Algorithm of Calculating the Maximum Permissible Speed of Traffic

Copyright © ITAC Algorithms and Architectures for Real-Time Control, Cancun, Mexico, 1998

ALGORITHM OF CALCULATING THE MAXIMUM PERMISSIBLE SPEED OF TRAFFIC

E. GAVRILOV *, N. ZHEMCHUGOV **

* professor, doctor of technical science, vice- rector of Kharkiv State Automobile & Highway Technical University, head of the Highway Design and Pioneering Chair, e-mail: [email protected] ** graduate engineer of the Highway Design and'Pioneering Chair

Abstract: The criterion and algorithm of calculating the maximum permissible speed of traffic on the automobile road in use are examined in the paper. The evaluation of the maximum permissible speed proceeds from technical, ergonomic and ecological requirements specified for the automobile road and traffic. The fulfillment of these requirements ensures the compatibility oftechnosphere and biosphere. Copyright © 1998 IFAC Keywords: Speed limit, performance criterion, the evaluation of the transportation system status, the algorithm of calculating the maximum permissible speed.

I. INTRODUCTION

2. THE CRITERION FOR LIMITING TRAFFIC SPEED

The expediency of limiting speed of traffic on automobile roads in use is based upon the necessity for increasing the road safety level, reducing the harmful effect produced by traffic on the environment and saving fuel. The maximum permissible speeds are different for each of the listed requirements since they are calculated on the basis of the operational condition analysis of different components of the ' man-automobile-traffic environment' system. The absolute prohibition of automobile traffic proves to be the best solution for the system on the whole. On the other hand, the very idea of limiting traffic speed is in conflict with the requirement of the most effective use of automobile roads for their designated purposes, which causes the increase of speed.

We see the way out of contradictoriness of local requirements in the transition to more important goals of traffic organization, i.e. to the requirements set by systems of the higher level. To our mind, such a goal can be ensuring the compatibility oftechnosphere and biosphere in the broad sense and compatibility of biosphere and the 'man -automobile-traffic environment' (MATE) system of human activity in the narrow sense. The structure of traffic environment includes a road and the part of biosphere the effect of which upon the system is not equal to zero. The result of the compatibility of technosphere and biosphere is viability of the MATE system. Therefore either the probability of the given system's destruction or the time of its existence (the system's life span) can serve as indexes of such compatibility. The task of automobile roads design and use and traffic organization is the creation of conditions ensuring the maximum time of the MATE system's existence. One of such conditions is a selection of the optimum behavior of the system's components.

The article suggests one of the possible ways of solving the conflict being considered. .

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The simplest mathematical model of behavior optimization for the system's components can be formulated as:

3. THE EVALUATION OF THE SYSTEM STATUS The description of the system status is very significant in formulating the defmed problem. For the quantitative evaluation of the status vector it is offered to use the complex index of adequacy of technosphere to biosphere in the following form (E.Gavrilov, 1997):

Suppose we have the 'man-automobile-traffic environment' system the status of which is determined by the n-dimensional vector F . The coordinates of the system's components status fh f2, .... fn increase with constant speeds U(uI , u2, ... u n ),Uj ~O . Biosphere is described by a

(3)

set of conditions C=(c\ , C2 ''''C n ) in which the system can be. In the i-th condition the effect is produced on the corresponding i- th coordinate of the system t; which reduces it at a speed Vj ,i.e. in the condition C the 'man-automobile-traffic environment' system moves at a speed Vj(uI,u2, ...Uj-Vj, ...v n ). At any moment the system can pass from one condition to another. No constraints are imposed.

where F \ is an index of adequacy provided that factors of interaction between technosphere and biosphere do not exceed the accepted values, b is an index which takes account of the action offactors exceeding the accept values, y is index of gain or attenuation of interacting factors.

Let's also consider the system existing if the vector F does not fall outside the limits of accepted values domain bounds of which are specified by the equation L( F)= O. At the bound of accepted values domain F = Fg ,where Fg is an accepted value of

The index of adequacy F \ is calculated on the basis of the formula:

the vector F. Values of controlled variables can lie only in the following range: t; ~ 0, i=l,n.

Ifja F _.o.::i-C-'-I_ _

n

I -

n

Ia i i-I

It is necessary to fmd the optimum behavior tactics

(the rule of conditions change) for the 'manautomobile-traffic environment' system that would maximize the time of the system's existence T, i.e. T~max.

where aj is a weight coefficient of the i- th coordinate of the status. The index 'b' is calculated on the basis formula:

(1)

then

according

to

at ct>i < ct>gi'

b = 0,

If accepted values domain of the vector F is bounded by a ndimensional ellipsoid and L(F) = ~) i 2k j -I = 0

(4)

m

1

b=I[1+(fi --)], atct>i ~ct>gi'

the

j=1

research conducted by I. Linnik (1996) the local rule of conditions change takes the following form:

a

(5)

j

where ct>j , ct>gi is an interaction factor and it's accepted value; m is a number of such components of the system for which ct>j ~ ct>gj.

