Nonlinear gating control for urban road traffic network using the network fundamental diagram

Abstract

This work proposes a nonlinear model predictive controller for the urban gating problem. The system model is formalized based on a research on existing models of the network fundamental diagram and the perimeter control systems. For the existing models, modifications are suggested: additional state variables are allocated to describe the queue dynamics at the network gates. Using the extended model, a nonlinear model predictive controller is designed offering a ‘non-greedy’ policy compared with previous, ‘greedy’ gating control designs. The greedy and non-greedy nonlinear model predictive control (NMPC) controllers are compared with a greedy linear feedback proportional-integral-derivative (PID) controller in different traffic situations. The proposed non-greedy NMPC controller outperforms the other two approaches in terms of travel distance performance and queue lengths. The performance results justify the consideration of queue lengths in dynamic modeling, and the use of NMPC approach for controller design. Copyright © 2014 John Wiley & Sons, Ltd.

Full text

Nonlinear gating control for urban road traffic network using the
network fundamental diagram
Alfréd Csikós
1
, Tamás Tettamanti
2
* and István Varga
2
1
Systems and Control Laboratory, Institute for Computer Sciences and Control, Hungarian Academy of Sciences, Kende
utca 13-17, 1111, Budapest, Hungary
2
Department of Control for Transportation and Vehicle Systems, Budapest University of Technology and Economics,
Stoczek utca 2., 1111, Budapest, Hungary
SUMMARY
This work proposes a nonlinear model predictive controller for the urban gating problem. The system model is
formalized based on a research on existing models of the network fundamental diagram and the perimeter con-
trol systems. For the existing models, modifications are suggested: additional state variables are allocated to
describe the queue dynamics at the network gates. Using the extended model, a nonlinear model predictive
controller is designed offering a ‘non-greedy’ policy compared with previous, ‘greedy’ gating control designs.
The greedy and non-greedy nonlinear model predictive control (NMPC) controllers are compared with a
greedy linear feedback proportional-integral-derivative (PID) controller in different traffic situations. The pro-
posed non-greedy NMPC controller outperforms the other two approaches in terms of travel distance perfor-
mance and queue lengths. The performance results justify the consideration of queue lengths in dynamic
modeling, and the use of NMPC approach for controller design. Copyright © 2014 John Wiley & Sons, Ltd.
KEY WORDS: urban traffic network; perimeter control; network fundamental diagram; nonlinear MPC
1. INTRODUCTION
Traffic-responsive control strategies have been intensively investigated for decades in the field of road
traffic engineering. This intention is particularly s igni ficant for dense urban traffic networks, because
dynamic measures may provide real-time solutions to everyday traffic congestion problems. Accord-
ingly, a plethora of new control methodologies for urban traffic have been introduced up to the present
time. A significant part of these methods practically apply link-level traffic states during operation, that
is, usually the link queues are intended to be minimized or balanced in order to maximize the capacity
of the signalized intersections within the network. Some interesting approaches of this concept are, for
example, Tettamanti et al. [1], Aboudolas et al. [2], de Oliveira and Camponogara [3], Lin et al. [4],
and Tettamanti et al. [5]. However, traffic demands of our days often exceed infrastructure capacities
in metropolitan areas leading to oversaturated traffic states. The control methods mentioned earlier
perform very efficiently as traffic-responsive strategies. At the same time, they are not able to deal with
extreme traffic conditions when demands extremely overpass the network capacity for a long period of
time. Therefore, the concept of the protected network (PN) has been highlighted recently as an efficient
solution to prevent traffic jams in certain networks. PN usually represents a city center or a dense urban
area that needs protection against insatiable demands during rush hours. The optimal traffic load of a
PN can be maintained by several techniques, for example, regulation, congestion charge, or gating
control. Normally, a mixture of these measures is needed to provide an effective and adaptive solution
of the problem. In this work, the gating control approach is analyzed.
*Correspondence to: Tamás Tettamanti, Department of Control for Transportation and Vehicle Systems, Budapest
University of Technology and Economics, Stoczek utca 2., 1111, Budapest, Hungary. E-mail: tettamanti@mail.bme.hu
Copyright © 2014 John Wiley & Sons, Ltd.
JOURNAL OF ADVANCED TRANSPORTATION
J. Adv. Transp. 2014
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/atr.1291

The traffic control of a PN is often related to the theory of the urban fundamental diagram, which
was first proposed by Godfrey [6]. The theory is called macroscopic fundamental diagram or network
fundamental diagram (NFD). This concept has been widely investigated during the past decades, for
example, Mahmassani et al. [7], Daganzo and Geroliminis [8], Helbing [9], Mazloumian et al. [10],
Geroliminis et al. [11]. Nevertheless, the applicable control methodologies have also appeared by
using the traffic lights along the boundary of the PN as controllable gates. Daganzo introduced a
control rule based on time dependent switching conditions [12]. In the paper of Aboudolas et al. [13],
the NFD modeling is integrated to a flow-transmission control for a set of subnetworks using an LQ
optimal controller. Hajiahmadi et al. [14] formulate the optimal control problem as a mixed integer lin-
ear opti mization problem, with two types of controllers: perimeter controllers and a switching controller
of fix-time signal plans. The paper of Hajiahmadi et al. [15] divides urban networks to homogeneous
regions, each modeled by macroscopic fundamental diagrams, and the problem of route guidance is
solved in a regional fashion by using a high-level NFD-based model used for prediction of traffic states
in the urban network. Geroliminis et al. [16] and Haddad et al. [17] are the most recent results investi-
gating the NFD-based perimeter control strategy with the use of model predictive control. The work of
de Jong et al. [18] analyzes the effect of different signal strategies within the PN on the shape of the
NFD, and proposes a control system, separating the control along the links at the boundary of the
PN, and inside the PN. Keyvan-Ekbatani et al. [19] provide a thorough description of the NFD model,
considering the actuation dynamics. A linear feedback regulator (PID control) is designed for the gating
problem using the suggested model dynamics. An interesting approach of perimeter control design is
presented in Keyvan-Ekbatani et al. [20], precluding the spreading of congestion though subnetworks.
Most recent research results cope with the heterogeneity of dynamics in NFD modeling; see, for
example, Ramezani et al. [21].
The existing works using th e NFD concept can be classified into two major classes. In the perimeter
control problem, several urban subnetworks are connected, and the control task is to optimize the
traffic flows among the subnetworks. The gating problem focuses on optimizing the traffic performance
of one protected network only. However, only the perimeter control considers the outside of a protected
network through other protected networks–gating control provides a greedy policy, improving the per-
formance of the protected network and neglecting the outside traffic conditions.
In our work, the dynamic modeling and control of the gating problem is developed by modeling the
dynamics of the vehicle queues at the controlled gates. On the one hand, by taking gate queues into
account, a less greedy control policy can be designed by considering the effects of gating on the
exterior of the PN as well. On the other hand, better performance can be achieved within the PN itself,
because extreme queue lengths may decrease network performance through the impeding or blocking
of the exits. Our approach also reflects on the nonlinear manner of the process by designing a nonlinear
model predictive control which is compared to the PID control approach, suggested by Keyvan-
Ekbatani et al. [19].
The paper is organized as follows. First, the mathematical model of the system is stated in Section 2.
Then, the control design is reviewed in Section 3. The description of the case study network and the
identification of network parameters are outlined in Section 4. Simulation results are presented in
Section 5. Finally, conclusions are drawn.
2. MODEL
2.1. Model equations
In this section, the modeling assumptions for the urban perimeter control system are formalized in
analytic equations. For the model, basically the same assumptions are taken as in by Keyvan-Ekbatani
et al. [22] as it is the most thorough model in the literature. However, certain differences are suggested.
As the control system is realized in a discrete time form, the model equations are directly stated in
discrete time.
The basic and most important rule that has to be satisfied by a traffic network is the conservation
law. For the protected network PN, the conservation of vehicles can be formalized as follows:
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

