Determining the Time-Dependent Trip Distribution in a Complex Intersection for Traffic Responsive Control

CI'P\11CII!

Holden B,l dl 'n

11' .\( Ctt11 11,d III 11,11 1' 1)4111.111'111 .... "lc·lll' h ·dt·l,tI R' Tldoll ' ,d (d '11ll,111\ )'1 ,"" )

DETERMINING THE TIME-DEPENDENT TRIP DISTRIBUTION IN A COMPLEX INTERSECTION FOR TRAFFIC RESPONSIVE CONTROL M. Cremer Fnchge bie / .He/3 · Ulld R ege lullgs/ec hll ik , T ec hn isc he U n/t'e rs /tat H a mburg· Har b urg. L ohbriigge r Kirc hstr . 65 , 2 050 H a mburg 80, Fed e ral R epublic of Ge rman ),

esc:~ E-:-.tr·y to &~-: J. exit of 2 for ;;r: efficieLt tr2ffic r':::.::;... o::..:i ·. . €: ~c. :_ tr-Ql of .:::ic;::c.lized i:~ '.:~~:::e c~ i o! ·..:; . :-:o·.·.'ever , in r.::any caSeS c:.c:~l: n c·. :.: .2:.:.Ct !)E:; [Ce2:oured directly b'..it r.'..ist be esti!:;ated fror. vehicle c O'.;!!t ::: "it tt, e er,tri es :'i!'.d tIle exits . P. r, e,.; rr.ethod for est i r~.atinb these flOl-i::: i s [.-r"ese:,ted LerE: Hhich not onl y evaluates the balance equations for ti,E: acc'..ir;ul:;.ted counts of enteri"G and leavinG volur:.es as previous methods have done but additionally r.akes use of an analys i s of the time sequences of enterir,e and leaving flol-ls by co rrelation techniques . The method i s 2pplied to "ir~u l ated and to real data alld it i s shown that fairly good estimates are obtained Hl'.ich are even capable to track time variations of the real flows . The cor.putational effo rt i s moderat e and may be done by a r:.icroprocesso r at t he s ite of th e i ntersect i on . :-: ·:..,.::tr·~~t .

'--C~Tl,,):

'T: '. ~

:·.:.::;!.it l;de ef ;:2.r'ti.:::.l flo·\·,IS

i:.ter::e ct io:·: i"

c,

::'o,,::,ic

~"'ro:·:·.

ir:for :~2ti c:,

KeYl-lords . Traffi c flol-l identifi cation ; origin - dest i nation mat rix ; int e r :::ec tio!J cor,t r ol ; corre lation methods .

r:;TPODUCTIOil

at all . For thi s rea so n a number of methods has been developed in th e past to estimate the partial floHs from each o ri g i n (o r entry) to each destination (o r exit) from some few volume measurements , prefera bly taken at the entries and the exits of the intersection . These me th ods use only the accumulated traffic volumes du ri ng the period of data col lection and omit that part o f the gathered information Hhich i s con tained in the spec ifi c pattern of the time se ries when tIle r:easurerr.ents are taken as a sequence of sampled data . Us i ng only the ac cumulated traffic counts fror: the Hhol e period of ob se rv"iti on lead s to a highly ullderdete r r.ined set of linear equatio~s f or the unknoloi n partial flows , w'1icr. c an be sa tisfieC by a large r:~ltipli c ity of solutions .

The capac i ty of a road netHork higllly depends on the throughput of its s i gnaliz ed intersect ions. Traff i c siGnal systems usually ope rate i n a time of day mode Hith a predetermi ned timing or i n a traffi c re s ponsive mode based on the measured volumes at some few key lo cat ions . ~oHever , as is knoHn from inve s ti gatio~s , traffic flol-l in a r oad netHork not only changes with respect to it s volumes but also Hit h re spect to it s structura l or i gindestinatio~ relatiors~1ips , and t!1es~ c tlanges occur fro ~:: hour to ~ :ou r "s well as fror: day to day . For eX2~, ple , it has been s i:cl-i:. by va n Vliet and ~illu~ s e~ ( 198 1) using a cor:pre~e~s ive data tase that these vari at ions are i~ fact conside rable . Consequen tly , a r:ore effic i e nt use of a road networ k can be e xpected , i f t he con trol st r ategy not only r eflects 50r:e r:e"i:ured volur:es but i s based on the k~o~l Ed~e of tt~e actual or i ; in - desti ! :at i o~ patter~ of its sig ~:alizet i:1ter sectior!s . It is 3 t~e

