Look-Ahead Platooning through Guided Dynamic Programming

Transcription

1 Look-Ahead Platooning through Guided Dynamic Programming GEORGE JITHIN BABU Master s Degree Project Stockholm, Sweden September 2012 XR-EE-RT 2012:035

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3 Abstract It has already been shown that look-ahead control can result in fuel savings up to 3 % and platooning can save around 7 %. The aim of this thesis is to combine these two strategies. Various strategies that can be used for this purpose are analysed. Fuel and time optimal state utilises the momentum or the stored energy of the truck and at the same time keeps the distance between the trucks as small as possible. This thesis presents a cooperative form of decentralised predictive controller to achieve this. Dynamic programming strategy, which is used to solve it, has a very high computational complexity due to the number of states and its possible values. To avoid this, an iterative dynamic programming algorithm which operates only on a small region of search space was developed. The search space is made flexible, which improves the runtime and performance of the algorithm. This resulted in a controller that decided whether platooning or look-ahead control or partly both should take the leading role in controlling the trucks for the best performance. The fuel and time optimal velocity profiles thus obtained could save 3.9% of fuel for a two platoon system on a real road simulation.

4 Acknowledgements The master s thesis on look-ahead platooning was done at the Pre-development of Intelligent Transportation Systems department (REPI), at Scania CV AB in Södertälje, Sweden, between March 2012 and September The thesis was supervised by the Automatic control department, at Royal Institute of Technology(KTH), in Stockholm, Sweden. I would like to use this opportunity to thank my supervisors Assad Alam and Kuo-Yun Liang for giving me the opportunity to do this project. Their support and guidance during the whole project was a great help, especially to identify various practical aspects of the problem. I would like to acknowledge Henrik Pettersson at Scania for his outstanding help, support, and knowledge. I also want to express my gratitude towards my examiner Karl Henrik Johansson at KTH. Along with thanks to everyone at the Pre-development department, at Scania for welcoming me and making every day enjoyable.

5 Contents Contents i Introduction 1 1 Introduction Background Problem Formulation Related Work Thesis Outline Truck Model Power Train Model Engine Mass Flow Model Aerodynamic Drag Simulation Model Control Strategy Analysis Velocity and Road Profiles Simple Look-Ahead Platooning Strategies Strategy Strategy Strategy Discussion Selection of Controller Look-Ahead Control Selection of Optimiser Dynamic Programming Constraints Criterion Search Space Algorithm Architecture Dynamic Programming 1 (DP1) i

6 ii CONTENTS Dynamic Programming 2 (DP2) Coordinator Extension Simulation Results Uphill Case 1: Heavier Follower Vehicle Case 2: Heavier Lead Vehicle Downhill Case 1: Heavier Follower Vehicle Case 2: Heavier Lead Vehicle Combination of Uphill and Downhill Case 1: Heavier Follower Vehicle Case 2: Heavier Lead Vehicle Extended Platoon Real Road Summary Conclusion 63 Bibliography 65

7 Chapter 1 Introduction The first truck rolled out in 1896, which had a 4 hp engine with two forward and reverse gears. Since then a succession of technological revolutions combined to create the modern trucks as we see today. Now trucks have become a necessity in the logistic chain and all products we use are transported by trucks at some point of time. Based on a study done by the Swedish Association of Road Haulage Companies in 2009, each year 406 million tonnes of goods are transported using trucks in Sweden, which accounts for 81 % of the total freight transport [1]. They have a unique role to play in today s transportation system which enable proper functioning of the economy at the same time assuring the quality of life we enjoy, which makes it essential for the growth of society. Today the fuel is widely available everywhere but the prices have become increasingly taxing on vehicle owners. The increase in the fuel price affects the heavy fleet providers with increased operational costs, which eventually affects everyone as transportation costs carry over to the goods and services. Based on the prediction of European Commission-Directorate General for Mobility and Transport, there will be a 20 % increase in the fuel price and 43 % increase in freight transport using trucks by 2030 [18]. This clearly shows an increase in the cost and number of trucks on the roads. An increase in the traffic naturally increases the emissions of the green house gases, which can have devastating effects in the long run. To ensure the quality of the living environment various governmental agencies around the world have set limits for carbon emissions. Directorate-General for Energy (European Commission) expects to reduce the green house gas emissions by 20 % and also save 20 % of energy[2]. The numbers point to the fact that energy efficient technology is a necessity for the survival in the future market scenario and to ensure a better future. The technological advancement both in hardware and software had been enormous in the past decade, which enables us to create a solution for fuel efficiency in trucks without driver intervention. 1

8 2 CHAPTER 1. INTRODUCTION Figure 1.1: Representations and notations of a three truck platoon. V 1, V 2 and V 3 are the velocities, d 12 and d 23 are the relative distances between the trucks. 1.1 Background Scania CV AB has estimated that the contribution of fuel consumption is 30 % of the operational costs for a heavy duty vehicle [3]. Apart from the cost increase, fuel consumption also results in increased emission. Since fuel cost is a big part of the total cost even a fuel saving as small as 1 % can result in significant savings, which in turn will increase the profit margins of the transport companies. Driving at a lower speed reduce the fuel consumption, which will increase the travel time. Thus the aim of reducing the fuel consumption, while maintaining the travel time, is challenging. One of the automation and energy efficient services that is under study now is platooning as illustrated in Figure 1.1. Platooning allows several trucks to travel at a small intermediate distance. The main advantage of this is to reduce the fuel consumption due to aerodynamic drag reduction caused by the truck ahead. According to the results published by Alam [3], the fuel consumption can be reduced by up to 7 % by maintaining a distance equivalent to 1 sec at 70 km/h between two identical trucks. A similar effect can be observed in bicycle racing or skating, where the contestants move in formation to use the reduction in air drag and thereby reducing the effort. The power to mass ratio of heavy trucks are often such that even a moderate slope can have a significant effect on the fuel consumption, if the speed is to be maintained. When driving in uphill and downhill, gravitational force is the major force when compared to frictional force and aerodynamic force. Depending on the mass and slope, significantly higher power is required to maintain a constant velocity or acceleration during uphill compared to a flat road. The trucks will accelerate during downhill without any fuel injection and will decelerate during steep uphills. The weight of the truck can be such that it cannot maintain its velocity during the uphill even with maximum power. Thus just by utilising the momentum or the stored energy in the truck there is a potential fuel saving with the knowledge of the terrain ahead. The control technique which uses the above concept is look-ahead control. The results published by Hellström and Ivarsson [10] and [11], showed a fuel saving of 3 % on a 120 km stretch without affecting the trip time much by using look-ahead control. The studies done by Hellström, Ivarsson and Alam [10], [11] and [3] had already shown that platooning and look-ahead control are capable of providing fuel efficiency. In platooning, the trucks in a platoon act as a single cooperative system

