A Density-Based Ramp Metering Model Considering Multilane ... · ResearchArticle A Density-Based Ramp Metering Model Considering Multilane Context in Urban Expressways LiTang,1,2

Research ArticleA Density-Based Ramp Metering Model Considering MultilaneContext in Urban Expressways

Li Tang12 Xia Luo23 Pengfei Zhai3 and Xunfei Gao3

1School of Automobile and Transportation Xihua University No 999 Jinzhou Road Chengdu Sichuan 610039 China2Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies Sipailou No 2 Nanjing 210096 China3School of Transportation and Logistics Southwest Jiaotong University No 111 Second Ring Road North Section 1Chengdu Sichuan 610031 China

Correspondence should be addressed to Li Tang tanglitrafficgmailcom

Received 27 October 2016 Accepted 17 January 2017 Published 2 March 2017

Academic Editor Gennaro N Bifulco

Copyright copy 2017 Li Tang et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As one of the most effective intelligent transportation strategies ramp metering is regularly discussed and applied all over theworld The classic ramp metering algorithm ALINEA dominates in practical applications due to its advantages in stabilizing trafficflow at a high throughput level Although ALINEA chooses the traffic occupancy as the optimization parameter the classic trafficflow variables (density traffic volume and travel speed) may be easier obtained and understood by operators in practice Thispaper presents a density-based ramp metering model for multilane context (MDB-RM) on urban expressways The field data oftraffic flow parameters is collected in Chengdu China A dynamic density model for multilane condition is developed An errorfunction represented by multilane dynamic density is introduced to adjust the different usage between lanes By minimizing theerror function the density of mainstream traffic can stabilize at the set value while realizing the maximum decrease of on-rampqueues Also VISSIM Component Object Model of Application Programming Interface is used for comparison of the MDB-RMmodel with a noncontrol ALINEA and density-basedmodel respectivelyThe simulation results indicate that theMDB-RMmodelis capable of achieving a comprehensive optimal result from both sides of the mainstream and on-ramp

1 Introduction

Building expressways or elevated expressways inside the cityto relieve severe congestion and its related problems is apopular strategy in many metropolitan areas of the worldHowever as urban planners must contend with limitedland resources increasing capacity simply by adding infras-tructure is not sustainable Thus Intelligent TransportationSystem (ITS) is introduced as an alternative approach totackle the issue of traffic jams [1] Although many effortshave been done to optimize traffic operations using ITStechniques the topic of motorway traffic regulation is still anopen field for researchers and for practical implementation[2]On-rampmetering is regarded as one of themost effectivetraffic control strategies to exploit the potential of supply inexpressways [3] Through the use of traffic lights local rampmetering corresponds to controlling the traffic flow merging

into the mainstream from the on-ramp so as to maximizethemainstream flow downstream of the on-ramp in responseto prevailing traffic conditions [4 5] Since it was first intro-duced on I-290 in Chicago in 1963 on-ramp metering hasbeen implemented on major expressways in many big citiesaround the world Examples include Route 101 in Los Angeles[6] Boulevard Peripherique in Paris [7 8] and the A10 WestMotorway in Amsterdam [9] Coordinated ramp meteringstrategies make use of measurements from an entire region ofthe network to control all metered ramps Coordinated rampmetering approaches include multivariable control strategies[10] optimal control strategies [11] and further heuristicalgorithms [12] A few coordinated ramp metering strategieshave been deployed recently in the field such as the I-25 Motorway in Denver [13] and the M1M3 Motorwayin Queensland [14] Another extensive control method isramp integration control metering which combines ramp

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 4909363 11 pageshttpsdoiorg10115520174909363

2 Mathematical Problems in Engineering

metering and other traffic control strategies such as variablespeed limits and route guidance so as to ameliorate trafficperformance both in motorways and in urban roads Duringthe last decades ramp integration control metering has beenwell developed The field tests are also well performed [15]

The current employed ramp metering algorithms largelyuse ALINEA the first local ramp metering control strategybased on straightforward application of classical feedbackcontrol theory so as to achieve and maintain maximumcapacity occupancy [16] Taking advantage of the feed-back mechanism ALINEA displays superior performancecompared with feedforward-based strategies (eg demand-capacity and occupancy strategies) in terms of smoothlyreacting to slight differences in occupancy and stabilizingtraffic flow at a high throughput level [17 18] ALINEA isdeveloped upon traffic occupancies the measurements ofwhich may not be readily related to the classic traffic flowvariables Thus a great number of efforts have been made toimprove ALINEA like GA-based ALINEA in 2002 [19 20]UP-ALINEA FL-ALINEA and UF-ALINEA in 2003 [21]AD-ALINEA and AU-ALINEA in 2004 [22] PI-ALINEA in2007 [23 24] IFT-tuned ALINEA in 2010 [25 26] and soforth In most of the improved algorithms traffic volumeis introduced as the set value in order to operate moreconveniently in practical applications However traffic flowdoes not uniquely characterize the traffic state (ie the samevalues of traffic volumemay indicate both light and congestedtraffic) as well as travel speed As a result a stream of scholarssuggested using density as the key objective of optimizingramp metering rates [27] Generally speaking this will bringabout benefits in two main aspects First the density ofthe mainstream traffic flow can stabilize at the expectedvalue like ALINEA Meanwhile the queue length in theon-ramp can be minimized as much as possible Secondand more significant compared with occupancy data theobtaining of which is relied on for the layout of undergroundloops the measurement of density is much easier and moreconvenient (eg using video recording or even unmannedaerial vehicles)

This paper focuses on modeling density-based rampmetering control for multilane expressways (MDB-RM) inurban areas The following contributions have been made adensity-based ramp metering model considering multilanesituations in real life is built A dynamic density model formultilane condition is developed on the basis of METANETmodel An error function represented by multilane dynamicdensity is introduced into the MDB-RM model to adjustthe different usage among lanes and minimize objectivefor optimization Real traffic data is collected from theSecond Ring Elevated Expressway in Chengdu It is usedfor parameter fixing and simulation VISSIM 7 is chosen asthe simulation platform running our proposed model andcompared models

