- Email: [email protected]
Nonlinear Control Applications on Integrating Freeway On-ramp Flow Meterings
NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
14th World Congress ofTFAC
Cvpyright © 1999 IFAC l4th Triennial World Congress, Beijing, P.R. China
Q-8e-Ol-6
NONLINEAR CONTROL APPLICATIONS ON INTEGRATING FREEWAY ON-RAMP FLOW METERINGS
Tang-Hsien Chang
Professor of Transportation Science, Tarnkang University P.O.Box 7-876, Taipei, Taiwan 10617 Fax:+886-2-26221I35; Email: thchang@ iml.im.tku.edu.tw
Abstract: Ramp metering control plays an important role in modem freeway traffic management, particularly during heavy traffic. Based on Pa}'11e's continuum traffic stream model, this study presents a nonlinear-integrated-responsive and closed-loop ramp metering control model. Simulation Tesults confirm that the proposed model is robust and efficient for freeway traffic control and, in doing so, maintains freeway operation at a high level of performance. With such control, freeway traffic can eliminate congestion. Copyright © 1999lFAC
KeY'Ivords: Traffic control, ramp input, nonlinear control systems, discrete event dynamic systems, control' engineering.
l.
INTRODUCTION
Traffic control should initially focus on understanding traffic characteristics, accounting for why many traffic stream models have been proposed. Flow, density and speed are generally considered to be three components for the representation of traffic streams. From a theoretical perspective, flolV is the product of density and speed. The correlation between density and speed forwards to negative, i.e. speed declines with an increasing density. However, both high speed and low speed causes low flow, also, despite of high or low density all correspond to low flow. This phenomenon indicates that high speed operation does not make a road efficient in traffic flow because the maximum flow occurs at a certain median speed. Therefore, in freeway management, ramp metering control (RMC) is necessary if the traffic demand is at peak because amitrarily allowing a vehicle onto the road would cause a traffic congestion. Many freeways around the world have implemented ramp metering control. However, those control methodologies are insufficient since a majority of them are pre-time control or single ramp controL To adapt to the varying traffic patterns, the most available form of metering control is the one with responsive and coordinated abilities and a closed-
loop algorithm. Among the notable investigations involving ramp metering control include Wattleworth and Berry (1965), Drew, Brewer, Buhr and Whitson (1969), Yagoda (1970), Wang (1972), Isaksen and Payne (1973), Wang and May (1973), Eldor and Adler (1977), Looze, Roupt, NHs, Sandell and Altans (1978), Papageorgiou (1980), Goldstein and Kumar (1982), Iida, Hasegawa, AsaJ..."Ura and Shao (1989), Papageorgiou, Blosseville and Habib (1990, 1991), Chang, Wu and Cohen (1994), Zhang, Ritchie and Recker (1996) and Zhang and Ritchie (1997). Although a few contributors recently presenled responsive, coordinated and closed-loop models, they arc linear models. This work pTesents a nonlinear model and applies it towards freeway integrated on-ramp metering control. The rest of this paper is OTganized as follows. Section 2 describes the dynamic traffic strcam model employed herein to represent traffic phenomenon. Section 3 presents the metering control model. Next, Section 4 summarizes the simulation results which verifY the proposed model's effectiveness. Concluding remarks are finaJly make in Section 5.
2.
DYNAMIC TRAFFIC STREAM MODEL
Constructing a ramp metering control model initially involves determining the traffic stream model.
8363
Copyright 1999 IF AC
ISBN: 008 0432484
NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
Traffic meam models can generally be classified as static types, or dynamic types. Owing to that our control is dynamic, a dynamic model should be selected. According to Traffic Flow Theory (Gerlough and Huber, 1975): . Oil ou du Speed FquallOn: - + u - = (1) ot ox dt .. E . op oq C ontlflUlty
quat/on; -
ot
+-
ox
= 0
(2)
Where x denotes location, t represents time, q is flow, u denotes speed,p represents density. However, q =pu (3) holds. Payne (1971) has investigated that du =-.!.{[ue (P)-ul-2::. 0P } (4) dt 'J" pox In which, lie (p) is the equilibrium speed at the condition of density p, r denotes the driver reaction time, and v represents the coefficient representing the driver's expectation relevant to the density ofthe next road segment. For practice, Isaksen and Payne (1973) resolved Payne's model to be the simplified difference equations: k
k+l
uJ
=u 1le -I'>.[.u/k
k
u l -U I _ 1 --..:;.--=-~-
14th World Congress ofTFAC
Ramp metering control attempts to adjust the flows of the controlled major road such that the road has maximum performance. The traffic scenario of a major road can be revealed on the basis of Eqn.(1)(2)(3), tIle continllollS type, or (5)(6)(7), the discrete type. For operations, a discrete type is acceptable. In general, using speed equation and concentration equation to represent/composite a traffic system equation is sufficient. Thus, for simplification, by substituting Eqn.(7) into (6), diminishing the variable of flow, and letting the net on-ramp flow be the control variable, then Eqn.(6) becomes: pt+ t
nl
R;+l =
1
(9)
1
For a segment of a fF.eway, by using Eqn.(5) and (8) tIle systcm equation can be written as:
[ ~Ul ]=~/.{[!I]+ [ I
~
nJ(x 1+ t -XI)
]Rf+1} (10)
ut+
(7)
qt:
Xl
sr+
1 - uf, 1>PI:= pf-rr - pr. If the Where ~ul = controlled freeway has N segments separated by ramps, i.e. 1~1,2,3, ... ,N, by Eqn.(lO) the whole system equation is presented as the nonlinear form: 6X ==!:it . [f(X) +Br] (11) or, separated as
Where k denotes the time state in step, J represents the road segment, l:f!::; N, ]V is the number of controlled road segments, and, xI : the location of the beginning point of segment I. n I : the number of lanes of segment t. M : the time duration of one step of operation. the flo""s passing through Xl at time. s~ : the off-ramp flows at
rt+ 1 -
I'>.xl = b.t 'g(X)
qr+ = pt-t"L
XI
(8)
nl (Xl+ 1 - XI)
and assume that
r{: the on-ramp flows at
k
R[+l
where
PI
1:
k
-nlPI u l
(xJ+l -xI)
I
+Ili
_1>t (u/"
O.5(XJ+l - Xl_I)
I:
k
= Pt + I'>.t n l - I P1-l uI- l
during k.
