Particle Filter for Estimating Freeway Traffic State in Beijing

Abstract

Freeway traffic state estimation is useful for intelligent traffic guidance, control, and management. The freeway traffic state is featured with rapid and dramatic fluctuations, which presents a strong nonlinear feature. In theory, a particle filter has good performance in solving nonlinear problems. This paper proposes a particle filter based approach to estimate freeway traffic state. The freeway link between the west of Peace Bridge and the west of San Yuan Bridge of the third ring in Beijing is used as the experimental object. According to the traffic characteristics and measurement mode of the link, the second-order validated macroscopic traffic flow model is adopted to set up the link model. The implementation steps of the particle filter for freeway traffic state estimation are described in detail. The estimation error analysis for the experiments proves that the proposed approach has an encouraging estimation performance.

1. Introduction

The increasing traffic flow is resulting in serious congestion of urban road network, which can decrease flow rate, delay travel time, increase fuel consumption and travel cost, and make negative environmental effect. Measures should be taken to alleviate traffic congestion. Traffic state estimation of freeway network is useful for traffic management, which involves estimating the traffic variables of the network based on available real-time traffic measurements [1]. A real-time Lagrangian traffic state estimator for freeway state has been proposed, which is considered to be more accurate and more computationally efficient than the Eulerian approach [2, 3]. However, the Lagrangian method is only appropriate for the sensing data obtained via GPS technology or other tracking devices providing position and velocity of individual vehicles. Chao Deng et al. present an approach using cluster analysis and multiclass support vector machine to estimate freeway traffic state. Historical traffic flow data are divided into clusters with different traffic states and the multiclass support vector machine is applied to identify the real-time traffic states [4]. Many previous researches in estimating aggregated traffic variables are based on applications of Kalman filter or extended Kalman filter [5–7]. Furthermore, most of them assume that noises obey Gaussian distribution and use linear models for state functions and observation functions. However, traffic state can fluctuate rapidly and dramatically in a short time, which indicates the strong nonlinear features of the freeway traffic state. Therefore, using Kalman filter may cause inaccurate estimation results and even divergent results. Particle filter is a powerful approximate solution to a general nonlinear problem or a non-Gaussian filtering problem [8, 9]. The basic idea of particle filter is that a posterior probability density function (PDF) of state can be represented by a set of particles with associated weights, and the estimation can be computed as the expected value of the discrete PDF [10, 11]. Currently only limited number of papers have discussed freeway traffic state estimation using particle filter algorithm. Mihaylova and Boel [12] proposed a particle filter (PF) to estimate freeway traffic based on the model of aggregated states and observations, and the investigations are conducted using the real traffic data from a Belgian freeway. Nicolae Marinica developed a particle filter (PF) state estimator using a platoon based model for urban traffic networks [13]. This paper uses the second-order validated macroscopic traffic flow model to evaluate a freeway link in Beijing according to its characteristics and proposes a particle filter method for estimating the freeway speed and density.

The rest of the paper is organized as follows. Section 2 describes a freeway traffic flow model in Beijing. Section 3 presents the design of the traffic state estimator based on the particle filter algorithm. In Section 4, several experiments are conducted to evaluate the particle filter estimation performance. Finally, the conclusions are summarized in Section 5.

2. Traffic Flow Modeling of a Freeway

2.1. Description of a Freeway

A freeway network can be represented as a directed graph. Bifurcations, junctions, on-ramps, and off-ramps of a freeway are represented as nodes, and a freeway stretch between two nodes is represented as a link in the graph [12]. A bidirectional freeway stretch is modeled as two opposite directional links. As shown in Figure 1, the freeway is divided into several links and each link is composed of several sections. In the paper, each link is assumed to have homogeneous geometric characteristics such as the number of lanes, slopes, and curvatures. Detectors are available only at some boundaries between sections. In Figure 1, 𝑞_𝑖 is the average number of vehicles at the boundary between sections 𝑖 and 𝑖+1, 𝑑_𝑖 is the average density in section 𝑖, 𝑣_𝑖 is the average speed in section 𝑖, 𝑠_𝑖 is the traffic flow into ramp 𝑖, and 𝑟_𝑖 is the traffic flow off ramp 𝑖. The whole state of the link 𝑚 at time 𝑘 is described by the vector 𝑥_𝑘=(𝑥^𝑇_1,𝑘,𝑥^𝑇_2,𝑘,…,𝑥^𝑇_𝑖,𝑘)^𝑇, and 𝑥_𝑖,𝑘=(𝑣_𝑖,𝑘,𝑑_𝑖,𝑘)^𝑇 is a local state vector of the 𝑖th section.

