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Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model

Shimao Fan, Michael Herty, Benjamin Seibold

arXiv:1310.8219v2physics.soc-phcs.NAmath.NAphysics.comp-ph

TL;DR

Macroscopic traffic models have had limited validation with traffic data, and ARZ drivers can have unrealistically varied stagnation densities. The paper fits GARZ models to historic fundamental-diagram data and compares predictions in a three-detector setup. GARZ yields the most accurate predictions overall, while optimal relaxation times are reported as 14–60 seconds, with some values larger.

  • Problem

    Macroscopic models have seen little validation with traffic data, and ARZ's different driver classes have vastly varying stagnation densities, although maximum density is a road property.

    Source: §1 · p. 2, §3.1 · p. 10

  • Method

    The paper fits GARZ parameter functions using historic fundamental-diagram data and compares macroscopic models with trajectory and sensor data in a three-detector test.

    Source: §1 · p. 2, §7 · p. 26

  • Results

    GARZ yields the most accurate predictions and best reproduces real data; optimal relaxation times lie in the range 14–60 seconds, with some values even larger.

    Source: §7 · p. 26

  • Takeaways & Limitations

    GARZ addresses ARZ's multiple-stagnation-density drawback and gives the most accurate predictions in the reported comparisons.

    Source: §3.1 · p. 10, §7 · p. 26

    For bounded-domain initial-boundary problems, inhomogeneous GARZ solutions do not in general converge to LWR solutions as τ → 0.

    Source: §2.5.3 · p. 8

Abstract

from arXiv · show

The Aw-Rascle-Zhang (ARZ) model can be interpreted as a generalization of the Lighthill-Whitham-Richards (LWR) model, possessing a family of fundamental diagram curves, each of which represents a class of drivers with a different empty road velocity. A weakness of this approach is that different drivers possess vastly different densities at which traffic flow stagnates. This drawback can be overcome by modifying the pressure relation in the ARZ model, leading to the generalized Aw-Rascle-Zhang (GARZ) model. We present an approach to determine the parameter functions of the GARZ model from fundamental diagram measurement data. The predictive accuracy of the resulting data-fitted GARZ model is compared to other traffic models by means of a three-detector test setup, employing two types of data: vehicle trajectory data, and sensor data. This work also considers the extension of the ARZ and the GARZ models to models with a relaxation term, and conducts an investigation of the optimal relaxation time.

1. Introduction

Macroscopic traffic models have seen little validation with traffic data compared with the widespread data use in microscopic and cellular models. This paper fits traffic models to historic fundamental-diagram data and tests their predictions using time-dependent data.

1. Introduction

  • The paper validates data-fitted first- and second-order macroscopic models against time-dependent trajectory and sensor data in a three-detector setup.The comparisons focus on second-order models, with first-order models included mainly as comparisons.Source: §1 · p. 2
  • The work designs the generalized Aw-Rascle-Zhang model, analyzes its mathematical properties, and systematically fits first- and second-order models from historic fundamental-diagram data.Source: §1 · p. 2
  • The validation compares predictive accuracy and traffic-state features, with PDE numerical errors kept negligibly small relative to model errors.The study also examines optimal relaxation time in second-order models with a relaxation term.Source: §1 · p. 2

2. Existing and New Macroscopic Traffic Models

Macroscopic traffic models share a continuity equation that conserves vehicles. Its fields represent density and velocity, aggregated across lanes when a road has multiple lanes.

2. Existing and New Macroscopic Traffic Models

  • The continuity equation is common to all macroscopic traffic models and expresses vehicle conservation.Source: §2 · p. 2
  • In the equation, ρ(x, t) is vehicle density and u(x, t) is the velocity field, with x denoting road position and t time.Source: §2 · p. 2
  • For roads with multiple lanes in one direction, the model aggregates lanes into scalar fields ρ and u.Source: §2 · p. 3

2.1. The Lighthill-Whitham-Richards Model and Flow Rate Functions.

The LWR model closes the traffic equations by relating velocity to density, yielding a first-order model with a flow-rate function. The paper also motivates second-order models, whose flow-rate choices face additional constraints.

2.1. The Lighthill-Whitham-Richards Model and Flow Rate Functions

  • LWR assumes velocity is a function of density, u = U(ρ), and defines flow rate as Q(ρ) = ρU(ρ), producing a first-order scalar conservation law.Source: §2.1 · p. 3
  • For second-order models, velocity U(ρ) must not be constant, since that would lose hyperbolicity; this rules out the Newell-Daganzo flux.The paper instead uses strictly concave flow-rate functions resembling its shape.Source: §2.1 · p. 3
  • Second-order models add an evolution equation for velocity; Aw-Rascle-Zhang pressure was proposed to remedy unrealistic solutions from the Payne-Whitham pressure.Source: §2.2 · p. 3

2.2. The Aw-Rascle-Zhang Model.

The ARZ model adds a velocity evolution equation and can include relaxation toward a desired velocity. Its homogeneous form has a limitation for all-stopped initial traffic, while the inhomogeneous form can yield more realistic predictions.

