3.1. Lateral control combined with sine function

During the lane-changing process in the coordination area, a vehicle’s lane change (LC) is decomposed into two independent movements: lateral and longitudinal. Laterally, the designed lane-changing trajectory takes into account the safety and comfort of driving. A sine function curve is used to determine the trajectory [24], which has the following two main advantages and ensures the smoothness of the continuous second derivative of the lane-changing trajectory.

The specific construction process of the LC trajectory is as Fig 2 shows, where is the longitudinal travel distance during the vehicle’s lane change process, and is the lane width. Based on the shape, speed, and acceleration characteristics of the lane-changing trajectory, the longitudinal and lateral axes of the lane are taken as the and coordinates, respectively, with as the starting point of the lane change, to establish a lane-changing trajectory model based on the sine function. The mathematical description equation of the lane-changing trajectory is as follows:

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Fig 2. Vehicle lane change trajectory.

https://doi.org/10.1371/journal.pone.0331986.g002

(1)

Based on the vehicle kinematic equations, the following expressions can be derived:

(2)

where is the total duration of the coordinated lane-changing process, and is the average longitudinal velocity during the coordinated lane-changing process. Combining Eq. (1) and (2), the trajectory equation of the ego vehicle during the lane-changing phase can be obtained,

(3)

By calculating the first and second derivatives of Eq. (3) respectively, the expressions for the lateral velocity and acceleration during the lane change can be obtained,

(4)

(5)

Based on Eq. (5), it can be inferred that the maximum lateral acceleration of the ego vehicle occurs at time , and the maximum lateral acceleration is,

(6)

Integrating Eq. (6) into Eq. (5) and (3), allows the lateral trajectory and acceleration during the LC to be described as the expressions that include the terms for the maximum lateral acceleration and the coordinated LC duration ,

(7)

(8)

Therefore, throughout the entire process in the coordination area, the lateral trajectory and acceleration of the ego vehicle during LC are as shown in Eq. (7) and (8). Since the ego vehicle changes lanes from the centerline of the original lane to the centerline of the target lane, its longitudinal displacement is exactly equal to the width of the lane. According to the road construction industry standards in China, the lane width is set to 3.75 meters in this paper. According to the sine function curve law, the ego vehicle moves to the boundary of the two lanes at time , and this time is considered as the moment when the ego vehicle just leaves the original lane and begins to enter the target lane.

3.2. Markov decision process model

Deep reinforcement learning algorithms are applied to solve sequential decision-making problems, which can be modeled within the framework of a Markov Decision Process (MDP). Research has shown that decision-making problems in autonomous driving can be regarded as an MDP problem [25]. MDP is a mathematical model for decision-making defined by the tuple , where is the state space, is the action space, describes the transition probabilities between states, is the reward function, and is the discount factor. In this model, an agent selects an action based on the current state of the environment, and the environment transitions to the next state as a result of the executed action. Then, the agent receives a scalar reward defined by the reward function . This process is repeated until the end of each episode. The goal of the DRL model is to learn an optimal policy that maps states to actions, maximizing the sum of future discounted rewards. The cumulative reward is calculated as follows,

(9)

where denotes the mathematical expectation, and is the k-th time point after the current time . This study aims to use the DRL model to solve the decision-making problem of autonomous driving in multi-lane mixed traffic scene at on-ramp merging. In conjunction with CSAT, the D3QN algorithm is used to train vehicles to successfully merge into the main road, to increasing the overall traffic volume of the road section. Vehicles interact with the environment and receive numerical rewards, which are used to learn the optimal driving strategy to maximize the expected return of the current state, as shown in Fig 3. The DRL method solves the decision-making problem by using the MDP framework. The model mainly consists of the state space 𝑆, action space 𝐴, and reward function 𝑅. The agent controls the vehicle, and during the driving process, it obtains the current road environmental state information from the VCS (where represents the number of steps in the current round), and then selects and executes actions according to the environmental state and its policy. The environment provides feedback to the vehicle in the form of rewards based on the current state and the vehicle’s motion, allowing the vehicle to improve its action strategy, and the environment transitions to the next state . The above iterative process is repeated until the system converges or reaches a preset number of iterations.

