A Method of Optimal Control for Class 8 Vehicle Platoons Over Hilly Terrain

Fuel is one of the most expensive operating costs for class 8 trucks, second only to driver wages [1]. Platooning technology has been shown to save significant amounts of fuel for a team of class 8 vehicles traversing level ground. Fuel savings while platooning are a strong function of the gap between vehicles, which is here defined by the time gap from the rearmost part of a lead vehicle to the nose of a follower. Under steady-state conditions and for vehicle gaps of less than 0.8 s, both leader and follower vehicles have been shown to save fuel, with the platoon team saving at least 7% fuel [2–4]. As the vehicle gap decreases and the number of vehicles in the platoon increases, the fuel savings will trend even higher [5–7].

Platooning saves fuel by reducing aerodynamic drag, which at constant speeds on the level ground often represents over half of the energy used by class 8 vehicles [8]. As shown in the energy balance from fuel energy to the wheels for a class 8 vehicle depicted in Fig. 1, fuel energy can also be allocated to kinetic and potential energy. There are no changes in kinetic and potential energy for a semitruck that is at a constant speed on level ground, but once speed or grade variations are introduced, kinetic and potential energy are stored and released according to the speed and elevation profiles a vehicle experiences. The kinetic and potential energy can be regarded as both a source and sink of mechanical energy which can be dissipated by the road loads or shed directly by active deceleration. Thus active deceleration, either by braking or retarding, is regarded as particularly wasteful, as it is an intentional loss of energy. It is clear that optimizing platooning in any conditions beyond constant speeds and level ground relies on optimizing the flow of kinetic and potential energy, and controlling how platooning dynamics will influence those loads. This starts with an understanding of the optimal speed profile for every single vehicle, which can then be applied in the context of cooperative platooning.

There is a wide body of research that aims to optimize the speed profile of a single-vehicle over terrain. studies in economical cruise control have found that constant speed is the optimal strategy for fuel consumption [9], provided the gradients meets one of the following conditions: (i) the gradient ascends slowly enough that it does not exceed vehicle power limitations or (ii) the gradient descends slowly enough as to not require braking. A given hill is said to be steep if it violates one of these road geometry constraints. Due to the limited power of heavy-duty vehicles, it is often impossible to maintain constant velocity on ascending grade. Additionally, higher mass vehicles accelerate more rapidly on downhill segments [10]. To overcome these issues, prior research has used a grade preview for look-ahead or model predictive control (MPC) to reduce the impact of steep hills on consumption. Under a top speed limit constraint, the resulting optimal solution preemptively slows down a truck before a steep downhill and conversely increases speed before a steep uphill [11]. Simulations of a real-time optimal cruise strategy using authentic road geometry claimed up to a 2.5% benefit in 2006 [12]. Further work utilizing MPC with a grade preview found up to a 3% benefit in simulation over rolling hills but was unable to replicate the results experimentally over mountainous road geometries with steep, long slopes [13]. It can be safely stated that deviations from optimal behavior stem from poor powertrain brake efficiency (as in the case of a downshift) or active deceleration.

The extension of fuel optimization over grade from single vehicles to platoons has universally used MPC control. The authors in Ref. [5] provide an overview of previous efforts to optimize a platoon's fuel economy over grade, which is summarized in Table 1 with some additions.

The solution for fuel-efficient platooning over grade shares many characteristics with the single-vehicle solution, with additional nuances. Due to the reduction in aerodynamic drag experienced by following vehicles, a 2014 doctoral dissertation found that a steep downhill requires additional braking for an identical following vehicle [10]. This issue is exacerbated when the platooning vehicles are of disparate mass, making the optimal platoon behavior complex. In the same work, several platooning controllers were proposed and simulated, including a cooperative controller—as platooning vehicles present the opportunity for cooperative control, sometimes referred to as centralized control. In such strategies, a lead vehicle chooses a speed profile that benefits the platoon as a whole, but may not necessarily benefit the lead vehicle. Simulation results showed that centralized control comes very close to realizing the hypothetical optimal fuel consumption reduction of a platoon, which would occur if both the fuel consumption reduction of optimal single-vehicle velocity profile and maximum possible air drag reduction of constant-spacing platooning was achieved. Theoretically, this hypothetical optimal fuel consumption is possible if a lead vehicle were to adopt a reverse platooning strategy, i.e., strictly maintain the spacing of the vehicle behind it [22]. In Ref. [10], following the simulation results, a constant-headway controller [set at $τ=1s$] was implemented and experimentally tested on a heterogeneous platoon. Over small gradients (i.e., not steep) the platooning vehicles achieved fuel savings from 3.9 to 6.5%. However, there were no observed fuel savings over steep terrain due to unwanted braking, re-emphasizing that active deceleration is a deviation from optimality.

A 2017 simulation study showed that fuel savings of up to 12% could be achieved when a two-layer MPC architecture was used instead of standard fixed-headway platoon controllers [17]. Interestingly, the energy balances in the study show that all road loads are essentially equal between the standard and improved controllers except for braking, further confirming that active deceleration is an important obstacle to fuel-optimal platooning.

In a 2019 thesis, simultaneous shifting was simulated on a production-intent two-truck platoon controller [20]. Tracking was greatly improved, but there was no change in fuel economy. The thesis also investigated a route-optimized gap growth controller, which used MPC on a follower to preemptively open a gap, lowering the amount of engine saturation. The route-optimized gap growth strategy demonstrated significant fuel savings over heavy grade relative to the nonpredictive controller.