N
PN
k þ 1ðÞ¼N
PN
kðÞþT
s
Q
in
kðÞþQ
d
kðÞQ
out
kðÞ½(1)
where k denotes the discrete time step index and T
s
in unit (h) is the discrete sample time step, in our
case, the signal cycle time. Practically, Equation (1) depicts the state variation during the time interval
[kT
s
,(k +1)T
s
]. The state is represented by N
PN
(k), the number of vehicles within the protected net-
work, given in passenger car equivalent (PCE) (the different types of road vehicles can be expressed
in the ratio of private car [23]). Q
in
kðÞ¼∑
n
in
j¼1
q
in;j
kðÞin unit (PCE/h) is the sum of the vehicle inflows
to the protected network, whereas n
in
denotes the number of controlled gates. Q
d
k
ðÞ
¼ ∑
n
d
j¼1
q
d;j
k
ðÞ
is
the sum of uncontrolled inflow in unit (PCE/h) with n
d
denoting the number of uncontrolled gates.
Q
out
kðÞ¼∑
n
out
j¼1
q
out;j
kðÞis the sum of outflow of vehicles from the protected network in unit (PCE/h),
with n
out
denoting the number of exit gates. Equation (1) does not contain any term concerning the inner
demand, that is, no sinks or sources are considered within the network.
The second basic law that is used for the model is the concept of urban NFD ([8]). In our model, the
NFD of the protected network describes the relationship between the total travel distance (TTD
PN
,in
units [PCEkm]) and the total time spent (TTS
PN
, in units [PCEh]) within the protected network
(during a discrete step). These traffic variables can be obtained by following the concept of Keyvan-
Ekbatani et al. [22], reformalized using Ashton [24]:
TTD
PN
kðÞ¼T
s
∑
n
link
j¼1
q
j
kðÞL
j
(2)
TTS
PN
kðÞ¼T
s
∑
n
link
j¼1
N
j
kðÞ (3)
n
link
denotes the number of links in the protected network. q
j
(k) is the traffic flow (PCE/h). N
j
(k) sums
the number of vehicles on link j. L
j
denotes the length of link j (in unit [km]).
It is important to notice that the exact knowledge of the complete operational fundamental diagram
assumes the measurements of traffic variables q
j
and ρ
j
(j =1,… n
link
) in unit (PCE/km), by loop detec-
tors. The measurement of the former variable is straightforward. Papageorgiou and Vigos [25] give a
formula for the estimation of ρ
j
. Accordingly, in our work, full information control is assumed, that is,
all traffic variables are available through measurements, and no measurement error is considered. Thus,
the number of vehicles in the protected network can be calculated as follows:
N
PN
kðÞ¼∑
n
link
j¼1
ρ
j
kðÞL
j
(4)
where ρ
j
(k) denotes the traffic density on link j. In the practice, N
PN
can be measured by appro priate
estimation, for example, Papageorgiou and Vigos [25], Vilgos et al. [26], Kulcsár et al.[27]. Using
(4), the knowledge of the complete operational fundamental diagram can be supposed.
The fundamental relationship can be stated as follows:
TTD
PN
kðÞ¼F TTD
PN
kðÞðÞþε kðÞ (5)
where F() denotes the nonlinear function of the complete operational NFD, fitted to historic measure-
ments. ε(k) denotes the fitting error, considered as noise in the system dynam ics.
The network model also assumes that the total outflow Q
out
(k) of the protected network is propor-
tional to TTD
PN
(k), satisfying the following equation:
Q
out
kðÞ¼Γ
TTD
PN
kðÞ
LT
s
(6)
where 0 ≤ Γ ≤ 1 is the network exit rate parameter, L is the average link length in the protected network.
Coefficient Γ can be fitted using the measurements of the to tal outflow of the network and TTD
PN
.Itis
important to notice, that for the modeling of network outflow, different modeling approaches can be
followed. For example, in Geroliminis [28] L represents the average trip length.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

The actuator dynamics of traffic lights at controlled gate j is described by Equation (7)
q
in;j
kðÞ¼β
j
q
g;j
k  τðÞ (7)
where β
j
is the portion of gated flow (q
g,j
) that enters the PN, (0 ≤ β
j
≤ 1) and τ is a time delay, caused
by the travel time needed for gated vehicles to approach the PN. In our work, this aspect is simplified
by assuming that the gating link is at the boundary of the protected network, and thus τ = 0 can be
substituted. Thus, the vehicle inflow to the PN:
q
in;j
kðÞ¼β
j
q
g;j
kðÞ (8)
Parameter β
j
can also be approximated by fitting to the measurements of q
in,j
and q
g,j
. Following the
notions of control system theory, q
in,j
is considered the intended control signal, which is ideally real-
ized and q
g,j
is considered the actuator signal, which is actually realized. As experienced during the test
runs, the realized inflows are different from the intended inflows. Thus, for the sake of completeness,
the actuator dynamics also have to be modeled.
In addition to the model outlined by Keyvan-Ekb atani et al. [22], the dynamics of queuing can be
formalized, too:
l
j
k þ 1ðÞ¼l
j
kðÞþT
s
q
dem;j
kðÞq
in;j
kðÞ