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Fig 3. MDP model for behavioral decision making in autonomous vehicles.

https://doi.org/10.1371/journal.pone.0331986.g003

3.2.1. State space.

The state space refers to the information that a vehicle can perceive and obtain during its motion. It includes the position and motion state of the ego vehicle, as well as the state information of the surrounding traffic vehicles. The ego vehicle is equipped with a radar with a range of 125 meters, which can collect information such as the position and speed of vehicles within the perception range [26]. As described in Fig 4, it illustrates the relative positions of the ego vehicle and the surrounding vehicles. When making lane selection decisions, vehicles need to extract features from the surrounding environmental information as state inputs. The proposed method considers the six closest vehicles on the main road within the perception range during the development process, including the three vehicles in front of the ego vehicle (f1, f2, f3) and the three vehicles behind the ego vehicle along the direction of the traffic flow (r1, r2, r3). If there are no vehicles within the radar range, zero padding is added to the state space. The state of each main road vehicle is defined as follows,

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Fig 4. Relative distance from surrounding vehicles.

https://doi.org/10.1371/journal.pone.0331986.g004

(10)

where is the state information of the ego vehicle, is the traffic density of each lane. , is the state information of the surrounding vehicles. Each piece of information 𝑆 includes seven state features. Taking as an example,

(11)

where indicates the presence or absence of a vehicle, and are the longitudinal and lateral positions of the vehicle, respectively. and are the longitudinal and lateral velocities of the vehicle. and are the cosine and sine values of the vehicle’s heading angle.

3.2.2. Action space.

In this model, the action space refers to all possible driving behaviors that a vehicle can exhibit in the environment. The action space is designed as follows: the vehicle’s acceleration control input is set within the range of [−4.5, 2.6] [27]. To ensure the safe driving of the vehicle on the main road, an action shielding mechanism is proposed. If the distance between vehicles is , an emergency braking action is triggered, setting the acceleration to , and a negative reward is returned from the environment. It is defined as ,

(12)

where is the length of the vehicle, assumed to be 5 meters.

The action set in the Markov Decision Process is:

(13)

where is to stay in the current lane, is to change lanes to the left, is to change lanes to the right. When a decision is made that the vehicle needs to perform a LC action, the lateral trajectory from Eq. (7) and the acceleration from Eq. (8) are executed.

3.2.3. Reward function.

Rewards are the responses of the environment to the actions of the agent, and the value of the reward is determined by the created reward function, which affects the effectiveness of the learning process and the achievement of expected results. In the on-ramp merging scene, to ensure the safe and efficient driving of vehicles on the road, the design of the reward function must take into account both driving efficiency and dangerous driving behaviors. This means that while rewarding efficient driving, dangerous driving must also be punished.

To enable the vehicle to learn safe driving while interacting with the environment, a safety reward function is designed. To teach the vehicle to avoid collisions, if it collides with other vehicles during lane changing and merging processes, then a penalty is applied, .

(14)

To optimize the overall travel time and encourage the vehicle to maintain the best speed, the efficiency reward function is defined as follows:

(15)

where is the current vehicle speed, and is the maximum expected speed, which is set according to the normal safe driving speed on highways in the real world. This reward provides a certain negative reward when the vehicle is driving too slowly or speeding, effectively avoiding unsafe driving behaviors.

Considering that frequent lane changes may reduce the overall traffic smoothness and increase safety risks, the LC reward function penalizes obviously unnecessary lane change behaviors. To encourage vehicles to choose the optimal lane and alleviate the fluctuations in the main traffic flow caused by the merging of ramp vehicles, the lane change reward function is defined as follows,

(16)(17)(18)

where is the overall traffic density at time , and is the traffic density of each lane. When a vehicle changes lanes and exacerbates the overall traffic density, it will receive a negative reward.

To assess the overall completion of the task, the successful merging of the ramp vehicle and its safe travel on the main road afterward are evaluated. If the ramp vehicle safely reaches the merging area, . If it safely exits the stabilization area, , and if it fails to reach or arrives at the destination after the timeout, .