~e cessa~j

prerequi s ite fer ti1is

To obta in still a unique

t~~ at

trsffic flo~s t~:ro~~ h 2. co ~: ­ er t~ro~~~ ~ part ef a net te c~ser~ed or deter~i!:ed ccn t~:e

~lex

i! 1tersect i o ~:

~ o rk

:a~

t i~~ o~sly

~y

o~ - li~e

~eaSGre~e=:ts .

i~troduce ~ore

~ 5ee

va~ Z~ylE:~

t~e

or l ess 2rt i trary additional assur:ptions or conditions. So~e of t!1e~ dEter~ine a sol~tio~ wti ct i s ::-:ir.i:.:al ',.Jit:: re spect to t::e nor:.-. of t:-:e :7".2 trix e le::-.e:~ts : e . [;. . Seil, '979 ) , otl~er try to find a sol~tic~ ~~i c ~ is r:ost likely ~ it~: res~e:t to sc~e ~easure o f likely~ood

ti :~e - ~e;e~:di~[ o ri€i:~ - desti~at i o~ ~a ­

tr i ces ef

solut ic~,

exi 2 ti~g ~eti 1ods

for

e x a~~le

1 ;50 ! e r '::a~_:-:-.-s~ s:.G ct:; E:~ S ,

cedu res

i~~olve

: ertai~

C!: t~:e 5truct~ r e

ho ~ e v er ,

a~d

~ ~~ i le vo l ~~e

data at cross - sections can easily be collected , ti:e partial flows fro~ sny e ::t ry to t!:e d iffer'ent e xi ts of an intersect i o~ can directly be r:easured only Hit h a rather expensive inst r umentation or even not

ot~ers ,

u~k~o ~ !~

19E~)

va ~i atles

or reduce

~i llu ~ se~ ,

"1922) . Sor.s ;:ro-

apric~i

of traffic by

~~d

ass~~;:tio~s

~ e~a~t

:Cc.rey

t~e nu~ber

setti ~-:g ~p

a

o~

;:ara~e ­

trized for~~lation of t~e proble~ Hit~ le ss degrees of freedor·. (Ki!"'by, 1979) . t-:os t of these a~d ot~er related ~et~ods are

14 I

Cremer

~1 .

142

~~ 5 ati ~ f

:t 0 ~':

t

::-ir'~~t,

E:)"

S:--. c

c s ~~

C::,

t r. 2t i s

E:: :"': : r~

~i': e

1

~ e :G ~ d ,

~r o :~

yi -:: l ci t ~ .~

t~E Y

~ e·.· e ~ ~ l

c :. J..~:

~;:.;: ~.

~ c i~.ts

~f

:-:i~:::=.~~

2c : ·; :i ~ ~. :

__ _ 1

·.·i e ~.

.,

.-=

::':. '~

- ..:..

.

~

.

... ..

--

-:: :''': -.; - . -

S:'J' '::: :==': E: ::',E: :', S~ C'';: t:'.e 2:::'''';~2. ­ ~·"'; 2 1:" t~: Cl-:.E ~ cl l...ti c ~-. .

~e~d

:c :~ : ~~ e~s

le

(~ ~~~: 2. ~~ = :. al

l e: . . l :.--: · ~ :JE a ::.s :-- ,: .. .; .:: i::.;: E:-:i::.E- :: ! f : :-- a:: i::-.;:.2. -:- ::'.e:'. : :::. :':' :Y', ~t. ::'_t; .=; it-::- c: :: s::: . <"j'

:c l:e c :i:~ ....

ss t.i::.atE: ti C ~2 S i~E

~i~ . . all::·,

:-.0 :' S

~~i ci . . s i£~a l

t: . . ~

~~ . . e y a !' E

j ~~ i~: . . s ~

~:

1c :', [ ti:-:-. e ::: -2 3:-, 5: :: :-' t :'.E ''''::',-

ar e : : e E ~E~ : c r s 5e : t i!"'.; .

t :' s~fi :