9 1.2. PROBLEM FORMULATION 3 Figure 1.2: Trucks moving in a platoon over an uphill and downhill. Truck 4 and 5 should maintain minimum distance between the heavy duty vehicles for maximum fuel efficiency. Truck 3 should let the gravity reduce the speed to a minimum acceptable level and truck 1 and 2 accelerate to the maximum speed using gravity. In order to reduce the fuel consumption, the distance between the trucks should be kept as small as possible. and all the trucks follows the lead or the reference truck, at the same time maintaining a safe distance between them. In general all the trucks in a platoon will have the same cruise control speed. On the other hand, in look-ahead control the speed of the truck varies depending on the terrain ahead but tries to maintain a speed close to cruise control speed. The aim of this thesis is study the combination of platooning and look-ahead control, and to formulate a control algorithm which can do it efficiently. A simple scenario for the look-ahead platooning is illustrated in Figure 1.2. The reduction in air drag coefficient with respect to the distance between the trucks is illustrated in Figure 1.3. It is necessary to design a strategy, which can use the gravitational force as much as possible at the same time reducing the distance between the trucks. The aim of this thesis is to formulate a look-ahead platooning strategy and evaluate its performance. There are several challenges that are involved in this problem, the most important one is the non-linearity of the system. A hybrid optimization has to be used to find the optimal strategy as gear can be changed only in discrete steps, i.e. optimizing a combination of discrete and continuous system. 1.2 Problem Formulation The aim of this thesis is to derive a time and fuel efficient strategy for a platoon with the knowledge of the terrain ahead. The strategy should also make sure that the trucks always maintains a safe distance between each other. The initial phase is to develop a strategy for a two vehicle platoon and is extended further to 3 vehicles. The vehicle platoon is described as shown in Figure 1.1 The following assumptions are made to simplify the algorithm 1. Speed control of the trucks are capable of maintaining a speed very close to the reference speed, difference is as low as 0.2 km/h.

10 4 CHAPTER 1. INTRODUCTION Figure 1.3: Reduction in drag coefficient Cd depending on the distance. [3] 2. Gear changes are made efficiently by the adaptive cruise control. 3. Linear engine fuel mapping near the operating region around 80 km/h. 4. The adaptive cruise controller is designed only to maintain a set speed and the set distance between the trucks. 5. A relative distance of 5 m between the trucks is considered a safe distance. To determine the optimal velocity for the whole platoon, the problem can be expressed as an optimal control problem and then the solution can be found. It is in essence an extension of the look-ahead control for single vehicle to multiple vehicles. The resulting algorithm should generate the optimal control law for the velocity of the truck with respect to both fuel and time. The aim of the optimisation is to reduce the total fuel consumption of the platoon for the whole trip. At the same time, the total trip time should not be more than the one which travels at the cruise control set speed. In addition to the time and fuel criterion, the platoon should maintain a safe distance between the two trucks. On top of this, an effort had to be made to make the algorithm as close as possible to real time in the development PC. Inputs to the look-ahead platooning system are cruising speed, mass, gear, position of the truck and relative distance between the trucks, which are obtained from sensors and the engine management unit. Apart from these, slope of the road ahead is also an input to the algorithm. Based on the position evaluated by the truck, the

11 1.3. RELATED WORK 5 slope is collected from a database. Each vehicle in a platoon communicates with other vehicles and exchanges information regarding their velocities, positions and other relevant information. 1.3 Related Work One of the earlier works on look-ahead control was done by Schwarzkopf and Leipnik [19]. Several studies were done on optimal fuel strategies using the information of the road ahead e.g. Fröberg [8], Hellström[11], Chang and Morlok [7]; based on an affine relation between the fuelling and engine torque. All the studies points to the fact that, fuel savings can be achieved by controlling the vehicles depending on the terrain ahead. These studies suggest various strategies like maintaining the speed if the slope is very small, enter the downhill with the minimum speed etc. The studies done by Huang and Bevly [13] proposed a solution to the look-ahead controller using a non-linear optimiser. A detailed study of look-ahead controller was done by Hellström and Ivarsson in [10] and [11]. A controller design using dynamic programming was proposed by them. The implementation of the look ahead controller in this thesis is based on [10] and [11]. Control of vehicles moving as a platoon was earlier studied by Levine and Athans [15]. Their studies concentrated on a centralised controller design. There were various studies later for vehicle platooning, focussing on different aspect of it e.g. improvement in traffic flow due to platooning by Varaiya [20]. Several strategies like centralised controller by Liang [16], decentralised and game theoretical approach by Alam [4] and [5], were proposed for maintaining a safe platoon. A distributed model predictive control was proposed by Kuwata and How in [14], which concentrates on fleet level objectives. This proposed a different way of representing the objectives of the trucks in the platoon. This thesis required the contributions from both the look-ahead and platooning projects. 1.4 Thesis Outline The model that is used for the design of the look-ahead platooning strategy is formulated in Chapter 2. Along with the formulation, the adaptations that were required for the algorithms are also explained in detail. Based on the model a prestudy is done to access the scope of the problem and is explained in Chapter 3. In this chapter various strategies are discussed and a requirement for the algorithm is formulated, which will act as guidelines for further development. A dynamic programming algorithm and a control strategy is developed in Chapter 4. This chapter also explains about various strategies that are implemented to make the algorithm faster and effective. The simulation and comparison of all the strategies are done in Chapter 5, and Chapter 6 concludes this thesis.

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13 Chapter 2 Truck Model To predict the behaviour of the truck it is required to have a model of the truck. This chapter explains the system models that are used for the development of the control algorithm. 2.1 Power Train Model The basic power train model is explained in detailed in [3], and is briefly explained below Engine A diesel engine is modelled in such a way to represent the look-ahead problem. Since the influence of combustion process and engine dynamics is less when compared to the vehicle dynamics, the simplified engine model is formulated as shown in (2.1), T e (ω e, u f ) T c = J e ω e (2.1) where T e is the torque generated due to the combustion after considering the engine friction, T c is the torque available at the clutch, J e is the moment of inertia of the engine,u f is the engine fuelling and ω e is the angular momentum at the clutch plate. For vehicle speeds between 75 km/h and 85 km/h the engine torque T e is modelled as shown in (2.2). The parameters in (2.2) was derived in [12] and these values were used directly in the algorithm. where u f is the engine fuelling which is bounded by where the upper limit u max is modelled by T e (ω e, u f ) = a 1 ω e + b 1 u f + c 1 (2.2) 0 < u f < u max (2.3) 7

14 8 CHAPTER 2. TRUCK MODEL u max = a 2 ω 2 e + b 2 ω e + c 2 (2.4) The torque generated by the engine is then transmitted through the clutch, transmission, drive shaft, differential and finally to the wheels. The gear shifts and the corresponding changes in torque and speed are assumed to be instantaneous. All the losses in the drive train are lumped in to the efficiency of transmission and final drive η t and η f respectively. similarly the moment of inertia of the drive train in lumped in to moment of inertia of the wheel J w. The torque transmitted to the wheel T w and the torque balance is given by T w = T c i t i f η t η f = J w ω w + rf b + T f (2.5) ω e = i t i f ω w (2.6) where i t and i f are the gear ratios of the transmission and the final drive, ω w is the angular velocity of the wheels, ω e is the angular velocity of the crank shaft, F b is the braking force, T f is the net tractive torque on the truck and r is the radius of the wheel. Longitudinal Forces on the Truck The free body diagram of a moving truck is shown in Figure 2.1. Based on this, using force balance motion model of the truck can be derived as m v = F f F r (α) F g (α) F d (v, d) (2.7) where m is the mass of the truck, α is the road angle, v is the velocity and d is the relative distance between the trucks. The models of the longitudinal forces F g, F f, F r and F d are explained in Table 2.1. The vehicle velocity is v = rω w (2.8) Table 2.1: Longitudinal Forces. Force Model F g (α),force due to gravity mg sin α F r (α), Rolling friction mgc r cos α F d (v, d), Aerodynamic drag 1 2 c d(d)a a ρ a v 2 F f, Tractive force T f r

