**1. Introduction**

Real-time traffic estimation has received increased attention with the introduction of advanced applications and technologies such as intelligent transportation systems (ITSs). Adaptive traffic signal controllers require real-time traffic state estimation to improve intersection performance, as real-time estimation plays a major role in capturing variations in traffic behavior (e.g., nonrecurrent changes). As inputs to traffic signal controllers, traffic state variables (e.g., travel time and traffic density) assist with green time allocation and help to enhance intersection performance by reducing traffic delays, vehicle emissions, and fuel consumption [1,2]. Several estimation techniques have been developed to estimate traffic state variables [3–7]. In some previous studies, the traffic state system has been treated as a linear system [4,7–10]. Other studies have considered the system as nonlinear [3,11–13]. For linear system models, a Kalman filter (KF) has been widely deployed to produce accurate estimates [4,7,10,14] due to its simplicity and applicability in the field. The KF assumes linear system transitions with a

Gaussian distribution for the probability density function (PDF) of the system and measurement noise. For nonlinear system models, an extended KF (EKF) has been utilized in estimation [11,15,16]. The EKF also assumes that the PDF distribution is Gaussian. The EKF is derived by linearizing the system using a Taylor series expansion by calculating the Jacobian expression. However, it was found that use of the EKF approach is only valid if the system is near linearity during the updating time [17], and thus, large errors may result from linearization. In addition, the task of deriving the Jacobian matrices may cause implementation difficulties [18]. A more robust nonlinear approach is a particle filter (PF), which has been frequently employed in the literature to handle nonlinear dynamic problems [12,19,20]. The PF approach is a Monte Carlo sequential solution that deals with nonlinear system transitions without the assumption of the PDF noise distribution [21,22]. In this paper, a PF approach is developed to estimate the traffic stream density along signalized intersection approaches using only connected vehicle (CV) data. Moreover, the paper compares the performance of the PF to the KF and adaptive KF (AKF) approaches.

Traffic density is defined as the number of vehicles per unit length on a specific roadway segmen<sup>t</sup> [23]. Estimating the traffic stream density is critical in the development of effective traffic controllers [24]. For instance in the case of freeways, identifying bottleneck locations in the early stages is critical in developing congestion mitigation strategies that include ramp metering, variable speed limits, and traffic routing. For signalized segments, the traffic density measures are crucial for either traffic signal performance [25–27] or traffic signal optimization [28–30]. Hence, traffic density measures must be precisely estimated to represent traffic demands at each signalized intersection approach. Once accurate measurements are obtained, efficient adaptive traffic signal controllers can be developed. However, determining traffic density is not a trivial task and cannot be directly measured in the field since it is a spatial measurement. Consequently, traffic stream density is typically based on estimations.

Previous research has utilized different data sources, such as stationary sensors (e.g., loop detectors), fused data (combining two distinct data sources), and CV data to estimate traffic stream density. Traffic density estimates can be measured using video detection systems, but this is difficult due to the high cost of the infrastructure and the limited visibility of roadway segments [4]. Time-occupancy measurements from loop detectors are used as an alternative data source to estimate the traffic density [31]. However, time-occupancy measurements only represent the temporal density estimates around the location of the detector. A recent study introduced a relationship between time-occupancy and space-occupancy to estimate traffic density by dividing the link into small segments and installing detectors on all of the small segments [32], but the installation cost is high. A more common way of estimating the traffic density is the use of the traffic flow continuity equation (input–output approach), which considers two traffic counting stations, one at the entrance and the other at the end of the link [33]. Vigos et al. [4] proposed a robust linear KF approach with at least three loop detectors to estimate the traffic density along signalized approaches. However, the implementation cost is high. Another study [34] employed two conventional loop detectors to estimate the traffic density using the flow continuity equation. The two loop detectors provide the estimation model with traffic flow and occupancy data. Bhouri et al. [35] proposed a KF approach to estimate the traffic stream density along a freeway segmen<sup>t</sup> using both loop detectors and a recorded film [35]. One commonality about the use of fixed sensors is that they are subject to detection failures and thus always produce errors in their data [36,37].