~ e~ : s !. ­

I n this ~aper & =:E~ ~e:tc d i: ;:re5e::: e~ 'vJhi c r. in co:-; tr3 s t t c t!:E: !~ett-:o c :3 SJ rark~ow~ Make s use o f t~ . . e i~f o rrati o ~: cc ~t a i~ ed

in t he ti~e s equ e ~ c es o f t ~e ~ea su red da t a . Fe r this the tra ffi c fl o ~ t~r ou s~ ar i r t e r s e c tion is trea t ed ~~ a c au sa l d~-~2~i c pro cess in whic~ the ti~e s eque nc e s of ertry flows generate the partly noi s e c orru~t e d time sequence s of exit flow s a s sy s t e ~ r es pon ses . In a re c ent paper, Cr eme r and Keller ( 1981) have taken a similar approac h and hav e proposed a dynami c recursive a l gorithm f o r the estimation of the unknown parameters . Their procedure has a closed loop stru c ture which g ives rise to a stability problem when the loop gain i s not chosen c arefully . Here we propose a new procedure which makes us e of the whole information contained i n th e process of time sequences of measured flows by correlation techniques . The procedure ir. volves no stability problem and can equally be app lied to obtain short time as well a s long time estimates for the unknown flow s . Moreover , an error term can be deduced from the estimated and the measured flows whi ch may be used as a measure for the quality of the solution . In the next section a precise statement of the problem i s g i ven together with a defi nition of the variables and parameters which are used in the model of the proces s of trip distribution . In Section 3 the estimation a l gor it hm is derived from the trip distri bution model. Section 4 presents some results which were obtained from numerous case studies using simulated data as well a s real data . Finally , a summary of t~e paper is given in Section 5 together with an outlook for future work . PROBL~t~

---....,,_ _-, <]c: ,

3 C>'

Fig . 1:

ConfiGu r a ti o:l o f a :. i! lte r ,cc:t i o:,

Here T is the sarlplin r:; i:-,te rv a l , ;·;Li cL :"l:ould be c hosen for an i nter s e c ti O!: il: the r' a :: ce e f some few minute s . If t\~e int e r :;ec tion L ,'i Cnal controlled , T i s appropri ate ly take r. a: an inteGer numbe r' of siz nal cyc l e c . T':e f;a l'=: meter 1 denotes the avera ge travel ti~e a vehicle needs to pa ss frorl an e ntry to ar. exit . (When the procedure i s appli ed to a net·work , it might bec ome ne :::esca ry to ci:oo::;e a longer sampling interval T a !:d t o introc 'J ce; indi vidual trave l times -l ij t "''- ····\J"""'~\;; ." " ,, 'L. .r"J i and exit j . ) ...,

1

I.,..

1

........

Taking the balan ce o f t~l e e r~tE ri! . ; ~!.d leaving vehi c l e s Eiv e s f o r eac :: t i:~: e ir:t~ r·~/ :::.l ,-

Yj a :)

f ..

L

i =1

1 .]

Since ea c h volume f .. :k l i s of the i - til e~te~i~~J~ J l~~E

: ~: )

~

STATEt':S~T

the psra~e t e ~ ~ co:.:,: ced by

W~l ere

To derive a mathematical model ef t~e proc es s of trip distribution we cor.si~er a cc~plex intersection wit~ ~ entries an~ r. exits as shown in Fig . 1. ~it~ respect to su c ~ ar. intersection t~e following variables are ir. -

~

troduce~ :

vclu~e wtic~

enters

e ~tra~ce

b ..

i

1 ~

:cr

\ ~: :

time interval : ~ - l)T < t < kT , wh ere i = 1, ... I..

whict l eaves exit j d ~ri~g interval (k- 1 )T + 1 < t < kT where j = 1, ... r

..

volu~e

ti~e

,. . )

-' j , '.

+

L,

i

"

1

'. '.

. :: .

-,

143

Determining the Time- Dependent Trip Distribution

.. ~ . .... ",~~:: .: ~:'. ~

l"av1:-.; volu~·."s y, ::.:) to , ,,::or J" » a:·.d tl',,, €:.t"r1r.; -: :.,l·~; :··.~':: .~; ~.: ) :c r::~:·",. ~ -: z~. - r o ·. : vec.tor q ' ( k) . -=-:.-:: ,:. : :::.' : ..... , '::':-::-:.C':. E:': :=;. r c ',) ''lEctor :>r tr~ns;:,o :': ':'-:2.::. -.. ;:-.E: :::.;.:: =-i -::C t:) So :-:-. :=.triz . ) Further:'.:~~ , ~i~ i:~~~' :~'~:~ ~~~~ ~xr: - ~3trix c~ s;lit co' ~ Y"

_

:(,~

:-'C'"

••• •

·: :"' :··:·~ i ·:<.: :

~:

:.

i

·.·.- 1::-.

-=le ~. E: :-:' :'::

i..' .: ) = ::; '

~:)

":':'i'; : ~:) .

,

.. .-

.

::: : ~:)

i:-.

