Coordinated ramp signal optimization framework based on time series flux-correlation analysis

Traffic flow prediction

Using large-scale historical traffic data for traffic flow prediction and solving traffic management, guidance and route planning problems have become a hot research topic today (Uras et al., 2020). In essence, traffic flow forecasting is the extraction of experience and knowledge from relevant historical data to estimate the future state. Previous studies can usually be divided into parametric and nonparametric methods (Zhang et al., 2018). Parametric methods provide simple estimates of future traffic conditions with low computational complexity. However, they are only applicable to specific traffic data conditions because changes in external conditions and the randomness and nonlinearity of traffic flow can impact prediction accuracy (Kong et al., 2020, 2021).

Time series analysis models use curve fitting and parameter estimation to predict future traffic flow information. The most typical method is the autoregressive integrated moving average (ARIMA). The ARIMA model adds an integral link to the autoregressive and sliding average models to eliminate short-term fluctuations in the time series. Many variants have been derived based on ARIMA, such as SARIMA (Williams & Hoel, 2003), which adds a periodic term to this base. This method is suitable for smoother traffic flows and is not sufficient for predicting complex traffic conditions.

Nonparametric models have unique advantages. However, these traditional machine learning methods require labeling data in model training. Furthermore, these methods capture traffic flow characteristics using artificial features, making it difficult to achieve desirable prediction results. Kumar, Parida & Katiyar (2013) used traffic volume, speed, density and time as input variables and used the artificial neural network for short time traffic flow prediction.

Many deep learning models have been proposed to solve the traffic flow prediction problem with the more expansive urban road sensing device arrangement and improved recognition accuracy. Stacked Autoencoder (SAE) (Lv et al., 2014) uses a hierarchical greedy algorithm to obtain spatio-temporal traffic flow characteristics. Recurrent neural networks have been widely used for short-time traffic flow prediction because of their ability to process arbitrary length inputs using memory units. LSTM and GRU (Cho et al., 2014) derived from RNN have shown better prediction performance. Zheng et al. (2020) proposed a convolutional LSTM neural network based on attention mechanism to extract the Spatio-temporal features of traffic flow. They also use Bi-LSTM module to extract the daily and weekly long-term features of traffic flow. Liu, Wang & Zhu (2018) used a gated CNN instead of LSTM to extract the temporal features of traffic flow and combined with CNN’s spatial features to predict the traffic flow. Besides, Traffic flow prediction with graph convolutional neural networks is becoming a trend (Shen, Zhao & Kong, 2021). Han proposed Dirgraph Convolutional Neural Network to tackle the congestion recognition problem through a new graph feature extraction method (Han et al., 2020).

In this article, we use the GRU neural network for traffic flow prediction. Firstly, we mainly forecast the downstream section’s traffic volume, so the traffic sequence’s temporal characteristic should be mainly concerned. Secondly, the traffic flow of the freeway is relatively stable. The variation of traffic flow in each section of the freeway mainly depends on the traffic flow entering and leaving the ramp, so we do not consider the traffic flow’s spatial characteristics. Finally, compared with other neural network methods, the design of GRU is more straightforward and meets our requirements for operability.

Time-series correlation

The correlation measurement between the vectors generally achieves through the distance matrix, including Euclidean distance (Berkhin, 2006) and Manhattan distance (Bakar et al., 2006). Conventional methods are generally used for two time-series data of the same length and different time lags. However, since there is necessarily a time lag between the traffic flow time series, there is no point-to-point correspondence between the two on the time axis. Researchers have come up with several ways to overcomes this shortcoming, such as dynamic time warping (Berndt & Clifford, 1994) and cross-correlation (Liao, 2005). In this article, cross-correlation is selected for correlation measurement. It has been applied to anomaly detection of key performance indicators (Li et al., 2018) and the classification of traffic smart card data (He, Agard & Trépanier, 2020). Meanwhile, it is also widely used in the field of time series clustering (Paparrizos & Gravano, 2015).

Su et al. (2019) first proposes the concept of flux-correlation. This research proposed an unsupervised method to determine the correlation of server KPI fluctuations as well as the direction of fluctuations. If the fluctuations of one series are correlated with the fluctuations of another series over a period of time, then define two series are flux-correlated. We compare the flux-correlation between ramps and bottleneck sections in this article to explore the correlation between flow fluctuations.

