Modified Model Predictive Control for Coordinated Signals along an Arterial under Relaxing Assumptions

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Introduction
Te arterial serves as a major roadway that connects adjacent urban functional areas.Signals are operated as a group to provide a good progression in the high-priority direction, known as the coordinated direction.Although the arterial is part of the trafc network, arterial coordinated signal control and regional signal control are two distinct techniques, with the most signifcant diference being that coordinated signal control along the arterial not only considers overall performance but also emphasizes coordination progression.
Arterial coordinated signal control techniques can be classifed into fxed-time, actuated, and adaptive control.In fxed-time control, a fxed green time is unable to accommodate demand fuctuations and to create a timing plan involves substantial investment and engineering judgment.Actuated control is an extension of fxed-time control that incorporates actuated logic, thus relying heavily on the quality of the timing plan.Furthermore, the issue of early return to green inevitably disrupts vehicle movement in the coordinated direction [1,2].In contrast to the aforementioned techniques, adaptive control typically involves the defnition of control objectives and objective functions, treating the adjustment of signal timing to real-time trafc states as an optimization problem.Adaptive control, as an advanced technology for arterial coordinated signal control, signifcantly reduces the need for engineering judgment in the control process.
Model predictive control (MPC) is an advanced process control method that has demonstrated its superiority through numerous industrial applications [3].Many attempts have been made to apply MPC to signal control, which falls within the scope of adaptive control by defnition [4].Te three fundamental elements of MPC are the predictive model, objective function, and rolling optimization [5].Te signal control problem can be transformed into a model-based rolling-horizon optimization problem [6,7], where the prediction horizon is composed of several future sample intervals, commonly the cycle length.Te prediction model, based on a macroscopic trafc fow model, deduces future trafc states from the current states and the future timing plan.Te objective function is constructed based on the trafc states within the prediction horizon.At every sampling instant, the optimization problem is solved online to generate a sequence of timing plans corresponding to the predicted horizon, and the frst timing plan in the sequence is executed, in a process referred to as rolling optimization.
Te MPC provides a robust framework for signal control, allowing cycle-by-cycle adjustments to varying trafc states.
Traditional MPC primarily targets the network level, relying on strong assumptions about the research environment, including one-way streets, disregarded pedestrian demand, and the absence of complex phase structures, which can enable easier and faster solutions.However, the resulting prediction models can signifcantly deviate from the realistic trafc environment, leading to inaccurate predictions of future trafc states, irrational timing plans, and potential implementation obstacles.For this purpose, this paper relaxes the assumptions and proposes a modifed model predictive control (MMPC) for coordinated signals along an arterial.Te main contributions are as follows: (1) By analyzing the typical assumptions in traditional MPC, this study proposes relaxed assumptions approximating the realistic trafc environment, addressing problem in the traditional MPC.(2) Te introduction of the transition-free ring-barrier structure makes the MMPC can maintain the ofset of the coordinated phases without transition, which provides a great vehicle progression for the coordinated movement.(3) A mechanism for estimating the number of vehicles for a phase is introduced, and this estimation is minimally infuenced by the timing plan.Tis modifcation allows the MMPC to more accurately perceive the trafc state.(4) Te introduction of the percent arrival before the end of green in MMPC prevents the wastage of green time caused by the inability of the infow to arrive, and eliminates the disturbance of this phenomenon on the estimation of the number of vehicles.(5) Te results of simulation experiments show that the MMPC method signifcantly improves the delay, number of stops, and total travel time while maintaining coordination compared to the traditional MPC method as well as other benchmark methods.

