# Graphical Optimization Method for Symmetrical Bidirectional Corridor Progression

Authors: Kai Lu; Shuyan Jiang; Yiming Zhao; Yongjie Lin; Yinhai Wang
Journal: Journal of Advanced Transportation
 (2021)

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## Abstract

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 coordinated 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.


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## Body

## 1. Introduction

Intersections are the road network nodes that frequently cause traffic disruption, severe delays, and accidents in a city [1]. Therefore, 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. The 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.The 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. The most classic models are the MAXBAND model proposed by Morgan and Little [5] and the MULTIBAND model constructed by Gartner et al. [6]. The 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]. The 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 [8–10]. 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.The 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.The 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). Then, 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. The corresponding progression coordination design process will also be presented in detail. The remainder of the article is organized as follows: Section3 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.


## 2. Materials and Methods

### 2.1. Design Process

The graphical method proposed can be solved by a normal step-by-step graphical process and programming. We define the horizontal scaling of the time-space diagram to convert the progression design speed of the road section to the corridor progression design speedV. Meanwhile, we also define the vertical scaling of the time-space diagram to realize the transformation of the optimization of the common signal cycle into the optimization of the corridor progression speed. The 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.Figure 1
The design flow chart of the proposed method.


### 2.2. Initialization

#### 2.2.1. Initial Common Signal Cycle

Let us suppose that there aren signalized intersections along the corridor and number the intersections in the ascending order in the outbound direction. Then, the pth (1 ≤ p ≤ n) intersection is defined as Ip in sequence along the outbound direction.The common signal cycle range will be determined by the signal cycle range of each intersection, denoted as [Cmin, Cmax]. The initial common signal cycle C1 can be valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax.


#### 2.2.2. Adjusted Distance and Speed

Letdp and vp be the actual distance and the progression design speed between intersection Ip and Ip + 1 (1 ≤ p ≤ n-1), respectively. Then, 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(b). Therefore, 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.Figure 2
The horizontal scaling of the time-space diagram. (a) Before the horizontal scaling. (b) After the horizontal scaling.
(a)(b)Therefore, to facilitate the diagrammatic design process, when the design speed of the road section is inconsistent with the corridor progression design speedV, we will convert the progression design speed of the road section to V by adjusting the intersection distance. The adjusted distance Dp between intersection Ip and intersection I1 can be calculated by the following equation:(1)Dp=V⋅∑k=1p−1dkvk.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 ofV 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.Figure 3
The vertical scaling of the time-space diagram. (a) Before the vertical scaling. (b) After the vertical scaling.
(a)(b)Then, we can first keep the common signal cycleC1 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 CB can be obtained.According to the corridor progression design speedV and the range of the common signal cycle [Cmin, Cmax], the adjusted speed V(i, j) obtained after the jth rotation transformation in the ith round should satisfy the following equation:(2)V⋅CminC1≤Vi,j≤V⋅CmaxC1.


#### 2.2.3. 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 intersectionI1, whereas backstepping coordination means that the red center point is consistent with the green center point of the coordinated phase at I1. The 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. Then, we define a Boolean variable Fp as the coordinated mode judgment factor. When Fp is 0, the optimal coordinated mode of the intersection Ip is synchronous coordination. When Fp is 1, the optimal coordinated mode of Ip is backstepping coordination.Algorithm 1: Initialization procedure.
Our first step is to define and calculate some basic parameters, and the initialization routine can be presented below.Initialization procedureStep0. Determine the range of the common signal cycle [Cmin, Cmax] and calculate the initial common signal cycle C1 = 0.5⋅(Cmin + Cmax).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 Dp 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).


### 2.3. The First Round of Rotation Transformation

The 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 atI1 as the reference point OL1 of the time-space diagram coordinate system. A horizontal line can be drawn from OL1 and rotated until the cotangent of its angle with the x-axis equals the corridor progression design speed V. This 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 L1 and then assign V to adjusted speed V(1, 1).The following steps have to be executed cyclically until the coordinated modes of all intersections have been determined.


#### 2.3.1. Primary Judgment Factor Calculation

The intersectionIp is selected as the current coordinated intersection during the first round’s pth (2 ≤ p ≤ n) rotation transformation. According to PTC line Lp-1 obtained by the last rotation transformation, the crossing point of Lp-1 and the timeline of Ip is marked as OLp as shown in Figure 4(a). The coordinated phase horizontal red center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as ORp. The coordinated phase horizontal green center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as OGp.Figure 4
The determination of the coordinated mode whenfp ≥ 0.
(a)(b)(c)Starting at the pointOL1, we can make a ray lRp (lGp) passing through the point ORp (OGp), and the corresponding adjusted speed VRp (VGp) satisfies the following:(3)VRpVGp=DpyRpDpyGp.The time difference between pointsOLp and ORp (OGp), which corresponds to the distance between the ordinate of points yLp and yRp (yGp), is recorded as TRp (TGp). Then, the primary judgment factor fp can be constructed according to TRp and TGp:(4)fp=TRp−TGp=yLp−yRp−yLp−yGp.


#### 2.3.2. Optimal Coordinated Mode Calculation

We definefR(p, q) (fG(p, q)) (1 < q < p) as a Boolean variable. When ray lRp (lGp) does not pass through the red interval of the intersection Iq (1 < q < p), fR (p, q) (fG (p, q)) is 0. When ray lRp (lGp) passes through the red interval of Iq, fR (p,q) (fG (p, q)) is 1. Also, the red interval crossing amount of lRp (lGp) in Iq can be defined as TR (p, q) (TG (p, q)).Whenfp ≥ 0, the situation is shown in Figure 4. If ray lGp does not pass through any red interval of the coordinated phase, all fG(p,q) equals 0. The current coordinated intersection can form a synchronous coordinated mode with I1, where Fp = 0. If ray lGp passes through any red interval of the coordinated phase while ray lRp does not pass through any red interval, as shown in Figure 4(b), the backstepping coordinated mode can be chosen, where Fp = 1. If both lGp and lRp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lGp is less than or equal to lRp, as shown in Figure 4(c), these two intersections can form a synchronous coordinated mode, where Fp = 0. Otherwise, the backstepping coordinated mode will be chosen as Fp = 1.Similarly, whenfp < 0, if ray lRp does not pass through any red interval of the coordinated phase, all fR (p,q) equals 0, Fp = 1. If ray lRp passes through any red interval of the coordinated phase while ray lGp does not pass through any red interval, then Fp = 0. If both lRp and lGp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lRp is less than or equal to lGp, Fp = 1. Otherwise, Fp = 0.


#### 2.3.3. PTC Line Optimization

If the current coordinated intersection has formed a synchronous coordinated mode withI1 and VGp is within the rotation range of the PTC line, we can define lGp as PTC line Lp and assign the value of VGp to the adjusted speed V(1,p), as shown in Figure 5(a). When VGp is out of the rotation range, we can keep the PTC line unchanged, that is, as shown in Figure 5(b), Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Figure 5
The PTC line updating under synchronous coordinated mode.
(a)(b)Similarly, if the current coordinated intersection has formed a backstepping coordinated mode withI1 and VRp is within the rotation range of the PTC line, we can define lRp as PTC line Lp and assign the value of VRp to V(1,p). When VRp is out of the rotation range, we can keep the PTC line unchanged. Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Algorithm2: First round rotation transformation procedure.
Our second step is to optimize the intersection coordinated modes, and the first round of the rotation transformation routine can be presented below.First Round Rotation Transformation ProcedureStep0. Determine the crossing point OLp, ORp, and OGp.First, letyLp=Dp/V1,p−1.Then, if modyLpC1/2,2=0, yRp equals C1⋅yLpC1/2/2+C1/2 and yGp equals C1⋅yLpC1/2/2, else yRp=C1⋅yLpC1/2/2−C1/2 and yGp=C1⋅yLpC1/2+1/2.Step1. Calculate the primary judgment factor fp as shown in Equation (4).Step2. Make a judgment of whether ray lRp and lGp pass through the red interval of any intersection Iq (1 <q < p) or not.IfDq/VRp−yRp,q≤0.5⋅C1⋅λq, fR(p, q) = 0, otherwise fR(p, q)= 1 and TRp,q=Dq/VRp−yRp,q−0.5⋅C1⋅λq.IfDq/VGp−yGp,q≤0.5⋅C1⋅λq, fG(p, q) = 0, otherwise fG(p, q)= 1 and TGp,q=Dq/VGp−yGp,q−0.5⋅C1⋅λq.yR(p, q) (yG(p, q)) is the ordinate of the green center point of Iq, which is closest to the crossing point of lRp (lGp) and the timeline of Iq, and λq is the green split of Iq.Step3. Determine the coordinated mode of Ip.Iffp≥0 and ∑q=2p−1fGp,q=0, or fp<0, ∑q=2p−1fRp,q>0 and ∑q=2p−1fGp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1 < max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 0.Iffp<0 and ∑q=2p−1fRp,q=0, or fp≥0, ∑q=2p−1fGp,q>0 and ∑q=2p−1fRp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1> max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 1.Step4. PTC line optimization.IfFp=0 and VGp is within the rotation range, calculate VGp as shown in Equation (3), and then let V1,p equal VGp, else if Fp=1 and VRp is within the rotation range, calculate VRp and then let V1,p equal VRp, else V1,p=V1,p−1.Step5. Make a judgment of whether p equals n or not. If p < n, let p equal p + 1 and then return to Step0.


