## 3 Proposed Coordinated Control Method
### Strategy 1: Green Phase Extension Strategy  
For the assumed intersection signal phases, consider the coordinated phase direction at the downstream intersection, which is the westbound through. The non-coordinated phases that do not overlap with their green phases are split into the following: the first turn (eastbound left-turn), the fifth turn (northbound through right), and the sixth turn (southbound through left). The control objective is to extend the green phase time for the westbound through-coordinated direction when a vehicle is detected to be arriving within the unit green extension time. This extension aims to increase the green bandwidth in that direction while simultaneously reducing the green phase time for the non-coordinated directions to balance the phase cycle duration.  
1. Determine the total amount of green phase time that can be reduced for each non-coordinated phase within the cycle while ensuring that the green phase time for each non-coordinated phase \(g^{i}_{L_{d},k,unco}\) is not less than its minimum green time \(g^{i,min}_{L_{d},k,unco}\) and the extended green phase time for the coordinated phase does not exceed the sum of the minimum green times \(g^{min}_{L_{d},k,co}\) for the non-coordinated phases plus the total available green phase time. That is, \[g^{max}_{L_{d},k,co}\leq g^{min}_{L_{d},k,co}+\sum_{i=1,5,6}\left(g^{i}_{L_{d},k,unco}-g^{i,min}_{L_{d},k,unco}\right)\] (9)  
Figure 6: Model assumptions for signal phases of the upstream and downstream intersections.  
Determine the extension time for a unit of green phase \(\Delta g^{le}_{unit}\) when the green phase time for the westbound through-coordinated phase at the downstream intersection reaches the initially pre-set green phase time, based on the detectors of the vehicle arrivals within the unit green extension time \(\Delta g^{le}_{unit}\), determining the extension amount \(g^{le}_{l_{d},k,co}\) for the green phase time of the coordinated phase in cycle \(k\). This value should not exceed the sum of the green phase time, which can be reduced. The green phase time for the coordinated phase after extension, \(g^{new}_{l_{d},k,co}\), is given by \[g^{new}_{l_{d},k,co}=g^{min}_{l_{d},k,co}+g^{le}_{l_{d},k,co}\] (10) 3. Sequentially reduce the green phase time \(g^{i}_{l_{d},k,unco}\) for each non-coordinated phase by the time increments defined by \(\Delta g=1\) s until the total reduction \(\sum\Delta g\) matches the extension amount for the green phase time of the coordinated phase \(g^{le}_{l_{d},k,co}\). Meanwhile, the following constraints ensure that the remaining green phase time \(g^{i}_{l_{d},k,unco}\) for each non-coordinated phase does not fall below the minimum green time \(\hat{g}^{i}_{l_{d},k,unco}\) to balance the phase cycle duration: \[\begin{cases}g^{i,re}_{l_{d},k,unco}=g^{i}_{l_{d},k,unco}-\Delta g\\ \ s.t.\sum\Delta g=g^{le}_{l_{d},k,co}\\ \hat{g}^{i,re}_{l_{d},k,unco}\geq\hat{g}^{i,min}_{l_{d},k,unco}\end{cases} i=1,5,6\] (11)  
After adjusting and allocating the phases at the downstream intersection, the phase cycle duration \(C^{T,new}_{l_{d},k}\) is updated and balanced by  
\[C^{T,new}_{l_{d},k}=g^{new}_{l_{d},k,co}+\sum_{i=1,5,6}g^{i,re}_{l_{d},k,unco} \tag{12}\]