The Fundamental Diagram

Abstract

In the previous chapter, the main variables in traffic flow modelling were introduced. In this chapter, we discuss how they are related: obviously, high speeds seldom occur together with low headways, similarly, low densities create room for high speeds. Traffic flow models are based on the assumption that there is some relation between these variables. The relation between distance and velocity was first studied by Greenshields (The photographic method of studying traffic behavior. In: Proceedings of the 13th annual meeting of the highway research board (1934), pp 382–399) and called the fundamental relation (or fundamental diagram) later. Therefore, Greenshields is often regarded as the founder of traffic flow theory, and the fundamental diagram is the first model in the genealogical tree of traffic flow models (see Page 15).

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In the previous chapter, the main variables in traffic flow modelling were introduced. In this chapter, we discuss how they are related: obviously, high speeds seldom occur together with short headways, similarly, low densities create room for high speeds. Traffic flow models are based on the assumption that there is some relation between these variables. The relation between distance and velocity was first studied by Greenshields (1934) and called the fundamental relation (or fundamental diagram) later. Therefore, Greenshields is often regarded as the founder of traffic flow theory, and the fundamental diagram is the first model in the genealogical tree of traffic flow models (see Page 15).

The reader of this chapter is introduced to the concept of the fundamental diagram. After reading this chapter, they will be able to explain the typical shape of a fundamental diagram and why and how the basic shape can be adapted to reflect observations, such as scattered data. Furthermore, the reader will be able to link characteristics of microscopic driving behaviour (e.g. low speed at small headway and vice versa, (non)-equilibrium, hysteresis, capacity drop and heterogeneity in driving behaviour) to the shape of a fundamental diagram. Finally, they will be able to reflect on the desired properties of a fundamental diagram and assess whether a given diagram satisfies the requirements.

2.1 High Densities, Low Speeds and Vice Versa

Common observations of traffic show that at high densities, such as in (heavy) congestion, speeds are low. Conversely, when there are few vehicles on the road, headways are large and speeds are high. This is partially due to simple human behaviour: drivers tend to chose a speed that is as high as possible, while still safe. Therefore, traffic flow models commonly use a decreasing—or at least non-increasing—relationship between density and speed.

Because of this decreasing density-speed relationship, the maximum flow (density × speed) occurs at some intermediate density and speed values. This gives rise to one of the main challenges in traffic flow modelling: when densities are low, flows increase with increasing densities, however, at high densities, an increased density leads to a reduction in flow. See Fig. 2.1 for an example. This is different from, for example water flow (in the same figure), which is incompressible and the speed of the flow (i.e. the flow rate) does not depend on the density (Fig. 2.2). The precise relationship between variables such as density, headway, speed and flow, is an important subarea of traffic flow research. The main insights developed over the decades are discussed in the following sections of this chapter.

Fig. 2.1

Traffic vs. water flow: increasing and decreasing flow vs. nondecreasing flow

Fig. 2.2

Comparison of water and granular flow: inflows versus outflows. (a) Water: outflow equals inflow, up to a maximum. (b) Granular: at low inflows, outflow equals inflow. When inflow is too high, particles ‘get stuck’ in bottleneck and outflow is low, lower than with a medium inflow

2.2 Shapes of the Fundamental Diagram

The original fundamental diagram proposed by Greenshields (1934), is linear in the spacing-velocity plane. However, his name has now been linked to the fundamental diagram that he proposed one year later (Greenshields 1935). This fundamental diagram is linear in the density-velocity plane and thus parabolic in the density-flow plane (see Fig. 2.3):

$$\displaystyle \begin{aligned} \begin{array}{rcl} V(\rho) = v_{\text{max}} - \frac{v_{\text{max}}}{\rho_{\text{jam}}} \rho, {} \end{array} \end{aligned} $$

(2.1)

with v _max the maximum speed and ρ _jam the jam density. Note the use of capital V to indicate that the speed is expressed as a function of density ρ. Therefore, V (ρ) is the density-speed fundamental diagram and later we will also encounter Q(ρ) as the density-flow diagram, V (s) as the spacing-speed diagram and S(v) as the speed-spacing diagram.

