This paper proposes a dynamic process to formulate the day-to-day evolution process of traffic flows, which is called “proportional-switch adjustment process with elastic demand (PSAP-ED)”. PSAP-ED extends Smith’s process (Smith in Transp Sci 18:259–304, 1984) from the case of fixed demand to the case of elastic demand and characterizes the process of approaching equilibrium states from non-equilibrium states in traffic networks. Several desirable properties of PSAP-ED are provided, such as the equivalence between user equilibrium and the stationary state of PSAP-ED, the existence and uniqueness of the stationary flow state, and the convergence of PSAP-ED. We also show that a proper fixed toll scheme can drive PSAP-ED to evolve towards system optimum states. When the demand functions are unknown, the trial-and-error procedure presented by Yang et al. (Transp Res B 38:477–493, 2004) can be modified and applied in the context of PSAP-ED. Furthermore, we develop a more efficient dynamic congestion toll scheme for PSAP-ED in the absence of demand functions. The properties of PSAP-ED and the effectiveness of toll schemes are demonstrated by a numerical example.

本文提出了一种动态过程来表述交通流的日常演化过程，称为“具有弹性需求的比例转换调整过程（PSAP-ED）”。PSAP-ED 将 Smith 的过程（Smith in Transp Sci 18:259-304, 1984）从固定需求的情况扩展到弹性需求的情况，并描述了交通网络中从非平衡状态接近平衡状态的过程。提供了PSAP-ED的几个理想性质，如用户平衡与PSAP-ED静止状态的等价性、静止流动状态的存在性和唯一性、PSAP-ED的收敛性等。我们还表明，适当的固定收费方案可以推动 PSAP-ED 向系统最佳状态发展。当需求函数未知时，Yang等人提出的试错程序。(Transp Res B 38:477–493, 2004) 可以在 PSAP-ED 的上下文中进行修改和应用。此外，我们在没有需求函数的情况下为 PSAP-ED 开发了一种更有效的动态拥堵收费方案。PSAP-ED 的特性和收费方案的有效性通过一个数值例子来证明。