Speed–density functional relationship for heterogeneous traffic data: a statistical and theoretical investigation
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Abstract
This study is an attempt to establish a suitable speed–density functional relationship for heterogeneous traffic on urban arterials. The model must reproduce the traffic behaviour on traffic stream and satisfy all static and dynamic properties of speed–flow–density relationships. As a first attempt for Indian traffic condition, two behavioural parameters, namely the kinematic wave speed at jam (C_{j}) and a proposed saturation flow (λ), are estimated using empirical observations. The parameter C_{j} is estimated by developing a relationship between driver reaction time and vehicle position in the queue at the signalised intersection. Functional parameters are estimated using Levenberg–Marquardt algorithm implemented in the R statistical software. Numerical measures such as root mean squared error, average relative error and cumulative residual plots are used for assessing models fitness. We set out several static and dynamic properties of the flow–speed–density relationships to evaluate the models, and these properties equally hold good for both homogenous and heterogeneous traffic states. From the numerical analysis, it is found that very few models replicate empirical speed–density data traffic behaviour. However, none of the existing functional forms satisfy all the properties. To overcome the shortcomings, we proposed two new speed–density functional forms. The uniqueness of these models is that they satisfy both numerical accuracy and the properties of fundamental diagram. These new forms would certainly improve the modelling accuracy, especially in dynamic traffic studies when coupling with dynamic speed equations.
Keywords
Heterogeneous traffic Speed–density model Kinematic wave speed Traffic flow CURE plots1 Introduction
Speed–density (v–k) relationship is straightforward and easy to explain when compared to other fundamental relationships. It is a one-to-one relationship between the driver behaviour and the number of vehicles present on the road. Speed–density relationship is also a part of traffic dynamics studies [1, 2, 3] to explore traffic flow patterns such as shock waves and queue lengths on highways and urban arterials. Selection of a suitable speed–density relationship influences the performance of macroscopic traffic flow models. A functional relationship is said to be accurate when it suitably represents the empirical data and satisfies all the properties of flow–speed–density relationships. This study is an attempt to analyse the existing speed–density functional forms for their numerical accuracy and properties. Further, new models have been proposed to overcome the limitations of the existing models.
Heterogeneous traffic mixes are the common sight of appearance in all the developing economies including India. The behaviour of heterogeneous traffic mix on urban arterials is described in various studies [4, 5, 6]. In brief, the traffic streams comprise small and highly manoeuvrable vehicles such as motorised two wheelers (MTW) and motorised three-wheelers (MThW). They continuously search for the gaps in the stream to move downstream even in the congested condition, which is described as creeping behaviour. Highway capacity is greatly affected by widely varying physical and dynamical characteristics of vehicles in addition to the absence of lane discipline. There is a significant difference between the behaviour of homogenous and heterogeneous traffic streams. Thus, it is interesting to study the relationship between traffic stream variables under heterogeneous traffic flow conditions and the functional forms that represent it.
The main objective of this paper is to evaluate the various functional forms using statistical techniques and the properties of flow–speed–density (q–v–k) relationships. Based on the results, new mathematical speed–density functional forms are proposed to improve the accuracy. This study considers only the single-regime-based models (continuously differentiable functions for the entire density range).
2 Literature review
2.1 Traffic stream models
Speed–density functional relations
Author | Functional form | Parameters |
---|---|---|
Linear | ||
Greenshields et al. [7] | \(v = v_{\text{f}} \left( {1 - \frac{k}{{k_{\text{j}} }}} \right)\) | \(v_{\text{f}} , k_{\text{j}}\) |
Drew [9] | \(v = v_{\text{f}} \left[ {1 - \left( {\frac{k}{{k_{\text{j}} }}} \right)^{m} } \right]\) | \(v_{\text{f}} ,k_{\text{j}}\), m |
Pipes [10] | \(v = v_{\text{f}} \left( {1 - \frac{k}{{k_{\text{j}} }}} \right)^{n}\) | \(v_{\text{f}} ,k_{\text{j}} ,n\) |
May and Keller [8] | \(v = v_{\text{f}} \left[ {1 - \left( {\frac{k}{{k_{\text{j}} }}} \right)^{m} } \right]^{n}\) | \(v_{\text{f}} ,k_{\text{j}} ,m,n\) |
Logarithmic | ||
Greenberg [12] | \(v = v_{\text{m}} \ln \frac{{k_{\text{j}} }}{k}\) | \(v_{\text{m}} ,k_{\text{j}}\) |
Exponential | ||