T:-.e:-, '..:e

oe ~.;re;osed

(6 )

co:-:trol of c. signa is i:~portant to know ~';e ti::.'= ·,'aryi:.-::: valc.;es ef the partial flows ~ , ;.: ) Ct ' ~t l e s.st t!lE::ir short tir.1e :::ear!s . ~~~e;er , ~~.ll~ t !le er ~ tE-rln~ and leaving ': 0:,1 :,.;::",5 q. ( v.), y. ( V.) can be :~;easu red easily by i:.~u ct lv e lood detectors or by other sen .:er·,: , t;:e ::.easure::.e:,t of the partial flows f .. , k) is often only possible with great tech :J:2,,1 effort or even i!:.possible . Th en :'Ie have 1~1>:: p'(;01er~ to de terr:lir:e the partial flows f, . or ~quivalently the sp lit parameters b . i f~d~ r:leasure~ents q. (k) , y.(k) using the J :~od e l equatio n (6) tOGethe ~ I-iith conditions ( 3 ) and (I~) .

~:r'

~!~:::f'fi ·:

~i~ ~ t

r·c.::~o!·. si ·: E-

i:.:e~se c: io~

i~

(! l )

Eqs . (4) a~d (E) give only linearly independent relations for the ,, ' r. u"v.nOl-:n para~.eters b i j (k) . T\,a t r.eans t~at we cannot solve t~is set of equations uniquely for the unkno w~ split coefficients bij(k) . On the other ha"d , it seer:lS to be less r.eaningful to adapt the traffic signal control in every cycle to the preceding par tial flow pattern as expressed by B(k), rather than to adapt it at tiffies to the short tilT.e IT.eans as represented by the ma trices ~ and B, i n Eq . (ll) . Therefore , we restate our p~oblem in the following way: given measurements of qi(k) over a not too long in terval KT , find the mean of the split coefficient matrix ~(k) and the matrix ~1 of that part of 8(k) which is correlated with a chosen drift function over the same interval.

~nfortu~ately, ~+~ - 1

£0

In the next section an algorithm for the est i ma tion of ~o and ~1 is derived which gives an approximate solution to this probler:l .

To proceed , we assume that measu rements for qi(k) and y · (k) are taken over a period of K sa~plin; in~ervals . Then we introduce the follOl-lil,g Ciean values

.9.0 Y -=-0

K

, K

L

(k)

(f)

'i.. ' (k)

(8)

~'

k =l K

, K

I

k =l

iJext 'de decor:lpose the tir:le dependent split para~Eter

~atrix

B(k)

~ + ~1

•

r (k )

+

6B(k)

(9 )

'..:here K

:::..a

~:

L

B;k)

( 10)

k =l

is tte ~atrix ef r:lean values of the coeffi c ient s bij : k) . T~e function r (k) ~odels a de tEr~i~istlc drift that is a mono tonic increase or decrease of the coefficients ti~ ( k) ·dithir. the ti~e interval 1 < k < K. r-7~) r,ay be taken as a rasp function 1 1 . Ctebyshev polyno~ial) , as a cosine half ~ ave or ~s t~:e first ialsh fu~ction with

= - 1 fo!~ 1 < ~ < ~ /2 a!~d r {k) = + 1 f~r ~: /= + < Y: <-~: . }B ( 1{) is the rer12ining ::.~!.., t ::.:'" S ( ~:) ·.';:~;ic:·. r'13s- zero r.;ean and is 1..10 -.:cr-rel:=.t2c ·.·:it~·. r' ~~ ) . D e;'E:ndi!"l~ or. t!~e cho i ce

ESTIMATION ALGORITHM It was already pointed out above that the mo del equations (6) together with conditions (3) and (4) don ' t give enough relations to determine the split parameters B(k) for a s i ngle sampl ing interval k uniquely. In the reformulat ion of the model equations (11) we have introduced matrices ~ and ~1 which are constant over the whole period k = 1, ... K. Now combining Eqs . (ll) for all sampling intervals , that is for k = 1, ... K, will provide a sufficient number of relations to determine the unknown matrices ~o and ~1 and by that way the short time means and possible drifts of the time variable split oarameters b lJ . . (k) . . It will turn out that the computational effort is reduced if the drift function r(k) is chosen to be the first Walsh function , i. e.

for k

1,

for k

K/2 + 1,

••• K/2 K

First, taking the mean of both sides of Eq . (ll) gives with respect to Eqs.(7), (8) and (10) :

r :~ )

~C~

r , t: ) ,

:::- :':;: S ~-. 5

:-'c, ~·

fU:-. ·::tioC':s

:C:;.". ' -;') t : ~E:

q

' .B --0

0:-'

+

K

0

+

(q '- q ' ) ·B - 2 - 1

-1

te regarded as a series ',.: i tL '::a lsh fu[:ctions .