Recent research has fused different data sources to estimate the traffic stream density along certain roadway sections, increasing the accuracy of the estimate over using just one data source [7,14]. Many works have employed the KF approach [14,38,39]. For instance, traffic flow data at the entrance and the exit of the roadway section observed from stationary sensors together with CV data were used to estimate the traffic density [38]. The CV data provided travel-time measurements to correct the prior estimate from the state equation. Another study has utilized fused loop and CV measurements to estimate the traffic density in a freeway section [19]. In that study, the authors derived the estimation model using the PF estimation approach, considering two sources of measurements: (1) loop detectors, and (2) fusing loop detectors and CVs. They obtained a 30% reduction in the mean absolute percentage error from the fused measurements compared to the measurements from loop detectors, demonstrating that more data sources produce more-accurate outcomes. However, the use of different data sources requires more computational cost in both time and memory as the data include both trivial (data that are not needed) and nontrivial (data that are needed) information.

Limited studies have used CV data as the only source of inputs to estimate the traffic stream density [7,9,10]. These studies developed the linear KF estimator approach. The CV data used were the number of CVs at the entry and at the exit of the tested roadway section, in addition to the travel time experienced for the CVs to traverse the tested section. Moreover, Aljamal et al. [7] demonstrated that treating the estimation interval time as a variable instead of a fixed value is mandatory when dealing with only CV data, as the variable approach always ensures that sufficient information is gathered from the CVs in every estimation interval. This approach enhances the accuracy of the estimation, especially for the scenarios with low CV level of market penetration (LMP) rate. The estimation time interval for this study is therefore defined as the time when an exact number of CVs (i.e., 5 vehicles) reach the end of the tested link.

Several researchers have employed the PF approach to improve traffic stream estimates for different transportation applications, including traffic flow [12,13], travel time [3,20], and traffic speed [40]. In one study, magnetic loop detectors were placed at the boundaries of the tested freeway section to estimate the traffic flow, and a PF estimation approach was developed using traffic flow and speed measurements [12]. In another study, Mihaylova et al. [13] developed two nonlinear approaches, an unscented KF and a PF, to produce real-time traffic flow estimates in a freeway network using data from stationary sensors. They found that the PF approach outperformed the unscented KF. Chen et al. [20] proposed a time series speed equation to estimate traffic speed. They claimed that the traffic system is nonlinear and thus presented two nonlinear approaches, a PF and an ensemble KF, using available speed measurements from loop detectors. They found that the PF approach is more accurate than the ensemble KF. Another study developed a PF estimation approach for travel-time predictions using real-time and historical data [3]. They used the historical data to generate particles as opposed to using a state-transition model. In addition, a comparison between the PF, KF, and k-nearest neighbor estimators found that the PF is the most accurate approach. CV data were employed to estimate the traffic speed and flow using the PF approach [40]. In that study, each link in the network was assumed to have base stations to retrieve and transfer the data. Results found that other data sources (e.g., loop detectors) should be incorporated with CVs to enhance the estimation performance. However, our recent study developed a KF approach, showing that the use of CV data alone is sufficient to obtain accurate results [7].

In summary, the existing literature shows that the PF has been widely used to address nonlinear systems and has been proven to outperform other nonlinear estimation techniques; however, to our best of knowledge in the application of traffic stream density estimation, only a few studies have applied the PF approach using data from stationary sensors and fusing data from different sources. In addition, no comparison between the PF and the linear KF has been reported. Therefore, the PF was adopted in this study. The primary objective of this study is to develop a nonlinear PF estimation approach to estimate the traffic stream density based solely on CV data on signalized approaches. Subsequently, we compare the PF approach to linear estimation approaches—namely, KF and AKF—to identify the best approach for the application of the traffic density estimation, given that no comparison has been reported in the literature between these filtering techniques. Consequently, this research will recommend a specific approach to estimate the traffic stream density. The proposed three approaches are employed to estimate the vehicle counts based solely on CV data. In addition, this study also investigates the sensitivity of the proposed estimation approaches to several factors, such as the LMP rate of the CVs, the initial conditions, and the number of PF particles.

The paper is organized as follows: Section 2 describes the problem formulation and the estimation approaches. Section 3 discusses the findings from applying the estimation approaches. Section 4 includes the conclusions of the study and the proposed future work.

#### **2. Problem Formulation and Estimation Approaches**

First, Section 2.1 formulates the research problem using a state-space model. Then, three different estimation approaches are described: the PF (Section 2.2.1), the KF (Section 2.2.2) and the AKF (Section 2.2.3).