Underwood [13] | \(v = v_{\text{f}} { \exp }\left( { - k/k_{\text{m}} } \right)\) | \(v_{\text{f}} ,k_{\text{m}}\) |
Drake et al. [14] | \(v = v_{\text{f}} { \exp }\left[ { - \frac{1}{2}\left( {\frac{k}{{k_{\text{m}} }}} \right)^{2} } \right]\) | \(v_{\text{f}} ,k_{\text{m}}\) |
Papageorgiou et al. [2] | \(v = v_{\text{f}} { \exp }\left[ { - \frac{1}{a}\left( {\frac{k}{{k_{\text{m}} }}} \right)^{a} } \right]\) | \(v_{\text{f}} ,k_{\text{m}} ,a\) |
Complex | ||
Newell [15] | \(v = v_{\text{f}} \left\{ {1 - { \exp }\left[ {\frac{ - \lambda }{{v_{\text{f}} }}\left( {\frac{1}{k} - \frac{1}{{k_{\text{j}} }}} \right)} \right]} \right\}\) | \(v_{\text{f}} ,k_{\text{j}} ,\lambda\) |
Del Castillo and Benitez [16] | Exponential curve \(v = v_{\text{f}} \left\{ {1 - { \exp }\left[ {\frac{{\left| {C_{\text{j}} } \right|}}{{v_{\text{f}} }}\left( {1 - \frac{{k_{\text{j}} }}{k}} \right)} \right]} \right\}\) | \(v_{\text{f}} ,k_{\text{j}} ,C_{\text{j}}\) |
Maximum sensitivity curve \(v = v_{\text{f}} \left\{ {1 - { \exp }\left[ {1 - { \exp }\left( {\frac{{\left| {C_{\text{j}} } \right|}}{{v_{\text{f}} }}\left( {\frac{{k_{\text{j}} }}{k} - 1} \right)} \right)} \right]} \right\}\) | ||
Lee et al. [17] | \(v = \frac{{v_{\text{f}} \left( {1 - \frac{k}{{k_{\text{j}} }}} \right)}}{{ 1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } }}\) | \(v_{\text{f}} ,k_{\text{j}} ,E,\theta\) |
Wang et al. [18] | \(v\left( {k,\theta } \right) = v_{\text{b}} + \frac{{v_{\text{f}} - v_{\text{b}} }}{{\left[ {1 + { \exp }\left( {\frac{{k - k_{\text{t}} }}{{\theta_{1} }}} \right)} \right]^{{\theta_{2} }} }}\) | \(v_{\text{f}} ,k_{\text{t}} ,v_{\text{b}} ,\theta_{1} ,\theta_{2}\) |
Logarithmic form of traffic stream model is introduced by Greenberg [12], which is derived using hydrodynamic principles. The parameters involved in this model are optimum speed (v_{m}) and k_{j}, and both are difficult to observe from the field data. Besides this, the model produces infinite speed at free flow conditions. In comparison, exponential forms in the literature such as [2, 13, 14] are robust in terms of representing empirical data and satisfying the properties of flow–speed–density relationships. Newell’s [15] model is derived from the nonlinear car following theories, and it uses proportionality factor \(\lambda\) in modelling traffic flow. The parameter \(\lambda\) is a function of the relative speed, the intervehicle distance which is estimated by drawing the tangent to speed vs spacing curve at v(t) = 0. Del Castillo and Benítez [16] believed that the traffic flow behaviour is strongly characterised by kinematic wave speed of vehicles at jam density (C_{j}). From the literature, it is observed that for various traffic facilities the C_{j} value is ranging from − 25 to − 15 km/h. They introduced two functional forms similar to Newell’s: one is the exponential curve (single exponential form) and the other one is a generalised sensitive curve (double exponential form). The estimation procedure for C_{j} and \(\lambda\) is discussed in Sect. 3.2.
Some recent developments are Lee et al.’s [17] rational model and Wang et al.’s [18] logistic model. Lee et al.’s [17] model is made up of the following four parameters v_{f}, k_{j}, E and θ; and the model is developed to capture the dynamic behaviour of the traffic flow occurring at the highway ramps. For the given facility, estimated values for the shape parameters E and θ are 100 and 4, respectively. Wang et al.’s [18] model is a 5 parameter logistic speed–density relationship which is developed using 100 stations data on GA 400 expressway in Atlanta. The data used in this modelling look asymptotic to the axis at upper and lower limbs. In the given formula, v_{f} and v_{b} are the upper and lower asymptotes of the curve. Here v_{b} is the average travel speed at saturation region (stop and go). In the given model, k_{t} is the inflection point where the curve turns from free flow to congested flow, θ_{1} is a scale parameter and θ_{2} is a lop-sidedness of the curve.
From the literature survey, it is believed that research on the development of suitable speed–density functional relationship for Indian traffic condition is limited. Recently, Thankappan and Vanajakshi [19] tried to evaluate different combinations of single and two-regime models for heterogeneous traffic data collected on urban roads. The study suggested that two-regime models are better in representing the speed–density data. However, the study has only considered simple linear and exponential models for evaluation purpose and the models are not evaluated for the properties of flow–speed–density relationships.
2.2 Properties of the speed–density and flow–density functional relationships
- (i)
\(v\left( k \right)_{k \to 0} = v_{\text{f}}\).
- (ii)
\(v^{\prime}\left( 0 \right) = 0\); i.e., vehicles move at free flow speed when interaction between vehicles is negligible.
- (iii)
\(v\left( k \right)_{{k \to k_{\text{j}} }} = 0\); i.e., vehicles stop at jam density.
- (iv)
\(0 < k \le k_{\text{j}} \); i.e., density varies from zero to maximum density.