### 2.4. The Second Round of Rotation Transformation

The 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 speedV(1,n) to V(2,0). Then, the beginning PTC line LB0 and the end PTC line LE0 of the outbound progression can be obtained. Then, the initial bandwidth b0 and ratio R0 can also be calculated. Meanwhile, the bottleneck intersections of the beginning PTC line are put into the intersection set SB0, and the bottleneck intersections of the end PTC line are put into the intersection set SE0. The calculation of related parameters will be introduced in the next section.We define a termination decision parameterFA, which is a Boolean variable. The following steps have to be executed cyclically until the termination decision parameter FA equals 1.


#### 2.4.1. Bottleneck Intersection Recognition

We assume that the obtained beginning PTC line, end PTC line, adjusted speed, bandwidth, and bandwidth ratio after themth rotation transformation in the second round are defined as LBm, LEm, V(2,m), bm, and Rm, respectively. The parameters obtained from the m-1th rotation transformation need to be used during the mth rotation transformation in the second round.According toFp, V(2,m-1), C1, and λp, we can draw the beginning and end PTC lines in the time-space diagram. We define the Boolean variables KB (m-1,p) and KE (m-1,p) as the judgment factors of the bottleneck intersection of the beginning and end PTC lines. When KB (m-1,p) equals 1, intersection Ip is the bottleneck intersection of the beginning PTC line after the m-1th rotation transformation, Ip∈SBm-1, and the corresponding bottleneck point is defined as PBp. When KE (m-1,p) equals 1, Ip is the bottleneck intersection of the end PTC line after the m-1th rotation transformation, Ip∈SEm-1, and the corresponding bottleneck point is recorded as PEp.Before the other calculation steps of themth rotation transformation in the second round, it is necessary to judge whether the sets SBm-1 and SEm-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 FA = 1.


#### 2.4.2. Conditions for Stopping the Rotation

Condition 1. ∃Ik∈(SBm-1∩​SEm-1), that is, there are both bottleneck points of the beginning and end PTC lines at intersection Ik. At this time, the ratio of the bandwidth Rm equals the split of the coordinated phase at Ik, which means that it has already reached the maximum value of the bandwidth ratio. Therefore, there is no need to continue the rotation transformation of the PTC line.Condition 2. ∃Ii∈SEm-1, Ij∈SBm-1, Ik∈SEm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 6
Condition 2 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)Condition 3. ∃Ii∈SBm-1, Ij∈SEm-1, Ik∈SBm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 7
Condition 3 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)


#### 2.4.3. Rotation Points and Direction Recognition

It is necessary to determine the rotation points and direction according to the relationship between the rotation transformation and the progression bandwidth during themth rotation transformation in the second round.When increasing the progression speed, that is,V(2,m) > V(2,m-1), the rotation point of LBm should fall at the most downstream bottleneck intersection in SBm-1, and the rotation point of LEm should fall at the most upstream bottleneck intersection in SEm-1, as shown in Figures 8(a) and 8(b). Therefore, when rotating the PTC line clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≤ k and ∀Il∈SBm-1 satisfies j ≥ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.Figure 8
The relationship between the rotation transformation and the progression bandwidth. (a)V2,m>V2,m−1andDm^>0. (b) V2,m>V2,m−1andDm^<0. (c) V2,m<V2,m−1andDm^>0. (d) V2,m<V2,m−1andDm^<0.
(a)(b)(c)(d)When reducing the progression speed, that is,V(2,m) < V(2,m-1), the rotation point of LEm should fall at the most downstream bottleneck intersection in SEm-1, and the rotation point of LBm should fall at the most upstream bottleneck intersection in SBm-1, as shown in Figures 8(c) and 8(d). Therefore, when rotating the PTC line counter-clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≥ k and ∀Il∈SBm-1 satisfies j ≤ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.D^m is defined as the abscissa difference between the bottleneck rotation points PBj and PEi in the mth rotation transformation in the second round, that is, D^m  = xj-xi. Then, the bandwidth bm can be calculated as follows:(5)bm=bm−1+Δbm=bm−1+D^m⋅1V2,m−1V2,m−1,where Δbm 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 variableKDm as the judgment factor of the rotation direction. When KDm equals 0, the rotation direction is the counter-clockwise rotation, whereas the rotation direction is the clockwise rotation when KDm equals 1.Scenario 1.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 1 should be determined as the counter-clockwise rotation, KDm = 0.Scenario 2.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 2 should be determined as the clockwise rotation, KDm = 1.


#### 2.4.4. Rotation Angle Calculation

Taking the bottleneck point of the end PTC linePEi and the bottleneck point of the beginning PTC line PBj 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.The green endpoint of the coordinated phase crossed by the end PTC lineLEm-1 at intersection Ik (1 ≤ k ≤ n) is defined as PFk, and the green start point of the coordinated phase crossed by the beginning PTC line LBm-1 at Ik (1 ≤ k ≤ n) is defined as PSk.The adjusted speedVFk (VSk) corresponding to the rotation line LFk (LSk) formed by connecting points PEi (PBj) and PFk (PSk) can be calculated by the following equation:(6)VFkVSk=xk−xiyFk−yEixk−xjySk−yBj.Here,yFk and yEi represent the ordinate of points PFk and PEi, respectively; ySk and yBj represent the ordinate of points PSk and PBj, respectively.IfKDm equals 0, we should connect points PEi and PFg (i + 1 ≤ g ≤ n) in turn to form the rotation line LFg. When the corresponding adjusted speed VFg is within the adjusted speed range, the eligible VFg is incorporated into the optional vehicle speed set, SVm, obtained after the mth rotation transformation. Meanwhile, PBj and PSh (1 ≤ h ≤ j-1) are connected to form the rotation line LSh. When the corresponding adjusted speed VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the maximum value in SVm 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).Figure 9
Rotation angle determination.
(a)(b)IfKDm equals 1, we should connect points PEi and PFg (1 ≤ g ≤ i-1) in turn to form the rotation line LFg. When VFg is within the adjusted speed range, the eligible VFg is incorporated into SVm. Meanwhile, points PBj and PSh (j + 1 ≤ h ≤ n) are connected to form the rotation line LSh. When VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the minimum value in SVm as V(2,m), as shown in Figure 9(b).According toV(2,m), the end PTC line LEm and the beginning PTC line LBm 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 SEm and SBm according to LEm and LBm.Algorithm3: Second round rotation transformation procedure.
Our third step is to optimize the common signal cycle and offsets, and the second round of the rotation transformation routine can be presented below.Second Round Rotation Transformation ProcedureStep0. Determine the bottleneck intersections and the set SBm-1 and SEm-1.According toFp, V(2,m-1), C1, and λp, draw the beginning and end PTC lines in the time-space diagram. If Ip is the bottleneck intersection of the beginning PTC line, let KBm−1,p=1 and add Ip in set SBm-1, else KBm−1,p=0. And if Ip is the bottleneck intersection of the end PTC line, let KEm−1,p=1 and add Ip in set SEm-1, else KEm−1,p=0.Step1. Make a judgment of whether the sets SBm-1 and SEm-1 meet any condition for stopping rotation.If∃Ik∈SBm−1∩SEm−1, or ∃Ii∈SEm−1,Ij∈SBm−1,Ik∈SEm−1 and i<j<k, or ∃Ii∈SBm−1,Ij∈SEm−1,Ik∈SBm−1 and i<j<k, then FA=1, finish the second round of rotation transformation. Otherwise, FA=0, proceed to Step 2.Step2. Determine the rotation direction.If∀Ik∈SEm−1,∀Il∈SBm−1,andk<l, let KDm=0, else if ∀Ik∈SEm−1,∀Il∈SBm−1,andk>l, let KDm=1.Step3. Determine the rotation angle.IfKDm=0, connect points PEi and PFg (i + 1 ≤ g ≤ n), points PBj and PSh (1 ≤ h ≤ j-1) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the maximum value in SVm as the updated adjusted speed V(2,m).IfKDm=1, connect points PEi and PFg (1 ≤ g ≤ i-1), points PBj and PSh (j + 1 ≤ h ≤ n) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the minimum value in SVm as the updated adjusted speed V(2,m).Step4. Let m = m + 1, and return to Step 0.


### 2.5. Scheme Generation

#### 2.5.1. Optimal Common Signal Cycle Calculation

The optimal adjusted speedVB 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 CB can be calculated by the following equation:(7)CB=C1⋅VBV.


#### 2.5.2. 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. The absolute offset of the intersectionIp (1 ≤ p ≤ n) is the beginning of the coordinated phase at Ip defined as Op. While the green center point of the coordinated phase at intersection I1 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:(8)Op=CB−0.5⋅CB⋅λp,1≤p≤nandFp=0,0.5⋅CB−0.5⋅CB⋅λp,1≤p≤nandFp=1.Algorithm4: Scheme generation procedure.
Our final step is to obtain the final optimized scheme, and the scheme generation routine can be presented below.Scheme Generation ProcedureStep0. Calculate the optimal common signal cycle CB as shown in Equation (7).Step1. Calculate the absolute offset Op 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.