2.2. The Aw-Rascle-Zhang Model

  • The hesitation function h(ρ) is assumed to have h′(ρ) > 0 and h(0) = 0; adding relaxation yields the inhomogeneous ARZ model.Source: §2.2 · p. 3
  • The equilibrium velocity Ueq(ρ) is the desired velocity, and τ is the relaxation time scale.Source: §2.2 · p. 3
  • The homogeneous ARZ model has no mechanism to make drivers move from an all-stopped state, so reasonable results are expected only near equilibrium.The inhomogeneous model includes relaxation and has potential to yield more realistic predictions.Source: §2.2 · pp. 3–4
  • In conservative form, the conserved variables are ρ and q = ρ(u + h(ρ)); Qeq(ρ) is the equilibrium curve toward which momentum density ρu relaxes.Source: §2.2 · p. 4
  • This paper fits h(ρ) and Ueq(ρ) from the same data, while the functions can in principle be chosen independently.Source: §2.2 · p. 4

2.3. Interpretation of ARZ as Generalization of LWR.

ARZ generalizes LWR by assigning each driver class an empty-road velocity and a corresponding velocity and flow curve. This family can produce widely varying stagnation densities, motivating a more general relationship between velocity, empty-road velocity, and density.

2.3. Interpretation of ARZ as Generalization of LWR.

  • The convected quantity w = u + h(ρ) is the empty-road velocity; hesitation h(ρ) reduces it to the actual vehicle velocity.The model associates w with each vehicle, and w is advected with the flow.Source: §2.3 · p. 4
  • ARZ represents the LWR curve as one member of a one-parameter family of velocity and flow-rate curves indexed by empty-road velocity w.The curves are vertical translations of the LWR velocity curve.Source: §2.3 · p. 4
  • Because ARZ adds hesitation to empty-road velocity, changes in w can produce substantially different densities at which flow stagnates.The paper argues maximum density should largely be a road property, not strongly dependent on drivers’ empty-road velocity.Source: §2.4 · p. 5

2.4. Generalized ARZ Model.

GARZ generalizes the homogeneous ARZ model by allowing a more general velocity function of density and empty-road velocity. The paper specifies physical and mathematical requirements for this function and describes its equilibrium relaxation and conservative form.

2.4. Generalized ARZ Model.

  • GARZ generalizes homogeneous ARZ by replacing its velocity relation with a generalized model in the generic second-order model class.The paper identifies this construction as a response to ARZ’s modeling shortcoming.Source: §2.4 · p. 5
  • The generalized velocity function must be nonnegative, decrease with density, increase with empty-road velocity, and yield strictly concave flow-rate curves.The model also imposes V(0,w) = w and V(ρ,0) = 0.Source: §2.4 · p. 5
  • Strict concavity of a flow-rate function Q(ρ) implies that its velocity U(ρ) = Q(ρ)/ρ decreases with density.This lemma links a condition on flow rate to a property required of velocity curves.Source: §2.4 · p. 5
  • The inhomogeneous GARZ model selects an equilibrium velocity curve from the family defined by V and generalizes the ARZ relaxation formulation.The paper specifies an equilibrium empty-road velocity weq.Source: §2.4 · p. 6
  • GARZ relaxation forms (9) and (10) are not equivalent, though they approach the same limit at different rates when ∂V/∂w(ρ,weq) > 0.The stated condition applies to the comparison of the two relaxation forms.Source: §2.4 · p. 6
  • The model is written conservatively with ρ and q = ρw as conserved variables, and Qeq(ρ) = ρUeq(ρ), Q(ρ,w) = ρV(ρ,w).Source: §2.4 · p. 6

2.5. Properties of the GARZ Model.

The GARZ properties section bounds the model’s state variables, defines inverse velocity mappings, and characterizes its waves and relaxation. While Cauchy-problem solutions converge toward LWR, bounded-domain boundary conditions can prevent that convergence even as relaxation time tends to zero.