Figure 1 

Freeway links and measurement points.

The progression from one sampling time to the next sampling time is described by the update equation 𝑥_𝑘+1=𝑓𝑥_𝑘,𝑝_𝑘,𝑞ⁱⁿ_𝑘,𝑣ⁱⁿ_𝑘,𝑞^out_𝑘,𝑣^out_𝑘,𝜉_𝑘, (1) where 𝑝_𝑘 denotes the vector of all time-varying parameters such as road conditions and the number of available lanes, 𝑞ⁱⁿ_𝑘 is the number of vehicles entering Section 1 during the interval [𝑡_𝑘,𝑡_𝑘+1), 𝑣ⁱⁿ_𝑘 is the average speed of these vehicles, 𝑞^out_𝑘 specifies the outflow at the speed 𝑣^out_𝑘 from section 𝑁, and 𝜉_𝑘 is a disturbance vector reflecting random fluctuation in the traffic states and the model error.

Noisy measurements of the average speed of vehicles crossing the boundary between section 𝑖 and section 𝑖+1 during the time interval [𝑡_𝑘,𝑡_𝑘+1) together with noisy measurements of the mean density of the vehicles are collected as the measurement data. The observation equation is given as 𝑦_𝑘+1=ℎ𝑥_𝑠,𝜂_𝑘,𝑠∈𝑡_𝑘,𝑡_𝑘+1, (2) where 𝑥_𝑠 is the state at time 𝑠 and 𝜂_𝑘 is the measurement noise.

2.2. Macroscopic Traffic Flow Model of a Freeway

The second-order validated macroscopic traffic flow model is employed to describe the dynamic behavior of traffic flow along a freeway stretch in terms of appropriate aggregated traffic variables such as traffic density, space average speed, and traffic flow [14]. Generally, a freeway stretch is divided into a number of 𝑁 sections. Assume that the length of each section is 𝐿_𝑖, 𝑖=1,2,…,𝑁, the time discretization is based on a time step 𝑇, and 𝑘(𝑘=0,1,2,…) is a discrete time index. For the section 𝑖, the stochastic nonlinear difference equations based on the second-order macroscopic traffic flow model are described as follows [14]: 𝑑_𝑖(𝑘+1)=𝑑_𝑖(𝑘)+𝑇𝜆_𝑖𝐿_𝑖×𝑞_𝑖−1(𝑘)−𝑞_𝑖(𝑘)+𝑟_𝑖(𝑘)−𝑠_𝑖(𝑘,𝑠_𝑖)(𝑘)=𝛽_𝑖(𝑘)𝑞_𝑖−1(𝑘),𝑣_𝑖(𝑘+1)=𝑣_𝑖(𝑘)+𝑇𝜏𝑣𝑑_𝑖(𝑘)−𝑣_𝑖(𝑘+𝑇)𝐿_𝑖𝑣_𝑖(𝑘)𝑣_𝑖−1(𝑘)−𝑣_𝑖(𝑘−)𝑛𝑇𝜏𝐿_𝑖𝑑_𝑖+1(𝑘)−𝑑_𝑖(𝑘)𝑑_𝑖(𝑘)−+𝜅𝛿𝑇𝜆_𝑖𝐿_𝑖𝑟_𝑖(𝑘)𝑣_𝑖(𝑘)𝑑_𝑖(𝑘)+𝜅+𝜉^𝑣_𝑖(𝑘),𝑣(𝑑)=𝑣_𝑓−1exp𝑎𝑑𝑑_cr^𝑎,𝑞_𝑖(𝑘)=𝑑_𝑖(𝑘)𝑣_𝑖(𝑘)𝜆_𝑖+𝜉^𝑞_𝑖(𝑘). (3)