### 2.6. 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]. The corridor being analyzed is an east-west roadway, and there are eight signalized intersections along the corridor. The direction from west to east is defined as the outbound direction, and the western-most intersection is selected as intersection I1. The schematic diagram of the selected corridor is shown in Figure 10. It has already known that each intersection adopts symmetrical phasing. The split of each intersection is 55%, 60%, 65%, 65%, 60%, 65%, 70%, and 50%, respectively. The optimization range of the common signal cycle is [60, 100] s, and the progression design speed is 40 km/h, which approximately equals 11 m/s.Figure 10
Structure diagram of the selected corridor.


### 2.7. Scheme Generation

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


#### 2.7.1. Initialization

The initial common signal cycleC1 is valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax, 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 C1 is unchanged, the adjusted speed range can be determined as [8.25, 13.75] m/s.


#### 2.7.2. The First Round of Rotation Transformation

The green center point of the coordinated phase at intersectionI1 is defined as the reference point OL1 of the coordinate system of the time-space diagram. A horizontal line from the reference point OL1 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 L1 and assign the value of V to adjusted speed V(1, 1). Then, the first rotation transformation in the first round is completed.The intersectionI2 is selected as the current coordinated intersection. According to PTC line L1 obtained by the first rotation transformation, the crossing point of the PTC line L1 and the timeline of I2 is marked as OL2. The nearest crossing point of the coordinated phase horizontal red (green) center line at I1, and the timeline of I2 is identified and recorded as OR2 (OG2), as shown in Figure 11(a). The time difference TR2 between points OL2 and OR2 is 8 s, and the time difference TG2 between points OL2 and OG2 is 32 s. Then, f2 = −24 s.Figure 11
The coordination process of the first round of rotation transformation. (a) The 2nd rotation transformation. (b) The 3rd rotation transformation. (c) The 4th rotation transformation. (d) The 5th rotation transformation. (e) The 6th rotation transformation. (f) The 7th rotation transformation. (g) The 8th rotation transformation. (h) The obtained time-space diagram.
(a)(b)(c)(d)(e)(f)(g)(h)RaylR2 does not pass through the red interval of the coordinated phase, and f2 < 0, and then, I2 can form a backstepping coordinated mode with I1, where F2 = 1.The corresponding adjusted speedVR2 of lR2 is 8.75 m/s, within the adjusted speed range [8.25, 13.75] m/s. Therefore, lR2 is defined as PTC line L2, and the value of VR2 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. Then, the coordinated mode betweenI1 and I3, I4, I5, I6, I7, and I8 can be determined, respectively. The coordination process is shown in Figures 11(b)–11(g).After the first round of rotation transformation,I2, I5, and I8 form a backstepping coordinated mode with I1, F2 = F5 = F8 = 1, whereas the intersections I3, I4, I6, and I7 form a synchronous coordinated mode with I1, F3 = F4 = F6 = F7 = 0. The 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). The corresponding bandwidth is 24 s, and the bandwidth ratio is 30.0%.


#### 2.7.3. The Second Round of Rotation Transformation

According to the adjusted speedV(1,8) finally obtained after the first round of rotation transformation, the end PTC line of vehicle band LE0 and the beginning PTC line of vehicle band LB0 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 SE0 = {I7}. Also, the bottleneck intersections of the beginning PTC line are put into the set SB0 = {I3}.Figure 12
The second round of rotation transformation.The setSB0 and SE0 do not meet any conditions for stopping rotation. Then, we have to complete the first rotation transformation in the second round.Because the only bottleneck intersection of the end PTC lineI7 is located downstream of the only bottleneck intersection of the beginning PTC line I3, it corresponds to the rotation direction judgment Scenario 2. Therefore, the PTC line can be rotated clockwise. Furthermore, the most upstream bottleneck point PE7 is selected as the rotation point of the end PTC line, whereas the most downstream bottleneck point PB3 is selected as the rotation point of the beginning PTC line.As shown in Figure12, we can connect points PE7 (PB3) and PFg (1 ≤ g ≤ 6) (PSh (4 ≤ h ≤ 8)) to form the rotation line LFg (LSh) and determine whether the corresponding adjusted speed VFg (VSh) is within the adjusted speed range. The calculation results are shown in Table 1.Table 1
The optional vehicle speed set during the first rotation transformation in the second round.
Bottleneck pointsThe start point/endpoint of the coordinated phaseAdjusted speed (m/s)Optional vehicle speedSV1PE7(2010, 268)PF1(0, 102)VF1 = 12.11VF1 ∈ SV1PF2(350, 144)VF2 = 13.39VF2 ∈ SV1PF3(750, 186)VF3 = 15.37VF3 ∉ SV1PF4(910, 186)VF4 = 13.41VF4 ∈ SV1PF5(1450, 224)VF5 = 12.73VF5 ∈ SV1PF6(1730, 266)VF6 = 140.00VF6 ∉ SV1PB3(750, 134)PS4(910, 134)VS4 = ∞VS4 ∉ SV1PS5(1450, 176)VS5 = 16.67VS5 ∉ SV1PS6(1730, 214)VS6 = 12.25VS6 ∈ SV1PS7(2010, 212)VS7 = 16.15VS7 ∉ SV1PS8(2280, 260)VS8 = 12.14VS8 ∈ SV1It can be seen from Table1 that SV1 = {VF1, VF2, VF4, VF5, VS6, VS8}. Then, select the minimum value VF1 in SV1 as the adjusted speed V(2, 1). Then, V(2, 1) is 12.11 m/s.According toV(2,1), the end PTC line LE1 and the beginning PTC line LB1 of the outbound progression can be drawn. Determine the bottleneck intersections set of the end PTC line SE1 = {I1, I7} and the beginning PTC line bottleneck intersections set SB1 = {I3}, according to LE1 and LB1. Then, the updating of the sets of bottleneck intersections is completed, as shown in Figure 13.Figure 13
Updating the sets of bottleneck intersections.The setsSE1 and SB1 meet one of the conditions for stopping rotation. At this time, ∃I1∈SE1, I3∈SB1, I7∈SE1, 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. Therefore, we can stop the further rotation transformation of the PTC line, and the second round of rotation transformation is finished.


#### 2.7.4. Scheme Generation

The optimal adjusted speedVB 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 CB is 88 s.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. The time-space diagram finally obtained is shown in Figure14. The corresponding progression bandwidth is 32.85 s, and the bandwidth ratio reaches 37.3%.Figure 14
The finally obtained time-space diagram.


#### 2.7.5. Programmatic Solution

The proposed solution logic steps can be solved graphically and programmed to quickly obtain an optimal signal coordination control scheme. The above case can be solved by python programming to obtain the optimal signal coordination control scheme. The experimental environment of the model solver is the Windows10 64-bit operating system, and the CPU is intel i5-6600 3.30 GHz. The 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.


### 2.8. Model Comparison

The 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 I1 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.Table 2
The comparison table of the coordination effects of various models.
ModelIntersectionI1I2I3I4I5I6I7I8Proposed graphical method (MAXBAND model)Coordinated modeS∗B∗S∗S∗B∗S∗S∗B∗Absolute offset (s)63.817.659.459.417.659.457.222.0Common signal cycle (s)88Bandwidth ratio (%)37.3Improved algebraic methodCoordinated modeS∗B∗S∗S∗B∗S∗S∗B∗Absolute offset (s)66.018.261.461.418.261.459.222.8Common signal cycle (s)91Bandwidth ratio (%)30.5S∗  = “synchronous coordinated mode” and B∗  = “backstepping coordinated mode”.Using the LINGO programming solution, the MAXBAND 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 Table2 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.


## 2.1. Design Process

The graphical method proposed can be solved by a normal step-by-step graphical process and programming. We define the horizontal scaling of the time-space diagram to convert the progression design speed of the road section to the corridor progression design speedV. Meanwhile, we also define the vertical scaling of the time-space diagram to realize the transformation of the optimization of the common signal cycle into the optimization of the corridor progression speed. The 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.Figure 1
The design flow chart of the proposed method.


## 2.2. Initialization

### 2.2.1. Initial Common Signal Cycle

Let us suppose that there aren signalized intersections along the corridor and number the intersections in the ascending order in the outbound direction. Then, the pth (1 ≤ p ≤ n) intersection is defined as Ip in sequence along the outbound direction.The common signal cycle range will be determined by the signal cycle range of each intersection, denoted as [Cmin, Cmax]. The initial common signal cycle C1 can be valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax.


### 2.2.2. Adjusted Distance and Speed

Letdp and vp be the actual distance and the progression design speed between intersection Ip and Ip + 1 (1 ≤ p ≤ n-1), respectively. Then, 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(b). Therefore, 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.Figure 2
The horizontal scaling of the time-space diagram. (a) Before the horizontal scaling. (b) After the horizontal scaling.
(a)(b)Therefore, to facilitate the diagrammatic design process, when the design speed of the road section is inconsistent with the corridor progression design speedV, we will convert the progression design speed of the road section to V by adjusting the intersection distance. The adjusted distance Dp between intersection Ip and intersection I1 can be calculated by the following equation:(1)Dp=V⋅∑k=1p−1dkvk.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 ofV 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.Figure 3
The vertical scaling of the time-space diagram. (a) Before the vertical scaling. (b) After the vertical scaling.
(a)(b)Then, we can first keep the common signal cycleC1 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 CB can be obtained.According to the corridor progression design speedV and the range of the common signal cycle [Cmin, Cmax], the adjusted speed V(i, j) obtained after the jth rotation transformation in the ith round should satisfy the following equation:(2)V⋅CminC1≤Vi,j≤V⋅CmaxC1.


