Optimal Speed Control of Automated Vehicles at Speed Reduction Zones
This article addresses the problem of controlling the speed of a number of automated vehicles before they enter a speed reduction zone on a freeway. We formulate the control problem and provide an analytical, closed-form solution that can be implemented in real time. The solution yields the optimal acceleration/deceleration of each vehicle under the hard safety constraint of rear-end collision avoidance. The effectiveness of the solution is evaluated through a microscopic simulation testbed and it is shown that the proposed approach reduces significantly both fuel consumption and travel time. In particular, for three different traffic volume levels, fuel consumption for each vehicle is reduced by 19-22% compared to the baseline scenario, where human-driven vehicles are considered, by 12-17% compared to the variable speed limit (VSL) algorithm, and by 18-34% compared to the vehicular-based speed harmonization (SPD-HARM) algorithm. Similarly, travel time is improved by 26-30% compared to the baseline scenario, by 3-19% compared to the VSL algorithm, and by 31-39% compared to the vehicular-based SPD-HARM algorithm.
In a rapidly urbanizing world, we need to make fundamental transformations in how we use and access transportation. We are currently witnessing an increasing integration of our energy, transportation, and cyber networks, which, coupled with the human interactions, is giving rise to a new level of complexity in transportation (Malikopoulos2015). As we move to increasingly complex transportation systems (Malikopoulos2016c), new control approaches are needed to optimize the impact on system behavior of the interplay between vehicles at different traffic scenarios. Intersections, merging roadways, speed reduction zones along with the drivers’ responses to various disturbances are the primary sources of bottlenecks that contribute to traffic congestion (Malikopoulos2016a). In 2015, congestion caused people in urban areas in US to spend 6.9 billion hours more on the road and to purchase an extra 3.1 billion gallons of fuel, resulting in a total cost estimated at $160 billion (Schrank2015). In the US, the average hours annually spent per commuter was estimated as 50 hours, having ranked the worst country worldwide (inrix_corporation_inrix_2016). In particular, in the most congested metropolitan areas including Los Angeles, CA (81 hours), Washington DC (75 hours) and San Francisco, CA (75 hours), every driver has spent more than three days in a congested traffic a year (inrix_corporation_inrix_2016).
Speed harmonization (SPD-HARM) is one of the major applications operated in the US towards reducing congestion. According to the US Department of Transportation (DOT), SPD-HARM is defined as a strategy that involves gradually lowering speeds before a heavily congested area to reduce the stop-and-go traffic that contributes to traffic flow break-down. The key of SPD-HARM is to mitigate loss of highway performance by preventing traffic break-downs and keeping bottlenecks operating at constant traffic feeds. The fundamentals of traffic flow theory reflect that the traffic break-down at the bottleneck can be prevented by progressively guiding the upstream traffic to equal the downstream traffic flow, so the upstream traffic smoothly runs into the downstream traffic and can pass through the bottleneck without disruptions. The control algorithms for SPD-HARM may include traditional intelligent transportation systems technologies (e.g., VSL and ramp metering) or use information provided to connected and automated vehicles (CAVs) to enforce speed on individual vehicle - the individual vehicle-based SPD-HARM can be viewed as an extreme form of VSL. It is important to note that the vehicular-based SPD-HARM should be differentiated from any type of cruise control system such as adaptive cruise control (ACC) or cooperative ACC (CACC), since the latter are intended to facilitate the inter-vehicular interactions by maintaining a given distance between the vehicles as opposed to a strategy for vehicles to approach a bottleneck.
In this paper, we develop a framework that will allow each vehicle to optimally control its speed before entering a bottleneck. The objective is to derive the optimal acceleration/deceleration of each vehicle within a “control zone” of appropriate length right before entering a “speed reduction zone.” The latter causes bottlenecks that builds up as vehicles exceeds the bottleneck capacity. By optimizing the vehicles’ acceleration/deceleration in the upstream, the time of reaching the speed reduction zone is controlled optimally, and thus the recovery time from the congested area is minimized. In addition, the vehicle can avoid getting into a stop-and-go driving mode, thereby conserving momentum and energy. Eliminating the vehicles’ stop-and-go driving aims at minimizing transient engine operation, and thus we can have direct benefits in fuel consumption (Malikopoulos2008b).