A 2020 thesis that built upon the work in Ref. [20] investigated the value of implementing an MPC controller on the lead vehicle instead of the following vehicle [21]. The designed a long-horizon predictive cruise controller was compared to a cruise controller with and without droop. Droop is a cruise control parameter that acts as a threshold allowance for speed error over grade. For instance, a downhill (or lower) drop of 3 mph would allow a vehicle to increase its speed by 3 mph on a downhill section before applying the retarder. For a single lead vehicle over heavy grade, long-horizon predictive cruise controller saved 1.8% fuel over a cruise controller with maximum droop, which itself saved 6% over the cruise controller with no droop. For a two-truck platoon with the leader using a no-droop cruise controller, the following vehicle required enough active deceleration that fuel consumption was increased relative to single-truck driving, in spite of aerodynamic savings. With observed savings of less than one percent for simultaneous shifting, the thesis also reaffirmed that simultaneous shifting has limited to no impact on fuel savings. From the prior research outlined above [10,17,20,21], it is clear that over steep gradients active deceleration has the potential to negate platooning aerodynamic benefits.

This study follows the development and experimental testing of two different platoon controllers: (i) a string-stable proportional-integral-derivative (PID) controller and (ii) a nonlinear model predictive control (NMPC) controller. The desire to develop a string-stable PID controller was motivated by velocity traces from a 2019 four-truck platoon test over the highway test track at the American Center for Mobility, which has a rolling grade of $±3%$⁠. Each subsequent truck in the platoon formation during those tests experienced higher acceleration demand than the preceding truck due to grade disturbances. Figure 2 exemplifies the grade-induced velocity amplification experienced by platooning test vehicles.

This phenomenon is known as string instability. As a result of string instability, significant braking was required and fuel consumption correspondingly increased, as is shown by the increasing aggressiveness in Fig. 3 [23].

While the NMPC controller developed in this paper is similar to those done in previous works, there are a few key distinctions. The first key distinction is that the sharing of information from the preceding vehicle to the following vehicle is not necessary. In this work, the following vehicle does use the lead vehicle's acceleration value to obtain knowledge of the preceding vehicle velocity profile. However, if this information is not available the following vehicle can still operate under a lead vehicle constant-velocity assumption. Additionally, this work does not take advantage of a platoon coordinator. The platoon coordinator is typically a low update rate process that assigns the optimal velocity trajectory to every vehicle in the platoon. The NMPC controller in this work instead uses a distributed approach with each vehicle planning their own independent velocity profile which also accounts for safety constraints.

While the PID controller is not expected to save as much fuel as the NMPC controller, it is more practical: the ease of implementation and computational speed are much higher, and the PID controller does not require a grade database. The remainder of this work is organized as follows: in Sec. 2, the cacc system hardware, software, and models are overviewed. In Secs. 3 and 4, the string-stable PID and look-ahead NMPC control strategies for lead and following vehicles are developed. In Sec. 5, simulation and experimental results for the newly developed control architectures are presented. Finally, Sec. 6 presents the conclusions and suggestions for future research.

Most traditional platooning control methods mainly rely on the transfer function from torque to velocity. This means that only the basic longitudinal equation of motion of the vehicle is needed in order to perform the necessary PID control synthesis. Because this work is focused not only on the relationship between engine torque and velocity but also the velocity and volumetric fuel flow rate, an additional relationship is needed. This section will introduce the longitudinal equation of motion and a three-link method of relating velocity to fuel rate. Additionally, a brief overview of the hardware and software of the system will be provided to allow for a better understanding of the physical implementation of the controllers developed in this work.

The most fundamental model for vehicle control is the differential equation that relates a vehicle's engine torque to vehicle acceleration. Figure 4 shows a basic powertrain model relating engine rotational speed ω_eng to wheel speed ω_wheel. In this figure, the subscripts eng, trans, ds, and diff stand for engine, transmission, driveshaft, and differential, respectively. This model assumes no transmission slip, no backlash, and no shaft compliance. These assumptions greatly simplify the driveline model without a substantial loss in model fidelity.

with J_eng and J_diff being the engine and differential inertia, respectively. Additionally, B_eng, B_diff, and B_trans are the damping values for the engine, differential, and transmission.

The relationships in Eqs. (1)–(3) are crucial for the development of the NMPC controller because they constitute the dynamic equation that is simulated forward in time. Now that the longitudinal dynamics of the vehicle have been derived, Secs. 2.1.2–2.1.4 will cover the specific relationships within the vehicle powertrain that provide the relationship between wheel speed and fuel consumption. This relationship will be covered in the three-link description motivated by work done by Ref. [13] and also used by Ref. [15].

where ω_wheel is the vehicle wheel speed in (rad/s) and ω_eng is in RPM. This equation again implicitly assumes no shaft compliance, backlash, or transmission slip. This model is verified in Fig. 6.

where τ_eng is the peak engine torque available at a given engine speed. Using least squares regression, the coefficients c₁–c₄ are solved for and the resulting polynomial is overlaid in Fig. 7.

where u_throttle is the normalized throttle position between 0 and 1. This provides a good approximation of the engine torque, as shown in Fig. 8, and is further validated in Sec. 2.1.4.