### 2.2.3. 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 intersectionI1, whereas backstepping coordination means that the red center point is consistent with the green center point of the coordinated phase at I1. The 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. Then, we define a Boolean variable Fp as the coordinated mode judgment factor. When Fp is 0, the optimal coordinated mode of the intersection Ip is synchronous coordination. When Fp is 1, the optimal coordinated mode of Ip is backstepping coordination.Algorithm 1: Initialization procedure.
Our first step is to define and calculate some basic parameters, and the initialization routine can be presented below.Initialization procedureStep0. Determine the range of the common signal cycle [Cmin, Cmax] and calculate the initial common signal cycle C1 = 0.5⋅(Cmin + Cmax).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 Dp 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).


## 2.2.1. Initial Common Signal Cycle

Let us suppose that there aren signalized intersections along the corridor and number the intersections in the ascending order in the outbound direction. Then, the pth (1 ≤ p ≤ n) intersection is defined as Ip in sequence along the outbound direction.The common signal cycle range will be determined by the signal cycle range of each intersection, denoted as [Cmin, Cmax]. The initial common signal cycle C1 can be valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax.


## 2.2.2. Adjusted Distance and Speed

Letdp and vp be the actual distance and the progression design speed between intersection Ip and Ip + 1 (1 ≤ p ≤ n-1), respectively. Then, 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(b). Therefore, 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.Figure 2
The horizontal scaling of the time-space diagram. (a) Before the horizontal scaling. (b) After the horizontal scaling.
(a)(b)Therefore, to facilitate the diagrammatic design process, when the design speed of the road section is inconsistent with the corridor progression design speedV, we will convert the progression design speed of the road section to V by adjusting the intersection distance. The adjusted distance Dp between intersection Ip and intersection I1 can be calculated by the following equation:(1)Dp=V⋅∑k=1p−1dkvk.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 ofV 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.Figure 3
The vertical scaling of the time-space diagram. (a) Before the vertical scaling. (b) After the vertical scaling.
(a)(b)Then, we can first keep the common signal cycleC1 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 CB can be obtained.According to the corridor progression design speedV and the range of the common signal cycle [Cmin, Cmax], the adjusted speed V(i, j) obtained after the jth rotation transformation in the ith round should satisfy the following equation:(2)V⋅CminC1≤Vi,j≤V⋅CmaxC1.


## 2.2.3. 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 intersectionI1, whereas backstepping coordination means that the red center point is consistent with the green center point of the coordinated phase at I1. The 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. Then, we define a Boolean variable Fp as the coordinated mode judgment factor. When Fp is 0, the optimal coordinated mode of the intersection Ip is synchronous coordination. When Fp is 1, the optimal coordinated mode of Ip is backstepping coordination.Algorithm 1: Initialization procedure.
Our first step is to define and calculate some basic parameters, and the initialization routine can be presented below.Initialization procedureStep0. Determine the range of the common signal cycle [Cmin, Cmax] and calculate the initial common signal cycle C1 = 0.5⋅(Cmin + Cmax).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 Dp 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).


## 2.3. The First Round of Rotation Transformation

The 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 atI1 as the reference point OL1 of the time-space diagram coordinate system. A horizontal line can be drawn from OL1 and rotated until the cotangent of its angle with the x-axis equals the corridor progression design speed V. This 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 L1 and then assign V to adjusted speed V(1, 1).The following steps have to be executed cyclically until the coordinated modes of all intersections have been determined.


### 2.3.1. Primary Judgment Factor Calculation

The intersectionIp is selected as the current coordinated intersection during the first round’s pth (2 ≤ p ≤ n) rotation transformation. According to PTC line Lp-1 obtained by the last rotation transformation, the crossing point of Lp-1 and the timeline of Ip is marked as OLp as shown in Figure 4(a). The coordinated phase horizontal red center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as ORp. The coordinated phase horizontal green center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as OGp.Figure 4
The determination of the coordinated mode whenfp ≥ 0.
(a)(b)(c)Starting at the pointOL1, we can make a ray lRp (lGp) passing through the point ORp (OGp), and the corresponding adjusted speed VRp (VGp) satisfies the following:(3)VRpVGp=DpyRpDpyGp.The time difference between pointsOLp and ORp (OGp), which corresponds to the distance between the ordinate of points yLp and yRp (yGp), is recorded as TRp (TGp). Then, the primary judgment factor fp can be constructed according to TRp and TGp:(4)fp=TRp−TGp=yLp−yRp−yLp−yGp.


### 2.3.2. Optimal Coordinated Mode Calculation

We definefR(p, q) (fG(p, q)) (1 < q < p) as a Boolean variable. When ray lRp (lGp) does not pass through the red interval of the intersection Iq (1 < q < p), fR (p, q) (fG (p, q)) is 0. When ray lRp (lGp) passes through the red interval of Iq, fR (p,q) (fG (p, q)) is 1. Also, the red interval crossing amount of lRp (lGp) in Iq can be defined as TR (p, q) (TG (p, q)).Whenfp ≥ 0, the situation is shown in Figure 4. If ray lGp does not pass through any red interval of the coordinated phase, all fG(p,q) equals 0. The current coordinated intersection can form a synchronous coordinated mode with I1, where Fp = 0. If ray lGp passes through any red interval of the coordinated phase while ray lRp does not pass through any red interval, as shown in Figure 4(b), the backstepping coordinated mode can be chosen, where Fp = 1. If both lGp and lRp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lGp is less than or equal to lRp, as shown in Figure 4(c), these two intersections can form a synchronous coordinated mode, where Fp = 0. Otherwise, the backstepping coordinated mode will be chosen as Fp = 1.Similarly, whenfp < 0, if ray lRp does not pass through any red interval of the coordinated phase, all fR (p,q) equals 0, Fp = 1. If ray lRp passes through any red interval of the coordinated phase while ray lGp does not pass through any red interval, then Fp = 0. If both lRp and lGp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lRp is less than or equal to lGp, Fp = 1. Otherwise, Fp = 0.


### 2.3.3. PTC Line Optimization

If the current coordinated intersection has formed a synchronous coordinated mode withI1 and VGp is within the rotation range of the PTC line, we can define lGp as PTC line Lp and assign the value of VGp to the adjusted speed V(1,p), as shown in Figure 5(a). When VGp is out of the rotation range, we can keep the PTC line unchanged, that is, as shown in Figure 5(b), Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Figure 5
The PTC line updating under synchronous coordinated mode.
(a)(b)Similarly, if the current coordinated intersection has formed a backstepping coordinated mode withI1 and VRp is within the rotation range of the PTC line, we can define lRp as PTC line Lp and assign the value of VRp to V(1,p). When VRp is out of the rotation range, we can keep the PTC line unchanged. Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Algorithm2: First round rotation transformation procedure.
Our second step is to optimize the intersection coordinated modes, and the first round of the rotation transformation routine can be presented below.First Round Rotation Transformation ProcedureStep0. Determine the crossing point OLp, ORp, and OGp.First, letyLp=Dp/V1,p−1.Then, if modyLpC1/2,2=0, yRp equals C1⋅yLpC1/2/2+C1/2 and yGp equals C1⋅yLpC1/2/2, else yRp=C1⋅yLpC1/2/2−C1/2 and yGp=C1⋅yLpC1/2+1/2.Step1. Calculate the primary judgment factor fp as shown in Equation (4).Step2. Make a judgment of whether ray lRp and lGp pass through the red interval of any intersection Iq (1 <q < p) or not.IfDq/VRp−yRp,q≤0.5⋅C1⋅λq, fR(p, q) = 0, otherwise fR(p, q)= 1 and TRp,q=Dq/VRp−yRp,q−0.5⋅C1⋅λq.IfDq/VGp−yGp,q≤0.5⋅C1⋅λq, fG(p, q) = 0, otherwise fG(p, q)= 1 and TGp,q=Dq/VGp−yGp,q−0.5⋅C1⋅λq.yR(p, q) (yG(p, q)) is the ordinate of the green center point of Iq, which is closest to the crossing point of lRp (lGp) and the timeline of Iq, and λq is the green split of Iq.Step3. Determine the coordinated mode of Ip.Iffp≥0 and ∑q=2p−1fGp,q=0, or fp<0, ∑q=2p−1fRp,q>0 and ∑q=2p−1fGp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1 < max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 0.Iffp<0 and ∑q=2p−1fRp,q=0, or fp≥0, ∑q=2p−1fGp,q>0 and ∑q=2p−1fRp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1> max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 1.Step4. PTC line optimization.IfFp=0 and VGp is within the rotation range, calculate VGp as shown in Equation (3), and then let V1,p equal VGp, else if Fp=1 and VRp is within the rotation range, calculate VRp and then let V1,p equal VRp, else V1,p=V1,p−1.Step5. Make a judgment of whether p equals n or not. If p < n, let p equal p + 1 and then return to Step0.


## 2.3.1. Primary Judgment Factor Calculation

The intersectionIp is selected as the current coordinated intersection during the first round’s pth (2 ≤ p ≤ n) rotation transformation. According to PTC line Lp-1 obtained by the last rotation transformation, the crossing point of Lp-1 and the timeline of Ip is marked as OLp as shown in Figure 4(a). The coordinated phase horizontal red center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as ORp. The coordinated phase horizontal green center lines at I1 have crossing points with the timeline of Ip and so we define the point closest to OLp as OGp.Figure 4
The determination of the coordinated mode whenfp ≥ 0.
(a)(b)(c)Starting at the pointOL1, we can make a ray lRp (lGp) passing through the point ORp (OGp), and the corresponding adjusted speed VRp (VGp) satisfies the following:(3)VRpVGp=DpyRpDpyGp.The time difference between pointsOLp and ORp (OGp), which corresponds to the distance between the ordinate of points yLp and yRp (yGp), is recorded as TRp (TGp). Then, the primary judgment factor fp can be constructed according to TRp and TGp:(4)fp=TRp−TGp=yLp−yRp−yLp−yGp.