I'>.x 1
In
which,
O=[O]N
= b.t . [b(X) +
X=[XI>X 2 ]T,
B=[O br,
(12)
br}
(13)
XI =[U 1 'U 2 ' .... ,u N
X2
f,
==[Pl'P2,····,PN f,
1
I'>.x 1 = [t.u l ,1>u2 , •.•. , llu N JT,
bll == - - - - nl (Xl+l - XI)
~""{2
b=
= [I'>.Pl , Ap2,····,APNJ
T
' r=[Rj,R2,····,RNJ
T
,
[hIlL,N' AX = [Ax Z,t.x2 ]T,
during k.
u; ;the actual speed on segment I at time k. pi :
the actual density on segment I at time k. The model incorporated by Eqn.(5), (6) and (7) is most effectively utilized for the description of dynamic traffic status. Figure 1 depicts the freeway scheme and relations of these variables in modeling. By assuming that each on-ramp and off-ramp is equipped "vith detectors for real-time flow measurement, the previous model can predict the traffic on the freeway. 3.
CONTROL MODEL MODELING
In system consideration, since Eqn.(l) is nonlinear, more accurate discrete expression for the system discrete speed equation involves expanding Eqn.(12) by Taylor series, unlike Eqn.(5). The fact that the second-order derivative of Eqn.(l2) exists, implies the following:
8364
Copyright 1999 IF AC
ISBN: 008 0432484
NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
XI (k
+ 1)
= XI (k)
+ &. g(X(k» +
l\,/2 {Og(X(k»)
2
8xI
. g(X(k) + eg(X(k» [b(X(k) + br(k)]}
(14)
OX2
Moreover, Eqn.(13) becomes x 2 (k + 1) = x 2 (k) + ~t· [h(X(k)) +br(k)] (15) if the first-order derivative is chosen. The above equations (14) and (15) now form a nonlineardiscrete-dynamie system describing freeway traffic with N segments. Consider a situation in which the metering control perfonnanee is set to maximize the flow rate on the major road. This is equivalent to minimizing the difference between real flow and ideal or maximal value. This is while assuming that the maximal flow occurs at a certain speed x 10 and density x 2o ' The performance function is then given as J =
.. T 2"1{[x[(k +l)-x 1o ] Q I [x](k+l)-xloJ+[X2(k+l)
- X20]"["
Qz[x2(k + 1) -
X20)
+
rT (k)Q3r(k)}
(16)
where, QI, Q2' Q 3 are the weights for speecl density and the control variable, respectively. The performance obviously makes the traffic approach to optimum at the next step by minimal energy. By substituting Eqn.(l4) and (15) into (16) and take the partial derivative J with the control variables, r3 J ( r3 r '" 0 , then the control law is obtained: r(k) = __1_[J..(og(X(k))b/Q]CO'g(X(k))b)+bTQ2b Ll.t' 4 c3x 2 O'X2
+~Q3rl)< 6t
{Ll.2t
(O'g~X(k))b?%[(Xl(k)-Xlo)+Ll.t. x2
14th World Congress ofTFAC
involved for the demonstration. Three simulations are executed respectively by (i) accomplishing RM"C with full demand at every ramp in peak hour, (ii) no RMC with the same demand condition, and (Hi) accomplishing RMC with few demand in off-peak. The run time of the three simulations is continuous for one hour. The traffic is described by Eqn.(14) and (15). The subsequent flows are calculated by Eqn.(3). In R1v[C, the feedback quantities in each step are given by Eqn.(l7). According to Haynes (1965), for a three-lane segment ofa freeway, q = 65.5p-0.179p 2 -80 (19) in which, the density is denoted in vehicles per mile. By Eqn.(l9), the following form is utilized for the term of u e (p) in function g(X): u e =-glp=105.39_L39p_26.67 (20) P where the unit of the density is converted in vehicles per kilo meter per lane. Eqn.(20) represents the real world traffic 011 the simulations and analysis. Based on Eqn.(20), the maximal flow should occur at U o = 52 kph corresponding to Po =- 38 vehl km.