2 MDB-RM Model

21 Dynamic Density Model for Multilane Condition SinceLighthill and Whitham published their famous dynamicmacroscopic traffic flow modeling paper [28] much work

has been done to provide mathematical and computationalapproaches for describing the characteristics of traffic flowIn this paper the dynamic density model is developed on thebasis of the META [29 30] and METANET [31 32] modelsproposed by Messmer and his collaborators Adopting andunifying the definitions used by Papageorgiou and Kotsialos(2002) [3] an expressway can be divided into119873 segments ofequal length 119871 The traffic density 120588(119896 + 1 119894) for each lane ina segment 119894 at time 119905 = (119896 + 1)119879 can be represented as

120588 (119896 + 1 119894) = 120588 (119896 119894) + 119879119871120582 [119902 (119896 119894 minus 1) minus 119902 (119896 119894)] (1)

where 119896 = 1 2 119870 is the discrete time index 119879 is sampletime which is typically equal to 5sdot sdot sdot 15 s 119902(119896 119894) is the numberof vehicles leaving segment 119894 during the time period [119896119879 (119896 +1)119879] divided by 119879 and 120582 is the number of lanes

Tan and Tang improved theMETAmodel in the aspect ofconsidering lane change behavior and additional factors suchas safety space between vehicles speed and route choiceThedensity model in multilane situations reads

120588119903119904 (119896 + 1 119894 119897) = 120588119903119904 (119896 119894 119897) + 119879119871 [119902119903119904 (119896 120593 (119894 119897))minus 119902119903119904 (119896 119894 119897) + 119906119903119904 (119896 119894 119897) minus 119890119903119904 (119896 119894 119897)]+ sum119910isin119862119894

119899119903119904119910119897 (119896 119894) minus sum119910isin119862119894

119899119903119904119897119910 (119896 119894) (2)

where 119903 is the origin of vehicle 119904 is the destination of vehicle120593(119894 119897) is the set of lanes adjacent to lane 119897 in the upstreamsegment 119894minus1120593(119894 119897) = (119894minus1sdot119910) |1minus119910| le 1 119910 isin 119862119894 119902119903119904(119896 119894 119897)is the number of vehicles leaving from lane 119897 of segment 119894 todownstream mainline during the time period [119896119879 (119896 + 1)119879]divided by 119879 119906119903119904(119896 119894 119897) is the number of vehicles mergingfrom on-ramp to lane 119897 of segment 119894 during the time period[119896119879 (119896+1)119879] divided by119879 119890119903119904(119896 119894 119897) is the number of vehiclesleaving through off-ramp from lane 119897 of segment 119894 during thetime period [119896119879 (119896 + 1)119879] divided by 119879 and 119899119903119904119910119897(119896 119894) is thenumber of vehicles changing from lane 119910 to lane 119897 of segment119894 during the time period [119896119879 (119896 + 1)119879]

Tanrsquos model gives an adequate description of real trafficflow conditions on freeways However excessively detaileddefinitions of origin and destination substantially increasethe difficulty of data acquisition something which is notnecessary in the ramp metering modeling Thus the densitymodel is reformulated based on the following assumptions

(i) The urban expressway is divided into 119873 segments ofequal length 119871The number of lanes in the expresswaystays the same

(ii) For segment 119894 there is at most one on-ramp or off-ramp connected with it

(iii) Vehicles from the on-ramp can only merge into theoutermost lane Vehicles from the mainstream canonly change to adjacent lanes Cross-lane change isnot allowed

To simplify themodel formula a two-lane situation is firstgiven Define the inner lane as lane 1 and the outer lane as

Mathematical Problems in Engineering 3

1 j Ni

On-ramp

Lane 1

Lane 2

Trac ow direction

L

w

middot middot middot middot middot middot middot middot middot

Segment j

q(k j minus 1 1)

q(k j minus 1 2)

q(k j 1)

q(k j 2)

120588(k j 1)

120588(k j 2)n21(k j) n21(k j)

r(k j)

q rampin(k)

Figure 1 Layout of segment 119894 = 119895 with an on-ramp

lane 2 When segment 119894 = 119895 that means no on-ramp or off-ramp is connected And when segment 119894 = 119895 this means anon-ramp is connected The layout of segment 119895 is shown asFigure 1 (since this paper focuses on the on-ramp meteringcontrol only the on-ramp layout is given the segment withan off-ramp can be deduced in the same process)

Thus the dynamic density of inner lane and outer lanecan be written as

120588 (119896 + 1 119894 1) = 120588 (119896 119894 1) + 119879119871 [119902 (119896 119894 minus 1 1) minus 119902 (119896 119894 1)+ 11989921 (119896 119894) minus 11989912 (119896 119894)]

120588 (119896 + 1 119894 2) = 120588 (119896 119894 2) + 119879119871 [119902 (119896 119894 minus 1 2) minus 119902 (119896 119894 2)+ 11989912 (119896 119894) minus 11989921 (119896 119894)] 119894 = 119895

120588 (119896 + 1 119894 2) = 120588 (119896 119894 2) + 119879119871 [119902 (119896 119894 minus 1 2) minus 119902 (119896 119894 2)+ 119906 (119896 119894) + 11989912 (119896 119894) minus 11989921 (119896 119894)] 119894 = 119895

(3)

where 119902(119896 119894 1) is the number of vehicles leaving from theinner lane of segment 119894 to downstream mainline during thetime period [119896119879 (119896 + 1)119879] divided by 119879 and 11989912(119896 119894) is thenumber of vehicles changing from inner lane to outer lane ofsegment 119894 during the time period [119896119879 (119896 + 1)119879]

Define 120578119910119897(119896 119894) as the ratio of the number of vehicleschanging from lane 119910 to lane 119897 of segment 119894 to the numberof vehicles in lane 119910 of segment 119894 during the time period[119896119879 (119896 + 1)119879] Thus (3) can be rewritten as

