A piecewise trajectory optimization model for connected automated vehicles: Exact optimization algorithm and queue propagation analysis
Introduction
Stop-and-go movements (Li et al., 2010) are almost inevitable experience in highway traffic due to intrinsic limitations in human driving behavior (Li, Ouyang, 2011, Li, Wang, Ouyang, 2012, Jiang, Hu, Zhang, Gao, Jia, Wu, 2015) and information access (Li et al., 2014a). Stop-and-go traffic is linked to a number of adverse impacts from highway traffic, including excessive fuel consumption, extra safety hazards, and increased travel delay. Among a number of potential solutions to stop-and-go traffic (e.g., variable speed limits (Lu and Shladover, 2014), ramp metering (Hegyi et al., 2005), merging traffic control (Spiliopoulou et al., 2009), and signal coordination (Day et al., 2010), the connected automated vehicle (CAV) technology has received increasing attention recently due to its capability of controlling vehicle trajectories and modifying driving behavior (Ma et al., 2016). Various studies have been conducted to utilize CAV to improve traffic smoothness and throughputs on both uninterrupted freeways and signalized arterials.
Studies on the freeway side focus on guiding vehicle trajectories for minimum speed oscillations and minimum conflicts in lane changes and merges. Van Arem et al. (2006) investigates traffic stability and efficiency at a merge point. Ahn et al. (2013) proposed a rolling-horizon model for an individual CAV control strategy that minimizes fuel consumption and emissions at different grades. Yang and Jin (2014) studied a vehicle speed control strategy to reduce vehicle fuel consumption and emissions. Wang, Daamen, Hoogendoorn, van Arem, 2014a, Wang, Daamen, Hoogendoorn, van Arem, 2014b proposed optimal control models to determine optimal accelerations of a platoon of CAVs to minimize a variety of objective cost functions in a rolling horizon manner. Later, Wang et al. (2016) investigated distributed CAV acceleration control methods to mitigate formation and propagation of moving jams.
Studies on the signalized arterial side concern the problem of coordinating and scheduling vehicle trajectories to avoid conflicts at crossing points while improving traffic performance measures. Some studies focus on scheduling of vehicles arrival and departure times at an intersection and aim to minimize stops and delay at the intersection. Li and Wang (2006) studied CAV scheduling and trajectory planning for a two-lane intersection, using spanning tree and simulation techniques. Dresner and Stone (2008) investigated a similar non-stop intersection problem and proposed a heuristic control algorithm that processes vehicles as a queuing system. Lee and Park (2012) proposed a nonlinear optimization model to optimize trajectories for CAVs approaching and passing a non-stop intersection. Zohdy and Rakha (2016) proposed a nonlinear optimization model that integrates an embedded car-following rule and an intersection communication protocol for non-stop intersection management. Other studies consider how to control vehicle trajectories in compliance with existing traffic signal timing at intersections. Trayford, Doughty, van der Touw, 1984a, Trayford, Doughty, Wooldridge, 1984b proposed to use speed advice to reduce fuel consumption for vehicles approaching an intersection. Later studies further investigated car-following dynamics (Sanchez et al., 2006), in-vehicle traffic light assistance (Iglesias, Isasi, Larburu, Martinez, Molinete, 2008, Wu, Boriboonsomsin, Zhang, Li, Barth, 2010), multi-intersection corridors (Mandava, Boriboonsomsin, Barth, 2009, Guan, Frey, 2013, De Nunzio, Canudas de Wit, Moulin, Di Domenico), scaled-up simulation (Tielert et al., 2010), and electric vehicles (Wu et al., 2015). These studies mainly concerned control of vehicle speeds but ignored acceleration detail, which however could cause significant errors in estimating fuel consumption and emissions and practical difficulties for real vehicles to follow these trajectories with speed jumps. To address this issue, Kamalanathsharma et al. (2013) considered acceleration detail in optimizing an individual vehicle trajectory. Li’s team (Zhou, Li, Ma, 2017, Ma, Li, Zhou, Hu, Park, 2017) proposed a parsimonious shooting heuristic to simultaneously optimize trajectories of a stream of CAVs approaching an intersection.
