A Control Theoretic Approach to Simultaneously Estimate Average Value of Time and Determine Dynamic Price for High-occupancy Toll Lanes

TL;DR

The paper addresses dynamic pricing for HOT lanes under uncertainty in driver value of time (VOT) by proposing a simple residual-capacity based point-queue model and a control-theoretic scheme to simultaneously estimate the average VOT $\pi(t)$ and compute the dynamic price $u(t)$. The core idea is an I-like estimator $\dot{\pi}(t)=K_1\lambda_1(t)-K_2\zeta(t)$ paired with $u(t)=\pi(t)w(t)+\frac{\ln\frac{q_1+q_2-C_1}{C_1-q_1}}{\alpha^*}$, ensuring convergence to the optimal state where $\lambda_1=0$ and $\zeta=0$, with Gaussian or exponential convergence depending on parameters. The authors provide analytical stability proofs and extensive numerical experiments showing robustness to stochastic demand and logit-model perturbations, demonstrating improved performance over Yin and Lou’s methods. The approach offers a practical, provably stable mechanism for real-time HOT-lane pricing that aligns with system-wide objectives and can adapt to uncertain driver behavior. The results have meaningful implications for congestion pricing implementation and dynamic toll design in real-world corridors.

Abstract

The dynamic pricing problem of a freeway corridor with high-occupancy toll (HOT) lanes was formulated and solved based on a point queue abstraction of the traffic system [Yin and Lou, 2009]. However, existing pricing strategies cannot guarantee that the closed-loop system converges to the optimal state, in which the HOT lanes' capacity is fully utilized but there is no queue on the HOT lanes, and a well-behaved estimation and control method is quite challenging and still elusive. This paper attempts to fill the gap by making three fundamental contributions: (i) to present a simpler formulation of the point queue model based on the new concept of residual capacity, (ii) to propose a simple feedback control theoretic approach to estimate the average value of time and calculate the dynamic price, and (iii) to analytically and numerically prove that the closed-loop system is stable and guaranteed to converge to the optimal state, in either Gaussian or exponential manners.

Figures (16)

• Figure 1: Exponential increase in the number of HOT lanes in the U.S.
• Figure 2: Illustration of the traffic system with HOT lanes
• Figure 3: Point queue representation of the traffic system with HOT lanes
• Figure 4: Block diagrams of two methods in yin2009
• Figure 5: Block diagram of the control system

Theorems & Definitions (2)

• Lemma 2.1
• Theorem 6.1