## 2.3.2. Optimal Coordinated Mode Calculation

We definefR(p, q) (fG(p, q)) (1 < q < p) as a Boolean variable. When ray lRp (lGp) does not pass through the red interval of the intersection Iq (1 < q < p), fR (p, q) (fG (p, q)) is 0. When ray lRp (lGp) passes through the red interval of Iq, fR (p,q) (fG (p, q)) is 1. Also, the red interval crossing amount of lRp (lGp) in Iq can be defined as TR (p, q) (TG (p, q)).Whenfp ≥ 0, the situation is shown in Figure 4. If ray lGp does not pass through any red interval of the coordinated phase, all fG(p,q) equals 0. The current coordinated intersection can form a synchronous coordinated mode with I1, where Fp = 0. If ray lGp passes through any red interval of the coordinated phase while ray lRp does not pass through any red interval, as shown in Figure 4(b), the backstepping coordinated mode can be chosen, where Fp = 1. If both lGp and lRp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lGp is less than or equal to lRp, as shown in Figure 4(c), these two intersections can form a synchronous coordinated mode, where Fp = 0. Otherwise, the backstepping coordinated mode will be chosen as Fp = 1.Similarly, whenfp < 0, if ray lRp does not pass through any red interval of the coordinated phase, all fR (p,q) equals 0, Fp = 1. If ray lRp passes through any red interval of the coordinated phase while ray lGp does not pass through any red interval, then Fp = 0. If both lRp and lGp pass through the red interval of the coordinated phase, and if the maximum red interval crossing amount of lRp is less than or equal to lGp, Fp = 1. Otherwise, Fp = 0.


## 2.3.3. PTC Line Optimization

If the current coordinated intersection has formed a synchronous coordinated mode withI1 and VGp is within the rotation range of the PTC line, we can define lGp as PTC line Lp and assign the value of VGp to the adjusted speed V(1,p), as shown in Figure 5(a). When VGp is out of the rotation range, we can keep the PTC line unchanged, that is, as shown in Figure 5(b), Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Figure 5
The PTC line updating under synchronous coordinated mode.
(a)(b)Similarly, if the current coordinated intersection has formed a backstepping coordinated mode withI1 and VRp is within the rotation range of the PTC line, we can define lRp as PTC line Lp and assign the value of VRp to V(1,p). When VRp is out of the rotation range, we can keep the PTC line unchanged. Lp is the same as Lp-1, and V(1,p) equals V(1,p-1).Algorithm2: First round rotation transformation procedure.
Our second step is to optimize the intersection coordinated modes, and the first round of the rotation transformation routine can be presented below.First Round Rotation Transformation ProcedureStep0. Determine the crossing point OLp, ORp, and OGp.First, letyLp=Dp/V1,p−1.Then, if modyLpC1/2,2=0, yRp equals C1⋅yLpC1/2/2+C1/2 and yGp equals C1⋅yLpC1/2/2, else yRp=C1⋅yLpC1/2/2−C1/2 and yGp=C1⋅yLpC1/2+1/2.Step1. Calculate the primary judgment factor fp as shown in Equation (4).Step2. Make a judgment of whether ray lRp and lGp pass through the red interval of any intersection Iq (1 <q < p) or not.IfDq/VRp−yRp,q≤0.5⋅C1⋅λq, fR(p, q) = 0, otherwise fR(p, q)= 1 and TRp,q=Dq/VRp−yRp,q−0.5⋅C1⋅λq.IfDq/VGp−yGp,q≤0.5⋅C1⋅λq, fG(p, q) = 0, otherwise fG(p, q)= 1 and TGp,q=Dq/VGp−yGp,q−0.5⋅C1⋅λq.yR(p, q) (yG(p, q)) is the ordinate of the green center point of Iq, which is closest to the crossing point of lRp (lGp) and the timeline of Iq, and λq is the green split of Iq.Step3. Determine the coordinated mode of Ip.Iffp≥0 and ∑q=2p−1fGp,q=0, or fp<0, ∑q=2p−1fRp,q>0 and ∑q=2p−1fGp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1 < max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 0.Iffp<0 and ∑q=2p−1fRp,q=0, or fp≥0, ∑q=2p−1fGp,q>0 and ∑q=2p−1fRp,q=0, or ∑q=2p−1fRp,q>0, ∑q=2p−1fGp,q>0 and max TGp,2,…,TGp,p−1> max TRp,2,…,TRp,p−1, the coordinated mode judgment factor Fp equals 1.Step4. PTC line optimization.IfFp=0 and VGp is within the rotation range, calculate VGp as shown in Equation (3), and then let V1,p equal VGp, else if Fp=1 and VRp is within the rotation range, calculate VRp and then let V1,p equal VRp, else V1,p=V1,p−1.Step5. Make a judgment of whether p equals n or not. If p < n, let p equal p + 1 and then return to Step0.


## 2.4. The Second Round of Rotation Transformation

The 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 speedV(1,n) to V(2,0). Then, the beginning PTC line LB0 and the end PTC line LE0 of the outbound progression can be obtained. Then, the initial bandwidth b0 and ratio R0 can also be calculated. Meanwhile, the bottleneck intersections of the beginning PTC line are put into the intersection set SB0, and the bottleneck intersections of the end PTC line are put into the intersection set SE0. The calculation of related parameters will be introduced in the next section.We define a termination decision parameterFA, which is a Boolean variable. The following steps have to be executed cyclically until the termination decision parameter FA equals 1.


### 2.4.1. Bottleneck Intersection Recognition

We assume that the obtained beginning PTC line, end PTC line, adjusted speed, bandwidth, and bandwidth ratio after themth rotation transformation in the second round are defined as LBm, LEm, V(2,m), bm, and Rm, respectively. The parameters obtained from the m-1th rotation transformation need to be used during the mth rotation transformation in the second round.According toFp, V(2,m-1), C1, and λp, we can draw the beginning and end PTC lines in the time-space diagram. We define the Boolean variables KB (m-1,p) and KE (m-1,p) as the judgment factors of the bottleneck intersection of the beginning and end PTC lines. When KB (m-1,p) equals 1, intersection Ip is the bottleneck intersection of the beginning PTC line after the m-1th rotation transformation, Ip∈SBm-1, and the corresponding bottleneck point is defined as PBp. When KE (m-1,p) equals 1, Ip is the bottleneck intersection of the end PTC line after the m-1th rotation transformation, Ip∈SEm-1, and the corresponding bottleneck point is recorded as PEp.Before the other calculation steps of themth rotation transformation in the second round, it is necessary to judge whether the sets SBm-1 and SEm-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 FA = 1.


### 2.4.2. Conditions for Stopping the Rotation

Condition 1. ∃Ik∈(SBm-1∩​SEm-1), that is, there are both bottleneck points of the beginning and end PTC lines at intersection Ik. At this time, the ratio of the bandwidth Rm equals the split of the coordinated phase at Ik, which means that it has already reached the maximum value of the bandwidth ratio. Therefore, there is no need to continue the rotation transformation of the PTC line.Condition 2. ∃Ii∈SEm-1, Ij∈SBm-1, Ik∈SEm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 6
Condition 2 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)Condition 3. ∃Ii∈SBm-1, Ij∈SEm-1, Ik∈SBm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 7
Condition 3 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)


### 2.4.3. Rotation Points and Direction Recognition

It is necessary to determine the rotation points and direction according to the relationship between the rotation transformation and the progression bandwidth during themth rotation transformation in the second round.When increasing the progression speed, that is,V(2,m) > V(2,m-1), the rotation point of LBm should fall at the most downstream bottleneck intersection in SBm-1, and the rotation point of LEm should fall at the most upstream bottleneck intersection in SEm-1, as shown in Figures 8(a) and 8(b). Therefore, when rotating the PTC line clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≤ k and ∀Il∈SBm-1 satisfies j ≥ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.Figure 8
The relationship between the rotation transformation and the progression bandwidth. (a)V2,m>V2,m−1andDm^>0. (b) V2,m>V2,m−1andDm^<0. (c) V2,m<V2,m−1andDm^>0. (d) V2,m<V2,m−1andDm^<0.
(a)(b)(c)(d)When reducing the progression speed, that is,V(2,m) < V(2,m-1), the rotation point of LEm should fall at the most downstream bottleneck intersection in SEm-1, and the rotation point of LBm should fall at the most upstream bottleneck intersection in SBm-1, as shown in Figures 8(c) and 8(d). Therefore, when rotating the PTC line counter-clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≥ k and ∀Il∈SBm-1 satisfies j ≤ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.D^m is defined as the abscissa difference between the bottleneck rotation points PBj and PEi in the mth rotation transformation in the second round, that is, D^m  = xj-xi. Then, the bandwidth bm can be calculated as follows:(5)bm=bm−1+Δbm=bm−1+D^m⋅1V2,m−1V2,m−1,where Δbm 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 variableKDm as the judgment factor of the rotation direction. When KDm equals 0, the rotation direction is the counter-clockwise rotation, whereas the rotation direction is the clockwise rotation when KDm equals 1.Scenario 1.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 1 should be determined as the counter-clockwise rotation, KDm = 0.Scenario 2.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 2 should be determined as the clockwise rotation, KDm = 1.


