Decongestion of Urban Areas with Hotspot Pricing

Abstract

The rapid growth of population in urban areas is jeopardizing the mobility and air quality worldwide. One of the most notable problems arising is that of traffic congestion which in turn affects air pollution. With the advent of technologies able to sense real-time data about cities, and its public distribution for analysis, we are in place to forecast scenarios valuable to ameliorate and control congestion. Here, we analyze a local congestion pricing scheme, hotspot pricing, that surcharges vehicles traversing congested junctions. The proposed tax is computed from the estimation of the evolution of congestion at local level, and the expected response of users to the tax (elasticity). Results on cities’ road networks, considering real-traffic data, show that the proposed hotspot pricing scheme would be more effective than current mechanisms to decongest urban areas, and paves the way towards sustainable congestion in urban areas.

Appendix: Optimal Traffic Redistribution Given Fixed Revenue

We want to compute the maximum fraction of vehicles, 1 − ϕ _i , that will avoid the junctions with the hotspot pricing by fixing the overall tax income for the city $\mathcal {P}$. This may happen when local authorities want to fix the economic effort of the drivers to improve the traffic conditions. This is equivalent to the following minimization problem:

$$ \min_{\{\phi_{i}\}}\left( \sum\limits_{i} \phi_{i} \sigma_{i} \right)\ \ \text{s.t.}\ \ \ \sum\limits_{i} \phi_{i} \sigma_{i} c_{i} =\frac{c_{0}}{\phi_{0}^{1/\mu}} \sum\limits_{i} \phi_{i}^{(\mu + 1)/\mu} \sigma_{i} = \mathcal{P}\,, $$

(5)

where σ _i is the amount of cars junction i receives before the taxing, c ₀ is the initial price to obtain a reduction of ϕ ₀, μ is the elasticity and c _i is defined in Eq. 4. The linear problem stand for the remaining cars that will cross the congested junctions after the taxing is applied and the restriction stands for the overall income produced by those cars.

We solve the minimisation problem using Lagrange multipliers. The objective function is

$$ L(\phi_{i},\lambda) = \sum\limits_{i} \phi_{i} \sigma_{i} - \lambda \left( \sum\limits_{i} {\phi_{i}}^{k_{2}}{\sigma_{i}} - k_{1} \right)\,, $$

(6)

where $k_{1} = \frac {\mathcal {P} \phi _{0}^{1/\mu }}{c_{0}}$ and $k_{2} = \frac {\mu +1}{\mu }$. Setting the gradient ∇L({ϕ _i },λ) = 0 we have:

$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \phi_{j}} &=& \sigma_{j} - \lambda k_{2} {\phi_{j}}^{k_{2}-1}\sigma_{j} = 0 \qquad\Longrightarrow \phi_{j} = \left( \frac{1}{\lambda k_{2}}\right)^{\frac{1}{k_{2}-1}} \end{array} $$

(7)

$$\begin{array}{@{}rcl@{}} \frac{\partial L}{\partial \lambda} &=& \sum\limits_{i} {\phi_{i}}^{k_{2}}\sigma_{i} - k_{1} = 0 ~\qquad\quad\Longrightarrow \sum\limits_{i} {\phi_{i}}^{k_{2}} \sigma_{i} = k_{1} \,. \end{array} $$

(8)

From Eq. 7 we see that all the ϕ _j are equal, i.e. independent of the node. Substituting Eqs. 7 into 8 we can obtain λ. Specifically,

$$\begin{array}{@{}rcl@{}} \left( \frac{1}{k_{2}\lambda}\right)^{\frac{k_{2}}{k_{2}-1}} \sum\limits_{i} {\sigma_{i}} = k_{1}\ \Longrightarrow \ \lambda = {k_{2}}^{-1} \left( {\frac{k_{1}}{\sum\limits_{i} \sigma_{i}}}\right)^{\frac{1-k_{2}}{k_{2}}}\,, \end{array} $$

(9)

which yields an homogeneous fraction

$$\begin{array}{@{}rcl@{}} \phi= \left( \frac{\mathcal{P} {\phi_{0}}^{1/\mu}}{c_{0} \sum\limits_{i} \sigma_{i}}\right)^{\frac{\mu}{\mu+1}}\,. \end{array} $$

(10)

The local reduction to be applied is given by Eq. 10, and the tax to apply to every congested junction is

$$\begin{array}{@{}rcl@{}} c = \left( {\frac{c_{0}^{\mu}\mathcal{P}}{\phi_{0}{\sum}_{i} \sigma_{i}}}\right)^{1/(\mu+1)}\,. \end{array} $$

(11)