Graphical Optimization Method for Symmetrical Bidirectional Corridor Progression

. The graphical progression method can obtain grand coordinated schemes with minimal computational complexity. However, there is no standardized solution for this method, and only a few related studies have been found thus far. Therefore, based on the in-depth discussion of the graphical optimization theory mechanism, a process-oriented and high-eﬃciency graphical method for symmetrical bidirectional corridor progression is proposed in this study. A two-round rotation transformation optimization process of the progression trajectory characteristic lines (PTC lines) is innovatively proposed. By establishing the updated judgment criteria for coordinated mode, the ﬁrst round of PTC line rotation transformation realizes the optimization of co-ordinated modes and initial oﬀsets. Giving the conditions for stopping rotation transformation and determining rotation points, rotation directions, and rotation angles, the second round of PTC line rotation transformation achieves the ﬁnal optimization of the common signal cycle and oﬀsets. The case study shows that the proposed graphical method can obtain the optimal progression eﬀect through regular graphing and solving, although it can also be solved by highly eﬃcient programming.


Introduction
Intersections are the road network nodes that frequently cause traffic disruption, severe delays, and accidents in a city [1]. erefore, corridor progression has always been an effective way to ensure traffic safety and efficiency at intersections [2,3]. Generally speaking, the solution methods of the corridor progression design scheme can be roughly divided into three types: model method, algebraic method, and graphical method.
In comparison, the model method can obtain multiple different ideal coordination design schemes. However, its modeling process is complex, requiring a long solution time for the sizeable calculated amount. e algebraic method, which can meet the requirements of corridor progression in various situations, has the advantages of good operability and robust reproducibility. However, practitioners often need to have a relatively complete theoretical knowledge of coordination planning. Meanwhile, the graphical method obtains coordination planning using time-space diagrams, illustrating the relationship between intersection spacing, signal timing, and vehicle movement. Although the graphical method is challenging to ensure the optimal corridor progression effect and measure the solution efficiency by manual drawing, it is suitable for engineering applications due to its slight computational complexity, strong operability, and intuitive reflection of the coordinated optimization process. e model method constructs a linear or nonlinear programming optimization model based on the relationships between the progression bandwidth and signal timing parameters, travel time, and progression boundary trajectory [4]. Moreover, the mixed-integer linear programming method is usually used to realize the optimal solution of signal timing parameters. e most classic models are the MAXBAND model proposed by Morgan and Little [5] and the MULTIBAND model constructed by Gartner et al. [6]. e MAXBAND model optimized common signal cycle, offsets, vehicle speed, and the order of left-turn phases to maximize the bandwidths, ensuring that most vehicles can drive through the downstream signal intersections without stopping [7]. e MULTIBAND model designed an individually weighted bandwidth for each directional road section, considering the traffic volumes and flow capacities. Extensive research has been constantly provided based on the MAXBAND and MULTIBAND models [8][9][10]. Optimizing the left-turn phase sequence of the MAXBAND model, Chang et al. [11] further established the MAXBAND-86 model. Lu et al. [12] introduced a bandwidth proration impact factor to improve the bandwidth maximization model further. Defining the signal phase sequence as a decision variable, Yang et al. [13] constructed three types of multipath progression models based on the MAXBAND model. Zhang et al. [14] established an asymmetrical multiband (AM-BAND) model and successfully achieved an excellent coordination effect. Zhang et al. [15] constructed the MAXBANDLA model, and numerical tests proved that the model could significantly reduce the number of stops and the average delay of vehicles on long corridors. Zhou et al. [16] built two multiobjectives uneven double cycles. By doing this, they allowed minor intersections to adopt uneven double cycling models to ensure the coordinated efficiency of minor intersections. Aiming to provide progression bands for all major O-D flows, Arsava et al. [17] constructed the OD-BAND model, which considered the significant turning traffic flows from and to cross streets. Arsava et al. [18] expanded the OD-BAND model to the OD-NETBAND model, solving the OD-based traffic signal coordination problem in multiarterial grid networks. e algebraic method utilizes the numerical calculation method to obtain the optimal corridor progression design schemes, realizing the comprehensive optimization of signal phases. Xu [19] expounded on the classical algebraic method, that is, to determine the optimal common signal cycle and offsets according to the ideal intersection distance that best matches the actual one. Moreover, this method is only applicable under symmetrical phasing. On this basis, the related research of the algebraic method has gradually gained traction. Lu et al. [20] first addressed the theoretical limitations of the classical algebraic method and improved the algebraic method for symmetrical corridor progression. Analyzing the advantages and disadvantages of the classical algebraic method, Wang et al. [21] further improved the calculation method of the bandwidths and the offsets and the location matching method of actual intersection and ideal intersection. As research progressed, the research focus of the corridor progression has been gradually expanded from the algebraic method under the symmetrical phasing to split phasing and asymmetrical phasing. Lu et al. [22] established a bidirectional algebraic method under the split phasing, which can adapt to different corridor conditions, such as asymmetric geometry, large left-turn traffic volume, uneven traffic flow. Ji and Song [23] proposed an asymmetrical bidirectional algebraic method and analyzed the effects of vehicle speed and queued vehicles during the red light. Using the speed transformation and phase combination method, Lu et al. [24] also constructed a bidirectional algebraic method under asymmetrical phasing, effectively solving each road section's asymmetric bidirectional distance and unequal speed. e graphical method establishes a set of easy-to-understand drawing rules and utilizes the transformation of basic elements on the time-space diagram to obtain ideal progression bandwidths. However, scholars have done little research on the graphical method of corridor progression until now. China Highway Association [25] defined the intuitive and straightforward graphical method as one of the earliest timing design methods for corridor progression. Xu [19] provided the basic solution idea of the graphical method, but there are still many deficiencies during the solution process. It lacked a specific design principle and well-defined rules for optimizing the common signal cycle and offsets. Lu and Cheng [26] introduced the concept of the NEMA phase based on the graphical method and optimized the phase sequence and offsets by the graphical way. Although an ideal bidirectional bandwidth was obtained, there was no amendment to the shortcomings of the existing graphical method.
In this study, we define the trajectory lines that reflect the unrestrained movement of characteristic vehicles in the progression as progression trajectory characteristic lines (PTC lines). en, based on the rotation transformation of the PTC line, a graphical method for symmetrical bidirectional corridor progression is proposed to improve further and optimize the overall process of the graphical method. e corresponding progression coordination design process will also be presented in detail. e remainder of the article is organized as follows: Section 3 introduces the design principle of the proposed method. Section 4 presents a case study, which helps to compare the coordinated optimization effect between the proposed method, the improved algebraic method, and the MAXBAND model. Concluding remarks are provided in the last section.

