the vehicle accordingly as a disturbance reaches. Therefore, if the traffic speed changes are detected before they reach the preceding vehicle, a prediction horizon is achieved. By estimating the preceding vehicle driver s behavioral parameters, the respective speed changes of the preceding vehicle can be predicted within the prediction horizon. For the near term applications, the V2V related system has to be able to work in low penetration rate scenarios. Although the Department of Transportation in US has required all new vehicles sold in US in the future to be equipped with vehicle communication systems, it is predicted by John A. Volpe National Transportation Systems Center that at least 9 years are needed to reach a 50% penetration rate after being mandated [24]. The speed prediction algorithm proposed in this paper doesn t require a high V2V penetration rate. The car-following behavioral parameter estimation of the preceding vehicle is realized by the speed and position information measured by the ego vehicle s onboard sensors, and a certain number of V2V-equipped vehicles sampling the traffic flow upper stream and broadcasting in real time. This system of optimal speed profile can work effectively during the process of V2V communication is being equipped by the traffic. 1.5 Organization of this Thesis The rest of this thesis is organized as follows. Chapter 2 describes the MPC algorithm to minimize the vehicle s fuel consumption in automated car-following scenarios. The longitudinal vehicle dynamic model and 8

23 fuel consumption rate estimation model used in this research is introduced. The development of the quadratic programming optimization problem and the optimization constraints are shown in this chapter. The algorithm is tested in several simulated scenarios, and the results are analyzed. Chapter 3 presents the preceding vehicle speed prediction algorithm. The traffic scenario of convoy driving is simulated using Intelligent Driver Model. To characterize the speed disturbance transmissions along the convoy in different traffic conditions, a driver behavioral car-following parameter estimation method is developed and the online data clustering technique is adopted. Prediction examples are shown. Chapter 4 introduces the leading vehicle speed prediction algorithm. A Finite State Machine (FSM) for driver s operation states on the gas pedal and the brake pedal is designed. Based on this FSM, a Speed-to-Action Decoder is developed to analyze the leading vehicle driver s driving style using its speed profile only. The driver s operations are formed into a Markov chain, and the leading vehicle s speed profile is predicted using Markov Model. Chapter 5 summarizes the results and advantages of the vehicle control algorithm and the speed prediction algorithms. Future researches on this project are also discussed in this chapter. 9

24 Chapter 2: Optimizing Vehicle Fuel Consumption using Model Predictive Control This chapter describes a Model Predictive Control (MPC) strategy to minimize the ego vehicle s fuel consumption by operating the vehicle s throttle and brake optimally. In this chapter, the control strategy is developed with the assumption that the preceding vehicle s future speed within the prediction horizon is perfectly predicted. The prefect prediction assumption helps to investigate the maximum fuel saving potential of the MPC strategy. In the next chapter, the preceding vehicle speed prediction method will be described in detail. 2.1 Introduction and Literature Review The idea of Model Predictive Control originated from late seventies, and has gained its popularity among industries and academia in the recent ten years. MPC does not refer to a specific control algorithm. It is a control idea that can be flexibly fitted into a wide range of control problems with a great variety of solutions. The idea of MPC is generally described as [25]: 1. The controlled system is modeled, and the system output can be explicitly predicted within the prediction horizon. 10

25 2. The potential control input sequence within the prediction horizon is first optimized based on the respective predicted system output, till the most desired control results are achieved in prediction. 3. Only the first step control of the whole control sequence is actually applied on the system, and the prediction horizon is moved forward for one step. For the vehicle s fuel consumption optimization, MPC is being widely adopted in this area. In [26], the authors model the factors of fuel economy, ride comfort, carfollowing ability, and safety into a total cost function with different weights assigned, and try to obtain the joint optimum. But those objectives are contradictory in most cases. For an ACC system, ride comfort sacrifices fuel economy [27] [28] [29]. And the fuel saving potential relies on the freedom in the car-following distances allowed. The weight assignment strategy will be very challenging for this approach. Moreover, safety should be an uncompromising first rule for an ACC system. Modeling the safety objective into the total cost function will increase the accident risk, as it requires a guaranteed well assigned cost function weights in any road conditions. But the weight assigning strategy is not introduced in this paper. In [30], the cost function is modeled into a similar way as [26], but with hard constraints on vehicle s car-following distance, acceleration, and speed to ensure vehicle safety. The preceding vehicle s speed is predicted based on the assumption that the preceding vehicle s state remains unchanged in the prediction horizon and can be approximated using last step state. Such prediction is only valid for a very short horizon and yields much error. Reference [31] describes a MPC strategy to minimize the vehicle fuel consumption in the stop-and-go scenarios between intersections. They use Continuation and Generalized Minimum Residual (C/GMRES) method to optimize the control input within 11

26 the prediction horizon, and claim that the algorithm is fast enough for real time onboard calculations. However, the author didn t model the vehicle s fuel consumption with respect to the engine s load. Instead, they approximate using polynomial function with respect to speed only, which too much simplified the fuel consumption estimation. Similar problem also happens in [32], which introduces a linear MPC based ACC system for reduced fuel consumption. In order to achieve the online computation speed, the authors simplify the vehicle fuel consumption calculation to a static fuel map which is only related with vehicle s speed and acceleration. Such simplification assumes the vehicle s load condition and road grade to be constant. The estimated vehicle fuel consumption using this static fuel map will lead to an unacceptable deviation from its real fuel consumption in actual operating conditions. Research in [26] [30] [31] [32] all have the drawback that the vehicle dynamics and fuel consumption rate are not modeled for a specific vehicle s actual parameters. In [26] [30] [31], even the influence of the gear selection is not considered. For the actual control of a vehicle s longitudinal dynamics and the engine s fuel supply in real load conditions, the preciseness of those models is not acceptable. The calculated control based on the imprecise models will result into much control deviations. The fuel consumption may not be really reduced, and driving safety may even suffer. 2.2 Vehicle Model Construction Thanks to Ford Motor Company s support, a full set of vehicle s transmission and engine parameters is provided to our group. In this research, a 2008 Lincoln MKS s longitudinal dynamic and fuel consumption map is modeled. The basic vehicle parameters are show in Table 2.1 [33]. 12