- (v)
\(0 \le v \le v_{\text{f}}\); i.e., speed varies between zero and maximum flow possible.
- (vi)
Speed decreases with density, i.e. \(v^{\prime}\left( k \right) < 0\).
The first dynamic property is that the kinematic wave speed \((C_{\text{j}} )\) of the traffic at jam condition must be a negative constant \((q^{ '} \left( k \right)_{{k \to k_{\text{j}} }} {\text{is}}\,{\text{a}}\,{\text{negative}}\,{\text{constant}})\). This is introduced by Del Castillo and Benítez [16], to represent shock propagation in saturation flow region. The second essential dynamic property is that flow–density (q–k) relation must be convex when traffic is approaching the jam density, which is necessary for producing stable shock waves at congested conditions. This is explained below. In Heydecker and Addison [20] terms, if the flow–density relationship is concave throughout its domain (i.e. when \(q^{\prime\prime}\left( k \right) < 0\)), then stable shock waves can only occur as transitions from low to high density. However, if the fundamental relationship has a subdomain within which it has a positive curvature (i.e. where \(q^{\prime\prime}\left( k \right) > 0\)), then stable start waves can arise when traffic accelerates from a region with density in this subdomain to a region of lower density.
3 Data collection and parameter estimation
One of the objectives of this study is to fit and evaluate speed–density functional forms for heterogeneous traffic data. Empirical data are the basis for development and validation of traffic flow models. The section describes traffic data collection and variable estimation procedure under non-lane based heterogeneous traffic environment. This section also presents vehicle composition and their physical dimensions and further parameter estimation from empirical data.
3.1 Empirical speed–flow data
Vehicle classes, physical dimensions and speed characteristics
Vehicle class | Vehicles included | Vehicle average dimensions (m) | Speed characteristics (km/h) | |||
---|---|---|---|---|---|---|
v _{free} | v _{cong} | v _{mean} | v _{σ} | |||
Car | Small car, SUV, van | 5.0 × 2.0 | 73.4 | 4.7 | 47 | 14.8 |
Motorised two wheeler | Scooter, moped | 1.8 × 0.6 | 65.2 | 7.4 | 46.5 | 12.8 |
Motorised three wheeler | Auto, LCV | 2.6 × 1.4 | 55.5 | 4.5 | 31 | 8.2 |
Heavy vehicles | Bus, truck | 10.3 × 2.5 | 52.3 | 3.5 | 29 | 9.0 |
3.2 Parameters from empirical observations
For fitting and evaluating the traffic stream models, parameters observed from the field data are required. The empirical observations are also used as initial parameters in optimisation tool. Since the behaviour of heterogeneous traffic is different to that of the homogenous traffic, estimating some of the parameters mentioned in Table 1 is difficult. For instance, jam density (k_{j}) is a function of vehicle headway maintenance, traffic composition and road width. Likewise, kinematic wave speed (C_{j}) is a function of vehicle length plus safety distance and driver reaction time. There will be a wide variation in the above-mentioned parametric values due to the following reasons. Vehicles are positioned close to each other in the same lane, and the gaps between large vehicles will be filled by the smaller ones. In addition, urban traffic composition in India shows that there are at least ten classes of vehicles observed on the roads where vehicle lengths vary from 1.8 m (MTW) to 10.3 m (Bus) and the safety distances maintained by these vehicles are very small. Therefore, jam density is not constant (suitable value will be considered for the typical composition) for a given road section. Moreover, the estimation of C_{j} will also be challenging. Now the parameter estimation procedure will be discussed under this section. It is believed that the macroscopic relationship of traffic flow is strongly characterised by some of the important parameters such as kinematic wave speed at jam density (C_{j}) and saturation flow parameter \(\left( \lambda \right)\). In microscopic scale, parameter \(\lambda\) is a function of relative speed and intervehicle distance. However, in macrolevel, it is a function of kinematic wave speed and jam density of vehicular flow. The parameter C_{j} is a disturbance propagation speed of the vehicles when density is approaching the jam density and it is a function of vehicle length plus safety distance and driver reaction time. It is a first attempt to estimate these parameters for Indian traffic condition. The detailed estimation procedure is given below.
3.2.1 Kinematic wave speed (C_{j}) estimation
It is well known that C_{j} value can be estimated by studying vehicle dynamics at the signalised intersection [16]. Stopping and starting waves can be observed at signals during the green and red time, and it resembles the vehicular behaviour at congestion region. Therefore, data regarding reaction time (t_{s}) of the different driver classes are obtained at Sri Aurobindo Marg signalised intersection (28°32′40.2″N 77°12′04.5″E) located in Delhi, India. Here the reaction time is an elapsed time between the start of the green time and start of the vehicle in a queue.
