Urban road traffic network control using traffic lights influences the route-choice of individual road users [Taale and van Zuylen, 2001]. The reason for this is that traffic lights influence the travel times on road sections and therewith the total travel times of different routes between an origin and a destination. The implication of this effect is that an urban traffic control strategy, that is designed based on knowledge of current and historical flows, may get out of date and become less efficient over time due to the changed route choice of road users when they get acquainted with the traffic dynamics.
The resulting equilibrium in route choice that appears in this way is commonly referred to as the user equilibrium [Wardrop and Whitehead, 1952]. The travel time on every used route from an origin to a destination is identical in the user equilibrium, and it is smaller than the travel time of the unused routes.
The efficiency loss of the signal controllers due to route choice may even occur quicker in the near future due to the proliferation of in-car technologies, such as, GPS naviga-tion devices with which more and more vehicles are equipped. An even stronger effect may be expected when automated vehicles that automatically navigate from origin to destination become widely available. Such systems may choose and adopt the best route for the individual road user based on knowledge of the current traffic situation, possibly combined with a prediction of the evolution of the traffic state over time. This may cause an even faster degradation of the overall network performance. Hence, an urban traffic control strategy has to take into account the impact of its control action onto the route choice, potentially leading to a more efficient user equilibrium or it has to explicitly control the route choice behavior so that a system optimum can be achieved.
In general, system optimality may imply that alternative (used) routes between a given origin-destination pair have different travel times. This implies that some vehicles may have shorter travel times compared to others so that the average travel time experienced by all the road users is optimal. Currently most drivers minimize their individual cost by choosing the route with the lowest travel time (if known), but in the future incentives may be given by monetary tolls or rewards for drivers for choosing the route that leads to the system optimum [Pigou, 2013]. In such systems the drivers will still minimize their individual costs, but now for the generalized (combined) monetary and travel time
costs. In a dynamic setting, pricing is discussed in the bottleneck model by Vickrey [1969]. A more operational incentive can be tradable driving rights [Xiao et al., 2013].
A common approach to optimize the network performance is predictive control. An advantage of a predictive control action is that it allows to account for the future impact of control actions. This is useful in a traffic network, since, changing the flows or route choices at one intersection affects other intersections at a later time instant. One of the major challenges of predictive control is that it may lead to a computationally complex optimization problem. This is a challenge because the computation time required to solve the optimization problem must be smaller than the real-time controller sampling time.
Considering the desired aspects of traffic control, the aim of this paper is the design of a control strategy that:
• optimizes the throughput of an urban road traffic network over a time horizon
• controls the average traffic flows at intersections which may be realized by traffic lights and the route choice of the traffic
• requires limited computation time, more specifically, the computation time has to be less than the real-time controller sampling time which is in the range of (several) minutes
Before detailing the research approach and contributions, first Section 5.1.1 discusses approaches to the combined dynamic traffic assignment and signal control problem.
After that, Section 5.1.2 details a specific sub-set of approaches, namely model-based optimization approaches.
5.1.1 Approaches to the combined dynamic traffic assignment and signal control problem
The problem of accounting for the impact of traffic signal control on route choice has drawn research attention for several decades. Already in 1974, Allsop [1974] dis-cussed the interaction effect of signal control and route choice. Taale and van Zuylen [2001] presented an overview of the literature on the combined traffic assignment and control problem. The approaches to solve the combined traffic assignment and con-trol problem can be divided into three categories, namely, 1) iterative procedures, 2) global optimization approaches, and 3) game-theoretic approaches that intend to solve the global optimization problem [Smith, 1985, Taale and van Zuylen, 2001].
In iterative approaches, the traffic signals settings are optimized for a given demand pattern. Next, the route choice of drivers is updated for the optimized signal settings, leading to a new demand pattern, and the process is repeated until convergence [Allsop and Charlesworth, 1977, Akc¸elik and Maher, 1977]. A comparison of several studies
that focused on the convergence properties of the iterative approach by Taale and van Zuylen [2001] indicated that a challenge of iterative approaches may be that they are not guaranteed to converge to a stable optimum despite the potential computation time gain.
Global optimization approaches address this issue by jointly optimizing the route choice and traffic signal settings [Taale and van Zuylen, 2001, Chen and Hsueh, 1997]. Due to the necessity to predict the impact of the traffic signals on the link travel times, a traffic assignment model is used. Hence, most of the combined traffic assignment and control approaches are of the model predictive control type. Recent developments in the area of model-based optimization approaches are discussed in the next section.
The first game-theoretic approaches to solve the optimization problem were presented in the 1980’s [Fisk, 1984]. The idea is that road users and traffic managers can be mod-eled as different decision makers that have different objectives [Gartner et al., 1980, Taale and van Zuylen, 2001]. Chen [1998] proposed a dynamic modeling framework where control strategies and assignment can be combined. The advantage of using game theory is that it does not require the explicit use of the evaluation of computation-ally expensive prediction models while still being able to realize similar performance according to Taale and van Zuylen [2001].
5.1.2 Model-based optimization approaches
A common approach to predict and optimize the (future) impact of the control action onto the overall network performance is model-based optimization, commonly known as model predictive control (MPC). Several researchers have proposed model-based optimization strategies for the combined dynamic traffic assignment and control prob-lem.
