1. Introduction
Over the last few decades, urban traffic congestion was an intractable problem for metropolises around the world, with negative follow-up effects on safety, economics, and the environment. Thus, various countermeasures were proposed to overcome the traffic congestion problem [
1]. Additionally, the speed of traffic flow also has a strong relevance for driving safety [
2]. Therefore, alleviating traffic pressure as well as reducing traffic accidents is an urgent problem that needs to be solved. With the development of technology, traffic signal control gradually became the main solution to alleviate these problems. An early work was reported in 1964 where a linear control algorithm was proposed to switch the traffic light under unsaturated conditions [
3]. Then, the representative systems were developed and widely used throughout the world include SCATS [
4], SCOOT [
5], and so on.
However, with the increase in vehicles, traffic dynamics became more complex and the above-mentioned control systems were found to have serious limitations in reaching high performance. Therefore, more algorithms were proposed. In [
6], a fuzzy-based intelligent traffic signal method was introduced for traffic signal control, which provided a convenient and economical way to improve existing traffic light infrastructure. Ref. [
7] proposed a cost-effective model estimating traffic pattern changes when adding newly constructed roads to existing facilities, which assisted decision-making processes. In [
8], the road network was divided into several agents and a distributed control scheme was then proposed. To accelerate congestion dissipation, this controller design process was based on a performance index aggregating the state of each agent and balancing their adjacent agents. In ref. [
9], to alleviate traffic congestion, a new control strategy was presented based on transition rules, proving that traffic signal timing optimization is an effective way to decrease congestion. In order to reduce the total travel time, ref. [
10] formulated a performance index constructed using time delay and signal control variables and gave an optimized solution by using a heuristic social learning-particle swarm optimization algorithm. The proposed method was evaluated via real traffic data and ensured real-time application. Ref. [
11] focused on the lane drop issue on freeways and designed an integrated variable speed limit and lane change control method to solve bottleneck congestion.
Additionally, considering the uncertainty of traffic conditions, a series of adaptive algorithms were also developed. Ref. [
12] gave an intensive study of the performance of an adaptive traffic-responsive strategy that adjusts traffic light parameters in an urban network to mitigate strong degradation of the infrastructure. In [
13], an adaptive multi-input and multi-output linear-quadratic regulator was designed based on a delay-related state space model. This proposed strategy was tested in a 35-intersection microscopic traffic simulation environment and proved to be more computationally feasible than existing centralized signal control methods, including deep Q learning-based methods, max-pressure, and self-organizing methods. Ref. [
14] proposed a method for evaluating the effect of regional traffic control based on genetic neural networks. At the same time, some researchers also studied the impact of pedestrians on traffic flow. In [
15], the concept of a dynamic traffic management system was extended by designing control strategies for pedestrian flow, learning the experience of a vehicle traffic management system. Ref [
16] developed a pedestrian-safety-aware traffic signal control strategy aiming to minimize vehicle traveling delay and reduce pedestrian crossing risk.
However, the aforementioned methods were mostly model-based, and it is difficult to accurately model traffic dynamics because of the randomness of demand, the difference in driver behavior, and the complexity of intersections. Therefore, model-based methods are difficult to apply in the actual process and their performance cannot be guaranteed. On the other hand, with the development of detection technology, massive traffic data were stored in everyday operations, providing a solid foundation for data-driven controller design [
17]. Thus, the so-called data-driven control methods point out a promising direction in future traffic control.
As an effective data-driven control method, model-free adaptive control (MFAC) gives an originally proposed framework to address a class of nonlinear non-affine systems based on a concept called pseudo-partial derivative [
18]. In order to treat different work scenarios, MFAC offers a compact/partial/full form dynamic linearization data model for controller design [
19,
20]. For the stability analysis, based on some acceptable assumptions, MFAC proves the convergence of the error dynamics by using a contraction mapping principle [
21]. Additionally, model-free adaptive iterative learning control (MFAILC) was developed to achieve perfect tracking performance when handling repetitive tasks [
22].