The final link to obtain fuel economy from wheel speed is the conversion of engine torque to fuel rate. To accomplish this, volumetric brake-specific fuel consumption (BSFC) is utilized. BSFC provides a functional relationship from engine power and engine speed to fuel rate. Because of friction, pumping losses, and heat transfer, this relationship is nonlinear. In order to generate the full BSFC map for a given engine, typically the vehicle must go to a dynamometer or the original equipment manufacturer (OEM) must provide it. Because this work was not supported by an OEM and access to a dynamometer was not possible, an alternative method was used.

Due to numerical issues in the optimization which caused software crashes, combined with and testing/time constraints, the linear model had to be used for the remainder of this work. The linear and nonlinear fuel rate models were evaluated in order to assess what level of accuracy is lost by using a linear BSFC assumption and to determine any disparity in optimization performance. Figure 9 displays the residuals of each model versus the CAN-recorded total fuel consumed. The nonlinear model clearly outperforms the linear model in this case. The main areas of improvement of the nonlinear model versus the linear model are at the boundaries of the engine operating conditions. At higher engine speeds when the nonlinear trends become more pronounced, the linear model will underestimate the fuel rate. Additionally, when the engine power demand is low, the linear model will overestimate the fuel rate.

While the linear model does have more error in predicting the magnitude of the fuel rate, it captures the general trends accurately. While it is possible that the nonlinear fuel model would generate a different optimal control output, it is not expected that there would be any real change in the overall performance of the system. Future work using a nonlinear fuel rate model is required to confirm this.

Two Freightliner M915A5 s with unloaded box trailers served as the test vehicles in this work. A Peterbilt 579 pulling a fully loaded trailer was used as the control vehicle. Fifth wheels were set to the rearmost position. A general overview of the trucks used during testing is provided in Table 3.

The range estimation and vehicle communications on the test vehicles use a lean sensor package of a 64-channel Delphi electronically scanning radar (ESR), Cohda MK5 on board unit (OBU) dedicated short-range communication radios, and Novatel PwrPak7 GPS receiver in conjunction with the vehicles' J1939 CAN networks. A central computer unites the system, shown in Fig. 10. Further detail on the control hardware setup can be found in Ref. [24].

All vehicles use a robotic operating system as the software platform. However, the software was written to be platform-agnostic. The PID controller is implemented in C++. The NMPC controller is also implemented in C++ and takes heavy advantage of the symbolic casadi interface. casadi is an automatic differentiation tool that generates all Jacobians and Hessians for the optimizer, which in this case was the software package ipopt. Due to casadi's symbolic language, the software package has the ability to update variables in real-time, such as vehicle drive ratio due to gear shifts and the associated vehicle dynamic changes. Further detail on the controller architectures and syntheses can be found in Secs. 4 and 5.

Results such as those shown in Fig. 2 demonstrate that designing a PID controller without respect to string stability will yield a controller that is insufficient to maintain adequate safety factors or fuel consumption. This motivates the need for a more advanced controller that can satisfy string stability constraints or greatly reduce the effects of disturbances on a platoon. While there are several methods to achieve this, the authors attempted a method with reduced implementation overhead and lower technical overhead. This method is called H-infinity optimal control.

What sets H-infinity control apart from other control methods such as MPC is that H-infinity control is generally applied to optimize control gains for a state-space or fixed-structure PID controller. Additionally, H-infinity methods can account for model uncertainty and sensor noise. The H-infinity optimization generates the control gains such that a chosen design criteria are optimally met. The job of the controls engineer, in this case, is to adequately select weightings on various system inputs and outputs such that the desired level of performance or robustness can be achieved. Once this optimization process is complete, the controller may run in real-time at any frequency that a traditional PID controller would run, with no online optimization libraries needed.

which states that the H-infinity norm for the transfer function between the lead vehicle acceleration, $ui+1$⁠, and the follower vehicle acceleration, u_i, must be less than one for all frequencies greater than zero. In a more applicable sense, this guarantees that, as long as the system model is accurate, the magnitude of the acceleration of every following truck will be smaller than the acceleration of the truck it is following. This meets the definition of string stability and allows acceleration perturbations to decay down the line of platooning vehicles.

Figure 11 introduces the total block diagram for the system. The system block D represents the communications delay of information being transmitted from vehicle to vehicle. This is characterized as a first-order Padé delay. Block K is the controller that the H-infinity process will optimize. In this case, a PID controller is selected. Block H is the system spacing policy. Finally, block G represents the transfer function from commanded acceleration to actual acceleration. Further details on this control structure can be found in the referenced paper.

The controller adapted from [25] was then evaluated in terms of the system's complementary sensitivity. String stability is typically evaluated through system complementary sensitivity defined by Eq. 9. The following analyses assume a linear time-invariant system. If a bode plot has a positive spectral frequency [dB > 0], then a velocity disturbance will grow for each vehicle. Figure 12 shows the system sensitivity of the H-infinity design versus the PID controller from Figs. 2 and 3. An interesting note on Fig. 12(b) is that since the spacing policy is specifically accounted for in the control synthesis, one can identify the closest feasible platooning distance allowed before losing string stability.

The H-infinity controller developed satisfies the system sensitivity condition. The controller was then simulated in TruckSim to validate that the developed controller performance matched that of Ref. [25] in response to velocity changes. As shown in Fig. 13, the implemented controller displays decreasing accelerations down the platoon when the leader changes velocity, indicating the controller is working as intended.

