The dynamical system approach to traffic assignment : the attainability of equilibrium and its application to traffic system management
by Bie Jing
Ph.D. Civil Engineering
xv, 168 leaves : ill. ; 30 cm
In this thesis we formulate the traffic assignment problem through a dynamical system approach. All exogenous factors are presumed to be constant over time and equilibrium is being pursued through a day-to-day learning process. Travellers’ knowledge of the network is represented by their perceived costs on individual routes. Route choice on a day is determined by the perceived route costs on that day. If the actual travel costs are identical to the perceived costs, then equilibrium is achieved; otherwise, travellers update their perceived costs for the route choice next day....[ Read more ]
In this thesis we formulate the traffic assignment problem through a dynamical system approach. All exogenous factors are presumed to be constant over time and equilibrium is being pursued through a day-to-day learning process. Travellers’ knowledge of the network is represented by their perceived costs on individual routes. Route choice on a day is determined by the perceived route costs on that day. If the actual travel costs are identical to the perceived costs, then equilibrium is achieved; otherwise, travellers update their perceived costs for the route choice next day.
The traffic dynamics can be formulated by a recurrence function of the vector of perceived costs. Fixed point in the dynamical system is equivalent to the stochastic user equilibrium in the static model. Equilibrium stability is analysed by a linearization of the dynamical system around equilibrium. Stability requires that the corresponding Jacobian matrix, when evaluated at the equilibrium point, has its eigenvalues (real or complex) all within the unit circle. Lyapunov function can also be utilized in stability analysis. Stability is important because unstable equilibrium is transient. In cases of instability, it is usually possible to shift the unstable equilibrium to a stable equilibrium by appropriately modifying the network.
Even for stable equilibrium, only points within its attraction basin are attracted to the equilibrium. Some topological analysis shows that the attraction basin of a stable equilibrium is always open. Furthermore, if all points in the state space are attracted to equilibria, one or another, then the boundary of the attraction basin for a stable equilibrium is formed by trajectories to unstable equilibria. Therefore, we can identify the exact range of attraction basins for stable equilibria by tracing back the dynamical evolution to unstable equilibria. Once this is done, the state space can be partitioned into a number of subsets, each representing the attraction basin of an equilibrium point. On this partition chart, given an initial point, the limit point of its trajectory is immediately known. If the exact range is not required, an estimate of the attraction basin can be generated by employing the Lyapunov function and performing an envelope search. The direction-based search method can also be used to obtain a rough approximation of the attraction basin.
Equilibrium is unattainable from points outside its attraction basin. In such cases, we can temporarily change the network and influence the direction of the traffic dynamics. If the temporary network alteration is properly done, after restoring the original network the dynamical evolution now at a different position may well converge to equilibrium. This transitional attainability of equilibrium enables traffic management agencies to direct traffic flow to the desired pattern, especially when there are multiple equilibrium points.
The impact of network changes is diverse; so are its implications for traffic management. If the traffic dynamics is not well understood, an improvident network modification may result in unintended consequences. In particular, we show that higher perception precision and reduction of travel cost may not help the system achieve a better equilibrium. Quite often the network modification may destabilize the traffic dynamics. It is then advisable to implement the modification in a sequential way, ensuring a smooth transition in each step and eventually leading to the modified network as planned.
The most important implication of network change for traffic system management is that temporary network change may have long term effects on the state of the system, a property which we call irreversibility. Particular attention should then be paid to irreversible temporary changes, whether planned or incidental. These changes, even though imposed on the system only temporarily, can permanently relocate the state of the system. Therefore it is essential that the planned changes should be well studied in advance and then carefully implemented, while remedies should be carried out for irreversible incidental changes.
Besides equilibrium, there are also cyclic and chaotic attractors. These non-equilibrium attractors share similar characteristics as equilibrium in terms of being the limit set of a trajectory. A cyclic attractor with a period of n days is an equilibrium point of the n-days-to-n-days mapping. Studying these attractors may help us understand the non-stationary flow in observed day-to-day traffic data. We also provide criteria for the case where equilibrium is the only type of attractor. When such criteria are satisfied we can eliminate the possibility of non-equilibrium attractors and focus instead on equilibrium attractors.
As an extension of the dynamical system approach which models the day-to-day traffic as a deterministic process, we also formulate the day-to-day evolution of traffic flow as a stochastic process. The stochastic process approach is more realistic in the sense that travellers’ route choice (and therefore traffic flow) over successive periods of similar network characteristics is not stationary but the realization of a stochastic process. The concept of equilibrium in the deterministic model is replaced by a stationary probability distribution over a collection of states.
In summary, the concept of dynamic equilibrium is formulated to replace the traditional static equilibrium, which only concerns the state of equilibrium. The dynamical approach in this thesis addresses the process of pursuing and obtaining equilibrium, i.e. how disequilibrium states evolve towards equilibrium. It enables the analysis on equilibrium stability and attainability. In particular, this thesis shows how the equilibrium’s attraction basin can be determined or estimated. Evolution starting from outside the attraction basin does not converge to the equilibrium. A temporary network alteration can then be made to divert the evolution to equilibrium. This implies that changes on the network, even temporary, can have long term effects on the system state. To avoid undesirable consequences, traffic management agency should assess the impact of planned network modification before implementing it.