3.2.2 Estimation of \(\lambda\) and other parameters
Parameter values from empirical data
Parameter | Free flow speed (v_{f}) (km/h) | Optimum speed (v_{m}) (km/h) | Kinematic wave speed (C_{j}) (km/h) | Average travel speed at saturation region (v_{b}) (km/h) | Jam density (k_{j}) (veh/km) | Optimum density (k_{m}) (veh/km) | Inflection point (k_{t}) | \(\lambda\) |
---|---|---|---|---|---|---|---|---|
Value | 65–70 | 25–30 | − 12.42 | 5–10 | 700–800 | 280–300 | 150–200 | 9000 |
4 Fitting and evaluation of traffic stream models
4.1 Model fitting and statistical evaluation
Model parameters and fitness values
Serial no. | Model | Fundamental parameters | Shape parameters | RMSE | ARE | ||
---|---|---|---|---|---|---|---|
v_{f} (km/h) | k_{j} (km/h) | Others | |||||
1 | Greenshields et al. | 64.57 | 596 | – | – | 4.687 | 0.312 |
2 | Drew | 68.68 | 619 | – | m = 0.85 | 4.612 | 0.335 |
3 | Pipes | 66.52 | 650 | – | n = 1.2 | 4.580 | 0.125 |
4 | May and Keller | 64.78 | 757 | – | m = 1.23, n = 2.0 | 4.312 | 0.113 |
5 | Greenberg | – | 900 | v_{m} = 25 km/h | – | 10.014 | 0.133 |
6 | Underwood | 73.60 | – | k_{m} = 339 veh/km | – | 4.904 | 0.101 |
7 | Drake et al. | 58.77 | – | k_{m} = 253 veh/km | – | 4.335 | 0.129 |
8 | Papageorgiou et al. | 61.59 | – | k_{m} = 260 veh/km | a = 1.7 | 4.177 | 0.100 |
9 | Newell | 65.00 | 750 | \(\lambda\) = 14,761 | 5.479 | 0.148 | |
10 | Del Castillo and Benitez | 62.00 | 891 | C_{j} = − 14 km/h | – | 5.994 | 0.120 |
11 | Lee et al. | 64.63 | 700 | – | E = 2.1, θ = 2.5 | 4.266 | 0.117 |
12 | Wang et al. | 65.00 | – | v_{b} = 9.64 km/h k_{t} = 200 | θ_{1} = 82.3, θ_{2} = 0.776 | 4.033 | 0.079 |
From the graphical and statistical measures (RMSE and ARE), it is observed that Wang et al.’s model is outperforming all the other models and the parameters also close to the empirical observations. It is followed by models of Papageorgiou et al., Lee et al. and May and Keller. However, barring Greenberg et al.’s model with the highest RMSE value and Drew’s model with the highest ARE value, it is somewhat difficult to choose the model on the basis of these statistics that can outperform other models. In this regard, cumulative residual (CURE) plots [23] have been used in assessing the models’ performance.
4.2 Cumulative residual plots
Statistical analysis and graphical presentation revealed that models involving a large number of parameters such as Wang et al.’s and Lee et al.’s are sound descriptors of empirical data. It is obvious that they resemble any kind of traffic phenomenon with some adjustments in boundary and shape parameter values. However, in Sect. 5 we will show that Lee et al.’s model accuracy can be improved further by introducing additional parameters. It is clear from the analysis that linear models of Greenshields et al., Drew and Pipes are poor in representing the data. While models of Greenbergs et al., Newell and Del Castillo can be part of multi-regime speed–density models due to their good estimation accuracy at high-density regions.
4.3 Theoretical investigation
On the basis of the statistical evaluation, we can converge on some models as the best candidates to represent the empirical data. This may, however, not be sufficient to ensure that these models are good in representing the behaviour of traffic flow. In this section, models will be evaluated for their static and dynamic traffic properties. Static properties of the model are derived from the fact that traffic flow is stationary and is always at equilibrium. However, properties such as the kinematic wave speed (C_{j}) and stable shock wave are related to the dynamic behaviour of the traffic flow and they are obtained from continuum theory of traffic flow. These properties equally hold good for both homogenous and heterogeneous traffic states. Interpretation of static and dynamic properties of the models is discussed below.