2.5.1. Regions of GARZ Variables and Inverse Velocity Functions.

  • The model restricts density and empty-road velocity to stated ranges and uses monotonicity of V to define inverse functions R and W.R recovers density from velocity and empty-road velocity; W recovers empty-road velocity from density and velocity.Source: §2.5.1 · p. 6
  • The admissible state region is invariant: GARZ dynamics do not generate densities or empty-road velocities outside the stated bounds.Source: §2.5.1 · p. 7

2.5.2. Characteristics and Associated Fields.

  • GARZ Riemann solutions connect states through a first-family shock or rarefaction and a second-family contact discontinuity; when wL < uR, vacuum is generated.Across first-family waves w is constant, while across second-family waves u is constant.Source: §2.5.2 · pp. 7–8
  • For the Cauchy problem, GARZ solutions converge to LWR solutions as t → ∞ and/or τ → 0.Smooth solutions relax because w tends to weq along characteristics; system shocks are also LWR shocks.Source: §2.5.3 · p. 8

2.5.3. Relaxation of GARZ to LWR.

  • On a bounded domain, GARZ solutions generally do not converge to LWR solutions as τ → 0 under the described boundary-value setup.At the outflow boundary, LWR preserves the constant state while GARZ uses the full boundary state and generates an intermediate state.Source: §2.5.3 · p. 8
  • In the illustrated right-boundary case, the GARZ shock moves into the domain, changing the initial state to the intermediate boundary state.The figure contrasts this with LWR, which preserves the constant state.Source: Figure 1 · p. 9

3. Data-Fitted Traffic Models

The section fits traffic-model functions to historic flow–density measurements, then extends the GARZ curve family to address the ARZ model’s driver-dependent stagnation densities and accommodate out-of-range velocity data.

3.1. Data-Fitting for the LWR and ARZ Models.

  • The LWR fit selects a three-parameter, strictly concave flow curve by least squares, while the ARZ family shifts that curve across driver types.The ARZ shifts produce different stagnation densities, although maximum density is a road property rather than a driver property.Source: §3.1 · p. 10, Figure 2 · p. 10

3.2. Data-Fitting for the GARZ Model.

  • Weighted least squares constructs GARZ curve families, with lower and upper curves chosen so 99.9% of measurements lie above and below them, respectively.The family is re-parameterized by empty-road velocity w; in the cases studied, the minimization has a unique solution and the velocity curves do not intersect.Source: §3.2 · pp. 11–12, Figure 3 · p. 11
  • Extreme weighted-quantile choices can produce unreasonable curves by fitting outliers, so the GARZ construction uses bounded lower and upper flow-rate curves.The specified βmin and βmax are 10^-4 and 1−10^-4.Source: §3.2 · p. 12

3.3. Domain Extension for the GARZ Model.

  • For initial or boundary velocities outside W’s domain, the extension projects velocity onto the lowest or highest curve while leaving density unchanged.This extends W constantly beyond its domain without changing its range, and the extended function is not invertible outside the original domain.Source: §3.2 · p. 12, §3.3 · p. 12

4. Numerical Methods

The numerical methods use a finite-volume discretization, with a semi-implicit scheme for second-order models to handle stiff relaxation and avoid a time-step restriction tied to τ.

4. Numerical Methods

  • The second-order models use a semi-implicit finite-volume scheme, treating hyperbolic terms explicitly and relaxation terms implicitly to avoid a Δt = O(τ) restriction.The grid spacing is chosen at Δx ≤ 50 cm so numerical approximation errors are much smaller than model errors.Source: §4 · p. 13
  • HLL approximate Riemann fluxes avoid the costly velocity-function inversion required by an exact GARZ Riemann solver.With fine grids and continuous initial and boundary conditions, velocity overshoots are reported as negligibly small.Source: §4 · p. 13
  • The GARZ momentum update solves a scalar nonlinear equation for each cell, reaching machine accuracy within a few Newton steps.The prior density update is explicit, so the updated density is known for the momentum step.Source: §4 · p. 13
  • The relaxation scheme approaches equilibrium projection as τ→0, while relaxation terms disappear in the homogeneous limit τ→∞.The semi-implicit update is equivalent to a forward Euler step for the homogeneous part followed by backward Euler for relaxation.Source: §4 · p. 14

5. Validation and Comparison of Models via Real Data

The section validates data-fitted traffic models against NGSIM trajectory data and RTMC sensor data using a three-detector highway test. It compares their predictions and errors, with particular attention to whether GARZ improves agreement with real traffic.