In (3), the variables and parameters are denoted as follows:(1)traffic density 𝑑_𝑖(𝑘) (in veh/km/lane) is the number of vehicles in section 𝑖 at time 𝑘𝑇 divided by the section length 𝐿_𝑖 and the number of lanes 𝜆_𝑖,(2)space average speed 𝑣_𝑖(𝑘) (in km/h) is the average speed of the vehicles in section 𝑖 at time 𝑘𝑇,(3)traffic flow 𝑞_𝑖(𝑘) (in veh/h) is the number of vehicles leaving section 𝑖 during the interval [𝑘𝑇,(𝑘+1)𝑇), divided by 𝑇,(4)𝑟_𝑖(𝑘) is the on-ramp inflow and 𝑠_𝑖(𝑘) is the off-ramp outflow (both in veh/h) in section 𝑖,(5)𝜏, 𝑛, 𝜅, 𝑎, and 𝜆 are the model parameters which are the same given values for all sections,(6)𝑣_𝑓 denotes the free flow speed, 𝑑_cr is the critical density, and 𝑎 is the exponent of the stationary speed equation,(7)𝜉^𝑣_𝑖(𝑘) and 𝜉^𝑞_𝑖(𝑘) reflect the model error, 𝜉^𝑣_𝑖(𝑘) denotes the zero-mean Gaussian white noise acting on the empirical speed equation, and 𝜉^𝑞_𝑖(𝑘) denotes the zero-mean Gaussian white noise acting on the approximate flow equation.

The detectors in the link entrance or exit can collect the speed data and density data. The speed measurement function is 𝑍^𝑣_inorout(𝑘)=𝑣_inorout(𝑘)+𝜉^𝑣_inorout(𝑘), (4) where 𝑍^𝑣_inorout(𝑘) is the measurement value of the average speed through entrance or exit during the interval [(𝑘−1)𝑇,𝑘𝑇) and 𝜉^𝑣_inorout(𝑘) is the measurement noise of the speed.

The measurement function of the density for the link entrance or exit is 𝑍^𝑑_inorout(𝑘)=𝑑_inorout(𝑘)+𝜉^𝑑_inorout(𝑘), (5) where 𝑍^𝑑_inorout(𝑘) is the measurement value of the mean density through entrance or exit during the interval [(𝑘−1)𝑇,𝑘𝑇) and 𝜉^𝑑_inorout(𝑘) is the measurement noise of the density.

3. Particle Filter for Freeway Traffic State Estimation

Nonlinear characteristics exist in the freeway traffic state, which makes it difficult to estimate the traffic state. Since particle filter is able to solve a general nonlinear problem in theory, particle filter is well studied to estimate the traffic state based on the second-order validated macroscopic traffic flow model.

3.1. Particle Filter Theory

The discrete-time stochastic model of a dynamic system is described as follows: 𝑥_𝑘=𝑓_𝑘𝑥_𝑘−1,𝑤_𝑘−1,𝑦_𝑘=ℎ_𝑘𝑥_𝑘,𝑣_𝑘, (6) where 𝑦_𝑘 is the observation vector at time 𝑘, 𝑥_𝑘 represents the state vector at time 𝑘, 𝑓_𝑘 is the nonlinear state transition function, and ℎ_𝑘 is the nonlinear observation function. The stochastic processes 𝑤_𝑘 and 𝑣_𝑘 represent the state noise process and the measurement noise process, respectively. The available information at time 𝑘 is the set of measurements 𝑍_𝑘={𝑦_𝑖∶𝑖=1,2,…,𝑘}. According to the Bayesian theory, all state information can be obtained from the posterior state distribution. Within the Bayesian framework, particle filter is used to sequentially update a priori knowledge about predetermined state vector 𝑥_𝑘 by using the measurements data 𝑍_𝑘. Suppose that the initial probability density function of the state vector is 𝑝(𝑥₀∣𝑍₀)=𝑝(𝑥₀); then we have [9, 10] 𝑝𝑥_𝑘∣𝑍_𝑘−1=𝑝𝑥_𝑘∣𝑥_𝑘−1𝑝𝑥_𝑘−1∣𝑍_𝑘−1𝑑𝑥_𝑘−1𝑝𝑥_𝑘,∣𝑍_𝑘=𝑝𝑦_𝑘∣𝑥_𝑘𝑝𝑥_𝑘∣𝑍_𝑘−1𝑝𝑦_𝑘∣𝑍_𝑘−1𝑝𝑦_𝑘,∣𝑍_𝑘−1=𝑝𝑦_𝑘∣𝑥_𝑘𝑝𝑥_𝑘∣𝑍_𝑘−1𝑑𝑥_𝑘. (7)(8)(9)