(9)
where l
j
in unit (PCE) denotes the queue length at gate j where trafficdemandq
dem,j
emerges (representing
nominal traffic flow intending to enter the network).
To sum up the differences between the proposed model and the one used by Keyvan-Ekbatani et al. [22],
in our work, a full information control system is supposed, that is, the complete operational fundamental
diagram is supposed to be known for modeling simplifications. Thus, following the notations of the
aforementioned work, for the operational fundamental diagram correction factors A = 1 and B =1 as
well as for the estimation error ε
1
= 0 are supposed. Apart from this, by placing the gates at the boundary
of the protected network, the delay of the actuator system is eliminated and τ = 0 is supposed.
The model, however, is extended by modeling the queues at the gates to optimize inflow allocation.
Also, a multiplicative uncertainty is applied to describe disturbances from uncontrolled gates.
2.2. State-space model
The dynamic model proposed in the previous section can be recasted into a nonlinear state-space sys-
tem framework as follows:
N
PN
k þ 1ðÞ
l
1
k þ 1ðÞ
…
l
n
in
k þ 1ðÞ
0
B
B
B
B
B
@
1
C
C
C
C
C
A
¼
N
PN
kðÞT
s
Γ
L
FN
PN
kðÞðÞ
l
1
kðÞ
…
l
n
in
kðÞ
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
þ
T
s
∑
n
in
j¼1
q
in;j
kðÞ
T
s
q
in;1
kðÞ
…
T
s
q
in;n
in
k
ðÞ
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
þ
T
s
∑
n
d
j¼1
q
d;j
kðÞ
T
s
q
dem;1
kðÞ
…
T
s
q
dem;n
in
k
ðÞ
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
(10)
that is, Equations (1) and (9) are tabulated into vector form. The first row of the vectors is deducted
from Equation (1) by using Equations (4) –(6). Moreover, from the second row to the last, the queue
dynamics is expressed based on Equation (9).
q
d;j
(j =1,…, n
d
) denotes the nominal traffic flow
reaching the network through uncontrolled gate j and q
dem,i
(i =1,…, n
in
) marks the traffic demand
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

emerging at controlled gate i. On the right hand side of the Equation (10), the second and third terms
include the effect of control inputs and disturbances, respectively.
2.3. System variables
As state variables, the vehicle number in the protected network N
PN
and queue lengths of the controlled
gates l
1
…l
n
in
are considered:
xkðÞ¼
N
PN
kðÞ
l
1
k
ðÞ
…
l
n
in
kðÞ
0
B
B
B
@
1
C
C
C
A
∈ℝ
n
in
þ1
(11)
The disturbances are collected in the following vector:
dkðÞ¼
q
d;1
kðÞ
…
q
d;n
d
k
ðÞ
q
dem;1
kðÞ
…
q
dem;n
in
kðÞ
0
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
A
∈ℝ
n
d
þn
in
(12)
with nominal traffic flow
q
d;j
at the uncontrolled gates and traffic demand q
dem;n
i
at the controlled gates.
The input vector is in the form
ukðÞ¼
q
in;1
kðÞ
…
q
in;n
in
kðÞ
0
B
@
1
C
A
∈ℝ
n
in
(13)
where q
in,j
, j =1,…, n
in
denotes the gated flow through gate j.
3. CONTROL DESIGN
In this section, the design of a control algorithm is outlined for the urban perimeter system. For the gate
signal control, the optimal input is obtained using the nonlinear model predictive control (NMPC)
method ([29]). The NMPC design is realized in two ways: the NMPC controller considers both the
inside and the outside of the PN when optimizing the control signals, whereas the greedy NMPC only
optimizes the traffic performance within the PN. The controllers are compared with the PID control,
suggested by Keyvan-Ekbatani et al. [19]. The details of the PID control design are given in the Appendix.
3.1. Control method: nonlinear model predictive control
Model predictive control (MPC) is a control method that calculates optimal input of a dynamic system
throughout a certain control horizon K, using predictions on future system dynamics (in our case, using
(10)) and future disturbances. Based on the predicted state, disturbance, and input values, predefined
objective functions are calculated, and optimized over the prediction horizon subject to constraints
on state and control variables. The optimal control input is obtained from the optimal input sequence
that minimizes the objective function. MPC is used in a rolling horizon manner, that is, optimization is
carried out and optimal control sequence is obtained in each sample step k, for a horizon of size K, but
only the first element of the control sequence is applied on the syst em, and the optimization is repeated
for the same horizon length K in each step. Calculation of the optimal input is possible analytically
(turning cost functions to linear programming and eventually recasting the problem as a state-feedback
control problem) or using online optimization tools (e.g. fmincon in the
MATLAB environment). A
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

further important aspect of MPC is that the nonlinear nature of the controlled system can be handled.
Accordingly, in this work, the NMPC technique of Grune and Pannek [29] is applied.
Due to the major advantages paraphrased text mentioned earlier, MPC has been widely investigated
for road traffic engineering as well, for example, Tettamanti et al. [1], Ab oudolas et al. [2], Tettamanti
et al. [30], Tettamanti et al. [5], Csikós et al. [31], Ramezani et al. [32] are a few relevant papers in the
field of MPC based traffic control.
3.2. Objective function of the optimal control
The controller aims to minimize cost function (14) over control horizon K:
JkðÞ¼∑
K
ℓ¼1
∥λ k þ ℓðÞ∥
2
2
þ w
1
∥μ k þ ℓðÞ∥
2
2
þ w
2
∥ν k þ ℓðÞ∥
2
2
→min (14)
where w
1
and w
2
are weighting parameters of the co ntrol aspects. For the optimal choice of con trol horizon
K, see the Appendix. The cost function contains performance criteria, and input oscillation weighting:
λ kðÞ¼N
PN
kðÞN
PN;crit
(15)
μ kðÞ¼l
1
kðÞ…l
n
in
kðÞ½ (16)
ν kðÞ¼ q
in;1
kðÞ…q
in;n
in
kðÞ

