The real-life traffic flow process of signalized surface street networks
can be naturally decomposed into the traffic flow and the control strategy
components. The first challenge is faced when modeling the traffic flow
process on the surface streets. Specifically, the mathematical
description of traffic dynamics can be obtained by a number of models that
have a direct effect on the complexity of the corresponding optimal
control problem. Moreover, in order for the mathematical model to be
considered as part of a constraint optimization problem the constraints
must be expressed by inequalities (linear or not). However, the
consistent mathematical description of the traffic flow process inevitably
includes conditional piece-wise functions. For example, the traffic flow
at the approach of a signalized intersection is a piece-wise function
whose range depends (is conditional) on the prevailing traffic conditions
and the signal indication. Expressing this function (or others of similar
form) as a set of constraints that are additionally linear with respect to
the corresponding variables is a non-trivial task. The practices
typically followed include ignoring this function by averaging the outflow
during green over the cycle length, or approximating it with inexact
representations, or manipulating it during the solution process. These
approaches result to modeling inconsistencies and solutions of
questionable quality due to the involved heuristics. On the other hand,
there are cases of such functions that are equivalently represented by a
Mixed Integer Model (MIM) i.e., a set of inequality constraints in both
continuous and discrete variables.
The next challenge appears in designing the control strategy model.
Surprisingly, while someone would expect the modeling of the control
strategies to be driven by the currently followed state-of-the-practice,
one discovers that even the most recent modeling approaches follow the
outmoded concept of single-ring controllers with cycles of fixed duration
for a single pair of conflicting movements. Moreover, despite the fact
that the aforementioned control strategy (or its variations) is the most
widely adopted and optimized over surface street networks of comparable
dimensions, the corresponding solution time that offers a qualitative
measure of the strategy performance is not reported. Similarly, for those
control strategy models that are solved as Mixed Integer Programming
Problems no relative information on the solution time is provided.
The general objective of this dissertation is to address the issues
associated with developing a model for signalized surface street networks
and solving the corresponding optimal control problem. In order to
accomplish this goal the following specific aims are fulfilled:
1. Based on analogies from the theory of mathematical logic we develop two
methodologies for transforming conditional piece-wise functions into an
equivalent MIM representation.
2. We demonstrate the potential of both methodologies by their application
to a number of conditional piece-wise functions that are found during the
process of developing a mathematical representation for signalized surface
street networks that is based either on the dispersion-and-storage or on
the cell transmission traffic flow models. For example, we have developed
MIM representations that describe the cases of the outflow at the approach
of a signalized intersection when assuming a 2-band (Green-Red) signal, or
when assuming a 3-band (Green-Yellow-Red) signal, etc.
3. We demonstrate the capability of both methodologies in analyzing the
structure of existing MIMs, which subsequently enables us to provide
improved (in terms of the variables and the constraints) representations.
4. We develop a control strategy model that describes a dual-ring,
8-phase, variable cycle controller, and we further propose an alternative
formulation that is based again on logic-based methods that could
potentially be useful (in terms of the solution time of the corresponding
problem) within the context of a customized branch-and-bound solution
algorithm.
5. We examine the performance of our control strategy under various
hypotheses both quantitatively (solution time) and qualitatively using the
CPLEX solver that is based on a branch-and-cut solution algorithm.
This dissertation demonstrates the potential of the optimal control
approach as a powerfull tool in solving the complex problem of optimizing
traffic signals for surface street networks.