15 2.1. POWER TRAIN MODEL 9 Figure 2.1: Longitudinal forces acting on a moving truck [3]. From (2.1), (2.6) and (2.7) m v = F f mgc r cos α mg sin α 1 2 c d(d)a a ρ a v 2 (2.9) (m + i2 t i 2 f η tη f J w r 2 + J e r 2 ) v = i T e (ω e, u f ) ti f η t η f F b (2.10) r (mgc r cos α + mg sin α c d(d)a a ρ a v 2 ) m t v = K e T e (ω e, u f ) F b + K r cos α + K g sin α + K d c d (d)v 2 (2.11) where m t = m + i2 t i 2 f η tη f J w r 2 + J e r 2, K e = i ti f η t η f r (2.12) K r = mgc r, K g = mg, K d = 1 2 A aρ a m t is called the total inertial mass The look-ahead horizon is defined in terms of distance S ahead of the truck. The models that were formulated above were with respect to time. In order to have a simpler algorithm it is required to convert these equations with respect to distance. We have that for any function f(x), the following holds df dt = df ds ds dt = df ds v (2.13) Fuel mass (2.14) and the truck model (2.11) can be modified accordingly. For solving the differential equations Euler s backward method was used.

16 10 CHAPTER 2. TRUCK MODEL Figure 2.2: Reduction in aerodynamic drag coefficient(c d ) depending on the relative distance [3] Mass Flow Model The mass flow is determined by the fuelling u f and the engine speed ω e. The fuel flow in g/s is m f = n cyl ω e u f (2.14) 2πn r where n cyl is number of cylinders and n r is number of crankshaft revolutions per cycle. When the truck is in neutral gear the fuel flow is equal to idle fuel flow, which is a constant. 2.2 Aerodynamic Drag Any moving vehicle experiences an aerodynamic drag. The main aim of the platooning is to achieve fuel savings due to the reduction in aerodynamic drag. This makes it necessary to model its reduction as a function of the distance between the trucks. From Table 2.1 the aerodynamic force is given as The drag coefficient c d (d) given by F d = 1 2 c d(d)a a ρ a v 2 (2.15) c d (d) = C d (1 fi(d) 100 ) (2.16)

17 2.3. SIMULATION MODEL 11 Figure 2.3: Simplified representation of the model that was used for simulation. Where C d is the drag coefficient for a truck without platooning. The change in c d (d) with respect to the spacing between the trucks is as shown in Figure 2.2. The function fi(d) is an approximation of the curves in Figure 2.2, where i represents the position of the truck in the platoon. f1(d) corresponds to the truck in front, f2(d) is the second truck and f3(d) is the last truck. The same linear function used in [16] is used here and is as shown in (2.17). f1(d) = d < d < 15 f2(d) = d < d < 80 (2.17) f3(d) = d < d < 80 otherwisef i(d) = Simulation Model The model that was derived above was used for developing the control strategy. For simulation a more detailed model was used. It was provided by Scania CV AB. A simplified block diagram of this model is shown in Figure 2.3. The model has a cruise controller that is used to maintain the speed of the truck at the set speed. The torque controller determines the amount of torque that is required for the speed determined by the cruise controller. The torque is then limited and filtered depending on the engine characteristics and limitations. The torque controller also determines the amount of braking torque required to maintain the speed. The combustion process is modelled by diesel engine model, which calculates the actual

18 12 CHAPTER 2. TRUCK MODEL torque generated and the corresponding engine speed. These calculations are based on the actual engine data. The torque thus generated is transmitted to the wheels through a series of gears, which is modelled using transmission model. The truck model simulates the behaviour of the truck. The input of the truck model is the sum of brake torque, transmitted engine torque and torque due to losses. The vehicle velocity is the output of the truck motion model and is the feedback signal to the cruise controller.

19 Chapter 3 Control Strategy Analysis The fuel and time optimal profile for a single vehicle based on the road slope is already presented in the work of Hellström and Ivarsson in [10] and [11]. The studies done by Alam and Liang in [4] and [16] had shown that there is a potential fuel saving and a controller was designed for a platoon. The purpose of this chapter is to analyse the effects of using the optimal fuel and time velocity profiles for each vehicle in a platoon. Apart from this, an analysis of various strategies that can be used is done here. The system under study here is a two truck platoon under suitable road conditions. The parameters of the trucks and the road profiles are selected such that there is a considerable difference between the fuel and time profiles of the trucks, making platooning almost impossible or less effective for the chosen vehicle configurations and road conditions. To simplify the analysis, an initial study was performed using a single point mass representation for the trucks. Such an approximation does not model the actual dynamics of the system, but it gives a fairly good idea about acceleration, deceleration and energy consumption. Apart from this, the losses like friction, aerodynamic drag and gravity are dependent on the mass, which make this approximation good enough. Once the initial analysis was done, the profiles were tested on a simplified truck model. Further development was based on these results. In addition to the simplified truck model, the trucks are considered as point objects, which are separated by relative distance only. Thus the look-ahead horizon of both trucks will be approximately same, provided the relative distance is small. 3.1 Velocity and Road Profiles Selection of suitable road profiles and the corresponding velocity profiles is crucial for a good analysis. In the studies done by Hellström, Ivarsson and Holma [10], [11] and [12], the fuel and time optimal velocities profiles for different road profiles were studied in detail. The speed profiles thus determined not only depends on the slope but also on the vehicle masses, losses and other truck parameters. Hence the optimum velocity profile will be different for different vehicles unless they are 13

20 14 CHAPTER 3. CONTROL STRATEGY ANALYSIS identical with same losses. When the trucks are moving as a platoon even identical trucks will have different aerodynamic losses, which results in slightly different optimal velocity profiles. The steady state values (speed on a level road) of these profiles are determined by the truck limitation, legislation or the adaptive cruise control. This makes the fuel and time optimal steady state speed for each truck in a platoon dependent only on the engine limitation, if the cruise control set speed is same for all. Thus for a platoon of trucks which differs only in their masses, each truck will have the same steady value for velocity profile. For example, if the cruise control set speed is 80 km/h, then the optimal velocity for the trucks on a flat road is 80 km/h. The difference in the profiles will appear only when there is a transition between different slopes or optimum steady velocities, as different trucks will have different optimum transition between the velocities. In Figure 3.1, each truck follows a different velocity profile due to the difference in their masses, until it reaches a steady state. Assume that the maximum allowable velocity change is from 75 km/h to 85 km/h. Then, for trucks with masses 35, 30 and 25 tonnes the final steady velocities are the same. It can be observed that in case of a 20 tonnes truck the final velocity is 83 km/h because, the weight of the truck is not enough to accelerate more (minimum allowable acceleration is.06 m/s 2 due to dynamic programming limitation). Assume 35 tonnes, 30 tonnes, 25 tonnes and 20 tonnes trucks are moving in a platoon, where each truck tries to maintain its optimum profile as shown in Figure 3.1. Then it can be observed that, during downhill a 35 tonnes truck always had a higher velocity when compared to the other trucks. This behaviour can either increase the relative distance, or the trucks become very close to each other and collide towards the end of the slope, depending on the position of the heavier truck. This makes it necessary to find another velocity profile, which is fuel and time optimal for the whole platoon. A combination of 30 or 35 tonnes and 20 tonnes truck has the considerable difference in the velocity profiles in the above range of masses. Figure 3.2 shows the relative distance between the trucks when a 30 tonnes truck is the lead vehicle and 20 tonnes is the follower vehicle. It can be seen that the distance between them is increased from 5 m to 25 m, which in turn increases the aerodynamic losses drastically. In the other case where 20 tonnes is in front of a 30 tonnes truck there will be a collision. Due to this difference in the individual velocity profiles 30 tonnes truck and 20 tonnes truck is selected for further study in this thesis. The same results can be shown for uphill also. Simple road profiles like uphill, downhill, uphill followed by downhill and downhill followed by uphill are used. To have reasonable impact on both 30 tonnes and 20 tonnes trucks without creating much system limitation, a slope of ±3% is used for the study. Aerodynamic losses were also considered while determining a suitable slope, since it should not lose its significance in the uphill or downhill when compared to gravitational force.