Then, for five segments, X20
eX I
oX 2
= Rt+ 1 +S1kT1
(18)
R 1k + l Er(k) solved from Eqn.(7). However, off-
ramp Dow ..,-[+1 should be estimated to complete Eqn.(l8). In general, the time series data J , .•• ,
{sf,
s1} of off-ramp flows can be obtained by
loop detectors which are installed at ramps. Based on the series and via Kalman Filter method, will
st"
be known. Nevertheless, r/,,+I has limitation: 0 s;:
rt+ 1 se, where C denotes the capacity of ramp I. 4.
= 38 [1 1 1 1 l]T are given. Herein, assume
qe 'f , 1 P < Po 1- e- qT [2000 + (2000 _ q)]e-[2000+(2()(J()-q)]T
"------"----;=~=;::__~---,
(17)
Based on Eqn.(9), the number of vehicles allowed to enter the freeway at ramp I during state k+ 1 is
sf
t1 1 I 1 1JT and
-qT
r'"
+b T %((x 2 (k) - "20) + bot ·h(X(k»)]J
rrhl
= 52
that all segments uniformly have three lanes. Where M is set with 60 seconds. v =19.2 (km2/hr) and .. =0.022 are provided (Goldstein and Kumar, 1982). The ramp capacity is defined at 1500 veh/hr. The OIl-ramp flow r in no RMC is calculated by the model (Drew, 1971):
g(X(k» + Ll.t 2 (og(X(k» g(X(k» + b'g(X(k» h(X{k»)]
2
X lo
MODEL EVALUATION
For applkation, the model proposed in the previous section should be verified and evaluated. The verification and evaluation are executed by simulations. A freeway with five segments is
l_e-[2
(21) .
If P > Po
where T is set as 5 seconds. Figure 2 summarizes the simulation results of accomplishing RM"C with full demand at every ramp in peak hour. Figure 3 results from no RMC. Figure 4 indicates that with few demand in off-peak. All simulations correspond to the initial conditions listed in Table 1. Each figure contains four graphs which illustrate (i) the flows of five ramps, (ii) the flows of five segments and their upstream of the controlled freeway, (iii) the speeds of five segments and their upstream, and (iv) the density. All the graphs are plotted with respect to time. According to Fig.2, the on-ramp flow in each ramp is at a high flow rate during the beginning control period and, tben, comes down since the initial condition is given at near saturatioll. Simulation results indicate that saturation occurs approximately at the flow 2000 veh/hr. The flows on all segments approach the value in other periods except at beginning. This phenomenon correlates with tile situation in which speed and density are accurately
8365
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
NONLlNEAR CONTROL APPLICATIONS ON INTEGRATING FREE...
reflected. Speed tends to 52 kph and density 38 vehlkm, the ideal values. The control is robust despite its unstable traffic at unconlroHed upstream segment SO. This finding means that the control model can maintain the traffic at high perfonnance, speed does not decline and avoid traffic jam during peak hours. However, Fig.3 illustrates a bad traffic situation. Obviously, no flow control on the ramps during peak periods causes traffic jam, in which concentration overloads, speed and flow decline, almost vanishing. Figure 4 reveals that first three ramps allow full access into the freeway dnring off-peak traffic. This situation indicates that, during off-peak, metering control on Ramp 1, 2 and 3 are unnecessary. After all, such a control is necessal}' for Ramp 4 and 5 to pertain the well level of traffic service. When vehicles fully pour in the freeway from first three ramps, the traffic gradually becomes heavy at downstreams sequentially. However, the traffic is stable and at high performance in all segments except the segment SO \....hich is uncontrolled. 5.
CONCLUSION
Ramp metering control is implemented in many freeway access tinting control. A well control model makes high performance of freeway operations. A coordinated-responsive and closed-loop system is more perfect than individual or pretime control. However, some improving spaces are available for conventional models. In this study, we derive a model which achieves the real philosophy inquired by freeway traffic control. The proposed model is nonlinear which is more p.recise to describe an actual environment than linear models. According to the verification and evaluation by simulations, the model proposed herein is robust and efficient for metering control. With such control, freeway traffic eliminates congestion. In according with the previous description, to implement this model, upstream approaching flow rate, flows on each on-ramp and off-ramp should be detected with real-lime. Based on the detected data and via the proposed traffic stream model, speed and density are converted and known. The optimal allowed flow rate on ramps is finally determined by the provided feedback gain. This allowed flow rate is generally achieved with the on/off timing sharing of a traffic signal. However, such a control is operated through a controller or a coordinated traffic control center. REFERENCES J. Wu and S.L Cohen (1994) . Integrated real-time ramp metering modeling for non-recurrent congestion: framework and preliminary results. Transportation Research Record,
Chang, G.L.,
1446,56-65.