( ) , tr.e ,"odel equation (c) traffic flow through an ~ay E writte~ i~ the following ~o

:·rocess

i::~Er~ectic:1 f'cr~.

~ay

-'0

i::-:. '..;i t ~ ·. C:-:.ety s :'.E \' ;;olyr:or..ials , Hi t:-.

; ·. a ~:~ c· !'i:: ·;. i~;~

~~ . l ? )

y , -=-0

1

i\

I

k =l

ll~ ' (k)

6B(k)

( 12)

~l .

144

q ' : ;.:) .. t< =

/2+:

.9. '

'./2 L

"

~

~ qi:t:)

:-./

..

.

(

~

~

~: ~

\

. I"'::i

I

assu~ptio~

a~~

.. /. =1

~.9. r. ;.:

'

..

-

-:J

that t~e ~eviati o n s 6bijlk) are uncorrelated , the last ter~ on the right side of (12) will be appro xi~ately zero and ~ay be neglected . Th i s gives a first equation i~volvinc the ~ean \'olu:::es

~~der

the

I k)

Cr eme r

, ' :-./ . '

• ..:....:; I

. :-:

~J.~ '

/.

; .

~

I

:

, -

\

I

~ • r ;: : '

are t!ic fir:ite irJt Ef' ve.l Ct'os.::; - cor r e l==.ti o!·; r:1a tr'icE;':; of t :'p2 ti:-:...~ '::t:qU€:~LE ~; ~q ·..:i t :-: ~y , 6q ',litL ~~ and ~q . f·\ ;-:) ',;ith ~q , re :::~e ct ivc: ly

(s ee Godfr ey , 19EC ) . l~Je..:e :~c:trices a~ \Je ll thE: f:: e 3.t! \i a lu €::3 .9.0 ' .9. 1' ~.:: ' and I o ' Ll' Y2 are eas ily co::.puted fro::: tl.e ::.easul'E:d ~olu~es .9. l k) and y (k) . 2:3

114 ) Lext , \,J e multiply the model equat i on 111 ) by r lk) and aga i n take the mean of both sides :

To pro cee d , we f:Ju l ti ply :",0'': ::q . ( 11) fror:: the left by the co lUlr.n 'v ecto r r l k) ' qlk) and then take the SUI:: ove r the vll, ole pe rIod once again. After s i~ilar al[eb r ai c ~anipulations as above and again neclecting ~umr::at ion terms over uncorrelated variables lead s to the following s e cond matrix equation

Iq '-q ' ) -2

1

+-

-1 K

L 6.9. ' I k ) . r I k) • 6B I k )

K k =l

~qr ~



--q yr

+

[2.qq - I .9.2-.9.1)(.9.2 '-.9.1 '~ ~1

(15) (19)

Here Yl ' and Y2 ' are defined s i milarly as .9.1 ' and.9.2 ' in Eq . ( 13) . Unde r th e reasonable assump tion that 6q i lk) · r l k) and 6bi jlk) are uncor related time sequences , the last term on the ri ght s ide of (15) may be negle cted again . Then we have a second equation

( 16) So far we have der i ved two equations for Bo and ~1 which take into account only balan ces o f the mean valu es of the volumes and ~ake no use o f the spec ifi c i nformation con tained i n t he time sequences of the volu~es . To do this , we multiply now Eq . ll l) fro~ the left by the column vector q (k) and th en take again the ~ean over the ,Ihol e period . Using Eq s . (14) and (16) and neglecting teres i n whi ch ti~e sequences 6q. Ik) and 6b .. Ik) are correlated leads after ~ome algeb r~~ c man i pulations to the ~at rix equation

':: 1

is de The cross - correlation matri x e -qyr f i ned a s K 4>

--qyr

L

6.9. l]r) . !Jy ' I k) . r ( k )