120588 (119896 + 1 119894 1)= 120588 (119896 119894 1) + 119879119871 [119902 (119896 119894 minus 1 1) minus 119902 (119896 119894 1)]+ 12057821 (119896 119894) 120588 (119896 119894 2) minus 12057812 (119896 119894) 120588 (119896 119894 1)

120588 (119896 + 1 119894 2)= 120588 (119896 119894 2) + 119879119871 [119902 (119896 119894 minus 1 2) minus 119902 (119896 119894 2)]+ 12057812 (119896 119894) 120588 (119896 119894 1) minus 12057821 (119896 119894) 120588 (119896 119894 2)

if 119894 = 119895120588 (119896 + 1 119894 2)= 120588 (119896 119894 2)+ 119879119871 [119902 (119896 119894 minus 1 2) minus 119902 (119896 119894 2) + 119906 (119896 119895)]+ 12057812 (119896 119894) 120588 (119896 119894 1) minus 12057821 (119896 119894) 120588 (119896 119894 2)

if 119894 = 119895(4)

And the dynamic density of on-ramp can be written as

120588ramp (119896 + 1 119894) = 120588ramp (119896 119894)+ 119879119908 [119902rampin (119896) minus 119903 (119896 119894)]

(5)

where 119902rampin(119896) is the number of vehicles entering on-rampduring the time period [119896119879 (119896 + 1)119879] divided by 119879 and 119908 isthe length of on-ramp

Furthermore for multilane situations define the totalnumber of lanes as 119883 Let 119897 represent the 119897th lane and 119897 =1 2 119883When 119897 = 1 it represents the inner lane andwhen119897 = 119883 it represents the outer lane The multilane dynamicdensity model reads as follows

4 Mathematical Problems in Engineering

For inner lane

120588 (119896 + 1 119894 119897) = 120588 (119896 119894 119897) + 119879119871 [119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897)]+ sum119910isin119862119894

120578119910119897 (119896 119894) 120588 (119896 119894 119910)minus sum119910isin119862119894

120578119897119910 (119896 119894) 120588 (119896 119894 119897)if 1 le 119897 le 119883 minus 1

(6)

For outer lane

120588 (119896 + 1 119894 119883)= 120588 (119896 119894 119883) + 119879119871 [119902 (119896 119894 minus 1 119883) minus 119902 (119896 119894 119883)]+ sum119910isin119862119894

120578119910119883 (119896 119894) 120588 (119896 119894 119910)minus sum119910isin119862119894

120578119883119910 (119896 119894) 120588 (119896 119894 119883) if 119894 = 119895120588 (119896 + 1 119894 119883)= 120588 (119896 119894 119883)+ 119879119871 [119902 (119896 119894 minus 1 119883) minus 119902 (119896 119894 119883) + 119906 (119896 119894)]+ sum119910isin119862119894

120578119910119883 (119896 119894) 120588 (119896 119894 119910)minus sum119910isin119862119894

120578119883119910 (119896 119894) 120588 (119896 119894 119883) if 119894 = 119895

(7)

22 Density-Based Ramp Metering Model for MultilaneExpressways To build up the MDB-RMmodel first we needto derive differential equations for the multilane dynamicdensity model For segment 119894 without a connected on-rampthat is 119894 = 119895 the variation of density for each lane isdetermined by outflow 119902(119896 119894 minus 1 119897) from segment 119894 minus 1 and119902(119896 119894 119897) from segment 119894 Define Δ119905 = Δ119896 sdot 119879 the variation ofdensity when 119894 = 119895 can be represented as

120588 (119896 + Δ119896 119894 119897) = 120588 (119896 119894 119897) + Δ119905119871 [[119902 (119896 119894 minus 1 119897)

minus 119902 (119896 119894 119897) + sum119910isin119862119894

119899119910119897 (119896 119894) minus sum119910isin119862119894

119899119897119910 (119896 119894)]]

(8)

Transposing (8) and taking limit of both sides we can get

limΔ119905=0

120588 (119896 + Δ119896 119894 119897) minus 120588 (119896 119894 119897)Δ119905= 119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897) + sum119910isin119862119894 119899119910119897 (119896 119894) minus sum119910isin119862119894 119899119897119910 (119896 119894)119871

(9)

Thus the differential equation for segment 119894 = 119895 reads120588 (119896 119894 119897) = d120588 (119896 119894 119897)

d119905 = 1119871 [[119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897)

+ sum119910isin119862119894

119899119910119897 (119896 119894) minus sum119910isin119862119894

119899119897119910 (119896 119894)]]

(10)

As for segment 119894 = 119895 the differential equation stays thesamewith 119894 = 119895 situation for inner lanes 1 le 119897 le 119883minus1 And forouter lane 119897 = 119883 the variation of density is determined by theoutflow 119902(119896 119894minus1 119897) from segment 119894minus1 119902(119896 119894 119897) from segment119894 and inflow 119906(119896 119894) from on-ramp Thus the variation ofdensity when 119894 = 119895 can be represented as

120588 (119896 + Δ119896 119894 119897) = 120588 (119896 119894 119897) + Δ119905119871 [[119902 (119896 119894 minus 1 119897)

minus 119902 (119896 119894 119897) + 119903 (119896 119894) + sum119910isin119862119894

119899119910119897 (119896 119894) minus sum119910isin119862119894

119899119897119910 (119896 119894)]]

(11)

Replicating the formula transformation like (9) thedifferential equation for segment 119894 = 119895 reads

120588 (119896 119894 119897) = d120588 (119896 119894 119897)d119905 = 1119871 [[

119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897)

+ 119903 (119896 119894) + sum119910isin119862119894

119899119910119897 (119896 119894) minus sum119910isin119862119894

119899119897119910 (119896 119894)]]

(12)