27 Table 2.1: Specifications for Lincoln MKS Specification Value Specification Value Mass (Kg) 1954 Cylinders alignment V6 Frontal Area (m 2 ) 2.77 Displacement (cm 3 ) 3726 Drag Coefficient 0.29 Maximum Torque (N m) 360 (4250rpm) Tire Radius (m) 0.36 Redline rpm Longitudinal Dynamic Vehicle Model The vehicle s longitudinal dynamic is decided by the vehicle s engine, road friction, aerodynamic resistance, road grade, and vehicle brake force [11]. where: m eq dv dt = F trac F roll F aero F grade F brake (2.1) F trac = ηγ R wh T e (2.2) F roll = mg cos (α) (r 0 + r 1 v) (2.3) F aero = 1 2 ρa fc d v 2 (2.4) F grade = mg sin (α) (2.5) m eq is the vehicle s equivalent mass, which includes the vehicle mass and the inertia of the powertrain s rotating parts; T e is the vehicle s engine torque; η is the transmission efficiency; γ is the total gear ratio; R wh is the wheel radius; α is the road grade; 13

28 v is the vehicle speed; r 0 and r 1 are the coefficients for a specific set of tyres and road surface; ρ is the air density; A f is the frontal area; C d is the air drag coefficient of the vehicle; F brake is the total brake force applied on the vehicle. Rearranging the above longitudinal dynamic equation ( ) [11]: v = 1 C 1 (γ) (C 2 (γ) T e C 3 v C 4 (α) F brake ) (2.6) where: C 1 (γ) = 1 m eq (2.7) C 2 (γ) = ηγ R wh (2.8) C 3 = 1 2 ρa fc d (2.9) C 4 (α) = mg (cos (α) r 0 + sin (α)) (2.10) Coefficient C 1 (γ) and C 2 (γ) are dependent on the gear selection, so as to model the acceleration difference caused by different gears. C 4 (α) is decided by the current road grade. Equation will be used in the optimization process in the later section Fuel Consumption Estimation Model Based on Willan s line approximation, Ozatay et al have developed a fuel consumption estimation model from the complex experimental model of Lincoln MKS s fuel consumption [11]: ṁ f = C 5 T e (t) v (t) + C 6 v(t) 2 + C 7 v (t) + C 8 (2.11) 14

29 γ (t) C 5 = (2.12) e H L R wh C 6 = P ( ) 2 loss V d γ (t) (2.13) e H L 4π R wh C 7 = P ( ) loss V d γ (t) (2.14) e H L 4π R wh C 8 = ṁ f,idle (2.15) where e is the average engine efficiency; H L is the lower heating values of fuel; P loss is the average engine friction term; V d is the displacement volume of the engine; ṁ f,idle is the engine s idle speed fuel consumption. The fuel consumption estimation model deployed in this research is closely related with vehicle s actual load condition. The vehicle s realtime fuel consumption is calculated based on the engine s output torque and the vehicle s current speed. The influence of gear selection is modeled, as the coefficients C 5, C 6 and C 7 are dependent on the gear the vehicle is driving on. 2.3 MPC Controller MPC Controller Design The vehicle models are first discretized into state space form with a sample time τ, so that the MPC controller can be designed in discrete time. The vehicle s longitudinal dynamic model in state space form is: x (k + 1) = Ax (k) + Bu (k) + D (2.16) 15

30 where x (k) is the vehicle s speed at the current time step; u (k) is the control applied on the vehicle at the current time step, which includes the vehicle s engine torque and total brake force: [ u (k) = T e (k) F brake (k) ] = [ u1 (k) u 2 (k) ] (2.17) A, B, and D are the coefficients at a certain gear: A = C 1 (γ) τc 3 C 1 (γ) [ ] B = τc2 (γ) C 1 τ (γ) C 1 (γ) = [ ] b 1 b 2 D = τc 4 (α) C 1 (γ) (2.18) (2.19) (2.20) Therefore, the vehicle s longitudinal dynamic states in a prediction horizon of p steps with a given control sequence is: X = Ãx (k) + BU + CD (2.21) where X = U = x (k + 1) x (k + 2). x (k + p) u 1 (k) u 2 (k) u 1 (k + 1) u 2 (k + 1). u 1 (k + p 1) u 2 (k + p 1) (2.22) (2.23) 16

31 Ã = A A 2. A p b 1 b Ab 1 Ab 2 b 1 b B = A p 1 b 1 A p 1 b 2 A p 2 b 1 A p 2 b 2 b 1 b 2 1 A + 1 C =. A p 1 + A p (2.24) (2.25) (2.26) Considering the objectives of fuel economy, ride comfort, car-following distances, and safety are more or less contradictory to each other, the MPC cost function in this research is designed in the single objective form of fuel economy only, with the other objectives fulfilled using constraints. Therefore, the cost function is to minimize the vehicle s total fuel consumption within the prediction horizon based on the vehicle s current state x (k): minj (x (k), U) = ] [u(k + j) T ex (k + j + 1) + C 6 x(k + j + 1) 2 + C 7 x (k + j + 1) + C 8 τ min p 1 j=0 (2.27) where e = [ C5 0 ] (2.28) Rearrange equation 2.27 into matrix form: minj (x (k), U) = ( U T EX + C 6 X T X + C 7 XI + p C 8 ) τ (2.29) where 17

32 E is a 2p p matrix, which is in the form: C C 5 0 E = C I is a p 1 identity matrix. (2.30) Substitute equation 2.21 into 2.29 and rearrange into quadratic form: minj (x (k), U) = U (E T B τ + B ) T B C 6 τ U+ U (EÃ T x (k) τ + E C dτ + 2 B T Ã C 6 x (k) τ + 2 B T C C6 Dτ + B ) T I C 7 τ +(ÃT Ã C 6 x 2 (k) τ + 2ÃT C C6 Dx (k) τ + C T C C6 D 2 τ + ÃT I C 7 x (k) τ + C T I C 7 dτ + pc 8 τ) = 1U T QU + R T U + N 2 (2.31) As matrix Q = E B τ + B T B C 6 τ is not symmetric, Q is substituted by Q ( = ) 1 2 Q + Q T, which is a commonly adopted technique. Equation 2.31 is the cost function used in the optimization process of MPC. Vehicle control U, which includes engine torque and total brake force at each time step in the prediction horizon, is the variable to be optimized Optimization Constraints Some constraints are applied during the optimization process so that the calculated control ensures the vehicle to follow the preceding vehicle safely, constantly, and comfortably. Safety Constraint A minimum car-following distance must be applied for safety. It should ensure the ego vehicle does not approach the preceding vehicle too closely. The constraint is 18

