Thanks Tobias. These links are really helpful.

You can also see this question

https://www.researchgate.net/post/Where_I_can_find_the_mathematical_formulation_of_Wiedemann_99_car_following_model

You can also see this question

https://www.researchgate.net/post/Where_I_can_find_the_mathematical_formulation_of_Wiedemann_99_car_following_model

Thanks Vincenzo.

I stumbled upon this discussion of the Wiedemann 99 Model and the related thread which is very fortunate: when searching material for my textbook "Traffic Flow Dynamics", I realized as well that not even the car-following component is completely documented anywhere. Now I am preparing a second edition and maybe this community can answer me some questions I have with this model.

(1) First it looks that the typical Wiedemann car-following diagram found at umpteenth places in the worldwide web does not represent the actual model equations as presented in Kayvan Aghabayk et al, "A NOVEL METHODOLOGY FOR EVOLUTIONARY CALIBRATION OF VISSIM BY MULTI-THREADING" (which is recommended here as probably the most authoritative source). Attached is a plot of the thresholds in a dv-gap diagram for the parameters given in this paper in form of a VISSIM screenshot. Comparing it with the widespread published diagrams (see Fig. 1 of this paper), I observe the following qualitative differences (and also big quantitative differences not discussed here):

* The line BX decreases with the approaching rate Delta v=v-v_lead while it increases in the published diagrams. The reason is that, in Eq. (2) of Aghabayk et al, the *minimum* of the own and the leading speed (plus some noise) is taken as the prefactor of the time gap CC1

* the line SDV is linear, not quadratic. Moreover, it crosses the line CLDV while, in the published diagrams, these lines are parallel

(2) In Fig. 2 of the Aghabayk et al (VISSIM screenshot), the units for the parameters CC3 [s], CC4 and CC5 [m/s], and CC6 [1/(ms)] are missing. Why the parameter CC6 has such a strange denominator 17000 (which could be eliminated without loss of generality?)

(3) There are three acceleration parameters CC7-CC9. However, CC8 and CC9 relate to free traffic flow, so the only acceleration for interacting traffic is CC7 which is a very small acceleration. It seems that all the acceleration functions in the many areas defined by the dv-gap diagram are undocumented (I do not assume that they are always +/- CC7, particularly not in the area between AX and BX). Any hints where to find them?

(1) The equations in the Aghabayk et al paper are used in Wiedemann 99. All published diagrams are for Wiedemann 74 - thus the differences.

(2) cc6 has this constant in the unit to allow a "simpler" number in the parameters dialog.

(3) The equations to generate the accelerations used in the various interaction states can be found in the Vissim code. There are many additional effects (e.g. lane changes fpr routes, parking, overtaking) and parameters (e.g. the min/max/desired acceleration/deceleration functions) which can affect this. At the moment, we have no time to write all this down in a readable form, sorry.

Dear Umair,

Do you have the copy of the article that was in your question? If so can you upload it here? Thank you.

Wiedemann, R. & Reiter, U., 1992. Microscopic traffic simulation: the simulation system MISSION, background and actual state. Project ICARUS (V1052) Final Report. Brussels, CEC, 2.

@Chandan, I downloaded this report from my university's library. Unfortunately, I can't share the file without permission.

@Lukas Kautzsch, Could you please confirm the units for CC6 please? Are they (1/ms) as Martin Treiber mentioned? Is the number 17000 used for any unit conversion?

The unit for CC6 is indeed 1/(m*s). The value is divided by 17000 and then multiplied with the squared following distance, then reduced by CC4 in order to determine CLDV. Similarly with CC5 for OPDV. As I said above, the number 17000 is in there only to allow the visible CC6 value to have less leading decimal zeros.

Maybe I should add that the units for CC3, CC4, CC5 and CC6 are always these metric units, regardless of the units selected for the display of other values in Vissim. That's why they are not shown in the dialog.

@Lukas Kautzsch Thanks for clarifying the units. I recently contacted Vissim technical support and I got the reply that the units for CC4 and CC5 are ft/s and CC6 is unitless. Kindly add the units in the Vissim driver behavior dialog window because this is really confusing especially for academics.

i am having one doubt about 74 model that it is defined as

 The distance d between two vehicles is computed using this

formula:

d = ax + bx

where ax is the standstill distance

bx = (bx _ add + bx _mult * z) * v

v is the vehicle speed [m/s]

z is a value of range [0,1] which is normal distributed around 0.5 with a

standard deviation of 0.15.

by this can we calibrate ax,bx_add,bx_mult  

if we know the distance between the vehicles

can we run by some optimization technique?

d = ax + bx is not the actual distance between two cars but a threshold where the behaviour changes. In principle, the complete model (not only the mentioned 4 parameters) can be estimated jointly by calibrating the model on suitable data (trajectory data including observed distances containing all relevant traffic states). However, because of the stochasticity in the model, this gives rise to stochastic objective functions and even defining a correct optimization goal is nontrivial (e.g., minimize the expected sum of squared errors, or maximize a log-likelihood)

i am having a doubt about stand still distance between vehicles, in some literature i had seen stand still even less than the length of the lead vehicle.

is it possible ???

You can set the standstill distance as a driving parameter value in Vissim.