Design Process.
e graphical method proposed can be solved by a normal step-by-step graphical process and programming. We define the horizontal scaling of the timespace diagram to convert the progression design speed of the road section to the corridor progression design speed V. Meanwhile, we also define the vertical scaling of the timespace diagram to realize the transformation of the optimization of the common signal cycle into the optimization of the corridor progression speed. e design process of the proposed graphical method is refined and organized as shown in Figure 1, which is mainly divided into four parts: initialization, the first round of rotation transformation, the second round of rotation transformation, and scheme generation.

Initial Common Signal
Cycle. Let us suppose that there are n signalized intersections along the corridor and number the intersections in the ascending order in the outbound direction. en, the pth (1 ≤ p ≤ n) intersection is defined as I p in sequence along the outbound direction. e common signal cycle range will be determined by the signal cycle range of each intersection, denoted as [C min , C max ]. e initial common signal cycle C 1 can be valued as the midpoint between the minimum common signal cycle C min and the maximum common signal cycle C max .

Adjusted Distance and Speed.
Let d p and v p be the actual distance and the progression design speed between intersection I p and I p + 1 (1 ≤ p ≤ n-1), respectively. en, we find that the optimal effect of progression coordination does not change in the time-space diagram as long as the travel time of the road section remains unchanged. For example, as shown in Figure 2, when we simultaneously change the progression design speed and distance of the road section highlighted by the blue and purple arrows in Figure 2(a), but keep the vehicle travel time unchanged, the progression bandwidth of the corridor remains unchanged in Figure 2 erefore, it can be said that the proposed method can be applied to coordinate corridors with inconsistent progression design speeds across road segments. We define it as the horizontal scaling of the time-space diagram. erefore, to facilitate the diagrammatic design process, when the design speed of the road section is inconsistent with the corridor progression design speed V, we will convert the progression design speed of the road section to V by adjusting the intersection distance. e adjusted distance  Journal of Advanced Transportation D p between intersection I p and intersection I 1 can be calculated by the following equation: On the other hand, we find that the optimal effect of progression coordination does not change in the time-space diagram as long as the product of V and C remains unchanged. For example, as shown in Figure 3, when we change the common signal cycle and corridor progression design speed but keep their product unchanged in Figure 3(a), the percentage of progression bandwidth of the corridor remains unchanged in Figure 3(b). We define it as the vertical scaling of the time-space diagram. en, we can first keep the common signal cycle C 1 unchanged and optimize the adjusted speed to find their optimal product value to obtain the best coordination effect. Finally, the final adjusted speed is adjusted to V, and the optimal common signal cycle C B can be obtained.
According to the corridor progression design speed V and the range of the common signal cycle [C min , C max ], the adjusted speed V (i, j) obtained after the jth rotation transformation in the ith round should satisfy the following equation:

