# Stochastic Model Predictive Control for Urban Traffic Networks
## 5 Simulation Results
On one hand, compared with the first two control steps, the input flow of link \(X_{i}\) (\(i=1,5,6\)) rapidly increases from the third control step. Meanwhile, vehicles entered into the link \(X_{1}\) and \(X_{7}\) increase quickly. On the other hand, the MPC-SQP did not make a proper allotment of green time to the link \(X_{1}\) and link \(X_{7}\) like the proposed model MPC-HIA did. Since there are constraints that are imposed on the queue lengths of both \(X_{1}\) and \(X_{7}\), both MPC-HIA and MPC-GA perform better than MPC-SQP. Figure 6 shows the comparison of the MPC-HIA, MPC-GA and the MPC-SQP on the green time that has been assigned to the link \(X_{1}\) and that of the link \(X_{7}\) at each control step. As shown in Figure 6, during the entire simulation period, the green time of the link \(X_{1}\) (and \(X_{7}\)) achieved by the proposed model MPC-HIA fluctuates within the minimum and maximum permissible green time (for example 30 s and 90 s). The reason for this may be due to the fact that the input flow \(d_{1}\) and the disturbance \(e_{7}\) are uncertain variables whose mean values vary over each control step (as shown in Figure 4). Furthermore, we can see that the proposed model MPC-HIA can adjust the green time assigned to link \(X_{1}\) (and \(X_{7}\)) according to the variational traffic demand (or disturbance) and the maximum allowable queue length of the link \(X_{1}\) (and \(X_{7}\)). This is mainly due to the fact that chance constraint programming was employed into the proposed MPC-HIA model to capture the uncertainties of the  
network. However, the green time of the link \(X_{1}\) (and \(X_{7}\)) obtained by the model MPC-SQP shows little changes between any two adjacent control steps. As a result, the queue length of the link \(X_{1}\) (and \(X_{7}\)) goes beyond the maximum allowable queue length since the fourth control step as shown in Figure 5. It is obvious that results reported in Figure 6 are consistent with those shown in Figure 5.  
Figure 7 shows the comparison of the total queue length (during the entire simulation period) of each link for the three models (MPC-HIA, MPC-GA and MPC-SQP). It is not hard to find that the the total lengths of the link \(X_{i}\) (\(i=3,4,5\)) achieved by both MPC-HIA and MPC-SQP are almost the same. Meanwhile, the total queue length of the link \(X_{i}\) (\(i=1,7\)) achieved by MPC-HIA is smaller than that obtained by MPC-SQP. However, for the link \(X_{i}\) (\(i=2,6,8\)), the situation is the reverse. For MPC-GA, a similar conclusion can be reached. The reason is that chance constraints have been imposed on the queue length of link \(X_{1}\) and that of link \(X_{7}\) in the proposed model MPC-HIA. Furthermore, the stochastic MPC model (MPC-HIA) has a better ability to deal with uncertainties than deterministic MPC models--both \(M_{1}\) (MPC-SQP) and \(M_{2}\) (MPC-GA). For each control step, the performances of the three models (MPC-HIA, MPC-GA and MPC-SQP), in terms of the total queue length of the network, are illustrated in Figure 8. It can be seen clearly that MPC-SQP performs better than MPC-HIA during the entire simulation period. This may be because, compared with MPC-SQP, there are chance constraints that are imposed on the queue lengths in MPC-SQP. However, MPC-HIA performs better than MPC-GA over the whole simulation period. This is perhaps due to the fact that the stochastic MPC model (MPC-HIA) has better performance than the deterministic MPC model \(M_{2}\) (MPC-GA) in terms of dealing with uncertainties, which are imposed on link \(X_{1}\) and link \(X_{7}\). Figure 5: Comparison of the queue length of \(X_{1}\) and \(X_{7}\) at each control step. In addition, to evaluate the performance of computational cost, we report the average computation time of ten independent runs for solving the MPC optimization problem with the three methods (MPC-SQP, MPC-GA and MPC-HIA). The computational time per simulation of MPC-SQP, MPC-GA and MPC-HIA are 1.653 s, 460 s and 16,932.5 s, respectively.