2018 21st International Conference on Intelligent Transportation Systems (ITSC) Maui, Hawaii, USA, November 4-7, 2018 Decentralized Optimal Control of Connected and Automated Vehicles in a Corridor Liuhui Zhao Member, IEEE, Andreas A. Malikopoulos, Senior Member, IEEE Abstract— In earlier work, we established a decentralized op- traffic operation center and received control actions from the timal control framework for coordinating online connected and center to improve network-wide traffic flow [11], [12]. automated vehicles (CAVs) in specific transportation segments, In earlier work, a decentralized optimal control framework e.g., urban intersections, merging roadways, roundabouts, and speed reduction zones. In this paper, we address coordination was established for coordinating online CAVs in different of CAVs in a corridor with multiple such scenarios and derive a transportation segments. A closed-form, analytical solution closed-form analytical solution that includes interior boundary without considering state and control constraints was pre- conditions. We evaluate the effectiveness of the solution through sented in [13], [14], and [15] for coordinating online CAVs simulation in VISSIM. The proposed approach reduces signif- at highway on-ramps, in [16] at two adjacent intersections, icantly both fuel consumption and travel time for the CAVs compared to the baseline scenario where traditional human- and in [17] at roundabouts. The solution of the unconstrained driven vehicles without control are considered. problem was also validated experimentally at the University of Delaware’s Scaled Smart City using ten robotic CAVs [18] in a merging roadway scenario. The solution of the optimal I. INTRODUCTION control problem considering state and control constraints was Urban intersections, merging roadways, roundabouts, and presented in [19] at an urban intersection without considering speed reduction zones along with the driver responses to rear-end collision avoidance constraint, and the conditions various disturbances [1] are the primary sources of bottle- under which the latter does not become active were presented necks that contribute to traffic congestion. Connectivity and in [20]. automation in vehicles provide the most intriguing oppor- In this paper, we consider coordination of a number of tunity for enabling users to better monitor transportation CAVs through a corridor. To ensure that no lateral collision network conditions and make better operating decisions. occurs, we impose interior constraints in our Hamiltonian Several research efforts have been reported in the literature analysis and derive the optimal solution throughout the entire proposing different approaches on coordinating CAVs at corridor. different transportation segments, e.g., urban intersections, The paper is organized as follows. In Section II, we merging roadways, roundabouts, and speed reduction zones, formulate the problem and provide the modeling framework. with the intention to improve transportation efficiency. In In Section III, we derive the analytical, closed form solution 2004, Dresner and Stone [2] proposed the use of the reserva- with interior constraints. In Section IV, we validate the tion scheme to control a single intersection of two roads with effectiveness of the analytical solution in a simulation envi- vehicles traveling with similar speed on a single direction on ronment and conduct a comparison analysis with traditional each road, i.e., no turns are allowed. Since then, numerous human-driven vehicles. Finaly, the concluding remarks and approaches have been reported in the literature [3]–[5], to discussion are provided in Section V. achieve safe and efficient control of traffic through intersec- II. PROBLEM FORMULATION tions including extensions of the reservation scheme in [2]. Some approaches have focused on coordinating vehicles at We consider a corridor (Fig. 1) that consists of three intersections to improve the traffic flow [6]–[8]. A detailed merging zones, e.g., two merging roadways and one urban discussion of the research efforts in this area that have been intersection. The corridor has a coordinator that can monitor reported in the literature to date can be found in [9]. the vehicles traveling along the corridor within a control Although previous research aimed at enhancing our un- zone (shown with a dashed box in Fig. 1). Note that the derstanding of improving emerging transportation systems coordinator serves as an information center which is able to just a few efforts have reported results on corridors [10] that collect vehicular data through vehicle-to-infrastructure and is include multiple intersections and merging roadways. More not involved in any decision on the vehicle operation. Road recently, a control framework was developed for a coordi- side units could be placed in each merging zone and used nated and integrated corridor