Possible Coordinated Mode.
To balance the effect of bidirectional corridor progression, the coordinated mode of each intersection must adopt synchronous coordination or backstepping coordination. Synchronous coordination means that the green center point of the coordinated phase is consistent with intersection I 1 , whereas backstepping coordination means that the red center point is consistent with the green center point of the coordinated phase at I 1 . e progressions in the outbound direction and the inbound direction have the characteristic of time symmetry, and then, only the outbound direction needs to be considered. en, we define a Boolean variable F p as the coordinated mode judgment factor. When F p is 0, the optimal coordinated mode of the intersection I p is synchronous coordination. When F p is 1, the optimal coordinated mode of I p is backstepping coordination.

e First Round of Rotation
Transformation. e optimization of the intersection coordinated mode can be realized by defining the rules of the first round of PTC line rotation transformation. First, we define the green center point of the coordinated phase at I 1 as the reference point O L1 of the time-space diagram coordinate system. A horizontal line can be drawn from O L1 and rotated until the cotangent of its angle with the x-axis equals the corridor progression design speed V. is process is defined as the first rotation transformation of the PTC line in the first round. We define the rotated ray as the initial PTC line L 1 and then assign V to adjusted speed V (1,1) . e following steps have to be executed cyclically until the coordinated modes of all intersections have been determined.

Primary Judgment Factor Calculation.
e intersection I p is selected as the current coordinated intersection during the first round's pth (2 ≤ p ≤ n) rotation transformation. According to PTC line L p-1 obtained by the last rotation transformation, the crossing point of L p-1 and the timeline of I p is marked as O Lp as shown in Figure 4 e green interval of coordinated phase e red interval of coordinated phase e green interval of coordinated phase Our first step is to define and calculate some basic parameters, and the initialization routine can be presented below.

Initialization procedure
Step0. Determine the range of the common signal cycle [C min , C max ] and calculate the initial common signal cycle Step1. Make a judgment of whether the progression design speeds of all road sections equal the corridor progression design speed or not. If there is an inequality, calculate the intersection adjusted distance D p for the corresponding road section as shown in Equation (1).
Step2. Determine the range of adjusted speed V (i, j) obtained after the jth rotation transformation in the ith round, as shown in Equation (2).
e red interval of coordinated phase e green interval of coordinated phase e range of adjusted speed Timeline of the intersection I p  ), which corresponds to the distance between the ordinate of points y Lp and y Rp (y Gp ), is recorded as T Rp (T Gp ). en, the primary judgment factor f p can be constructed according to T Rp and T Gp : as a Boolean variable. When ray l Rp (l Gp ) does not pass through the red interval of the inter- ). When f p ≥ 0, the situation is shown in Figure 4. If ray l Gp does not pass through any red interval of the coordinated phase, all f G(p,q) equals 0. e current coordinated intersection can form a synchronous coordinated mode with I 1 , where F p � 0. If ray l Gp passes through any red interval of the coordinated phase while ray l Rp does not pass through any red interval, as shown in Figure 4(b), the backstepping coordinated mode can be chosen, where F p � 1. If both l Gp and l Rp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of l Gp is less than or equal to l Rp , as shown in Figure 4(c), these two intersections can form a synchronous coordinated mode, where F p � 0. Otherwise, the backstepping coordinated mode will be chosen as F p � 1.
Similarly, when f p < 0, if ray l Rp does not pass through any red interval of the coordinated phase, all f R (p,q) equals 0, F p � 1. If ray l Rp passes through any red interval of the coordinated phase while ray l Gp does not pass through any red interval, then F p � 0. If both l Rp and l Gp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of l Rp is less than or equal to l Gp , F p � 1. Otherwise, F p � 0.

PTC Line
Optimization. If the current coordinated intersection has formed a synchronous coordinated mode with I 1 and V Gp is within the rotation range of the PTC line, we can define l Gp as PTC line L p and assign the value of V Gp to the adjusted speed V (1, p) , as shown in Figure 5(a). When V Gp is out of the rotation range, we can keep the PTC line unchanged, that is, as shown in Figure 5(b), L p is the same as L p-1 , and V (1, p) equals V (1, p-1) .
Similarly, if the current coordinated intersection has formed a backstepping coordinated mode with I 1 and V Rp is within the rotation range of the PTC line, we can define l Rp as PTC line L p and assign the value of V Rp to V (1,p) . When V Rp is out of the rotation range, we can keep the PTC line unchanged. L p is the same as L p-1 , and V (1, p) equals V (1, p-1) .