Drew, D.R. (1971). Traffic Flow Theory and Control. McGraw-HiIl Book Company, NY 10036. Drew, D.R., K. A . Brewer, J. R Buhr and R.H. Whitson (1969). Multilevel approach to the design of a freeway control system. Highway Research Record, 279,70-86. Eldor, M. and H. Adler (1977). Post-optimality analysis methodology for freeway on-ramp controL Transportation Research Record, 644, 51-53. Gerlough, D.L. and M.I. Huber (1975) . Traffic Flow Theory, TRB Special Report 165, National Science Council, Washington D.e. Goldslcin, N.B. and K.S.P. Kumar (1982). A decentralized control strategy for freeway regulation. Transportation Research, ]6B(4), 279-290. Raynes, 1.J. (1965). Some considerations of vehicular density on urban freeways. Highway Research Record, 199, 59-80. Iida, Y., T . Hasegawa, Y. Asakura and C.F. Shao (1989). A fonnulation of on-ramp traffic control system with route guidance for urban expressway. Proceedings of IFAC Control Computers, Communications in Transportation, Paris, France. Isakscn, L. and H.J. Payne (1973). Sub-optimal control of linear systems by augmentation with application to freeway traffic regulation. IEEE Transactions on Automatic Control, 18(3),210-219. Looze, D .P., P.K. Roupt, R. Nils, JR. Sandell and M. Athans (1978). On decentralized estimation and control with application to freeway ramp metering. IEEE Transactions on Automatic Control, 23(2), 268-275 .
Papageorgiou, M. (1980). A new approach to timeof-day control based on a dynamic freeway traffic model. Transportation Research, 14B(5),349-360. Papageorgioll, M., J.M. Blosseville and RS. Habib. (1990). Modelling and real-time control on the southern part of boulevard peripherique in Paris: Part I Modelling and Part II Coordinate on-ramp metering. Transportation Research, 24A(5), 345370.
Papageorgiou, M., H.S. Habib and 1.M. Blosseville (1991). ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, 1320, 58-64. Payne, H.1. (1971). Models of freeway traffic and control, In: Mathematical Models of Public Systems (G.A. Bekey, Ed.), Simulation Council. Proc. Ser., 1(1),51-61. Wang, C.F. (1972) . On a ramp flow assignment problem. Transportation Science, 6, 114-130. Wang, J.1. and A.D. May (1973). Computer model fOT optimal freeway on-ramp control. Transportation R esearch Record, 469, 16-25. Wattleworth J.A. and D.S. Beny (1965). Peak-period control of a freeway system-some theoretical investigations. Highway Research Record, 89, 11-25. Yagoda, H.N. (1970). The dynamic control of
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Copyright 1999 IF AC
ISBN: 0 08 043248 4
NONLINEAR CONTROL APPLICATIONS ON INTEGRATING FREE. ..
14th World Congress ofIFAC
tation R esearch, SC(5), 273-286. Zhang, H.M., S.O. Ritchie and W.W. Recker (1996) . Some general results on the optimal ramp control problem. Transportation Research , 4C(2), 5]-69.
automotive traffic at a freeway entrance ramp. Automatica, 6, 385-393. Zhang, H.M. and S.G. Ritchie (1997). Freeway Tamp metering artificial neural networks. Transp or-
Basic data of the simulation case
Table 1 Segment 0 (SO) (upstream)
Segment I
Segment 2
Segment 3
Segment 4
Segment 5
Segment
(SI)
(S2)
(S3)
(S4)
(SS)
Len~h(km)
----
1.50
1.70
1.00
1.80
2 .60
Peak Hour Initial Flow (veh/hr/lane)
1980
1910
1850
1850
1720
1730
Segment
q~
0
...
1
I
/-1
pi
k
Pt ••• Plk
ui
u 1k
k SI
J.,
t
-+
U f- 1
Xl~
Xo
q1
qf
k I
qt.
-
k
pf u Ik
Pl+l k
U'+l
p~ uNk
Xl
s~ J.,t r~
r k
1
Fig. 1.
Freeway layout and relevant variables foa.vs on m ad
lows on ramp"
.100
1000 900 \ $ 00
Segment N
1+1
2tl5O
P
I
700 500
I
500
S5
1750
40
5{J
sa
1700
0
10
;D
30
lime (min)
time (rnin )
s poeed co road
den5ity 00 reed
'0
50
00
75
" 70
65
i1
~
SS
2fj
24 50
0
10
20
:l(J
22 40
50
&l
0
timoe (min,
Fig. 2.
10
2()
30 time (rni n)
'0
5D
60
Simulation result ofRMC during peak hours
8367
Copyright 1999 lFAC
ISBN: 0 08 043248 4
NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
14th World Congress ofTFAC
'Dow-s on road
lI.ows- 00 ramps
250
2500
200
2000
i
1500
TI
~
1000
500
0
0
10
20
30
SO
40
time (min)
time (min,
speed en rood
density on rood
time (mn)
time (rnln)
60
80
Fig. 3.
Simulation result of without RMC during peak periods
to'.llison r(Bd ~0r-----r-----~--
1500
20
30
40
-
iD
BD
50
-
--------
Sll
20
30
~ime(min)
5~eed
__----__----__-----,
40
50
60
time (min)
density 00 rood
.....
_------..--------
S2
20 --
-
~3-
-
-
____
-
-'\. . . . . ..r \'\,.., . -. . . . . . . ...;./ v/"""
-J
'-
/
SO
15
55L---~----~----~----~----~----->
o
10
20
J.(J 11me (mln)
Fig. 4.