@Narayana, The standstill distance is the distance between the front bumper of the following vehicle and the rear bumper of the lead vehicle. So, yes, it can be less than the length of lead vehicle.

Ah, I see what you mean now. Sometimes, the distance is given as gross value (including the length of the leading vehicle), sometimes as net value (as defined by Umair). When comparing the actual distance with the desired distance, both need to be gross values or both net values, of course.

@umair durrani i agree your point

but i had seen stand still distance(front bumper to front bumper) is less than length of lead vehicle

can we say in staggered following behavior is it acceptable for heterogeneous traffic??

@Narayana, can you share where you found such use of standstill distance?

@Lukas Kautzsch, Is CC6 angular velocity? Are the alternative units for CC6 radians per second? Please confirm.

Dear all,

I have done some analysis of the Wiedemann models since I want to

present the car-following part of them in full pedagogical detail in the new edition of my

textbook (Traffic Flow Dynamics). Below are some points that

the readers of this thread might find interesting:

(1) quantitative s-dv plots for concrete parameter sets: copies of

the classical but qualitative "Wiedemann plot" can be found everywhere

"out there". But what is the quantitative truth for the Wiedemann74 and

Wiedemann99 models? Since I could not find them, I plotted them for

specific parameter values, together

with the meaning of the threshold parameters. Notice that, seemingly, the

classical diagram is only valid for Wiedemann74.

(i) Wiedemann74_boundaries_vl20.png :

Wiedemann74 model boundaries as presented in the original paper for a

leading speed v_l=20 m/s, without driver-dependent and general

randomness. Since I could not find parameter values, I just assumed

realistic ones:

# Parameters # meaning [units]

AXadd=1.5 # minimum gap [m]

BXadd=3.5 # sqrt(v) dependence, corresp. to IDM's s1/sqrt(v0) [sqrt(s/m)]

EXadd=2. # range factor max steady-state gap/min steady-state gap

# "between 1.5 and 2.5 times the minimum following distance"[1]

CX=20. # (2*dv''(gap))^{-0.5}=CXconst*CXadd of the paper [sqrt(m*s)]

CX_CLDV=0.8*CX # ad-hoc assumption from published graphics [sqrt(m*s)]

OPDVadd=1.5 # ratio of perceived negative/positive approaching rates [1]

(ii) Wiedemann99_boundaries_v20_param1.png :

specification as given in Aghabayk et al.,

A NOVEL METHODOLOGY FOR EVOLUTIONARY CALIBRATION OF VISSIM BY MULTI-THREADING

c0=1.5 # minimum gap [m] [1.5m]

c1=0.9 # minimum time gap [s] [0.9s]

c2=8. # absolute space difference between min and max gap [m] [4m]

c3=-8. # negative steepness -d(gap)/d(dv) of the SDV curve [s] [-8s]

c4=-1.35 # (i) negative SDV speed difference for maximum steady-state gap

# (ii) negative hypothetical CLDV speed difference for zero gap [m/s][-0.35m/s]

c5=0.35 # positive hypothetical OPDV speed difference for zero gap [m/s] [0.35m/s]

c6=11.44/17000. #2*curvature |dv''(gap)| of CLDV and OPDV curves [1/(ms)][11.44/17000/m/s]

In this plot, I changed the space difference c2 between minimum and

maximum steady-state gap from the default 4m to 8m and c4 from -0.35m

to 1.35m to get a bigger (and better visible) following

regime.

(iii) Wiedemann99_boundaries_v20_orig.png:

original VISSIM defaults as given in the aforementioned paper.

Notice the small car-following regime

whose width, in the gap direction, is given by c2=4m.

(2) comments on calibration

generally, the effects of model parameters on the quantitative model

properties of a nonlinear model are coupled, so it is generally not possible to estimate a

model parameter separately from the others. Furthermore, some

parameter combinations (particularly, Wiedemann99's c6 and c5) give nearly the

same boundaries separating regimes with different acceleration

functions and therefore can hardly be calibrated separately.

More generally, what you can estimate depends on the data (Wiedemann99):

* if there are only steady-state data, the CLDV and OPDV parameters c5,c6 cannot

be estimated ("are not identified")

* if there is only free traffic, neither of c1-c6 can be estimated

* if there is only highly dynamical traffic, the AX, BX and SDX parameters

c0-c2 cannot be estimated

* if there is no stopped traffic, AX cannot be estimated.

In these cases, the default values for the not identified parameters

are just fine.

(3) Interpretation of c6 (CC6) of Wiedemann99

CC6 is, in fact, approximatively proportional to the rate of change of the visual

angle as following simple derivation shows: Given a

leader of width w at an eye-to-back-of-the-follower distance d which

can be approximated by the bumper-to-bumper gap s. Then, the visual

angle theta caused by the leader's width is given by

theta=2*arctan(0.5*w/s) approx w/s

and the angular rate of change (notice that the approaching rate dv=-d/dt(s)):

d/dt(theta)=-w/s^2 d/dt(s)=w*dv/s^2=w*CC6.

Notice that the last expression is only valid for the OPDV curve since

the CLDV curve has an additional dv-offset of -CC4.

Finally:

If you have any information of the different acceleration functions of

Wiedemann99, I would like to get informed.