33 described as below: τ 0 0 τ τ τ τ τ where: x (k + 1) x (k + 2). x (k + p) P ego (k) P pre (k + 1) P pre (k + 2). P pre (k + p) d min (2.32) P ego (k) is the ego vehicle s position at the current time step; [ Ppre (k + 1) P pre (k + 2) P pre (k + p) ] T is the preceding vehicle s positions at each time step of the prediction horizon; d min is the minimum car-following distance allowed. Substitute the vehicle state matrix [ x (k + 1) x (k + 2) x (k + p) ] T in 2.32 with calculated vehicle states within the prediction horizon based on the current state: ) T (Ãx (k) + BU + C D + I P ego (k) P pre I d min (2.33) Rearrange the inequality 2.33: T BU P pre I d min I P ego (k) T Ãx (k) T C D (2.34) Inequality 2.34 is the safety constraint applied on the vehicle s throttle and brake control. Maximum Car-following Distance Constraint The ego vehicle should not fall too far behind the followed preceding vehicle. This makes the ego vehicle follow the same vehicle constantly, and ensures the ego vehicle have nearly the same total trip time as the followed vehicle. ) T (Ãx (k) + BU + C D + I P ego (k) P pre I d max (2.35) 19

34 where: d max is the maximum car-following distance allowed. Rearrange the inequality 2.35: T BU P pre + I d max + I P ego (k) + T Ãx (k) + T C D (2.36) Inequality 2.36 is the maximum car-following constraint applied on the vehicle s throttle and brake control. Ride Comfort Constraint To make the system more acceptable to the drivers, the system should provide a comfortable car-following experience. The vehicle s throttle and brake should be controlled without heavy accelerations and jerks. Therefore, the comfort constraints are defined from two aspects: (1) Heavy accelerations and brakes are not allowed. a dece max a a acce max (2.37) In this research, the maximum acceleration rate a acce max is set at 1m/s 2, and the maximum deceleration rate a dece max is set at 2m/s 2. Write inequality 2.37 into vehicle state space form: x (k + 1) x (k) x (k + 2). x (k + 1). 1 I a τ acce max x (k + p) x (k + p 1) x (k + 1) x (k) x (k + 2). x (k + 1). 1 I a τ dece max x (k + p) x (k + p 1) (2.38) 20

35 Denote: Ã 2 = A 0 A 1. A p b 1 b B 2 = A p 2 b 1 A p 2 b 2 A p 3 b 1 A p 3 b A C 0 2 =. A p 2 + A p A 0 And write inequality 2.38 into the the form with respect to control: ( ) ) ( ) B B2 U I τa acce max (Ã Ã 2 C C2 D ( ) ) ( ) B B2 U I τa dece max + (Ã Ã 2 + C C2 D (2.39) (2.40) (2.41) (2.42) (2) Jerky accelerations are not allowed. The gaining of acceleration should be gentle, with throttle gradually applied. T e T d max (2.43) where T d max is the maximum torque increasing rate allowed, which is set at 50Nm/s in this research. Write inequality 2.34 into the form with respect to control U: (S T 1 S T 2 ) U I T d max (2.44) where S T 1 and S T 2 are (p 1) 2p matrixes: S T 1 = S T 2 = (2.45) (2.46) 21

36 Vehicle Performance Constraints The solution domain is also limited by the vehicle s engine and brake performance: For the engine s torque output: 0 T e 360Nm (2.47) For the total brake force can be applied on the vehicle: 2.4 Problem Convexity 0 F brake 10000N (2.48) Before selecting an optimization algorithm, the convexity of the problem should be first checked. Judging from the constraints, the problem domain is very likely to be not convex. As the problem is a car-following problem, the domain is dependent on the preceding vehicle s speed profile in the prediction horizon. The ego vehicle s position should be controlled within the range allowed by the minimum car-following distance and the maximum car-following distance in the prediction horizon. Therefore, unless the preceding vehicle is driving at a constant speed, the problem domain cannot be convex. Fig.2.1 shows one example when the domain is nonconvex. We are also curious about the convexity of the cost function, which is equation 2.31, when the constraints are convex. Although the cost function is in quadratic form, the convexity judgement of the cost function cannot be done simply from the Hessian of the cost function. Because the vehicle is only operated with positive torque and brake force, the solution domain can be at most the positive part of the full domain of real number. Therefore, even though the Q matrix in 2.31 is neither 22

37 1000 Position Constraints in the Prediction Horizon Position(m) Minimum Car following Distance Maximum Car following Distance Time(s) Figure 2.1: An Example of the Position Constraints positive definite nor negative definite, it cannot be concluded that the cost function is neither convex nor concave. The following section checks the convexity of the cost function using the basic definition of convexity [34]: For a function f: R n R, if dom f is a convex set and x 1, x 2 dom f with 0 λ 1, f (λx 1 + (1 λ) x 2 ) λf (x 1 ) + (1 λ) f (x 2 ) (2.49) then function f is convex. As the cost function of this problem contains 2p variables, where p is the number of prediction steps, it s not very possible to directly check the convexity using the original cost function. The dimension of the cost function is first scaled down using 23