In addition, MFAC was widely used in many fields as well, such as underwater vehicle manipulators, torque control of asynchronous motors, unmanned surface vehicles, cable-driven robots, and multi-agent systems [
23,
24,
25,
26,
27,
28]. The feature of independence of mathematical models also leads to the application intelligent transportation field. To ease traffic pressure in a macroscopic sense, [
29] provided a novel data-driven constrained model-free adaptive predictive control scheme for the multi-region urban perimeter control problem, where the merits of the MFAC method and model predictive control approach were combined. Similarly, Ref. [
30] proposed a model-free adaptive iterative learning perimeter control (MFAILPC) scheme based on data collected from past days. By mining the repetitive operation pattern via historical traffic data, the MFAILPC improves performance iteratively and the learning gain is tuned adaptively along the iterative axis. To overcome the strong coupled characteristic of the traffic system, ref. [
31] proposed a novel data-driven strategy called decentralized estimation and decentralized MFAC method for the multi-region perimeter control problem, with the key advantage that it can only use traffic data, instead of the traffic model.
In the intersection level, ref. [
32] designed a MFAC algorithm for signal intersections control problem, to achieve queue length equalization, obtain less delay, and increase travel efficiency. Moreover, ref. [
33] proposed a novel distributed model-free adaptive predictive control (D-MFAPC) approach for multi-region urban traffic networks. Different from the traditional design process, D-MFAPC uses the dynamic linearized data models instead of mathematical traffic models as the prediction model in the distributed control design. Additionally, the formulated control problem was finally solved by an alternating direction method with a multipliers-based approach. Motivated by the similarity and repeatability of the traffic flow, ref. [
34] proposed a data-driven MFAILC urban traffic control strategy, in which the urban traffic dynamics is dynamically linearized along the iteration axis. Then, an iterative compensation algorithm was added to the MFAILC to handle the data dropout in the real traffic network system.
Although the above MFAC methods studied the intersections control, they cannot be applied to general multi-phase intersections directly, and they show certain limitations in self-regulation ability facing fast-varying traffic flow. So, in order to solve the above-mentioned issues, an improved method needs to be further studied. Inspired by the symmetric phenomenon, one promising way is to adjust the green time making the traffic flow into a symmetric state. The balanced flow in the intersection further reduces time delay and releases the congestion. Additionally, enlightened by the strong approximating capability of artificial neural networks, this paper tried to construct a hierarchical frame to improving the parameters adjusting process.
The main contributions of this work are summarized as follows:
- (1)
Based on the full-format dynamic linearization (FFDL), a novel MFAC traffic signal control scheme was proposed for multi-phase intersections. The raised scheme combines data-driven prediction technique with symmetrical queuing equalization rules in order to balance the pressure of each phase.
- (2)
A two-layer parameters tuning framework was designed for the MFAC controller, aiming to deal with the fast-changing demand. In the first layer, radial basis function neural network (RBFNN) was used to just two key parameters in the second layer (i.e., η(k) and μ(k)) based on the error function. Then, the two adjusted parameters drove the projection algorithm to estimate pseudo partitioned Jacobian and give a prediction of queuing length.
- (3)
A variable cycle mechanism was added to the above algorithm to make it work for different traffic patterns and further reduce the time loss of vehicles. Finally, the proposed method was tested on the micro traffic flow simulation platform compared with other three control methods. The simulation results showed the superiority of the proposed method.
The remainder of the paper is organized as follows:
Section 2 describes the problem formulation and preliminaries, and multi-phase traffic signal controller design is introduced in
Section 3.
Section 4 presents the applying steps of the proposed algorithm. In
Section 5, simulations are given to prove the effectiveness of the proposed algorithm. The conclusions are given in
Section 6.