Most studies on using CAVs to smooth traffic are essentially centered on a vehicle trajectory optimization problem. Simply speaking, this problem determines the optimal shapes for interdependent vehicle trajectories constrained by their boundary conditions, physical limits and safety risks. However, this problem in a general form is very complex and difficult to solve due to several computational challenges. First, each trajectory is essentially an infinite-dimensional object since every point along it can be a variable, and thus this problem deals with an infinite number of decision variables. Second, the optimization objective often involves highly non-linear components such as fuel consumption and emissions. Third, problem constraints can be quite complex due to vehicle interactions (e.g., two consecutive vehicles have to maintain a safe headway all the time) and boundary conditions (e.g., vehicles can only pass an intersection during a green light). Directly solving this problem, even a quite simple version, requires quite some computational resources and sophistication in algorithm design (Von Stryk, Bulirsch, 1992, Wei, Liu, Li, Zhou, 2016). Instead of solving the original trajectory problem, Li’s team (Zhou, Li, Ma, 2017, Ma, Li, Zhou, Hu, Park, 2017) opted to investigate a reduced problem where a trajectory is broken into a small number of quadratic sections and only a few acceleration levels are used to control the overall smoothness of the stream of vehicle trajectories. Although this reduced problem may not necessarily solve the true optimal solution to the original problem, it can yield a stream of trajectories with appealing overall smoothness and performance measures that much outperform the benchmark case without trajectory smoothing. Further, this simplification enables discovery of elegant theoretical properties and development of an efficient sub-gradient-based optimization algorithm for real-time applications.
Despite the breakthroughs from this previous work, there still remain a number of fundamental challenges in CAV trajectory optimization. First, the trajectory optimization method based on the shooting heuristic still relies on a numerical algorithm that does not ensure solution optimality and may need many iterations to converge. Second, one may intuitively think that since trajectory smoothing always leads to longer acceleration and deceleration distances for vehicles (though with milder acceleration magnitudes), it shall always yield a longer queue propagation or spillback. However, this intuition has not been systematically and analytically verified. Third, the previous work only focuses on a signalized intersection and applications in other types of highway segments remain to be investigated.
This paper aims to address these challenges by investigating a further simplified trajectory optimization model. This simplified model confines each trajectory to consist of no more than five quadratic sections. This simplification is based on the observation that a trajectory in the optimal shooting heuristic solution likely has a small number of quadratic sections, probably because more sections postulate more frequent accelerations, less smooth trajectories, and thus a sub-optimal solution. Further, this simplified model assumes that all vehicles arrive at the same speed. This is a reasonable assumption for cases when the upstream traffic is well controlled in a similar manner. While the new model preserves the main features of the shooting heuristic (e.g., yielding overall smooth trajectories) only with these minor simplifications, it has a number of appealing theoretical and algorithmic properties that were not found in the previous shooting heuristic. We discover elegant theoretical relationships between a general objective function and its associated variables and constraints. These findings enable development of an analytical solution algorithm that efficiently solves the exact solution to this simplified problem. This optimal solution has an elegant physical interpretation, i.e., stretching the trajectories as far as the deceleration parts of these trajectories are about to spill back upstream and making the acceleration and deceleration magnitudes as close as possible. This analytical exact algorithm could be consider as a core module to trajectory optimization problems on both freeway (e.g., speed harmonization, shock wave dampening, and merging control; see Ghiasi, Ma, Zhou, Li, Wang, Daamen, Hoogendoorn, van Arem, 2016, Yang, Jin, 2014, Ahn, Rakha, Park, 2013, Letter, Elefteriadou, 2017) and signalized arterial (e.g., joint signal and trajectory optimization or non-stop intersection control; see Zhou, Li, Ma, 2017, Ma, Li, Zhou, Hu, Park, 2017, Sun, Zheng, Liu, 2017, Yang, Guler, Menendez, 2016, Zohdy, Rakha, 