38 the property [34]: if function f 1 f m are all convex functions, then their nonnegative weighted sum f = w 1 f 1 + w m f m is convex. As equitation 2.27 is the sum of vehicle fuel consumption at each step with positive weight τ, the convexity check can start with the fuel consumption calculation equation at any one single step as in equation If the fuel consumption calculation equation at one step is convex, then the total cost function of the prediction horizon is convex. J k+j = u(k + j) T ex (k + j + 1) + C 6 x(k + j + 1) 2 + C 7 x (k + j + 1) + C 8 (2.50) (j [0, p 1] Z) where x (k + j + 1) refers to the vehicle s next step speed, which can be substituted by the vehicle speed calculation equation with respect to control as x (k + j + 1) = b 1 T (k + j) + b 2 B (k + j) + Ax (k + j) + D (2.51) Equation 2.52 is the substituted and rearranged equation, which is related with the current control of throttle, brake, and current speed only. As the vehicle s current speed is not a variable can be controlled, equation 2.52 is a cost function with respect to throttle T (k + j) and brake B (k + j) only. Therefore, turn to check the convexity of equation 2.52 for a given speed x (k + j). J k+j = (C 5 b 1 + C 6 b 2 1) T (k + j) 2 + (C 5 b 2 + 2b 1 b 2 C 6 ) T (k + j) B (k + j) + (C 5 A + 2b 1 AC 6 ) T (k + j) x (k + j) + (C 5 D + 2b 1 DC 6 + C 7 b 1 ) T (k + j) +C 6 b 2 2B(k + j) 2 (2.52) + 2C 6 Ab 2 B (k + j) x (k + j) + (2b 2 DC 6 + C 7 b 2 ) B (k + j) +C 6 A 2 x(k + j) 2 + (2C 6 AD + C 7 A) x (k + j) + C 6 D 2 + C 7 D + C 8 Then the vehicle s actual parameters are substituted into equation Because the parameters are Ford Motor Company s confidential data, the parameters are represented by a, b,, l: J k+j = at (k + j) 2 + bt (k + j) B (k + j) + ct (k + j) x (k + j) + dt (k + j) +eb(k + j) 2 + fb (k + j) x (k + j) + gb (k + j) + hx(k + j) 2 + ix (k + j) + l 24 (2.53)

39 Then apply the convexity definition: χ 1 = [ T1 B 1 ] S, χ 2 = [ T2 B 2 ] S, λ [0, 1] f [(1 λ) χ 1 + λχ 2 ] (1 λ) f (χ 1 ) λf (χ 2 ) = (λ 1) λ [ a(t 1 T 2 ) 2 + e(b 1 B 2 ) 2 + b (T 1 T 2 ) (B 1 B 2 ) ] [ = (λ 1)λ ((T1 T a 2 ) + b (B 2a 1 B 2 ) ) 2 + 4ae b 2 (B 4a2 1 B 2 ) 2] (2.54) [ ((T1 T 2 ) (B 1 B 2 )) (B 1 B 2 ) 2] = (λ 1)λ a Because T 1 T 2 [ 360, 360], B 1 B 2 [ 10000, 10000], equation 2.54 is not always less than zeros or larger than zero. Therefore, the cost function 2.50 is not convex or concave in the solution domain. However, the vehicle s throttle and brake are not applied at the same time in normal driving in real conditions. The convexity should then be checked when either throttle or brake is zero. When brake is kept constant, equation 2.54 becomes: f [(1 λ) χ 1 + λχ 2 ] (1 λ) f (χ 1 ) λf (χ 2 ) = (λ 1) λa(t 1 T 2 ) 2 (2.55) which is always larger than zero. When throttle is kept constant, equation 2.54 becomes: f [(1 λ) χ 1 + λχ 2 ] (1 λ) f (χ 1 ) λf (χ 2 ) = (λ 1) λe(b 1 B 2 ) 2 (2.56) which is also always larger than zero. Therefore, as long as the vehicle s throttle and brake are not operated simultaneously, the single step cost function 2.52 is convex, and the total cost function of the prediction horizon 2.31 is convex. 2.5 Quadratic Programming Optimization As the cost function 2.31 is in quadratic form and subjects to several inequality linear constraints (2.34, 2.36, 2.42, 2.44, 2.47 and 2.48), the optimization part in MPC 25

40 is a quadratic programming problem. Because the car-following distance constraints (2.34 & 2.36) are not convex under most conditions, nonconvex quadratic programming optimization method should be deployed. Active-set method is adopted in this research. Active-set method is an iterative method containing two-phases: In the first phase, a feasible point is found inside the domain limited by the constraints, with the optimization objective not considered. In the second phase, the objective is optimized within the feasible area by computing a sequence of feasible iterates [35]. The detailed algorithm is not described in this paper. In this research, the MATLAB solver Active-set of toolbox quadprog is used. 2.6 Algorithm Tests and Improvements The MPC based fuel consumption optimization algorithm is first tested under two basic scenarios. Then based on the test results, the algorithm is further improved Basic Scenario Tests The developed algorithm is tested with a deceleration-to-acceleration scenario and an acceleration-to-deceleration scenario. The ego vehicle started following the preceding vehicle at the initial distance of 80 meter and the same initial speed with the preceding vehicle. The minimum car-following distance allowed is 40 meter, the maximum distance allowed is 120 meter. The ego vehicle is driving on the 5th gear in the simulation. Fig.2.2 and fig.2.3 shows the test results with a 15 seconds prediction horizon. 26

41 Velocity profiles of the preceding vehicle and the ego vehicle 32 Speed(m/s) Preceding Ego Time(s) Distance(m) Car following Distance Time(s) 350 Torque Brake Engine Torque(Nm) Time(s) Brake Force(N) Time(s) Figure 2.2: Test Results with 15 Seconds Horizon in the Deceleration-to-Acceleration Scenario In the deceleration-to-acceleration scenario, the preceding vehicle has a rather heavy brake and then resumes the original cruising speed of 30m/s. With a prediction horizon of 15 seconds, the MPC controlled ego vehicle can slow down 15 seconds earlier, which allows the ego vehicle to decelerate with much lower rate. The ego vehicle decelerates mainly due to air-resistance, road friction, and transmission resistance in this process. Merely 100N brake force is applied, which is almost negligible for a vehicle s dynamic. No torque is given from the engine during this process. By slowing down ahead of time, the car-following distance is enlarged, thus allowing the ego vehicle not to decelerate to a low speed as the preceding vehicle does. A lot less energy is wasted during this optimized deceleration process. In the acceleration process, the ego vehicle also accelerates with a much lower rate than the preceding 27

42 vehicle, which again helps saving more fuel. During the 80 seconds process, the preceding vehicle consumes g fuel. In comparison, the MPC controlled ego vehicle consumes g, which has a 15.02% fuel saving advantage. In the acceleration-to-deceleration scenario, the MPC controlled ego vehicle is also able to accelerate and decelerate with much lower rate by planing ahead of time. The allowed car-following distance is much utilized during the early accelerations and decelerations. During the 160 seconds process, the preceding vehicle consumes g fuel. In comparison, the MPC controlled ego vehicle consumes g, which has a 6.33% fuel saving advantage. Velocity profiles of the preceding vehicle and the ego vehicle 42 Preceding 40 Ego Speed(m/s) Time(s) Distance(m) Car following Distance Time(s) 350 Torque Brake Engine Torque(Nm) Time(s) Brake Force(N) Time(s) Figure 2.3: Test Results with 15 Seconds Horizon in the Acceleration-to-Deceleration Scenario 28