The same controller was then implemented on the vehicles in Table 3 in C++. As shown in Fig. 14, the implemented controller displays decreasing accelerations down the platoon when the leader changes velocity, indicating the controller is working as intended and that string stability is guaranteed for a nonhomogeneous class 8 vehicle platoon. The official fuel results over the proposed test area will be described in Sec. 6.

Owing to their lack of look-ahead, classical PID platooning control methods are limited in their ability to minimize the controller saturation and active deceleration that occurs over steep hills. For this reason, NMPC is an attractive alternative to classical methods. NMPC allows for a nonlinear vehicle model to be propagated forward, with the inclusion of any future knowledge about external disturbances such as grade. This section presents both the overall NMPC architecture and the synthesis of both the leader and follower control architectures.

Nonlinear model predictive control allows for the minimization of a cost associated with a defined process. Unlike PID control, NMPC does not require a linear model. NMPC takes in the system model as a differential equation and simulates the system model forward in time, determining future states of the vehicle and updating cost functions as necessary. The differential equation used in this work is defined in Eq. (1). Not only does this differential equation capture air drag as a nonlinear term and allow the air-drag coefficient to be changed as a function of headway, but it also incorporates the effects of grade on a vehicle.

In order to allow vehicles to use grade to their advantage, the grade profile of the vehicle route must already be known. By predefining a grading map that contains the East–North position coordinates of road-grade along a defined local coordinate frame, it is possible to include the upcoming grade in the simulated system model. This approach was applied in this work, where a vector of grade points is provided to the optimizer, allowing a velocity profile to be generated that minimizes fuel consumption with respect to upcoming terrain.

Typical control design criteria call for a controller to be 5–10× faster than the dominant dynamics of a system. Due to this criterion, 20 Hz was selected as the desired control rate for this application. This means that some form of inner-loop control is necessary in order to reliably achieve 20 Hz control.

A nested control structure was developed to achieve the desired 20 Hz control. The optimal torque from the optimizer is taken as a feedforward torque, and the optimal velocity profile is taken as a reference velocity from an inner-loop velocity regulator. Figure 15 displays the control architecture described above. Inputs to the optimizer are torque, grade, braking, and the vehicle's current velocity. The outputs are the desired torque, braking, and velocity.

The main objective of the lead vehicle optimizer is to generate a velocity trajectory that to minimizes fuel consumed over some predefined terrain. This is a relatively straightforward task given the models already defined in this article and can be broken down into two distinct parts: propagation of the relevant states and minimization of the objective function.

For state x₁ the derivative is x₂. The derivative for state x₂ is less straightforward but was previously derived as the longitudinal vehicle equation of motion Eq. (1). The states are then propagated forward by using the Runge–Kutta fourth-order method. Direct multiple shooting is used to solve for the optimal control horizon and thus it is necessary for the initial conditions of the optimal-control problem to be defined, as well as the cost function.

where Q₁ and Q₂ are both tunable weights to yield the desired performance. If Q₂ were zero, the controller would aggressively track a desired set cruise control speed. By increasing Q₂ however, the system begins to behave in a way that allows the velocity to fluctuate over rolling hills, saving fuel. An example of this would be the NMPC controller allowing a vehicle to coast over its desired set velocity while going down a hill so that the vehicle carries more kinetic energy into the next hill, requiring less fuel to power itself back up to speed, which was noted in prior work. This cost function is used for all single-vehicle optimal platooning in this work, which is referred to as EcoCruise here. The chosen weights for Q₁ and Q₂ are listed in Table 4.

The final parameter to list is the look-ahead distance used. Since the longest half-hill on the experimental test route is shorter than 1500 m the look-ahead distance for the EcoCruise platform for the remainder of the work is set to 1500 m. Details regarding the half-hill lengths of the testing route are provided in Sec. 5.1.

Vehicle-to-vehicle (V2V) communication allows for the sharing of vehicle states such as velocity and acceleration. This allows for the vehicle headway to be propagated as the difference between the leader and follower velocities. Also, due to the following vehicle receiving the leader's acceleration, a first-order decay is applied to the leader acceleration and the velocity profile of the leader is generated using a decaying acceleration model. Finally, the following vehicle's acceleration is propagated in the same manner as EcoCruise, where the engine torque can be optimized to minimize Eq. (16).

Simulation testing played an important role in the development of the EcoCruise controller. As this work differs from other previous work conducted in this field, a plethora of cost-functions and weightings were tested in order to pick out the desired performance. Many of these cost functions produced undesirable performance. As a benchmark for desirable performance in the simulated environment, results from Ref. [13] were used.

The results from prior art show that when approaching a decline, it is most optimal for a vehicle to coast below the setpoint velocity. Once the vehicle is on the downhill segment, the vehicle allows the grade to accelerate it back up to and possibly higher than the target velocity. Figure 16 represents the prior work juxtaposed with the EcoCruise controller. The desired velocity for EcoCruise was selected at 25 m/s. With the knowledge of the upcoming grade, the optimizer allows the vehicle to slow down before entering the downhill section, which matches prior art. It should be noted that because of computational inefficiencies in the optimizer-simulink interface, as well as computation power limitations, the EcoCruise controller was only simulated at 2 Hz with a zero-order-hold used for all time steps in between. This is what causes the generated control signal to be” noisy”.