Thus the differential equations of dynamic density con-sidering lane change behavior in multilane expressways canbe summarized as

120588 (119896 119894 119897) = 1119871 [[119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897) + sum

119910isin119862119894

119899119910119897 (119896 119894)

minus sum119910isin119862119894

119899119897119910 (119896 119894)]]if 119894 = 119895 or 119894 = 119895 1 le 119897 le 119883 minus 1

120588 (119896 119894 119897) = 1119871 [[119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897) + 119903 (119896 119894)

+ sum119910isin119862119894

119899119910119897 (119896 119894) minus sum119910isin119862119894

119899119897119910 (119896 119894)]]if 119894 = 119895 119897 = 119883

(13)

And the differential equation of dynamic density for anon-ramp (see (5)) can be written as

120588ramp (119896 119894) = 1119908 [119902rampin (119896) minus 119903 (119896 119894)] (14)

Mathematical Problems in Engineering 5

Section L

Direction of trac ow

Camera

A B

n

Space headway ln+1n

n + 1

Figure 2 Layout scheme of data collection

In consideration of the fact that there is an obviousdifference in the usage of innerouter lanes [33 34] the errorfunction 119869(119896) is introduced in the MDB-RM model to adjustthis difference while keeping the mainstream density closeto the expected value like ALINEA and lowering the queuelength at the same time Hence

119869 (119896) = 119883sum119897=1

120582 (119897) 1003816100381610038161003816120588 (119896 119894 119897) minus 120588119888 (119894 119897)1003816100381610038161003816 + 120582ramp120588ramp (119896) (15)

where 120588119888(119894 119897) is the expected density of lane 119897 in segment119894 120588ramp(119896) is the density of on-ramp at time 119896119879 120582(119897) is theweight function of lane 119897 120582ramp is the weight function of on-ramp and sum120582(119897) + 120582ramp = 1

To minimize 119869(119896) homogeneous linear differential equa-tion of first order is defined as

119901119869 (119896) + 119869 (119896) = 0 119869 (119896) = 119869 (0) 119890minus119901119896119879 (16)

Further the derivative 119869(119896) of error function 119869(119896) can bewritten as

119869 (119896) = 119883sum119897=1

120582 (119897) sdot 120588 (119896 119894 119897) + 120582ramp 120588ramp (119896) (17)

Overall the MDB-RMmodel can be written as

119903 (119896 119894) = 119871 sdot 119908120582 (119883) sdot 119908 minus 120582ramp sdot 119871 119869 (119896) minus 1119871119883sum119897=1

120582 (119897)

sdot [119902 (119896 119894 minus 1 119897) minus 119902 (119896 119894 119897)] minus 1119871119883minus1sum119897=1

[120582 (119897) minus 120582 (119897 + 1)]sdot [120578119897+1119897 (119896 119894) sdot 120588 (119896 119894 119897 + 1) minus 120578119897119897+1 (119896 119894) sdot 120588 (119896 119894 119897)]minus 120582ramp sdot 119902rampin (119896)119908

(18)

3 Data Spectrum

31 Data Collection Validation of the proposed MDB-RMmodel is conducted by simulator VISSIM 7 The data forsimulation is collected from a section of the Second RingElevated Expressway in Chengdu a large city in southwestChina Video recording is applied for data acquisition A HDcamera was set up in the BRT station of Taoxi Road and therecording work lasted from 1700 to 1840 onMay 2 2016Thelayout scheme of data collection is shown in Figure 2

Wu discusses how to extract traffic flow characters fromtraffic video recordings (eg ldquospeed-densityrdquo and ldquospeed-space headwayrdquo) in his series of field studies [35ndash37] Basedon Wursquos idea we draw ldquosection travel time-time headwayrdquodata directly from the video record in the first place and thenobtain the vehicle speed data according to section travel timeand set section length 119871

Define 119860 as origin and 119861 as destination of the roadsection In the data extraction process the expressway issimplified as a one-way roadwith two lanes Assume that 119905119860119899 isthe time when vehicle 119899 arrived at119860 and 119905119861119899 is the time when 119899arrived at119861Then the timewhen the next vehicle 119899+1 arrivedat 119861 is 119905119861119899+1 Thus the section travel time of vehicle 119899 can berepresented as 119905119881119899 = 119905119861119899 minus 119905119860119899 The time headway of vehicle 119899+1can be represented as 119905119863119899+1 = 119905119861119899+1 minus 119905119861119899

Further the section speed V119899 can be written as

V119899 = 119871119905119881119899 =119871119905119861119899 minus 119905119860119899 (19)

When the head of vehicle 119899 arrives at 119861 the location ofvehicle 119899+1 in the same lane can be shown in Figure 2 Definethe space headway between vehicle 119899 and vehicle 119899 + 1 as119897119899+1119899 Thus the space headway 119897119899+1119899 can be approximatelyrepresented as the product of the section speed V119899+1 and thetime headway 119905119863119899+1 shown as

119897119899+1 = V119899+1 sdot 119905119863119899+1 = V119899+1 sdot (119905119861119899+1 minus 119905119861119899) (20)

6 Mathematical Problems in Engineering

According to the relationship between space headway anddensity the density of vehicle 119899 + 1 can be represented as

120588119899+1 = 1000119897119899+1 (21)

Although time counting of the recordings is operatedmanually the video playback can be slowed down to thelevel of a frame (the video recording lasts 100 minutes whichis equal to 150000 frames) The error of section travel timerecording is controlled in 3 frames That is for a vehicletraveling at speed of 50 kmh the relative error of speed is lessthan 6329 which is acceptable for our research purposeMoreover since the data is collected during the peak hour ofdaily traffic most vehicles were traveling at a relatively lowspeed The data processing in level of a frame ensures theaccuracy of the traffic flow data

32 Data Description After data extraction two principlesare applied for data filtering first the section speed of thevehicle cannot be higher than 100 kmh (80 kmh is the speedlimit on the Second Ring Elevated Expressway) seconddensity cannot be greater than 150 vehkm Finally 3385 pairsof ldquosection travel time-time headwayrdquo and ldquosection speed-densityrdquo are obtained