 q
in;1
k  1ðÞ…q
in;n
in
k  1ðÞ

(17)
During the design of the control input, the primary control aim is the optimization of traffic perfor-
mance, thus the maximization of throughput (the total traveled distances) of the network. This goal is
fulfilled if the number of vehicles in the protected network is near its critical value:
N
PN;crit
¼ argmax F TTS
PN
ðÞ (18)
Thus, where the maximal TTD
PN
(the capacity of the network) can be obtained. At the same time, the
queues at the controlled gates should be minimized, too. Formalizing these goals, a regulator control
problem is described, with N
PN,crit
as set point for the number of vehicles through Equation (15), and
zero for queue lengths by applying Equation (16). The controller also needs to minimize the oscillation
in input values. Therefore, it is additionally involved in Equation (14) through Equation (17).
3.3. Constraints
The following constraints were set for the system variables.
Disturbances
Exploiting that the horizon length is chosen as K = 3, constant disturbance values can be considered
throughout the control horizon. Thus, for each sample step k:
q
d;j
k þ ℓðÞ¼q
d;j
kðÞ (19)
q
dem;l
k þ ℓ
ðÞ
¼ q
dem;l
k
ðÞ
(20)
for ∀ j ∈ {1, …, n
d
}, for ∀ l ∈ {1, …, n
in
}, and for ∀ ℓ ∈ {1, …, K} control horizon step.
Inputs
The optimal input is expressed in vehicle flows through the gates. However, the control is realized
via traffic lights, and concurring traffic demands need to be satisfied in a traffic light cycle as well.
Thus, the maximal admissible control input is defined using the maximal green light value for the
direction z → y, that is feeding the PN through the traffic light:
q
max
in;j
¼ g
max;zy
S
cap
(21)
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

where g
max,zy
denotes the maximal admissible green length for direction z → y, if this is the direction
entering the network at gate j. S
cap
denotes the maximal flow capacity through traffi c intersections
(practically the saturation flow). In our case, S
cap
= 0.6.
States
The link queues of the controlled gates have to be kept under the capacity of the links:
0 ≤ l
j
k þ ℓðÞ≤ l
max
j
(22)
for ∀ j ∈ {1, …, n
in
}, and for ∀ ℓ ∈ {1, …, K} control horizon step.Also, a theoretic maximum of N
PN
can be stated as well:
N
PN
k þ ℓ
ðÞ
≤ N
max
(23)
for ∀ j ∈ {1, …, n
in
}, and for ∀ ℓ ∈ {1, …, K} control horizon step. N
max
can be stated based on obser-
vations. In our case, N
max
= 600 PCE.
4. CASE STUDY
4.1. Network description
The proposed model is applied to a test network created in Vissim simulator. The model network is a
part of the sixth district located in downtown Budapest. Here, all streets are one-way streets, and the
average link length is 0.143 km, thus the concept of the network fundamental relationship can be
applied for the network. The protected network is separated by a dashed line in Figure 1. The network
can be entered via six controlled and six uncontrolled gates, and escaped via nine exit gates. Three of
the exit gates are conflicted by gate queues (thus, outflow can be impeded by extreme long queues at
these exits).
4.2. Identification of network parameters
Firstly, the network fundamental diagram (NFD) is identified for the model network. For this end,
simulations were run with different traffic demands, representing the low demands and rush hours as
well. Attention was paid to catch the hysteretic behavior of the system dynamics, resulting different
Figure 1. Network layout.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

paths during unloading, and loading the system. The result of the 25 different, 2-h-long simulations
and the fitted fundamental function F() of Equat ion (5) are plotted on Figure 2.
The best fit is obtained by a fifth-order polynomial function:
F TTSðÞ¼1:23910
9
TTS
5
þ 1:51610
6
TTS
4
þ 3:21410
4
TTS
3
 0:1989 TTS
2
þ 67:93TTS
(24)
The second step is the identification of model parameter Γ based on the relationship in Equation (6).
This is carried out by a linear regression between the measurement data of Q
out
and TTS
PN
. Using that
the average network link length is L = 0.143 km, the linear regression results in Γ = 0.7516. Outflow
from the network is modeled by the following formula:
Q
out
¼ 0:0584  TTD
PN
(25)
The regression is illustrated by Figure 3.
0 100 200 300 400 500 600
0
1000
2000
3000
4000
5000
6000
Total time spent in PN [PCE h/h]
Total travel distance in PN [PCE km/h]
Polynomial fit
Loading dynamics
Unloading dynamics
Figure 2. Network fundamental diagram of the protected network, that is, the variation of TTD together with TTS.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
0
50
100
150
200
250
300
350
400
Total travel distance in PN [PCE km/h]
Outflow from the PN [PCE/h]
Linear fit
Loading dynamics
Unloading dynamics
Figure 3. Linear regression for parameter Γ, that is, relationship between total outflow and TTD
PN
.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

The third parameter to be identified is β
j
of Equation (7). This parameter describes the relationship
between the intended and realized inflow through a gate. In our approach, an average parameter β is
considered representing β
j
for each j gate. The measurement data and linear regression are plotted in
Figure 4. The result of the fitting is β = 0.971.
5. SIMULATION
The developed controllers are compared in two case studies representing realistic traffic situations. In
the first case study, a rush hour scenario is featured with constant traffic load through the boundaries of
the PN, whereas in the second study changing conditions are simulated, with increased traffic load
during a short period.
Four different control strategies are compared in the case studies:
• Fixed-time strategy: the basic control scenario, a non-adaptive signal plan that is normally applied
for the intersections by the traffic control center. Constant green times are used.
• PID control: the control scenario suggested by Keyvan-Ekbatani et al. [19]. The tuning of the controller
is outlined in the Appendix.
• Greedy NMPC control: a controller that does not take into account the outside of the PN. It is designed
based on Section 3, minimizing cost function (14) with the weighting parameters: w
1
=0,w
2
= 0.8.
• NMPC control: a controller that punishes gate queues as well. Its design is similar to the Greedy
NMPC control apart from considering the effect of the queues in the cost function during the control
input optimization by adding a non-zero weight to the cost parameter μ(k). Thus, it uses the weighting
parameters with the following values: w
1
= 0.6, w
2
= 0.8.
0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
Intended inflow Q
g
[veh/h]
Realized inflow Q
in
[veh/h]
Linear regression β⋅Q
g
Measured data Q
in
Figure 4. Linear regression for parameter β, that is, relationship between realized and intended inflows.
Figure 5. The closed-loop simulation environment.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