Multi-ramp coordination signal optimization

In addition to ramp signal optimization, other methods include the mainline variable speed limit. However, because there are few mainline variable speed limit applications, the actual deployment is difficult and the safety risks are high. Therefore, MPC control strategy is mostly used at present, which is a typical nonlinear optimal control method that can predict the control effect to achieve control optimality, most commonly by combining the mainline variable speed limit method and the ramp control method to achieve synergistic control of both (Hegyi, De Schutter & Hellendoorn, 2005; Van de Weg et al., 2019).

Multi-ramp coordination signal optimization methods can be mainly divided into model-based and model-free methods (Papageorgiou & Kotsialos, 2002). The optimal coordination method is realized based on the traffic flow model. According to the real-time traffic flow information, the control rate is solved by taking the shortest travel time and waiting time as the control objectives and the mainline capacity and ramp queue length as the constraints. Traffic flow models such as the Payne model (Chang & Li, 2002), the cell transmission model (Chen, Lin & Jiang, 2017; Meng & Khoo, 2010; Schmitt, Ramesh & Lygeros, 2017) and METANET (Dabiri & Kulcsar, 2017; Frejo & Camacho, 2012; Kontorinaki, Karafyllis & Papageorgiou, 2019) are widely used. Meshkat applied a quantitative hierarchical model to ramp coordination signal optimization for the first time (Meshkat et al., 2015). Chen added real-time OD information based on Meshkat to determine the priority of on-ramp flow through the total vehicle travel distance (Chen et al., 2019b).

In the field of the model-free method, most researches focus on various heuristic algorithms, such as HELPER (Lipp, Corcoran & Hickman, 1991), SWARM (Paesani et al., 1997) and HERO (Papamichail et al., 2010b). In meanwhile, many researchers adopt a data-driven approach to optimize ramp signals. Chen uses large-scale traffic data and integrates external weather factors to analyze and model the evolution pattern of traffic congestion. The signal duration adjusts dynamically through dynamic congestion threshold classification and congestion mode clustering. For the first time, the analysis of big traffic data was applied to ramp signal optimization, with more careful considerations and more consistent with the actual situation (Chen et al., 2019a). Zhang uses large-scale vehicle trajectory data to extract the vehicle on-ramp behavior pattern, trace the source of congestion formation, and optimize the signal duration of multiple ramps (Zhang et al., 2019).

We combine the dynamic traffic flow correlation based on the traditional heuristic method. The excess traffic demand is assigned by tracing the distribution weights of the congestion sources.

Data pre-processing

Time series data is a set of observations, and time series are generally discrete collections of discrete points that contain temporal relationships in contrast to other vectors. A set of traffic flow time series data is continuous observed values collected by a loop detector according to equally spaced time stamps. A set of time series can be represented as $Q=\left[q_1,q_2,...,q_i,...,q_n\right]$, where $q_i$ is the observed traffic flow value based on the time index value. But the raw data is not organized and needs to be filtered, repaired and smoothed. After data cleaning, it will become serial data.

The raw traffic data is the traffic information of each highway’s detection point at a certain moment. We choose roads with complete data. Make sure that there is the least amount of missing data in two consecutive month time periods. The final data for the experiment is obtained and some examples are shown in Table 1 below.

Although the quality of the data obtained through screening is high, noise and missing data inevitably occur, which can negatively affect traffic flow prediction results. Many situations can cause missing and abnormal traffic data, such as data anomalies caused by sensor failures and missing data during data transmission. Noise points are irregularly distributed in the data set, and the judgment of noise points is mainly based on the distribution of data before and after the noise points. If a point is significantly higher or lower than the data before and after, or the value is higher than the normal situation, we judge it as a noise point and remove it. After processing the noise points, it will produce discontinuity of time series, i.e. missing data. In addition, the data itself may miss some points of data because of equipment failure and other reasons, so it is significant for data repairing. The processing methods for missing data are divided into short-time missing and long-time missing. Short-time missing refers to missing data around 5 min. We directly use the last time point instead. Long-time missing refers to the absence of multiple consecutive time points, and we take the average value of the historical data for the same period as a substitute. Finally, the data are smoothed based on the use of the locally estimated scatterplot smoothing method. The processed data is smoother and more continuous than the raw data, which is consistent with the real situation and helps the neural network’s training, and improves the neural network’s prediction performance.

This article uses neural network and time series methods to analyze dynamic traffic flows. In order to ensure real-time optimization, we first make a short-term prediction of the traffic flow, because accurate traffic flow forecasting is a key part of the overall framework. This article uses GRU neural network for online traffic flow prediction. The neural network can better describe the randomness and nonlinearity of traffic flow (Fu, Zhang & Li, 2016) and get more accurate prediction results compared to linear models such as ARIMA. GRU was proposed by Cho et al. (2014) and is very similar to LSTM, but is simpler to compute compared to LSTM. Figure 2 shows the structure of GRU neural network.