21 3.1. VELOCITY AND ROAD PROFILES 15 Figure 3.1: Fuel and time optimal velocities for different masses of trucks with the same slope ignoring the position and air drag reduction in a platoon. Figure 3.2: Relative distance between 20 tonnes and 30 tonnes truck with the altitude as shown in Figure 3.1.

22 16 CHAPTER 3. CONTROL STRATEGY ANALYSIS 3.2 Simple Look-Ahead Platooning Strategies Three strategies are analysed in this section. Based on the results from this analysis a suitable strategy for control was decided. One is that both the trucks follow the same velocity profile, in most cases the velocity profile of the heaviest vehicle or the vehicle with least powerful engine. The second strategy is to allow both the trucks to continue with their individual optimal velocities. Another strategy was to completely ignore the optimal velocity profiles and create a profile using a PI controller. All the cases are tested with the approximated vehicle model as described above and then to confirm the findings they were verified in a more accurate truck model. The actual truck behaviour was much different from these two models but it gave a fairly good idea which was enough during this stage Strategy 1 Consider a two truck scenario. One of the trucks in the platoon is selected as the reference truck. Both trucks in the platoon will move with the fuel and time optimal velocity of the reference truck. Thus the trucks are driven with a common velocity, which is considered as the platoon velocity. The reference truck is the one with lowest speed limitation. The speed limitation occur due to the weight or due to a relatively less powerful engine. In any case the weakest truck will be considered as the reference truck. For the study, same engine was chosen for both the trucks, hence the weakest truck is the heaviest one. In this strategy the velocity profile will be the optimal for the heaviest or the reference truck and the lighter one will follow, which assures that the platooning is effective and safe as the distance between them is maintained. The optimal velocity for a platoon of 20 and 30 tonnes truck in an uphill and downhill are shown in Figure 3.3. A small kink that can be observed in the optimal velocity of a 30 tonnes truck along a downhill is due to the discretisation of the model. The fuel and time optimal velocity profile is the one which requires the least amount of braking and fuel. In case of downhill, the common velocity is higher than the optimal velocity for the lighter truck along the downhill. This means that lighter vehicle must spend a little amount of fuel to attain this acceleration. Along the downhill it is optimal to use the gravitational force to accelerate and maintain the speed of the truck. In this situation fuel saving due to platooning is not significant. The advantage of platooning during a downhill is that gravitational force can result in slightly higher velocity and acceleration as the losses are reduced. In other words the platoon velocity profile is optimal only for the heavy vehicle, i.e.optimal fuel, time and brake. In case of uphill, even though the velocities are different, the platoon velocity profile for the lighter will result only in slightly higher fuel consumption when compared to the optimal case. There is some saving as the distance between the them

23 3.2. SIMPLE LOOK-AHEAD PLATOONING STRATEGIES 17 Figure 3.3: Common velocity profile for a platoon of 20 tonnes and 30 tonnes truck, compared with the fuel and time optimal velocity of the 20 tonnes trucks in an uphill and downhill. is kept small. This points to the fact that there is a possibility of another velocity profile which is time and fuel optimal for the whole platoon. This strategy performed well and could result in fuel and time savings when compared to adaptive cruise control strategy. The main disadvantage of this method is that it does not consider the second vehicle, the optimal profile is generated based on the weakest truck. The velocity profile will be the optimal only for one truck and might not be for the others. Apart from this, the second truck might not be able to follow the velocity profile accurately due to its engine limitation Strategy 2 In this strategy, each vehicle in the platoon moves with its fuel and time optimal velocity. In order to make sure that the trucks do not collide and distance between them is small on a level road, an attempt to design a controller for relative distance was done. Unlike the previous strategy it requires different profiles depending on the position of the vehicle in the platoon. The implementation of this strategy using a proportional integral(pi) controller was not very effective. The controller had an oscillatory behaviour. The calibration of the PI parameters was difficult as the relative distance changed very fast. This main disadvantage of this controller was that there was no information for the controller regarding the preferred variation of the relative distance. Without this information, the PI controller had the same controlling effect depending on the calibration, irrespective of the slope. The strategy works in case of similar trucks in which case the velocity profiles are similar. A similar strategy is used in the next chapter, where the problem is expressed as an

24 18 CHAPTER 3. CONTROL STRATEGY ANALYSIS optimal control problem and solution is found, hence this analysis was not done in depth Strategy 3 In this strategy, a profile was generated such that, the trucks were decelerated using gravitational force during uphill till it reached the minimum velocity. Similarly, the trucks were accelerated during downhill without using any fuel till it reached the maximum velocity. These behaviour has the least fuel consumption. Also in this strategy, the previously determined optimal velocity profiles were not used. A proportional integrating(pi) controller was used to control the speed of the vehicle. Along with this velocity controller, another proportional integral derivative(pid) controller was added to control the relative distance. The relative distance controller was necessary to make sure that the trucks always keeps a safe distance between them. In short the controller can be described as When slope is zero the cruise set speed and the relative distance were maintained at the set values When the truck enters a downhill and before the exit of an uphill, the fuel was cut-off, so that vehicle moves under gravitational force (either accelerate or decelerate) until it reaches the maximum or minimum velocity. Whenever the relative distance crosses acceptable limits, a correction from the PID controller was added to the velocity controller and the distance was maintained. To make sure that the travelling time was least affected, the velocity was maintained at the set value for the first half of the slope during an uphill. The truck decelerated only towards the end of the slope. Controller was not intelligent, it operated based on the value of current velocities, slope and relative distance. The velocity for such a controller is shown in Figure 3.4. It can be observed that there is an intermittent braking and acceleration. 3.3 Discussion All the above analysis were done on the assumption that the trucks were point objects, i.e. the look-ahead horizon is the same for both vehicles. In practice the trucks are separated by a distance, truck length plus relative distance. Hence both the trucks have slightly different slopes, as the slope changes are gradual this approximation is good if the relative distance is small. For the analysis, the controllers generated the truck velocity for each time step and outputs were represented with