5.1. Simulation environment and setup
For the simulations, the microscopic traffic simulator Vissim is utilized [33]. Through the COM
interface ([34]), Vissim is connected to
MATLAB, which is used for the online optimization of green
times. The optimal control input is calculated using the nonlinear model predictive control algorithm.
The closed-loop simulation environment is shown in Figure 5. At the end of each cycle, the traffic
0
1000 2000 3000 4000 5000 6000
7000
0 1000 2000 3000 4000 5000 6000
7000
0
1000 2000 3000 4000 5000 6000
7000
30
40
50
60
70
Disturbance flow
[PCE/cycle]
0
50
100
Traffic flow through
controlled gates [PCE/cycle]
Simulation time [sec]
0 1000 2000 3000 4000 5000 6000
7000
0
1000 2000 3000 4000 5000 6000
7000
Simulation time [sec]
0 1000 2000 3000 4000 5000 6000
7000
Simulation time [sec]
Fixed time control PID control NMPC control Greedy NMPC control
(a)
0
200
400
600
Number of vehicles in PN
0
50
100
Vehicles stuck outside PN
Fixed time control PID control NMPC control Greedy NMPC control
(b)
0
2000
4000
6000
TTD in PN
[PCEkm/h]
Fixed time control PID control NMPC control Greedy NMPC control
0
1000
2000
3000
TTD outside PN
[PCEkm/h]
(c)
Figure 6. Results of case study 1.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

Table I. Aggregated results of case study 1.
Parameter Controller Abs. value Improvement vs. fix-time control
Vehicles let into PN (PCE) Fix-time control 2733 —
PID control 4052 +48.26%
Greedy NMPC control 4134 +51.26%
NMPC control 4623 +69.15%
Avg. number of vehicles in PN (PCE) Fix-time control 382.1 —
PID control 152.3 60.1%
Greedy NMPC control 137.8 63.9%
NMPC control 148.5 61.1%
Avg. TTD in PN (PCE km/h) Fix-time control 2242.7 —
PID control 4079.6 +81.9%
Greedy NMPC control 4421.8 +97.2%
NMPC control 4178.0 +86.4%
Vehicles stuck outside PN (PCE) Fix-time control 6094 —
PID control 4166 31.6%
Greedy NMPC control 4081 33.0%
NMPC control 3362 44.8%
PN, protected network; NMPC, nonlinear model predictive control; PCE, passenger car equivalent.
0 1000 2000 3000 4000 5000 6000 7000
2000
2500
3000
3500
4000
4500
5000
Traffic demand [PCE/h]
Simulation time [sec]
Controlled gates
Non−controlled gates
Figure 8. Case study 2: Traffic load at controlled and uncontrolled gates.
0 50 100 150 200 250 300 350 400 450 500 550
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
TTD in PN [PCEkm/h]
Number of vehicles in PN
Fixed time control
PID control
NMPC control
Greedy NMPC control
Figure 7. Network fundamental diagram of case study 1.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

measurements of the states are updated in MATLAB. After minimizing the cost function, the new control
signals returned to the traffic simulator.
For the simulations, the following parameters are set. The sampling time is chosen as the signal
controller cycle time, thus T
S
= 90 s. For the optimization horizon, K = 3 is used (for the justification,
see the Appendix). The simulation length of the case studies is equally 7200 s. Concerning the control
0 1000 2000 3000 4000 5000 6000 7000
0 1000 2000 3000 4000 5000 6000 7000
0 1000 2000 3000 4000 5000 6000 7000
0 1000 2000 3000 4000 5000 6000 7000
0 1000 2000 3000 4000 5000 6000 7000
50
60
70
80
90
Disturbance flow
[PCE/cycle]
0
50
100
Traffic flow through
controlled gates [PCE/cycle]
Simulation time [sec]
Simulation time [sec]
0 1000 2000 3000 4000 5000 6000 7000
Simulation time [sec]
Fixed time control PID control NMPC control Greedy NMPC control
0
200
400
600
Number of vehicles in PN
0
50
100
150
Vehicles stuck outside PN
Fixed time control PID control NMPC control Greedy NMPC control
0
2000
4000
TTD in PN
[PCEkm/h]
0
1000
2000
3000
TTD outside PN
[PCEkm/h]
Fixed time control PID control NMPC control Greedy NMPC control
(a)
(b)
(c)
Figure 9. Results of case study 2.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

input, upper, and lower bounds are applied, that is, green times constraints are considered: g
max
=56s,
g
min
=10s.
5.2. Simulation results
In order to make a thorough comparison of the PID controller and the NMPC design, two scenarios are
simulated. In case study no. 1, rush hour traffic is modeled with constant traffic load, a sum of
3600 PCE/h is loaded at the controlled perimeter gates. The traffic demand through the uncontrolled
gates are considered disturbances, their sum is plotted in Figure 6 alongside with the controlled input
values.
The performance and system state values are plotted in Figure 6b and 6c. The results of the fixed
time control scenario show that without an adaptive control, a traffic jam emerges within the protected
network and very low traffic performance is present in the PN from 4000 s onwards. The PID, greedy
NMPC, and NMPC controllers are able to prevent congestion, and they all act as expected. The PID
controller solves the task by eliminating the traffic jam; however. it has high settling time (it reaches
the optimal vehicle number after 2000 s only around 5000 s). PID control is outperformed both by
the greedy NMPC and the NMPC design. Throughout the whole horizon, the NMPC controller adapts
much better to disturbance oscillations than the PID controller. The greedy NMPC provides higher
performance inside the PN with slightly higher TTD values, but on the expense of slightly higher
queues at the perimeter gates. The aggregated results of Table I confirm that the queue lengths are
considerably shorter in the NMPC case, thus it can be stated that the modeling of gate queues and
the optimization of green times considering queue lengths is reasonable.
The NFD plots of the different controllers (Figure 7) clearly show that the different controllers
remain in different intervals of state-domain. The fixed time controller has the widest interval, as its
use leads to congested vehicle numbers. The oscillation of the PID control can be observed on the
NFD plot as well, as it roves a fairly wide interval around N
PN,crit
. The NMPC controllers have the
smallest working state interval; however, the greedy NMPC is biased toward lowe r N
PN
values.
In case study no. 2, also a rush hour situation is modeled; however, with changing traffic demands
from 2400 to 4800 s. The traffic demand is plotted in Figure 8. This scenario provides the opportunity
to analyze the reaction of the controllers of moderate changes in the input demand and the distur-
bances. A desired behavior of a controller can be simply described in terms of performance: during
higher traffic demands, the controllers should maintain the prescribed vehicle number in the PN, and
also minimize the queues at the controlled gates.
The profile of the distu rbance changes is plotted in the upper axis of Figure 9a. The lower axis
shows the input values of the controllers. The higher traffic loads give an oppor tunity to analyze the
Table II. Aggregated results of case study 2.
Parameter Controller Abs. value Improvement vs. fix-time control
Vehicles let into PN (PCE) Fix-time control 2413 —
PID control 3213 +33.15%
Greedy NMPC control 2930 +21.4%
NMPC control 3040 +25.98%
Avg. number of vehicles in PN (PCE) Fix-time control 382.7 —
PID control 192.3 49.7%
Greedy NMPC control 161.4 57.8%
NMPC control 175.9 54.0%
Avg. TTD in PN (PCE km/h) Fix-time control 2120.2 —
PID control 3987.4 +88.1%
Greedy NMPC control 4213.4 +98.7%
NMPC control 4132.1 +94.9%
Vehicles stuck outside PN (PCE) Fix-time control 6724 —
PID control 4648 30.9%
Greedy NMPC control 4960 26.2%
NMPC control 4583 31.8%
PN, protected network; NMPC, nonlinear model predictive control; PCE, passenger car equivalent.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