The input and output structure of GRU is similar to a typical recurrent neural network. The hidden state of memory cells is computed in the following formulas:

(1) $$z_t=\sigma (W_z\cdot [h_{t-1},x_t])$$

(2) $$r_t=\sigma (W_r\cdot [h_{t-1},x_t])$$

(3) $${\tilde{h}}_t=\tanh (W\cdot [r_t\ast h_{t-1},x_t])$$

(4) $$h_t=(1-z_t)\ast h_{t-1}+z_t\ast {\tilde{h}}_t$$

For the current node, the input values are the current input x^t and the hidden state h^t−1 containing the information of the previous node, and the output values are the output y^t of the current node and the hidden state h^t containing the information of this node. GRU first obtains the gating states z and r by two input values, where z is the gate that controls the update and r is the gate that controls the reset. Then it calculates the candidate hidden layer h^t, which represents the new information at the current moment, and controls the retention of the previous memory with r. Finally, both forgetting and remembering steps are performed simultaneously, using the update gate z to control the amount of information that needs to be forgotten from the previous moment’s hidden layer h^t−1, and the amount of memory for the currently hidden layer information h^t.

Traffic flow analysis

Once the predicted values of the mainline and ramp flow are obtained, we measure the flux-correlation between flow-series, to find the relationship between mainline and ramps. In this article, we choose the cross-correlation algorithm (Paparrizos & Gravano, 2015) to measure the correlation, which is a similarity measure for time-lagged time series of traffic flow. The value of cross-correlation ranges between [−1, 1], while closing to 1, indicating a strong correlation and a strong negative correlation close to −1. For the time series X = (x₁, x₂, …, x_n) and Y = (y₁, y₂, …, y_n), the reciprocal method holds Y stationary so that X slides along Y. For each slide s of X, calculate their inner product as shown in the following equation.

(5) $$X=(x_1,x_2,\dots ,x_{\mathrm{n}})$$

(6) $$Y_s=\{\begin{array}{cc}(\stackrel{|s|}{\overbrace{0,\dots ,0}},y_1,y_2,\dots ,Y_{n-s})& S\ge 0\\ (y_{1-s},\dots ,y_{n-1},y_n,\underset{|s|}{\underbrace{0,\dots ,0}})& S<0\end{array}$$

For all possible slides s, the inner product CC (X, Y) is computed as the similarity between the two time series X and Y. The equation is shown below.

(7) $$\mathrm{C}\mathrm{C}(X,Y)=\{\begin{array}{cc}\sum _{i=1}^{n-s}{x}_{s+i}\cdot y_i& s\ge 0\\ \sum _{i=1}^{n+s}{x}_i\cdot y_{i-s}& s<0\end{array}$$

The cross-correlation is the maximum value of the inner product, which represents the similarity between X and Y under optimal phase sliding s conditions. Because under optimal sliding conditions, pattern-similar X and Y are exactly aligned, the internal product of both is maximal. Therefore, the Cross-correlation method overcomes the phase sliding problem and compares the shape similarity of two-time series. In practice, the normalized value of Cross-correlation is usually used to limit the range to be within [−1, 1], where 1 means strong correlation and −1 means that they are completely opposite. Besides, a positive NCC means that two series are in the same direction, while a negative NCC means that when one series tends to increase, the other tends to decrease and vise versa. The Eq. (8) defines NCC.

(8) $$NCC(X,Y)=\underset{s}{max}({\displaystyle \frac{CC(X,Y)}{{\Vert X\Vert }^2\cdot {\Vert Y\Vert }^2})}$$

Generally, the distance between the bottleneck and ramp determines the weight matrix. In this article, we combine several factors to determine the weight matrix, including correlation, predicted ramp flow and distance. We measure the cross-correlation between the predicted flow-series of bottleneck and upstream ramps. If the value is greater than 0, it means that the ramps positively correlated with the bottleneck. The magnitude of the coefficient represents the degree of correlation. The value of correlation represents by NCC, together with the physical distance d(i, j) between the ramp and the bottleneck, and the predicted ramp traffic flow q(j) are taken as feature parameters of the weight matrix. If the number of correlations is less than 0, it means that the ramp flow is negatively correlated with the bottleneck flow, so the correlation has the opposite effect. Therefore, we only consider the physical distance and the predicted ramp flow as the feature parameters of the weight matrix.