25 3.4. SELECTION OF CONTROLLER 19 Figure 3.4: Velocities and the corresponding relative distance of a 30 tonnes truck following a 20 tonnes truck in a 2% uphill followed by a 2% downhill. respect to time. In other words, time was chosen as the reference. This controller is good only if both the trucks are moving with the same velocity. If the trucks moves with a different velocity, the time taken by the trucks to travel the same distance will be different. This can influence the performance of the controller. One observation from the calibration was that relative distance responded faster compared to the velocity of the trucks. Thus there was a fast and a slow controller controlling the same system. This resulted in oscillations, when the same PID parameters were used for different cases. The performance was better when the velocity profiles were similar. The analysis points to the fact that there was energy savings when the relative distance was kept small. This suggest that an intelligent controller with preview information should be more fuel efficient than a simple PID controller. 3.4 Selection of Controller As an optimal velocity has to be determined based on the slope of the road ahead, a predictive control strategy is required. The controller can either be a centralised controller or a decentralised one. A centralised controller has all the information regarding the trucks in the platoon. The space-time complexity of such a controller increases exponentially as the number of trucks in a platoon increases. The advantage of this method is that, as it has all the state information of the trucks, such a controller can provide an optimal solution, which has the best performance the platoon can achieve. However, the increase in the space and time complexity makes it impossible to use this in case of a larger platoon. Another disadvantage is that

26 20 CHAPTER 3. CONTROL STRATEGY ANALYSIS if a centralised controller is designed for a two truck platoon, adding a third truck will require a recomputation of the controllers. Decentralised controller design on the other hand, has one controller for each truck. Each controller will have information regarding the state of the corresponding truck, and has only limited knowledge about the other trucks. In short, a controller that is unaware of the limitations, capabilities and dynamics of the other trucks. Hence, in case of a decentralised strategy, each controller uses this limited knowledge of the other trucks to generate its optimal strategy. This is a major drawback for this strategy. The advantage is that, it is easy to add additional trucks to a platoon, and can be applied to a larger platoon also. In case of a platoon it is more meaningful to design a decentralised controller, which enables addition or removal of trucks from the platoon. The controllers communicate with each other regarding their relative distance and velocity. The velocity profile determined for a truck gives a good indication about its capabilities, limitations and dynamics, which are significant for the given condition. Hence, a decentralised controller will be designed, which cooperates with the other controllers to design an optimal strategy.

27 Chapter 4 Look-Ahead Control In this chapter a predictive control strategy resulting in fuel and time optimal velocity profile is developed. The idea behind the controller is to use the information that is available about the truck and the terrain to predict the behaviour of the system for a specific distance ahead. The concept is shown in Figure 4.1. The prediction can result in many behaviours with varying performance like high fuel saving and decreased trip time, low fuel savings and decreased trip time, no savings in trip time or fuel etc. The predictive controller that has to be derived will select one behaviour which maximises the fuel saving and minimises the trip time. In simple terms, predictive controller searches all possible combinations of inputs and selects the one which maximises the profit, which also implies that with the size and dimensions of the search space the time and code complexity will also increases drastically. The look-ahead platooning problem is represented as an optimal control problem and is solved using standard methods. This chapter explains the formulation of an optimisation algorithm from the basic concepts. Figure 4.1: An overview of look-ahead platooning. The controller will decide the velocity and gear at B, which can have the minimum fuel while maintaining the travelling time for the look-ahead horizon. 21

28 22 CHAPTER 4. LOOK-AHEAD CONTROL 4.1 Selection of Optimiser There are several methods in which the optimal control problem can be solved. It is very important to select the most suitable one as each method has it own advantage and disadvantage. One method is to linearise the system model and the performance measures are incorporated in a quadratic cost function. For this problem an analytical solution can be found using Pontryagin s principle. This method offers a fast way of computing the optimal control solution. In case of constrained input and states the solution has to be determined numerically. The computational complexity increases linearly with the dimension of the states, which is an advantage of this method. The disadvantage is that it can lead to multiple solutions in some cases, one among the solutions is optimal. This method can only be applied to deterministic systems. Another method is to use dynamic programming introduced by Bellman, which is basically searching the entire search space for an optimal solution. It also offers much more flexibility when compared to the previous method and can be used to optimise without linearising the system. This method always finds the optimal solution if it exists, but the major drawback is that the computational complexity increases exponentially with the states. This makes it impractical for higher dimensional problem. The system equation of the look-ahead platooning problem can be represented as shown below. x i+1 = f(x i, u i, slope) (4.1) where u i denote inputs and x i represents the states of the system and i represents the discreet time steps. The optimal control algorithm should find the fuel and time optimal velocity for the type road ahead such that the constraints are satisfied. The type of road is represented using the slope, which can be considered as a constrained input that can take only one particular value for a segment of road. As the values can be obtained in real time through GPS, modelling the slope variation is not meaningful. One method is to consider slope as a noise in the system and determine an analytical solution. To ensure a realistic prediction, the noise in such cases are usually white making its modelling easier. But in this case the slope does not show the properties of a noise. The behaviour of the slope clearly denotes that a white noise approximation is not reasonable. This means the system is no longer deterministic without slope as the input and an analytical solution cannot be determined unless a way is found to incorporate slope also in to the system. When dynamic programming is used to solve the optimal control problem such constraints can be avoided. Thus dynamic programming was chosen instead of Pontryagin s maximum principle.

29 4.2. DYNAMIC PROGRAMMING 23 Figure 4.2: Dynamic Programming. 4.2 Dynamic Programming Dynamic programming is an approach of solving complex problems by splitting it in to sub-problems, which are simpler and is explained in the work of Bellman [6]. A simple representation of dynamic programming is as shown in Figure 4.2. Here, the whole problem is broken down into a sequence of decision steps over time. Each block in Figure 4.2 represents a state and the arrows represent the transition of the states along various stages or steps over time. Every such transition is associated with a cost and the decision is taken based on this cost till it reaches the final stage. Dynamic programming is a structured way to find a sequence of states to reach the final state with minimum cost. The cost function is used to evaluate the performance of the solution and is a function of states and inputs. The cost function is shown in (4.2) where x represents the state vector, u represents the input vector and N is the number of stages. J = N 1 i=0 f(x i, u i ) (4.2) 4.3 Constraints The objective is to minimise the total fuel consumption of the platoon and the total trip time for the road profile ahead. The velocity should be within the limits shown

30 24 CHAPTER 4. LOOK-AHEAD CONTROL below v min < v < v max (4.3) where v max is set by the engine limitation or legislation and v min is set by the allowable deviation from the set speed. The brakes are assumed to be powerful enough and no limitations are considered. Apart from the speed limitation; the torque, engine speed and fuel are also limited by the maximum or minimum values specified by the engine characteristics. Possible gear shifts are also limited along with the velocity. The gear should be such that the engine speed should be within the limits set by the engine specifications. In addition to the system limitations, an additional constraint is added to the acceleration to ensure driver comfort. A deviation ± 5 km/h from the set speed is allowed, i.e. if the set speed is 80 km/h the maximum and minimum speed are respectively 85 and 75 km/h. 4.4 Criterion The main objective of the optimisation is to minimise the fuel and reduce or maintain the trip time. In addition to this, there were others objectives like driver comfort and safety. The cost function was formulated to represent these objectives as shown in (4.4), where the look-ahead horizon is divided into N steps each of length l. J = N 1 k=0 ζ in (4.4) is called the transition cost and is given by ζ(m k, t k, d k, g k, δv k ) (4.4) m k [ ] t k ζ = 1 α β γ ɛ d k g k δv k (4.5) In (4.5) α, β, γ and ɛ are constants and m k, t k, d k, g k and δv k are fuel mass, time, relative distance, gear and change in velocity respectively for k th stage. The constants represents the trade off made between the fuel savings, trip time and driver comfort for optimal behaviour. When compared to the other objectives no compromises were made for the safety and engine limitations. The fuel consumption(m k ), time(t k ) and change in velocity(δv) for a step of size l are given by (4.6) where m f is the fuel flow in g/sec and v is the velocity of the truck. m k = l 0 l 1 v m f ds, t k = 0 1 v ds δv = v k v k 1 (4.6)