effect of the possible conflicts of the network exits and controlled gates. The phenomenon is clearly
visible as TTD performance between 3000 and 4000 s drastically decreases, compared with the reduc-
tion in case study 1.
The traffic performance Figure (9b and 9c) attest to the same behavior of controllers as scenario 1:
shorter settling time of the NMPC controllers, and also better reactions to disturbance oscillations.
During the period of higher demands (2400–4800 s), the better performance of the NMPC controllers
is clearly visible: it is able to maintain the optimal number of vehicles in PN resulting in better TTD
values. Also, gate queues are lower with less oscillations in the NMPC and greedy NMPC case. The
0 50 100 150 200 250 300 350 400 450 500 550
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
TTD in PN [PCEkm/h]
Number of vehicles in PN
Fixed time control
PID control
NMPC control
Greedy NMPC control
Figure 10. Network fundamental diagram of case study 1.
Figure 11. Comparison of PID, Greedy nonlinear model predictive control (NMPC), and NMPC performances
(considering TTD
PN
and vehicles stuck outside protected network) - case study 1.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

superiority of the NMPC controller to the greedy NMPC controller is clearly manifested again in
shorter queues with similar TTD performance. For the aggregated results of case study 2, see Table II.
The NFD plot of case study 2 (Figure 10) again shows the different working intervals of
the controllers.
5.2.1. Comparison: greedy versus non-greedy NMPC control
The most important performances of the addressed control problem are the TTD values and the number
of vehicles stuck outside the network. For the case studies, performance graphs of the three controllers
are highlighted in Figures 11 and 12. The superiority of the NMPC and greedy NMPC controllers to the
PID controller are clearly visible. The graphs also show that the greedy and non-greedy NMPC control-
lers provide similar TTD performances in the PN; however, a better exterior performance is present by
involving the queue lengths in the objective function.
Figure 12. Comparison of PID, Greedy nonlinear model predictive control (NMPC), and NMPC performances
(considering TTD
PN
and vehicles stuck outside protected network)—case study 2.
Table III. Comparison: nonlinear model predictive control (NMPC) versuss greedy NMPC control.
Parameter Greedy NMPC NMPC Improvement of NMPC
Case study 1 Vehicles let into PN (PCE) 4134 4623 +11.8%
Avg. number of vehicles in PN (PCE) 137.8 148.5 +7.8%
Avg. TTD in PN (PCE km/h) 4421.8 4178.0 5.5%
Vehicles stuck outside PN (PCE) 4081 3362 17.6%
Case study 2 Vehicles let into PN (PCE) 2930 3040 +3.8%
Avg. number of vehicles in PN (PCE) 161.4 175.9 +8.9%
Avg. TTD in PN (PCE km/h) 4213.4 4132.1 1.9%
Vehicles stuck outside PN (PCE) 4960 4583 7.6%
PN, protected network; NMPC, nonlinear model predictive control; PCE, passenger car equivalent.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

The relative differences in aggregated performances (Table III) confirm that with a small loss in
TTD performance, signi ficant improvements can be reached in queue lengths and thus the exterior
performance.
The aforementioned results represent single simulation runs for the specified case studies. However,
to prove the performance of the tested methods, a representative number of runs is needed. For this
end, the performance results of 10 simulation runs are averaged for each case study. During the runs,
the same excitation is set in Vissim for input and disturbance demands; however, with different
random seeds. The simulation results of the aggregated variables (Tables IV and V) show that the same
level of performance can be experienced regardless the random seed of the simulations.
6. CONCLUSION
Based on the modeling results by Keyvan-Ekbatani et al. [22], a novel nonlinear system model is
proposed for the urban gating problem: additional state variables are allocated to describe the queue
Table IV. Average aggregated results of simulation runs for case study 1.
Parameter Controller Abs. value Improvement vs. fix-time control
Vehicles let into PN (PCE) Fix-time control 2757 —
PID control 4034 +46.3%
Greedy NMPC control 4128 +49.7%
NMPC control 4609 +67.17%
Avg. number of vehicles in PN (PCE) Fix-time control 378.4 —
PID control 154.6 59.1%
Greedy NMPC control 138.9 63.3%
NMPC control 149.3 60.5%
Avg. TTD in PN (PCE km/h) Fix-time control 2268.3 —
PID control 4074.3 +79.6%
Greedy NMPC control 4423.9 +95.0%
NMPC control 4173.2 +83.9%
Vehicles stuck outside PN (PCE) Fix-time control 6071 —
PID control 4183 31.1%
Greedy NMPC control 4090 32.6%
NMPC control 3379 44.3%
PN, protected network; NMPC, nonlinear model predictive control; PCE, passenger car equivalent.
Table V. Average aggregated results of simulation runs for case study 2.
Parameter Controller Abs. value Improvement vs. fix-time control
Vehicles let into PN (PCE) Fix-time control 2397 —
PID control 3257 +35.18%
Greedy NMPC control 2943 +22.7%
NMPC control 3053 +27.28%
Avg. number of vehicles in PN (PCE) Fix-time control 386.3 —
PID control 194.5 50.3%
Greedy NMPC control 160.7 58.4%
NMPC control 176.1 54.4%
Avg. TTD in PN (PCE km/h) Fix-time control 2113.8 —
PID control 3980.3 +88.3%
Greedy NMPC control 4201.7 +98.6%
NMPC control 4131.3 +95.4%
Vehicles stuck outside PN (PCE) Fix-time control 6738 —
PID control 4616 31.5%
Greedy NMPC control 4943 26.6%
NMPC control 4568 32.2%
PN, protected network; NMPC, nonlinear model predictive control; PCE, passenger car equivalent.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