Considering the above factors, the total reward function is defined as,

(19)

where 𝑤 is the weight for each reward component. After trial and error, the optimal values for , , , and are 0.1, 0.2, 0.1, and 0.05, respectively. In this study, the observation space, state space, and reward function are selected based on the optimal results from preliminary experiments, where different options were examined.

3.3. IDS algorithm

In reinforcement learning, it is necessary to estimate the value for each state, but for many states, it is not necessary to estimate the value for each action. The Dueling Double Deep Q-Network (D3QN) adds a Dueling Network Architecture(DNA) to both the main network and the target network on the basis of the Double Deep Q-Network (DDQN) [28]. The DNA separates the state value and the advantage of the state-action pair, allowing for a more accurate assessment of the Q-values. The uniform random sampling used in Deep Q-Networks (DQN) [29], DDQN, and Dueling DQN [30] algorithms is not optimal. During the interaction with the environment, the experience replay unit continuously stores experience samples for model training, and these data may be kept in the experience replay unit indefinitely. Frequent replay of these experiences can make the agent aware of correct or incorrect actions, thereby constantly correcting them. However, the importance of different experience samples varies. Since the samples in the experience replay unit are constantly being updated, if a small number of samples are taken uniformly at random from the experience replay unit as model input, some samples with higher importance may not be fully utilized and may even be directly overwritten, leading to reduced model training efficiency.

To improve the training efficiency of the model, this paper adopts prioritized experience replay [31], which increases the probability of extracting more important samples. It uses a proportional prioritization mechanism that defines the priority of experience,

(20)

where is a small constant number that ensures samples with a TD-error almost equal to zero also have a lower probability of being extracted.

Based on this, the probability of sampling experience can be defined by Eq. (22),

(21)

where is the size of the replay experience unit. controls the degree of prioritization used. implies uniform sampling, while implies greedy strategy sampling.

Due to the frequent replay of experiences with high temporal difference (TD) errors and the overly frequent access to certain states, the lack of diversity in experience can lead to overfitting in network training. Therefore, importance sampling weights can be used for correction, as shown in Eq. (23),

(22)

where represents the probability of sampling experience , represents the minimum sampling probability, and the parameter represents the degree of correction.

The loss function of the Q-network is defined in Eq. (24),

(23)

where represents the target Q-value at time , and represents the output Q-value of the Q-network.

3.3.1. Target network soft update.

Traditional Deep Q-Network often use hard updates, where the target network is updated once every stable steps. This paper proposes an alternative soft update idea to ensure that the target network is updated in every iteration, which is equivalent to a network update interval of 1. Soft update uses a convex combination of the current network parameters and the target network parameters to update the network, i.e., the parameters of the target network will be updated using the current network parameters according to Eq. (25).

(24)

In this way, the parameters of the target network change slightly, and the calculated target label values change relatively smoothly. Then, even if the target network is updated in every iteration, the algorithm can maintain a certain degree of stability. The smaller the soft update interval factor, the more stable the algorithm is. An appropriate soft update interval factor can make the DQN algorithm training both stable and fast.

3.3.2 Dynamic epsilon-greedy decay method.

In the D3QN algorithm presented in this paper, a dynamic greedy strategy is employed [32]. During the training phase, traditional reinforcement learning algorithms use a fixed greedy factor , which select actions randomly with probability and choose the best action with probability . If is too small, the environment will not be sufficiently explored in the early training stages, leading to over-exploitation or insufficient exploration. If is too large, even in the later stages of training when the agent has fully explored the environment, there will be a significant increase in the probability of randomly selecting actions instead of choosing the optimal action, which contradicts the principle of finding the optimal policy. Therefore, a dynamic epsilon-greedy method with an exponential decay is used to address this issue, as shown in Eq. (26).

(25)

where represents the number of training episodes, represents the designed offset, and is a variable that changes according to the environment. This method ensures more exploration at the uncertainty of the environment in the early stages of training, and solve the exploration-exploitation dilemma of deep reinforcement learning.