Literature Review
Numerous studies have been conducted on the application of MPC to trafc signal control, with a particular focus on the network level.Researchers have attempted to increase the operational efciency of MPC by investigating control architectures, macroscopic trafc fow models, and assumptions, to make it applicable to larger and more complex trafc networks.MPC can be categorized into distributed and centralized architectures, depending on the control architecture.Te advantage of centralized architecture is that the control center can optimize the problem globally to fnd the global optimal solution, and determine the timing plans for all of the intersections.However, faced with large-scale trafc networks, the computational complexity of centralized architectures increases dramatically, resulting in unacceptable runtimes.Conversely, distributed architectures can signifcantly decrease the overall runtime by decomposing the problem into multiple subproblems and allocating the computations to the signal controllers, thereby reducing the communication load and computational complexity of the key nodes [8,9].However, its performance is inferior to that of the centralized architecture [10][11][12][13].Each architecture has its own specifc focus and suitable applications.Te adoption of a centralized architecture in arterial trafc systems is common, feasible, and necessary [13,14].
Te macroscopic trafc fow model encompasses the researchers' understanding of the realistic trafc environment.It plays a crucial role in the prediction model, directly impacting the performance and computational complexity of MPC.Among such models, MPC commonly employs the store-and-forward model (SFM) and cellular transport model (CTM) [14].A key characteristic of SFM is its ability to model trafc fows using a simplifed mathematical description, eliminating the need for discrete variables during optimization.Tis model has paved the way for optimization problems with polynomial complexity and has found practical applications in realistic networks [15,16].In contrast to SFM, CTM divides links into smaller segments, enabling a more precise representation of non-uniform trafc states within every segment.Smaller segments necessitate shorter sampling intervals, which have minimal impact in trafc networks, but increase model complexity [17].Terefore, SFM is considered more suitable than CTM for trafc networks.
Assumptions are employed to simplify and abstract the complex nature of the realistic trafc environment, aiming to facilitate the solution of MPC.Te assumptions that follow are common in MPC in the feld of signal control, either explicit or implicit, as shown in Table 1.However, these assumptions are excessively strong, and they signifcantly increase the deviation of the model from realistic trafc.Although some assumptions can be relaxed through simple extensions based on existing research, it should be noted that to achieve complex phase structures and ofset transitions in existing studies, it inevitably requires structural modifcations.Te purpose of these assumptions is to eliminate nonessential confounding factors from the research, enabling a more focused investigation of the core problem.However, in the case of network-level MPC, current research places a greater emphasis on the methodology, often customizing assumptions to conform to it.Terefore, the objective of this paper is to reassess the role of assumptions in signal control and restore a realistic trafc environment for arterials.
To summarize, traditional network-level MPC methods are deemed unsuitable for arterials.Due to the key characteristics of arterials, i.e., a limited number of intersections and simple relationships, they have low requirements for control architectures and macroscopic trafc fow models.Hence, the increase in computational complexity resulting from the relaxation of typical assumptions is acceptable.We aim to restore the realistic trafc environment of arterials by relaxing typical assumptions, thereby facilitating the implementation of MPC.

Assumptions
Te assumptions proposed in this paper, along with the diferences from typical assumptions, are presented in Table 2. Assumptions 1-3 each play a role in approximating the trafc environment, corresponding to trafc channelization, trafc demand, and signal controller types, respectively.Assumptions 4-6 represent the fundamental understanding of the realistic trafc environment by MMPC control techniques, corresponding respectively to ofset confguration, turning ratio, and vehicle passage conditions.
For convenience, in this paper, the coordination direction is set from west to east.

Base Traffic Predictive Model
Te trafc fow prediction model, which encompasses various constraints and trafc fow models, serves as a key component of MPC.Under typical assumptions, signal controllers operate in a stage-based manner, accommodating only two or four phases.Te link is chosen as the control object, and the number of vehicles (referred to as queues in some studies) within the link is considered the state variable.Te prediction of the number of vehicles within the link is then conducted using the time constraints of every stage, the storage capacity constraints of the link, and the trafc fow model.Because the stage-based structure is not compatible with the NEMA standard, Wang and Abbas proposed an MPC method for NEMA-compliant signal controllers [27].A trafc fow prediction model is established in this method, with phase as the control object, by introducing the concept of virtual phase links.By modifying this model, we create the basic trafc fow prediction model of this paper, which incorporates constraints related to green time, the number of vehicles, and the storeand-forward model.Table 3 summarizes the important variables used in this paper.