To illustrate the traffic features of the Second Ring Ele-vated Expressway time-varying diagrams of traffic volumespeed and density are separately plotted shown as Figure 3The traffic volume data is obtained on the basis of the rate offlow for 20-second intervals instead of the flow-speed-densityformula The number of passing vehicles is counted overevery 20 seconds to reflect short-term fluctuations of trafficin as much detail as possible [38 39] As seen in Figure 3operation features of the Second Ring Elevated Expresswayduring the peak hour of the observed day can be concludedas follows

(i) In the first 30 minutes during the data collectionthe traffic is mostly under free flow condition Thusthe traffic flow fluctuation is relatively large while thespeed and density are stable The traffic jams in thistime period are nonrecurrent and temporary such asthe breakdowns observed in the 21stndash23rdminute andthe 25ndash27th minute

(ii) Viewed from the speed time-varying diagram vehiclespeed has a significant drop from the 39th minutewhich implies the start of recurrent congestion It isalso observed that the congestion lasts from the 41st tothe 63rd minute Most vehicles travel at 10ndash20 kmhand density during this time internal is relativelyhigh As for traffic volume though the fluctuationdecreases the 20-second rate of flow in some intervalsdecreases since merging vehicles increase the conges-tion in the mainstream (such as the flow rate of 0 atthe 43rd minute)

(iii) Evacuation of vehicles in the mainstream starts fromthe 64th minute and the speed rises back to 60 kmhin a very short time The main reason is the decreaseof both vehicle arrival and downstream traffic flow

(there is an obvious trough during the 64ndash67thminute in time-varying diagram of flow rate) Due tothe rebound of vehicles the speed drops again duringthe 69ndash85th minute as shown in speed time-varyingdiagram

(iv) The traffic recovers to free flow after the 85th minuteMore randomness of vehicle arrival which is shownas the large vibration in flow rate time-varying dia-gram can be seen during this time period The speedof vehicles is generally higher than the average andthe density descends to the same level with the first30 minutes

While Figure 3 reveals traffic flow features of the investi-gated road on the other hand the consistency of the trafficvariation tendency indicated by the volume-speed-densitydata also proves the reliability of the collected dataMoreoverwe translate the ldquosection speed-densityrdquo data into ldquotrafficvolume-densityrdquo data and plot the scatter diagram to repre-sent the relationship between these traffic flow parametersin Figure 4 In this way the effectiveness of our field data isfurther supported

Taking density variation in multilane situations intoaccount the error of traffic volume in the whole road cannotbe neglected Thus vacancy parameter 120582 is introduced to fixvolume 1198761015840 as follows

1198761015840 = 120582V119904 sdot 120588 = 120582V119904 1000ℎ (22)

The value of 120582 is related to both the level of congestionin real situations and the counted number of vehicles In thispaper we put 120582 = 077 based on former experience

Figure 4 shows the relationship between speed and den-sity as well as volume and density in terms of a scatterdiagram Regression analysis is conducted for the ldquospeed-densityrdquo data using the Greenshields Underwood andGreenburg models separately The result indicates that allthe models are statistically significant and that the Under-wood model fits best which is identical to another researchoutcome in Beijing [40] The basic trend of the ldquovolume-densityrdquo diagram can also be supported by Kerner andRehbornrsquos conclusion in 1996 [41] In his empirical study withlarge amounts of field data Kerner points out that underlow density situation (free flow condition) the ldquovolume-densityrdquo can be simplified as a curve and a two-dimensionalarea under middlehigh density situation (congestion) Italso explains why the linearity is not very significant inFigure 4(b) since free flow condition in the investigated roadrarely occurs during the peak hour

4 Simulation and Discussion

41 Simulation The simulation is conducted for twomonodirectional homogeneous lanes of the freewaymainstream with a single on-ramp The interval distance 119871between the mainstream detectors and the interval distance119908 between on-ramp detectors are set equal to 05 and03 km respectively The time interval 119879 is set equal to30 s for all control scenarios Four evaluation scenarios

Mathematical Problems in Engineering 7

1 5 8 12 15 18 21 24 28 31 35 38 41 44 47 50 54 57 60 63 68 71 75 78 81 84 88 92 95Time

DensityMean density

020406080

100120140160

Den

sity

(veh

km

)

1 5 8 12 15 18 21 24 28 31 35 38 41 44 47 50 54 57 60 63 68 71 75 78 81 84 88 92 95Time

0102030405060708090

Spee

d (k

mh

)

SpeedMean speed

2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70 74 78 82 86 90 94Time

0500

10001500200025003000350040004500

Flow rateMean ow rate

Tra

c vol

ume (

veh20

s)

Figure 3 Time-varying diagram of traffic volume speed and density

are tested that is noncontrol ALINEA DB-RM (density-based ramp metering model) and MDB-RM The ALINEAalgorithm implemented is the ALINEA module based onVisVAP VISSIM COM (Component Object Model) of API(Application Programming Interface) is used to implementthe MDB-RM and DB-RM control scenarios An optimaloccupancy threshold 03 is calibrated to achieve the optimalperformance of ALINEA

The field data presented in Section 3 is converted intotraffic counts as inputs for the simulator The traffic flowsof the mainstream and on-ramp are set as 2185 vehh and620 vehh respectively The expected densities of lane 1 and

lane 2 are separately set as 120588119888(119894 1) = 63 vehkm and 120588119888(119894 2) =69 vehkmMoreover define 120582(1) = 04 120582(2) = 045 120582ramp =015 and 119896 = 095We run the simulation with five random seeds and the

time length of each simulation investigation is 1 hour Theresults are evaluated by comparing the mean speed of themainstream mean travel time of the mainstream and on-ramp mean vehicle delay of the mainstream and on-rampand the mean queue length of the on-ramp Table 1 shows themain output data of simulation

