*2.4.2. The MOBIL lane-changing model*

Many microsimulation models have been developed in the last few decades. The choice of a suitable microsimulation model mainly relies on the traffic features of interest, e.g. during freeflowing traffic or congestion. For bridge-loading applications, the microsimulation model should be able to reproduce the range of traffic states likely to occur on a bridge. Once calibrated, microsimulation enables the modelling of a large number of congestion events (and the subsequent identification of extreme loading events), which would be extremely difficult

Microsimulation models divide into *car-following* (single-lane) and *lane-changing* (multi-lane) models. While in short-span load models lane-changing manoeuvres can be neglected, they play an important role in the context of long-span bridge loading. In fact, when overtaking is allowed, the car-truck mix for congestion is likely to be different to that for free traffic, since car drivers do not feel comfortable following trucks As traffic slows down and therefore tend to overtake them [49]. This typically results in longer truck-only platoons in congested traffic

The *Intelligent Driver Model* (IDM) is a car-following model which has been shown to successfully replicate observed multi-lane congestion patterns on several motorways With few parameters using a single-lane traffic stream made up of identical vehicles with deterministic parameters [36, 50]. The IDM has also been proven able to replicate observed single-vehicle trajectory data [51–55]. The motion of each vehicle is simulated through an acceleration

( ) <sup>4</sup> <sup>2</sup> \*

ì ü D

There are five parameters in this model (*v*0, *s*0, *T, a, b*) to capture driver

î þ (5)

(4)

0 dv(t) v t s (t) a 1 dt v s(t) é ù æ ö æ ö =- - ê ú ç ÷ ç ÷ è ø è ø ë û

the current gap to the front vehicle and *s*\*(*t*) the *desired minimum gap*, given by:

v(t) v(t) s t s max Tv t ;0 2 ab

=+ + í ý

in which *s0* is termed the *minimum jam* (bumper-to-bumper) *distance, T* the *safe time headway*, Δ*v*(*t*) the speed difference between the current vehicle and the vehicle in front and *b* the

behaviour, which are relatively easy to measure. For simulation purposes, the length of the

 The desired minimum gap s\* is limited to be greater than or equal to the minimum jam distance s0, as noted in Eq. (5); otherwise, an inconsistent driver behaviour in multi-lane scenarios may be generated when the front vehicle is faster [18,

( ) ( ) \* 0

in which *a* is termed the *maximum acceleration, v*0 the *desired speed, v*(*t*) the current speed, *s*(*t*)

to record in the real world.

34 Structural Bridge Engineering

than in free-flowing traffic.

function:

*comfortable deceleration*.

6

56].

6

*2.4.1. The Intelligent Driver Model*

The MOBIL lane-changing model has been proposed in [59], to which the reader is referred For a detailed description. An overview of the model is given here, whereas details about the application of MOBIL to bridge loading can be found in [60].

**Figure 3** depicts a lane change event, in which the subscript *c* refers to the lane-changing vehicle, *o* refers to the old follower (in the current lane) and *n* to the new one (in the target lane). All the accelerations, current and proposed (i.e. before and after the lane change), can be calculated according to the IDM given in Eqs. (4) and (5).

**Figure 3.** Vehicles involved in a lane-changing manoeuvre (adapted with permission from [59]).

A lane change occurs if both *incentive* and *safety* criteria are fulfilled. For a *slow-to-fast* lane change, the incentive criterion is expressed as follows:

<sup>7</sup> The traffic stability is the response of the traffic flow to a perturbation, e.g. a braking vehicle. If the traffic flow is unstable, then a perturbation will break the flow down and propagate into a stop-and-go wave (such as those shown in **Figure 1**).

$$
\widetilde{a\_c}(\mathbf{t}) - a\_c(\mathbf{t}) > \Delta a\_{th} + \Delta a\_{bias} + \mathbf{p} \left( a\_n(\mathbf{t}) - \widetilde{a\_n}(\mathbf{t}) \right) \tag{6}
$$

This means that the acceleration advantage − in performing a lane change must be greater than the sum of the *acceleration threshold* Δ*a*th, which prevents overtaking with a marginal advantage, the *bias acceleration* Δ*a*bias, which acts as an incentive to keep in the slow lane, and the imposed disadvantage to the new follower in the fast lane − , weighted through a *politeness factor p*, to account for the driver aggressiveness. On the other hand, the incentive criterion for a *fast-to-slow* lane change is as follows:

$$
\Delta \overline{a\_c}(\mathbf{t}) - a\_c(\mathbf{t}) > \Delta a\_{th} - \Delta a\_{bias} + \mathbf{p} \left[ \left( a\_n(\mathbf{t}) - \overline{a\_n}(\mathbf{t}) \right) + \left( a\_o(\mathbf{t}) - \overline{a\_o}(\mathbf{t}) \right) \right] \tag{7}
$$

In this case, the acceleration advantage − must be greater than the sum of the acceleration threshold Δ*a*th, minus the bias acceleration Δ*a*bias (which acts as an incentive to move back to the slow lane), plus the disadvantage imposed to both new follower *n* in the slow lane and to the current follower *o* in the fast lane, weighted through the politeness factor *p*. 8

Finally, the safety criterion limits the imposed deceleration to the follower *n* in the target lane to the *safe braking* value *b*safe:

$$
\tilde{a}\_{\imath}\left(t\right) \cong -b\_{\ast\ast\ast} \tag{8}
$$

#### **2.5. Summary**

Traffic data for bridge-loading applications is typically collected at high-speed WIM stations. Free-flowing traffic measurements are unbiased, reliable and now commonly available. Generally speaking, if traffic information is available at a point detector, such as a WIM station, it is possible to accurately reconstruct vehicle positions from single-vehicle data only when traffic characteristics do not change significantly. This is the usual case in free-flowing traffic and therefore applicable to short-span bridges.

On the contrary, there is a shortage of data during congestion, mainly Due to current technological limitations. In addition, the analysis of traffic data can pose some issues: in fact, a vehicle's speed is likely to fluctuate, e.g. as a result of *stop-and-go waves*, making the estimation of vehicle positions from point measurements problematic. The use of spatial detectors (such as cameras) over a stretch of road allows the collection of vehicle positions during congestion, without resorting to estimation. Although cameras are the best solution from a theoretical point of view, they are not often used for several practical reasons.

<sup>8</sup> Eq. (7) is based on [61] and preferred to the formulation in [59], which does not include the disadvantage to the new target follower an−ãn in the fast-to-slow lane change.

Traffic microsimulation provides a valuable tool for long-span bridge loading, as it is capable of reproducing realistic congested scenarios from free-flowing traffic measurements. Load effects can be computed directly from the actual spatial distribution of vehicles, thus avoiding any inaccuracy due to the estimation of vehicle positions from point measurements.
