# Stochastic Model Predictive Control for Urban Traffic Networks
## 5 Simulation Results  
In this paper, we perform simulation experiments on the road network consisting of four intersections. Figure 3 illustrates the basic layouts of the test network. It includes eight links, and four among them ( x1, x2, x5, x6) are input sources where the traffic flow enters the test road network. The lengths of the four input links ( x1, x2, x5, x6) are the same and are equal to 600 m, while about 700 m for the rest links (x3, x4, x7, x8). As shown in Figure 3, each intersection has only two phases. Furthermore, assume that the offset between any two intersections is equal to zero. Table 1 shows some basic parameters that will be used in the simulation. The turning rates and input traffic flow are illustrated in Table 2 and Figure 4, respectively. The initial state (the number of vehicles on each link at the beginning of the first control step) of the test network is randomly generated. For instance, \(x_{i}\in[\alpha\cdot V_{n},\beta\cdot V_{n}]\)\((i=1,2,\cdots,8)\) denotes that the number of vehicles on the link \(X_{i}\)\((i=1,2,\cdots,8)\) is randomly generated in the interval \([\alpha\cdot V_{n},\beta\cdot V_{n}]\). Herein, \(\alpha=0.01\), \(\beta=0.9\). According to the length of each link and the average length of each vehicle (as defined in Table 1), we set \(V_{n}=120\) for the link \(X_{i}\)\((i=1,2,5,6)\) and \(V_{n}=140\) for the rest links. Herein, \(V_{n}\) denotes the maximum number of vehicles that can be accommodated in a link. For simplicity, we assume that uncertainties only exist on link \(X_{1}\) and link \(X_{7}\). Furthermore, we assume that the uncertain traffic flow \(d_{1}\) and disturbance \(e_{7}\) are both normally distributed. Aiming to demonstrate the performance of the proposed HIA, the traffic flow \(d_{1}\) and the disturbance \(e_{7}\) shown in Figure 4 are, respectively, assumed to be the mean value of the input flow of \(X_{1}\) and the disturbance of \(X_{7}\). Furthermore, \(d_{1}\) and \(e_{7}\) are both estimated by historical data. As shown in Table 3, the covariance matrix of \(d_{1}\) and \(e_{7}\) are defined as \(\Sigma_{1}\) and \(\Sigma_{2}\), respectively. Note that, for computational simplicity, we assume the covariance matrix \(\Sigma_{1}\) and \(\Sigma_{2}\) remain unchanged during the whole simulation period. Since previous studies [20; 21; 22] have shown that it seems reasonable to select the prediction horizon \(P=3\), the experiment results reported in this study are all achieved under the condition that \(P=3\). As a result, the covariance matrix \(\Sigma_{1}\) and \(\Sigma_{2}\) are three-dimensional matrices. For the measure of solution quality, we report the queue length and the green time of each link in the test network over a simulation period of 20 min (that is six simulation steps). For the sake of simplicity, let MPC-SPQ denote a deterministic MPC model (\(M_{1}\)) that was solved by Sequential Quadratic Programming (SQP), and MPC-GA denotes a deterministic MPC model (\(M_{2}\)) that was solved by Genetic Algorithms (GA). In order to demonstrate the performance of the proposed stochastic MPC model that was solved by HIA (abbreviated as MPC-HIA), the performance is compared with MPC-SPQ and MPC-GA. Firstly, the objective functions both of \(M_{1}\) and \(M_{2}\) are the same and defined as \(\min\limits_{\mathbf{u}}f(\mathbf{x},\mathbf{u})\) and the Equation (18). In addition, except that there are no chance constraints imposed on the queue length (as defined in Equation (12)), the rest constraints of  
Figure 4: Input traffic flow and disturbance of the test network. Figure 3: Test network with signalized intersections.