Model Matematik Lalu Lintas

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M O D U L K E 6 P E R T E M U A N K E 9

R E N I K A R N O K I N A S I H , S . T . , M . T . U N I V E R S I T A S M E R C U B U A N A

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 _{Penggunaan sistem kontrol traffic light pada lalu lintas}

belum memberikan prioritas berupa nyala lampu hijau lebih lama pada jalur-jalur yang lebih padat penggunanya.

 _{Hal tersebut dapat menyebabkan antrian panjang pada}

sebuah ruas.

 _{Sehingga bila dilihat secara gambaran besar, sistem}

kontrol traffic light yang ada ternyata belum maksimal.

 Belum adanya informasi yang bisa kita andalkan untuk

memberikan prioritas atau mengambil keputusan.

 Bahkan sering kali justru sistem kontrol traffic light lah

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 Dalam simulasi traffic light, untuk menganalisa arus

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Model Greenshield

 Berguna untuk membantu peneliti dibidang

transportasi dalam memahami arus tanpa hambatan.

 Model ini memberikan rumusan matematika arus

kendaraan sebagai fungsi dari kepadatan lalu-lintas, serta arus kendaraan sebagai fungsi dari kecepatan kendaraan.

 _{Akan tetapi, model greenshields tidak dapat}

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Interrupted Flow

 Arus dikatakan memiliki hambatan, jika arus

lalu-lintas terhenti secara periodik yang disebabkan oleh rambu-rambu lalulintas.

 Arus yang berhambatan itu memerlukan

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Teori Antrian

 _{Teori antrian dapat digunakan untuk menganalisis arus}

lalu-lintas melalui pendekatan sebuah persimpangan jalan yang dikontrol oleh rambu lalu-lintas.

 Namun teori ini harus memiliki asumsi bahwa arus

kendaraan harus dalam keadaan rapi dan tidak fleksibel.

 _{Sehingga diperlukan metode yang dapat digunakan untuk}

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Computational Fluid Dynamic (CFD)

 Adalah metode yang digunakan untuk menganalisis

aliran fluida atau air, namun seiring perkembangan zaman, metode ini mulai diterapkan di bidang engineering, salah satunya adalah transportasi

 _{CFD juga dapat memainkan peran penting dalam}

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How CFD Works?

 _{CFD menganalisa aliran fluida dengan cara pemodelan}

matematika (persamaan diferensial parsial), metode numerik (diskritisasi dan solusi teknik) dan perangkat lunak (pemecah, pra-dan utilitas postprocessing)

 _{CFD menggunakan sudut pandang Eulerian, yakni bukan}

memandang kendaraan secara individual dalam aliran, tetapi memandang arus lalu lintas sebagai aliran sederhana yang didistribusikan secara terus menerus, dengan melihat kesenjangan atau selang yang konsisten antara jumlah mobil dengan panjang jalan yang dikenal sebagai density.

 _{Penekanan metode CFD adalah pada aliran secara}

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 CFD memungkinkan untuk melakukan eksperimen

berupa perhitungan numerik dan simulasi komputer. Sedangkan metode yang digunakan untuk menghitung waktu nyala lampu lalu lintas adalahaturan Manual Kapasitas Jalan Indonesia (MKJI)

 Computational Fluid Dynamic merupakan metode yang

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Greenshiled’s Linear Model (1935)

 Macroscopic stream models represent how the

behaviour of one parameter of traffic flow changes with respect to another.

 Most important among them is the relation between

speed and density. The first and most simple relation between them is proposed by Greenshield.

 Greenshield assumed a linear speed-density

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 _{The equation for this relationship is shown below.}

………. (1)

 _{Where v is the mean speed at density , vf is the free speed}

and kj is the jam density. T

 _{This equation (1) is often referred to as the Greenshields'}

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 Once the relation between speed and flow is

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 Now substituting equation 1 in equation 2, we get

……….. (3)

 Similarly we can find the relation between speed and

flow. For this, put in equation 1 and solving, we get

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 This relationship is again parabolic and is shown in

figure 2.

 _{Once the relationship between the fundamental}

variables of traffic flow is established, the boundary conditions can be derived.

 The boundary conditions that are of interest are jam

density, freeflow speed, and maximum flow.

 _{To find density at maximum flow, differentiate}

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 Denoting the density corresponding to maximum flow as k₀,

 Therefore, density corresponding to maximum flow is half the

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 Thus the maximum flow is one fourth the product of

free flow and jam density. Finally to get the speed at

maximum flow, v0, substitute equation 5 in equation 1 and solving we get,

……… (6)

 Therefore, speed at maximum flow is half of the free

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Calibration of Greenshield's model

 Inorder to use this model for any traffic stream, one

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 Let the linear equation be y = ax = b such that y is

density, k and x denotes the speed v. Using linear regression method, coefficients a and b can be solved as,

 (7)

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 Alternate method of solving for b is,

……(9)

 where xi and yi are the samples, n is the number of

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Problem

 For the following data on speed and density,

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Solution

 Denoting y = v and x = k, solve for a and b using

equation 8 and equation 9. The solution is tabulated as shown below.

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 Step 2: From equation 9, define b = ….. And a =……  Step 3: Define the linear regression from Step 2

above

v =……. (10)

 Here vf = …….. and vf/kj= …….. This implies, =

……/…….. = …….. veh/km

 The basic parameters of Greenshield's model are free

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 To find maximum flow, use equation 6

q max = …… veh/hr

 _{Density corresponding to the speed 30 km/hr can be}

found out by substituting in equation 10. i.e, 30 = 40.8 - 0.2 k

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G R E E N B E R G ’ S A N D U N D E R W O O D ’ S

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

In Greenshield's model, linear relationship

between speed and density was assumed. But in

field we can hardly find such a relationship

between speed and density.



Therefore, the validity of Greenshields' model was

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Greenberg's logarithmic model

(1959)

 Greenberg assumed a logarithmic relation between

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 This model has gained very good popularity because

this model can be derived analytically. (This derivation is beyond the scope of this notes).

 However, main drawbacks of this model is that as

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Underwood's exponential model

 Trying to overcome the limitation of Greenberg's

model, Underwood put forward an exponential model as shown below.

……… (12)

 Where vf The model can be graphically expressed as

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 _{In this model, speed becomes zero only when density}

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Pipes' generalized model

 Further developments were made with the

introduction of a new parameter (n) to provide for a more generalised modelling approach. Pipes proposed a model shown by the following equation.

………(13)

 When is set to one, Pipe's model resembles

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Multiregime models

 _{All the above models are based on the assumption that the same}

speed-density relation is valid for the entire range of densities seen in traffic streams.

 _{Therefore, these models are called single-regime models.}

However, human behaviour will be different at different densities. This is corraborated with field observations which shows different relations at different range of densities.

 _{Therefore, the speed-density relation will also be different in}

different zones of densities.

 _{Based on this concept, many models were proposed generally}

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Resources

 https://www.civil.iitb.ac.in/tvm/1100_LnTse/

503_lnTse/plain/plain.html

 _{Khisty, J & Lall, K. 2003.}_{Dasar-Dasar Rekayasa}

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S E E Y O U I N T H E N E X T C H A P T E R !