Investigation of the Influence of Macroscopic Traffic Flow
         Models in Real – Time Feedback OD Flow Estimation
                                                 Anburuvel Arulanantham
                                    Candidate for the Degree of Master of Engineering
                                           Supervisor: Prof. Takashi Nakatsuji
                            Division of Engineering and Policy for Cold Regional Environment

Introduction                                                            D st denote the origin flow from r and destination flow
                                                                        towards s respectively. The superscript t denotes the
Most researchers have recommended real-time Origin-
                                                                        measurement time interval. Since the number of OD
Destination (OD) matrix as an essential input data for
                                                                        flows to be estimated (n x m) are greater than the number
traffic analysis. This has given rise to consistent real-
                                                                        of available equations (n or m), the problem is reduced
time OD matrix estimation techniques.
                                                                        as an indeterminacy problem.
    Researchers have been concentrating on indirect OD
                                                                            The next issue is formulating a relationship between
flow estimation methods using less expensive traffic data               OD flows and link traffic measurements. Researchers
collected from traffic surveillance systems. Because of                 have explicitly modeled this relationship through
the limited information available from data sources (data
                                                                        assignment matrix can be written as follows:
scarcity), OD flow estimation becomes challenging.                                               t
    Due to data scarcity, the number of unknown OD                                        xlt   a rs ,T q rs  wlt
                                                                                                    t       T
flows to be estimated drastically increases. Therefore                                          t T
deriving OD flows has a high probability of being                                     t
biased. An efficient way to solve this problem is to relate             where xl represents the measured traffic flows at link l.
the derived OD flows to a measurement variable. Several                   t
                                                                        a rs,T is the assignment proportions of OD flow between
approaches have treated this problem by introducing
assignment matrix and a macroscopic model. However                      pair rs departing its origin during time interval T and
the relationship is still not well depicted. Establishing               observed at link l. q rs denotes the OD flow between OD
such a relationship enables implementing feedback                                                                                             t
process by extracting information from the measurement                  pair rs departed its origin during time interval T. wl
data.                                                                   represents the traffic flow measurement error at link l.
    In order to provide solutions to the identified critical            However the indirect dependency of travel times on OD
issues the present study proposes a real-time OD flow                   flows makes the relationship nonlinear. Pueboobpaphan
estimation scheme. The scheme begins with deriving OD                   proposes an assignment matrix free OD flow estimation
flows with a traffic distribution model. The derived OD                 scheme formulated as follows:
flows are then mapped to measurement variables by                                         ˆ          ˆ ˆ                   ~
                                                                             ~ t 1  f [O t k ,....O t 1 , b t k ,.....b t 1 ]   t 1 (4)
means of modified Payne model based on the Godunov
                                                                             xl             r         r        rs           rs         l
scheme. This mapping tool captures the real nature of                         ~ t 1 represents the estimated traffic measurement
                                                                        where x   l
traffic flow sufficiently and provides accurate mapping
                                                                        at link l. The function f [.] denotes the nonlinear mapping
between OD flows and measurement variables. Utilizing
                                                                        scheme between OD flows and link traffic
the original Payne model to perform traffic simulation                                               t       t
has been criticized by several researchers. To admit those              measurements. brs and O r represent the vector of OD
criticisms, this study adopts the Godunov solution                      proportions and the vector of origin flows respectively.
scheme to treat macroscopic models. Finally, to extract
the information from the measurement data and to
                                                                        lt 1  denotes the measurement error at link l.
implement feedback estimation, Unscented Kalman                         Pueboobpaphan expresses the nonlinear relationship
Filter (UKF) is utilized. The UKF allows the use of any                 through Cell Transmission Model (CTM). Conducting a
sophisticated models or software.                                       traffic simulation with CTM using OD proportions and
                                                                        origin flows link traffic measurements were estimated.
                                                                        Though CTM provides some unique advantages over the
Statement of problem                                                    other first order macroscopic models, it still has all the
                                                                        limitations of the first order model. On the other hand
Conventional approaches estimate real-time OD matrix                    assuming input origin flows are noise free and available
for a traffic network with n origins and m destinations                 throughout the period of analysis may result in erroneous
using the real-time data on origin flows and destination                estimations.
flows. The relationship among them can be given as                                The last issue is implementing feedback
follows:                                                                estimation to keep the estimation consistent with
 m                                     n                                measurement data. The traffic distribution and
 qri  Ort

i 1
                   (1)                q
                                      j 1
                                             js    Dst       (2)       macroscopic models cannot be analytically expressed.
                                                                        Moreover, the nonlinear mapping between OD flows and
         t                                                t             link traffic measurements must be preserved. Due to
where qri denotes OD flow between OD pair ri. O r and                   these constraints standard Kalman Filter (KF) or