40
50
60 time (min)
Simulation result ofRMC during off-peak traffic
8368
Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress ofTFAC
Cvpyright © 1999 IFAC l4th Triennial World Congress, Beijing, P.R. China
Q-8e-Ol-6
NONLINEAR CONTROL APPLICATIONS ON INTEGRATING FREEWAY ON-RAMP FLOW METERINGS
Tang-Hsien Chang
Professor of Transportation Science, Tarnkang University P.O.Box 7-876, Taipei, Taiwan 10617 Fax:+886-2-26221I35; Email: thchang@ iml.im.tku.edu.tw
Abstract: Ramp metering control plays an important role in modem freeway traffic management, particularly during heavy traffic. Based on Pa}'11e's continuum traffic stream model, this study presents a nonlinear-integrated-responsive and closed-loop ramp metering control model. Simulation Tesults confirm that the proposed model is robust and efficient for freeway traffic control and, in doing so, maintains freeway operation at a high level of performance. With such control, freeway traffic can eliminate congestion. Copyright © 1999lFAC
KeY'Ivords: Traffic control, ramp input, nonlinear control systems, discrete event dynamic systems, control' engineering.
l.
INTRODUCTION
Traffic control should initially focus on understanding traffic characteristics, accounting for why many traffic stream models have been proposed. Flow, density and speed are generally considered to be three components for the representation of traffic streams. From a theoretical perspective, flolV is the product of density and speed. The correlation between density and speed forwards to negative, i.e. speed declines with an increasing density. However, both high speed and low speed causes low flow, also, despite of high or low density all correspond to low flow. This phenomenon indicates that high speed operation does not make a road efficient in traffic flow because the maximum flow occurs at a certain median speed. Therefore, in freeway management, ramp metering control (RMC) is necessary if the traffic demand is at peak because amitrarily allowing a vehicle onto the road would cause a traffic congestion. Many freeways around the world have implemented ramp metering control. However, those control methodologies are insufficient since a majority of them are pre-time control or single ramp controL To adapt to the varying traffic patterns, the most available form of metering control is the one with responsive and coordinated abilities and a closed-
loop algorithm. Among the notable investigations involving ramp metering control include Wattleworth and Berry (1965), Drew, Brewer, Buhr and Whitson (1969), Yagoda (1970), Wang (1972), Isaksen and Payne (1973), Wang and May (1973), Eldor and Adler (1977), Looze, Roupt, NHs, Sandell and Altans (1978), Papageorgiou (1980), Goldstein and Kumar (1982), Iida, Hasegawa, AsaJ..."Ura and Shao (1989), Papageorgiou, Blosseville and Habib (1990, 1991), Chang, Wu and Cohen (1994), Zhang, Ritchie and Recker (1996) and Zhang and Ritchie (1997). Although a few contributors recently presenled responsive, coordinated and closed-loop models, they arc linear models. This work pTesents a nonlinear model and applies it towards freeway integrated on-ramp metering control. The rest of this paper is OTganized as follows. Section 2 describes the dynamic traffic strcam model employed herein to represent traffic phenomenon. Section 3 presents the metering control model. Next, Section 4 summarizes the simulation results which verifY the proposed model's effectiveness. Concluding remarks are finaJly make in Section 5.
2.
DYNAMIC TRAFFIC STREAM MODEL
Constructing a ramp metering control model initially involves determining the traffic stream model.
8363
Copyright 1999 IF AC
ISBN: 008 0432484
NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
Traffic meam models can generally be classified as static types, or dynamic types. Owing to that our control is dynamic, a dynamic model should be selected. According to Traffic Flow Theory (Gerlough and Huber, 1975): . Oil ou du Speed FquallOn: - + u - = (1) ot ox dt .. E . op oq C ontlflUlty
quat/on; -
ot
+-
ox
= 0
(2)
Where x denotes location, t represents time, q is flow, u denotes speed,p represents density. However, q =pu (3) holds. Payne (1971) has investigated that du =-.!.{[ue (P)-ul-2::. 0P } (4) dt 'J" pox In which, lie (p) is the equilibrium speed at the condition of density p, r denotes the driver reaction time, and v represents the coefficient representing the driver's expectation relevant to the density ofthe next road segment. For practice, Isaksen and Payne (1973) resolved Payne's model to be the simplified difference equations: k
k+l
uJ
=u 1le -I'>.[.u/k
k
u l -U I _ 1 --..:;.--=-~-
14th World Congress ofTFAC
Ramp metering control attempts to adjust the flows of the controlled major road such that the road has maximum performance. The traffic scenario of a major road can be revealed on the basis of Eqn.(1)(2)(3), tIle continllollS type, or (5)(6)(7), the discrete type. For operations, a discrete type is acceptable. In general, using speed equation and concentration equation to represent/composite a traffic system equation is sufficient. Thus, for simplification, by substituting Eqn.(7) into (6), diminishing the variable of flow, and letting the net on-ramp flow be the control variable, then Eqn.(6) becomes: pt+ t
nl
R;+l =
1
(9)
1
For a segment of a fF.eway, by using Eqn.(5) and (8) tIle systcm equation can be written as:
[ ~Ul ]=~/.{[!I]+ [ I
~
nJ(x 1+ t -XI)
]Rf+1} (10)
ut+
(7)
qt:
Xl
sr+
1 - uf, 1>PI:= pf-rr - pr. If the Where ~ul = controlled freeway has N segments separated by ramps, i.e. 1~1,2,3, ... ,N, by Eqn.(lO) the whole system equation is presented as the nonlinear form: 6X ==!:it . [f(X) +Br] (11) or, separated as
Where k denotes the time state in step, J represents the road segment, l:f!::; N, ]V is the number of controlled road segments, and, xI : the location of the beginning point of segment I. n I : the number of lanes of segment t. M : the time duration of one step of operation. the flo""s passing through Xl at time. s~ : the off-ramp flows at
rt+ 1 -
I'>.xl = b.t 'g(X)
qr+ = pt-t"L
XI
(8)
nl (Xl+ 1 - XI)
and assume that
r{: the on-ramp flows at
k
R[+l
where
PI
1:
k
-nlPI u l
(xJ+l -xI)
I
+Ili
_1>t (u/"
O.5(XJ+l - Xl_I)
I:
k
= Pt + I'>.t n l - I P1-l uI- l
during k.