3.3.3. Algorithm description.

When the agent is ignorant of the environment in the early stages of learning, the DQN chooses the most advantageous action at each iteration, such as the action with the maximum value function. Then, the agent will be able to find a suboptimal policy in a short period of time with high learning efficiency. However, there is always a problem of overestimation. The DQN decouples the selection of the target Q-value action and the calculation of the target Q-value in two steps to eliminate the problem of overestimation.

In this paper, the corresponding problem in the DDQN algorithm are not selected according to the optimal policy, instead of a dynamic greedy strategy based on Eq. (26) is used. For the sample importance selection, the previously experienced replay method is used to extract experience samples from the experience replay unit, increasing the utilization rate of important samples and accelerating the training speed of the neural network. Then, Eq. (24)is used as a soft update to ensure that the target network is updated in every iteration, making the DQN algorithm training both stable and fast. A deep neural network is used to approximate the Q-function instead of traditional tabular value recording, which allows the D3QN algorithm to solve large-scale reinforcement learning problems with complex environmental states. The loss function of the deep neural network is shown in Eq. (23). The gradient backpropagation of the neural network updates all parameters of the Q-network. Finally, a competitive network structure is introduced to optimize the neural network, thereby optimizing the algorithm, as shown in Algorithm 1.

Algorithm 1. IDS Algorithm

1:  Input: minibatch , action advantage parameter , state value parameter , budget , update factor

2:  Initialize replay memory , , Q-Network parameters

3:  For to :

4:   Observe , and choose At (St) with dynamic

5:  Store transition in M with maximal priority

6:  update state

if mod

7:  For to

8:   Sample transition

9:   Compute importance-sampling weight

10:  Calculate the current target Q-value

11:  Use the mean square error loss function to backpropagate to update the Q-Network parameters

12:  Recalculate TD-error

13:  Update transition priority

14:  Soft update target network

15:  end for

16: end for

The architecture that integrates the vehicle coordination system and IDS on-ramp merging strategy proposed in this paper is shown in Fig 5. Agents acquire all vehicle information from the road network environment through VCS, and the IDS lane-changing strategy makes corresponding actions. If maintaining the original lane, the IDM car-following model is used directly; otherwise, a lane change is initiated, with the trajectory based on a sine function curve. Ultimately, a complete control strategy is formed and executed on the ego vehicle.

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Fig 5. Complete Architecture Combining VCS and IDS.

https://doi.org/10.1371/journal.pone.0331986.g005

4.1. Experimental setup

The simulation environment in this study is based on the Simulation of Urban Mobility (SUMO) driving simulator [33], as shown in Fig 1 and Fig 6, the mainline consists of three lanes, with a coordination area of 400 meters, a merging area of 100 meters, and a stabilization area of 100 meters. CHVs follow the Intelligent Driver Model (IDM), capable of basic car-following and collision avoidance [34]. The initial speed range of the merging vehicles is [5,25] . When merging into the mainline, vehicles must travel safely for another 100 meters in the stabilization area to complete the task. The speed limit on the mainline is 30 . New vehicles entering the scene are assigned vehicle types of CAV or CHV using a random seed. Traffic flow of low density is 800(veh/lane/h), medium density is 1000(veh/lane/h), and high density is 1200 (veh/lane/h).

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Fig 6. SUMO On-Ramp Traffic Scenario.

https://doi.org/10.1371/journal.pone.0331986.g006

The experimental model has a three-layer neural network structure. The number of neurons in the input layer is the dimension of the environmental state. The number of neurons in the output layer is set to the dimension of environmental actions for all models. Rectified Linear Unit (ReLU) is used as the activation function. To facilitate experimental comparison, the hyperparameters of the comparative algorithms and the proposed algorithms are the same, and the hyperparameter settings for the reinforcement learning algorithm are shown in Table 1.

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Table 1. Algorithm Parameter Setting.

https://doi.org/10.1371/journal.pone.0331986.t001

4.2. Performance of different methods

To evaluate the performance of the proposed VCS + IDS strategy, comparative experiments were conducted with methods D3QN, DDQN, and VCS+DDQN. The success of vehicles merging into the main road and reaching their destination was assessed. In the experiments, each simulation lasted for a time period of 600 seconds, and each method was iterated for more than 6000 episodes, with data recorded every 20 episodes to verify their performance.

