Model Matematik Lalu Lintas
Model Matematik Lalu
Lintas
MODUL KE 6
PERTEMUAN KE 9
RENI KARNO KINASIH, S.T.,M.T.
UNIVERSITAS MERCU BUANA
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
yang membuat kemacetan pada sebuah ruas.
Dalam simulasi traffic light, untuk menganalisa arus
kendaraan pada sebuah ruas, terdapat beberapa
metode yang bisa diterapkan. Diantaranya metode
Greenshield,
teori
antrian
dan
metode
Computational Fluid Dynamic
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
mengatasi kerumitan yang dihasilkan oleh kondisi
arus yang memiliki hambatan.
Interrupted Flow
Arus dikatakan memiliki hambatan, jika arus lalu-
lintas terhenti secara periodik yang disebabkan oleh
rambu-rambu lalulintas.
Arus
yang
berhambatan
itu
memerlukan
pemahaman teori antrian, yang sepenuhnya
merupakan model terpisah dari model arus lalu
lintas
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
mengatasi permasalahan ini . Salah satunya adalah metode
Computational Fluid Dynamic. Computational Fluid
Dynamic (CFD) adalah metode yang digunakan untuk
menganalisis aliran fluida atau air
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
mengatur waktu sinyal, sesuai dengan kondisi lalu
lintas, sehingga akan menjamin arus lalu lintas
seragam bahkan ketika tingkat aliran tinggi
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
keseluruhan atau sistem dan bukan pada individu kendaraan.
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
digunakanuntuk dapat menganalisis keadaan arus
antrian pada ruas jalan. Sedangkan aturan MKJI
digunakan untuk mendapatkan nilai kapasitas jalan,
waktu nyala lampu lalu lintas dan derajat kejenuhan
Model – Model Makroskopis
Arus Lalu Lintas
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
relationship as illustrated in figure 1 to derive the
model.
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'
model. It indicates that when density becomes zero, speed
approaches free flow speed (ie. V vf when k 0).
Once the relation between speed and flow is
established, the relation with flow can be derived.
This relation between flow and density is parabolic in
shape and is shown in figure 3. Also, we know that
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
………………….. (4)
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
equation 3 with respect to and equate it to zero. ie.,
Denoting the density corresponding to maximum flow as k0,
Therefore, density corresponding to maximum flow is half the
jam density Once we get , we can derive for maximum
flow, qmax. Substituting equation 5 in equation 3
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
speed.
Calibration of Greenshield's model
Inorder to use this model for any traffic stream, one
should get the boundary values, especially free flow
speed (vf) and jam density (kj).
This has to be obtained by field survey and this is
called calibration process.
Although it is difficult to determine exact free flow
speed and jam density directly from the field,
approximate values can be obtained from a number
of speed and density observations and then fitting a
linear equation between them.
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)
(8)
Alternate method of solving for b is,
……(9)
where xi and yi are the samples, n is the number of
samples,
and are the mean of xi
and yi respectively.
Problem
For the following data on speed and density,
determine the parameters of the Greenshields'
model. Also find the maximum flow and density
corresponding to a speed of 30 km/hr.
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.
Step 1
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
flow speed and jam density and they are obtained as
….. kmph and ……… veh/km respectively.
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
Therefore, k = ………. veh/km
Other macroscopic
stream models
GREENBERG’S AND UNDERWOOD’S
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
questioned and many other models came up.
Prominent among them are Greenberg's
logarithmic model, Underwood's exponential
model, Pipe's generalized model, and multiregime
models. These are briefly discussed below.
Greenberg's logarithmic model
(1959)
Greenberg assumed a logarithmic relation between
speed and density. He proposed,
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
density tends to zero, speed tends to infinity. This
shows the inability of the model to predict the speeds
at lower densities.
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
in figure 5 is the free flow speed and k0 is the
optimum density, i.e. the densty corresponding to
the maximum flow.
In this model, speed becomes zero only when density
reaches infinity which is the drawback of this model.
Hence this cannot be used for predicting speeds at
high densities.
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
Greenshields' model. Thus by varying the values of , a
family of models can be developed.
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
called multi-regime models. The most simple one is called a tworegime model, where separate equations are used to represent
the speed-density relation at congested and uncongested traffic.
Resources
https://www.civil.iitb.ac.in/tvm/1100_LnTse/
503_lnTse/plain/plain.html
Khisty, J & Lall, K. 2003. Dasar-Dasar Rekayasa
Transportasi, 3rd Edition. Prentice Hall.