As Eq. (9) shows, w(i, j) represents the weight of the j-th ramp for the i-th bottleneck. d(i, j) and q(j) are normalized parameters. $\mu$ represents the correlation limitation, less than which means weak correlation between the time series, while a greater set shows strong correlation. This value is generally intermediate, in this Equation is 0.5. $\alpha$, $\beta$ represents the weight of the temporal correlation coefficient, in the weak correlation condition, correlation shows light impact, while the strong correlation condition has a greater impact.

(9) $$w(i,j)=\{\begin{array}{cc}{\alpha }_1\cdot d(i,j)+{\beta }_1\cdot q(j)& \mathrm{N}\mathrm{C}\mathrm{C}\in [-1,0]\\ {\alpha }_2\cdot \mathrm{N}\mathrm{C}\mathrm{C}(i,j)+{\beta }_2\cdot d(i,j)+(1-{\alpha }_2-{\beta }_2)\cdot q(j)& \mathrm{N}\mathrm{C}\mathrm{C}\in (0,\mu ]\\ {\alpha }_3\cdot \mathrm{N}\mathrm{C}\mathrm{C}(i,j)+{\beta }_3\cdot d(i,j)+(1-{\alpha }_3-{\beta }_3)\cdot q(j)& \mathrm{N}\mathrm{C}\mathrm{C}\in (\mu ,1]\end{array}$$

To explore the factors that are most relevant to the traffic states, (Yu et al., 2019) used the gray correlation analysis to measure the similarity between the flow, occupancy, density curves, and the speed curves. Grey correlation determines the degree of association between two factors based on how similar or dissimilar the trends are. It is often used to determine the relative strength of a project influenced by other factors. In this article, we compare the bottleneck speed curves with correlation value, the flow value of the on-ramps and the distance to optimize the parameters in Eq. (9).

For $Att{r}_{i,j}=(a_{1,j},a_{2,j},...,a_{k,j}),a_{i,j}\in \left\{ncc,inflow,distance\right\}$, we can obtain the degree of association $GR{C}_{i,j}=(gr{c}_{1,j},gr{c}_{2,j},...,gr{c}_{k,j}),{\sum }_{i=1}^{k}gr{c}_{i,j}=1$, and then the influence degree of ramp inflow and downstream speed. The influence degree shows how much influence will the factor affects downstream speed as Eq. (10), where the rs means road section number and k represents the factor number.

(10) $$\mathrm{I}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{D}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}{\mathrm{e}}_{\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}}={\displaystyle \frac{1}{rs}\sum _{i=1}^k\sum _{j=1}^{rs}gr{c}_{i,j}}$$

Equation (11) calculates the final weight matrix. Where m is the number of bottlenecks, this weighting matrix will be used in signal coordination optimization to assign the excess traffic demand, in order to ensure the rationality of the traffic reduction at each ramp.

(11) $$W_{ij}=\left|\begin{array}{ccccc}w(1,1)& & \cdots & & w(1,m)\\ & \ddots & \cdots & \ddots \\ 0& ⋮& w(i,j)& ⋮& ⋮\\ & 0& \cdots & \ddots & \\ 0& & 0& & w(n,m)\end{array}\right|$$

Coordinated signal optimization

The coordinated signal optimization method proposed in this article implements a dynamic weight matrix between the ramp and mainline. The application of this matrix in the bottleneck algorithm is mainly reflected in the calculation of the coordinated metering rate. The bottleneck algorithm first determines whether the road segment downstream of the ramp is a bottleneck based on the real-time occupancy or density of the road segment. When there are no bottlenecks in the multi-ramp area, the local metering rate is applied to each on-ramp. When a bottleneck generates in a road segment, the excess traffic demand for each bottleneck generated needs to be allocated to each ramp. The more restrictive value of the coordinated and local metering rates is taken, and the final ramp metering rate is obtained by considering the ramp queue length limitation.

Figure 3 shows a segment of multi-ramp road section. Initially, the local metering rate $r_L$ of each ramp at k time is calculated according to Eq. (12). The K represents metering parameter, and $\hat{\rho }$ is the critical density of ramp downstream road section, $\rho (k-1)$ is average density of downstream road section at last time step.

