M. Hajiahmadi, J. Haddad, B. De Schutter, and N. Geroliminis, "Optimal hybrid perimeter and switching plans control for urban traffic networks," IEEE Transactions on Control Systems Technology, vol. 23, no. 2, pp. 464-478, Mar. 2015.
Since centralized control of urban networks with detailed modeling approaches is computationally complex, developing efficient hierarchical control strategies based on aggregate modeling is of great importance. The dynamics of a heterogeneous large-scale urban network is modeled as R homogeneous regions with the macroscopic fundamental diagrams (MFDs) representation. The MFD provides for homogeneous network regions a unimodal, low-scatter relationship between network vehicle density and network space-mean flow. In this paper, the optimal hybrid control problem for an R-region MFD network is formulated as a mixed integer nonlinear optimization problem, where two types of controllers are introduced: (i) perimeter controllers, and (ii) switching signal timing plans controllers. The perimeter controllers are located on the border between the regions, as they manipulate the transfer flows between them, while the switching controllers influence the dynamics of the urban regions, as they define the shape of the MFDs and as a result affect the internal flows within each region. Moreover, to decrease the computational complexity due to the nonlinear and non-convex nature of the optimization problem, we reformulate the problem as a mixed integer linear programming (MILP) problem utilizing piecewise affine approximation techniques. Two different approaches for transformation of the original model and building up MILP problems are presented, and the performances of the approximated methods along with the original problem formulation are evaluated and compared for different traffic scenarios of a two-region urban case study.