e Second Round of Rotation Transformation.
e final optimization of the common signal cycle and the offsets will be determined upon completing second round of PTC line rotation transformation.
After the first round of rotation transformation, the coordinated modes of all signalized intersections have been determined. We assign adjusted speed V (1,n) to V (2,0) . en, the beginning PTC line L B0 and the end PTC line L E0 of the outbound progression can be obtained. en, the initial bandwidth b 0 and ratio R 0 can also be calculated. Meanwhile, the bottleneck intersections of the beginning PTC line are put into the intersection set S B0 , and the bottleneck intersections of the end PTC line are put into the intersection set S E0 . e calculation of related parameters will be introduced in the next section.
We define a termination decision parameter F A , which is a Boolean variable. e following steps have to be executed cyclically until the termination decision parameter F A equals 1.

Bottleneck Intersection Recognition.
We assume that the obtained beginning PTC line, end PTC line, adjusted speed, bandwidth, and bandwidth ratio after the mth rotation transformation in the second round are defined as L Bm , L Em , V (2, m) , b m , and R m , respectively. e parameters obtained from the m-1th rotation transformation need to be used during the mth rotation transformation in the second round.
According to F p , V (2, m-1) , C 1 , and λ p , we can draw the beginning and end PTC lines in the time-space diagram. We define the Boolean variables K B (m-1,p) and K E (m-1,p) as the judgment factors of the bottleneck intersection of the beginning and end PTC lines. When K B (m-1,p) equals 1, intersection I p is the bottleneck intersection of the beginning PTC line after the m-1th rotation transformation, I p ∈ S Bm-1 , and the corresponding bottleneck point is defined as P Bp .
When K E (m-1,p) equals 1, I p is the bottleneck intersection of the end PTC line after the m-1th rotation transformation, I p ∈ S Em-1 , and the corresponding bottleneck point is recorded as P Ep .
Before the other calculation steps of the mth rotation transformation in the second round, it is necessary to judge whether the sets S Bm-1 and S Em-1 meet the conditions for stopping rotation. If the conditions for stopping rotation are unsatisfied, then it must be continued to complete the mth rotation transformation until the bottleneck intersections meet the conditions of stopping rotation, where F A � 1.

Conditions for Stopping the Rotation
, that is, there are both bottleneck points of the beginning and end PTC lines at intersection I k . At this time, the ratio of the bandwidth R m equals the split of the coordinated phase at I k , which means that it has already reached the maximum value of the bandwidth ratio. erefore, there is no need to continue the rotation transformation of the PTC line. Condition 2. ∃ I i ∈ S Em-1 , I j ∈ S Bm-1 , I k ∈ S Em-1 , and i < j < k, that is, there are two bottleneck intersections of the end PTC line located upstream and downstream of a bottleneck intersection of the beginning PTC line, respectively. e corresponding bandwidth will decrease when we increase the progression speed and rotate the PTC line clockwise, as shown in Figure 6(a). Furthermore, when we reduce the progression speed and rotate the PTC line counter-clockwise the corresponding bandwidth will decrease, as shown in Figure 6(b). erefore, the further rotation transformation of the PTC line has to be finished. Condition 3. ∃ I i ∈ S Bm-1 , I j ∈ S Em-1 , I k ∈ S Bm-1 , and i < j < k, that is, there are two bottleneck intersections of the beginning PTC line located upstream and downstream of a bottleneck intersection of the end PTC line, respectively. e corresponding bandwidth will decrease when we increase the progression speed and rotate the PTC line clockwise, as shown in Figure 7(a). Furthermore, when we reduce the progression speed and rotate the PTC line counter-clockwise, the corresponding bandwidth will decrease, as shown in Figure 7(b). erefore, the further rotation transformation of the PTC line has to be finished.
Step2. Make a judgment of whether ray l Rp and l Gp pass through the red interval of any intersection I q (1 <q < p) or not.
If q) ) is the ordinate of the green center point of I q , which is closest to the crossing point of l Rp (l Gp ) and the timeline of I q , and λ q is the green split of I q .
Step3. Determine the coordinated mode of I p .
If F p � 0 and V Gp is within the rotation range, calculate V Gp as shown in Equation (3), and then let V (1,p) equal V Gp , else if F p � 1 and V Rp is within the rotation range, calculate V Rp and then let V (1,p) Step5. Make a judgment of whether p equals n or not. If p < n, let p equal p + 1 and then return to Step0.  8(b). erefore, when rotating the PTC line clockwise, we should select intersections I i ∈ S Em-1 , I j ∈ S Bm-1 to ensure that ∀ I k ∈ S Em-1 satisfies i ≤ k and ∀ I l ∈ S Bm-1 satisfies j ≥ l. en, the bottleneck point P Ei and P Bj should be taken as the rotation points.    8(d). erefore, when rotating the PTC line counter-clockwise, we should select intersections I i ∈ S Em-1 , I j ∈ S Bm-1 to ensure that ∀ I k ∈ S Em-1 satisfies i ≥ k and ∀ I l ∈ S Bm-1 satisfies j ≤ l. en, the bottleneck point P Ei and P Bj should be taken as the rotation points.  D m is defined as the abscissa difference between the bottleneck rotation points P Bj and P Ei in the mth rotation transformation in the second round, that is, D m � x j -x i . en, the bandwidth b m can be calculated as follows: where Δb m is the bandwidth increment resulting from the transformation of the progression speed from V (2,m-1) to V (2,m) . We define a Boolean variable K Dm as the judgment factor of the rotation direction. When K Dm equals 0, the rotation direction is the counter-clockwise rotation, whereas the rotation direction is the clockwise rotation when K Dm equals 1.

