Freeway work zone can easily lead to traffic congestion which has detrimental effects on travel time and safety. By using the detected traffic states near work zone area, variable speed limit (VSL) control system has been widely studied to improve traffic mobility and safety. However, occurrence of sensor fault can cause traffic state deviation and system degradation. Therefore, this study presents a fault-tolerant VSL control system for freeway work zone. A traffic flow model was first designed to analyze the traffic dynamics. Then a sliding mode controller for VSL was designed based on the previous study. By comparing the estimated traffic states from Kalman filter with the observed states from the observer, the developed fault diagnosis can detect the sensor fault and reconfigure the controller. The proposed system was evaluated under a realistic freeway work zone environment in traffic simulator SUMO. The results show the proposed system can achieve fault tolerance and improve traffic mobility and safety under faultfree and sensor-fault scenarios.
al., 2006). Since the physical redundancy demands additional components to be installed, it is
hardly affordable to use such fault tolerance due to its extra cost in a large scale system like
freeway in this study. However, the analytical redundancy is analyzed and extracted from the
mathematical system model, whereby the dependability of a system is improved with little
additional cost. Therefore, the analytical redundancy is mainly discussed and studied.
Various studies have developed traffic data imputation methods to overcome the issue of
traffic sensor faults. A tensor decomposition based method was developed to recover the
missing traffic data
The rest of this paper is organized as follows. The system framework is first described. Based on the designed traffic flow model, a sliding mode controller is introduced for VSL control. Then Kalman filter is employed to estimate the traffic states. After the design of the observer, fault diagnosis scheme is discussed. Afterwards, the developed system is evaluated with analytical results. Finally, conclusions and future work are discussed.
The framework of the designed fault-tolerant VSL control system is shown in Figure 1. The sliding mode controller is designed to generate the speed limit signal in order to track the traffic state reference and improve traffic condition near work zone area. Then this speed limit signal is incorporated in the traffic flow model to affect the evolution of the traffic states. With traffic measurements from stationary sensors and probe sensors, Kalman filter is employed to improve the accuracy of the traffic states estimation. Traffic measurements are also sent to the designed observer to generate the observed traffic states as the redundant information. Then both estimated traffic states from Kalman filter and observed traffic states from the observer are fed into the fault diagnosis to detect whether a sensor fault occurs and reconfigure the controller accordingly. The details of each component in Figure 1 are presented in the following sections.
Traffic State Reference
Sliding Mode Controller
Speed Limits
Control Signal Model Reconfiguration
Signal
Traffic Flow
Model Fault Diagnosis
Traffic State Measurement
Traffic State
Estimation Kalman
Filter
Observer
The layout of freeway work zone with lane closure is shown in Figure 2. The mix traffic flow with conventional and connected vehicles moves towards the right side. Connected vehicles used in this study have the ability to transmit their positions and speeds during the travel while conventional vehicles do not have the communication ability. The driver behavior between conventional vehicles and connected vehicles is assumed to be the same. Traffic states are detected using stationary sensors and probe sensors. Stationary sensors are installed at fixed locations to detect traffic states and connected vehicles are utilized as probe sensors. In the case of stationary sensors, two traffic sensors TS 1 and TS 2, which are located immediate upstream of the work zone and within work zone area respectively, are used. It can be seen that congestion can easily occur when traffic demand exceeds the work zone capacity.
TS 1
TS 2
Emergency
Lane Work Zone Conventional
Vehicle Connected Vehicle Traffic Sensor To analyze the traffic dynamics near work zone area, the layout of freeway work zone area is partitioned into multiple segments as shown in Figure 3. The traffic flow rate qi and qi1 are the inflow and outflow for segment i. The length of each segment, average speed and density in segment i are expressed as Li , vi and i respectively. Traffic sensor TS 1 and TS 2 are located inside acceleration zone and work zone respectively. One speed limit sign is installed at the beginning of the VSL control zone to control vehicle speed within VSL control zone. Another speed limit sign is installed at the start of the acceleration zone to allow vehicles to accelerate and travel through the work zone area with maximum free flow speed. In this study, we assume no congestion happens downstream of the work zone.
Based on the conservation law
Li
where Ts is the sample time interval and k stands for the discrete time step k. Since the
objective of VSL controller is to control the traffic density of the acceleration zone
, ( j 2 (k))} 1 Ts v2 (k) 2 (k 1) L2 3(k 1) Ts v2 (k) L3
1 LT30s v3(k) 32((kk)) LT2s 0q2u (k) 12((kk))
1 0
32((kk 11)) 0 1 LT3s v3(k) 32((kk)) 2
Ts (q2u (k) Cb )
L
Ts Cb
L3
12((kk))
Equation (4) and (5) are derived under without capacity drop and with capacity drop scenarios
respectively. The average speed v2 (k ) and v3 (k ) are detected by connected vehicles.
