A Non-homogeneous Time Mixed Integer LP

1 As urban traffic congestion is on the increase worldwide, it is critical to maximize capacity and 2 throughput of existing road infrastructure through optimized traffic signal control. To this end, we 3 build on the body of work in mixed integer linear programming (MILP) approaches that attempt to 4 jointly optimize traffic signal control over an entire traffic network and specifically on improving 5 the scalability of these methods for large numbers of intersections. Our primary insight in this work 6 stems from the fact that MILP-based approaches to traffic control used in a receding horizon con7 trol manner (that replan at fixed time intervals) need to compute high fidelity control policies only 8 for the early stages of the signal plan; therefore, coarser time steps can be employed to “see” over 9 a long horizon to preemptively adapt to distant platoons and other predicted long-term changes 10 in traffic flows. To this end, we contribute the queue transmission model (QTM) which blends 11 elements of cell-based and link-based modeling approaches to enable a non-homogeneous time 12 MILP formulation of traffic signal control. We then experiment with this novel QTM-based MILP 13 control in a range of traffic networks and demonstrate that the non-homogeneous MILP formula14 tion achieves (i) substantially lower delay solutions, (ii) improved per-vehicle delay distributions, 15 and (iii) more optimal travel times over a longer horizon in comparison to the homogeneous MILP 16 formulation with the same number of binary and continuous variables. 17 Guilliard, Sanner, Trevizan, and Williams 2 INTRODUCTION 1 As urban traffic congestion is on the increase worldwide with estimated productivity losses in the 2 hundreds of billions of dollars in the U.S. alone and immeasurable environmental impact (1), it is 3 critical to maximize capacity and throughput of existing road infrastructure through optimized traf4 fic signal control. Unfortunately, many large cities still use some degree of fixed-time control (2) 5 even if they also use actuated or adaptive control methods such as SCATS (3) or SCOOT (4). 6 However, there is further opportunity to improve traffic signal control even beyond adaptive meth7 ods through the use of optimized controllers (that incorporate elements of both adaptive and actu8 ated control) as evidenced in a variety of approaches including mixed integer (linear) program9 ming (5, 6, 7, 8, 9, 10), heuristic search (11, 12), queuing delay with pressure control (13) and 10 linear program control (14), to scheduling-driven control (15, 16), and reinforcement learning (2). 11 Such optimized controllers hold the promise of maximizing existing infrastructure capacity by 12 finding more complex (and potentially closer to optimal) jointly coordinated intersection policies 13 in comparison to heuristically-adaptive policies such as SCATS and SCOOT. However, optimized 14 methods are computationally demanding and often do not guarantee jointly optimal solutions over 15 a large intersection network either because (a) they only consider coordination of neighboring in16 tersections or arterial routes or (b) they fail to scale to large intersection networks simply for com17 putational reasons. We remark that the latter scalability issue is endemic to many mixed integer 18 programming approaches to optimized signal control. 19 In this work, we build on the body of work in mixed integer linear programming (MILP) ap20 proaches that attempt to jointly optimize traffic signal control over an entire traffic network (rather 21 than focus on arterial routes) and specifically on improving the scalability of these methods for 22 large urban traffic networks. In our investigation of existing approaches in this vein, namely exem23 plar methods in the spirit of (7, 9) that use a (modified) cell transmission model (CTM) (17, 18) for 24 their underlying prediction of traffic flows, we remark that a major drawback is the CTM-imposed 25 requirement to choose a predetermined homogeneous (and often necessarily small) time step for 26 reasonable modeling fidelity. This need to model a large number of CTM cells with a small time 27 step leads to MILPs that are exceedingly large and often intractable to solve. 