4.3.1 Static properties of the model
Validation of static properties
Model | Model static properties | ||||
---|---|---|---|---|---|
Free flow property \(v\left( k \right)_{k \to 0} = v_{\text{f}}\) | Independent property \(v^{\prime}\left( 0 \right) = 0\) | Jam density property \(v\left( k \right)_{{k \to k_{\text{j}} }} = 0\) | Speed range \(0 \le v \le v_{\text{f}}\) | Slope property \(v^{\prime}\left( k \right) < 0\) | |
Greenshields et al. | √ | x | √ | √ | √ |
Drew | √ | √ | √ | √ | √ |
Pipes | √ | x | √ | √ | √ |
May and Keller | √ | √ | √ | √ | √ |
Greenberg | x | x | √ | x | √ |
Underwood | √ | x | x | x | √ |
Drake et al. | √ | √ | x | x | √ |
Papageorgiou et al. | √ | √ | x | x | √ |
Newell | √ | √ | √ | √ | √ |
Delcastillo and Benítez | √ | √ | √ | √ | √ |
Lee et al. | √ | x | √ | √ | √ |
Wang et al. | √ | √ | x | x | √ |
4.3.2 Kinematic wave speed property (C _{j})
Validation of dynamic properties
Model | Model dynamic properties | |
---|---|---|
Kinematic wave speed property \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} \,{\text{is}}\,{\text{a}}\,{\text{negative}}\,{\text{constant}}\) | Stable shock wave property \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\) | |
Greenshields et al. | \(q^{\prime}\left( k \right) = v_{\text{f}} \left( {1 - \frac{2k}{{k_{\text{j}} }}} \right)\), \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - v_{\text{f}}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - 2v_{\text{f}} }}{{k_{\text{j}} }} < 0\) |
Drew | \(q^{\prime}\left( k \right) = v_{\text{f}} \left[ {1 - \frac{{\left( {m + 1} \right)k^{m} }}{{k_{\text{j}}^{m} }}} \right]\), \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - mv_{\text{f}}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - m\left( {m + 1} \right)v_{\text{f}} }}{{k_{\text{j}} }}\); for m > 0, \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} < 0\) |
Pipes | \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - nv_{\text{f}} \left( {1 - \frac{k}{{k_{\text{j}} }}} \right)^{n - 1}\); \({\text{for }}n > 1,q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} \to 0\) | \(q^{\prime\prime}\left( k \right) = \frac{{nv_{\text{f}} }}{{k_{\text{j}} }}\left( {1 - \frac{k}{{k_{\text{j}} }}} \right)^{n - 2} \left[ {\left( {n + 1} \right)\frac{k}{{k_{\text{j}} }} - 2} \right]\), \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\) when \(k > \frac{2}{{\left( {n + 1} \right)}}k_{\text{j}}\) |
May and Keller | \(q^{\prime}\left( k \right) = \left\{ {v_{\text{f}} \left[ {1 - \left( {\frac{k}{{k_{\text{j}} }}} \right)^{m} } \right]^{n - 1} \left[ {1 - \left( {1 + nm} \right)\left( {\frac{k}{{k_{\text{j}} }}} \right)^{m} } \right]} \right\}\); \({\text{for }}m > 1,n > 1\), \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} \to 0\) | \(q^{\prime\prime}\left( k \right) = \frac{{mnv_{\text{f}} }}{{k_{\text{j}}^{m} }}\left[ {1 - \left( {\frac{k}{{k_{\text{j}} }}} \right)^{m} } \right]^{n - 2} \left[ {\left( {mn + 1} \right)\left( {\frac{k}{{k_{\text{j}} }}} \right)^{n} - \left( {1 + m} \right)} \right]\), \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0 \,{\text{when}}\, k > \left( {\frac{1 + m}{mn + 1}} \right)^{n} k_{\text{j}}\) |
Greenberg | \(q^{\prime}\left( k \right) = v_{\text{m}} \left[ {\ln \left( {\frac{{k_{\text{j}} }}{k}} \right) - 1} \right]\), \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - v_{\text{m}}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - \frac{{v_{\text{m}} }}{{k_{\text{j}} }} < 0\) |
Underwood | \(q^{\prime}\left( k \right) = \left[ {v_{\text{f}} {\text{exp}}\left( {\frac{ - k}{{k_{\text{m}} }}} \right)} \right]\left( {1 - \frac{k}{{k_{\text{m}} }}} \right)\); for \(k \to \infty , q^{\prime}\left( k \right) = - 0\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{v_{\text{f}} }}{{k_{\text{m}} }}{ \exp }\left( {\frac{{ - k_{\text{j}} }}{{k_{\text{m}} }}} \right)\left[ { - 2 + \frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right]\), \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\) |
Drake et al. | \(q^{\prime}\left( k \right) = \left\{ {v_{\text{f}} {\text{exp}}\left[ {\frac{ - 1}{2}\left( {\frac{k}{{k_{\text{m}} }}} \right)^{2} } \right]} \right\}\left[ {1 - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{2} } \right]\); for \(k \to \infty , q^{\prime}\left( k \right) = - 0\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{v_{\text{f}} k_{\text{j}} }}{{k_{\text{m}}^{2} }}\left\{ {{ \exp }\left[ {\frac{ - 1}{2}\left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{2} } \right]} \right\}\left[ { - 3 + \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{2} } \right] > 0\) |