Scenario 1.
If ∀ I k ∈ S Em-1 , and ∀ I l ∈ S Bm-1 satisfy k < l, that is, all the bottleneck intersections of the end PTC line are located upstream of any bottleneck intersection of the beginning PTC line, then D m > 0. At this time, if the PTC line is rotated clockwise, V (2,m) > V (2,m-1) , the obtained bandwidth after the rotation will decrease according to Equation (5). If the PTC line is rotated counter-clockwise, V (2,m) < V (2,m-1) , the obtained bandwidth after the rotation will increase according to Equation (5). erefore, the rotation direction in Scenario 1 should be determined as the counter-clockwise rotation, K Dm � 0.

Scenario 2.
If ∀ I k ∈ S Em-1 , and ∀ I l ∈ S Bm-1 satisfy k > l, that is, all of the bottleneck intersections of the end PTC line are located downstream of any bottleneck intersection of the beginning PTC line, then D m < 0. At this time, if the PTC line is rotated clockwise, V (2,m) > V (2,m-1) , the obtained bandwidth after the rotation will increase according to Equation (5). If the PTC line is rotated counter-clockwise, V (2,m) < V (2,m-1) , the obtained bandwidth after the rotation will decrease according to Equation (5). erefore, the rotation direction in Scenario 2 should be determined as the clockwise rotation, K Dm � 1.

Rotation Angle Calculation.
Taking the bottleneck point of the end PTC line P Ei and the bottleneck point of the beginning PTC line P Bj as extreme points, we can calculate the rotation angle formed with other designated crossing points and determine the adjusted speed comprehensively after the rotation transformation. e green endpoint of the coordinated phase crossed by the end PTC line L Em-1 at intersection I k (1 ≤ k ≤ n) is defined as P Fk , and the green start point of the coordinated phase crossed by the beginning PTC line L Bm-1 at I k (1 ≤ k ≤ n) is defined as P Sk . e adjusted speed V Fk (V Sk ) corresponding to the rotation line L Fk (L Sk ) formed by connecting points P Ei (P Bj ) and P Fk (P Sk ) can be calculated by the following equation: Here, y Fk and y Ei represent the ordinate of points P Fk and P Ei , respectively; y Sk and y Bj represent the ordinate of points P Sk and P Bj , respectively.
If K Dm equals 0, we should connect points P Ei and P Fg (i + 1 ≤ g ≤ n) in turn to form the rotation line L Fg . When the corresponding adjusted speed V Fg is within the adjusted speed range, the eligible V Fg is incorporated into the optional vehicle speed set, S Vm , obtained after the mth rotation transformation. Meanwhile, P Bj and P Sh (1 ≤ h ≤ j-1) are connected to form the rotation line L Sh . When the corresponding adjusted speed V Sh is within the adjusted speed range, the eligible V Sh is incorporated into S Vm . en, we have to select the maximum value in S Vm as the adjusted speed V (2,m) determined by the mth rotation transformation of the vehicle PTC line in the second round, as shown in Figure 9(a).
If K Dm equals 1, we should connect points P Ei and P Fg (1 ≤ g ≤ i-1) in turn to form the rotation line L Fg . When V Fg is within the adjusted speed range, the eligible V Fg is incorporated into S Vm . Meanwhile, points P Bj and P Sh (j + 1 ≤ h ≤ n) are connected to form the rotation line L Sh . When V Sh is within the adjusted speed range, the eligible V Sh is incorporated into S Vm . en, we have to select the minimum value in S Vm as V (2,m) , as shown in Figure 9(b).
According to V (2,m) , the end PTC line L Em and the beginning PTC line L Bm in the outbound direction can be obtained after the rotation transformation. Before entering the next rotation transformation in the second round, the sets of the bottleneck intersections can be updated as S Em and S Bm according to L Em and L Bm .