### 2.4.4. Rotation Angle Calculation

Taking the bottleneck point of the end PTC linePEi and the bottleneck point of the beginning PTC line PBj 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.The green endpoint of the coordinated phase crossed by the end PTC lineLEm-1 at intersection Ik (1 ≤ k ≤ n) is defined as PFk, and the green start point of the coordinated phase crossed by the beginning PTC line LBm-1 at Ik (1 ≤ k ≤ n) is defined as PSk.The adjusted speedVFk (VSk) corresponding to the rotation line LFk (LSk) formed by connecting points PEi (PBj) and PFk (PSk) can be calculated by the following equation:(6)VFkVSk=xk−xiyFk−yEixk−xjySk−yBj.Here,yFk and yEi represent the ordinate of points PFk and PEi, respectively; ySk and yBj represent the ordinate of points PSk and PBj, respectively.IfKDm equals 0, we should connect points PEi and PFg (i + 1 ≤ g ≤ n) in turn to form the rotation line LFg. When the corresponding adjusted speed VFg is within the adjusted speed range, the eligible VFg is incorporated into the optional vehicle speed set, SVm, obtained after the mth rotation transformation. Meanwhile, PBj and PSh (1 ≤ h ≤ j-1) are connected to form the rotation line LSh. When the corresponding adjusted speed VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the maximum value in SVm 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).Figure 9
Rotation angle determination.
(a)(b)IfKDm equals 1, we should connect points PEi and PFg (1 ≤ g ≤ i-1) in turn to form the rotation line LFg. When VFg is within the adjusted speed range, the eligible VFg is incorporated into SVm. Meanwhile, points PBj and PSh (j + 1 ≤ h ≤ n) are connected to form the rotation line LSh. When VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the minimum value in SVm as V(2,m), as shown in Figure 9(b).According toV(2,m), the end PTC line LEm and the beginning PTC line LBm 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 SEm and SBm according to LEm and LBm.Algorithm3: Second round rotation transformation procedure.
Our third step is to optimize the common signal cycle and offsets, and the second round of the rotation transformation routine can be presented below.Second Round Rotation Transformation ProcedureStep0. Determine the bottleneck intersections and the set SBm-1 and SEm-1.According toFp, V(2,m-1), C1, and λp, draw the beginning and end PTC lines in the time-space diagram. If Ip is the bottleneck intersection of the beginning PTC line, let KBm−1,p=1 and add Ip in set SBm-1, else KBm−1,p=0. And if Ip is the bottleneck intersection of the end PTC line, let KEm−1,p=1 and add Ip in set SEm-1, else KEm−1,p=0.Step1. Make a judgment of whether the sets SBm-1 and SEm-1 meet any condition for stopping rotation.If∃Ik∈SBm−1∩SEm−1, or ∃Ii∈SEm−1,Ij∈SBm−1,Ik∈SEm−1 and i<j<k, or ∃Ii∈SBm−1,Ij∈SEm−1,Ik∈SBm−1 and i<j<k, then FA=1, finish the second round of rotation transformation. Otherwise, FA=0, proceed to Step 2.Step2. Determine the rotation direction.If∀Ik∈SEm−1,∀Il∈SBm−1,andk<l, let KDm=0, else if ∀Ik∈SEm−1,∀Il∈SBm−1,andk>l, let KDm=1.Step3. Determine the rotation angle.IfKDm=0, connect points PEi and PFg (i + 1 ≤ g ≤ n), points PBj and PSh (1 ≤ h ≤ j-1) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the maximum value in SVm as the updated adjusted speed V(2,m).IfKDm=1, connect points PEi and PFg (1 ≤ g ≤ i-1), points PBj and PSh (j + 1 ≤ h ≤ n) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the minimum value in SVm as the updated adjusted speed V(2,m).Step4. Let m = m + 1, and return to Step 0.


## 2.4.1. Bottleneck Intersection Recognition

We assume that the obtained beginning PTC line, end PTC line, adjusted speed, bandwidth, and bandwidth ratio after themth rotation transformation in the second round are defined as LBm, LEm, V(2,m), bm, and Rm, respectively. The parameters obtained from the m-1th rotation transformation need to be used during the mth rotation transformation in the second round.According toFp, V(2,m-1), C1, and λp, we can draw the beginning and end PTC lines in the time-space diagram. We define the Boolean variables KB (m-1,p) and KE (m-1,p) as the judgment factors of the bottleneck intersection of the beginning and end PTC lines. When KB (m-1,p) equals 1, intersection Ip is the bottleneck intersection of the beginning PTC line after the m-1th rotation transformation, Ip∈SBm-1, and the corresponding bottleneck point is defined as PBp. When KE (m-1,p) equals 1, Ip is the bottleneck intersection of the end PTC line after the m-1th rotation transformation, Ip∈SEm-1, and the corresponding bottleneck point is recorded as PEp.Before the other calculation steps of themth rotation transformation in the second round, it is necessary to judge whether the sets SBm-1 and SEm-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 FA = 1.


## 2.4.2. Conditions for Stopping the Rotation

Condition 1. ∃Ik∈(SBm-1∩​SEm-1), that is, there are both bottleneck points of the beginning and end PTC lines at intersection Ik. At this time, the ratio of the bandwidth Rm equals the split of the coordinated phase at Ik, which means that it has already reached the maximum value of the bandwidth ratio. Therefore, there is no need to continue the rotation transformation of the PTC line.Condition 2. ∃Ii∈SEm-1, Ij∈SBm-1, Ik∈SEm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 6
Condition 2 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)Condition 3. ∃Ii∈SBm-1, Ij∈SEm-1, Ik∈SBm-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. The 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). Therefore, the further rotation transformation of the PTC line has to be finished.Figure 7
Condition 3 for stopping the rotation. (a)V2,m>V2,m−1. (b) V2,m<V2,m−1.
(a)(b)


## 2.4.3. Rotation Points and Direction Recognition

It is necessary to determine the rotation points and direction according to the relationship between the rotation transformation and the progression bandwidth during themth rotation transformation in the second round.When increasing the progression speed, that is,V(2,m) > V(2,m-1), the rotation point of LBm should fall at the most downstream bottleneck intersection in SBm-1, and the rotation point of LEm should fall at the most upstream bottleneck intersection in SEm-1, as shown in Figures 8(a) and 8(b). Therefore, when rotating the PTC line clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≤ k and ∀Il∈SBm-1 satisfies j ≥ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.Figure 8
The relationship between the rotation transformation and the progression bandwidth. (a)V2,m>V2,m−1andDm^>0. (b) V2,m>V2,m−1andDm^<0. (c) V2,m<V2,m−1andDm^>0. (d) V2,m<V2,m−1andDm^<0.
(a)(b)(c)(d)When reducing the progression speed, that is,V(2,m) < V(2,m-1), the rotation point of LEm should fall at the most downstream bottleneck intersection in SEm-1, and the rotation point of LBm should fall at the most upstream bottleneck intersection in SBm-1, as shown in Figures 8(c) and 8(d). Therefore, when rotating the PTC line counter-clockwise, we should select intersections Ii∈SEm-1, Ij∈SBm-1 to ensure that ∀Ik∈SEm-1 satisfies i ≥ k and ∀Il∈SBm-1 satisfies j ≤ l. Then, the bottleneck point PEi and PBj should be taken as the rotation points.D^m is defined as the abscissa difference between the bottleneck rotation points PBj and PEi in the mth rotation transformation in the second round, that is, D^m  = xj-xi. Then, the bandwidth bm can be calculated as follows:(5)bm=bm−1+Δbm=bm−1+D^m⋅1V2,m−1V2,m−1,where Δbm 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 variableKDm as the judgment factor of the rotation direction. When KDm equals 0, the rotation direction is the counter-clockwise rotation, whereas the rotation direction is the clockwise rotation when KDm equals 1.Scenario 1.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 1 should be determined as the counter-clockwise rotation, KDm = 0.Scenario 2.
If∀Ik∈SEm-1, and ∀Il∈SBm-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). Therefore, the rotation direction in Scenario 2 should be determined as the clockwise rotation, KDm = 1.