Green Time Constraints.
Te NEMA standard defnes the organization of phases using rings and barriers [33], which also impose green time constraints.In the ring-barrier phase structure, a ring represents a sequence of conficting phases, while a barrier indicates the point at which the phases in each ring must end simultaneously.For example, in a conventional four-leg intersection, there are four through phases and four protected left-turn phases, which are numbered as depicted in Figure 1.Moreover, according to NEMA standards, right-turn movements are typically permitted and are combined with through movements [33].Te ring-and-barrier diagram, as illustrated in Figure 2, imposes the following constraints on S i,Kj (n): where Te start time of a timing plan is introduced and defned as the start of the frst phase in the phase sequence.As the cycle length represents the duration of a complete phase sequence, a recursive relationship arises: ( Te ofset reference point is utilized to establish the relationship between the coordinated phases along the arterial.Te ofset reference selected in this paper is the beginning of the frst coordinated phase green, as defned ofcially in NTCIP 1202 [33].OR i (n) is given by Te minimum green and maximum green constraints are applied to G i,Kj (n) as follows: Vehicle demand is the sole consideration in trafc demand, and pedestrian crossing demand is not taken into account [6,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] 3 Trafc signal controllers can only accommodate two or four phases and do not support complex phase structures (e.g., phase overlap) [28][29][30][31][32] 4 Te ofset of every intersection is set to 0, and the ofset transition is not taken into account [20,21,23,28,31] 5 Te turning ratio at every approach remains constant and known, with turning vehicles uniformly distributed over the link [6,17,20,27] 6 Vehicles entering the link can pass the stop line during the current cycle [6,16,17,26,27] Journal of Advanced Transportation

Constraints on Number of Vehicles.
Te number of vehicles for a phase serves as the state variable of MPC, refecting the trafc state of that phase, and is subject to a constraint: Because vehicles occupy similar lengths in every lane, the storage capacity of the phases depends on their corresponding zones, which include the approach lanes and upstream links.In contrast to the approach lanes, an upstream link serves as a shared zone for all of the phases in the approach.Te storage capacity is commonly allocated as where r i,Kj is replaced by the given turning rate for phase Kj at the i th intersection; L i,Kj and N i,Kj , respectively, represent the length and number of lanes of the designated zones for phase Kj at the i th intersection; superscripts (•) app and (•) link indicate that a designated zone corresponds to an approach lane or upstream link, respectively; and L car is the average length of a vehicle.Similar to (9), X i,Kj (n), the initial number of vehicles for a phase, is calculated based on r i,Kj and the number of vehicles detected in the approach lanes and upstream links.

Store-and-Forward Model.
Te key idea of the storeand-forward model is vehicle conservation; this means that the future state is determined by both the current state and the change of state.Tus, the dynamics of phase Kj is given by the conservation equation: Cycle length of the n th cycle along the arterial Cycle length of the n th cycle for the major and minor street phases at the i th intersection EG i,Kj (n) Efcient green time of the n th cycle for phase Kj at the i th intersection G i,Kj (n) Green time of the n th cycle for phase Kj at the i th intersection Min G i,Kj , Max G i,Kj Minimum and maximum green, respectively, for phase Kj at the i th intersection N h , N p Historical and predictive horizon, respectively OR i (n) Ofset reference point of the n th cycle for the i th intersection QS i,Kj Queue service time for phase Kj at the i th intersection r i,Kj Vehicle distribution ratio for phase Kj at the i th intersection R, Y Red clearance interval and yellow change interval, respectively S i,Kj (n) Split time of the n th cycle for phase Kj at the i th intersection SC i,Kj Storage capacity for phase Kj at the i th intersection SFR i,Kj Saturation fow rate for phase Kj at the i th intersection STP i (n) Start time of the timing plan of the n th cycle for the i th intersection Infow and outfow of the n th cycle for phase Kj at the i th intersection X i,Kj

Journal of Advanced Transportation
Te green time and saturation fow rate are commonly used to simplify the calculation of the outfow for the phase in order to avoid exponential growth in computational complexity.However, maintaining the saturation fow rate for the entire green time period is challenging.Terefore, the independent variable EG i,Kj (n) is introduced to represent the efcient green time [17,27], such that Te infow is classifed into diferent cases depending on the presence or absence of upstream signals within the study area.V in i,Kj (n) is given by where PS in i,Kj is the set of intersections and phases from which the outfows directly enter phase Kj at the i th intersection; t x,Ky,i is the turning rate from phase Ky from the x th to i th intersection; and AR i,Kj is the rate of vehicles entering phase Kj at the i th intersection.