Papageorgiou et al. | \(q^{\prime}\left( k \right) = \left\{ {v_{\text{f}} {\text{exp}}\left[ {\frac{ - 1}{a}\left( {\frac{k}{{k_{\text{m}} }}} \right)^{\text{a}} } \right]} \right\}\times\left[ {1 - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{a} } \right]\); for \(k \to \infty , q^{\prime}\left( k \right) = - 0\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{v_{\text{f}} k_{\text{j}}^{a - 1} }}{{k_{\text{m}}^{2} }}\left\{ {{ \exp }\left[ {\frac{ - 1}{a}\left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{a} } \right]} \right\}\left[ { - \left( {a + 1} \right) + \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{a} } \right]\); for \(k_{\text{j}} > k_{\text{m}} \left( {a + 1} \right)^{{\frac{1}{a}}}\), \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\) |
Newell | \(q^{\prime}\left( k \right) = v_{\text{f}} \left\{ {1 - { \exp }\left[ {\left( {\frac{ - \lambda }{{v_{\text{f}} }}} \right)\left( {\frac{1}{k} - \frac{1}{{k_{\text{j}} }}} \right)} \right] - \frac{\lambda }{{v_{\text{f}} k}}\left[ {{ \exp }\left( {\frac{ - \lambda }{{v_{\text{f}} }}\left( {\frac{1}{k} - \frac{1}{{k_{\text{j}} }}} \right)} \right)} \right]} \right\}, q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - C_{\text{j}}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - \lambda^{2} }}{{v_{\text{f}} k_{\text{j}}^{3} }}{ \exp }\left[ {\frac{ - \lambda }{{v_{\text{f}} }}\left( {\frac{1}{{k_{\text{j}} }} - \frac{1}{{k_{\text{j}} }}} \right)} \right] = \frac{{ - \lambda^{2} }}{{v_{\text{f}} k_{\text{j}}^{3} }} < 0\) |
Delcastillo and Benítez | \(q^{\prime}\left( k \right) = v_{\text{f}} \left\{ {1 - { \exp }\left[ {\left( {\frac{{C_{\text{j}} }}{{v_{\text{f}} }}} \right)\left( {1 - \frac{{k_{\text{j}} }}{k}} \right)} \right]\left[ {1 + \left( {\frac{{C_{\text{j}} k_{\text{j}} }}{{v_{\text{f}} k}}} \right)} \right]} \right\}\), for \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - C_{\text{j}}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - C_{\text{j}}^{2} }}{{v_{\text{f}} k_{\text{j}} }}{ \exp }\left[ {\frac{{C_{\text{j}} }}{{v_{\text{f}} }}\left( {1 - \frac{{k_{\text{j}} }}{{k_{\text{j}} }}} \right)} \right] = \frac{{ - C_{\text{j}}^{2} }}{{v_{\text{f}} k_{\text{j}} }} < 0\) |
Lee et al. | \(q^{\prime}\left( k \right) = \frac{{v_{\text{f}} \left\{ {\left[ {1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } } \right]\left( {1 - \frac{2k}{{k_{\text{j}} }}} \right) - \left[ {\frac{E\theta }{{k_{\text{j}} }}\left( {k - \frac{{k^{2} }}{{k_{\text{j}} }}} \right)\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta - 1} } \right]} \right\}}}{{\left[ {1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } } \right]^{2} }}, {\text{for }}q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - v_{\text{f}} }}{1 + E}\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{2v_{\text{f}} }}{{k_{\text{j}} \left( {1 + E} \right)}}\left( {\frac{E\theta }{1 + E} - 1} \right)\), for \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{0.45v_{\text{f}} }}{{k_{\text{j}} }} > 0\) |
Wang et al. | \(q^{\prime}\left( k \right) = v_{\text{b}} + \frac{{v_{\text{f}} - v_{\text{b}} }}{{\left[ {1 + { \exp }\left( {\frac{{k - k_{\text{t}} }}{{\theta_{1} }}} \right)} \right]^{{\theta_{2} }} }} - \frac{{k\left( {v_{\text{f}} - v_{\text{b}} } \right)\theta_{2} { \exp }\left( {\frac{{k - k_{\text{t}} }}{{\theta_{1} }}} \right)}}{{\theta_{1} \left[ {1 + { \exp }\left( {\frac{{k - k_{\text{t}} }}{{\theta_{1} }}} \right)} \right]^{{\theta_{2} + 1}} }}, {\text{for}}\, k \to \infty , q^{\prime}\left( k \right) = - \infty\) | \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{ - \left( {v_{\text{f}} - v_{\text{b}} } \right)\theta_{2} { \exp }\left( {\frac{{k_{\text{j}} - k_{\text{t}} }}{{\theta_{1} }}} \right)}}{{\theta_{1} \left[ {1 + { \exp }\left( {\frac{{k_{\text{j}} - k_{\text{t}} }}{{\theta_{1} }}} \right)} \right]^{{\theta_{2} + 1}} }}\left\{ {2 + \frac{k}{{\theta_{1} }}\left[ {1 - \frac{{\theta_{2} + 1}}{{1 + { \exp }\left( {\frac{{k_{\text{j}} - k_{\text{t}} }}{{\theta_{1} }}} \right)}}} \right]} \right\}\), for given values \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} < 0\) |
4.3.3 Stable shock wave property
This property can be analysed by obtaining the second order derivative of the flow equation with respect to density or by using graphical representation; i.e., convexity (positive sign) must be observed when curve approaching jam density. From the literature, it is observed that this property is important in describing some of the nonlinear behaviour of traffic flow such as hysteresis, platoon dispersion and density oscillations observed in the congested condition. The analysis results are set out in Table 6, where models satisfying the stable shock wave property are pointed out with a sign greater than zero and otherwise less than zero.