Taale and Hoogendoorn [2013] proposed a framework for real-time integrated and anticipatory traffic management that is somewhere in between iterative and global op-timization approaches. The framework is similar to iterative approaches, but when optimizing the signal settings, the impact of the control settings on the route choice behavior is explicitly considered using a traffic flow model. In this way, better con-vergence is expected. The computation time of the iterative procedure may still be high. Abdul Aziz and Ukkusuri [2012] used the cell transmission model (CTM) as a prediction model in an MPC framework for optimization of the signal settings assum-ing system-optimal route choice behavior. The authors optimize the phase selection using a mixed-integer linear programming problem (MILP) but they do not optimize the route choice. Challenges of this approach are that the computation time is high due to the use of the MILP and that the linear formulation leads to violation of the first-in-first-out (FIFO) principle for the different destination-oriented flows in the cells.
Le et al. [2013] proposed an optimization approach based on a multi-class variant of the CTM. This means that it is similar to the CTM, except that the shock wave speed
of spillback is not modeled. The authors aggregate the traffic dynamics to (several) tens of seconds so that the discontinuous nature of the signal settings does not have to be considered. In this way, the authors are able to formulate a quadratic program-ming problem to optimize the flows at the intersections and the destination-oriented turn rates. Due to the use of (a variant of) the CTM in combination with a quadratic optimization problem, the approach may – similarly as in Abdul Aziz and Ukkusuri [2012] – violate the FIFO principle. Hence, the approach will only provide correct results when applied to networks with a single destination. Another weakness of the model used is that it does not reproduce the shock wave speed of spillback, leading to a performance degradation in oversaturated traffic conditions [van de Weg et al., 2016].
Li et al. [2015] optimize the route guidance and traffic signal settings using a space-phase-time hyper network. The authors decompose the problem into two sub-problems with different properties. Hence, the traffic signal optimization problem is optimized based on a phase-time network considering aggregated dynamics. The route guidance for individual vehicles is solved based on the link travel times. This procedure is re-peated until convergence. An advantage of this work is the level of detail considered – i.e., the explicit inclusion of signal timings, and control of route decisions of indi-vidual road users. However, this also leads to a very complex model that runs with a resolution of 1 second.
This brief overview shows that there exist different model-based optimization ap-proaches, each having its own advantages and challenges. However, to the best knowl-edge of the authors, an approach that can optimize the throughput in all traffic regimes – i.e. the undersaturated, saturated, and oversaturated regimes – using a computation-ally efficient optimization procedure does not exist yet. In this paper, a link is in the undersaturated regime when the queue can fully clear when given green, it is in the saturated regime when the queue does not clear when given green and neither spills back to upstream intersections, and it is in the oversaturated regime when the queue that spills back and cause blocking of upstream intersections.
5.1.3 Research approach and contributions
The aim of this paper is to develop an algorithm for the joint optimization of intersec-tion flows and route decisions that is computaintersec-tionally efficient and is able to improve the network throughput in all traffic regimes. In order to reach this goal, the following simplifications are made. First of all, a 100% compliance rate to the optimized route choice is assumed. In practice, this may be realized using, for instance, (monetary) incentives as discussed above. Second, the traffic dynamics are aggregated to several (tens of) seconds. Due to this, the intersection flows are continuous and signal timings are not explicitly considered so that the objective function of the optimization problem is differentiable everywhere with respect to the control variables. Hence, gradient-based solvers can be used, which are generally faster when compared to gradient-free
solvers when the problem size is not too large. Third, it is assumed that the origin-destination demands are known and that no noise or uncertainties affect the system.
Hence, in practice a module may be needed to predict the demand. It must be noted that both the quality of the optimization and the demand predictions may influence the controller performance. Because this paper aims at studying the quality of the op-timization, simulations are carried out in a controlled environment that are primarily focused on investigating the impact of the controller on the throughput and computa-tion time.
In the light of these design considerations, this paper proposes an efficient model pre-dictive control strategy based on the link transmission model (LTM) for the combined optimization of intersection flows and route choice. The LTM is used because it is a more computationally efficient model when compared to segment-based models, such as the CTM, as shown in van de Weg et al. [2016]. Due to the non-linear nature of the optimization problem, an efficient optimization algorithm of the sequential linear programming type is proposed [Marcotte and Dussault, 1989]. The idea behind the algorithm is that first the non-linear model is used to predict the traffic state trajec-tories for a candidate control signal. Based on that prediction, an analytic approxi-mation of the model linearization is determined which is used to formulate a linear optimization problem which is solved, giving a new control signal. Next, the candi-date control signal and the optimized control signal are used as a search direction in a line-search optimization algorithm giving a new candidate control signal. Finally, it is tested whether the control signal found satisfies the stopping criteria. If not, the process is repeated. Compared to the research discussed above, advantages are the use of the LTM, the improved, analytic approximation of the model linearization, and the use of the line-search algorithm.
The contributions of this paper are:
• Design of an MPC strategy using the LTM for optimization of both intersection flows and routing decisions (Section 5.3)
• Design of an efficient optimization algorithm based on analytic approximation of the model linearization (Section 5.3)
• Formulation of an analytic approximation of the model linearization around an operating point (Section 5.3.3)
• Evaluation of the approach in terms of the trade-off between realized throughput and CPU time used by comparing with different variants of the SLP algorithm and a numerical linearization based optimization algorithm within a simulation environment (Section 5.4)
The paper is structured as follows. First, Section 5.2 details the traffic flow prediction model that is used. Second, Section 5.3 introduces the optimization algorithm and the
analytic linearization of the model. Section 5.4 evaluates the approach using simula-tions. Section 5.5 concludes the paper. 5.B provides an overview of the variables used in the paper.