43 The fuel saving advantage in the acceleration-to-deceleration scenario is much smaller than in the deceleration-to-acceleration scenario. There are three reasons account for that: (1)The simulated acceleration-to-deceleration scenario has much longer cruising time, and there is not much fuel saving potential when the vehicle is cruising. (2)In the acceleration-to-deceleration scenario, the preceding vehicle decelerates a lot more gentle than in the deceleration-to-acceleration scenario. In the deceleration-to-acceleration scenario, the preceding vehicle had a 10m/s deceleration in 7 seconds. But in the acceleration-to-deceleration scenario, the preceding vehicle has a 10m/s deceleration in 15 seconds, which means the preceding vehicle is driving in a more economical way. (3)In the deceleration-to-acceleration scenario, the preceding vehicle accelerates immediately after a heavy brake, which does not cause the carfollowing distance to reduce much. The minimum distance from the ego vehicle to the preceding vehicle is still about 75m away after the hard brake. Thus, the ego vehicle doesn t have to change its speed much. However, in the acceleration-to-deceleration scenario, the preceding vehicle maintains the speed after a fast acceleration, causing the ego vehicle have to also speed up a lot to maintain the car-following distance. As can be seen from the car-following distance figure in fig.2.3, the maximum carfollowing distance has been fully utilized. Therefore, the fuel saving advantage is strongly dependent on the speed scenario. Simply using some extreme scenarios to measure the fuel saving advantage doesn t make much sense. And that is one common problem among fuel consumption optimization papers. It is more reasonable to evaluate using more complicated scenarios over a longer period, which will be done in the following section. 29

44 2.6.2 Eliminating Unnecessary Brakes It can be noticed that in both scenarios, a rather significant brake is applied at the beginning, which is at about 1800N. It is caused by the relationship of vehicle s fuel consumption with speed, and the limit of the prediction horizon. Based on the fuel consumption calculation equation (2.11), the vehicle consumes less fuel at a lower speed. And because of MPC s framework, the vehicle s carfollowing is optimized only within the prediction horizon. Therefore, the minimum fuel consumption is generated by driving at the lowest speed allowed by the carfollowing distances. Fig.2.4 shows the optimized results given within the prediction horizon. Velocity(m/s) Velocity in the Prediction Horizon Ego Vehicle Front Vehicle Time(s) Torque(Nm) Throttle in the Prediction Horizon Time(s) 1000 Position in the Prediction Horizon 5000 Brake in the Prediction Horizon Position(m) Ego Vehicle Position Position Constraints Time(s) Brake(N) Time(s) Figure 2.4: The Optimized Control within the Prediction Horizon 30

45 On some scenarios when there is more than enough space in the car-following distance constraints that allow the ego vehicle to coast through, throttle is not applied at all. Brake is applied at the beginning of the prediction horizon to reduce the vehicle initial speed, so that the vehicle can coast exactly to the maximum car-following distance allowed at the end of the prediction horizon. This way maximizes the space given by the car-following distance, and thus minimizes the fuel consumption by coasting at the lowest possible speed. However, such strategy is only advantageous within the prediction horizon. For a longer trip, the advantage gained within the horizon will be out-weighted as the horizon moves forward. In the newer horizon, the vehicle will start with a lower speed, and this lower speed cannot make the vehicle stay in the car-following distance constraints by only coasting. Then the vehicle will have to accelerate harder so as to keep within the constraints. The optimization process can only optimize the control of each prediction horizon, not the whole trip. Overall fuel saving potential can be sacrificed by depleting the maximum fuel saving potential of each prediction horizon. The solution is to force the algorithm not to apply brake only for the aim of saving fuel. Before the optimization process, first calculate the vehicle s terminal position with respect to the preceding vehicle at the end of the prediction horizon by coasting. The initial coasting speed is the current vehicle speed. If the terminal position is inside the maximum car-following distance constraint and larger than the minimum car-following distance, then change the terminal constraint of the maximum car-following distance to the vehicle s terminal position. Otherwise, the car-following distances are not changed. This prevents the algorithm from applying unnecessary 31

46 brake to only deplete the fuel saving potential within the horizon, but not interfering with the normal distance keeping. Fig.2.2 and fig.2.3 shows the test results with same 15 seconds prediction horizon in the same scenarios, but with unnecessary brakes prevented. For the decelerationto-acceleration scenario, the total fuel consumption of the MPC controlled ego vehicle is reduced from g to g. For the acceleration-to-deceleration scenario, the total fuel consumption of the MPC controlled ego vehicle is reduced from g to g. And no brake is applied at all in the both scenarios, which means the later processes of the two scenarios are also benefiting from removing the unnecessary brakes. Therefore, the overall performance is improved by modifying the algorithm s shortsighted decisions. Velocity profiles of the preceding vehicle and the ego vehicle 32 Speed(m/s) Preceding Ego Time(s) Distance(m) Car following Distance Time(s) 350 Torque 100 Brake Engine Torque(Nm) Time(s) Brake Force(N) Time(s) Figure 2.5: Test Results with Unnecessary Brake Prevented in the Deceleration-to- Acceleration Scenario 32

47 Velocity profiles of the preceding vehicle and the ego vehicle 42 Preceding 40 Ego Speed(m/s) Time(s) Distance(m) Car following Distance Time(s) 350 Torque 100 Brake Engine Torque(Nm) Time(s) Brake Force(N) Time(s) Figure 2.6: Test Results with Unnecessary Brake Prevented in the Acceleration-to- Deceleration Scenario Complex Speed Profile Test The algorithm is tested with a rather complex preceding vehicle speed profile, where the preceding vehicle, which is driven by a human driver without any assistance, experiences several accelerations and decelerations with different rates and periods. Different length of prediction horizons are tested, including 5s, 10s, 15s, 20s, and 25s, so as to evaluate the influence of prediction horizon on the results. The results are shown in fig.2.7, 2.8, 2.9, 2.10, and table.2.2. As is shown in fig.2.7, the MPC controlled ego vehicle is much less sensitive to the preceding vehicle s speed changes with longer prediction horizons. Longer prediction horizons make the algorithm plan the vehicle s speed profile earlier before speed 33