31 4.5. SEARCH SPACE 25 Table 4.1: Range and resolution of the variables. Variable Quantisation level Range Number of discrete values Velocity 0.2 km/h km/h 51 Relative distance 0.4 m 3-80 m 193 Gear Search Space The search space is an imaginary space where the solution to the optimal control problem lies and whose dimensions are the states and control signals. In dynamic programming approach, the continuous state and control variables are discretised, i.e. the whole search space can be considered as a finite multidimensional grid. The granularity of the grid determines the accuracy and complexity of the algorithm. As the grid becomes finer, the complexity increases and at the same time the approximation is more closer to the actual problem. Thus a suitable grid size has to be chosen, which reduces the complexity as much as possible at the same time approximating the problem effectively. For look-ahead platooning there are 2 states and 2 control signals, velocity and relative distance are the states and gear and fuel flow are the control signals. This means the search space is a 4 dimensional grid space. Dynamic programming searches in this 4 dimensional space for a solution for the problem. In standard approach of the dynamic programming the outputs are the control signals, i.e. fuel flow and gear. Unlike the standard approach, in this case the output of the algorithm consists of a state and a control signal, which are gear and the velocity. The fuel flow, which is the other control signal can be derived from these two. The fuel flow is proportional to the engine torque and engine speed. This selection of output prevents the interaction of look-ahead controller with the controllers inside the engine management unit and adaptive cruise controller. The engine management unit and the adaptive cruise controller will have controllers, which governs the torque and fuel flow so that engine is operating efficiently and effectively. Hence it is not a good strategy to use torque or the fuel flow as the control signal. Even though the algorithm evaluated both the velocity and gear profiles, only the velocity profile is given to the adaptive cruise controller, the optimal gear is maintained automatically by the adaptive cruise control. The gear profile determined by the algorithm is used to calculate the fuel consumption, which is required for the evaluation purpose. To analyse the complexity of the program, a study was required regarding the size of the search space. Quantisation level and range of values were determined based on previous studies on look-ahead control, platooning, constraints and experiments. Fuel flow was calculated from velocity, relative distance and the gear, which will be explained later in this chapter. The number of discrete levels can be calculated from the range and quantisation levels, these are shown in Table 4.1. As the fuel can be calculated from velocity, relative distance and gear; dimension corresponding to fuel is ignored in the search space. Thus the initial search space is

32 26 CHAPTER 4. LOOK-AHEAD CONTROL Figure 4.3: Architecture of the algorithm. For simplicity the coordinator is represented as two blocks, based on its function. reduced to a 3 dimensional space. Table 4.1 shows that the size of this 3 dimensional grid space is 51 x 193 x 3. This means that the algorithm has to search combinations of states in each step. There are 100 such steps each of length 20 m for each look-ahead horizon (the look-ahead horizon is 2 km with step length of 20 m). This makes the time complexity of the algorithm very high. One method to reduce this computational complexity is to use an approximation of the cost function by using neural networks. At any stage, the cost of transition depends on the road slope at that stage. This makes it difficult to design such a network, which could perform the dynamic programming with reduced computational complexity. The resulting algorithm could still have high complexity. Hence the standard implementation of the dynamic programming was chosen. In order to have a practical solution, the standard dynamic programming algorithm had to be modified to make it faster. 4.6 Algorithm Architecture For look-ahead platooning, a look-ahead distance of 2 km is selected and each step is of length 20 m. Based on the slope of this 2 km road ahead, the algorithm calculates the fuel and time optimal velocity and the gear profile. As explained in the previous section the standard dynamic program algorithm can be very expensive when run time is considered. Hence an algorithm was developed which could perform dynamic programming much faster than the conventional one. This algorithm uses only a part of the search space to search for a solution and relies on the repeated run of the controller to converge to the optimal solution. The convex cost function makes

33 4.6. ALGORITHM ARCHITECTURE 27 sure that the algorithm converges to the optimal solution. The architecture of the algorithm is as shown in Figure 4.3. The algorithm has two different dynamic programming algorithms, DP1 and DP2, with slightly different cost functions. The first one to be executed is DP1. It generates a fuel and time optimal velocity profile without considering the other truck. In other words DP1 is look-ahead control. As the dynamic programming ignores the presence of the other truck, the velocity profile generated for each truck need not satisfy the necessary safety requirements. To incorporate safety requirements, a coordinator is added to generate feasible solutions as input to DP2. The coordinator makes sure that the trucks have a safe distance between them, when moving in a platoon, in other words feasible solution. DP2 generates the final fuel and time optimal velocity profile for the whole platoon. Depending on the variation of the output velocities, the search space is limited by the search space limiter block Dynamic Programming 1 (DP1) DP1 is a look-ahead controller, which determines the fuel and time optimal velocity profile for a single truck irrespective of the other vehicles in the platoon. The obtained velocity profile might not be even feasible when the second truck is considered. This velocity profile acts as the base on which the algorithm starts. The algorithm is similar to the work done on look-ahead control by Hellström [11] and Ivarsson [10]. DP1 has velocity as state with gear and fuel flow as control signals. The only difference in the cost function when compared to (4.4) is that it does not have the term for relative distance. The cost function used in DP1 is shown in (4.7) J = N 1 k=0 ζ in (4.7) is a function is given by (4.8) ζ(m k, t k, g k, δv k ) (4.7) m k [ ] t ζ = 1 α γ ɛ k g k δv k (4.8) Search Space Consider that the look-ahead horizon is divided in to N stages each of length l. At any point of time the status of the truck under consideration is its velocity v and gear g. The search space of any stage k comprises of all possible velocities V k and gears G k bounded by the constraints explained in the previous section. It is denoted by the set S k. S k can be represented as shown in (4.9), where ω min and ω max are the engine speed limits.