Thank You
SEE YOU IN THE NEXT CHAPTER!
Lintas
MODUL KE 6
PERTEMUAN KE 9
RENI KARNO KINASIH, S.T.,M.T.
UNIVERSITAS MERCU BUANA
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
yang membuat kemacetan pada sebuah ruas.
Dalam simulasi traffic light, untuk menganalisa arus
kendaraan pada sebuah ruas, terdapat beberapa
metode yang bisa diterapkan. Diantaranya metode
Greenshield,
teori
antrian
dan
metode
Computational Fluid Dynamic
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
mengatasi kerumitan yang dihasilkan oleh kondisi
arus yang memiliki hambatan.
Interrupted Flow
Arus dikatakan memiliki hambatan, jika arus lalu-
lintas terhenti secara periodik yang disebabkan oleh
rambu-rambu lalulintas.
Arus
yang
berhambatan
itu
memerlukan
pemahaman teori antrian, yang sepenuhnya
merupakan model terpisah dari model arus lalu
lintas
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
mengatasi permasalahan ini . Salah satunya adalah metode
Computational Fluid Dynamic. Computational Fluid
Dynamic (CFD) adalah metode yang digunakan untuk
menganalisis aliran fluida atau air
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
mengatur waktu sinyal, sesuai dengan kondisi lalu
lintas, sehingga akan menjamin arus lalu lintas
seragam bahkan ketika tingkat aliran tinggi
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
keseluruhan atau sistem dan bukan pada individu kendaraan.
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
digunakanuntuk dapat menganalisis keadaan arus
antrian pada ruas jalan. Sedangkan aturan MKJI
digunakan untuk mendapatkan nilai kapasitas jalan,
waktu nyala lampu lalu lintas dan derajat kejenuhan
Model – Model Makroskopis
Arus Lalu Lintas
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
relationship as illustrated in figure 1 to derive the
model.
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'
model. It indicates that when density becomes zero, speed
approaches free flow speed (ie. V vf when k 0).
Once the relation between speed and flow is
established, the relation with flow can be derived.
This relation between flow and density is parabolic in
shape and is shown in figure 3. Also, we know that
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
………………….. (4)
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
equation 3 with respect to and equate it to zero. ie.,
Denoting the density corresponding to maximum flow as k0,
Therefore, density corresponding to maximum flow is half the
jam density Once we get , we can derive for maximum
flow, qmax. Substituting equation 5 in equation 3
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
speed.
Calibration of Greenshield's model
Inorder to use this model for any traffic stream, one
should get the boundary values, especially free flow
speed (vf) and jam density (kj).
This has to be obtained by field survey and this is
called calibration process.
Although it is difficult to determine exact free flow
speed and jam density directly from the field,
approximate values can be obtained from a number
of speed and density observations and then fitting a
linear equation between them.
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)
(8)
Alternate method of solving for b is,
……(9)
where xi and yi are the samples, n is the number of
samples,
and are the mean of xi
and yi respectively.
Problem
For the following data on speed and density,
determine the parameters of the Greenshields'
model. Also find the maximum flow and density
corresponding to a speed of 30 km/hr.
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.
Step 1
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
flow speed and jam density and they are obtained as
….. kmph and ……… veh/km respectively.
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
Therefore, k = ………. veh/km
Other macroscopic
stream models
GREENBERG’S AND UNDERWOOD’S
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
questioned and many other models came up.
Prominent among them are Greenberg's
logarithmic model, Underwood's exponential
model, Pipe's generalized model, and multiregime
models. These are briefly discussed below.
Greenberg's logarithmic model
(1959)
Greenberg assumed a logarithmic relation between
speed and density. He proposed,
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
density tends to zero, speed tends to infinity. This
shows the inability of the model to predict the speeds
at lower densities.
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
in figure 5 is the free flow speed and k0 is the
optimum density, i.e. the densty corresponding to
the maximum flow.
In this model, speed becomes zero only when density
reaches infinity which is the drawback of this model.
Hence this cannot be used for predicting speeds at
high densities.
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
Greenshields' model. Thus by varying the values of , a
family of models can be developed.
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
called multi-regime models. The most simple one is called a tworegime model, where separate equations are used to represent
the speed-density relation at congested and uncongested traffic.
Resources
https://www.civil.iitb.ac.in/tvm/1100_LnTse/
503_lnTse/plain/plain.html
Khisty, J & Lall, K. 2003. Dasar-Dasar Rekayasa
Transportasi, 3rd Edition. Prentice Hall.
Thank You
SEE YOU IN THE NEXT CHAPTER!