5. Simulation
5.1. Simulation Platform
The simulation was conducted on SUMO (Simulation of Urban Mobility), an open-source microscopic traffic simulation software. It supports the user-defined control algorithm obtaining real-time traffic information. The test was developed by Python using the TraCI (Traffic Control Interface) interface, while controlling the vehicle states and signal light states in real time. It is noteworthy that Algorithm 1 was also presented in the SUMO environment. Thus, one can easily recurrent or transplant the method into the traffic signal controller in the field test. Please see the official website for more details of SUMO.
The simulation intersection is shown in
Figure 4 with four phases. There were three lanes in each direction: straight, left turn and right turn, and the roads were set as 600 m, the detectors in each phase were set as 500 m (i.e., the queuing length can be measured completely). The simulation time was 19,800 s.
The test platform versions were SUMO 1.15.0 and Python 3.10.0. Additionally, the CPU was 11th Gen Intel(R) Core (TM) i7-11800H @ 2.30GHz.
5.2. Low Traffic Demand
The flow rates of vehicles used in the simulations are shown in
Figure 5, which were the actual values with fluctuations in a certain range based on the values in
Table 1. The traffic flow of arriving vehicles was subjected to the Formula (29):
where
ñ(
k) and
n(
k) represent the actual flow rates and basic flow rates as
Table 2, and
U represents a random variable subjecting to the uniform distribution. For example, in the time interval 0–6600 s, the vehicle base arrival rate of Phase 2 was 1/10. So, the actual vehicle arrival rate of this phase was [1/10 − 1/25, 1/10 + 1/25] = [3/50, 7/50], i.e., the number of arrived vehicles was a random number between [6 and 14] for 100 s.
- (1)
Fixed timing control (FC)
In the fixed time control strategy, the green time and cycle were set to be constants as g1 = 31, g2 = 30, g3 = 29, g4 = 30, yy = 3, C = 132, and the cycle number Kf = 150.
- (2)
Linear control (LC)
See more details of the control flow diagram in [
3]. Based on the cut-and-trial method, parameters were chosen as
α1 =
α2 =
α3 =
α4 = 2;
β1 =
β2 =
β3 =
β4 = 0.1;
gmin = 15,
gmax = 60.
- (3)
Variable-period queuing feedback control (VQF)
In this strategy, the green time for each phase was calculated by the equation , with parameters gi(1) = 30, C (1) = 132, yy = 3, gmin = 15, i = 1, 2, 3, 4.
- (4)
FFDL control based on queuing feedback (FFDL-QF)
For comparison purse, FFDL-QF strategy (without the two-layer tuning part) comprises of (6)–(9) and (14)–(15) were used, where gi(1) = 30, ∆gi(1) = 1, yy = 3, C = 132, a = 0.9, b = 0.1, η = 0.01, μ = 0.1, li(1) = 0, ∆li(1) = 1, , , I is the matrix with element 1, and the cycle number KF = 150, gmin = 15, i = 1, 2, 3, 4.
- (5)
The proposed algorithm control (FFDL-RBFNN)
For the proposed method, the parameters were chosen as gi(1) = 30, ∆gi(1) = 1, yy = 3, C(1) = 132, a = 0.9, b = 0.1, li(1) = 0, ∆li(1) = 1, yi(1) = 0, yi*(1) = 0, ∆yi(1) = 1, α = 0.75, β = 0.5, , , , , I is the matrix with element 1, O is the matrix with element 0, vp = [−1, −2, 3, 2, 0], σp = 2, gmin = 15, i = 1, 2, 3, 4, and T = 1000.
The dimension of
in the calculation process was 4 × 20, consisting of five matrices whose dimensions were 4 × 4. To show the estimating process and avoid the overlap, five elements
φ111,
φ112,
φ113,
φ114, and
φ115 (i.e., the elements in the first row and first column of
Φ1(
k) −
Φ5(
k)) were selected for drawing in
Figure 6.
Figure 7 shows the histograms of green times for each phase,
Figure 8 shows the values of
η and
μ under low traffic demand, and
Figure 9 shows the histograms of the sum of the maximum queuing lengths of the four phases in each cycle for the different control strategies under low traffic demand.