(12) $$r_L(j,k+1)=r_L(j,k)+K\cdot [\hat{\rho }-\rho (k)]$$

Defining the difference between the entering flow (including the upstream inflow from the mainline and the flow entering the entrance ramp), and exiting flow (including the downstream outflow from the mainline and the flow exiting the exit ramp) as the excess demand for that segment, see Eq. (13).

(13) $$\mathrm{\Delta }d(i,k+1)=q_{in}(i,k)+q_{on}(i,k)-q_{out}(i,k)-q_{off}(i,k)$$

If the average density in a control cycle is higher than the threshold (Eq. (14)); at the same time the excess demand is more than zero (Eq. (15)), there is a risk of breakdown, defining such a segment as a bottleneck.

(14) $$\rho \left(i,k\right)>{\rho }_{\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}}\left(i\right)$$

(15) $$\mathrm{\Delta }d(i,k+1)>0$$

It is necessary to calculate the coordinated metering rate to eliminate bottlenecks, and the coordinated metering rate is calculated as shown in the Algorithm 1.

Every bottleneck needs to reduce the excess demand $\mathrm{\Delta }d(i,k+1)$, when $\mathrm{\Delta }d(i,k+1)<0$, the bottleneck is eliminating, $q_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(i,k+1)=0$. Eq. (17) shows the volume reduction of every ramp, $W_{ji}$ is weight matrix. The max volume reduction is selected to calculate the coordinate metering rate in Eq. (18), based on metering rate r(j, k) last time step.

(16) $$q_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(i,k+1)=\mathrm{\Delta }d(i,k+1)$$

(17) $$q_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(i,j,k+1)=q_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(i,k+1)\cdot W_{ji}$$

(18) $$r_C(j,k+1)=r(j,k)-\underset{i}{max}[q_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}(i,j,k+1)]$$

The system metering rate adopts the more restrictive of coordinate metering rate and local metering rate as Eq. (19) shows.

(19) $$r^{\mathrm{\prime }}(j,k+1)=min[r_L(j,k+1),r_C(j,k+1)]$$

If the ramp metering rate approaches zero, it will cause a ramp spillback, thus affect surface traffic. Therefore, the metering rate needs to be constrained based on the queue length of the ramp. In Eq. (20), $a(j,k)$ is the arrival rate of vehicles entering the entrance ramp, $\hat{\omega }(j)$ is the capacity of the j-th ramp, $\omega (j,k)$ is the queue length of the j-th on-ramp at the previous time step, and T refers to a time step. The metering rate can be obtained by Eq. (21).

(20) $$r_{\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{u}\mathrm{e}}(j,k+1)=a(j,k)-{\displaystyle \frac{1}{T}[\hat{\omega }(j)-\omega (j,k)]}$$

(21) $$r(j,k+1)=max[r^{\mathrm{\prime }}(j,k+1),r_{\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{u}\mathrm{e}}(j,k+1)]$$

The green duration is calculated based on the metering rate and sent to each signaler in Eq. (22). Where C(j) indicates the signal cycle duration and r_s(j) indicates the saturation flow rate.

(22) $$g(j,k+1)={\displaystyle \frac{r(j,k+1)}{r_s(j)}\times C(j)}$$

Traffic Flow Prediction

The dataset used in this article is collected from the California PEMS platform, which provides 5-min intervals of freeway mainline loop detector data (both speed and flow) and on-ramp passing data. We select the flow data for the northbound upstream mainline and on-ramp of the I110 freeway from 1 January 2020 to 26 February 2020, which has four on-ramps with a total length of approximately 3.8 KM. The detector deployment locations show in Fig. 4.

Combined with the local traffic flow characteristics, it can be found that the traffic flow is at its peak during 15:00–18:00, which is suitable for applying ramp control. Ramp control is not sufficient when the traffic flow is low, smooth traffic flow can be achieved using no control or simple demand capacity control (Papageorgiou & Kotsialos, 2002). We plotted the average traffic flow curve of the bottleneck road section over 2 months in Fig. 5.

The red curve in the figure indicates the average traffic flow on the road section during the 2 months. The blue part indicates the interval formed by the maximum and minimum traffic volumes. It can be seen from the graph that the traffic flow starts to increase at 6:00 and stays at a high level for a long time afterward. The interval we selected is the time segment with relatively high traffic volume, which helps verify the effectiveness of the signal optimization scheme. Based on the above findings, flow data from 15:00 to 18:00 on 13 February 2019 upstream of the mainline and the four ramps were selected as inputs, as shown in Fig. 6.