Particle filter uses Monte Carlo techniques and sequence importance sampling (SIS) methods to solve (7). The posterior PDF is computed based on 𝑁 samples from an important distribution function, as follows, 𝑝𝑥_𝑘∣𝑍_𝑘=^𝑁_𝑖=1𝑤^𝑖_𝑘𝛿𝑥_𝑘−𝑥^𝑖_𝑘,𝑤^𝑖_𝑘=𝑤^𝑖_𝑘∑^𝑁_𝑖=1𝑤^𝑖_𝑘,𝑤^𝑖_𝑘=𝑤^𝑖_𝑘−1𝑝𝑦_𝑘∣𝑥^𝑖_𝑘𝑝𝑥^𝑖_𝑘∣𝑥^𝑖_𝑘−1𝑞𝑥^𝑖_𝑘∣𝑥^𝑖_𝑘−1,𝑦_𝑘, (10)(11)(12) where 𝑤^𝑖_𝑘 is the weight of each particle and satisfies 0≤𝑤^𝑖_𝑘≤1 and 𝑞 is an important distribution function.

Particle filter has good effects in solving both nonlinear and non-Gaussian applications, which is suitable for estimating the nonlinear freeway state.

3.2. Application to Freeway Traffic State Estimation

The application steps of freeway traffic state estimation based on particle filter are as follows.

(1) Initialization. Set the iteration variable 𝑘=0. The particle set {𝑥^𝑖₀,𝑤^𝑖₀} (𝑖=0,1,2,…,𝑁, 𝑁 is the number of particles) is obtained based on the initial distribution 𝑝(𝑥₀).

(2) Particle Generation. For 𝑖=1,2,…,𝑁, 𝑥^𝑖_𝑘 is sampled from 𝑝(𝑥_𝑘∣𝑥^𝑖_𝑘−1,𝑍_𝑘) according to (3).

(3) Weight Computation. Equation (12) is complex to compute the weight, and usually the weight is updated by the following equation when the new measurement data are obtained by the detector in the section: 𝑤^𝑖_𝑘=𝑤^𝑖_𝑘−1×𝑝𝑦_𝑘∣𝑥^𝑖_𝑘. (13) The normalized weight 𝑤^𝑖_𝑘 is computed using (11).

(4) Resampling. Drawing a random sample 𝑢_𝑖 from the uniform distribution over (0,1]. If 𝑢_𝑖 satisfies ∑^𝑚−1_𝑗=1𝑤^𝑗_𝑘<𝑢_𝑖≤∑^𝑚_𝑗=1𝑤^𝑗_𝑘, the 𝑚th particle is resampled as follows: 𝑥^𝑖_𝑘=𝑥^𝑚_𝑘,𝑤^𝑖_𝑘=𝑤^𝑚_𝑘. (14)(5) State Estimation. The approximate posterior PDF is computed using (10); then the state is estimated by 𝑥_𝑘=^𝑁_𝑖=1𝑤^𝑖_𝑘𝑥^𝑖_𝑘. (15)(6) 𝑘 Is increased by Itself. If the end condition of algorithm is not satisfied, the algorithm goes to step (2) again; otherwise, it ends.

4. Freeway State Estimation Experiments

4.1. Experimental Object

A freeway between the west of Peace Bridge and the west of San Yuan Bridge of third ring in Beijing is used as an experimental link. The link has no ramps. The length of the link is 1476 m. The link is composed of 3 sections with 3 lanes and the length of each section is 556 m, 475 m, and 445 m, respectively. The measurement detectors are located in the boundaries between the first section and the third section. The detectors can collect the traffic flow, speed, and density of the link every 2 min.

One example of the entrance traffic state during 3 hours between 8:00 and 11:00 AM on July 23, 2010, is shown in Figure 2. We know that the traffic flow varied between 40 veh/2 min and 60 veh/2 min, the speed varied between 32 km/h and 50 km/h, and the density varied between 23 veh/km and 58 veh/km. In most of the time, the traffic state varied smoothly except for a rapid change at 8:20 AM.

Figure 2 

Traffic state example of exit in the link.