The contributions of this paper are: (1) the problem formulation to control optimally the speed of a number of automated vehicles cruising on a freeway before they enter a speed reduction zone, (2) the analytical, closed-form solution of this problem along with a rigorous analysis that reveals the conditions under which the rear-end collision avoidance constraint does not become active within the control and speed reduction zones, and (3) bringing the gap between the existing algorithms and the real-world implications by providing an optimal solution that can be implemented in a vehicle in real time. We should emphasize that our approach focuses on controlling optimally the vehicle before its entry in the speed reduction zone and not on inter-vehicle interactions, and threfore, it is fundamentally different from the ACC and CACC approaches.
I-B Literature Review
Traditionally, the SPD-HARM has been realized through variable message signs (VMS), variable speed limit (VSL) and the rolling SPD-HARM (a.k.a., pace-car technique) (roberts_i-70_2012). Both VMS and VSL systems employ the display gantries mounted along roadways to deliver messages or control schemes. Another method for achieving SPD-HARM is the rolling SPD-HARM, which uses designated patrol vehicles entering the traffic to hold a traffic stream at a lower speed, and thus, traverse the congestion area smoothly while mitigating shock waves.
The application of SPD-HARM has been mainly evolved through VSL which appeared to be more effective and efficient than VMS and the rolling SPD-HARM (robinson_examples_2000; roberts_i-70_2012). Current VSL methods employ a proactive approach in which they apply a control action beforehand and then anticipate the behavior of the system (traffic) (khondaker_variable_2015). Even though this proactive approach has made VSL a popular method over the years, it is likely to find a sub-optimal solution since it is based on a heuristic approach (frejo_global_2012).
SPD-HARM methods can be broadly categorized into the (1) reactive approach and (2) proactive approach. The reactive approach initiates the operation at a call upon a queue detected, and it uses immediate traffic condition information to determine the control strategy for the subsequent time interval. While the reactive approach allows to remedy the bottleneck with real-time feedback operations, it has limitations related to time lag between the occurrence of congestion and applied control (khondaker_variable_2015). In contrast, the proactive approach has the capability of acting proactively, while anticipating the behavior of traffic flow (khondaker_variable_2015). Thus, it can predict bottleneck formations before they even occur, while potential shock waves can be resolved by restricting traffic inflow. In addition, the nature of predictions of proactive VSL methods allows for a systematic approach for network-wide coordination which supports system optimization, whereas reactive approach is restrained to a localized control logic.
I-B1 Reactive Speed Harmonization
The first field implementation of SPD-HARM was the VSL system in the German motorway A8 corridor in Munich extented to the boundary of Salzburg, Austria in 1965 (schick_influence_2003). During the early 1960s, US implemented SPD-HARM using VMS on a portion of the New Jersey Turnpike (robinson_examples_2000). These SPD-HARM systems required human interventions to determine the messages or speed limits based on the conditions such as weather, traffic congestion and construction schedules. Since 1970s, advances in sensor technologies and traffic control systems allowed the SPD-HARM to automatically operate based on the traffic flow or weather conditions using various types of detectors. The earlier VMS and VSL implementations were often at the purpose of addressing safety issues under work zone areas or inclement weather conditions (robinson_examples_2000). In 2007, SPD-HARM started focusing on improving traffic flow mobility. The VSL systems implemented in the M42 motorway at Birmingham, UK, and Washington State Department of Transportation (brinckerhoff_active_2008), use algorithms which are automatically activated based on pre-defined threshold of flow and speed measured by detectors embedded in the pavement. The systems display the lowered speed limit within a “control zone” of a pre-defined length (mott_macdonald_ltd._atm_2008).
There has been a significant amount of VSL algorithms proposed in the literature to date. A reactive approach for VSL to improve safety and mobility at work-zone areas (park_development_2003; lin_exploring_2004) outperformed the existing VSL algorithms, especially with the traffic demand fluctuations (park_development_2003). A simulation-based study show that the performance of VSL is a function of the traffic volume levels (juan_simulation_2004). After reaching a particular traffic volume level, the benefit can become more apparent, and therefore, VSL needs to be integrated with ramp metering control (juan_simulation_2004). Another VSL algorithm, which implemented in the Twin Cities Metropolitan area, can identify the moving jam based on the deceleration rate between adjacent spots (kwon_minnesota_2011). The field evaluation showed the reduction in average maximum deceleration by 20% over the state-of-the-art in that area while improved the vehicle throughput at the bottleneck areas. Through its evolution, reactive SPD-HARM has consistently showed improvements in many aspects such as reliability, safety and environmental sustainability by providing adequate feedback to the dynamic traffic conditions. However, the capability of reactive control is limited as it can be only effective after a bottleneck occurs while it mainly depends on heuristics.