In this article, the GRU neural network is used for traffic flow prediction. The best lag time can minimize the prediction error of the model, and we use the maximum lag time of 60 min. It is more reasonable to use 12 historical values to predict the latter value, and fewer historical values lead to more deviations in the prediction results. More historical values lead to long computation time and are prone to overfitting. In addition to the lag time, we also need to choose the best parameters for the deep neural network model. We use a two-layer GRU neural network with 64 hidden cells in each layer, a batch size of 256, and a max epoch of 600. The training set uses forty-eight days of data (84.2% of the total number of days), the test set uses the remaining nine days of data (15.8%).

We compare the GRU neural network with two other methods, SAEs and LSTMs, respectively. We use the rooted mean squared error (RMSE) and mean absolute percentage error (MAPE) as the metric to evaluate these methods. When the predicted value exactly matches the true value, RMSE is equal to 0, indicating a perfect model. RMSE can be calculated as follows:

(23) $$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{m}\sum _{i=1}^m{(y_i-{\hat{y}}_i)}^2}}$$

where $y_i$ is the ground truth and ${\hat{y}}_i$ is the prediction value, m represents the total number of predictions.

Mean absolute percentage error of 0% indicates a perfect model, while a MAPE greater than 100% indicates a flawed model. MAPE can be calculate as follows:

(24) $$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{100\mathrm{\%}}{m}\sum _{i=1}^m\left|{\displaystyle \frac{{\hat{y}}_i-y_i}{y_i}}\right|}$$

where $y_i$ is the ground truth and ${\hat{y}}_i$ is the prediction value, m represents the total number of predictions.

The Table 2 shows the prediction results of various methods. It can be seen that in the RMSE, both LSTM and GRU are much better than SAEs. There is little difference between LSTM and GRU. In the MAPE column, GRU performs the optimal prediction effect.

Figure 7 shows how the predicted value compares with the observed value and other methods. It can be seen from the figure that all curves fit the observed data. During the low-flow phase, the SAEs were predicted to be substantially higher than the true values. And the LSTM and GRU are much closer to the true values.

SUMO platform

Simulation of Urban Mobility (Lopez et al., 2018; Krajzewicz et al., 2012) used in this article is a free and open-source traffic system simulation software compared with the above microsimulation platforms. It can realize microscopic control of traffic flow. Each vehicle’s route on the specific road can be planned individually. Furthermore, it has a mature code control module, which makes data analysis more convenient. The entire SUMO-based simulation framework shows in Fig. 8.

We model multi-ramp section in SUMO, and entering the traffic flow data during the morning peak period. We use the IDM model as car following model. The IDM model (Treiber, Hennecke & Helbing, 2000) is by far the most complete and simple accident-free theoretical heeling model, which belongs to the class of expectation measures. The IDM model is described in Eqs. (25) and (26).

(25) $$v_{\alpha }^{\mathrm{\prime }}=a\left[1-{\left({\displaystyle \frac{v_{\alpha }}{v_0}}\right)}^{\delta }-{\left({\displaystyle \frac{s^{\ast }\left(v_{\alpha },\mathrm{\Delta }{v}_{\alpha }\right)}{s_{\alpha }}}\right)}^2\right]$$

(26) $$s^{\ast }\left(v,\mathrm{\Delta }v\right)=s_0+s_1\sqrt{{\displaystyle \frac{v}{v_0}}}+T^{\alpha }v+{\displaystyle \frac{v\mathrm{\Delta }v}{2\sqrt{ab}}}$$

The vehicle acceleration $v_{\alpha }^{\mathrm{\prime }}$ can be obtained by minimum vehicle distance $s^{\ast }\left(v,\mathrm{\Delta }v\right)$, velocity, vehicle clearance from the vehicle in front $s_{\alpha }$ and expected velocity $v_0$. In Eq. (26), T represents safety time step, $ab$ represent the max acceleration and the expected deceleration ability of vehicles. $\mathrm{\Delta }v$ is speed difference with the car in front, $s_0$ and $s_1$ are distance with congested traffic. For model parameter calibrations, see Table 3 below.

Besides, the channel change model is also very critical. The vehicle lane change model we have chosen is LC2013 (Erdmann, 2015).

In conjunction with the application in the related literature (Papamichail et al., 2010a), the common evaluation metrics are Average Travel Time (ATT), which indicates the travel time of mainline vehicles; Average Waiting Time (AWT), which indicates the waiting time of ramp vehicles. Besides, the density and speed of the bottleneck locations are selected as evaluation metrics, which can reflect the ability to alleviate congestion at the bottleneck locations.