I'>.x 1
In
which,
O=[O]N
= b.t . [b(X) +
X=[XI>X 2 ]T,
B=[O br,
(12)
br}
(13)
XI =[U 1 'U 2 ' .... ,u N
X2
f,
==[Pl'P2,····,PN f,
1
I'>.x 1 = [t.u l ,1>u2 , •.•. , llu N JT,
bll == - - - - nl (Xl+l - XI)
~""{2
b=
= [I'>.Pl , Ap2,····,APNJ
T
' r=[Rj,R2,····,RNJ
T
,
[hIlL,N' AX = [Ax Z,t.x2 ]T,
during k.
u; ;the actual speed on segment I at time k. pi :
the actual density on segment I at time k. The model incorporated by Eqn.(5), (6) and (7) is most effectively utilized for the description of dynamic traffic status. Figure 1 depicts the freeway scheme and relations of these variables in modeling. By assuming that each on-ramp and off-ramp is equipped "vith detectors for real-time flow measurement, the previous model can predict the traffic on the freeway. 3.
CONTROL MODEL MODELING
In system consideration, since Eqn.(l) is nonlinear, more accurate discrete expression for the system discrete speed equation involves expanding Eqn.(12) by Taylor series, unlike Eqn.(5). The fact that the second-order derivative of Eqn.(l2) exists, implies the following:
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NONLINEAR CONTROL APPLTCA nONS ON INTEGRATING FREE ...
XI (k
+ 1)
= XI (k)
+ &. g(X(k» +
l\,/2 {Og(X(k»)
2
8xI
. g(X(k) + eg(X(k» [b(X(k) + br(k)]}
(14)
OX2
Moreover, Eqn.(13) becomes x 2 (k + 1) = x 2 (k) + ~t· [h(X(k)) +br(k)] (15) if the first-order derivative is chosen. The above equations (14) and (15) now form a nonlineardiscrete-dynamie system describing freeway traffic with N segments. Consider a situation in which the metering control perfonnanee is set to maximize the flow rate on the major road. This is equivalent to minimizing the difference between real flow and ideal or maximal value. This is while assuming that the maximal flow occurs at a certain speed x 10 and density x 2o ' The performance function is then given as J =
.. T 2"1{[x[(k +l)-x 1o ] Q I [x](k+l)-xloJ+[X2(k+l)
- X20]"["
Qz[x2(k + 1) -
X20)
+
rT (k)Q3r(k)}
(16)
where, QI, Q2' Q 3 are the weights for speecl density and the control variable, respectively. The performance obviously makes the traffic approach to optimum at the next step by minimal energy. By substituting Eqn.(l4) and (15) into (16) and take the partial derivative J with the control variables, r3 J ( r3 r '" 0 , then the control law is obtained: r(k) = __1_[J..(og(X(k))b/Q]CO'g(X(k))b)+bTQ2b Ll.t' 4 c3x 2 O'X2
+~Q3rl)< 6t
{Ll.2t
(O'g~X(k))b?%[(Xl(k)-Xlo)+Ll.t. x2
14th World Congress ofTFAC
involved for the demonstration. Three simulations are executed respectively by (i) accomplishing RM"C with full demand at every ramp in peak hour, (ii) no RMC with the same demand condition, and (Hi) accomplishing RMC with few demand in off-peak. The run time of the three simulations is continuous for one hour. The traffic is described by Eqn.(14) and (15). The subsequent flows are calculated by Eqn.(3). In R1v[C, the feedback quantities in each step are given by Eqn.(l7). According to Haynes (1965), for a three-lane segment ofa freeway, q = 65.5p-0.179p 2 -80 (19) in which, the density is denoted in vehicles per mile. By Eqn.(l9), the following form is utilized for the term of u e (p) in function g(X): u e =-glp=105.39_L39p_26.67 (20) P where the unit of the density is converted in vehicles per kilo meter per lane. Eqn.(20) represents the real world traffic 011 the simulations and analysis. Based on Eqn.(20), the maximal flow should occur at U o = 52 kph corresponding to Po =- 38 vehl km.