I-B2 Proactive Speed Harmonization
The necessity of a systematic approach for preventing adverse impacts from impending shock waves eventually led to the development of the proactive VSL. The proactive VSL approach was first proposed by (alessandri_optimal_1998) adopting Kalman Filter aimed at estimating impending traffic status based on the time-series traffic measurements (welch_introduction_2001). Given the estimated traffic flow, the proactive VSL approach derives a control policy that minimizes various cost functions (e.g., average travel time, summation of square densities of all sections). Although this effort initiated prediction-based VSL systems, the prediction using a time-series approach is not robust, especially under unexpected traffic flow disturbances, since it heavily relies on the empirical patterns.
A pioneering effort in developing a proactive VSL system was made by (hegyi_optimal_2003) using model predictive control (MPC). The key aspect of that work is that it prevents traffic breakdown by decreasing the density of approaching traffic rather than focusing on reducing the speed variances. Using MPC, which enabled a network-wide optimization, a series of VSL systems can be coordinated for system-wide optimization that eventually aims at preventing upstream delays. Another MPC-based proactive VSL system was proposed in (lu_new_2010) focusing on creating a discharge section immediate upstream of the bottleneck to regulate traffic flow into the bottleneck that remains close to its capacity. With the intention to influence the motorway mainstream, a traffic flow control approach was proposed in (carlson_optimal_2010), which adapted a discrete-time dynamic control method using a suitable feasible-direction algorithm (papageorgiou_feasible_1995), that can yield feedback control policies. These approaches have showed substantial improvements in vehicle throughput, safety, equity, and driver acceptance through microscopic simulation studies (carlson_optimal_2010; hegyi_optimal_2003; hegyi_model_2005; hegyi_optimal_2005). However, there are significant challenges in practical applications associated with computational requirements. Using shock wave theory, another VSL algorithm was proposed in (hegyi_specialist:_2008) that predicts future traffic evolution based on the different traffic states along the freeway segments. By identifying the location of the front boundaries of shock waves and the active speed limits, the algorithm maximizes the discharge rate at the bottleneck (hegyi_specialist:_2008).
The performance of SPD-HARM can vary depending on the control approach, characteristics of the topology, and driving behaviors. The potential travel time improvements through SPD-HARM have been debatable during peak hours (kwon_minnesota_2011; roberts_i-70_2012). However, it has been widely agreed that SPD-HARM aims to increase vehicle throughput at the bottleneck. It has been shown that the vehicle throughput can be increased by 4-5% via VSL system (mott_macdonald_ltd._atm_2008) and by 5-10 % via rolling SPD-HARM implemented in European countries with significant benefits in safety since personal injury crashes reduced about 30-35% (fuhs_synthesis_2010). The environmental impacts of SPD-HARM were also substantial demonstrating reduction in vehicle emissions by 4-10% (depending on the pollutants) (fuhs_synthesis_2010), and fuel consumption by 4% (mott_macdonald_ltd._atm_2008).
Although previous research reported in the literature has aimed at enhancing our understanding of SPD-HARM algorithms and demonstrating significant improvements, deriving in real time an optimal solution for each individual vehicle under the hard safety constrains still remains a challenging control problem. In this paper, we address the problem of controlling the speed of a number of automated vehicles before they enter a speed reduction zone on a freeway. We formulate the control problem and provide an analytical, closed-form, optimal solution that can be implemented in real time. The solution yields the optimal acceleration/deceleration of each vehicle in the upstream, and thus it controls the time that each vehicle enters the speed reduction zone. Furthermore, we provide the conditions under which the rear-end collision avoidance constraint does not become active at any time within the control and speed reduction zones. By controlling the time of reaching the speed reduction zone, the recovery time from the congested area is reduced which in turn leads to the increase in average speed, and eventually, the travel time. In addition, the vehicles will avoid getting into a stop-and-go driving mode, thereby conserving momentum and energy. The unique contribution of this paper hinges on the following three aspects: (1) the formulation of the problem of controlling the speed of a number of automated vehicles before they enter a speed reduction zone on a freeway in Section II, (2) a rigorous analysis that reveals the conditions under which the rear-end collision avoidance constraint does not become active through the optimal solution in Section III, and (3) the proposed optimal control framework bridges the gap between the existing algorithms and the real-world implications by providing an optimal solution that can be implemented in a vehicle in real time.
I-C Organization of the Paper
The remaining paper proceeds as follows. In Section II we introduce the model, present the assumptions of our approach and formulate the optimal control problem. In Section III, we provide the control framework, derive an analytical closed-form solution, and discuss the conditions under which the rear-end collision avoidance constraint does not become active. Finally, we provide simulation results in Section IV and concluding remarks in Section V.