To further analyze the key evaluation indexes the meantravel times of the mainstream and on-ramp are counted

8 Mathematical Problems in Engineering

0 20 40 60 80 100 120 140 160Density (vehkm)

0

10

20

30

40

50

60

70

80Sp

eed

(km

h)

Measured valueGreenshields

GreenburgUnderwood

(a) ldquoSpeed-densityrdquo scatter diagram

0

500

1000

1500

2000

2500

3000

3500

4000

0 20 40 60 80 100 120 140 160

Tra

c vol

ume (

veh)

Density (vehkm)

(b) ldquoTraffic volume-densityrdquo scatter diagram

Figure 4 Scatter diagram of traffic flow parameterM

ean

trav

el ti

me o

f on-

ram

p (s

)

250

200

150

100

50

0

NoncontrolALINEA

DB-RMMDB-RM

600 900 1200 1500 1800 2100 2400 2700 3000

Time (s)

80

70

60

50

40

30

20

10

0600 900 1200 1500 1800 2100 2400 2700 3000

Time (s)

Mea

n tr

avel

tim

e of t

he m

ains

tream

(s)

NoncontrolALINEA

DB-RMMDB-RM

Figure 5 Mean travel time of the mainstream and on-ramp

from the 600th to the 3000th second in every 60 seconds52 samples are obtained from the simulator for each controlscenario The variation trends are drawn in Figure 5

Moreover in order to prove the advantages of MDB-RMmodel in populationZ-test is conducted formean travel timeof the mainstream and on-ramp 119885-statistic yields

119885 = (1198831 minus 1198832) minus (1205831 minus 1205832)radic119878121198991 + 119878221198992 (23)

where119883119894 represents the mean value of sample 119894 120583119894 representsthe mean value of population 119894 (in null hypothesis 1205831 = 1205832)1198781198942 represents variance of sample 119894 Since the sample sizesof the above four simulation schemes are all equal to 52

mean travel time of vehicles in the mainstream and on-ramp can be approximately regarded as normal distributionCompare the output results of MDB-RM model with thosefrom noncontrol ALINEA and DB-RMmodel respectivelyAssume that using MDB-RM model can statistically reducemean travel time of vehicles both in the mainstream and inon-ramp In the 95 percent confidence interval the criticalvalue of Z-test is equal to 1645 The relevant parameters andresults of the 119885-test are shown in Table 2

As seen from Table 2 for mean travel time of on-rampthe 119885-statistics are all smaller than the critical 119885-value (ie1645) Thus the null hypothesis cannot be rejected whichindicates that mean travel time of on-ramp is significantlydecreased when using theMDB-RMmodel As for the resultsin the mainstream we can also conclude that the MDB-RMmodel has a significant advantage over other models except

Mathematical Problems in Engineering 9

Table 1 Main output data of simulation

Index Noncontrol ALINEA DB-RM MDB-RMMean output Change rate () Mean output Change rate () Mean output Change rate ()

Mean speed of themainstream(kmh)

306 335 948 329 752 331 817

Mean travel time ofthe mainstream (s) 605 567 minus628 573 minus529 579 minus430Mean travel time ofon-ramp (s) 1673 1732 353 1542 minus783 1525 minus885Mean vehicle delayof the mainstream(s)

247 225 minus891 231 minus648 229 minus729Mean vehicle delayof on-ramp (s) 854 956 1194 812 minus492 805 minus574Mean queue lengthof on-ramp (m) 1073 1175 951 935 minus1286 929 minus1342

Table 2 Relevant parameters and results of the 119885-testEvaluation index Parameter Noncontrol ALINEA DB-RM MDB-RM

Mean travel time ofthe mainstream

Mean (119883119894) 605 567 573 579Variance (1198781198942) 5292 3162 4392 2122Sample size 119899119894 52 52 52 52119885-statistic minus340 221 083 mdash

Mean travel time ofon-ramp

Mean (119883119894) 1673 1732 1542 1525Variance (1198781198942) 25562 19562 18642 10912Sample size 119899119894 52 52 52 52119885-statistic minus383 minus668 minus058 mdash

forALINEAThis is not a surprise since our primary objectiveof modeling is to improve the efficiency of on-ramp control

42 Discussion Generally speaking all these three controlmethods (ALINEA DB-RM and MDB-RM) are capable ofincreasing the mainstream speed and reducing the main-stream travel time Yet ALINEA best performs in terms ofincreasing the mainstream speed and saving time On theother hand compared with ALINEA and DB-RM the MDB-RM model displays significant improvements in the aspectsof mean travel time mean vehicle delay and mean queuelength of the on-ramp Also Figure 5 suggests that the curvesof mean travel time in both the mainstream and the on-rampare the smoothest when using the MDB-RM model This ismainly because the design principle of the MDB-RM modelis concerned with not only the on-ramp density but also thedifference of multilane density in the mainstream somethingwhich is absent in the DB-RM model Meanwhile the on-ramp queue length has not been taken into consideration inALINEA

The main objective of the MDB-RM model is to keepthe lane density around the expected value This explainswhy the MDB-RM model reaches a similar achievement

to the ALINEA and DB-RM In total MDB-RM DB-RMand ALINEA improve the mean speed of the mainstreamby 817 752 and 948 respectively Additionally themean travel time of the mainstream decreases by 52943 and 628 separately in each scenario The reason whyless improvement in the mainstream is observed when usingthe MDB-RM and DB-RM models is mainly caused by thesynchronous consideration of both the mainstream flow andthe on-rampqueueMoreover the gaps between their optimalresults are relatively small

However observing from the on-ramp side the improve-ments achieved by the MDB-RM model are noteworthyMean travel time of the on-ramp is reduced by 867Additionally there is a 729drop inmean vehicle delay in themainstream A 557 and a 1242 decrease of mean vehicledelay andmean queue length of the on-ramp respectively arealso obtained when using the MDB-RM model By contrastusing ALINEA will increase by 353 the on-ramp meantravel time along with an 1194 increase of mean vehicledelay and a 951 increase formean queue length of on-ramp