(20)

k =l

Eqs . ( 14) , (16) , (17) and (1 9) establ i sh a system of 2 · n · ( ~ + 1) linear equations for the 2 · n · ~ unknown co effi c ie~t s of Eo and Bl ' Due to the neglect i ng of te l' :::s i:-. the abo~e develop~ent the se equatio~s are ce~erally i ncon s i::tent , tl:at :-:-:e3.!: ,s ',;2 have :-:-:or' E equa tion s tr:ar. unk no\·if. par 3.:":', eter' : . Ir, t:~i2 :2:S we ca~ deter~iJ le a u:l iqu s ~i~i ~u~ :-:or~ least - s quare s 501:.;:io :-: f ::. !' ~ s':lC 3 i ty sol -ji:~,£ tr.e cO L:pled se t of ~q1Jat ior;-i 2ppl),, i~~ th e co:-:cept of ; e ::er3.1i =ed i::~er ses o f >:oore - Pe:', r ose ( fo:-' de tc. i:' .: see ==.::'0 a.:-!d ~-:itr a , 1 j7!) . Co :~;:>:..;ti~,:: .3c ~~ti ·:) : -, s for' ~o a~':j ~ 1 si ~~lta:leou~ly fr o~ t~,e :ouple d 3e~ of equatio:is "'i O'lld i:·.v,)l·. .· E ':.!~E:: i: ·;·~'er·s i :) :1 o f 5. 2~ x ~ ~ - ~atrix ~ ~:i:~-: i: s :ill s co ~.:idera ~ le co ~putatio:l al effort . ~ o ~e~er , ss C2~ be see~ fro~ i ~spection of t~e eq~ ati o :1s , :oupli :: ; cO::'.es up tr.rou';~j t!-,e :-:-.a':.!"ii: ~qqr' 3. :-;d tile vector (32 - .9.1) . :; 0\'; it roll~)'.·; s fr0:-:-, t~eir ~efi~itio~s by ~qs . : 13) a~d : lE ) that t~ese coefficients are rat~~ e r 5~all co ~parEd wit~: the other coeffic ie:-;t.s i!', t::e eCjuat i o:~s as lon~ as tte ~ea~ vsl~e s a::d t~:e co '/aria~ces

of t".e ( 17 )

K

~u c~ .

volT:e.:: ::]i :;: ) d::;:-' t vari too t!:is case it ~i;t: ~E j~5t i~i e d to

i:~;:;ut I~

Determ i n in z the Time-Dependent Tr ip Distr i but i on ~eglect

the

the coupl ing terms which results i n decoupled set of equations :

PC:S CLTS

followi~g

..

1...0

,

q

,

(21a)

--'0

(21b)

Since the right s i de of (21a , b) and (22a , b) has the same form , solving this set of equa ti ons needs the i nversion of only one m x m - mat rix . Define the followi ng (m+ 1)x m - matrix

x

145

The est i mat i or. procedure was tested ir nume rous case st~dies ~i th si~ulated data 3 S well as wi th real data . Some of the results wil l be presented now . A.

Test wit h s imu la t ed data

To test the estimation algori th m traffi c flow was s i mulated through an i r.tersec ti on wit h five entr ies and five exits as shown i n fig . 1 . The variances of the random variations 6qi (k) of the en te ri ng volumes as well a s the variances of the variations of the split parameters 6b ij (k) were chosen accord i ng to real observat i ons . To study the tra ck i ng capab ility of the e s ti ma tes , the short time means of t he real parameters b 13 and b 14 were a lte r ed according to a cos ine half wave . The r esults o f a simulation run o ver a total period of 120 samp ling i ntervals are shown in fig . 2 .

(23) .a r---------------------------------------,

then estimates for the unknown matrices B and ~1 will be computed from Eqs . (21a,b)-0 and (22a , b) by the following formulas:

0 8 ~------------------------------------~

(24)

B

-0

24

-y 2 '--y 1 ']

( X ' X) - 1 . X ' •

[

48

72

96

(25)



-qyr

At th is point two remark s sho uld be made . Be caus e of the yarious~simplifications we made the mat rice s ~ and ~1 as computed from Eqs . (22) a nd (23) are only est i mates for th e co rresponding real mat ri ces . The qualit y of these est i ma tes mi ght be tested by re gard ing the error between the measu red~leaving volumes Yi (k) and the volumes yjJk) whi~h are computed when the matr i ces ~o and ~1 found above are inserted in the pro cess - equa tior. ( 11) . In the der i vation o f the for mu l a s abo ve , t he neglec ting of the uncorrelated terms is the mo re justifi ed the l onger the whole i nte r val K· T i s chosen . However , i n order to track time var yi ng changes of the split parameters ~ithout too much delay it might be advisable to make t he i nterval K· T not too long. It has to be the subject o f future in vestiga tions us i ng real data to find out a su i table s iz e o f K.

fig . 2 : Spl i t paramete~s b 13 (k) and b14(k )( - ) and estimates b and 814 (--- ) 13 for

~ach

~o ,

~1

computat i on of new estimation value s a peri od of K = 24 samp ling interva l s was f ound to be a su itable cho i ce . Beginning wit h t he 24 t h s a mp li ng interval new esti mates were compu te d after every 12th samp li ng i nterva l. The parameters shown i n F~g . 2 are ele ments o f the~est i mated ma trix ~, wh il e t he elements o f ~1 were sma ll numbers because of the r elat ively slow variat i on of the real parameters . While t he es timate s are mostly nea r t he real value s when we use K = 24 sampled measurements , the estimat i on of ~ became ever. better us ing all 120 measurements of the total i nterval . Tab l e 1 preser.ts a compar i son of real and estimated mean of ~ .