Ii Problem Formulation
Ii-a Modeling Framework
We address the problem of optimally controlling the speed of vehicles cruising on a freeway (Fig. 1) that includes a speed reduction zone of a length . The speed reduction zone is a bottleneck that builds up as vehicles exceeds the bottleneck capacity. The freeway has a control zone right before the speed reduction zone, inside of which the vehicles need to accelerate/decelerate optimally, in terms of fuel consumption, so as to enter the speed reduction zone with the appropriate speed. Therefore, the speed of the potential queue built-up inside the reduction zone is controlled, and thus the congestion recovery time can be optimized. The distance from the entry of the control zone until the entry of the speed reduction zone is .
We consider a number of automated vehicles in each lane, where is the time, entering the control zone (Fig. 1). Let be the queue in one of the lanes inside the control zone. The dynamics of each vehicle are represented by a state equation
where , , are the state of the vehicle and control input, is the time that vehicle enters the control zone, and is the value of the initial state. For simplicity, we model each vehicle as a double integrator, e.g., and , where , , and denote the position, speed and acceleration/deceleration (control input) of each vehicle inside the control zone. Let denote the state of each vehicle , with initial value , taking values in the state space . The sets , and , are complete and totally bounded subsets of . The state space for each vehicle is closed with respect to the induced topology on and thus, it is compact.
We need to ensure that for any initial state and every admissible control , the system (1) has a unique solution on some interval , where is the time that vehicle enters the speed reduction zone. To ensure that the control input and vehicle speed are within a given admissible range, the following constraints are imposed:
where , are the minimum and maximum control inputs (maximum deceleration/ acceleration) for each vehicle , and , are the minimum and maximum speed limits respectively. For simplicity, in the rest of the paper we do not consider vehicle diversity, and thus, we set and .
To ensure the absence of any rear-end collision of two consecutive vehicles traveling on the same lane, the position of the preceding vehicle should be greater than or equal to the position of the following vehicle plus a safe distance , which is a function of speed. Thus, we impose the rear-end safety constraint
where denotes the vehicle which is physically immediately ahead of in the same lane, and is the average speed of the vehicles inside the control zone at time .
In the modeling framework described above, we impose the following assumptions:
The vehicles do not change lanes inside the control and speed reduction zones.
Each vehicle cruises inside the speed reduction zone with the imposed speed limit, .
Each vehicle has proximity sensors and can measure local information without errors or delays.
We briefly comment on the above assumptions. The first and second assumptions are intended to enhance safety awareness, but they could be modified appropriately, if necessary. The third assumption may be a strong requirement to impose. However, it is relatively straightforward to extend our results in the case that it is relaxed, as long as the noise in the measurements and/or delays are bounded. For example, we can determine the uncertainties of the state of the vehicle as a result of sensing or communication errors and delays, and incorporate these into the safety constraints.
Ii-B Optimal Control Problem Formulation
We consider the problem of deriving the optimal acceleration/deceleration of each vehicle to control the speed of the vehicles in a freeway (Fig. 1), under the hard safety constraints to avoid rear-end collision. The potential benefits of the solution of this problem are substantial. By controlling the speed of the vehicles in the upstream or tighten the inflow traffic, the speed of queue built-up decreases, and thus the congestion recovery time is also reduced. Even though the speed of each vehicle is reduced, the throughput at the speed reduction zone is maximized. Moreover, by optimizing the acceleration/deceleration of each vehicle, we minimize transient engine operation, thus we can have direct benefits in fuel consumption (Malikopoulos2008b) and emissions since internal combustion engines are optimized over steady state operating points (constant torque and speed) (Malikopoulos2010a).
When a vehicle enters the control zone, it receives some information from the vehicle which is physically located ahead of it.
For each vehicle when it enters a control zone, we define the information set as
where are the position and speed of vehicle inside the control zone, and , is the time targeted for vehicle to enter the speed reduction zone. Each vehicle has proximity sensors and can observe and/or estimate the information in without errors or delays (Assumption 3). Note that once vehicle enters the control zone, then immediately all information in becomes available to : are read from its sensors and can also be computed at that time based on the information the vehicle receives from as described next.