The uphill driving scenario was also investigated while using the same dissertation as the baseline for comparison. In this scenario, it is deemed optimal by the reference case to begin accelerating before the hill and to allow the vehicle to slow down below its desired velocity before cresting the hill. There is an added benefit to this optimal approach. Because the power demand on the engine is not rapidly increasing, gear shifts can be avoided, helping save even more fuel. The EcoCruise system displays the expected behavior. The optimizer begins accelerating the vehicle before the start of the uphill segment and the vehicle avoids the need to downshift (Fig. 17).

Unfortunately, due to software package integration issues with simulink and trucksim, the optimal-follower software was unable to be tested in the simulation environment. To accommodate some level of software testing and tuning before running on a closed track, data was collected from a two-truck manually driven platoon over the planned test path. Using the robotic operating system's rosbag playback features, the compiled C++ optimization software was tested. By analyzing the output of the prediction horizon for both the vehicle speed and torque, reasonable cost-function weightings were generated to begin tuning on the physical system. The chosen weights for the following vehicle were then set to the values in Table 5.

The final parameter to be decided was the look-ahead distance for the optimal following vehicle. A key factor in deciding the look-ahead distance was the accuracy of a first-order acceleration decay model or a constant-velocity model for the lead vehicle. Both assumptions get worse the further out that the following vehicle predicts. A look-ahead distance of 100 m or approximately 4 s was chosen for the following vehicle. This reduced the size of the optimization problem drastically from the 1500 m look-ahead used on the EcoCruise platform and allows the optimization to generate new optimal velocity and torque profiles at approximately 10–15 Hz. By allowing the optimizer to run at a higher frequency, a poor constant-velocity assumption is less likely to have a negative impact on the optimal following platooning method.

This section overviews the full development and test cycle that the EcoCruise and optimal following system underwent. Some of the first development tests were performed in simulink, utilizing the trucksim-simulink interface. The trucksim/matlab simulation environment is what drove the preliminary development of the cost function used for EcoCruise. After the EcoCruise development, the optimal following software was adopted and tested in a hybrid manner between simulation and on-road testing. Finally, the full system was run on highway US-280 in Alabama, and the results are presented and analyzed herein.

A 15.4 mi (24.8 km) loop on highway US-280 was selected as the test section for the work. For consistent analysis, the data was bounded to portions of the test loop for all runs, approximately 8 km on both the east- and westbound segments. A map of the test loop can be seen in Fig. 18.

The experiments were designed in the spirit of SAE J1321 type II fuel testing [26]. All three trucks were warmed up for three laps of the test loop or roughly 1 h. The control vehicle ran concurrently with the test vehicles at a distance greater than 1500 ft (457 m) behind the test vehicles to avoid “naturalistic” platooning. The control vehicle used a traditional cruise control with no droop independent of the platoon to provide a constant baseline.

In Figs. 19 and 20, the grade over the eastbound and westbound sections of the testing loop is compared to the national activity-weighted average from Ref. [27]. For the eastbound section, the half-hill length is shorter and the grade is steeper than the national average. As a result, the first half of the test cycle is deemed to be more demanding than the national average, both in the steepness and frequency of hills. In contrast, for the westbound section the half-hill length is generally longer and the grade flatters than the national average. As a result, the second half of the test cycle is deemed to be less demanding than the national average. Because the two test sections more or less bound the national average, the results can be reasonably expected to represent upper and lower limits of the average national results if this study were theoretically conducted over the entire national highway network.

The experimental testing occurred on April 12 and 13 of 2021. External test conditions for each day are presented in Table 6. The weather was taken from the weather station KALSALEM, located 0.52 mi (0.85 km) from the turnaround point. The variation in temperature and speed is higher for April 12 due in large part to the longer sample period.

Because the experiments were conducted on real roads, surrounding traffic could not be controlled. The results assume that the influence of surrounding traffic on fuel consumption was relatively constant throughout the course of the tests. No significant qualitative difference in traffic density or flow was observed by the testing personnel, and because the set speed of 55 mph is below the posted speed limit of 65 mph, there were no attempts by drivers to cut into the platoon. The influence of surrounding traffic on a two-truck platoon was investigated in-depth in Ref. [4], which found that dynamic passing events had a small impact on lead truck consumption, but not on the follower.

Time and funding constraints limited the amount tests for this investigation. As a result, the only configurations for which three replicates were collected were the fixed-spacing platooning led by cruise control and NMPC optimal platooning led by EcoCruise. These were hypothesized as the worst and best-case scenarios and thus were prioritized. Table 7 shows the test matrix as completed for this work, and the nomenclature used throughout the remainder of this section. EcoCruise and cruise control refer to the lead vehicle strategy, with the abbreviations “ec” and “cc,” respectively. Optimal, fixed-spacing, and H-infinity refer to the follower vehicle control strategy, with the abbreviations “opt,” “fx,” and “hinf,” respectively. The fixed-spacing runs use the H-infinity control design, but use a range error rather than a time-gap error. There was little to no difference in observed performance, as is noted in Sec. 5.6.3.

One important way to compare different cruise control profiles over the same test sections is travel time. Travel time in this work is defined as the time it takes the vehicle to travel from the test starting position to the test end position illustrated in Fig. 18. It is extremely important that any new cruise control methodology does not save fuel merely by having the vehicle travel at a lower overall speed, therefore increasing travel time. Not only would this be an undesirable controller, but travel time is particularly important for scheduling deliveries and practical real-world trucking.