34 28 CHAPTER 4. LOOK-AHEAD CONTROL Figure 4.4: Limited velocity search space. S k = {(v, g) v V k, g G k }, where (4.9) V k = {v v min < v < v max } G k = {g ω max < ω(v, g) < ω min } The time complexity of the algorithm depends on the total number of grids in the search space. The lower the number, the faster the algorithm will be, but this affects the accuracy of the results. If the search space can be reduced without losing the optimal solution, this will make the algorithm faster and more closer to real time. An algorithm was developed for this purpose, which runs before the dynamic programming. The current velocity of the truck is used to limit the set of possible initial states. To avoid terminal effects, the final velocities are also limited. The terminal velocity is limited to 80 km/h (set speed) whenever the engine is powerful enough to maintain it. With the initial and terminal velocities limited using the model, the search space is traversed forward and backward at the same time. This removes the velocities which are not feasible, thereby downsizing the search space. In most cases where the engine is not limiting the possible velocities, the search space looks as shown in Figure 4.4. Terminal Cost A terminal cost was introduced in order to avoid terminal effects of the dynamic programming algorithm completely. This was necessary to avoid the terminal effects at the same time downsizing the search space. Terminal effect is the tendency of

35 4.6. ALGORITHM ARCHITECTURE 29 the algorithm to choose the lowest possible velocity for the last horizon, which has the least fuel consumption. The terminal cost is a reflection of the future fuel consumption of the truck. In short terminal cost represents an approximate amount of fuel that is required to accelerate the truck to the set speed from the current speed. Thus, all the velocities less than the set velocity has an additional cost. This additional cost make the velocities below set velocity less attractive for the dynamic programming, when compared to the velocities above the set velocity. Algorithm At any stage k, the fuel flow that is required for a transition from (v, g) S k to (v, g) S k+1 is determined. The fuel flow thus determined is used instead of using predefined discrete fuelling levels. The resolution of the fuelling level depends on the resolution of the velocity. The equations and more details are available in the work of Fröberg and Nielsen [9]. The advantage of this approach is that, the discretisation of the fuel flow is not required. If a feasible fuel flow cannot be determined for the transition, then the transition is either impossible or it requires a valid braking signal. In the former case, the cost is set to infinity and in the later one, fuel flow is set to zero. The benefit of this method over the standard one is that the control signal need not be discretised, hence no interpolation also. The quantisation level of the fuel flow is indirectly dependent on the granularity of the velocity search space. Let ζ (i,p) (j,q) k denotes the transition cost when the truck velocity and gear is changed from (v i, g p ) to (v j, g q ). J k (v i, g p ) is the cost function of k th stage corresponding to v i and g p. Then algorithm is as shown below. 1. The cost function of the N th stage is J N = J N, where J N is the terminal cost. 2. k = N Calculate J k (v i, g p ) = min (ζ (i,p) (j,q) (v j,g q k + J k+1 (v j, g q )), where (v i, g p ) S k ) S k+1 4. k = k-1 and continue step 3 till k = J 0 is the optimal cost. The optimal velocity profile was derived using a forward pass on the search space using the results of step 3 and 4 starting from the initial state. Calibration The output of the algorithm depends on the calibration of the parameters in the cost function. The calibration in other words is to make the necessary trade off between fuel consumption, time and driver comfort. It is not possible to have the lowest

36 30 CHAPTER 4. LOOK-AHEAD CONTROL trip time and fuel consumption at the same time, which makes it very important to have the right values for the parameters. The time parameter α has to be set such that the optimal velocity is equal to the set velocity (80 km/h) when the look-ahead horizon has a zero slope. The parameter γ is to discourage gear changes as much as possible and ɛ is to smoothen the velocity profile Dynamic Programming 2 (DP2) DP2 is responsible for look-ahead platooning. The main difference when compared to DP1 is that DP2 also considers the relative distance between the trucks. Hence it has velocity and relative distance as state and gear and fuel flow as control signals. Along with the optimal velocity and gear profile, DP2 also generates an optimal relative distance profile which is used internally. The cost function is as shown in (4.4). Search Space Consider that the look-ahead horizon is divided in to N stages each of length l. At any point of time the status of the truck under consideration is its velocity v, gear g and relative distance d. When compared to the search space of DP1 the only difference is that DP2 has an additional dimension for relative distance between the trucks. Thus the search space is a 3 dimensional space consisting of velocity, gear and relative distance. The search space can be represented as in (4.10), where d min and d max are the minimum and maximum acceptable relative distance. S k = {(v, g, d) v V k,g G k d D k } where, (4.10) V k = {v v min < v < v max }, D k = {d d min < d < d max }, and G k = {g ω max < ω(v, g) < ω min } The increase in the time complexity with the increase in dimension of the search space was drastic. This is one of the main drawbacks of dynamic programming. The same search space downsizing explained for DP1, was also used in DP2. DP2 took a long time to search the entire search space for an optimal solution. In order for the algorithm to execute in reasonable time, further downsizing of the search space had to be performed. The search space was further reduced by removing the velocities and relative distances, which are less likely to be optimal. The search space for the current step was determined based on the reference velocity and relative distance profile. The reference velocity and relative distance profiles were the optimal profiles for the previous time step, which was generated either by DP1 or DP2 determined by the coordinator. The new search space is defined as a small region above and below this reference velocity and distance. In most of the cases the fuel and time optimal profiles were within this region. In cases where the optimal profiles were not within this range, the algorithm converged to

37 4.6. ALGORITHM ARCHITECTURE 31 Figure 4.5: Downsizing of velocity search space. Figure 4.6: Downsizing of relative distance search space. the optimal solution during the subsequent runs as the cost function is convex. The concept behind this method is taken from differential dynamic programming [17]. Downsizing of the velocity and relative distance space are shown in Figure 4.5 and 4.6 respectively. Thus the algorithm had to search only a small portion of the actual search space, which drastically improves the time complexity. To make the downsizing algorithm more effective; the entire horizon was divided into 4 overlapping segments, based

38 32 CHAPTER 4. LOOK-AHEAD CONTROL Figure 4.7: Splitting and further downsizing of velocity search space. on the observations made on the variation of the profiles between subsequent runs. Depending on the variance and the maximum variation of the velocity and relative distance between subsequent runs, the width of the region also varies among the segments. This is illustrated in the Figure 4.7. Relative Distance Disretisation It was explained in the previous sections that the accuracy of the algorithm depends on the granularity of the search space, in other words the quantisation level. From Table 4.1 it can be seen that the quantisation level chosen for relative distance is 0.4 m. The quantisation level chosen should be such that the discrete model captures the actual dynamics of the system. The discrete model with grid size 0.4 m completely fails to capture slow changes in the distance, and such slow changes happens very frequently during the simulation. One solution to this problem is to reduce the grid size to model even the slow changes, but this increases the time complexity drastically. Thus another solution had to be found for improving the performance of the discrete model without reducing the quantisation levels. In the standard case, a discrete state is denoted using a single value; this single value is the discrete level corresponding to that state. The accuracy of the model was improved, without affecting the time complexity, by using a different representation of discrete states. In this case, each discrete state was represented using a pair of values. One of which is the discrete level corresponding to the state and the other one is the deviation of this discrete level from the actual state. Both the values were used for the calculation. Thus the loss in accuracy due discretisation was compensated up to an acceptable limit due to presence of the error component

39 4.6. ALGORITHM ARCHITECTURE 33 Figure 4.8: Accuracy of the relative distance model. along with the value. Even though the discrete states are stored as pair of values, for the purpose of simplicity only discrete levels are plotted. The error is used only for internal calculations. The result is shown in the Figure 4.8, it can be observed that the predicted values of the relative distance and the actual values are very close for both slow and fast changes. The quantisation level of 0.4 m for the relative distance along with the modified representation made sure that the effective quantisation level is much smaller than 0.4 m without increasing the time complexity. Algorithm The approach followed in DP1 was a backward approach where the algorithm starts from the last horizon. In general for a look-ahead horizon, the uncertainty of all the states grows towards the last horizon, i.e. there is more certainty for the values towards the first horizon. The search space has to make sure that the uncertainty is not bigger than the search space itself. In case of DP2 the uncertainty of the relative distance was higher than the modified search space itself for some stages in some cases. To deal with this uncertainty a forward approach was adopted for DP2, where the algorithm starts from the first horizon. In order to improve the performance a reference follower was also added to the cost function, which will be active only when the truck enters a level road. During the execution of DP2, it was observed that many states for relative distance in the downsized search space was impractical. To improve the performance further, the relative distance search space of each stage was limited further depending on the results of the previous stage. This resulted in considerable reduction in