48 25 20 Vehicle Velocity Profiles Front vehicle 5s horizon 10s horizon 15s horizon 20s horizon 25s horizon Velocity(m/s) Time(s) Figure 2.7: Speed Profiles with Different Prediction Horizons 120 Car following Distance Profiles Distance(m) s horizon 10s horizon 15s horizon 50 20s horizon 25s horizon Time(s) Figure 2.8: Car-following Distance Profiles with Different Prediction Horizons 34

49 Vehicle Engine Torque 5s horizon 10s horizon 15s horizon 20s horizon 25s horizon Torque(Nm) Time(s) Figure 2.9: Engine Torque Profiles with Different Prediction Horizons Vehicle Brake Force 5s horizon 10s horizon 15s horizon 20s horizon 25s horizon Force(N) Time(s) Figure 2.10: Brake Force Profiles with Different Prediction Horizons 35

50 Table 2.2: Vehicle Fuel Consumption of Different Prediction Horizons Horizon Length Fuel Consumed (gram) Fuel Saving Advantage Preceding Vehicle seconds % 10 seconds % 15 seconds % 20 seconds % 25 seconds % changes happen. Therefore, the space allowed by car-following distances is more fully utilized with longer prediction horizons, which is proved in fig.2.8. The amplitude of the vehicle s engine torque and brake force is also significantly reduced as the prediction horizon enlarges, which are illustrated in fig.2.9 & fig Less energy is required in the acceleration processes, and less energy is wasted in the deceleration processes. The total fuel consumption and fuel saving advantage are calculated and listed in table.2.2. It shows that longer prediction horizon leads to better fuel economy in this algorithm. However, as the prediction horizon increases, the magnitude of the cost function and constraints increase respectively, thus requiring more calculation time. The simulation was run by 64-bit MATLAB(2013) on a laptop, with a 2.40GHz Core i CPU and 4.00 GB RAM. The average calculation time for different prediction horizons was counted by MATLAB and is listed in table.2.3. Thanks to the high efficiency nature of quadratic programming, the calculation time can satisfy the requirement of real time calculation. 36

51 Table 2.3: Average Calculation Time of Different Prediction Horizons 5 seconds 10 seconds 15 seconds 20 seconds 25 seconds Time(Second) The fuel saving advantage and average calculation time of different prediction horizons are plotted in fig The fuel saving advantage increases with the prediction horizon logarithmically, while the average calculation time increases with the prediction horizon exponentially. A prediction horizon of 10 seconds can achieve a significant fuel saving advantage of more than 10%, with a very low calculation time of merely 56 milliseconds. The fuel saving potential has almost saturated after the prediction horizon is longer than 20 seconds, but with calculation time fast increasing. Therefore, a very long prediction horizon is not necessary. A prediction horizon between 10 seconds and 20 seconds can achieve a balance between control results and calculation efficiency. 2.7 Multiple MPC Controlled Vehicle Convoy The scenario of several MPC controlled vehicles following one after another as a convoy is simulated. The leading vehicle speed profile, vehicle initial states, and the car-following constraints are the same as those in the previous section. The prediction horizon of the vehicles are all 15 seconds. And the vehicles are identical. Vehicle convoys of four vehicles and ten vehicles are simulated and evaluated. 37

52 Average Calculation Time (Seconds) Fuel Saving Advantage (%) Prediction Horizon Length (Seconds) Figure 2.11: Fuel Saving Advantage and Average Calculation Time Four Vehicle Convoy Simulation The simulated results show that the following vehicles velocities will converge to the same velocity as the first MPC controlled vehicle, and form a traffic platoon. Fig.2.12, 2.13, 2.14 and 2.15 show the simulation results of a convoy driving of four MPC controlled vehicles. The first MPC controlled vehicle follows the leading vehicle, and adjusts its car-following distance through the whole process to achieve optimized fuel consumption. For the 2nd, 3rd and 4th MPC controlled following vehicles, after 40 seconds they converge to the same speed as the 1st MPC controlled vehicle, and their car-following distances stay at the maximum car-following distance constantly. The four vehicles engine torque profiles and brake force profiles are almost identical. Therefore, a vehicle platoon is formed when having several vehicles controlled by the 38

53 25 20 Velocity profiles of the leading vehicle and the following vehicles Leading vehicle 1st Following vehicle 2nd Following vehicle 3rd Following vehicle 4th Following vehicle Speed(m/s) Time(s) Figure 2.12: Speed Profiles of 4 MPC Controlled Vehicles Convoy 130 Distance to the Preceding Vehicle Distance(m) st Following vehicle 2nd Following vehicle 3rd Following vehicle 4th Following vehicle Time(s) Figure 2.13: Car-following Distance Profiles of 4 MPC Controlled Vehicles Convoy 39

54 Engine Torque(Nm) The Vehcile Engine Torque Profiles 1st Vehicle 2nd Vehicle 3rd Vehicle 4th Vehicle Time(s) Figure 2.14: Engine Torque Profiles of 4 MPC Controlled Vehicles Convoy Brake Profiles of the Vehicles 1st Vehicle 2nd Vehicle 3rd Vehicle 4th Vehicle 2500 Brake Force(N) Time(s) Figure 2.15: Brake Force Profiles of 4 MPC Controlled Vehicles Convoy 40

55 same algorithm following one after another, even though there is no V2V communication to coordinate their speed. Table.2.4 shows the fuel consumption of each vehicle and the fuel saving advantages over the leading vehicle. It can be noticed that for a relatively smaller convoy, the fuel saving advantage actually increases as the number of following vehicle increases. Table 2.4: Each Vehicle s Fuel Consumption of a 4 MPC Vehicle Convoy Vehicle # Fuel Consumed (gram) Fuel Saving Advantage Leading Vehicle st Following Vehicle % 2nd Following Vehicle % 3rd Following Vehicle % 4th Following Vehicle % Ten Vehicle Convoy Simulation The dimension of the convoy is increased to ten MPC controlled vehicles and is simulated. The results are shown in fig.2.16 and fig In the beginning of the whole process, because of the rather long convoy, the vehicles behind would rather slow down to still and wait until the distances become close to the maximum carfollowing distance. After 50 seconds, the following vehicles converge to the same speed and maintain an almost constant car-following distance, but with some velocity oscillations happening in the convoy. Fig.2.18 shows the details of the speed profile of the convoy. The oscillations start from the 7th vehicle, and is enlarged on the following vehicles. Table.2.5 lists each vehicle s fuel consumption within the convoy. 41