Fig. 2.3

The Greenshields fundamental diagram. (a) Density-speed. (b) Density-flow

2.2.1 Fundamental Diagrams in Macroscopic Models

The model tree shows that since the 1930s, many other shapes of fundamental diagrams have been proposed, mostly for use in combination with macroscopic models. The Daganzo (1994) fundamental diagram is probably the most widespread due to its simplicity. It is bi-linear (triangular) in the density-flow plane (see Fig. 2.4):

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q(\rho) &\displaystyle =&\displaystyle \begin{cases} v_{\text{max}} \rho &\displaystyle \mbox{if }\rho < \rho_{\text{crit}} \\ \frac{\rho_{\text{crit}} v_{\text{max}}}{\rho_{\text{jam}}-\rho_{\text{crit}}}\left( \rho_{\text{jam}} - \rho \right) &\displaystyle \mbox{if }\rho \geq \rho_{\text{crit}} \end{cases} {} \end{array} \end{aligned} $$

(2.2)

with v _max, v _crit the maximum and critical speed, respectively, ρ _jam, ρ _crit, the jam and critical density, respectively. A single parameter $w=\frac {\rho _{\mathit {crit}} v_{\mathit {max}}}{\rho _{\mathit {jam}}-\rho _{\mathit {crit}}}$ is sometimes used to denote the congestion wave speed. Figure 2.5 shows the Smulders fundamental diagram (Smulders 1990), which is a combination of the previous two: it is parabolic for low densities and linear for high densities (parabolic-linear):

$$\displaystyle \begin{aligned} \begin{array}{rcl} Q(\rho) &\displaystyle =&\displaystyle \begin{cases} v_{\text{max}} \rho - \frac{v_{\text{max}}-v_{\text{crit}}}{\rho_{\text{crit}}} \rho^2 &\displaystyle \mbox{if }\rho < \rho_{\text{crit}} \\ \frac{\rho_{\text{crit}} v_{\text{max}}}{\rho_{\text{jam}}-\rho_{\text{crit}}}\left( \rho_{\text{jam}} - \rho \right) &\displaystyle \mbox{if }\rho \geq \rho_{\text{crit}} \end{cases} {} \end{array} \end{aligned} $$

(2.3)

Fig. 2.4

The Daganzo fundamental diagram. (a) Density-speed. (b) Density-flow

Fig. 2.5

The Smulders fundamental diagram. (a) Density-speed. (b) Density-flow

It still debated what is the best shape for a fundamental diagram, and how that relates to the applications. This has led to the development of even more shapes such as the exponential and power fundamental diagrams, named after their shape (del Castillo 2012). These are further generalized into a generic model, which also includes the bilinear fundamental diagram for certain parameter choices. The main advantage of the generic model (and also the exponential and power models), lies in the fact that it can be expressed in a single formula, i.e. not consisting of two branches that need to be defined separately such as in the bi-linear or parabolic-linear fundamental diagram. Furthermore, by restricting the choice of the invertible function ϕ and the model parameters, the models satisfy the requirements that will be discussed later (Sect. 2.3). The generic model is as follows:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat q(\hat \rho) &\displaystyle =&\displaystyle b + (a-b)\hat \rho - \phi^{-1}(\phi(a\hat \rho) + \phi(b(1-\hat \rho)) - \phi(0)) {} \end{array} \end{aligned} $$

(2.4)

with parameters a > 0, b > 0. $\hat \rho =\rho /\rho _{\text{jam}}$ and $\hat q=q/q_0$ are introduced to make the variables dimensionless. q ₀ is a reference flow and not directly related to a certain traffic flow property.

If the function ϕ is chosen appropriately, then the generic model leads to a realistic and useful fundamental diagram. Examples for ‘sound’ functions ϕ and parameters are the power function (see Fig. 2.6):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi(\rho) = \rho^\theta {} \end{array} \end{aligned} $$

(2.5)

with θ > 1, a = v _free∕w, b = 1 and q ₀ = wρ _jam, and exponential function (see Fig. 2.7):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi(\rho) = e^{\alpha \rho} - 1 {} \end{array} \end{aligned} $$

(2.6)

with α > 0, $a=\frac {v_{\text{free}}}{w(1-e^{-\alpha b})}$, $b=\frac {1}{1-e^{-\alpha a}}$ and q ₀ = wρ _jam.

Fig. 2.6

The Power function fundamental diagram, with shape parameter θ = 5. (a) Density-speed. (b) Density-flow

Fig. 2.7

The Exponential fundamental diagram, with shape parameter α = 2. (a) Density-speed. (b) Density-flow

2.2.2 Fundamental Diagrams in Microscopic Models

The fundamental diagrams discussed above, are mostly applied in macroscopic traffic flow models. However, many microscopic traffic flow models also include a fundamental diagram. In this case, it is usually ‘hidden’ in the formulation of the car-following model. Furthermore, in microscopic models, spacing and speed are often used as main variables, and therefore it is more natural to express the fundamental diagram in these terms. As an example, the Optimal Velocity Model (OVM, Sect. 3.2.1, (Bando et al. 1995)) includes a fundamental diagram with a hyperbolic tangent function in the spacing-speed plane:

$$\displaystyle \begin{aligned} \begin{array}{rcl} V(s) &\displaystyle =&\displaystyle c_1 \big( \tanh [ c_2 (s-c_3) ] + c_4 \big) {} \end{array} \end{aligned} $$

(2.7)

c ₁, c ₂, c ₃ and c ₄ all nonnegative scaling parameters, see Fig. 2.8. The parameters are not straightforward to interpret, but Fig. 2.9 gives some indications of their relevance.

