VIEWS: 9 PAGES: 4 CATEGORY: Technology POSTED ON: 2/6/2010 Public Domain
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 (3) 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 T 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 t i 1 (1) q j 1 t 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 1 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 ) , ( i1 ) ] t (7) 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 ( i1 ) 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: t 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 t t terms of shortest travel time ( u rs ) between OD pair rs, it 1 it (Qit1 Qit rit sit ) ; (9) xi which is given by t t t t vt it1 it ; (10) t vit 1 vit [Ve ( it ) vit ] vi [vi 1 vit ] srs u rs 1 t (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 2 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 ˆt 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. t 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. uncorrelated. 3 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. ˆ 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, 4