56 The fuel saving advantage starts to decrease after the 7th vehicle, on which the speed oscillations start to happen. 25 Speed profiles of the leading vehicle and the following vehicles 20 Speed(m/s) Time(s) Figure 2.16: Speed Profiles of a 10 MPC Controlled Vehicles Convoy Such phenomenon is also known as slinky effect, which refers to the control errors being amplified along the traffic stream, and the convoy may lose its string stability as the number of vehicles increases [36] [37]. It has been pointed out that using V2V communication to broadcast the leading vehicle s velocity and acceleration to the other vehicles within the convoy can effectively avoid the slinky effect [38]. And there are some control laws have been developed to control the vehicles within the convoy to form a stable platoon [36] [39]. For the near term traffic condition considered in this research, it s not likely to have the situation when more than six vehicles equipped with the proposed system meet and form a convoy, and the slinky effect does not 42

57 130 Distance to the Preceding Vehicle Distance(m) Time(s) Figure 2.17: Car-following Distance Profiles of a 10 MPC Controlled Vehicles Convoy Speed profiles of the leading vehicle and the following vehicles Speed(m/s) Time(s) Figure 2.18: Speed Profile Details of the 10 Vehicles Convoy 43

58 Table 2.5: Each Vehicle s Fuel Consumption of a 10 MPC Vehicle Convoy Vehicle # Fuel Consumed (gram) Fuel Saving Advantage Leading Vehicle st Following Vehicle % 2nd Following Vehicle % 3rd Following Vehicle % 4th Following Vehicle % 5th Following Vehicle % 6th Following Vehicle % 7th Following Vehicle % 8th Following Vehicle % 9th Following Vehicle % 10th Following Vehicle % happen for a short convoy of less than six vehicle. When we have reached a high penetration rate, it will be essential to have controllers designed for eliminating the slinky effect. 44

59 Chapter 3: Speed Prediction of the Preceding Vehicle This chapter introduces an algorithm to predict the speed of the preceding vehicle, which is required by the MPC based vehicle fuel consumption optimization system. 3.1 Introduction and Literature Review For improving traffic speed, capacity, and energy efficiency, there have been many prediction algorithms developed. The research in [40] predicts drivers operating speed at certain parts of road, and the influence of road structure is mainly investigated. In [41], a speed prediction algorithm for the upcoming journey on a given route is developed using a neural network based traffic model. The neural network is trained with historical data of the route, and can predict the speed profile of the route for at most 30 minutes based on the detected traffic dynamic changes. The authors of [42] cluster the traffic states from different road point into time variant clusters, so as to evaluate the adjacent road sections influence on each other. Then the traffic conditions are predicted online based on the recent cluster information through neural networks. In [43], different approaches were used to predict the vehicle speed for a short term on a specific road, including multiple-regression analysis, time-series forecasting, artificial neural networks, and Kalman filtering. The vehicle s speed at some certain spots of the road is predicted and compared. The results show that 45

60 different approaches have their advantages in different traffic conditions. In [24], the authors propose a vehicle location prediction algorithm in low V2X penetration rate scenarios. Their algorithm inserts an initial number of estimated non-connected vehicles into the queue, compare the estimated connected vehicle s position with the reported position, then reduce or increase the number of estimated non-connected vehicles till the estimated position coincide with real position. The research is focused on estimating the positions of the non-connected vehicles to predict the traffic density and overall vehicle queue speed. Other studies [40][41][42][43][24] are more about predicting the overall traffic flow speed, or an expected vehicle speed in a certain traffic condition from a relatively macroscopic view. There are also research efforts on more microscopic speed prediction for different scenarios. The authors in [31] introduce a prediction method of the preceding vehicle s acceleration when approaching a traffic light. The prediction is based on a measured average relationship of braking rate with speed and distance to stop. Such prediction methods cannot fit to a particular driver, and the influence of the traffic condition is not considered. The speed prediction in [44] is designed for a similar scenario. The method first estimates the vehicle s driving configuration of the current intersection traffic condition, then predict the vehicle s velocity for a horizon of three seconds using Random Forest Regressors. Studies presented in [31][44] are more focused on predicting the vehicle s stop-and-go behaviors in low speed range. In [45],the authors propose a vehicle dynamic model that can accurately predict the acceleration profile for a vehicle within a convoy. The model can work in large speed range with estimated parameters for speed prediction. But the parameter estimation method is not developed in the paper. The speed prediction method described in [46] is 46

61 aimed at working in the exactly same scenario as in this chapter. And the authors also use Intelligent Driver Model (IDM) to model vehicle behaviors. A particle-filter based Expectation-Maximization algorithm is adopted to do parameter estimations. However, the authors estimate only one of the 5 parameters of IDM, which is the time headway parameter, and set the other 4 parameters as fixed values to do speed prediction. In the current work introduced in this chapter, 4 parameters of IDM are estimated. The designed preceding vehicle speed prediction algorithm is based on V2V communication. It can be implemented when there is only a low portion of vehicles on the road can communicate via V2V. Those V2V vehicles are required to broadcast some basic safety information in realtime, which includes the recent speed profiles and position profiles. The preceding vehicle s speed and position is measured by onboard sensors of the ego vehicle, and this vehicle does not have to be able to V2V communicate. Therefore, the system requirement for the speed prediction algorithm is rather low. It does not need a high V2V penetration rate, and it requires those V2V vehicles no more than broadcasting their speed and position information. Such characteristics make the preceding vehicle speed prediction algorithm presented in this thesis more practical for near-term production vehicles. 3.2 Scenario description and modeling Scenario Description The tracked and predicted situation presented in this thesis involves a single lane road, where each vehicle follows its preceding vehicle and forms a convoy. Since the upstream disturbances of speed in the traffic propagate via the following vehicles 47