Extended Kalman Filter (EKF) cannot be used because                 drawbacks in Logit model are less influential in this
they cannot accommodate such complex models. EKF, a                 traffic distribution process.
well known tool that deals with nonlinear systems
applies a linearization to posterior mean and covariance            Macroscopic traffic flow model
of nonlinear systems to first-order Taylor series                       This section provides the solution for the second
expansion. This approximation may introduce large                   issue, mapping derived OD flows to link traffic
errors in estimation and sometimes divergence of the                measurements. Macroscopic traffic flow models consider
filter. Moreover, EKF needs computation of Jacobians.               the aggregate behavior of vehicles mostly based on
This is possible only if the functions are continuous.              hydrodynamic theory. Because of computational
           In order to overcome the problems discussed              efficiency, they are preferred for large network studies.
above, a solution scheme is proposed. Details are                   However due to the inconsistency in the underlying
elaborated in the next section.                                     theory, macroscopic traffic flow models are severely
                                                                    criticized by several researchers. Among them,
                                                                    Daganzo’s is the most radical one.
Solution scheme
                                                                    Payne model Having the criticisms in mind, the
Joint OD flow estimation                                            second order Payne model introduced by Payne and later
    In order to deal with the first issue mentioned above           modified by Cremer is chosen to perform the traffic
Kamide introduces an approach treating origin flow as               simulation in this study. The drawbacks of the original
an unknown variable. His approach embeds joint OD                   Payne model, which Daganzo especially points out, are
flow estimation scheme into Pueboobpaphan’s                         treated using the Godunov scheme. Lebacque notes that
assignment matrix free method.                                      the Godunov scheme is the best first order scheme to
    The present study also considers the origin flows as            compute solutions for macroscopic models.
unknown variables. But the other essential input data the
destination flow is assumed error free and available                Godunov scheme The Godunov scheme proposes a
throughout the period of analysis. Including destination            time-space dicretization to solve partial differential
flow and model parameters as unknown variables makes                equations in macroscopic models by converting them
the scheme more advanced. However estimating many                   into    approximate    finite   difference    equations.
unknown variables simultaneously in a dynamic                       Furthermore to compute boundary flow between cells
approach is much more complicated.                                  Godunov scheme adopts solutions to the generalized
    Since the present study is in the initial stage, the OD         Riemann problem which defines the boundary flow
estimation is limited to freeway network where no path              based on local demand-supply concept. The local
choices exist. Therefore two stage scheme is reduced to             demand function holds the maximum possible outflow
one stage which only deals with traffic distribution                the upstream can transfer to downstream. Similarly, the
related issues.                                                     local supply function holds the maximum possible
                                                                    inflow the downstream can receive. The boundary flow
Traffic distribution model Traffic distribution                     is then computed by selecting the minimum between the
process is performed to reduce unknowns by computing                local demand and the local supply, as shown below:
OD flows, provided that origin flows and destination                           Qit =Min [ (  it ) , (  i1 ) ]
flows are known and fixed. Assuming that the traffic                              t
flow is distributed according to the transportation cost            where Qi represents the traffic flow from cell i to cell
                                                                    (i+1). (  i ) and (  i1 ) denote local demand and
between OD pairs, an optimization problem is devised:                                    t                     t
                        m    n
        min Z (q)   s rs q rs
                         t    t
                                                         (5)        local supply respectively.
                       s 1 r 1                                             Applying Godunov scheme, Payne model
where  Z (q) stands for the optimization process of OD              equations can be written in finite difference
                                                                    approximations as follows:
flow q. s rs denotes the transportation cost between OD                   qit   it vit ;                                                                    (8)
pair rs at time interval t. Transportation cost is defined in
terms of shortest travel time ( u rs ) between OD pair rs,               it 1   it               (Qit1  Qit  rit  sit ) ;                            (9)
which is given by
                                                                                         t                        t t t                   vt  it1   it ; (10)
                t                                                      vit 1  vit       [Ve (  it )  vit ]      vi [vi 1  vit ] 
        srs  u rs 1
                                                       (6)                                                        xi                     xi  it  
The shortest travel time and hence transportation cost is           where length of each cell is x i .  i and v i denote
                                                                                                                                     t              t
updated each time interval with the macroscopic model.
However before performing traffic simulation the                    traffic density and space mean speed at cell i
                                                                                              t            t
shortest travel times remain unknown. To avoid this                 respectively. s i and ri stand respectively for on-ramp
problem the shortest travel time in the previous time               and off-ramp flows. T is the reaction time.
interval (t-1) is considered as the transportation cost of
the current time interval (t). One may argue that the               Ve ( it )   represents      speed-density                                  equilibrium
adopted distribution process implicitly depends on Logit            relationship. τ, ν, κ are model parameters.
model. However it is assumed that the persistent