40 34 CHAPTER 4. LOOK-AHEAD CONTROL the distance search space. Along with this, for the cases in which the width of the search space was found too small, additional values for the states were added. Thus making the search space flexible and improving the overall performance. At any stage k, let the status of the system be represented by x i k which comprises of velocity, gear and relative distance (v, g, d). Let ζ i j k denote the transition cost when the truck velocity, gear and relative distance is changed from (v, g, d) in stage k to (ṽ, g, d) in stage k + 1. J k (x i k ) is the cost function of kth stage corresponding to x i k. Then algorithm is as shown below. 1. The cost function of the 1 st stage is J 1 = k = 2 3. Calculate J k (x i ) = 4. k = k+1 and continue step 3 till k = N. min (ζ j i x j k + J k 1 (x j )), where x i S k S k 1 5. The cost function of the N th stage is J N + J N, where J N is the terminal cost. 6. min(j N ) is the optimal cost. The optimal velocity profile was derived using a backward pass on the search space using the results of step 3 and 4 starting from the optimal final state to initial stage. The optimal velocity profile will be determined iteratively. The velocity profile thus determined will be optimal in the reduced search space on which the dynamic programming is done. The cost function is designed to be convex with only one minima, which is at the optimal velocity. The set velocity is the optimal velocity on a level road. Once the relative distance is added to the cost function, this property changes slightly. This is because unlike the previous case, now the cost function also depends on the velocity of the other truck in a two truck platoon. The resulting cost function tries to keep the relative distance equal to 5 m, which is considered as a safe distance between trucks in a platoon. Along with the relative distance, the cost function also tries to maintain the optimal or set speed, depending on the slope. The resulting total cost function will still be convex, but the minimum cost need not be for the set or the optimal speed, it always depends on the velocity profile of the second truck. The reason is that each controller assumes that the other controller had already performed its best. Hence, it tries to match its profile with the profile of the other controller in terms of relative distance. In other words the convexity of the cost function also depends on the other truck. But new minimum will be very close to the optimal velocity. This is explained with an example for a platoon travelling over a level road.

41 4.6. ALGORITHM ARCHITECTURE 35 When the velocity profile of the second truck is the set speed, then the total function is convex and the minimum cost corresponds to the set velocity. But if the second truck have a velocity profile close to the set speed, then the cost function has the minimum cost, when the velocity of both the trucks are equal. The reason is that, if the velocity of the truck had to be changed, then there is a corresponding change in the relative distance also, which is not preferred. The reference follower tries to make the cost minimum at the set speed on a flat road, by reducing the dependency with the other truck. In case of slopes, this is handled by the coordinator. Calibration The disadvantage of determining the search space using the previous optimal profiles is that, it can lead to reduced performance, the algorithm might breakdown completely or it might have an oscillatory behaviour. In order to avoid this, the width (width is the distance between the minimum and maximum values for each step) of the search space had to be high enough. As the width increases the time complexity increases and algorithm becomes more accurate and, as the width decreases the performance and time complexity also reduces. The values of the parameters were similar to the ones used in DP1. Parameter for relative distance has to be calibrated here. Since there is a decrease in the fuel consumption, there is a tendency to reduce the relative distance to a minimum. So if the relative distance is left unattended the trucks will eventually collide, as this is the most fuel optimal strategy but unsafe. To prevent this, a reference relative distance was chosen. The weights were calibrated such that the cost function of the relative distance look like the letter V as shown in the Figure 4.9. The lower tip of the V represents the reference distance and the cost increases as the distance increases from the reference distance. In order that fuel and time have more importance the weights assigned to distance was very small in the range Even though the value of the weight is small it had significant impact. A discounted weight was used for the acceleration and the relative distance, i.e. a higher weight was used for the last horizon and the magnitude of the weight reduced towards the first horizon Coordinator Coordinator acts as a supervisor for the dynamic programming algorithms. The input to the coordinator is the output of DP1 and DP2. The coordinator at the output of DP1 checks the feasibility of the velocity profiles when applied to a platoon. The velocity profiles from DP1 were used for DP2, if the trucks could maintain a safe distance between them when moving in a platoon. If the profiles are not feasible, the coordinator will decide on a velocity profile. This profile ensures a safe distance between the trucks moving in a platoon. This is done by determining the fuel and time cost of all the combinations of DP1 profiles and then choose a pair

42 36 CHAPTER 4. LOOK-AHEAD CONTROL Figure 4.9: Cost function for relative distance. The blue dot indicates the cost when the relative distance is 4 m. with least cost. The profiles selected by the coordinator ensures a safe distance and the velocity profile could be followed by the truck. As each DP2 algorithm considers the profile of the other truck as the best profile, it completely ignores the fact that the other truck s velocity profile might not be optimal or it can change. Hence DP2 algorithm tries to match its velocity profile according to the other truck s profile. This makes the relative distance very dominant in the cost function even though its weight is very low. The main role of the coordinator is to reduce this behaviour of DP2 and make it more independent of the other truck s profile. Thus, ensuring that the profiles are not deviating from the optimal ones. It compares the fuel and time cost of the DP2 velocity profile with the output of DP1 (with all the combinations of DP1 profiles). If one combination of DP1 profile can keep a safe distance and has a minimum cost, then it is selected over the DP2 profile as the next reference profile. As the coordinator increases the overhead of the algorithm, the execution was executed once in every m. The switching logic ensures that there is no jump in the velocity profile given to the cruise controller of the truck. In short, the coordinator can be considered as an evaluating function which measures the performance of the velocity profiles, and resets DP2 if necessary. A simple block diagram is as shown in the Figure 4.10 The coordinator indirectly affects the terminal velocity profiles by making it more smooth. The evaluating function of the coordinator does not have brake as a parameter, hence the resulting velocity might result in more braking than that is expected from the dynamic programming algorithm. The coordinator can also be considered as a logic that mixes the output of DP1 and DP2 to obtain good

43 4.7. EXTENSION 37 Figure 4.10: Coordinator architecture. performance. The switching logic, which decides when to switch between DP1 and DP2, is crucial for the effectiveness of the algorithm. The coordinator also prevents the algorithms from getting stuck due to second truck as explained in the previous section by reseting the reference profile. 4.7 Extension The algorithm was extended for a platoon consisting of 3 trucks. The architecture is same as shown in Figure 4.3. There is one more DP1 and DP2 blocks and the coordinator now handles all the six dynamic programming algorithms. The cost function also slightly differs in case of extended platooning. The middle vehicle has an additional factor. The cost function is as shown in (4.11), where d (2) k is the relative distance between the middle vehicle and the vehicle behind and d k is the relative distance to the vehicle in front. ζ = [ ] 1 α β γ ɛ ψ m k t k d k g k δv k d (2) k Cost function remains the same as in (4.5) for the first and last vehicle. (4.11)