The analysis shows that the models of Greenshileds et al., Drew, Greenberg, Newell, Del Castillo and Benitez, and Wang et al. are strictly concave and hence unable to satisfy the stable shock wave property. Therefore, it is difficult to explain some of the important traffic flow phenomena, for instance, hysteresis originated in the non-stationary region of the flow–density fundamental diagram. Further, models of Underwood, Drake et al., Papageorgiou et al. and Lee et al. have positive curvature in their subdomain; therefore, they can produce stable shock waves when the vehicles accelerate from a region with high density to a lower density. However, mathematical analysis (Tables 5, 6) revealed that none of the existing models satisfies all the properties of the speed–flow–density relationship.
5 Proposed speed–density models
The limitation of the existing models in satisfying all the properties encouraged us to develop a new speed–density model [Eq. (11)]. The essential requirement for the functional form is that it must satisfy all the properties of the fundamental diagram with a less numerical error. In addition to the new model, a modification has been proposed to the Lee et al.’s model (Eq. (12)) to overcome the deficiencies.
5.1 The proposed model
- 1.
The empirical speed–density data show some asymptotic behaviour at the free flow and jam conditions, and also data show a smooth decline in speed values after a critical point. The selection of reciprocal exponential forms is the better representation of this kind of speed–density relationship. The functional form must be continuous and differentiable.
- 2.
The parameter free flow speed (v_{f}) is the property of freely moving traffic and jam density (k_{j}) is the property of queueing traffic. Critical density (k_{m}) gives an idea on where the traffic condition is changing from free flow state to congested state. These parameters (v_{f}, k_{j} and k_{m}) convey some physical meaning.
- 3.
The choice of shape parameters will help in replicating the shape of the data.
- 4.
Importantly, the model must also satisfy all the properties of fundamental diagrams: \(v\left( k \right)_{k \to 0} = v_{\text{f}}\), \(v\left( k \right)_{{k \to k_{\text{j}} }} = 0\), \(v^{\prime}\left( 0 \right) = 0\), \(v^{\prime}\left( k \right) < 0\), \(q^{\prime}\left( k \right)_{{k \to k_{\text{j}} }} = - C_{\text{j}}\), \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\).
Equation (11) gives the general form of the model. Here, a and b are shape parameters.
- 1.
At k = 0, v = v_{f}.
- 2.
At k = k_{j}, v = 0.
- 3.
\(v^{\prime}\left( k \right) = v_{\text{f}} \frac{{\left[ { - \left( {1 + a} \right){\text{e}}^{{ - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} }} \left( {\frac{{k^{a} }}{{k_{\text{m}}^{1 + a} }}} \right)} \right]}}{{\left[ {1 - {\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} } \right]}}\); at k = 0, \(v^{\prime}\left( k \right) = 0 \forall a > 0\) satisfies the independent property.
- 4.
\(q^{\prime}\left( k \right) = v_{\text{f}} \frac{{\left[ {{\text{e}}^{{ - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} }} - \left( {1 + a} \right)\left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} {\text{e}}^{{ - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} }} - {\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} } \right]}}{{\left[ {1 - {\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} } \right]}} \) ; at \(k \to \infty ,\) \(q^{\prime}\left( k \right) = - v_{\text{f}} \left[ {\frac{{{\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} }}{{1 - {\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} }}} \right]\) produces constant negative wave speed.
- 5.
\( q^{\prime\prime}\left( k \right) = v_{\text{f}} \frac{{\left\{ {\left[ { - \left( {1 + a} \right)\frac{{k^{a} }}{{k_{\text{m}}^{1 + a} }}} \right] - \left[ {\left( {1 + a} \right)^{2} \frac{{k^{a} }}{{k_{\text{m}}^{1 + a} }}{\text{e}}^{{ - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} }} } \right] + \left[ {\left( {1 + a} \right)^{2} \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} \frac{{k^{a} }}{{k_{\text{m}}^{1 + a} }} .{\text{e}}^{{ - \left( {\frac{k}{{k_{\text{m}} }}} \right)^{1 + a} }} } \right]} \right\}}}{{\left[ {1 - {\text{e}}^{{ - \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} }} }\right]}}\); at k = k_{j}, \( \left( {\frac{{k_{\text{j}} }}{{k_{\text{m}} }}} \right)^{1 + a} > \frac{a + 2}{a + 1} \), for a > 0, \( {\text{and}}\, q^{\prime \prime } \left( k \right) > 0 \), produces stable shock waves.
This model satisfies all the properties of flow–speed–density relationships. The model fitness is also good at ARE = 0.096 and RMSE = 4.172 except some deviance at the bottom tail. Kinematic wave speed value for given parameter values is − 8 km/h.
5.2 Modified Lee et al.’s model
- 1.
At k = 0, v = v_{f}.
- 2.
At k = k_{j}, v = 0 for a > 0.
- 3.
\(v^{\prime}\left( k \right) = \frac{{\left\{ {\left[ {1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } } \right]\left[ {\frac{{ - av_{\text{f}} }}{{k_{\text{j}} }}.\left( {\frac{k}{{k_{\text{j}} }}} \right)^{a - 1} } \right]} \right\} - \left\{ {v_{\text{f}} \left[ {1 - \left( {\frac{k}{{k_{\text{j}} }}} \right)^{a} } \right]\frac{E\theta }{{k_{\text{j}} }}\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta - 1} } \right\}}}{{\left[ {1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } } \right]^{2} }}\), at k = 0, \(v^{\prime}\left( k \right) = 0\) which satisfies the independent property.