Optimal Common Signal Cycle Calculation.
e optimal adjusted speed V B can be calculated according to the final PTC line obtained after the rotation transformations in the second round. According to the vertical scaling of the time-space diagram, the optimal common signal cycle C B can be calculated by the following equation:

Offset Calculation.
When the coordinated mode of each intersection and the optimal common signal cycle have been determined, combined with the known split distribution scheme, the green interval of the coordinated phase and absolute offset of each intersection can be calculated. e absolute offset of the intersection I p (1 ≤ p ≤ n) is the beginning of the coordinated phase at I p defined as O p . While the green center point of the coordinated phase at intersection I 1 is defined as the offset reference point with a value of zero, the absolute offset of each intersection can be calculated by the following equation:

Case Study.
To facilitate the comparative analysis of the optimization effect of the graphical method proposed in this paper, we cite a case study that has been used in many books and papers [19,20,25]. e corridor being analyzed is an east-west roadway, and there are eight signalized intersections along the corridor. e direction from west to east is defined as the outbound direction, and the western-most intersection is selected as intersection I 1 . e schematic diagram of the selected corridor is shown in Figure 10. It has already known that each intersection adopts symmetrical phasing. e split of each intersection is 55%, 60%, 65%, 65%, 60%, 65%, 70%, and 50%, respectively. e optimization range of the common signal cycle is [60, 100] s, and  Step1. Make a judgment of whether the sets S Bm-1 and S Em-1 meet any condition for stopping rotation.
If ∃ I k ∈ (S Bm−1 ∩ S Em−1 ), or ∃ I i ∈ S Em−1 , I j ∈ S Bm−1 , I k ∈ S Em−1 and i < j < k, or ∃ I i ∈ S Bm−1 , I j ∈ S Em−1 , I k ∈ S Bm−1 and i < j < k, then F A � 1, finish the second round of rotation transformation. Otherwise, F A � 0, proceed to Step 2.
Step2. Determine the rotation direction.
If K Dm � 0, connect points P Ei and P Fg (i + 1 ≤ g ≤ n), points P Bj and P Sh (1 ≤ h ≤ j-1) in turn, calculate the corresponding V Fg and V Sh as shown in Equation (6), add the eligible V Fg and V Sh in set S Vm , then select the maximum value in S Vm as the updated adjusted speed V (2,m).
If K Dm � 1, connect points P Ei and P Fg (1 ≤ g ≤ i-1), points P Bj and P Sh (j + 1 ≤ h ≤ n) in turn, calculate the corresponding V Fg and V Sh as shown in Equation (6), add the eligible V Fg and V Sh in set S Vm , then select the minimum value in S Vm as the updated adjusted speed V (2,m).
Step4. Let m � m + 1, and return to Step 0. ALGORITHM 3: Second round rotation transformation procedure. the progression design speed is 40 km/h, which approximately equals 11 m/s.

Scheme Generation.
e proposed method can obtain the scheme through a limited number of rule-guided drawings, and it is suitable for engineering applications. e related process can be shown as follows:

Initialization.
e initial common signal cycle C 1 is valued as the midpoint between the minimum common signal cycle C min and the maximum common signal cycle C max , which is 80 s. Because the progression design speeds of all the road sections are 11 m/s, there is no need to adjust the intersection distance. When the initial common signal cycle C 1 is unchanged, the adjusted speed range can be determined as [8.25, 13.75] m/s.