## 2.4.4. Rotation Angle Calculation

Taking the bottleneck point of the end PTC linePEi and the bottleneck point of the beginning PTC line PBj 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.The green endpoint of the coordinated phase crossed by the end PTC lineLEm-1 at intersection Ik (1 ≤ k ≤ n) is defined as PFk, and the green start point of the coordinated phase crossed by the beginning PTC line LBm-1 at Ik (1 ≤ k ≤ n) is defined as PSk.The adjusted speedVFk (VSk) corresponding to the rotation line LFk (LSk) formed by connecting points PEi (PBj) and PFk (PSk) can be calculated by the following equation:(6)VFkVSk=xk−xiyFk−yEixk−xjySk−yBj.Here,yFk and yEi represent the ordinate of points PFk and PEi, respectively; ySk and yBj represent the ordinate of points PSk and PBj, respectively.IfKDm equals 0, we should connect points PEi and PFg (i + 1 ≤ g ≤ n) in turn to form the rotation line LFg. When the corresponding adjusted speed VFg is within the adjusted speed range, the eligible VFg is incorporated into the optional vehicle speed set, SVm, obtained after the mth rotation transformation. Meanwhile, PBj and PSh (1 ≤ h ≤ j-1) are connected to form the rotation line LSh. When the corresponding adjusted speed VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the maximum value in SVm 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).Figure 9
Rotation angle determination.
(a)(b)IfKDm equals 1, we should connect points PEi and PFg (1 ≤ g ≤ i-1) in turn to form the rotation line LFg. When VFg is within the adjusted speed range, the eligible VFg is incorporated into SVm. Meanwhile, points PBj and PSh (j + 1 ≤ h ≤ n) are connected to form the rotation line LSh. When VSh is within the adjusted speed range, the eligible VSh is incorporated into SVm. Then, we have to select the minimum value in SVm as V(2,m), as shown in Figure 9(b).According toV(2,m), the end PTC line LEm and the beginning PTC line LBm 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 SEm and SBm according to LEm and LBm.Algorithm3: Second round rotation transformation procedure.
Our third step is to optimize the common signal cycle and offsets, and the second round of the rotation transformation routine can be presented below.Second Round Rotation Transformation ProcedureStep0. Determine the bottleneck intersections and the set SBm-1 and SEm-1.According toFp, V(2,m-1), C1, and λp, draw the beginning and end PTC lines in the time-space diagram. If Ip is the bottleneck intersection of the beginning PTC line, let KBm−1,p=1 and add Ip in set SBm-1, else KBm−1,p=0. And if Ip is the bottleneck intersection of the end PTC line, let KEm−1,p=1 and add Ip in set SEm-1, else KEm−1,p=0.Step1. Make a judgment of whether the sets SBm-1 and SEm-1 meet any condition for stopping rotation.If∃Ik∈SBm−1∩SEm−1, or ∃Ii∈SEm−1,Ij∈SBm−1,Ik∈SEm−1 and i<j<k, or ∃Ii∈SBm−1,Ij∈SEm−1,Ik∈SBm−1 and i<j<k, then FA=1, finish the second round of rotation transformation. Otherwise, FA=0, proceed to Step 2.Step2. Determine the rotation direction.If∀Ik∈SEm−1,∀Il∈SBm−1,andk<l, let KDm=0, else if ∀Ik∈SEm−1,∀Il∈SBm−1,andk>l, let KDm=1.Step3. Determine the rotation angle.IfKDm=0, connect points PEi and PFg (i + 1 ≤ g ≤ n), points PBj and PSh (1 ≤ h ≤ j-1) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the maximum value in SVm as the updated adjusted speed V(2,m).IfKDm=1, connect points PEi and PFg (1 ≤ g ≤ i-1), points PBj and PSh (j + 1 ≤ h ≤ n) in turn, calculate the corresponding VFg and VSh as shown in Equation (6), add the eligible VFg and VSh in set SVm, then select the minimum value in SVm as the updated adjusted speed V(2,m).Step4. Let m = m + 1, and return to Step 0.


## 2.5. Scheme Generation

### 2.5.1. Optimal Common Signal Cycle Calculation

The optimal adjusted speedVB 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 CB can be calculated by the following equation:(7)CB=C1⋅VBV.


### 2.5.2. 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. The absolute offset of the intersectionIp (1 ≤ p ≤ n) is the beginning of the coordinated phase at Ip defined as Op. While the green center point of the coordinated phase at intersection I1 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:(8)Op=CB−0.5⋅CB⋅λp,1≤p≤nandFp=0,0.5⋅CB−0.5⋅CB⋅λp,1≤p≤nandFp=1.Algorithm4: Scheme generation procedure.
Our final step is to obtain the final optimized scheme, and the scheme generation routine can be presented below.Scheme Generation ProcedureStep0. Calculate the optimal common signal cycle CB as shown in Equation (7).Step1. Calculate the absolute offset Op 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.


## 2.5.1. Optimal Common Signal Cycle Calculation

The optimal adjusted speedVB 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 CB can be calculated by the following equation:(7)CB=C1⋅VBV.


## 2.5.2. 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. The absolute offset of the intersectionIp (1 ≤ p ≤ n) is the beginning of the coordinated phase at Ip defined as Op. While the green center point of the coordinated phase at intersection I1 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:(8)Op=CB−0.5⋅CB⋅λp,1≤p≤nandFp=0,0.5⋅CB−0.5⋅CB⋅λp,1≤p≤nandFp=1.Algorithm4: Scheme generation procedure.
Our final step is to obtain the final optimized scheme, and the scheme generation routine can be presented below.Scheme Generation ProcedureStep0. Calculate the optimal common signal cycle CB as shown in Equation (7).Step1. Calculate the absolute offset Op 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.


## 2.6. 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]. The corridor being analyzed is an east-west roadway, and there are eight signalized intersections along the corridor. The direction from west to east is defined as the outbound direction, and the western-most intersection is selected as intersection I1. The schematic diagram of the selected corridor is shown in Figure 10. It has already known that each intersection adopts symmetrical phasing. The split of each intersection is 55%, 60%, 65%, 65%, 60%, 65%, 70%, and 50%, respectively. The optimization range of the common signal cycle is [60, 100] s, and the progression design speed is 40 km/h, which approximately equals 11 m/s.Figure 10
Structure diagram of the selected corridor.


## 2.7. Scheme Generation

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


### 2.7.1. Initialization

The initial common signal cycleC1 is valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax, 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 C1 is unchanged, the adjusted speed range can be determined as [8.25, 13.75] m/s.


### 2.7.2. The First Round of Rotation Transformation

The green center point of the coordinated phase at intersectionI1 is defined as the reference point OL1 of the coordinate system of the time-space diagram. A horizontal line from the reference point OL1 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 L1 and assign the value of V to adjusted speed V(1, 1). Then, the first rotation transformation in the first round is completed.The intersectionI2 is selected as the current coordinated intersection. According to PTC line L1 obtained by the first rotation transformation, the crossing point of the PTC line L1 and the timeline of I2 is marked as OL2. The nearest crossing point of the coordinated phase horizontal red (green) center line at I1, and the timeline of I2 is identified and recorded as OR2 (OG2), as shown in Figure 11(a). The time difference TR2 between points OL2 and OR2 is 8 s, and the time difference TG2 between points OL2 and OG2 is 32 s. Then, f2 = −24 s.Figure 11
The coordination process of the first round of rotation transformation. (a) The 2nd rotation transformation. (b) The 3rd rotation transformation. (c) The 4th rotation transformation. (d) The 5th rotation transformation. (e) The 6th rotation transformation. (f) The 7th rotation transformation. (g) The 8th rotation transformation. (h) The obtained time-space diagram.
(a)(b)(c)(d)(e)(f)(g)(h)RaylR2 does not pass through the red interval of the coordinated phase, and f2 < 0, and then, I2 can form a backstepping coordinated mode with I1, where F2 = 1.The corresponding adjusted speedVR2 of lR2 is 8.75 m/s, within the adjusted speed range [8.25, 13.75] m/s. Therefore, lR2 is defined as PTC line L2, and the value of VR2 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. Then, the coordinated mode betweenI1 and I3, I4, I5, I6, I7, and I8 can be determined, respectively. The coordination process is shown in Figures 11(b)–11(g).After the first round of rotation transformation,I2, I5, and I8 form a backstepping coordinated mode with I1, F2 = F5 = F8 = 1, whereas the intersections I3, I4, I6, and I7 form a synchronous coordinated mode with I1, F3 = F4 = F6 = F7 = 0. The 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). The corresponding bandwidth is 24 s, and the bandwidth ratio is 30.0%.


### 2.7.3. The Second Round of Rotation Transformation

According to the adjusted speedV(1,8) finally obtained after the first round of rotation transformation, the end PTC line of vehicle band LE0 and the beginning PTC line of vehicle band LB0 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 SE0 = {I7}. Also, the bottleneck intersections of the beginning PTC line are put into the set SB0 = {I3}.Figure 12
The second round of rotation transformation.The setSB0 and SE0 do not meet any conditions for stopping rotation. Then, we have to complete the first rotation transformation in the second round.Because the only bottleneck intersection of the end PTC lineI7 is located downstream of the only bottleneck intersection of the beginning PTC line I3, it corresponds to the rotation direction judgment Scenario 2. Therefore, the PTC line can be rotated clockwise. Furthermore, the most upstream bottleneck point PE7 is selected as the rotation point of the end PTC line, whereas the most downstream bottleneck point PB3 is selected as the rotation point of the beginning PTC line.As shown in Figure12, we can connect points PE7 (PB3) and PFg (1 ≤ g ≤ 6) (PSh (4 ≤ h ≤ 8)) to form the rotation line LFg (LSh) and determine whether the corresponding adjusted speed VFg (VSh) is within the adjusted speed range. The calculation results are shown in Table 1.Table 1
The optional vehicle speed set during the first rotation transformation in the second round.
Bottleneck pointsThe start point/endpoint of the coordinated phaseAdjusted speed (m/s)Optional vehicle speedSV1PE7(2010, 268)PF1(0, 102)VF1 = 12.11VF1 ∈ SV1PF2(350, 144)VF2 = 13.39VF2 ∈ SV1PF3(750, 186)VF3 = 15.37VF3 ∉ SV1PF4(910, 186)VF4 = 13.41VF4 ∈ SV1PF5(1450, 224)VF5 = 12.73VF5 ∈ SV1PF6(1730, 266)VF6 = 140.00VF6 ∉ SV1PB3(750, 134)PS4(910, 134)VS4 = ∞VS4 ∉ SV1PS5(1450, 176)VS5 = 16.67VS5 ∉ SV1PS6(1730, 214)VS6 = 12.25VS6 ∈ SV1PS7(2010, 212)VS7 = 16.15VS7 ∉ SV1PS8(2280, 260)VS8 = 12.14VS8 ∈ SV1It can be seen from Table1 that SV1 = {VF1, VF2, VF4, VF5, VS6, VS8}. Then, select the minimum value VF1 in SV1 as the adjusted speed V(2, 1). Then, V(2, 1) is 12.11 m/s.According toV(2,1), the end PTC line LE1 and the beginning PTC line LB1 of the outbound progression can be drawn. Determine the bottleneck intersections set of the end PTC line SE1 = {I1, I7} and the beginning PTC line bottleneck intersections set SB1 = {I3}, according to LE1 and LB1. Then, the updating of the sets of bottleneck intersections is completed, as shown in Figure 13.Figure 13
Updating the sets of bottleneck intersections.The setsSE1 and SB1 meet one of the conditions for stopping rotation. At this time, ∃I1∈SE1, I3∈SB1, I7∈SE1, 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. Therefore, we can stop the further rotation transformation of the PTC line, and the second round of rotation transformation is finished.