:1 . Cremer

146 ~s~l

a~d

esti~ate~

~ea~

spl i t ;ar s -

~~. ~ t srs ~0

•~ 1

.~v

:.f"',

1~

o2C.

. 3::;

• ..,v

. CC

."

1

le:

. OC

.

. CC

. jG

. 41

. Gj

. 00

esti -

. 00

.09

. 44

. 32

. 10

. Vv

r~:=.l ::-. a~ri:·:

£0

iv

d::: v

.

.:.;

-:

18 . v~

. ~-

I

. '"

;

,

~

. :JI....,. ~

IV

r-.ated

. 06

. OC

. 21

. 22

. 1-+5

r.:atrix

. 21

. 10

. 00

. 21

. 48

~ (1: =120)

. 37

. 4S

.06

. 00

. 12

.2 2

. 22

. 40

.1 1

.00

B.

Test with real measu reme nts

To test the efficiency of the proposed procedure when applied to real measurements , the entering volumes qi(k), the leaving volumes Yj(k) and the partial volumes fij(k) were collec ted at an i ntersection i n the city of Munich during a total pe ri od of 100 sampl i ng intervals of l ength T = 70 sec . The con fi gu ration of the intersection is shown in Fig . 3 together with the time sequences of the ente ring volumes . The a l gorithm was fed by the sequences of entering and leaving volumes qi(k) and Yj(k) while the partial volumes fij (k) were used to compute the real spl i t parameters for comparison . Though this is not a very complex i nte r section , determining the partia l flows is sti ll a nont ri vial probler. s i nce Eqs.(6) and (4) provid e only five l i nearly independent relations for seven unknown sp lit parameters .

.------------------

Fi~.

4 : Split para~ets~s t~l(~:l 3~d b2~~~) ar,d estir.:ate: b21 a~d b 24 (-- - )

(-)

I n Fig . 4 the tir:.e sequences of parar.,eters b 21 (k) and b24 (k) an: S:,O',I); tOt;e tLel' "'ii t,: ti',eir est imat es as cor:pc;ted fro~'. the ale:ori thc-o using at al' Y given tir:e the ,: = 20 last vo lur.:e samp l es . In Table 2 th e real and the est i mated sp li t ;:,atrix Eo are t;i 'Jer: as computed fror.: the total int erva l. :Tlle ele ments of matr ix ~ 1 wer e rather sma ll numbers i n th i s case . )

TABLE 2

r eal matrix ~o

estimated mat rix .§.o (K =100)

Real and e stimat ed mean spl it para r:.eters ~

0 .0

0 .0

O. sg

0 .11

0 .11

0 .0

0 . 08

0 . 81

0 . 86

0 .0

0 .0

0.14

0 .0

0 .0

0 . 8S

0 . 15

0 . 14

0 .0

O. OS

O. 7E

0 . 77

0 .0

0 .0

0 . 23

It c a n b~ se e !: fror:: 13::1e 2 t:-::2t t:".E: c.lzo 1"i tl ~ rr:: produces est i r:a tes for' the ur.:·::~.o '..; ~

split parar:.eters ~hic h differ fror.: the real va l ues by less than 0 . 09 . This accurac y ~ay be regarded to be suffi cient for t~: e ;urpose of traffi c r esponsive contl'ol 2t sL::;·alized i ntersect i or:s ,

---------------K 5': lac Fi g . 3 :

Configuration of real data coIl ec tior,

A ~e ~ dyna~i c ~ett1od for t~~e e2ti~:atiot1 o~ ttle distribution of partial ~lc~s ~i tti ~ 2 co;,-;p l e x intersection ::as tee!: prese::tec . T:~e basic id ea i s ~ot on ly to Lse bala~ce equa ti ons of the accG~ulated traffi c COU ~1tS bu t to extract f ur ti':er iEfor~:atior. fro~: t::e causal dependellcy betweeZl tile ti~e v2ryi~~ sequellces of e~try volu~es al1d ti1E ti~e 56 -