Overall the simulation results demonstrate that theMDB-RM model outperforms the DB-RM model in everyevaluation index and they also reach almost the same

10 Mathematical Problems in Engineering

effectiveness for improving the mainstream traffic conditionscompared with the most well-known algorithm ALINEAwhile behaving much better than ALINEA in the on-rampsection Thus we can conclude that the proposed model iscapable of achieving a comprehensive optimal result fromboth sides of the mainstream and on-ramp

5 Conclusion

In this paper an MDB-RM model aimed at realizing thesame control effect as ALINEA as well as improving on-ramptraffic condition is developed Dynamic density is chosenas the optimal index due to its convenient accessibility inpractical area The error function is introduced to adjust thedensity difference between lanes In this way lane changebehavior is included in the multilane dynamic density-basedmodel With the field traffic flow data collected in Chengdusimulation for the MDB-RM model is conducted on theplatformofVISSIMCOMandVisVAP alongwith three otherscenarios that is noncontrol ALINEA and DB-RM Thesimulation result shows that the MRD-RM model is capableof reducing the queue length of on-ramp while keeping thetraffic volume of the mainstream close to capacity

Furthermore the simulation results indicate that theMRD-RM model outperforms ALINEA with 11ndash22 in theevaluation indexes for the on-ramp while comprehensivelysurpassing the DB-RM model The simulation results alsoillustrate the potential of usingMRD-RMas an effective alter-native in addressing congestion caused by merging traffic

On the other hand when the traffic demands of boththe mainstream and the on-ramp grow largely especiallywhen capacity is exceeded there will be a tradeoff betweenreducing ramp queue andmaximizing the mainstream trafficflow How to balance these two sides and reach systemoptimization requires further discussion

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This study is supported by the Sichuan Provincial ScientificResearch Innovation Team Program (17TD0035) SichuanProvincial Science and Technology Program (2017ZR0032)Chengdu Science and Technology Project (2015-RK00-00227-ZF) 2015 Natural Science Key Foundation of XihuaUniversity (no Z1520315) and the Open Research Subject ofKey Laboratory of Vehicle Measurement Control and SafetyXihua University (szjj2016-014 szjj2015-044)

References

[1] M-H Ma Q-F Yang S-D Liang and Z-L Li ldquoIntegratedvariable speed limits control and ramp metering for bottleneckregions on freewayrdquoMathematical Problems in Engineering vol2015 Article ID 313089 17 pages 2015

[2] G N Bifulco G E Cantarella F Simonelli and P VelonaldquoAdvanced traveller information systems under recurrent traffic

conditions network equilibrium and stabilityrdquo TransportationResearch Part B Methodological vol 92 pp 73ndash87 2016

[3] M Papageorgiou and A Kotsialos ldquoFreeway ramp meteringan overviewrdquo IEEE Transactions on Intelligent TransportationSystems vol 3 no 4 pp 271ndash281 2002

[4] M Papageorgiou E Kosmatopoulos I Papamichail and YWang ldquoALINEAmaximises motorway throughputmdashan answerto flawed criticismrdquo Traffic Engineering and Control vol 48 no6 pp 271ndash276 2007

[5] J Fang ldquoPerformance tuning of coordinated active traffic con-trol algorithm simultaneously improving corridor safety andmobility performancesrdquoMathematical Problems in Engineeringvol 2014 Article ID 130804 11 pages 2014

[6] California Department of Transportation 2013 RampMeteringAnnual Report httpwwwdotcagovdist07resourcesrampmeteringdocs2013 RMAR Nov2014pdf

[7] H Hadj-Salem J M Blosseville and M PapageorgiouldquoALINEA A local feedback control law for on-rampmetering areal-life studyrdquo inProceedings of the 3rd International Conferenceon Road Traffic Control pp 194ndash198 London UK May 1990

[8] H Hadj-Salem M M Davee and J M Blosseville ldquoALINEAun outil de regulation drsquoacces isole sur autorouterdquo ReportINRETS 80 Institut de Recherche des Transports ArcueilFrance 1988

[9] S Smulders and F Middelham Isolated Ramp Metering Real-Life Study in the Netherlands Deliverable 7a Project CHRIS-TIANE (V1035) DRIVE Office Brussels Belgium 1991

[10] M Papageorgiou J-M Blosseville and H Haj-Salem ldquoMod-elling and real-time control of traffic flow on the southern partof boulevard peripherique in Paris part II coordinated on-ramp meteringrdquo Transportation Research Part A General vol24 no 5 pp 361ndash370 1990

[11] L Zhang andD Levinson ldquoOptimal freeway ramp controlwith-out origin-destination informationrdquo Transportation ResearchPart B Methodological vol 38 no 10 pp 869ndash887 2004

[12] L Lipp L Corcoran and G Hickman ldquoBenefits of centralcomputer control for the denver rampmetering systemrdquo Trans-portation Research Record vol 1320 pp 3ndash6 1990

[13] H Haj-Salem N Farhi and J P Lebacque ldquoField evaluationresults of new isolated and coordinated ramp metering strate-gies in Francerdquo IFAC Proceedings Volumes vol 45 no 6 pp378ndash383 2012

[14] L Faulkner F Dekker D Gyles L Papamichail and M Papa-georgiou ldquoEvaluation of HERO-Coordinated ramp meteringinstallation at M1 and M3 freeways in Queensland AustraliardquoTransportation Research Record vol 2470 pp 13ndash23 2014

[15] A Alessandri A Di Febbraro A Ferrara and E PuntaldquoOptimal control of freeways via speed signalling and rampmeteringrdquo Control Engineering Practice vol 6 no 6 pp 771ndash780 1998

[16] M Papageorgiou H Hadj-Salem and F Middelham ldquoALINEAlocal ramp metering summary of field resultsrdquo TransportationResearch Record no 1603 pp 90ndash98 1997