28 Our primary insight in this work stems from the fact that MILP-based approaches to traffic 29 control used in a receding horizon control manner (that replan at fixed time intervals) need to 30 compute high fidelity control policies only for the early stages of the signal plan; therefore, coarser 31 time steps can be employed to “see” over a long horizon to preemptively adapt to distant platoons 32 and other predicted long-term changes in traffic flows. This need for non-homogeneous control in 33 turn spawns the need for an additional innovation: we require a traffic flow model that permits non34 homogeneous time steps and properly models the travel time delay between lights. To this end, 35 we might consider CTM extensions such as the variable cell length CTM (19), stochastic CTM 36 (20, 21), CTM extensions for better modeling freeway-urban interactions (22) including CTM 37 hybrids with link-based models (23), assymmetric CTMs for better handling flow imbalances in 38 merging roads (24), the situational CTM for better modeling of boundary conditions (25), and 39 the lagged CTM for improved modeling of the flow density relation (26). However, despite the 40 widespread varieties of the CTM and usage for a range of applications (27), there seems to be no 41 extension that permits non-homogeneous time steps as proposed in our novel MILP-based control 42 approach. 43 For this reason, as a major contribution of this work to enable our non-homogeneous 44 time MILP-based model of joint intersection control, we contribute the queue transmission model 45 Guilliard, Sanner, Trevizan, and Williams 3 (a) q 1 q 7 q 9 p l6 : EW NS t : NS n : 5 1 3 2 4 6 7 8 5.3 0.0 2.0 1.0 4.1 5.8 6.5 8.8 t : 0.5 1.0 2.1 1.0 1.2 0.7 2.3 d l6,EW : 4.3 0.0 1.0 2.1 0.0 3.1 4.3 4.3 4.3 d l6,NS : 0.0 0.0 1.0 1.0 1.0 1.0 0.5 1.2 3.5 0 max  0 0 max EW EW EW NS NS NS (b) FIGURE 1 (a) Example of a real traffic network modeled using the QTM. (b) A preview of different QTM model parameters as a function of non-homogeneous discretized time intervals indexed by n. For each n, we show the following parameters: the elapsed time t, the non-homogeneous time step length t, the cumulative duration d of two different light phases for l 6 , the phase p of light l 6 , and the traffic volume of different queues q linearly interpolated between time points. There is technically a binary p for each phase, but we abuse notation and simply show the current active phase: NS for north-south green and EW for east-west green assuming the top of the map is north. Here we see that traffic progresses from q 1 to q 7 to q 9 according to light phases and traffic propagation delay with non-homogeneous time steps only at required changepoints. We refer to the QTM model section for precise notation and technical definitions. (QTM) that blends elements of cell-based and link-based modeling approaches as illustrated and 1 summarized in Figure 1. The QTM offers the following key benefits: 2 • Unlike previous CTM-based joint intersection signal optimization (7, 9), the QTM is 3 intended for non-homogeneous time steps that can be used for control over large horizons. 4 • Any length of roadway without merges or diverges can be modeled as a single queue 5 leading to compact QTM MILP encodings of large traffic networks (i.e., large numbers of 6 cells and their associated MILP variables are not required between intersections). Further, 7 the free flow travel time of a link can be modeled exactly, independent of the discritizaiton 8 time step, while CTM requires a further increased discretization to approach the same 9 resolution. 10 • The QTM accurately models fixed travel time delays critical to green wave coordination 11 as in (5, 6, 8) through the use of a non-first order Markovian update model and further 12 combines this with fully joint intersection signal optimization in the spirit of (7, 9, 10). 13 In the remainder of this paper, we first formalize our novel QTM model of traffic flow 14 with non-homogeneous time steps and show how to encode it as a linear program for computing 15 traffic flows. Next we proceed to allow the traffic signals to become discrete phase variables that 16 are optimized subject to a delay minimizing objective and standard minimum and maximum time 17 Guilliard, Sanner, Trevizan, and Williams 4 constraints for cycles and phases; this results in our final MILP formulation of traffic signal control. 1 We then experiment with this novel QTM-based MILP control in a range of traffic networks and 2 demonstrate that the non-homogeneous MILP formulation achieves (i) substantially lower delay 3 solutions, (ii) improved per-vehicle delay distributions, and (iii) more optimal travel times over a 4 longer horizon in comparison to the homogeneous MILP formulation with the same number of 5 binary and continuous variables. 6 THE QUEUE TRANSMISSION MODEL (QTM) 7 A Queue Transmission Model (QTM) is the tuple (Q,L, ~ t, I), where Q and L are, respectively, 8 the set of queues and lights; ~ t is a vector of size N representing the homogeneous, or non9 homogeneous, discretization of the problem horizon [0,T] and the duration in seconds of the n-th 10 time interval is denoted as t n ; and I is a matrix |Q| ⇥ T in which I i,n represents the flow of 11 vehicles requesting to enter queue i from the outside of the network at time n. 12 A traffic light ` 2 L is defined as the tuple ( min