(2)

The index 'y' is calculated on the basis of the formula:

In line with this rule, the control action maximizing the time T must be directed to such a component of the 'man-automobile-traffic environment' system for which the product k f v is a minimum. Therefore the solution of the problem of limiting speed of traffic on automobile roads can be executed on the basis of status analysis only of that component of the MATE system for which the product k i f; Vi is also a minimum. If those products are equal then limiting speed of traffic should be executed proceeding from the requirements of all components of the system.

I S y=-UYj,at S>I,

S J=1

y=O,

atS=O,

where S is a number of components which enhance or reduce load on biosphere at the expense of their joint operation;

192

(6)

4. THE CALCULATION OF THE MAXIMUM PERMISSIBLE SPEEDS OF TRAFFIC

(7) In view of everything stated above the maximum permissible speed of traffic on a road in use can be calculated on the basis of the condition 'Pi is function of f; which adopt numerical values in view of the direction of mutual effect of components (the value 'Pi can be signed plus as well as minus). Functions 'Pi are calculated by experiment.

F=Fg

(12)

or n

(13)

IfiUi =3 . i=l

Coordinates of the status of the system's components and their weight coefficients are calculated on the basis of factor of interaction:

If coordinates of the status of the system's components are evaluated on the basis of traffic speeds then in view of(13) the maximum permissible speed will be calculated on the basis of the formula :

(8)

where <1>hi is a normal value of a factor of interaction (the functional norm).

where V gL is the maximum permissible speed of traffic by indexes of the MATE system status; Vgn Vgo> Vgc are the maximum permissible speeds of traffic by indexes of the status of the main, automobile and traffic environment accordingly.

By a normal value of a factor of interaction we mean it's functional optimum value, i.e. the value of a factor complying to the highest degree with tasks and conditions of the functioning of the i- th component of the system.

The maximum permissible speed of traffic based upon the status of the man is calculated on the basis of the formula (E. Gavrilov 1988):

The evaluation of coordinates of the status can also be executed by the results of the functioning of the system's components. In the latter case V V '

f= 1

hi

V · U · = _ hI

='l:.vmt + Vmt (1- Hm -H mo ) 3

r

'

(15)

at Hmo < Hm < (Hm +r), where V mt is the maximum possible speed of the automobile under standard conditions (the automobile 's values from the registration certificate); r = Hmi< - Hmo ; Hmi< is the maximum entropy of the man 's perception field at the traffic rate equal to the road 's traffic capacity; Hmo is the maximum entropy of the man 's perception field at the traffic rate equal to zerro; Hm is the maximum entropy of the man 's perception field at the calculated traffic rate,

Let's suppose that at the bound of accepted values domain of the vector F the actual value of the factor of interaction is equal to the accepted value, i. e. <1>i =

~gi,

(10)

Then in view of (8), (9) and (10) the accepted value of the vector of the MATE system status is equal to :

'"

V

> (Hm +r),

gl

where V, Vhi and V gi are actual, normal and accepted (permissible) speeds of traffic for the i- th component of the system.

Fa

= 3" Vmt , at Hm

gr3

(9)

V .'

1

2

Vgr

= - 3--3

(11) (16)

IUi i=l

m is a number of data carriers within the perception field of the man driving the automobile.

193

Q = 0.0548 · Mx . PT · x ·a ·PT ,

The maximum pennissible speed of traffic based upon the automobile status is calculated proceeding from the theory of interaction between the automobile and the road. Thus, at driving along narrow curves in the plan: Vga = .. 127R(!!±i n )

,

(22)

where Mx is the molecular weight of a pollutant, glmole; x is the content of pollutants, %; a is a coefficient of excess of air; PT is the fuel density, glsm 3 ; PT is the fuel consumption, 1/100 km.

(17)

where R is the radius of the curve; in is the cross slope, fraction of one; J.! is a coefficient of the lateral force (J.!=0.15).

The fuel consumption is calculated on the bases of the fonnula:

1 2 P(= - [Aik+B V+D(Ga
At curves in the plan at the limited visual range

(23)

dV + O.077k s V ± 0.113 G a - )], dt 2

(18)

where II is the indicator coefficient of efficiency; ik is the gear ratio; Ga is the calculated weight of the automobile;
where

A = 7.95a l V h io ; Hh Pt rk

La the visual range, LB = 2·, RB B is the width of the roadbed.