The penetration rate of CAVs was referenced from paper [20], which provided average speed, percentage difference in speed, and corresponding merging area and main road traffic flow under different CAV penetration rates. The average speed increased with the increase of penetration rate. When the CAV penetration rate is less than 60%, the impact on the average speed of the main road and merging speed is small, while the impact is significant when the CAV penetration rate is high (about 60%). Therefore, the CAV penetration rate in this experiment is 60%.

For comparison, three density of vehicle flow (800, 1000, 1200 veh/lane/h) and three demand splits (80−20, 65−35, 50−50) were predefined. Please note that this is the number of vehicles per lane per hour, and the specified demand split is to divide the total capacity. For example, when the traffic density is 800 (veh/lane/h) and the demand split is 50−50, the actual total traffic density of the main road lane and on-ramp lane is 3200 (veh/lane/h), the traffic volume of the on-ramp lane is 1600 (veh/lane/h), and the remaining three main road lanes have a total traffic volume of 1600 vehicles, with an average traffic volume of about 533 (veh/lane/h) per lane.

Let the traffic volumes of each downstream lane in the merging area be denoted as q1, q2 and q3. Then, define the downstream lane imbalance factor u as follows , which is always greater than or equal to 1. A larger imbalance factor may cause traffic chaos downstream of the merging area; therefore, we choose a relatively smaller imbalance factor.

The simulation results for different assessment scenarios are shown in Fig 7, where each table contains a scenario with a specific demand split. In terms of the imbalance factor for downstream lanes, DDQN performs much worse than VCS+DDQN because vehicles in the DDQN strategy cannot comprehensively obtain information about surrounding vehicles, indirectly affecting their driving strategies. The performance of DDQN and D3QN is always lower than when VCS is available, indicating that VCS helps balance traffic volumes across different lanes. DDQN and D3QN perform poorly in complex scenarios and cannot effectively cope with the challenges brought by increased traffic density and changes in demand splits.

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Fig 7. Simulation Results of Different Demand Split.

https://doi.org/10.1371/journal.pone.0331986.g007

VCS + IDS consistently outperforms other scenarios under three different demand splits, especially when the demand split is 50−50. Due to the increased traffic volume from the ramp, the impact on the main road lanes is increased, forcing an increase in the imbalance factor for all strategies. However, VCS + IDS remains relatively stable. Despite the increasing traffic volume requirements, the imbalance factor tends to remain relatively stable, with fluctuations not exceeding 0.03, ensuring a better balance of traffic volumes between lanes, maximizing the utilization of each lane, and demonstrating its high stability in complex traffic environments. Balancing the demand split (from 80−20–50−50) increases the complexity of traffic flow distribution, leading to an increase in the imbalance factor. However, VCS + IDS can still maintain a low imbalance factor value, indicating its significant advantage in balancing traffic flow and reducing traffic chaos.

Overall, VCS + IDS demonstrates stronger robustness and optimization capabilities under high-density and complex traffic conditions through intelligent strategies and coordination mechanisms. It is more effective when the merging area becomes congested, generating better downstream traffic.

This article presents the average traffic speed under different strategies for completing tasks, as shown in Fig 8. VCS + IDS achieved the maximum average speed under various density of vehicle flow. At a traffic density of 800 (veh/lane/h), the speed reached 19.11 m/s, significantly improving overall traffic efficiency by 11.36%, 10.85%, 4.88%, and 8.21% compared to DDQN, VCS+DDQN, and D3QN, respectively. At a traffic density of 1000 (veh/lane/h), the speed can also reach 18.11 m/s, enhancing the average speed by 12.41%, 10.56%, 6.03%, and 8.70% compared to DDQN, VCS+DDQN, and D3QN under the same traffic density. Traditional DDQN and D3QN methods showed limited improvement in traffic efficiency. The integration of VCS+DDQN methods, however, resulted in increased average speeds, with improvements of 5.68%, 6.57%, and 4.27% at different density of vehicle flow compared to DDQN, with the most significant enhancement of 6.57% at 1000 (veh/lane/h). The results indicate that the proposed VCS and IDS can enhance traditional DRL algorithms, as VCS provides strategies that further consider the most appropriate timing for lane changes, thereby reducing congestion rates.