Observing
Figure 9 and
Table 3, it can be seen that the queuing lengths and time loss in FC were much longer in this random traffic flow. FFDL-QF and VQF (with 162 cycles running) showed better performance in turn. Finally, the proposed FFDL-RBFNN (with 163 cycles running) had an optimal control effect with shortest average queuing lengths of each cycle (42.22% less than the FC) and less average time loss of vehicles (31.21% less than the FC) under low traffic demand.
5.3. High Traffic Demand
Table 4 shows the basic vehicle arrival rate under high traffic demand. In the high demand case, the traffic flow still followed Equation (29), while the base arriving rate was increased.
Figure 10 shows the actual values with fluctuations in the certain range [−1/25, 1/25], which were used to simulate traffic uncertainty in the real system.
- (1)
Fixed timing control (FC)
In the high demand case, after ten times tests, we manually found the following relative good signal timing: g1 = 29, g2 = 28, g3 = 34, g4 = 29, yy = 3, C = 132, and the cycle number Kf = 150.
- (2)
Linear control (LC)
Parameters were the same as the low traffic demand case.
- (3)
Variable-period queuing feedback control (VQF)
Parameters were the same as the low traffic demand case.
- (4)
FFDL control based on queuing feedback (FFDL-QF)
Parameters were the same as the low traffic demand case.
- (5)
The proposed algorithm control (FFDL-RBFNN)
Parameters were the same as the low traffic demand case.
Figure 10 represents actual number of vehicles arriving in each phase per cycle under high traffic demand.
Figure 11 shows the changes in the values of
φ11L.
Figure 12 depicts the histograms of green times for four phases in each cycle.
Figure 13 shows the values of
η and
μ under high traffic demand. Additionally,
Figure 14 shows the histograms of queuing lengths of each cycle of different control strategies under high traffic demand.
The result is summarized in
Table 5. In this case, FC performed poorly against high demand and fast-changing flow. This usually happened if the signal time was not adjusted according to the change of the area traffic flow after a long time. FFDL-QF and VQF improved a certain level compared with FC, but there still was room to promote. Observing the results in
Table 5 and
Table 3, it can be seen that LC was effective under low traffic demand. However, it did not perform well under high traffic demand. The main reason for this was that it was designed based on the simplified linear relationship of each phase, and only analyzed under on the unsaturated condition (note that the high demand case may cause an oversaturated condition). The proposed FFDL-RBFNN produced the shortest average queuing lengths of each cycle (58.00% less than the FC) and less average time loss of vehicles (54.78% less than FC), which showed its superiority in a high demand fast-changing traffic flow. In addition, one may also notice that the cycle number of FFDL-RBFNN was 141 (original
Kf = 150), which indicates the cycle tended to increase in a high demand case.
6. Conclusions
In this paper, a novel MFAC-based symmetrical signal timing design was proposed by combining FFDL and RBFNN. To treat with fast-changing traffic flow in the real world, a two-layer tuning framework was designed, with BRFNN in the first layer continuously accelerating the PPJM estimation process in the second layer. Then, the variable cycle algorithm was further introduced to explore the possibility of single intersection signal control. In order to draw as close as possible to the real traffic flow, this paper set up two simulation scenarios: low traffic demand and high traffic demand superposed with fast-changing random traffic flow. Simulation tests on the SUMO showed that the proposed method outperformed the other four chosen control strategies (FC, LC, FFDL-QF, and VQF).
However, there were still some limitations. This study did not analyze the impact of buses, non-motor vehicles, and pedestrians on traffic flow in detail, which can be added to traffic control for specific analysis in the future. Additionally, the future work included two directions. Firstly, the two-layer tuning frame should be extended from a single intersection to a multi-intersection region. Secondly, an algorithm choosing the initial parameters can be developed. As a priority, the research group is making an attempt to apply the proposed method to real intersection signal control systems.