Figure 21 shows box plots of the travel times broken out by type of run. The EcoCruise runs trended slightly faster for the eastbound travel direction, and for the westbound runs the travel times were all roughly equivalent. The furthest outlier is only 1.25% less than the median of all runs, which is not caused for concern. To counteract potential issues with travel time, the fuel results will be calculated on a consumption per time basis.

As was discussed in the introduction, for a given vehicle, a hill is classified as steep when its demands exceed a vehicle's power limits (steep uphill) or when it forces a vehicle to actively decelerate (steep downhill). In this section, uphill saturation is investigated on the testing loop. Due to the high mass and limited power of class 8 vehicles, many hills are classified as steep. Using a constant road load assumption of 150 kW, which was estimated from the data during small gradients, the power required to maintain a constant velocity over the eastbound and westbound sections was calculated. The result of the calculation shows that, on the more challenging eastbound section, there are 13 discrete instances that the power demands exceed the rated power of the heavier vehicle. On the flat westbound section, there are no instances where rated power is exceeded, except at the very end of the test section. Figure 22 shows the calculated power requirements and the max available power at that engine speed during the cruise control reference run for the lead vehicle.

Figure 23 shows the same calculations for the lighter follower vehicle. Due to its reduced mass relative to the lead vehicle, the power demand intensity of the hills is less severe, as expected.

This section aims to elaborate further on the negative effects of the power limitations mentioned in Sec. 5.2. A key takeaway of Sec. 5.2 is that due to power limitations, there are some sections of the road in which the vehicle cannot produce enough power to maintain a set velocity. Since PID controllers cannot look ahead and see the terrain coming, the power output of the engine cannot be optimized to minimize saturation issues. Alternatively, because the NMPC model has a built-in engine model and uses information about the road ahead, it can more intelligently traverse the hills without running into as many saturation issues. Figure 24 provides an example of the difference between the H-infinity and optimal following controllers over the same section of terrain.

While the commanded torque value is limited before it is sent to the engine, Fig. 24 displays the torque value that the H-infinity and NMPC command before any saturation is applied. This helps graphically illustrate the benefit of using the look-ahead approach. The H-infinity controller will actively decelerate on the downhill sections of terrain to reduce the velocity and maintain the desired headway. This is very inefficient especially with a steep uphill section of road immediately following. To compensate for this, a large commanded torque is requested and often cannot be realized. The NMPC controller, however, with knowledge of power limitations and upcoming terrain, allows the vehicle to carry more of its kinetic energy through the downhill to aid in the following climb uphill. This greatly reduced the torque and power demanded and results in a drastically reduced fuel rate.

The section of terrain highlighted in Fig. 24 captures the key distinction between the H-infinity Controller and NMPC controller in practice. The NMPC controller allows for more efficient use of the following vehicles potential energy and uses it to help reduce the amount of power required by the engine to climb hills immediately following a decline. Sections 5.4–5.7 will formalize the approach to analyzing the fuel data collected over the entire test runs and will present the final fuel savings results.

The statistical significance calculation methodology for fuel results in this study used the SAE J1321 standard [26] as a reference. A brief overview is provided here for continuity. A ratio of test vehicle to control vehicle fuel consumption from the same time is calculated first for each test run, called the T/C ratio. Then T/C ratios for one configuration are compared to T/C ratios from another configuration, usually a baseline of some kind. The T/C comparison process is as follows:

1. Obtain T/C ratios by dividing the fuel consumption for the truck in question by the fuel control vehicle consumption for every test segment

   $T/Ci,j=Test Vehicle ConsumptionjControl Vehicle Consumptionjfor configuration i$

   (24)

   where i denotes a different configuration of the test, e.g., EcoCruise leader with a follower using NMPC, and j denotes the iteration number of the test.

2. Select two sets of tests from i for comparison, calling one the baseline and the other the Test.

3. Compare the T/C variances between the j iterations of the baseline and test sets using AN F test with a significance level of 0.95.

4. If the variances are equal for the baseline and test T/C ratios, compare the T/C ratios and calculate the 95% two-sided confidence intervals of the baseline and test sets using a t-test for equal variances. If the variances are not equal, use a t-test for unequal variances. This significance level is also selected to be 0.95.

5. Depending on the result of the t-test (specifically, its p-value), the difference in mean T/C for the Test and the Baseline is deemed either significant or insignificant. Results are expressed as a percent improvement over the baseline ± the two-sided confidence interval.

The fuel consumption numbers in this study were collected by logging the volumetric CAN fuel rate signal which was integrated over the test area. The calculation was then performed using volume consumed per hour as the fuel consumption input to the T/C ratio. The conclusions herein were derived utilizing the total fuel volume consumed in the aforementioned calculations.

Visualization of the lead truck T/C ratios is shown in Fig. 25. The raw fuel results for the leader from the experiment are in Table 10 of the Appendix.

It is not expected that the following vehicle's control strategy had a significant influence on the lead truck's aerodynamic fuel benefit in this work. As can be seen in Table 8, the mean headway is approximately the same between platooning run types. Furthermore, the 1.4 s/35 m platooning headway used in this study is outside of the regime where lead vehicles begin to experience an appreciable reduction in drag (less than 0.8 s/23 m). Assuming any reduction in aerodynamic drag for the platooning truck is approximately equal during all platooning runs enables all cruise control runs to be compared to all runs using EcoCruise, regardless of whether the follower was using PID or NMPC control.