Then, for five segments, X20
eX I
oX 2
= Rt+ 1 +S1kT1
(18)
R 1k + l Er(k) solved from Eqn.(7). However, off-
ramp Dow ..,-[+1 should be estimated to complete Eqn.(l8). In general, the time series data J , .•• ,
{sf,
s1} of off-ramp flows can be obtained by
loop detectors which are installed at ramps. Based on the series and via Kalman Filter method, will
st"
be known. Nevertheless, r/,,+I has limitation: 0 s;:
rt+ 1 se, where C denotes the capacity of ramp I. 4.
= 38 [1 1 1 1 l]T are given. Herein, assume
qe 'f , 1 P < Po 1- e- qT [2000 + (2000 _ q)]e-[2000+(2()(J()-q)]T
"------"----;=~=;::__~---,
(17)
Based on Eqn.(9), the number of vehicles allowed to enter the freeway at ramp I during state k+ 1 is
sf
t1 1 I 1 1JT and
-qT
r'"
+b T %((x 2 (k) - "20) + bot ·h(X(k»)]J
rrhl
= 52
that all segments uniformly have three lanes. Where M is set with 60 seconds. v =19.2 (km2/hr) and .. =0.022 are provided (Goldstein and Kumar, 1982). The ramp capacity is defined at 1500 veh/hr. The OIl-ramp flow r in no RMC is calculated by the model (Drew, 1971):
g(X(k» + Ll.t 2 (og(X(k» g(X(k» + b'g(X(k» h(X{k»)]
2
X lo
MODEL EVALUATION
For applkation, the model proposed in the previous section should be verified and evaluated. The verification and evaluation are executed by simulations. A freeway with five segments is
l_e-[2
(21) .
If P > Po
where T is set as 5 seconds. Figure 2 summarizes the simulation results of accomplishing RM"C with full demand at every ramp in peak hour. Figure 3 results from no RMC. Figure 4 indicates that with few demand in off-peak. All simulations correspond to the initial conditions listed in Table 1. Each figure contains four graphs which illustrate (i) the flows of five ramps, (ii) the flows of five segments and their upstream of the controlled freeway, (iii) the speeds of five segments and their upstream, and (iv) the density. All the graphs are plotted with respect to time. According to Fig.2, the on-ramp flow in each ramp is at a high flow rate during the beginning control period and, tben, comes down since the initial condition is given at near saturatioll. Simulation results indicate that saturation occurs approximately at the flow 2000 veh/hr. The flows on all segments approach the value in other periods except at beginning. This phenomenon correlates with tile situation in which speed and density are accurately
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NONLlNEAR CONTROL APPLICATIONS ON INTEGRATING FREE...
reflected. Speed tends to 52 kph and density 38 vehlkm, the ideal values. The control is robust despite its unstable traffic at unconlroHed upstream segment SO. This finding means that the control model can maintain the traffic at high perfonnance, speed does not decline and avoid traffic jam during peak hours. However, Fig.3 illustrates a bad traffic situation. Obviously, no flow control on the ramps during peak periods causes traffic jam, in which concentration overloads, speed and flow decline, almost vanishing. Figure 4 reveals that first three ramps allow full access into the freeway dnring off-peak traffic. This situation indicates that, during off-peak, metering control on Ramp 1, 2 and 3 are unnecessary. After all, such a control is necessal}' for Ramp 4 and 5 to pertain the well level of traffic service. When vehicles fully pour in the freeway from first three ramps, the traffic gradually becomes heavy at downstreams sequentially. However, the traffic is stable and at high performance in all segments except the segment SO \....hich is uncontrolled. 5.
CONCLUSION
Ramp metering control is implemented in many freeway access tinting control. A well control model makes high performance of freeway operations. A coordinated-responsive and closed-loop system is more perfect than individual or pretime control. However, some improving spaces are available for conventional models. In this study, we derive a model which achieves the real philosophy inquired by freeway traffic control. The proposed model is nonlinear which is more p.recise to describe an actual environment than linear models. According to the verification and evaluation by simulations, the model proposed herein is robust and efficient for metering control. With such control, freeway traffic eliminates congestion. In according with the previous description, to implement this model, upstream approaching flow rate, flows on each on-ramp and off-ramp should be detected with real-lime. Based on the detected data and via the proposed traffic stream model, speed and density are converted and known. The optimal allowed flow rate on ramps is finally determined by the provided feedback gain. This allowed flow rate is generally achieved with the on/off timing sharing of a traffic signal. However, such a control is operated through a controller or a coordinated traffic control center. REFERENCES J. Wu and S.L Cohen (1994) . Integrated real-time ramp metering modeling for non-recurrent congestion: framework and preliminary results. Transportation Research Record,
Chang, G.L.,
1446,56-65.
