Traffic Signal Control of a Road Network Using MILP in the MPC Framework

Abstract

This paper investigates the significance of a traffic signal control scheme that simultaneously adjusts all signal parameters, i.e., cycle time, split time and offset, in a road network. A novel framework of model predictive control (MPC) is designed that overcomes the limitations of other MPC based traffic signal control strategies, which are mostly restricted to control only split or green time in a fixed cycle ignoring signal offset. A simple macroscopic model of traffic tailored to MPC is formulated that describes traffic dynamics in the network at a short sampling interval. The proposed framework is demonstrated using a small road network with dynamically changing traffic flows. The parameters of the proposed model are calibrated by using data obtained from detailed microscopic simulation that yields realistic statistics. The model is transformed into a mixed logical dynamical system that is suitable to a finite horizon, and traffic signals are optimized using mixed integer linear programming (MILP) for a given performance index. The framework makes the signals flexibly turn to red and green by adapting quickly to any changes in traffic conditions. Results are also verified by microscopic traffic simulation and compared with other signal control schemes.

Appendix

1.1 6 Mixed Logical Dynamical Systems

Many real world problems, such as traffic control of an urban network considered in this paper, fall in the class of hybrid dynamical systems that involve the interaction of continuous dynamics and discrete dynamics. The nonlinear nature of the prediction model of hybrid systems, due to inclusion of discrete dynamics as finite state machines, makes most optimization problems difficult to solve in real-time. The framework of MLD systems is effective to transform dynamics, logic and constraints of such a system into an integrated model [28]. The transformation into the MLD systems allows us to develop numerically tractable solution schemes for hybrid dynamic optimization problems as mixed-integer linear and quadratic programs, which can be efficiently solved by the existing solvers [29].

In general, there are three basic steps to derive the MLD form of a hybrid system, as proposed in [28]. The first step is to associate a binary variable, say δ∈{0,1}, with a logical statement, say f(x) ≤ 0, that can be either true or false, in such a way that δ = 1 if and only if the statement holds true. The association $[f(x) \le 0]\Leftrightarrow [\delta =1]$ can be equivalently stated by the following inequalities

$$ \begin{cases} f(x) \le f_{max}(1-\delta),\\ f(x) \ge \varepsilon + (f_{min}-\varepsilon)\delta, \\ \end{cases} $$

(16)

where f _max , f _min and ε are the upper or maximum bound of f, the lower or minimum bound of f and a small positive number expressing machine tolerance, respectively. The second step is to replace the product of a linear function and a binary variable by introducing an auxiliary real variable z = δf(x), which satisfies $[\delta = 0]\Rightarrow [z=0]$ and $[\delta = 1]\Rightarrow [z = f(x)]$. Equivalently, z = δf(x) is uniquely specified through the four mixed integer linear inequalities given by

$$ \begin{cases} z \le f_{max} \delta ,~~~~~\\ z \ge f_{min} \delta,\\ z \le f(x)-f_{min} (1-\delta) ,~~~~~\\ z \ge f(x) -f_{max} (1-\delta). \end{cases} $$

(17)

The final step is to obtain the MLD system by including binary and auxiliary variables in a linear time-invariant (LTI) discrete time dynamic system and combining equalities and inequalities derived from the hybrid system, which in general can be expressed as

$$\begin{array}{@{}rcl@{}} x(k+1)=\mathbf{A}x(k)+\mathbf{B}_{1}u(k)+\mathbf{B}_{2}\delta(k)+\mathbf{B}_{3}z(k), \end{array} $$

(18)

$$\begin{array}{@{}rcl@{}} \mathbf{E}_{2}\delta(k)+\mathbf{E}_{3}z(k)\le \mathbf{E}_{1}u(k)+\mathbf{E}_{4}x(k)+\mathbf{E}_{5},~~~~~ \end{array} $$

(19)

where $x \in \mathbb {R}^{n_{c}} \times \{0,1\}^{n_{l}}$ is a vector of continuous and binary states, $u \in \mathbb {R}^{m_{c}} \times \{0,1\}^{m_{l}}$ is an input vector, δ∈{0,1}^r _l is an auxiliary binary vector and $z \in \mathbb {R}^{r_{c}}$ is a auxiliary continuous variable, and A, B ₁, B ₂, B ₃, E ₁, E ₂, E ₃, E ₄, and E ₅ are matrices and a vector of suitable dimensions.