The time that the vehicle will be entering the speed reduction zone is restricted by the imposing rear-end collision constraint. Therefore, to ensure that (3) is satisfied at we impose the following condition
where is the speed of the vehicle at the time that enters the speed reduction zone, and it is equal to the speed, , imposed inside the reduction zone (Assumption 2). Thus, the condition (5) ensures that the time that vehicle will be entering the speed reduction zone is feasible and can be attained based on the imposed speed limits inside the control zone. In addition, for low traffic flow where vehicle and might be located far away from each other, there is no compelling reason for vehicle to accelerate within the control zone just to have a distance from vehicle at the time that vehicle enters the speed reduction zone. Therefore, in such cases vehicle can keep cruising within the control zone with the initial speed that entered the control zone at The recursion is initialized when the first vehicle enters the control zone, i.e., it is assigned . In this case, can be externally assigned as the desired exit time of this vehicle whose behavior is unconstrained. Thus the time is fixed and available through . The second vehicle will access to compute the times . The third vehicle will access and the communication process will continue with the same fashion until the vehicle in the queue access the .
We consider the problem of minimizing the control input for each vehicle from the time it enters the control zone until the time that enters the speed reduction zone under the hard safety constraint to avoid rear-end collision. The optimal control problem can be as to minimize the L norm of the control input
with initial and final conditions: , , and . Note that we have omitted the rear end safety constraint (3) in the problem formulation above. The analytical solution of the problem including the rear-end collision avoidance constraint may become intractable for real-time implementation. However, in the following section, we provide the conditions under which the rear-end collision avoidance constraint does not become active at any time in assuming it is not active at . Note that we can guarantee rear-end collision avoidance at time based on (5).
Iii Solution of the optimal control problem
For the analytical solution and real-time implementation of the control problem (6), we apply Hamiltonian analysis. In our analysis, we consider that when the vehicles enter the control zone, none of the constraints are active. However, this might not be in general true. For example, a vehicle may enter the control zone with speed higher than the speed limit. In this case, we need to solve an optimal control problem starting from an infeasible state.
The Hamiltonian function is formulated as follows
where and are the costates, and is a vector of Lagrange multipliers with
The Euler-Lagrange equations become
with given initial and final conditions , , , and . The necessary condition for optimality is
To address this problem, the constrained and unconstrained arcs need to be pieced together to satisfy the Euler-Lagrange equations and necessary condition of optimality. The analytical solution of (6) without considering state and control constraints was presented in earlier papers (Rios-Torres2015; Rios-Torres2; Ntousakis:2016aa) for coordinating in real time CAVs at highway on-ramps and (ZhangMalikopoulosCassandras2016) at two adjacent intersections.
When the state and control constraints are not active, we have Applying the necessary condition (15), the optimal control can be given , The Euler-Lagrange equations yield and From the former equation we have and from the latter , where and are constants of integration corresponding to each vehicle . Consequently, the optimal control input (acceleration/deceleration) as a function of time is given by
Substituting the last equation into the vehicle dynamics equations (1), we can find the optimal speed and position for each vehicle, namely
where and are constants of integration. These constants can be computed by using the initial and final conditions. Since we seek to derive the optimal control (16) in real time, we can designate initial values and , and initial time, , to be the current values of the states and and time , where . Therefore the constants of integration will be functions of time and states, i.e., , and . It follows that (17) and (18), along with the initial and terminal conditions, can be used to form a system of four equations of the form , namely
where is specified by (5), and is the imposed speed limit at the speed reduction zone. Hence, we have
where contains the four integration constants , , , . Thus, (16) can be written as
Since (19) can be computed in real time, the controller can yield the optimal control in real time for each vehicle , with feedback provided through the re-calculation of the vector in (20). Similar results can be obtained when the state and control constraints become active (Malikopoulos2017).
This analytical solution, however, does not include the rear-end collision avoidance constraint. Thus, we investigate the conditions under which the rear-end collision avoidance constraint does not become active for any two vehicles and at any time in assuming they are not active at .
Without loss of generality and to simplify notation, we reset the time at , i.e., . Thus, at , vehicle has traveled a distance inside the control zone and has a speed . Similarly, at , vehicle just entered the control zone with an initial speed . From (5), the position of vehicle will be at and will have the speed imposed at the speed reduction zone, . Similarly, at the vehicle will be at the entry of the speed reduction zone, so its position will be and its speed will be the speed imposed at the speed reduction zone, .
Since the state and control constraints are not active, the control input, speed and position for each vehicle are given by (16) - (18). Substituting the conditions at and for the vehicles and in (19), and by letting , , and we have
where is the speed of vehicle at time , is the speed at the reduction zone; and , and , are the constants of integration that can be computed from (20) where
We know for For all , we have
Substituting the constants of integration, (32) becomes
By rearranging the terms in the last equation we have
Since we can multiply both sides by , hence