4.2. Experiments and Analysis

The state vector 𝑥_𝑘 and measurement vector 𝑍_𝑘 are computed, respectively, as follows: 𝑥_𝑘=𝑣₁(𝑘),𝑑₁(𝑘),𝑣₂(𝑘),𝑑₂(𝑘),𝑣₃(𝑘),𝑑₃(𝑘)^𝑇,𝑍_𝑘=𝑍^𝑣_in(𝑘),𝑍^𝑑_in(𝑘),𝑍^𝑣_out(𝑘),𝑍^𝑑_out(𝑘). (16) The parameters of traffic model in Section 2 are set as follows.

According to the experimental object in Section 4.1, each section of the freeway has 3 lanes, and the length of each section is about 500 m. The data detection period is 2 min. In daytime, the free flow speed of the freeway is about 80 km/h and the critical density is about 50 veh/km. Therefore, some of the parameters can be set as 𝜆_𝑖=3, 𝐿_𝑖=500 m, 𝑇=120 s, 𝑣_𝑓=80 km/h, and 𝑑_cr=50 veh/km. Based on the prior knowledge, other parameters can be set as 𝜏=20 s, 𝑛=45 km²/s, and 𝜅=15 veh/km/lane.

According to the data analysis in Figure 2, the initial state is set to be 𝑥₀=[42,40,50,35,48,35]^𝑇. (17) The number of particles 𝑁 is set to be 500.

The freeway speed and density are estimated using the particle filter algorithm in the Section 3. The estimation results for the speed and the density of each section are, respectively, shown in Figures 3 and 4. In the horizontal axis of each figure, the step time interval is 2 min which is equal to the data collecting interval of the detectors. According to Figures 3 and 4, the estimation trend is similar to the true trend.

Figure 3 

Speed estimation for three sections.

Figure 4 

Density estimation for three sections.

The indices of relative error (RE), mean square error (MSE), and root mean square relative error (RMSRE) are utilized to evaluate estimation performance.

The indices of RE, MSE, and RMSE are, respectively, defined as follows: 𝑥RE=^𝑖−𝑥_𝑖𝑥_𝑖×100%,MSE=1𝑛^𝑛_𝑖=1𝑥_𝑖−𝑥_𝑖²,RMSRE=1𝑛^𝑛_𝑖=1𝑥_𝑖−𝑥_𝑖𝑥_𝑖², (18) where 𝑥_𝑖 represents the true state and 𝑥_𝑖 represents the estimation state.

The REs of speed estimation and density estimation for each section are, respectively, shown in Figures 5 and 6. In the horizontal axis of each figure, each step time interval is 2 min.

Figure 5 

REs of speed estimation for three sections.

Figure 6 

REs of density estimation for three sections.

According to Figures 5 and 6, the RE is in [−10%, +10%] and varies smoothly in most of the time. Only at several time intervals, the absolute RE is more than 15%.

The MSE and RMSRE of speed estimation and the MSE and RMSRE of density estimation for each section are shown in Table 1. The maximum MSE difference of speed estimation among the sections is 1.8675 and the maximum RMSRE difference of speed estimation among the sections is 0.0156. The maximum MSE difference of density estimation among the sections is 1.0273 and the maximum RMSRE difference of density estimation among the sections is 0.0062. The mean MSE of the speed estimation is 3.7136, the mean RMSRE of the speed estimation is 0.0750, the mean MSE of the density estimation is 2.6859, and the mean RMSRE of the density estimation is 0.0751. Therefore, the estimation effect is desirable.

Table 1 

MSE and RMSRE of estimation of density and speed.

In theory, the resampling procedure in particle filter has a complexity of 𝑂(𝑁); 𝑁 is the number of sample variables generated from the uniform distribution [11]. In fact, the computation time can be reduced in the particle filter implementation by using the multithread programming technology.

5. Conclusion

This paper proposes an approach to estimate the freeway traffic state based on the particle filter algorithm. The freeway traffic is modeled by the second-order validated macroscopic traffic flow model. A freeway link between the west of Peace Bridge and the west of San Yuan Bridge of third ring in Beijing is used as an experimental object and real traffic data are used in the experiments. The mean square error of the speed estimation is 3.7136 and the mean square error of the density estimation is 2.6859. The results suggest that particle filter is valid and effective in freeway traffic state estimation. In addition, the proposed approach for traffic state estimation is modular, and therefore different traffic models can be used in different sections on a freeway link. In the future research, we will use parallel computation to improve the performance of the particle filter for estimating freeway state.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is sponsored by the National 863 Program (2011AA110303) and the Fundamental Research Funds for the Central Universities (2013JBM052).