Drew, D.R. (1971). Traffic Flow Theory and Control. McGraw-HiIl Book Company, NY 10036. Drew, D.R., K. A . Brewer, J. R Buhr and R.H. Whitson (1969). Multilevel approach to the design of a freeway control system. Highway Research Record, 279,70-86. Eldor, M. and H. Adler (1977). Post-optimality analysis methodology for freeway on-ramp controL Transportation Research Record, 644, 51-53. Gerlough, D.L. and M.I. Huber (1975) . Traffic Flow Theory, TRB Special Report 165, National Science Council, Washington D.e. Goldslcin, N.B. and K.S.P. Kumar (1982). A decentralized control strategy for freeway regulation. Transportation Research, ]6B(4), 279-290. Raynes, 1.J. (1965). Some considerations of vehicular density on urban freeways. Highway Research Record, 199, 59-80. Iida, Y., T . Hasegawa, Y. Asakura and C.F. Shao (1989). A fonnulation of on-ramp traffic control system with route guidance for urban expressway. Proceedings of IFAC Control Computers, Communications in Transportation, Paris, France. Isakscn, L. and H.J. Payne (1973). Sub-optimal control of linear systems by augmentation with application to freeway traffic regulation. IEEE Transactions on Automatic Control, 18(3),210-219. Looze, D .P., P.K. Roupt, R. Nils, JR. Sandell and M. Athans (1978). On decentralized estimation and control with application to freeway ramp metering. IEEE Transactions on Automatic Control, 23(2), 268-275 .
Papageorgiou, M. (1980). A new approach to timeof-day control based on a dynamic freeway traffic model. Transportation Research, 14B(5),349-360. Papageorgioll, M., J.M. Blosseville and RS. Habib. (1990). Modelling and real-time control on the southern part of boulevard peripherique in Paris: Part I Modelling and Part II Coordinate on-ramp metering. Transportation Research, 24A(5), 345370.
Papageorgiou, M., H.S. Habib and 1.M. Blosseville (1991). ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, 1320, 58-64. Payne, H.1. (1971). Models of freeway traffic and control, In: Mathematical Models of Public Systems (G.A. Bekey, Ed.), Simulation Council. Proc. Ser., 1(1),51-61. Wang, C.F. (1972) . On a ramp flow assignment problem. Transportation Science, 6, 114-130. Wang, J.1. and A.D. May (1973). Computer model fOT optimal freeway on-ramp control. Transportation R esearch Record, 469, 16-25. Wattleworth J.A. and D.S. Beny (1965). Peak-period control of a freeway system-some theoretical investigations. Highway Research Record, 89, 11-25. Yagoda, H.N. (1970). The dynamic control of
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14th World Congress ofIFAC
tation R esearch, SC(5), 273-286. Zhang, H.M., S.O. Ritchie and W.W. Recker (1996) . Some general results on the optimal ramp control problem. Transportation Research , 4C(2), 5]-69.
automotive traffic at a freeway entrance ramp. Automatica, 6, 385-393. Zhang, H.M. and S.G. Ritchie (1997). Freeway Tamp metering artificial neural networks. Transp or-
Basic data of the simulation case
Table 1 Segment 0 (SO) (upstream)
Segment I
Segment 2
Segment 3
Segment 4
Segment 5
Segment
(SI)
(S2)
(S3)
(S4)
(SS)
Len~h(km)
----
1.50
1.70
1.00
1.80
2 .60
Peak Hour Initial Flow (veh/hr/lane)
1980
1910
1850
1850
1720
1730
Segment
q~
0
...
1
I
/-1
pi
k
Pt ••• Plk
ui
u 1k
k SI
J.,
t
-+
U f- 1
Xl~
Xo
q1
qf
k I
qt.
-
k
pf u Ik
Pl+l k
U'+l
p~ uNk
Xl
s~ J.,t r~
r k
1
Fig. 1.
Freeway layout and relevant variables foa.vs on m ad
lows on ramp"
.100
1000 900 \ $ 00
Segment N
1+1
2tl5O
P
I
700 500
I
500
S5
1750
40
5{J
sa
1700
0
10
;D
30
lime (min)
time (rnin )
s poeed co road
den5ity 00 reed
'0
50
00
75
" 70
65
i1
~
SS
2fj
24 50
0
10
20
:l(J
22 40
50
&l
0
timoe (min,
Fig. 2.
10
2()
30 time (rni n)
'0
5D
60
Simulation result ofRMC during peak hours
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14th World Congress ofTFAC
'Dow-s on road
lI.ows- 00 ramps
250
2500
200
2000
i
1500
TI
~
1000
500
0
0
10
20
30
SO
40
time (min)
time (min,
speed en rood
density on rood
time (mn)
time (rnln)
60
80
Fig. 3.
Simulation result of without RMC during peak periods
to'.llison r(Bd ~0r-----r-----~--
1500
20
30
40
-
iD
BD
50
-
--------
Sll
20
30
~ime(min)
5~eed
__----__----__-----,
40
50
60
time (min)
density 00 rood
.....
_------..--------
S2
20 --
-
~3-
-
-
____
-
-'\. . . . . ..r \'\,.., . -. . . . . . . ...;./ v/"""
-J
'-
/
SO
15
55L---~----~----~----~----~----->
o
10
20
J.(J 11me (mln)
Fig. 4.
40
50
60 time (min)
Simulation result ofRMC during off-peak traffic
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ISBN: 008 0432484