2016, Guler, Menendez, Meier, 2014, Li, Elefteriadou, Ranka, 2014b, Kamalanathsharma, Rakha, et al., 2013) sides that have mostly relied on numerical and heuristic algorithms in the past. Numerical studies are conducted to verify the solution efficiency and quality compared with the existing approach and illustrate applications of the proposed model to signalized highways and non-stop intersections. Although the proposed longitudinal trajectory control can be directly implemented in future one-lane roadways (e.g., CAV managed lanes Ghiasi et al., 2017a), to demonstrate the robustness and applicability of the proposed model in multi-lane highway traffic, the algorithm is numerically tested on a multi-lane highway and on a stochastic vehicle arrival time setting. Further, to examine the intuitive conjecture that traffic smoothing leads to longer queue propagation, we investigate a homogeneous yet representative case and find analytical conditions for this conjecture to fail. Interestingly, we find that trajectory smoothing may not always cause longer queue propagation but instead may mitigate queue propagation with appropriate settings. This theoretical finding has valuable implications to joint optimization of queuing management and traffic smoothing in complex transportation networks. Note that the focus of this study is planning near-optimum trajectories for a platoon of vehicles with sufficient safety spacing, whereas real-time control (e.g., at the sub-second resolution) to ensure vehicles to follow the planned trajectories (while always maintaining sufficient safety gaps supported with a fail safe mechanism) is out of the scope of this study.
This paper is organized as follows. Section 2 describes the original trajectory optimization problem and proposes a simplified model. Section 3 investigates relevant theoretical properties of the simplified model and proposes an exact analytical solution algorithm accordingly. Section 4 presents numerical examples that test the solution algorithm efficiency and illustrate applications of this algorithm. Section 5 presents a homogeneous special case and analyzes the relationship between trajectory smoothing and queue propagation. Section 6 concludes this paper and briefly discusses future research directions.
Section snippets
Original formulation
This study investigates the pure CAV traffic scenario where all vehicles are controllable CAVs rather than mixed traffic including both CAVs and human-driven vehicles. Although the pure CAV traffic scenario is not likely prevailing soon, certain special infrastructures, such as managed lanes (Ghiasi et al., 2017a), may possibly be implemented at a local scale in the near future. Further, knowledge on pure CAV traffic performance provides a benchmark to the full potential of the CAV based
Solution approach
This section analyzes the structure of STO and aims to find an exact solution approach to this problem. Section 3.1 investigates certain theoretical properties on how the variable values affect the STO objective and the constraints. Based on these theoretical results, Section 3.2 proposes an exact analytical algorithm to solve the optimal solution to STO.
Numerical examples
This section conducts numerical examples to test the solution efficiency of the proposed algorithm and the application of this trajectory optimization model. Section 4.1 reports the solution times of the PSA for different instances and concludes that this proposed algorithm has appealing computational efficiency for real-time applications. The proposed trajectory optimization model can actually be applied to a general highway segment under different control strategies. For illustration
Queuing propagation analysis
Intuitively, it may be easy to arrive at a conjecture that traffic smoothing would cause vehicles queued (or slowing down) at more upstream locations, or even cause further queue spillback. This section will investigate this conjecture by rigorously analyzing a special case of the studied problem with homogeneous settings. We assume that the entry headway and the exit headway between every two vehicles is the same, i.e.,
Conclusion
This paper investigates a trajectory smoothing problem for a general one-lane highway segment with pure CAVs and provides elegant theoretical insights and efficient algorithmic methods. Inspired by previous studies from Co-author Li’s research team, this problem is simplified to one where each vehicle’s trajectory is approximated with no more than five pieces of consecutive quadratic functions and all trajectories share identical acceleration and deceleration rates in the same platoon. This
Acknowledgment
This research is supported by the U.S. National Science Foundation through Grant CMMI#1453949.
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