5.1. Treatment of Data.

  • NGSIM fields are reconstructed from trajectories using a 25-meter kernel and smoothed at the boundaries in 15-second intervals; RTMC sensor data are spline-interpolated and initialized for five minutes before validation.Source: §5.1 · pp. 14–15

5.2. Quantification of Model Errors.

  • The error measure combines density and velocity deviations after normalization, addressing their different units and avoiding low-density amplification associated with scaling by the data norms.Source: §5.2 · p. 15
  • For normalization ranges, the method excludes densities below 5 veh/km/lane and outliers, then sets upper-density and velocity boundaries using 99.9% cutoffs.Source: §5.2 · p. 15, Figure 4 · p. 16

5.3. List of Models.

  • The comparison includes linear interpolation, data-fitted LWR, ARZ using a family of curves from the LWR fit, and GARZ using a weighted least-squares fit.Source: §5.3 · pp. 16–17

5.4. Model Validation with the NGSIM Trajectory Data.

  • GARZ is the best model in every reported Table 2 case, across the NGSIM and RTMC datasets and congested and non-congested conditions.Source: Table 2 · p. 20
  • On NGSIM, GARZ captures density and velocity evolution better than LWR and ARZ with a much smaller, though persistent, delay; ARZ captures velocity better than LWR despite a 30–60-second delay.Source: §5.4 · pp. 18, 20
  • Interpolation has average errors similar to LWR and ARZ on the short NGSIM segment, but its immediate transmission of boundary oscillations makes its predictions more oscillatory; it performs visibly worse on the longer RTMC segment.Source: §5.4 · pp. 20–21, §5.5 · p. 23

5.5. Model Validation with the RTMC Sensor Data.

  • On RTMC data, second-order models outperform LWR on low-density days, while GARZ deteriorates less than ARZ on high-density days; LWR and ARZ predict congestion onset five minutes late, versus two minutes for GARZ.Source: §5.5 · p. 23

6. Inhomogeneous ARZ and GARZ Models

The paper tests inhomogeneous ARZ and GARZ models by adding relaxation toward the LWR equilibrium curve and selecting relaxation times by predictive error. The relaxation term improves ARZ in some settings but does not improve GARZ in the tested data.

6. Inhomogeneous ARZ and GARZ Models

  • With boundary data, the limiting second-order model solutions are LWR solutions with boundary conditions different from those used for the LWR model in this study.The stated convergence limits are time tending to infinity at fixed τ, or τ tending to zero for fixed final time.Source: §6 · p. 23
  • The study varies τ from milliseconds to days because historic fundamental-diagram data provide no canonical way to determine it; each test selects the value with the smallest model error.The inhomogeneous models reuse the data-fitted functions of their homogeneous counterparts and add τ as their only new parameter.Source: §6 · p. 24
  • Across four plotted tests, both inhomogeneous models have a finite, positive error-minimizing relaxation time.The study uses one RTMC day with time-averaged error and three NGSIM intervals with spatio-temporally averaged error.Source: §6 · pp. 24–25, Figure 10 · p. 24
  • Error-minimizing relaxation times are about 25 s for ARZ, about 50 s for GARZ on NGSIM, and about 150 s for GARZ on RTMC.The ARZ minimum for NGSIM 4:00–4:15 pm is weakly pronounced and less reliable; the RTMC–NGSIM GARZ difference may reflect a flat minimum and sensor-data aggregation and imprecision.Source: §6 · p. 25
  • The relaxation term noticeably improves ARZ accuracy, especially at high densities, but hurts GARZ accuracy without significant gains over homogeneous GARZ.For ARZ, relaxation toward the LWR curve partly ameliorates its unrealistic spread in flow-rate curves at large densities.Source: §6 · pp. 25–26, Table 3 · p. 25

7. Conclusions

The conclusions identify GARZ as the most accurate model in the comparisons and describe how accuracy varies with congestion and relaxation time. They also note a free-flow limitation that GARZ, as studied here, does not address.

7. Conclusions

  • GARZ yields the most accurate predictions and best reproduces the real data among the compared models.The comparison includes macroscopic first- and second-order models and an interpolation predictor, using vehicle-trajectory and stationary-sensor data.Source: §7 · p. 26
  • ARZ outperforms LWR at low densities, but its accuracy deteriorates with congestion, consistent with its multiple stagnation densities.Interpolation can have errors similar to or slightly lower than LWR and ARZ, while producing more oscillatory, less sharp predictions and performing less well on longer road segments.Source: §7 · p. 26
  • Relaxation can further improve second-order model accuracy when τ is well chosen, but a too-small τ can make an inhomogeneous model less accurate than its homogeneous counterpart.Source: §7 · p. 26
  • Optimal relaxation times are reported in the range 14–60 seconds, with some larger values; a possible explanation is that empty-road velocity, representing general driving behavior, relaxes rather than actual vehicle velocity.Source: §7 · p. 26
  • The GARZ model studied here does not allow a unique flow-rate–density relationship in the free-flow regime, which may matter in some applications.A companion paper describes achieving that relationship in the GARZ framework by collapsing the fundamental-diagram curves into a single function in free flow.Source: §7 · pp. 26–27
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