- 4.
\(q^{\prime}\left( k \right) = \frac{{v_{\text{f}} \left[ {1 - \left( {a + 1} \right)\left( {\frac{k}{{k_{\text{j}} }}} \right)^{a} } \right]}}{{1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } }} - \frac{{\frac{{v_{\text{f}} E\theta }}{{k_{\text{j}}^{\theta } }}\left( {k^{\theta } - \frac{{k^{a + \theta } }}{{k_{\text{j}}^{a} }}} \right)}}{{\left[ {1 + E\left( {\frac{k}{{k_{\text{j}} }}} \right)^{\theta } } \right]^{2} }}\); at k = k_{j}, \(q^{\prime}\left( k \right) = \frac{{ - av_{\text{f}} }}{1 + E}\) is always less than zero for a > 0, E > 0. Therefore, kinematic wave speed is a negative constant.
- 5.
\(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} = \frac{{v_{\text{f}} }}{{k_{\text{j}} }}\left[ {\frac{E\theta a}{{\left( {1 + E} \right)^{2} }} + \frac{E\theta a}{1 + E} - a\left( {a + 1} \right)} \right]\), for E > 0, θ > 0, a > 0; \(q^{\prime\prime}\left( k \right)_{{k \to k_{\text{j}} }} > 0\) produces stable shock waves.
The modified Lee et al.’s model satisfies all the properties of the flow–speed–density relationships with the least error. The RMSE and ARE values for the model are 4.158 and 0.0822, respectively. For given parameter values, the kinematic wave speed value is − 21.96 km/h.
Parameters and estimation accuracy of proposed models
Serial no. | Model | Fundamental parameters | Shape parameters | RMSE | ARE | ||
---|---|---|---|---|---|---|---|
v_{f} (km/h) | k_{j} (km/h) | Others | |||||
1 | Proposed model | 62.9 | 850 | k_{m} = 360 veh/h | a = 0.60 | 4.172 | 0.096 |
2 | Modified Lee et al’s. model | 63.5 | 900 | – | E = 10.30, \(\theta = 2.14\), a = 4 | 4.158 | 0.082 |
6 Limitations of this study
The present study mainly focused on evaluating all the existing single-regime models using two criteria: one is fitting empirical speed–density data and the second is properties of speed–flow–density relationships. It is important to point out that all the analyses and conclusions of this study are conducted based on data collected for a short period of time on a single road section. This may count as a limitation of the study.
7 Conclusions and future scope
- 1.
Empirical v–k relationship of heterogeneous traffic observed on urban arterials revealed some interesting facts: (i) Dependence of speed on density is diminished as density approaching zero, i.e. \(v^{\prime}\left( k \right)_{k \to 0} = 0,\) and it is very small compared to homogeneous traffic section. The length of this non-dependency region is generally a function of number of lanes, type of facility and composition of vehicles. (ii) Capacity of the stream is observed to be very high due to the effective utilisation of the road width (this behaviour is possibly attributed to non-lane discipline and the presence of small sized vehicles). (iii) Large variation in highway capacity values can be seen in q–v and q–k plots. Capacity value is ranging from 8000 to 11,000 veh/h. The variation in capacity is observed due to vehicle composition and their respective selection of speeds. (iv) Large deviations (nonlinear behaviour) can be observed in the q–k curve at the congested region. This behaviour is attributed to varying vehicle dynamics and their selection of safety headways.
- 2.
As a first attempt for Indian traffic conditions, some of the behavioural parameters such as kinematic wave speed (C_{j}) and saturation flow parameter (λ) are determined using empirical observations. The typical value of the parameter C_{j} is − 12.42 km/h and the value is less than that of homogenous traffic case [16]. Jam density values for given facilities are difficult to estimate and they approximately vary between 700 and 800 veh/km. The saturation flow parameter value depends on kinematic wave speed and jam density, and the value estimated in the present study is 9000.
- 3.
In addition to RMSE and ARE, we employed CURE plots to evaluate the overall performance of the models. They revealed some interesting facts: (i) Models of Greenberg, Newell, and Del Castillo and Benitez can be part of multi-regime speed–density models due to their good estimation accuracy at high-density regions. (ii) Linear models of Greenshields et al., Drew and Pipes are poor in representing the data. (iii) Models involving a large number of parameters, for instance, Wang et al.’s model, are sound descriptors of empirical data.
- 4.
The study set out several static and dynamic properties of q–k–v relationships for which the models are examined and compared. The study concludes that none of the existing functional forms can fulfil many of the properties.
- 5.
Two new speed–density functional forms are proposed. From the analysis, it can be concluded that both the proposed models satisfy the numerical accuracy and the fundamental diagram properties. These new forms would be able to improve the model predictions, especially in continuum traffic modelling when in couple with dynamic speed equations.
- 6.
Further, there is a scope for improving the proposed v-k relationships by taking model bias and variance into account. In future, it is planned to collect data on more arterial sections to check the compatibility of the proposed models.
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