dynamics at the protected network gates. Using the extended model, a ‘non-greedy’ control approach
is suggested to conside r the exterior of the protected area as well: the reduction of queue lengths mit-
igates the risk of exterior congestions. The control strategy is analyzed through different case studies,
and apart from the comparison of the greedy and non-greedy gating approach, the controller perfor-
mances of the NMPC and the PID approach (given in Keyvan-Ekbatani et al. [19]) are also compared.
Performance results clearly show that the gating problem should not be restricted to TTD optimization
within the PN. Significant improvements can be reached in gate queue lengths with only a minor loss
in inner TTD performance. The simulation results also show that the NMPC provides better perfor-
mance (due to the nonlinear characteristics of the process) than the PID, which is a simple linear con-
trol approach.
Concerning the practical use, it has to be noted that traffic control centers of our days are usually
designed to perform network wide control, which also indicates the construction of appropriate mea-
surement system (e.g. detector measurements on each link), that is, full information control. Therefore,
the proposed method can be considered as efficient traffic control candidate for field implementation.
Further research is planned with emphasis on the uncertain manner of the process, which may arise
from several points. In this particular case, the oscillation of disturbances can be considered as an un-
certainty throughout the optimization horizon. Moreover, the analysis of the network size on controller
performance is also planned as part of future research.
7. LIST OF ABBREVIATIONS
COM Component Object Model
NFD network fundamental diagram
NMPC nonlinear model predictive control
PCE passenger car equivalent
PID proportional-integral-derivative
PN protected network
TTD total travel distance
TSS total time spent
ACKNOWLEDGEMENTS
The authors acknowledge Center for Budapest Transport for providing traffic volume data and fixed-
time signal plans. This work is connected to the scientific program of EITKIC-12-1-2012-0001 project
(supported by the Hungarian Government, managed by the National Development Agency, financed by
the Research and Technology Innovation Fund), and TÁMOP-4.2.2.C-11/1/KONV-2012-0012: Smarter
Transport project (supported by the Hungarian Government, co-financed by the European Social Fund).
REFERENCES
1. Tettamanti T, Varga I, Kulcsár B, Bokor J. Model predictive control in urban traffic network management. Proceed-
ings of the 16th Mediterranean Conference on Control and Automation, Ajaccio, Corsica, 2008; 1538–1543. DOI:
10.1109/MED.2008.4602084
2. Aboudolas K, Papageorgiou M, Kosmatopoulos E. Store-and-forward based methods for the signal control problem
in large-scale congested urban road networks. Transportation Research Part C: Emerging Technologies 2009;
17:163–174. DOI:10.1016/j.trc.2008.10.002.
3. de Oliveira LB, Camponogara E. Multi-agent model predictive control of signaling split in urban traffic networks.
Transportation Research Part C: Emerging Technologies 2010; 18:120–139. DOI:10.1016/j.trc.2009.04.022.
4. Lin S, De Schutter B, Xi Y, Hellendoorn H. Efficient network-wide model-based predictive control for urban traffic
networks. Transportation Research Part C: Emerging Technologies 2012; 24:122–140. doi:10.1016/j.trc.2012.02.003.
5. Tettamanti T, Luspay T, Kulcsar B, Peni T, Varga I. Robust control for urban road traf fi c networks. IEEE Transac-
tions on Intelligent Transportation Systems 2013, Accepted. DOI: 10.1109/TITS.2013.2281666
6. Godfrey J. The mechanism of a road network. Traffic Engineering and Control 1969; 11(7):323–327.
7. Mahmassani H, Williams J, Herman R. Performance of urban traffic networks. 10th International Symposium on
Transportation and Traffic Theory. Amsterdam, The Netherlands, 1987; 1–20.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