Fig. 2.8

The fundamental diagram of the Optimal Velocity Model (OVM). (a) Spacing-speed. (b) Density-speed. (c) Density-flow

Fig. 2.9

The optimal velocity function used in OVM, with indications of the interpretation of the parameters

The Intelligent Driver Model (IDM, Sect. 3.2.1, (Treiber et al. 2000)) includes a fundamental diagram that expresses the (equilibrium) headway as a function of the speed:

$$\displaystyle \begin{aligned} \begin{array}{rcl} S(v) &\displaystyle =&\displaystyle \left( s_{{\text{jam}}} + T v \right) \left[ 1- \left(\frac{v}{v_{\text{free}}}\right)^\delta \right]^{-1/2} {} \end{array} \end{aligned} $$

(2.8)

with s _jam the jam spacing and T the minimum time headway, v _free the free flow velocity (desired maximum speed) and δ the acceleration exponent, see Fig. 2.10.

Fig. 2.10

The fundamental diagram of the Intelligent Driver Model (IDM). (a) Spacing-speed. (b) Density-speed. (c) Density-flow

2.3 Properties and Requirements

The previous section suggests that there are many different possible shapes of fundamental diagrams, and a few more will be introduced in the next section. Some are more popular than others, however, there is some agreement on basic requirements for the fundamental diagrams.

2.3.1 Requirements

We refer to Fig. 2.11 for an illustration of some of the properties of and requirements for fundamental diagrams. The three most important requirements for their shape are as follows:

• 1. A finite maximum speed exists and it is reached when density approaches zero: $V(\rho )_{\lim \rho \rightarrow 0} = v_{\text{max}}$, with v _max finite.

• 2. A finite maximum density exists and at this density the speed is zero: V (ρ _jam) = 0, with ρ _jam finite.

• 3. When density increases, speed does not increase, and when density decreases speed does not decrease: $\frac {{\mathrm {d}} V}{{\mathrm {d}} \rho } \leq 0$ for all feasible densities ρ ∈ (0, ρ _jam].

Fig. 2.11

Example of a fundamental diagram and its properties

2.3.2 Properties: Capacity, Free Flow and Congestion

When the requirements are combined with the definition of flow (q = ρv), we also find that the flow is zero at the density extremes ρ = 0 and ρ = ρ _jam. Furthermore, there is a maximum flow, known as the capacity flow, at some density between both extremes. The corresponding density and speed are called the critical density (ρ _crit) and the critical speed (v _crit), respectively. This also splits the fundamental diagram into two branches: (1) a free flow branch with densities below critical, velocities above critical and increasing flow for increasing density and (2) a congestion branch with densities above critical, velocities below critical and a decreasing flow for increasing density.

2.3.3 Additional Requirements

In some cases, additional requirements are proposed:

• 1. The fundamental diagram must define speed as a unique function of density.

• 2. The fundamental diagram must be continuous.

• 3. The Q(ρ) fundamental diagram must be concave, or even strictly concave.

The first additional requirement excludes fundamental diagrams that allow different speeds for the same density, such as those used to model hysteresis (see Sect. 2.4). The second additional requirement excludes fundamental diagrams with a discontinuity around capacity, such as those used to model a capacity drop (see Sect. 2.4). The third additional requirement excludes Q(ρ) fundamental diagrams that are linear for a certain portion of the density domain (they are not strictly concave) and those that are convex for a certain portion of the density domain. This includes some of the fundamental diagrams in the next section, but the strict concavity requirement also excludes widespread ones such as the bi-linear and the linear-parabolic fundamental diagrams.

The main argument behind all these requirements originates from the possibility of non-unique solutions to simple problems. More specifically, when a fundamental diagram that does not satisfy the requirements is applied in a simple macroscopic model to calculate traffic states, the solution could be non-unique, i.e. multiple solutions to the same problem exist. For example, it is possible that a simple problem describing the growth of the queue upstream from a traffic light, does not have a unique solution. This issue will be explored deeper in Sect. 4.1.2.