2.7.2.
e First Round of Rotation Transformation. e green center point of the coordinated phase at intersection I 1 is defined as the reference point O L1 of the coordinate system of the time-space diagram. A horizontal line from the reference point O L1 can be drawn, and then, it is rotated until the cotangent of its angle with the x-axis equals the corridor progression design speed V. We can define the rotated ray as the initial PTC line L 1 and assign the value of V to adjusted speed V (1,1) . en, the first rotation transformation in the first round is completed. e intersection I 2 is selected as the current coordinated intersection. According to PTC line L 1 obtained by the first rotation transformation, the crossing point of the PTC line L 1 and the timeline of I 2 is marked as O L2 . e nearest crossing point of the coordinated phase horizontal red (green) center line at I 1 , and the timeline of I 2 is identified and recorded as O R2 (O G2 ), as shown in Figure 11 Ray l R2 does not pass through the red interval of the coordinated phase, and f 2 < 0, and then, I 2 can form a backstepping coordinated mode with I 1 , where F 2 � 1.
e corresponding adjusted speed V R2 of l R2 is 8.75 m/s, within the adjusted speed range [8.25, 13.75] m/s. erefore, l R2 is defined as PTC line L 2 , and the value of V R2 is assigned to the adjusted speed V (1,2) . Up to now, the second rotation transformation in the first round is finished.
Repeat the above steps to complete the 3rd to 8th rotation transformation in the first round in turn. en, the coordinated mode between I 1 and I 3 , I 4 , I 5 , I 6 , I 7 , and I 8 can be determined, respectively. e coordination process is shown in Figures 11(b)-11(g).
After the first round of rotation transformation, I 2 , I 5 , and I 8 form a backstepping coordinated mode with I 1 , F 2 � F 5 � F 8 � 1, whereas the intersections I 3 , I 4 , I 6 , and I 7 form a synchronous coordinated mode with I 1 , F 3 � F 4 � F 6 � F 7 � 0. e adjusted speed is 11.40 m/s, and the time-space diagram obtained after the first round of rotation transformation is shown in Figure 11(h). e corresponding bandwidth is 24 s, and the bandwidth ratio is 30.0%.

2.7.3.
e Second Round of Rotation Transformation. According to the adjusted speed V (1,8) finally obtained after the first round of rotation transformation, the end PTC line of vehicle band L E0 and the beginning PTC line of vehicle band L B0 of the outbound progression can be drawn, as shown in Figure 12. Meanwhile, the bottleneck intersections of the end PTC line are put into the set S E0 � {I 7 }. Also, the bottleneck intersections of the beginning PTC line are put into the set S B0 � {I 3 }. e set S B0 and S E0 do not meet any conditions for stopping rotation. en, we have to complete the first rotation transformation in the second round.
Because the only bottleneck intersection of the end PTC line I 7 is located downstream of the only bottleneck intersection of the beginning PTC line I 3 , it corresponds to the rotation direction judgment Scenario 2. erefore, the PTC Our final step is to obtain the final optimized scheme, and the scheme generation routine can be presented below. Scheme Generation Procedure Step0. Calculate the optimal common signal cycle C B as shown in Equation (7).

Step1. Calculate the absolute offset O p of each intersection as shown in Equation (8).
Step2. Make the time-space diagram to check the coordination effect of the final optimized scheme obtained by the proposed graphical optimization method.        line can be rotated clockwise. Furthermore, the most upstream bottleneck point P E7 is selected as the rotation point of the end PTC line, whereas the most downstream bottleneck point P B3 is selected as the rotation point of the beginning PTC line. As shown in Figure 12, we can connect points P E7 (P B3 ) and P Fg (1 ≤ g ≤ 6) (P Sh (4 ≤ h ≤ 8)) to form the rotation line L Fg (L Sh ) and determine whether the corresponding adjusted speed V Fg (V Sh ) is within the adjusted speed range. e calculation results are shown in Table 1.
It can be seen from Table 1 S6 , V S8 }. en, select the minimum value V F1 in S V1 as the adjusted speed V (2,1) . en, V (2, 1) is 12.11 m/s. According to V (2,1) , the end PTC line L E1 and the beginning PTC line L B1 of the outbound progression can be drawn. Determine the bottleneck intersections set of the end PTC line S E1 � {I 1 , I 7 } and the beginning PTC line bottleneck intersections set S B1 � {I 3 }, according to L E1 and L B1 . en, the updating of the sets of bottleneck intersections is completed, as shown in Figure 13. e sets S E1 and S B1 meet one of the conditions for stopping rotation. At this time, ∃ I 1 ∈ S E1 , I 3 ∈ S B1 , I 7 ∈ S E1 , that is, there are two bottleneck intersections of the end PTC line located upstream and downstream of a bottleneck intersection of the beginning PTC line, which satisfies Condition 2 for stopping the rotation. erefore, we can stop the further rotation transformation of the PTC line, and the second round of rotation transformation is finished.

Scheme Generation.
e optimal adjusted speed V B is 12.11 m/s, according to the final PTC line obtained after the second round of rotation transformation. Combined with the progression design speed V, it can be calculated that the optimal common signal cycle C B is 88 s.
Time (     According to the coordinated mode of each intersection and the optimal common signal cycle obtained after the two rounds of rotation transformation, combined with the green signal ratio distribution scheme, the green interval of the coordinated phase and absolute offset of each intersection can be calculated. e time-space diagram finally obtained is shown in Figure 14. e corresponding progression bandwidth is 32.85 s, and the bandwidth ratio reaches 37.3%.