Meanwhile, the stationary sensors TS 1 and TS 2 can be used to detect the traffic flow. Thus
the measurement equation is derived as
where and j represent the backward wave speed and jam density respectively. u1(k ) is
the speed limit of VSL control zone. Cb is the work zone bottleneck capacity which equals to
2/3 of full road capacity C as one of three lanes is closed shown in Figure 3. The capacity
drop phenomenon occurs when a queue forms upstream of the work zone
The sliding mode controller, which is a non-linear control strategy for VSL control, can
generate switching control signal to drive the traffic state to the desired equilibrium state with
certain convergence rates. The objective of the designed controller is to stabilize the traffic
density 2 at the bottleneck critical density cb Cb / vf where v f is the free flow speed.
Through this stabilization, the capacity drop is avoided and the maximum flow rate can be
achieved. The designed controller is based on the previous study
s(k) c[cb 2 (k)]
where c is a constant nonzero parameter. The reaching process, which is designed as Equation
(9) is utilized to drive the traffic state towards the surface
s(k 1) s(k ) Ts sgn(s(k )) Tsqs(k ) where both and q are positive constant parameters. The term Tsqs(k ) can make the traffic state move towards the surface at exponential convergence rate, while the term Ts sgn(s(k )) can make the traffic state converge to equilibrium state with finite time steps. Therefore, the traffic density 2 (k 1) can be derived from the perspective of the controller design by combining the Equation (8) at time step k+1 with Equation (9). On the other hand, the traffic density 2 (k 1) can also be obtained from the perspective of the traffic dynamics in Equation (3) which contains the control signal u1(k ) . Thus combining these two densities 2 (k 1) gives a function g of speed limit control signal u1(k ) as (7) (8) (9) u1(k) g( 2 (k), v2 (k), s(k)) (10) The stability of the designed controller is guaranteed with the condition that 0 , q 0 and 2 Tsq
0 which is proved in the previous study
To apply the speed limit control in real world, three speed constraints are considered: 1) only discrete speed limits between maximum legitimate speed limit vmax and minimum speed limit vmin with the increment speed v are displayed; 2) the continuous speed limit is rounded up to its closed discrete speed limit; 3) the maximum speed difference is limited to vmax between two consecutive control time step.
Kalman filter is a dynamic estimation algorithm with time update and measurement update
P(k 1| k) A(k)P(k) A(k)T Q(k) K (k) P(k 1| k)C (k)T (C (k)P(k 1| k)C (k)T R(k))1 x(k 1) x(k 1| k) K (k)( y(k) C(k) x(k 1| k))
P(k 1) (I K (k)C(k))P(k 1| k) With the one step prediction of state x(k 1| k) and state covariance P(k 1| k) , the filter gain K (k ) is updated. By incorporating the error between the measurements and one step prediction, the state x(k 1) along with the state covariance P(k 1) at next time step are updated. Then this recursive process is performed to update and estimate the traffic density with new measurements at each time step.
The observer can provide the redundant traffic state to achieve fault tolerance. It can be seen that traffic density 2 (k ) is critical to the design of sliding mode controller. Consequently, a faulty traffic sensor TS 1, which can provide the measurement of 2 (k ) via the detection of traffic flow, can greatly affect the VSL control. Therefore, the observer should provide the observed traffic density 2o (k) when traffic sensor TS 1 fails.
Observability. When TS 1 fails, Equation (6) is modified as 3(k) C o (k) x(k) 2 (k) where C o (k ) [0 1] and 2 (k ) is the Gaussian noise with zero mean and variance R2 (k ) . Then the observability matrix Ono,drop and Odrop for without capacity drop and with capacity drop scenarios respectively are derived as When v2 (k ) 0 , Ono,drop has full rank. Accordingly, traffic density 2 (k ) is observable when capacity drop does not occur, while 2 (k ) is non-observable with capacity drop. Observer Design. In the case of the scenario without capacity drop, since 2 (k ) is observable, the observed density 2o (k) can be obtained using the Kalman recursive algorithm which acts as an observer instead of a filter. By replacing C (k) , y(k) and R(k) in Equation (13)-(15) with C o (k ) , 3 (k ) and R2 (k ) respectively, the Kalman recursive process is performed using (11) (12) (13) (14) (15) (16) Equation (11)-(15) to obtain the observed o (k) . When capacity drop occurs, the recursive 2 process cannot be used as 2 (k ) is non-observable. However, the error of one-step prediction does not diverge with capacity drop. Therefore, an open loop estimator with one-step prediction using Equation (11) is utilized to obtain the observed traffic density o (k) . 2
The observer-based method is employed to design the fault diagnosis. By utilizing the
observer discussed in Section 2.4, the designed fault diagnosis can detect the sensor fault, and
then reconfigure the VSL controller when a fault occurs. This study focuses on the stationary
sensor fault, specifically traffic sensor TS 1 which is critical to the design of VSL controller.