When all five EcoCruise runs are treated as the test set and all six cruise control platoon runs are treated as baseline, the calculated fuel savings for the eastbound direction using the outlined methodology are $14.0±1.9%$⁠. This means the EcoCruise saved 14% fuel relative to the cruise control, plus or minus 1.9%. The savings are statistically significant and are shown in Fig. 26(a).

For the westbound direction, the calculated fuel savings were $0.0±4.7%$⁠. It is clear that the lead truck did not save any appreciable amount of fuel by using EcoCruise on the westbound test section. Figure 26(b) shows the results visually.

The T/C ratios for the follower vehicle are shown visually in Fig. 27. The raw fuel results for the follower from the experiment are in Table 9 of the Appendix.

First, fixed-spacing following where the lead vehicle used cruise control is compared to optimal following where the lead vehicle used EcoCruise. This is hypothesized to represent the worst-case scenario versus the best-case scenario, respectively. When all three valid optimal following runs led by EcoCruise are treated as the test set and all three fixed-spacing platoons led by cruise control runs are treated as a baseline, the calculated fuel savings for the eastbound direction using the outlined methodology are $20.4±4.3%$⁠. This means that optimal following behind EcoCruise saved 20.4% fuel relative to using fixed following behind cruise control, plus or minus 4.3%. The savings are statistically significant and shown in Fig. 28(a).

For the westbound direction, comparing optimal following led by EcoCruise to fixed following led by cruise control yielded fuel savings of $1.0±5.0%$⁠, shown in Fig. 28(b). Because the confidence interval is much larger than the perceived savings, the result cannot be called significant. The insignificant difference attained by using NMPC following on the flatter westbound section is noteworthy because it reveals the efficiency gain of the optimal following controller comes from a more efficient driving style over steep terrain. When on flat terrain, there is no perceived benefit to using EcoCruise or optimal following on flat terrain versus traditional H-infinity or PID platooning methods.

Another interesting comparison that can be made is optimal following versus fixed spacing when both are led by cruise control. There were only two optimal runs led by cruise control, which weakens the strength of the conclusions for this comparison. Taking optimal following behind cruise control as the test set and fixed following behind cruise control as the baseline set, the following vehicle saved 21.9$±5.3%$ fuel on the eastbound section, which is shown in Fig. 29(a). This is nearly the same savings as the EcoCruise/optimal following configuration. Indeed, taking EcoCruise/optimal following as test and cruise control/optimal following as baseline yields no significant difference in fuel savings between the two.

On the westbound section for cruise control/optimal following versus cruise control/fixed following, the calculations indicate $−5.6±7.6%$ savings, which is shown in Fig. 29(b). A negative percent savings would indicate an increase in fuel consumption, but the confidence interval is large enough to dismiss the finding as inconclusive, possibly due to the scarcity of data points for optimal following led by cruise.

Especially compared to NMPC performance, no differences were perceived in the performance of the fixed spacing versus timegap spacing policies (represented as fx and hinf in Table 7). Thus, a comparison between the PID following strategies and optimal following strategies is made, taking all three EcoCruise-led PID following (two fixed and one timegap) results as the baseline and the three EcoCruise-led optimal following results as the test set. Over the eastbound section, EcoCruise/optimal following saved $9.2±6.9%$ over EcoCruise/PID following, though the confidence interval is large relative to other results, see Fig. 30(a). Westbound results in Fig. 30(b) show no significant difference, at $3.0±3.4%$⁠.

The influence of leader dynamics on a follower running PID control was calculated by treating the four PID following cruise control runs as a baseline set and the three PID following EcoCruise runs as the test set. When the lead truck was running EcoCruise instead of cruise control, the PID-controlled follow vehicle netted fuel savings of $11.8±5.4%$ on the eastbound section, see Fig. 31(a). For the westbound section results shown in Fig. 31(b) no difference was observed, with the calculations indicating savings of $−1.3±5.2%$⁠.

Only one valid single truck run was collected due to budgetary and time constraints. Thus, no confidence interval can be calculated for the platoons with respect to single-truck driving. However, a T/C ratio can still be calculated if the one single-truck run is taken as the baseline set, and under the bold assumption that the single sample represents the true mean, the difference between standalone driving fuel consumption and platooning fuel consumption can be calculated, making no statements on the statistical significance of the results.

The percent difference from baseline for the lead truck running cruise control while in the platoon was 0.7% eastbound and 1.2% westbound. When the lead truck leads the platoon with EcoCruise, the percent difference from baseline is was 14.6% eastbound and 1.2% westbound.

The percent difference from baseline for PID following cruise control was −1.1% eastbound and 12.6% westbound. In contrast, the percent difference from baseline for PID following EcoCruise consumption was 10.8% eastbound and 11.5% westbound. Finally, for optimal following, regardless of leader strategy the fuel savings were 19.6% eastbound and 11.9% westbound. Blocking results by leader control strategy for optimal following resulted in an only minor change to the percent difference.

The goal of this study was to experimentally investigate the performance of two candidate controllers for fuel-efficient cruise control and platooning over grade. Testing over real roads was conducted, complete with a control truck for reference.