Furthermore, the slope of a realistic Q(ρ) fundamental diagram corresponds with the propagation speed of information at the corresponding density. In particular: the slope at zero density Q′(0) equals the maximum vehicle speed and the slope at jam density Q′(ρ) equals the congestion wave propagation velocity − w. This is, for example, the speed at which the front of the queue propagates upstream after a traffic light turns green. This all relates to the characteristic speed, which will also be discussed in more detail in Sect. 4.1.2.

2.4 Scatter in the Fundamental Diagram

Observed density-flow plots usually show a wide scatter, see Fig. 2.12. Much of the scatter can be explained by non-equilibrium traffic conditions (Zhang 1999; Laval 2011; Schnetzler and Louis 2013). When vehicles accelerate or decelerate, or when their headways change, their (and their drivers’) behavior may be different from what may be observed when all of these variables are constant. E.g. when headways increase, a relatively high speed may still be perceived as ‘safe’ and thus acceptable. Zhang defines traffic to be in equilibrium if over a sufficiently long time (t) and road length (space x), velocity and density do not change: ∂v∕∂t = 0, ∂ρ∕∂t = 0, ∂v∕∂x = 0 and ∂ρ∕∂x = 0. Only points in the scatter plot that satisfy these criteria can be used to fit the fundamental diagram. Furthermore, lane changes also influence speeds and densities, and traffic can be considered to be out of equilibrium when vehicles are moving from one lane to the other. Certain branches of the family of fundamental diagrams in the model tree try to explain scatter in different ways, regardless of biased observations.

Fig. 2.12

Scatter in an observed density-flow plot, also showing a capacity drop (picture adapted from Calvert et al. (2016))

Scatter is mostly explained through vehicle properties and driver behaviour:

Capacity drop and hysteresis:

Scatter can be (partially) explained by a capacity drop: just before the onset of congestion, the outflow out of a bottleneck is known to be higher than in congestion, see Figs. 1.4 and 2.13a. The capacity drop has been explained by a low acceleration rate of vehicles leaving congestion, while vehicles decelerate at a higher rate when entering congestion (Edie 1961; Cassidy and Bertini 1999). Similar explanations including differences in acceleration and deceleration lead to fundamental diagrams with hysteresis (Newell 1965; Treiterer and Myers 1974; Zhang 1999), see Fig. 2.13b.

Fig. 2.13

Fundamental ‘relations’ based on scatter in observations. (a) Fundamental diagram with capacity drop. (b) Fundamental diagram with hysteresis. (c) 3-phase fundamental ‘relation’: lines and gray area are admissible traffic states. (d) 3-dimensional fundamental diagram with multi-class approach. (e) Fundamental diagram with capacity drop explained through difference in net time headway τ

Heterogeneity:

An other approach to varying capacities is introduced in multi-class models. The fundamental diagram takes into account heterogeneity among vehicles and drivers. For example, trucks may be slower than cars, but occupy more space, leading to a lower jam density when there are many trucks. Therefore, the flow is a function of both the density of cars and the density of trucks. For example, Chanut and Buisson (2003) propose a three-dimensional fundamental diagram, see Fig. 2.13d. The figure shows that if truck densities are relatively high, capacity is low.

From a different perspective, it has been argued that observations show too much scatter to derive a unique fundamental diagram from (Kerner and Rehborn 1996; Kerner 2009). It is proposed to use a three-phase approach characterized by the existence of three phases, one of them featuring wide scatter in the density-flow plane, see Fig. 2.13c. As a result, the maximum flow (capacity) of a road may vary over time. The idea of stochastic capacity is also explored by Srivastava and Geroliminis (2013); Calvert et al. (2016). The fundamental diagram including capacity drop and hysteresis are non-unique: in a region around capacity the flow is not uniquely defined by the density, but also depends on previous traffic states. Therefore, for a unique solution of the model when it is applied in a dynamic setting, additional assumptions on the transitions between the branches are needed (Zhang 2001).

The approach to including heterogeneity in the fundamental diagram by Chanut and Buisson (2003) makes very clear how the traffic throughput not only depends on density, but also depends on the composition of traffic. It is found that a large fraction of trucks, which—at least in Europe—drive at low speeds results in a lower flow. While, if there are only fewer trucks, cars and possibly also trucks drive at higher speeds. This approach leads to a unique flow, when densities of each type of vehicle are given. The core ideas have been implemented in many multi-class macroscopic traffic flow models, which are discussed in Chap. 4.

Schnetzler and Louis (2013) take a similar approach: explaining scatter through differences between cars and trucks. However, they also include a capacity drop, explaining this through differences in net time headway. This results in a fundamental diagram with a congestion branch (and congestion capacity) dependent on the net time headway, see Fig. 2.13e.