There are different types of sensor faults. For example, the fault with the detection of zero
flow rate is the leading fault among the malfunctions in loop detectors
After the detection of a sensor fault, the fault diagnosis can reconfigure the controller. Without sensor fault, the estimated density can be used in the design of VSL controller. However, when a sensor fault occurs, the estimated density will greatly deviate from the actual density. To maintain the control system performance, the fault diagnosis can reconfigure the controller by replacing the estimated density with the observed density.
The developed fault-tolerant VSL control for freeway work zone was evaluated on a 4.8 km
(3 mi) freeway segment of I-15S in California, US. A work zone was set up from State PM
(postmile) 40.2 to State PM 37.6 on July 22, 2016. This freeway segment with the work zone
is shown in Figure 4. The blue line represents the simulated freeway segment, while the red
line shows part of the work zone where one of three lanes was closed. The freeway network
was built in traffic simulator SUMO. The microscopic model was calibrated and validated
using realistic traffic measurements from California freeway database
The developed fault-tolerant control system was evaluated under three scenarios: 1) without
control; 2) with fault-tolerant control but no sensor fault occurs; 3) with fault-tolerant control
but TS 1 fails during the simulation. The simulation was first run for 5 minutes as the
warmup period with low traffic demand. This 5-minute traffic data was discarded. Then 1 h
simulation was conducted. During this 1 h simulation, an average low demand 1500 veh/h
was first generated until the time 400 s, then there was an average high demand 3600 veh/h
lasting for 1100 s, followed by low demand 1500 veh/h for the rest 2100 s. An artificial
sensor fault occurred at the time step 1200 s under scenario 3.
The work zone throughputs under three scenarios are illustrated in Figure 5. The actual
throughput in dash line is collected by TS 2 every sample time inteval, while the throughput
trend in solid line is presented using moving average technique. Without VSL control, the
capacity drop occurs in Figure 5a with a throughput of about 3000 veh/h during the high
demand. In contrast, throughputs under scenario 2 and 3 stay around 3200 veh/h which is the
work zone capacity under high demand. Though the throughput slightly drops for a short time
near time step 900 s due to the initiation of VSL, the overall throughput succesfully maintains
at the work zone capacity. The similar performance between Figure 5b and 5c after 1200 s
when the sensor fault occurs shows the effectiveness of fault-tolerant control.
The residual and traffic density at acceleration zone under scenario 2 and 3 are presented in
Figure 6 and 7 respecitvely. It can be seen that the developed system shows the ability to
achieve density estimation and fault tolerance. Under scenario 2, the residual stays below the
threshold all the time without false alarm. The density estimation is compared with the ground
truth measurement from the simulator in Figure 6b. An accurate density estimation is
achieved with the RMSE of 3.9 veh/km. In Figure 7a, the residual exceeds the threshold at the
time step 1200 s when the sensor fails. Then the density estimation greatly deviates from the
measurment in Figure7b. However, the observed traffic density in blue line comes into effect
and replaces the corrupted density estimation. Thus the fault tolerance can be achieved.
The travel time T and probablity of time-to-collision p are presented in Table 1. Only the
travel time upstream of the work zone is considered. 1.5 s is used as the minimum time to
avoid the collision
To ensure the effectiveness of VSL control and avoid system degradation when a sensor fault occurs, a fault-tolerant VSL control system for freeway work zone is developed in this study. Accurate traffic density estimation is achieved by employing Kalman filter. Meanwhile, the sliding mode controller for VSL control can avoid the capacity drop and improve traffic mobility and safety for freeway work zone. Furthermore, the developed fault diagnosis can not only detect a sensor fault but also reconfigure the controller accordingly. The designed 9 fault-tolerant VSL control shows the ability to consistently improve the traffic condition near freeway work zone area even with a sensor fault.
In this study, only the stationary sensor fault is considered. However, the faults of connected vehicles such as the communication delay and corrupted samples can also affect the VSL control performance. Probe sensor faults will be considered in the future. Meanwhile, the impacts of different connected vehicle market penetration rates will also be studied. Mobile applications as another type of probe sensors will be considered for fault detection as well.