62 with oscillations of amplitude and delays[23], it is feasible to predict the preceding vehicle s speed changes if the disturbances can be captured before they arrive at the preceding vehicle. Fig.3.1 is a brief illustration of the preceding vehicle speed prediction scenario. The V2V vehicle in front of the ego vehicle within the convoy is regarded as the leading vehicle, which broadcasts its real-time speed and position information. From this V2V vehicle, the speed disturbances propagating along the convoy can be detected earlier. The vehicle immediately ahead of the ego vehicle, which is Vehicle 4 in the figure, is denoted as the preceding vehicle. The preceding vehicle s speed and position information is measured and recorded by the ego vehicle. Figure 3.1: The Traffic Scenario. Other than the leading vehicle and the preceding vehicle, the traffic condition in between is not visible to the ego vehicle, which is shown in fig.3.2. The number of vehicles between the leading vehicle and the preceding vehicle is unknown, and it s not likely to estimate the behavioral parameters of those vehicles. However, it can be known to the ego vehicle that there exists an unknown number of vehicles in between, relaying the perturbations in traffic flow. In the case where no vehicle is between the leading vehicle and the preceding vehicle, the preceding vehicle driver will follow free road behaviors, and the transmission of speed perturbations is discontinued. 48

63 The existence of vehicles between the leading vehicle and the preceding vehicle can be deduced by analyzing the recent speed profiles of the leading vehicle and the preceding vehicle, which will be researched in my future work. This study focuses on the scenarios of speed prediction when there is at least one vehicle in between, though the number of vehicles is not directly detectable. Figure 3.2: Traffic Condition from the View of the Ego Vehicle Traffic Modeling The traffic flow on a single lane road is simulated using the intelligent driver model(idm) developed by Treiber, Hennecke, and Helbing in 2000[47]. The IDM is a microscopic car-following model that was first developed to describe the traffic flow in congested traffic conditions, but the model structure makes it also accurate enough to simulate free road traffic. The IDM describes the practical crash-free responses of the drivers with respect to the speed difference and the car-following distance to the front vehicle[48]. It has the major advantages of having controllable stability properties, demonstrating smooth transitions in vehicle dynamics, and using parameters with solid physical meanings[49]. Equation 3.1 and 3.2 show the dynamics of the αth 49

64 vehicle within a convoy modeled by IDM [47][49]: dv α dt = a α ( 1 ( vα v 0α ) δ ( ) ) s 2 (v α, v α ) s α (3.1) where v α is the speed of the αth vehicle; s (v α, v α ) = s 0 + v α T α + v α v α 2 a α b α (3.2) v α is the approaching rate to the front vehicle: v α = v α v α 1 ; s α is the car-following distance: s α = x α 1 x α l α 1, of which x α is the position of the αth vehicle, and l α 1 refers to the length of the vehicle in front of it. δ is the acceleration exponent, which is usually set at 4. For modeling a specific vehicle s car-following behaviors, the IDM uses five parameters: with; with; T α is the desired time headway of the αth vehicle; a α is the maximum acceleration rate that the αth vehicle driver feels comfortable b α is the maximum deceleration rate that the αth vehicle driver feels comfortable s 0α is the gap between vehicles in complete stand-still traffic jams. v 0α is the desired speed of the αth vehicle, which is the speed it would traffic on free road; A 600-second scenario of a convoy traveling on a single lane road in different traffic conditions is simulated in MATLAB using IDM. There are five vehicles in the convoy. The leading vehicle s speed profile is assigned, and the other four vehicles follow one after another, following behavior modeled by IDM. The model parameter 50

65 sets change during the process to represent the influence of different traffic conditions on the drivers behaviors. The simulated convoy travels through the traffic conditions of normal, wet, normal, and then congested conditions. In wet conditions, the desired time headway T and gap distance s 0 are larger, while the comfortable maximum acceleration rate a and comfortable maximum deceleration rate b are smaller. In congested conditions, the desired time headway T and gap distance s 0 are smaller, while the comfortable maximum acceleration rate a and comfortable maximum deceleration rate b are larger. The parameter set of each traffic condition is listed in table 3.1. The vehicles within the convoy are of same parameter sets. Table 3.1: Driver Parameters in Different Traffic Conditions Normal Wet Normal Congested s 0 (m) T (s) a (m/s 2 ) b (m/s 2 ) v 0 (m/s) Time 0-210s s s s Fig.3.3 shows the simulated vehicles speed profiles using IDM. The time when traffic condition transition happens is marked with the red line. 3.3 Preceding vehicle speed prediction General Description of the Algorithm The general idea of the speed prediction algorithm is to estimate the characteristics of speed perturbations propagating from the leading vehicle to the preceding vehicle. 51

66 st Vehicle 2nd Vehicle 3rd Vehicle 4th Vehicle 5th Vehicle 18 Speed(m/s) Time(s) Figure 3.3: Speed Profiles of the Simulated Vehicle in the Convoy Then, the influence of a given speed perturbation on the preceding vehicle can be calculated immediately as it acts on the leading vehicle. The transmission/propagation of speed perturbations requires a convoy as the medium, of which the parameters should be estimated. The convoy can be viewed as a black box. The inputs of the black box, which are the leading vehicle s speed and position, are known to the ego vehicle. And the outputs of the black box, which are the preceding vehicle s speed and position, are also measurable to the ego vehicle. The structure inside the black box can be modeled with high flexibility, as long as the modeled black box gives accurate output with respect to the same input. Therefore, some imaginary vehicles can be inserted between the leading vehicle and the preceding vehicle to form an imaginary convoy. The imaginary convoy only serves to transmit the speed disturbance to the ego vehicle. The dynamics of the vehicles inside the convoy is of no concern. The number of imaginary vehicles in the 52

67 convoy can be any reasonable value, because the parameters of each imaginary vehicles are flexible and can adapt to the difference. For example, fig.3.4 shows the case when the number of vehicles between the leading vehicle and the ego vehicle is guessed at five, while there are actually four vehicles in between as in fig.3.1. Those imaginary vehicles are designed identical for estimation simplicity. The 5th imaginary vehicle s behavior should be the same as the preceding vehicle, to serve as the prediction of the preceding vehicle. Figure 3.4: The Created Imaginary Convoy Car-following Parameter Estimation Initialization The imaginary convoy is modeled using IDM, and the parameters of each imaginary vehicle of the convoy are estimated in realtime. The recent speed and position profiles of the leading vehicle and the preceding vehicle are used as training samples. Within each estimation cycle, the imaginary vehicles velocities and positions are initialized with evenly distributed speed differences and gaps. For the i th imaginary vehicle: P i0 = P Lead0 i PLead0 P P receding0 n v i0 = v Lead0 i vlead0 v P receding0 n (3.3) (3.4) 53