De t erm i n i ng t h e Time - Dependen t Tri p Di s tri bu ti on er ::',-2

E: i t \·cl·J~. es . =y ~ : :i3 '.. ;s." i: to c tai:~ a u~~i~~~ scl~tio~ for :. '2 ; : ·.;·:~.C;·I; ~. ;:=.rt 21 fl:;-..;3 frc~:. t:',E: =-': =L!.~tl'2 .. .~:::.,:-.._; ~-.-= ::~€:~. t.:; -..;i t. .c~..::. :::':--. ~; 2cci tic;'":c.l pr-E: :. _·~·; : :.~~':,i::~ . :': :,r'2c; ': '2~', t:',c 3.1;:~r-·it:.::. i .:: :5;::: ij-=~;:,::,:,:,,:

:

;c~=i~l~

:::.'~:!.-s

c:'"'

1-'.7

i:-=:.:i;"'::i~. ;

·.'::.~i=:i 'J :-.':

c:f' t :'; '2 Jri:;i:--.-

effi:ie~:j of t~~s alcori~ :.,:: :=.;;;;li:s':i::;:--..:: ·.·; it : ~ '::l ~:.' .... ',·,lit:: ;,c:::.l js.'CE:. .

~

is

::.~ej

~e~:~5:r~:e~ ~2

It ~il l t6 ~~ . 6 ;;rc~le~ ~f ~~:~rs i :....·-=.:t.i:=at€: :',0 -..; t: '.E: :-: . c~:.·:;c ·..i O;·;":~ u~:der

i ~;eti~~ :c~~itic=:s

li~e

'.-l el :' s..::

~_ ;~is:

'..; ~. ~:--.

s;:;:.,lis:

~:::.t~rat i c!:

.

.':'.:-:ct: '. sr' s'.,.;:":s:t ~f :\,;: ·. .; :6 i : ";-2.:ti:::::::.tio : -. ~ ~ o~ld be t~ . -2 i~e~ti~i~:::.:ic~ c ~ fl o ~~ ~ :. r J ~C~ .

.i :ro;: ~~:: -

=.ei1 , :: . ( 1979) , ·v"o~ .

:< =.:.tri:·::~. cdclle

~eil.str8!:. e ! 1

::e: . . Dissert2 tic:} . Cs.r~ :i ,

:-:. ,

ZU!~ :::rr.~i ttlu :-!Z

a:..:.:; CUtr'scr'.:iitts:-:-:E::ssur;-

:-:E~:dric;·:sO: 1 ,

'~r:i versi th

t

?:3rl~r \jr.e .

C. a:ld Sidd:"larta:"! , !< .

(195 1) . .c. ;';et':od for dire c t esti :~3tio:l of cri;i:l/destin3tion trip =atri ces . Transpn . 3cier-:ce , 1:; , 32 - 1.,9 . Cre!:.E:r , t·; . a:ld ;·:eller , r' . (1921) . CY!'3:::ic idertif i catior of flows fro~ traffic counts at co~plex intersection s . Prepr . of thE: 5th Ir.t . Syr:p . or, Transporta.tTOtl and Traffic Theory , Toronto , Canada , pp . 199 - 209 . Godfrey , K. P. (1980) . Cor r elat i on Methods . .l\utor~at i ca , 16 , 527 - 534 . i:irby , !l . R. (1979) . Part i al [~a trix techniques . Traff . Enb . and Control , 20 , 422 - 428 .

?20 ,

C.; . a:.d :':itra , S . ~ .. :e!~er21izEd i~ ­ verse of :::atri ces and i ts appl i cat i on! . John ·..;iley & SO:lS , :;e;·; York , Lo;.dor. , Sid"ey , Tor·o·~to , 1971 , cp . 50 - 5': .

Sar.::'.er , G. , Zelle , F'ort~ cr.rE:ib

~: .,

Schechtner , O. ·;1922 ) .

'!;E: ei:1er >;3tri>: der ':e r -

~e~ r sbezie~ur.gen ~ittels Ouerschnitts zdhlunben . StraBe:1verkehrstechr.ik , 26 , ~eft

1.

Van Vliet , D. and ';; i llur-,set1 , K. G. (1981) . Va li da t i on of the t' E2 r.1ode l fo r est i ma t i on tr i p Matri ces fror.1 traffic counts . Prepr . of the 8th Int . Symp . on Trans po r tat i on and Tr affic Theory, Toronto , Canada , pp . 191 - 198 . Van Zuylen , H. J . and Willumsen , L. G. (1980) . The r.1ost likely trip ~13tr i x est i mated from traffic counts . Transpn . ?esearch , ~ 281 - 293 .