B = 0.69b l Vh In io . 2

Hh Pt rk 100 D= , Hh Pt lltr

At concave vertical curves the maximum pennissible speed is: (19)

Vga = . 13aR ,

Vh is the working volume of engine cylinders; Hh is the lowest heat of fuel combustion; Ih is the piston stroke; io is the axle ratio; lltp is a coefficient of efficiency of transmission; rk is the radius of the automobile's wheel rolling; al and b l are experimental coefficients,

where 'a' is the centripetal acceleration 2 (a"" 0.5- 0.7 m/s ). At grades 'i' (up to 2%) ending with horizontal section Vga =

. 127~

'

5) .

(20)

£

At the convex change of slope with mating slopes 'il ' and'i2'

al "" 45 kPa for Otto engines, al"" 48 kPa for diesel engines, b l "" 13 kPa· s·m- I for Otto engines, I b l"" 16 kPa· s·m- for diesel engines. The maximum pennissible emission Qg at the level of a residential construction is calculated on the basis of the solution of Bozanke- Pirson differential equation in the fonn :

The maximum pennissible speed based upon the traffic environment status is defined as the speed at which the running exhaust emission of pollutants produced by the automobile reaches the maximum pennissible value. In it's turn, the latter one is calculated on the basis of the maximum allowable concentration of pollutants at the border of a sanitary zone or at the level of a res idential construction. The running exhaust emission Q (g!km) of pollutants is calculated on the basis of the fonnula (N . Govorushenko 1990):

VB PX . (PDK-C)TI·I at x< (I ,- I) T Q g -

1,000

exp[-h /(P · X)]'

"

(24)

Qg

194

= YBp · X . (PDK-C
exp[-h /(P . X)] '

, T -

- ).

where PDK is the maximum allowable concentration of the pollutant in the atmosphere; Cel> is the background concentration of the pollutant; y B is the speed of the wind perpendicular to the road direction; X is the distance from the pollution source to the calculated point; h is the altitude of the automobile ' s exhaust pipe above the traffic road; P is a coefficient of the effect of pollutant's diffusion angle in the vertical plane; 13 is the distance from the pollution source to the construction; 1r is the width of the winded aerodynamic shadow of the construction; le is a coefficient of the effect of the construction, le = 1+ 0.044(X-I,+ It) + 0.0013(X- 13+ 1,)2; I is the length of the calculated section of the road; T I is the time of covering I km of the way, T= 3,600 N .

5. CONCLUDING REMARKS The given algorithm of calculation the maximum permissible speed of traffic on the automobile road i use is practically implemented as a computer program for Windows 95 written in the C++ programming language. The results of calculating by the given program make it possible to calculate the maximum permissible speed of traffic for automobiles of different types and transport on the whole. In the data base traffic environment is represented by 36 data carriers including: geometrical characteristics of the road, opposing automobiles and automobiles gang in the same direction, road signs, indexes, marking- out, planting of trees and gardens, equipping the road, etc. Speeds of traffic are calculated with due account of patterns of driver's behavior on a road. Tht: given paper extends the ideas suggested at the 8th IF AC Symposium ' Transportation System'.

REFERENCES Gavrilov E., Turenko A. (I 997). Traffic efficiency and human factor . In: Preprints of the 8th IFACIIFIP/IFORS Symposium ' Transportation System', Chania, Greece. (Papageorgion M, Pouliezos A.(Ed.», pp. 1268- 1270.

Thus, in view of (22),(23) and (24) under the set traffic conditions the condition of calculating the maximum permissible speed of traffic on the basis of traffic environment takes the following form : D .0.077 .k.s.y3 +B . i£ . y2 + +(A ik +D · G a ·
R ·3 600 ·11 '

z

. =0

,

Gavrilov E., Alekseev 0 ., Tumanov B. (1988). Computer in designing automobile roads. EMK HE Publishing House Kiev, Ukraine ..

(25)

Govorushenko N. (I 990). Fuel saving and toxicity reducing for automobile transport. 'Transport' Publishing House, Moscow, Russia.

where R = Y6 · p . X . (PDK - Co <1» . L . A·1,000 exp[ -h /(p . X)] , z = 0.0548· Mx ·Pt

·X

·u.

Linnik 1. (1996). The optimization model for volumes of work at reducing fuel consumption and amount of pollutants. In: The Bulletin of Kharkiv State Automobile & Highway Technical University. The 4th issue, RIO KhSAHTU, Kharkiv, Ukraine, pp. 28- 31.

The solution of the equation (25) for Y makes it pissible to calculate the maximum permissible speed of traffic Yge on the basis of the traffic environment status.

195