Feedback estimation                                                              Update Step: By getting the feedback from the
    As discussed as the third issue in previous section,                         corresponding link data update the state estimate
                                                                                        ˆ        ~
the feedback estimation ensures that the computation                                    Ort 1  Ort 1  K ( ~lt 1  xlt 1 )
                                                                                                              x                     (13)
process is consistent with measurements. As mentioned
earlier traffic distribution model and macroscopic traffic                           The Kalman Gain (K) is used to update the predicted
flow model cannot be represented in analytical form.                             origin flow.
Unscented Kalman Filter (UKF) is thus preferred for
feedback estimation which can accommodate any
simulation model or software in the OD estimation
scheme. The UKF is a derivative free approach
applicable for both continuous and discontinuous
functions. It has the same computational effort as EKF
but it produces accurate results.
        Mihaylova has recently shown that particle filter
(PF) produces more accurate outputs than UKF. A                                   FIGURE 1 Proposed real-time OD estimation scheme
dispute may arise here that, using PF which can treat
system state with any probability distribution may
produce accurate results since it relies on the Monte                            Case study
Carlo sampling approach. But the deterministic sampling
approach in UKF consumes less computation time than
                                                                                 Data description
PF, which requires a large number of sample points to be
                                                                                 The numerical evaluation study uses traffic data on
produced in order to obtain accurate output.
                                                                                 inflows and outflows, collected in November 1, 1994,
                                                                                 from Matsubara line of the Hanshin freeway in Japan. In
The unscented Kalman filter In UKF the state
                                                                                 addition to that 15 minute detector data on link flows and
variable is assumed Gaussian random variable which
                                                                                 flow speeds are available at eight different locations of
undergoes unscented transformation producing a set of
                                                                                 the freeway stretch. The study section depicted in figure
“sigma points” that represent the given distribution of
                                                                                 2 is 11.220 km long was divided into 28 segments for the
the state variable. When the derived sigma points
                                                                                 purpose of traffic simulation. It composed of 15 feasible
propagates through a nonlinear system, captures the
                                                                                 OD pairs.
posterior mean and covariance of original state variable
to the second order Taylor series expansion, it preserves
nonlinearity of the estimation.