8. Daganzo CF, Geroliminis N. An analytical approximation for macroscopic fundamental diagram of urban traffic.
Transportation Research Part B 2008; 42(9):771–781. DOI:10.1016/j.trb.2008.06.008.
9. Helbing D. Derivation of a fundamental diagram for urban traffic flow. The European Physical Journal B 2009; 70(2):
229–241. DOI:10.1140/epjb/e2009-00093-7.
10. Mazloumian A, Geroliminis N, Helbing D. The spatial variability of vehicle densities as determinant of urban net-
work capacity. Philosophical Transactions of the Royal Society A 2010; 368(1928):4627–4647. DOI:10.1098/
rsta.2010.0099.
11. Geroliminis N, Zheng N, Ampountolas K. A three-dimensional macroscopic fundamental diagram for mixed bi-
modal urban networks. Transportation Research Part C: Emerging Technologies 2013; 42:168–181.
12. Daganzo CF. Urban gridlock: macroscopic modeling and mitigation approaches. Transportation Research Part B
2007; 41(1):49–62. DOI:10.1016/j.trb.2006.03.001.
13. Aboudolas K, Geroliminis N. Perimeter and boundary flow control in multi-reservoir heterogeneous networks.
Transportation Research Part B: Methodological 2013; 55:265–281.
14. Hajiahmadi M, Haddad J, De Schutter J, Geroliminis N. Optimal hybrid macroscopic traffic control for urban re-
gions: perimeter and switching signal plans controllers. Proceedings of the 2013 European Control Conference
(ECC). Zürich, Switzerland, 2013; 3500–3505.
15. Hajiahmadi M, Knoop VL, De Schutter B, Hellendoorn H. Optimal dynamic route guidance: a model predictive ap-
proach using the macroscopic fundamental diagram. Proceedings of the 16th International IEEE Annual Conference
on Intelligent Transportation Systems (ITSC 2013), 2013; 1022–1028. The Hague: The Netherlands. DOI: 10.1109/
ITSC.2013.6728366
16. Geroliminis N, Haddad J, Ramezani Ghalenoei M. Optimal perimeter control for two urban regions with macro-
scopic fundamental diagrams: a model predictive approach. IEEE Transactions on Intelligent Transportation Sys-
tems 2013; 14(1):348–359.
17. Haddad J, Ramezani Ghalenoei M, Geroliminis N. Cooperative traffic control of a mixed network with two urban
regions and a freeway. Transportation Research Part B: Methodological 2013; 54:17–36.
18. de Jong D, Knoop VL, Hoogendoorn SP. The effect of signal settings on the macroscopic fundamental diagram and
its applicability in traffic signal driven perimeter control strategies. Proceedings of the 16th International IEEE Annual
Conference on Intelligent Transportation Systems (ITSC 2013), The Hague, The Netherlands, 2013; 1010–1015. DOI:
10.1109/ITSC.2013.6728364
19. Keyvan-Ekbatani M, Papageorgiou M, Papamichail I. Urban congestion gating control based on reduced operational
network fundamental diagrams. Transportation Research Part C
2013; 33:74–87. DOI:10.1016/j.trc.2013.04.010.
20. Keyvan-Ekbatani M, Yildirimoglu M, Geroliminis N, Papageorgiou M. Traffic signal perimeter control with multi-
ple boundaries for large urban networks. Proceedings of the 16th International IEEE Annual Conference on Intel-
ligent Transportation Systems, The Hague, 2013.
21. Ramezani M, Haddad J, Geroliminis N. Integrating the dynamics of heterogeneity in aggregated network modeling
and control. Transportation Research Board 93rd Annual Meeting, No. 14-0710, 2014.
22. Keyvan-Ekbatani M, Kouvelas A, Papamichail I, Papageorgiou M. Exploiting the fundamental diagram of urban
networks for feedback-based gating. Transportation Research Part B 2012; 46:1393–1403. DOI:10.1016/j.
trb.2012.06.008.
23. Lay MG. Handbook of Road Technology. Spon Press: Abingdon, UK, 2009.
24. Ashton WD. The Theory of Traffic Flow. Spottiswoode. Ballantyne and Co. Ltd.: London, 1966.
25. Papageorgiou M, Vigos G. Relating time-occupancy measurements to space-occupancy and link vehicle-count.
Transportation Research Part C 2008; 16(1):1–17. DOI:10.1016/j.trc.2007.06.001.
26. Vilgos G, Papageorgiou M, Wang Y. Real-time estimation of vehicle-count within signalized links. Transportation
Research Part C: Emerging Technologies 2008; 16(1):18–35. DOI:10.1016/j.trc.2007.06.002.
27. Kulcsár B, Varga I, Bokor J. Constrained split rate estimation by moving horizon. Proceedings of the 16th IFAC
World Congress, Prague, Czech Republic, 2004.
28. Geroliminis N, Daganzo CF. Existence of urban-scale macroscopic fundamental diagrams: Some experimental find-
ings. Transportation Research Part B: Methodological 2008; 42(9):759–770.
29. Grune L, Pannek J. Nonlinear Model Predictive Control: Theory and Algorithms. Springer: London, 2011.
30. Tettamanti T, Varga I, Péni T, Luspay T, Kulcsár B. Uncertainty modeling and robust control in urban traffic. Pro-
ceedings of the 18th IFAC World Congress, Milan, 2011; 14910–14915.
31. Csikós A, Varga I, Hangos KM. Freeway shockwave control using ramp metering and variable speed limits. Pro-
ceedings of the 21st Mediterranean Conference on Control and Automation, Platanias-Chania, Greece, 2013,
1569–1574. DOI: 10.1109/MED.2013.6608931
32. Ramezani M, Haddad J, Geroliminis N. Macroscopic traffic control of a mixed urban and freeway network. 12th
Swiss Transportation Research Conference, Ascona, Switzerland, 2012.
33. Wiedemann R. Simulation des Straßenverkehrsflusses. Schriftenreihe des Instituts für Verkehrswesen der
Universität Karlsruhe 8. The Karlsruhe Institute of Technology: Karlsruhe, Germany, 1974.
34. Tettamanti T, Varga I. Development of road traffic control by using integrated Vissim-Matlab simulation environ-
ment. Periodica Polytechnica ser. Civil. Eng. 2012; 56:43–
49. DOI:10.3311/pp.ci.2012-1.05.
35. Aboudolas K, Papageorgiou M, Kouvelas A, Kosmatopoulos E. A rolling-horizon quadratic-programming approach
to the signal control problem in large-scale congested urban road networks. Transportation Research Part C:
Emerging Technologies 2010; 18:680–694. DOI:10.1016/j.trc.2009.06.003.
A. CSIKÓS ET AL.
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr

APPENDIX
PID control
For the PID control, the same approach is followed as by Keyvan-Ekbatani et al. [19]. First, the model
is linearized around the setpoint TTS
PN
, then a PID controller is designed based on the linear model.
The controller is stated as follows:
Q
in;j
kðÞ¼Q
in;j
k  1ðÞK
P
TTS
PN
kðÞTTS
PN
k  1ðÞðÞ
þK
I
TTS
PN;crit
 TTS
PN
kðÞ

(A1)
where K
P
and K
I
are the control design parameters, obtained by manual tuning. The design resulted in
the following values: K
P
= 0.4, K
I
= 0.05. Q
in;j
kðÞ¼∑
N
j
j¼1
q
in;j
kðÞ, the sum of the controlled traffic flows
through the perimeter. The controller provides the value of Q
in,j
(k), from which q
in,j
(k) values are split
proportionally to the corresponding l
j
(k) queue lengths.
Horizon length
The horizon length is usually chosen so that the overall condition of the system can be represented by
specific performance parameters, aggregated through the control horizon. By increasing the horizon K,
the performance usually improves in the MPC problems. Nevertheless, the amelioration of the perfor-
mance by longer horizon is not straightforward because of the model uncertainty. This might lead to
uncertain predictions where only conservatively chosen control variables can minimize the objective
function.
In our work, the horizon length is chosen to be equal to a nominal travel time within the network.
For the case study, the longest route through the network is 1.2k m. Considering an average cruising
speed of 16 km/h, the route is completed in 270 s. Thus, with a cycle length of 90 s, a control horizon
of K = 3 steps covers the route of an individual vehicle. The effect of different horizon lengths is pro-
vided in Table A1 with emphasis on the controller performance. The figures of the table are the aver-
age values of simulation runs for scenario 1 and 2. Clearly, best results can be obtained by using 3 or 4
for horizon length. K = 3, however, is preferred as TTD is considered the primary performance, and the
CPU’s computation time is also lower using this horizon length. It is worth to notice that the best result
at K = 3 coincides with the suggestion of Aboudo las et al. [35]: ‘a satisfactory optimization horizon K
should be in the order of the time needed to travel through the network’.
Simulations are performed on a personal computer with Intel Core I3 2.4-GHz CPU and 3 GB of
RAM.
Table A1. Simulation results with different horizon lengths.
Horizon length 2 3 4 5 6
Average TTD in PN (PCE km/h) 4053.1 4196.1 4213.9 4093.1 4114.2
Vehicles stuck outside PN (PCE) 4395.7 4557.2 4419.8 4723.4 4812.7
Computation time of a step (s) 5.24 9.32 16.01 22.45 29.32
PN, protected network; PCE, passenger car equivalent.
NONLINEAR GATING CONTROL FOR URBAN ROAD TRAFFIC NETWORK USING THE NFD
Copyright © 2014 John Wiley & Sons, Ltd. J. Adv. Transp. 2014
DOI: 10.1002/atr