management in a mixed traffic to transmit data between vehicles and the coordinator. Thus, environment, where CAVs send information to a centralized the coverage of the coordinator is flexible and the length of corridor could be extended in the presence of connected This research was supported by ARPAE’s NEXTCAR program under the infrastructure. award number DE-AR0000796. Let N(t) 2 N be the number of CAVs in the corridor at The authors are with the Department of Mechanical Engineering, Uni- + versity of Delaware, Newark, DE 19716 USA (emails: [email protected]; time t 2 R and M 2 M be the number of merging zones [email protected].) along the corridor where lateral collisions may occur. When 978-1-7281-0322-8/18/$31.00 ©2018 IEEE 1252 To ensure that the control input and vehicle speed are within a given admissible range, the following constraints are imposed. ui;min ≤ ui(t) ≤ ui;max; and 0 f (2) 0 ≤ vmin ≤ vi(t) ≤ vmax; 8t 2 [ti ; ti ]; where ui;min, ui;max are the minimum deceleration and maximum acceleration for each vehicle i = 1;:::;N(t) 2 N, and vmin, vmax are the minimum and maximum speed limits respectively. To ensure the absence of rear-end collision of two con- secutive vehicles traveling on the same lane, the position of the preceding vehicle should be greater than or equal to Fig. 1. Corridor with connected and automated vehicles. the position of the following vehicle plus a predefined safe distance δi(t). Thus we impose the rear-end safety constraint a vehicle enters the control zone of the corridor, it broadcasts 0 f si(t) = ξi · (pk(t) − pi(t)) ≥ δi(t); 8t 2 [ti ; ti ]; (3) its origin-destination (OD) to the coordinator. Then, the coordinator assigns a unique number i = 1;:::;N(t) 2 N where si(t) 2 Si denotes the distance of vehicle i from the (Fig. 1) that serves as an identification of the vehicles within vehicle k which is physically located ahead of i. Relate the the control zone. The policy through which the sequence minimum safe distance δi(t) as a function of speed vi(t), that each vehicle crosses each merging zone throughout the 0 f corridor may be the result of a higher level optimization δi(t) = γi + ρi · vi(t); 8t 2 [ti ; ti ]; (4) problem, which is not addressed in this paper. In what where γ is the standstill distance, and ρ is minimum time follows, we will adopt a specific scheme for determining i i gap that vehicle i would maintain while following another this sequence of the vehicles crosses each merging zone vehicle. throughout the corridor but we emphasize that our analysis is not restricted to this sequence. B. Policy for Vehicle Sequence Crossing the Merging Zones To avoid any possible lateral collision while the vehicles In the modeling framework described above, we assume crossing the merging zones, once a vehicle i enters the that the vehicles traveling in the corridor do not change control zone, it computes the time that will be entering each lanes except to make necessary turns. When a vehicle i merging zone which is discussed in the next Section. Let t0 i enters the control zone of the corridor it computes the time be the initial time that vehicle i enters the control zone of the mj m t for each merging zone j based on the following three corridor, t j be the time that vehicle i enters the merging i i subsets: 1) S contains all vehicles share the same route zone j, j 2 M, and tf be the time that vehicle i exits the last i i with vehicle i, 2) L contains all vehicles that travel in the merging zone along its route. In what follows, we provide a i same lane in merging zone j with vehicle i but travel from policy for the vehicle sequence crossing the merging zones different routes (i.e., may have rear-end collision with vehicle in the corridor. The focus is on the lower level control i in merging zone j, but lateral collision in the immediate problem that yields for each vehicle the optimal control input upstream merging zone j −1), and 3) C contains all vehicles (acceleration/deceleration) to achieve the assigned tmj using i i from different entry links in merging zone j that may have a given policy that designates tmj (upon arrival of CAV i at i lateral collision with vehicle i. the entry of the control zone). mj To ensure that (3) is satisfied at ti , we first impose the A. Vehicle Model and Constraints following condition For simplicity, each vehicle is modeled as a double inte- m m p (t) p (t) p (t) t j = max min t j +ρ ; j +t0 ; j +t0; j +t0 ; grator i k i i 0 i i vmin vi vmax (5) p_i = vi(t) mj (1) where tk is the time when the vehicle k enters merging v_i = ui(t) zone j, k 2 Si, pj(t) is the distance from the entry point of 0 where pi(t) 2 Pi, vi(t) 2 Vi, and ui(t) 2 Ui denote the the control zone until the entry of merging zone j, and vi is position, speed and acceleration/deceleration (control input) the initial speed of vehicle i when it enters the control zone of each vehicle i in the corridor.