Dynamic OD matrix estimation Many past
approaches note that the OD estimation and prediction
process can be formulated as a state-space model. As
mentioned already by setting up origin flows as state
variable the following formulations were carried out to
implement dynamic OD estimation scheme.
Prediction Step: The state equation is defined as
                                                                                      FIGURE 2 Schematic layout of the study area
autoregressive process as follows:
       ~        ˆ
       Ort 1  Ort  rt                        (11)                            Implementation
where O r represents the origin flow vector comprised
                                                                                 During the numerical study, four different OD estimation
                                                                                 schemes were set up in order to evaluate the estimation
of origin flows departed from origins r =1 to n.                                 potential. Scheme A is the proposed OD estimation
Ort 1 represents the predicted origin flow vector.  rt is                      scheme in this study which utilizes the macroscopic
the modeling noise vector with Gaussian distribution                             model in scheme A1 and the microscopic model in
                                                                                 scheme A2. Scheme B is Pueboobpaphan’s proposal.
zero mean and r covariance.
                                                                                 Scheme B engages the macroscopic model in scheme B1
Definition of measurement equation is completely                                 and the microscopic model in scheme B2.
different from conventional approaches given as:                                  SchemeA1: Macroscopic model, Origin flow
             ~ t 1  f [O t 1 ]   t 1                                        SchemeA2: Microscopic model, Origin flow
             xl           r          l           (12)
                                                                                  SchemeB1: Macroscopic model, OD proportions
Here f [.] denotes the macroscopic traffic flow model                             SchemeB2: Microscopic model, OD proportions
mapping discussed earlier. ~l denotes estimated link
                           x t 1                                                    The microscopic model used here is the Excess
                                                                                 Critical Speed (ECS) model proposed by Gurusinghe et
traffic data.     t 1 represents measurement noise vector                      al. based on stimulus-response concept. In addition to the
with       Gaussian        distribution      zero      mean            and       relative speed stimulus in the GM model a headway-
   t 1
    l      covariance.     Note     that        t
                                                     and      l
                                                                t 1
                                                                       are       dependant stimulus term was added.


          3 (a) OD flows for OD pairs 11 and 13                         3 (b) OD flows for OD pairs 15 and 16

        FIGURE 4 (a) RMSE values of OD flows                         FIGURE 4 (b) Computation time in seconds

    Each scheme is tested separately with the data from         illustrated in figure 4 (b). In the perspective of
9:00 a.m. to 9:00 p.m. starting with the predefined             computation time the OD estimation scheme proposed in
inputs. The schemes A1 and A2 assume measured origin            this study requires the least time which is very much
flows during time interval 9:00-9:15 a.m. as the initial        appropriate for online applications where less
origin flows and the covariance (O) as 2 % of the              computation time is highly appreciated.
average of measured origin flows. Similarly destination
flows are also given the measured values during the
same time interval. The initial shortest travel times are       Conclusions
assumed as the free flow travel times between OD pairs.
    In contrast the schemes B1and B2 assumes initial OD             This study proposes a real-time OD matrix estimation
proportions to be equally distributed and covariance            scheme to efficiently manipulate the limited data
 (b) as 1/10000. The initial origin flows are given the        obtained from the traffic surveillance systems. The
measured values during time interval 9:00-9:15 a.m. The         scheme composed of traffic distribution model and an
                                                                appropriate macroscopic traffic flow model subject to
measurement noises (  ) for both link traffic flows and
                                                                feedback estimation based on unscented Kalman filter.
flow speeds were set as 5 % of the measured values.                 The traffic distribution model solves the data scarcity
                                                                problem and estimates OD flows accurately satisfying
                                                                the provided constraints.
Results                                                             The modified Payne model based on the Godunov
                                                                scheme perfectly relates the derived OD flows to link
The estimated OD flows in 15-minute intervals are               traffic measurements (link flow and flow speed). The
illustrated along with the measurements in figure 3. The        presence of Godunov scheme ensures the projection of
results are generally satisfactory, clearly showing the         real-traffic efficiently while attending the crucial
potential of the proposed OD estimation scheme.                 criticisms on Payne model.
    To closely observe the estimation potential between             Extracting link traffic measurement data and
the four schemes root-mean-square (RMSE) was                    implementing UKF interacted feedback process
computed and depicted in figure 4 (a). The definition of        maintains the OD flow estimation complies with real-
RMSE is as follows:                                             time measurements.
                    1 N                                              Another merit of the proposed scheme is less
        RMSE            ( zi  zi )2 ;
                   N  1 i 1
                                 ˆ                  (14)
                                                                computation time, obviously indicates that the scheme is
                                                                very much appropriate for online traffic control
where N denotes the total number of entities used for           applications.
comparison. z i and z i denote respectively the actual              Because of the time constraints some topics are left
value of the variable and the corresponding estimation          unattended. First is considering destination flows and
outputs.                                                        model parameters as unknown variables and performing
    The RMSE values also clearly show that the                  a simultaneous optimization of OD flows, traffic states
proposed method in this study (scheme A1) has good              and model parameters. Finally extending this approach
potential to estimate OD flows.                                 to urban traffic networks where route choices need to be
    Furthermore, to address the computational efficiency        considered.
related issues the time of computation was estimated,


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