Programmatic Solution.
e proposed solution logic steps can be solved graphically and programmed to quickly obtain an optimal signal coordination control scheme. e above case can be solved by python programming to obtain the optimal signal coordination control scheme. e experimental environment of the model solver is the Win-dows10 64-bit operating system, and the CPU is intel i5-6600 3.30 GHz. e optimal signal coordination control scheme obtained by programming is the same as the graphical way, with a solution time of only 2.3 × 10 −3 s.

Model Comparison.
e MAXBAND model [7] and the improved algebraic method [20] can also be used to design the corridor progression schemes for the same case, and the calculation results are shown in Table 2. As for the coordinated mode between intersection I 1 and other intersections, the graphical method proposed in this study is consistent with the improved algebraic method and the MAXBAND model. As for the bandwidth ratio, the proposed graphical method can obtain 37.3% in both coordination directions, which is higher than the improved     algebraic method. Moreover, it also achieves the global optimal coordination effect, the same as the MAXBAND model. Using the LINGO programming solution, the MAX-BAND model can be solved in 3 s. Meanwhile, the proposed graphical method obtains the global optimal scheme in 2.3 × 10 −3 s, which means the solving efficiency is more than a thousand times higher than the MAXBAND model.
It can be seen from Table 2 that for the symmetrical corridor progression design, the method proposed in this study has the advantages of solid operability, solving simplicity, and a small amount of calculation. It can obtain an ideal coordination effect and be solved by a rule-guided graphical method with a limited number of steps.

Conclusions
In summary, a symmetrical bidirectional corridor progression method based on graphical optimization theory is established in this study. According to the internal relationship between the PTC lines rotation transformation and the progression bandwidth, a step-by-step optimization process of symmetrical corridor progression based on the rotation transformation of the PTC lines is proposed. After a two-round rotation transformation of the PTC line, the comprehensive optimization of the intersection coordinated mode, common signal cycle, and offsets are realized.
In the first round of PTC line rotation transformation, based on the existing graphical method designing flow, we have innovatively added a process for judging whether the PTC line passes through the red interval of the intersections with a determined coordinated mode or not, achieving a more optimized choice of coordinated mode at each intersection. Meanwhile, we have innovatively proposed the second round of PTC line rotation transformation. In the second round of PTC line rotation transformation, the regular bandwidth pattern during the rotation transformation of the PTC line is explored. Innovative rules for rotation transformation stop conditions and selection of the rotation points and direction are given to achieve the reoptimization of the common signal cycle. e proposed graphical method can be used to obtain a signal coordination control scheme by a rule-guided graphical method and be programmed. It can be seen from the case study that the bandwidth ratio obtained by the first round of rotation transformation of the proposed graphical method is 30.0%, with the optimization of the coordinated mode and the initial offsets. en, through the second round of rotation transformation, the reoptimization of the common signal cycle and the offsets increases the bandwidth ratio to 37.3%. Finally, the effect of the corridor progression control scheme, which is better than the improved algebraic method, is ultimately the same as the global optimal result of the MAXBAND model with a more than a thousand times higher solving efficiency. e graphical method proposed in this study has the advantages of intuitive readability, strong operability, and is particularly suitable for engineering applications. e horizontal and vertical scaling of the time-space diagram is one of the few attempts to standardize a graphic solution to corridor traffic coordination control. However, extending the graphical method to the corridor progression under the asymmetrical phasing and constructing a more applicable graphical method for corridor progression will be important for future follow-up research. Notations a p : e ideal distance between intersection I p and I 1 b m : e bandwidth obtained after the mth rotation transformation in the second round C: e common signal cycle C 1 (C B ): e initial (optimal) common signal cycle C max (C min ): e maximum (minimum) common signal cycle d p : e actual distance between intersection I p and I p + 1 (1 ≤ p ≤ n-1) d p ′ : e adjusted distance between intersection I p and I p + 1 (1 ≤ p ≤ n-1) D p : e adjusted distance between the nonbenchmark intersection I p and the benchmark intersection I 1 D m : e difference between the abscissa of the bottleneck rotation point of the beginning and end PTC line in the mth rotation transformation in the second round f p : e primary judgment factor for the coordinated mode of intersection I p S * � "synchronous coordinated mode" and B * � "backstepping coordinated mode". 16 Journal of Advanced Transportation f G(p,q) (f R(p,q) ): e judgment factor of whether ray l Gp (l Rp ) passes through the red interval of the intersection I q (1 < q < p) or not F p : e coordinated mode judgment factor of intersection I p F A : e termination decision parameter of the second round of rotation transformation I p : e pth signalized intersection along the corridor in the outbound direction K Dm : e rotation direction judgment factor in the mth rotation transformation in the second round K E (m-1,p) (K B(m-1,p) ): Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.