### 2.7.4. Scheme Generation

The optimal adjusted speedVB 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 CB is 88 s.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. The time-space diagram finally obtained is shown in Figure14. The corresponding progression bandwidth is 32.85 s, and the bandwidth ratio reaches 37.3%.Figure 14
The finally obtained time-space diagram.


### 2.7.5. Programmatic Solution

The proposed solution logic steps can be solved graphically and programmed to quickly obtain an optimal signal coordination control scheme. The above case can be solved by python programming to obtain the optimal signal coordination control scheme. The experimental environment of the model solver is the Windows10 64-bit operating system, and the CPU is intel i5-6600 3.30 GHz. The 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.


## 2.7.1. Initialization

The initial common signal cycleC1 is valued as the midpoint between the minimum common signal cycle Cmin and the maximum common signal cycle Cmax, 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 C1 is unchanged, the adjusted speed range can be determined as [8.25, 13.75] m/s.


## 2.7.2. The First Round of Rotation Transformation

The green center point of the coordinated phase at intersectionI1 is defined as the reference point OL1 of the coordinate system of the time-space diagram. A horizontal line from the reference point OL1 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 L1 and assign the value of V to adjusted speed V(1, 1). Then, the first rotation transformation in the first round is completed.The intersectionI2 is selected as the current coordinated intersection. According to PTC line L1 obtained by the first rotation transformation, the crossing point of the PTC line L1 and the timeline of I2 is marked as OL2. The nearest crossing point of the coordinated phase horizontal red (green) center line at I1, and the timeline of I2 is identified and recorded as OR2 (OG2), as shown in Figure 11(a). The time difference TR2 between points OL2 and OR2 is 8 s, and the time difference TG2 between points OL2 and OG2 is 32 s. Then, f2 = −24 s.Figure 11
The coordination process of the first round of rotation transformation. (a) The 2nd rotation transformation. (b) The 3rd rotation transformation. (c) The 4th rotation transformation. (d) The 5th rotation transformation. (e) The 6th rotation transformation. (f) The 7th rotation transformation. (g) The 8th rotation transformation. (h) The obtained time-space diagram.
(a)(b)(c)(d)(e)(f)(g)(h)RaylR2 does not pass through the red interval of the coordinated phase, and f2 < 0, and then, I2 can form a backstepping coordinated mode with I1, where F2 = 1.The corresponding adjusted speedVR2 of lR2 is 8.75 m/s, within the adjusted speed range [8.25, 13.75] m/s. Therefore, lR2 is defined as PTC line L2, and the value of VR2 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. Then, the coordinated mode betweenI1 and I3, I4, I5, I6, I7, and I8 can be determined, respectively. The coordination process is shown in Figures 11(b)–11(g).After the first round of rotation transformation,I2, I5, and I8 form a backstepping coordinated mode with I1, F2 = F5 = F8 = 1, whereas the intersections I3, I4, I6, and I7 form a synchronous coordinated mode with I1, F3 = F4 = F6 = F7 = 0. The 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). The corresponding bandwidth is 24 s, and the bandwidth ratio is 30.0%.


## 2.7.3. The Second Round of Rotation Transformation

According to the adjusted speedV(1,8) finally obtained after the first round of rotation transformation, the end PTC line of vehicle band LE0 and the beginning PTC line of vehicle band LB0 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 SE0 = {I7}. Also, the bottleneck intersections of the beginning PTC line are put into the set SB0 = {I3}.Figure 12
The second round of rotation transformation.The setSB0 and SE0 do not meet any conditions for stopping rotation. Then, we have to complete the first rotation transformation in the second round.Because the only bottleneck intersection of the end PTC lineI7 is located downstream of the only bottleneck intersection of the beginning PTC line I3, it corresponds to the rotation direction judgment Scenario 2. Therefore, the PTC line can be rotated clockwise. Furthermore, the most upstream bottleneck point PE7 is selected as the rotation point of the end PTC line, whereas the most downstream bottleneck point PB3 is selected as the rotation point of the beginning PTC line.As shown in Figure12, we can connect points PE7 (PB3) and PFg (1 ≤ g ≤ 6) (PSh (4 ≤ h ≤ 8)) to form the rotation line LFg (LSh) and determine whether the corresponding adjusted speed VFg (VSh) is within the adjusted speed range. The calculation results are shown in Table 1.Table 1
The optional vehicle speed set during the first rotation transformation in the second round.
Bottleneck pointsThe start point/endpoint of the coordinated phaseAdjusted speed (m/s)Optional vehicle speedSV1PE7(2010, 268)PF1(0, 102)VF1 = 12.11VF1 ∈ SV1PF2(350, 144)VF2 = 13.39VF2 ∈ SV1PF3(750, 186)VF3 = 15.37VF3 ∉ SV1PF4(910, 186)VF4 = 13.41VF4 ∈ SV1PF5(1450, 224)VF5 = 12.73VF5 ∈ SV1PF6(1730, 266)VF6 = 140.00VF6 ∉ SV1PB3(750, 134)PS4(910, 134)VS4 = ∞VS4 ∉ SV1PS5(1450, 176)VS5 = 16.67VS5 ∉ SV1PS6(1730, 214)VS6 = 12.25VS6 ∈ SV1PS7(2010, 212)VS7 = 16.15VS7 ∉ SV1PS8(2280, 260)VS8 = 12.14VS8 ∈ SV1It can be seen from Table1 that SV1 = {VF1, VF2, VF4, VF5, VS6, VS8}. Then, select the minimum value VF1 in SV1 as the adjusted speed V(2, 1). Then, V(2, 1) is 12.11 m/s.According toV(2,1), the end PTC line LE1 and the beginning PTC line LB1 of the outbound progression can be drawn. Determine the bottleneck intersections set of the end PTC line SE1 = {I1, I7} and the beginning PTC line bottleneck intersections set SB1 = {I3}, according to LE1 and LB1. Then, the updating of the sets of bottleneck intersections is completed, as shown in Figure 13.Figure 13
Updating the sets of bottleneck intersections.The setsSE1 and SB1 meet one of the conditions for stopping rotation. At this time, ∃I1∈SE1, I3∈SB1, I7∈SE1, 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. Therefore, we can stop the further rotation transformation of the PTC line, and the second round of rotation transformation is finished.


## 2.7.4. Scheme Generation

The optimal adjusted speedVB 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 CB is 88 s.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. The time-space diagram finally obtained is shown in Figure14. The corresponding progression bandwidth is 32.85 s, and the bandwidth ratio reaches 37.3%.Figure 14
The finally obtained time-space diagram.


## 2.7.5. Programmatic Solution

The proposed solution logic steps can be solved graphically and programmed to quickly obtain an optimal signal coordination control scheme. The above case can be solved by python programming to obtain the optimal signal coordination control scheme. The experimental environment of the model solver is the Windows10 64-bit operating system, and the CPU is intel i5-6600 3.30 GHz. The 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.


## 2.8. Model Comparison

The 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 I1 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.Table 2
The comparison table of the coordination effects of various models.
ModelIntersectionI1I2I3I4I5I6I7I8Proposed graphical method (MAXBAND model)Coordinated modeS∗B∗S∗S∗B∗S∗S∗B∗Absolute offset (s)63.817.659.459.417.659.457.222.0Common signal cycle (s)88Bandwidth ratio (%)37.3Improved algebraic methodCoordinated modeS∗B∗S∗S∗B∗S∗S∗B∗Absolute offset (s)66.018.261.461.418.261.459.222.8Common signal cycle (s)91Bandwidth ratio (%)30.5S∗  = “synchronous coordinated mode” and B∗  = “backstepping coordinated mode”.Using the LINGO programming solution, the MAXBAND 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 Table2 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.


## 3. 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 re-optimization of the common signal cycle.The 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. Then, 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.The graphical method proposed in this study has the advantages of intuitive readability, strong operability, and is particularly suitable for engineering applications. The 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.



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*Source: 7649214-2021-11-02.xml*