Problem Set

Assessment of the Fundamental Diagrams

Consider the fundamental diagrams discussed in this chapter, in particular:

• 1. The Greenshields fundamental diagram (parabolic in Q(ρ))

• 2. The Daganzo fundamental diagram (bi-linear in Q(ρ))

• 3. The Smulders fundamental diagram (parabolic-linear in Q(ρ))

• 4. The OVM fundamental diagram (hyperbolic tangent in V (s))

• 5. The IDM fundamental diagram

• 6. The power function fundamental diagram

• 7. The exponential fundamental diagram

2.1

Do the fundamental diagrams satisfy the requirements in Sect. 2.3.1? Does this depend on the parameter values, and if so, how?

2.2

For which fundamental diagrams are the critical density and critical speed explicitly defined as parameters? For the other models: what are the critical density, critical speed and capacity? How do they depend on other parameters?

2.3

Do the first five fundamental diagrams satisfy the additional requirements in Sect. 2.3.1? Does this depend on the parameter values, and if so, how?

Power and Exponential Fundamental Diagram

Consider the power and exponential fundamental diagram as discussed in Sect. 2.2.

2.4

Choose either of these fundamental diagrams and:

• 1. Draw the fundamental diagram using your method of choice (by hand or using a computer program like Octave or Matlab).

• 2. Change the parameters and see how the fundamental diagram changes.

• 3. Which parameter settings recover the bi-linear (Daganzo) fundamental diagram?

For better insight into the interpretation of the parameters, it can be useful to rewrite the fundamental diagram (2.4) such that it again expresses the flow as a function of the density (instead of the dimensionless flow as a function of the dimensionless density).

2.5 (Advanced)

Rewrite (2.4) expressing flow as a function of density, i.e. find Q(ρ).

del Castillo (2012) proves that when the following conditions all hold, then the generic model (2.4) is strictly concave:

• ϕ(0) ≥ 0

• ϕ′≥ 0

• ϕ″ ≥ 0

We encourage the interested reader to study this proof.

2.6 (Advanced)

Proof that the power and exponential fundamental diagrams are strictly concave.

Heterogeneity in the Fundamental Diagram

Consider the multi-class fundamental diagram as introduced in Chanut and Buisson (2003). It applies a Smulders fundamental diagram, with maximum speed parameter v _max unequal for cars (v _max,car) and trucks (v _max,truck). Furthermore, the other parameters of the fundamental diagram depend on the car density ρ _car (the average number of cars per unit length) and the truck density ρ _truck (the average number of trucks per unit length):

$$\displaystyle \begin{aligned} \begin{array}{rcl} \rho_{\text{crit}} = \beta \rho_{\text{jam}} \quad\mbox{and} \quad\rho_{\text{jam}} = \frac{\rho_{\text{car}} + \rho_{\text{truck}}}{L_{\text{car}}\rho_{\text{car}} + L_{\text{truck}}\rho_{\text{truck}}} {} \end{array} \end{aligned} $$

(2.9)

with L _car and L _truck the gross vehicle lengths of cars and trucks, respectively. I.e. L _car is the front to front distance between 2 cars in a queue (at standstill). Note that the critical speed v _crit is equal for both classes and independent of the traffic state.

2.7

Use these parameter values v _max,car = 30 m/s, v _max,truck = 25 m/s, v _crit = 20 m/s, β = 0.2, L _car = 6 m, L _truck = 18 m to calculate jam density ρ _jam and critical density ρ _crit in the following situations:

• (a) only cars on the road

• (b) only trucks on the road

• (c) 10% trucks, 90% cars

• (d) 50% trucks, 50% cars

2.8

Draw the axes for a density-speed fundamental diagram with on the horizontal axis the total density (cars and trucks), on the vertical axis speed.

• Draw the fundamental diagrams of cases (a) and (b).

• Add the fundamental diagrams for both cars and trucks for cases (c) and (d).

• Under which conditions do the car and truck fundamental diagrams overlap? I.e., when are the speeds of the cars equal to those of trucks?

2.9

Answer the questions below using the graph from the previous problem.

• Under which traffic conditions do the car and truck fundamental diagrams overlap? I.e., when are the speeds of the cars equal to those of trucks?

• Is this realistic? Why (not)?

Fundamental Diagrams Explaining Scatter

Consider the fundamental diagrams that explain scatter in Sect. 2.4.

2.10 (Advanced)

Reflect on whether these models satisfy the requirements in Sect. 2.3.1.