A linear controller was implemented under the hypothesis that if the acceleration of each subsequent vehicle in a platoon could be dampened, then the platoon would become string stable, and fuel economy would be improved. However, the presence of steep hills led to frequent saturation of that controller, and its performance suffered on the hilly eastbound test section. Prior research indicated that where steep hills occurred, both velocity control and especially platooning control would benefit from grade look-ahead. Novel decentralized NMPC controllers were therefore designed and implemented for both the leading and following vehicle, which were referred to as the EcoCruise and optimal following controllers, respectively.

The fuel consumption of the two platooning and cruise controllers was compared directly. On the eastbound test section, the lead truck saved 14% fuel when running EcoCruise instead of cruise control, $±1.9%$⁠. Had droop been implemented on the cruise controller, the savings would almost certainly be lower, as was seen in Ref. [21]. The worst-case scenario for the following vehicle was PID control behind a leader running cruise control. The PID-controlled follower saved 11.8% fuel when the leader switched from cruise control to EcoCruise. This result stresses the importance of a leader's velocity trace in a platoon, especially when using linear control methods for spacing control. Interestingly, regardless of the leadership strategy, if the follower was using the optimal following, it was saved approximately 20% fuel versus the worst-case PID following cruise control.

For the westbound section across all controller types, no statistical difference was found in fuel consumption. That indicates that in this study, the controllers did not materially influence the fuel consumption on the westbound test section, and furthermore, the westbound section serves as a good reference for determining the aerodynamic benefit of the platoon. This does not mean that platooning was of no benefit.

It must be re-emphasized that only one single truck test was collected and the conclusions presented with respect to single truck performance make no claims of significance. That said, there are underlying trends within the baseline results that still have much scientific value.

Conclusions for leader platooning versus single truck:

1. The lead truck had only a trivial improvement in fuel consumption on the eastbound section from platooning when operating on cruise control.

2. EcoCruise greatly outperformed cruise control on the eastbound section, with an indicated 15% improvement.

3. Both EcoCruise and cruise control platooning were 1.2% improved on the westbound section versus standalone driving, though it is unclear if this is platooning benefit or a coincidental result within experimental error due to lack of replicates.

Conclusions for follower platooning versus single truck:

1. Using PID control when the leader is running cruise control appears to have negated the aerodynamic benefit of platooning on the eastbound section.

2. When the same PID-controlled follower is led by EcoCruise, however, the follower truck appears to save fuel again, at an indicated 11%.

3. When the following truck uses the optimal following, regardless of the leadership strategy, the follow vehicle indicated nearly 20% fuel savings on the eastbound section.

4. Indicated savings of the follower on the westbound section were very consistent and provide a good indication of the fuel savings due to aerodynamic benefit. During these tests, it was near 12%.

The H-infinity PID controller has not deemed a suitable solution for fuel-efficient platooning over the grade in the eastbound section. Its losses due to active deceleration and power saturation negated the aerodynamic benefit of platooning, as Secs. 5.2 and 5.3 indicate. However, if the lead vehicle was using EcoCruise, the H-infinity controller performed acceptably. It is easy to implement, has low computational requirements, and does not need a database of road grade. The NMPC greatly outperformed it, however, and where the grade begins to approach or intensify like that of the eastbound section, using the NMPC is recommended. The NMPC controller showed great potential to save fuel over hilly terrain, both for single truck velocity control and especially for platooning gap control.

The results in this paper were generated utilizing trucks from 2009 which lack aerodynamic fairings and have six-speed automatic transmissions with a torque converter. If the results were collected with more modern trucks, the fuel savings would likely take different absolute values. However, it is unlikely that there would be changes to the following general trends:

1. NMPC allows the aerodynamic benefit of platooning to occur over moderate grade variations

2. The lead velocity trace is highly important for a PID-based platoon

3. Over small gradients, aerodynamics dominate the fuel savings, and the control strategy does not appreciably affect fuel so long as active deceleration is minimal

It was claimed in Sec. 5.1 that the grade encountered on the eastbound and westbound test sections in this work bound the national average grade profile, and that the results herein reasonably represent upper and lower limits of the theoretical national average results. However, with the results being so drastically different between the two sections, expected behavior for grade distributions falling between the eastbound and westbound sections is of interest. The authors believe that the very short and steep hills on the eastbound test section represent an ideal use case for the NMPC controller. As half-hills lengths grow longer, NMPC must surely use a longer control horizon to continue to leverage its advantage over classical methods, which may eventually reach a limit in processing power. Even if processing power is sufficient to select the optimal control horizon for longer hills, one can contrive a very long steep downhill section where no amount of preemptive slowing before the hill would prevent top speed limits from being reached. Thus active deceleration becomes necessary for a steep downhill that is long enough, regardless of a control strategy. This is conversely true for long, steep uphill sections, where a downshift and its associated penalty in fuel consumption will eventually become necessary. This behavior is piecewise and merits further investigation. As for steepness, for distributions that are less steep than the eastbound test section, there will be: (i) less need for active deceleration on downhills, and (ii) less need for gear shifting to satisfy power demands. This behavior is expected to be somewhat continuous until the hills in question are no longer classified as “steep,” at which point results will likely follow those seen on the westbound section in this work.

• U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) (Award Number DE-EE0008470; Funder ID: 10.13039/100006134).

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.