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Fig 8. Average Traffic Speed under Different Strategies.

https://doi.org/10.1371/journal.pone.0331986.g008

From Table 2, it can be observed that the frequent lane changes in DDQN are also one of the reasons for the low overall traffic efficiency. As the number of vehicles per hour increases, i.e., as traffic density increases, the role of lane changing in improving traffic efficiency becomes more significant. DDQN and IDS equipped with VCS reduced the frequency of lane changes by 33.66% and 69.30% respectively under a traffic flow of 800 (veh/lane/h), by 35.33% and 73.89% under a traffic flow of 1000 (veh/lane/h), and by 47.20% and 76.98% under a traffic flow of 1200 (veh/lane/h). The results indicate that in congested traffic conditions, interactions between vehicles become more frequent. VCS + IDS, when implementing lane change strategies, takes into account the traffic volume of surrounding lanes, significantly improving overall efficiency.

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Table 2. Lane change frequency for different methods (times/(veh*km)).

https://doi.org/10.1371/journal.pone.0331986.t002

As shown in Table 3, the collision rates under different strategies are presented. The results indicate that VCS + IDS achieved lower and relatively stable collision rates across various traffic densities, with the lowest being 0.69%, enhancing the safety of lane-changing behavior. With the increase in traffic density, the collision rates for traditional D3QN and DDQN gradually increased: by 0.87% and 1.04% respectively under a traffic flow of 800 (veh/lane/h), by 0.32% and 1.97% respectively under a traffic flow of 1000 (veh/lane/h), and by 1.03% and 3.44% respectively under a traffic flow of 1200 (veh/lane/h). These results clearly demonstrate that VCS + IDS achieved the best performance in the on-ramp merging scenario with mixed traffic on multiple lanes, proving the effectiveness of the proposed VCS and IDS methods, ensuring the driving safety of the mainline and merging vehicles on the road.

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Table 3. Collision rate for different strategies (%).

https://doi.org/10.1371/journal.pone.0331986.t003

As shown in Table 4 and Fig 9, the completion rate of on-ramp merging under different traffic densities is presented. DDQN and IDS equipped with VCS improved the completion rate by 11.13% and 6.42% respectively under a traffic flow of 800 (veh/lane/h), by 9.1% and 5.95% respectively under a traffic flow of 1000 (veh/lane/h), and by 19.45% and 11.08% respectively under a traffic flow of 1200 (veh/lane/h). From Fig 9, it can be observed that the merging rate of DDQN at the on-ramp is generally moderate, with no significant increasing trend as the number of training episodes increases. In contrast, for DDQN and IDS with VCS, the completion rate is initially low, but it gradually increases as the number of training episodes increases, eventually stabilizing around 4000 episodes.

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Table 4. On-ramp merging completion rate for different methods (%).

https://doi.org/10.1371/journal.pone.0331986.t004

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Fig 9. Merging completion rate for different methods at 1200(veh/lane/h).

https://doi.org/10.1371/journal.pone.0331986.g009

As shown in Table 5 and Fig 10, DDQN and D3QN did not perform very well in terms of task completion rate under different traffic densities. One possible reason is that traditional DRL struggles with multi-lane scenarios due to the complexity of environmental factors, which agents cannot effectively process, leading to increased lane changes and collisions, thereby indirectly affecting the task completion rate. In contrast, DDQN and IDS equipped with VCS improved the completion rate by 11.64% and 8.47% respectively under a traffic flow of 800 (veh/lane/h), by 13.37% and 10.41% respectively under a traffic flow of 1000 (veh/lane/h), and by 24.37% and 10.79% respectively under a traffic flow of 1200 (veh/lane/h). This further demonstrates the effectiveness of VCS and IDS. These results indicate that the proposed methods are applicable to different scenarios and traffic densities, exhibiting better generalization and robustness.

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Table 5. Task completion rate for different methods (%).

https://doi.org/10.1371/journal.pone.0331986.t005

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Fig 10. Task completion rate for different methods at 1200(veh/lane/h).

https://doi.org/10.1371/journal.pone.0331986.g010