[17] P Masher D W Ross and P J Wong Guidelines for Design andOperation of RampControl Systems Stanford Research Institute1975

[18] H M Koble and V S Samant Control Strategies in Response toFreeway Incidents Federal Highway Administration Offices ofResearch amp Development Traffic Systems Division 1980

[19] X Yang L Chu andWRecker ldquoGA-based parameter optimiza-tion for the ALINEA ramp metering controlrdquo in Proceedings

Mathematical Problems in Engineering 11

of the 5th IEEE International Conference on Intelligent Trans-portation Systems (ITSC rsquo02) pp 627ndash632 IEEE SingaporeSeptember 2002

[20] L Chu and X Yang ldquoOptimization of the ALINEA ramp-metering control using genetic algorithm with micro-simulationrdquo in Proceedings of the Transportation ResearchBoard Annual Meeting vol 37 pp 2918ndash2924 2003

[21] E Smaragdis and M Papageorgiou ldquoSeries of new local rampmetering strategiesrdquo Transportation Research Record no 1856pp 74ndash86 2003

[22] E SmaragdisM Papageorgiou andEKosmatopoulos ldquoAflow-maximizing adaptive local rampmetering strategyrdquoTransporta-tion Research Part B Methodological vol 38 no 3 pp 251ndash2702004

[23] Y Wang and M Papageorgiou ldquoLocal ramp metering in thecase of distant downstream bottlenecksrdquo in Proceedings of theIEEE Intelligent Transportation Systems Conference (ITSC rsquo06)September 2006

[24] Y Wang M Papageorgiou J Gaffney I Papamichail and JGuo ldquoLocal rampmetering in the presence of random-locationbottlenecks downstream of a metered on-ramprdquo in Proceedingsof the 13th International IEEE Conference on Intelligent Trans-portation Systems (ITSC rsquo10) vol 2178 pp 1462ndash1467 MadeiraIsland Portugal September 2010

[25] R Chi Z Hou S Jin D Wang and J Hao ldquoA data-driveniterative feedback tuning approach of ALINEA for freewaytraffic ramp metering with PARAMICS simulationsrdquo IEEETransactions on Industrial Informatics vol 9 no 4 pp 2310ndash2317 2013

[26] S Jin Z Hou R Chi and J Hao ldquoA data-driven control designapproach for freeway traffic ramp metering with virtual refer-ence feedback tuningrdquo Mathematical Problems in Engineeringvol 2014 Article ID 936531 8 pages 2014

[27] W-B Gao J Wu and J Zou ldquoOn-ramp metering algorithmbased on real-time densityrdquo Journal of Transportation SystemsEngineering and Information Technology vol 12 no 2 pp 150ndash155 2012

[28] M J Lighthill and G B Whitham ldquoOn kinematic waves II Atheory of traffic flow on long crowded roadsrdquo Proceedings of theRoyal Society Mathematical Physical and Engineering Sciencesvol 229 no 1178 pp 317ndash345 1955

[29] M Papageorgiou B Posch and G Schmidt ldquoComparison ofmacroscopic models for control of freeway trafficrdquo Transporta-tion Research Part B vol 17 no 2 pp 107ndash116 1983

[30] M Papageorgiou J-M Blosseville and H Hadj-SalemldquoMacroscopic modelling of traffic flow on the BoulevardPeripherique in Parisrdquo Transportation Research Part B vol 23no 1 pp 29ndash47 1989

[31] A Messmer and M Papageorgiou ldquoMETANET a macroscopicsimulation program for motorway networksrdquo Traffic Engineer-ing amp Control vol 31 no 9 pp 466ndash470 1990

[32] A Kotsialos M Papageorgiou C Diakaki Y Pavlis and FMiddelham ldquoTraffic flow modeling of large-scale motorwaynetworks using the macroscopic modeling tool METANETrdquoIEEE Transactions on Intelligent Transportation Systems vol 3no 4 pp 282ndash292 2002

[33] Z-X Lin Z Wu C-H Yang and X-Q Zheng ldquoA studyon traffic flow models based on measure video of YananExpressway of Shanghairdquo Chinese Journal of Hydrodynamicsvol 25 no 5 pp 683ndash693 2010

[34] Z Wu M Guo X Zheng et al ldquoMeasured data analysis ofurban Expressway and research on traffic flowmodelsrdquo Chinese

Journal of Theoretical and Applied Mechanics vol 42 no 4 pp789ndash797 2010

[35] Z Wu H Zhu and N Jia ldquoMeasuring method study of thetraffic flow model parameters based on video recording ofexpressway trafficrdquo Journal of FudanUniversity (Nature Science)vol 47 no 2 pp 147ndash152 2008

[36] Z Lin ZWu andC Yang ldquoA study on traffic flowmodels basedon measure video of yanan expressway of Shanghairdquo ChineseJournal of Hydrodynamics vol 25 no 5 pp 683ndash693 2010

[37] ZWuMGuoX Zheng et al ldquoMeasured data analysis of urbanexpressway and research on traffic flowmodelsrdquoChinese JournalofTheoretical and AppliedMechanics vol 42 no 4 pp 789ndash7972010

[38] M J Cassidy and R L Bertini ldquoSome traffic features at freewaybottlenecksrdquo Transportation Research Part B Methodologicalvol 33 no 1 pp 25ndash42 1999

[39] R L Bertini and M T Leal ldquoEmpirical study of traffic featuresat a freeway lane droprdquo Journal of Transportation Engineeringvol 131 no 6 pp 397ndash407 2005

[40] X Sun H Lu and J Wu ldquoResearch on speed-flow-densityrelationship model of Beijing urban expresswayrdquo HighwayEngineering vol 37 no 1 pp 43ndash48 2012

[41] B S Kerner and H Rehborn ldquoExperimental properties ofcomplexity in traffic flowrdquo Physical Review E vol 53 no 5 pp4275ndash4278 1996

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of