Transition-Free Ring-Barrier Structure
Te transition is a necessary process of changing from one timing plan to another in the arterial [33][34][35].It plays a key role in maintaining progression opportunities to the coordinated movement.Te transition is commonly completed within one to fve cycles, and frequent adjustments of the timing plan may result in a situation where the negative impacts of the transition outweigh the benefts of the new timing plan [33,34].
Te cycle length is consistent among all of the intersections in the arterial.Terefore, according to the conventional defnition of cycle length, the start time of the timing plan at every intersection remains constant, and the ofset reference points are determined through equations ( 5) and (6).Transition is required to adjust the start time of the timing plan and thereby maintain the relative relationship between the ofset reference points along the arterial.

C (n) C (n-1) C (n+1) STP i (n)
STP i (n+1) 6 Journal of Advanced Transportation However, in the transition-free structure, the cycle length is redefned as the duration between two ofset reference points, thereby preserving the relative relationship.Te ofset reference points exhibit a recursive relationship: For simplicity, the phases within the n th cycle are denoted as light parts, as illustrated in Figure 3. Tis includes the phase between the ofset reference points on the ring with the coordinated phase and between the start time of the timing plan on the other ring.When the coordinated phase is leading, the start time of the timing plan aligns with the ofset reference point.Terefore, the distinction from existing research lies in the case where the coordinated phase lags behind.
STP i (n) is given by Te original cycle length constraints for major streets and intersections, i.e., (1) and ( 3), require respective modifcations as follows:

Vehicle Distribution Ratio
Being a shared zone for all of the phases in the approach, the storage capacity, vehicles, and infow should be distributed to these phases from the upstream link.Currently, there are two common ways to obtain the vehicle distribution ratio.One is to directly utilize the given turning rate, such as in typical assumption 5, but this ignores the stochastic nature of trafc demand.Te other uses the number of departing vehicles, but this ignores the infuence of timing plans.
For the real-time detection of phase demand, the phase vehicle weight W i,Kj (n) is constructed based on the number of departing and queued vehicles.Tis considers both the undersupply and oversupply of green time, thereby making trafc demand estimation nearly independent of timing plans.Te phase vehicle weight is where ND i,Kj (t1, t2) is the number of departing vehicles during time period [t1, t2] for phase Kj at the i th intersection and NQ i,Kj (n) is the maximum number of queuing vehicles during the n th cycle for phase Kj at the i th intersection, which can be obtained from detectors.Note that when calculating has not been reached, the current system clock can be used as a substitute.
Te expressions for r i,Kj (n) and its estimator ri,Kj are where PS app i,Kj is the set of phases in the same approach at phase Kj and n c is the current cycle number.

Percent Arrival before End of Green
In the store-and-forward model, the green time directly impacts the outfow, which is determined by the current number of vehicles and the infow, with a theoretical upper limit.
Current research usually assumes that all of the infow will be able to pass the stop line within the cycle, i.e., typical assumption 6.When the phase will display green or when the infow will arrive within the cycle is not taken into account.Tis assumption contradicts reality and is replaced by assumption 6.
Te percent arrival before the end of green (α i,Kj (n)) represents the partial infow that can pass the stop line before the end of green in the n th cycle for phase Kj, expressed as a percentage of the total infow.It assumes that the infow can travel at the speed limit without being Journal of Advanced Transportation afected by other vehicles.Tis variable can precisely constrain the number of vehicles, but it requires much data, such as timing plans of the current and adjacent intersections, which signifcantly increases the complexity of the solution.To simplify the calculation, the future percentage is predicted by utilizing the percent arrival of the past N h cycles: where NV i,Kj (n, t) is the cumulative number of vehicles passing in the upstream section for phase Kj at the i th intersection from STP i (n) to time t; AEG i,Kj (n) is the actual end of green for phase Kj at the i th intersection; tt is the travel time of vehicles from the upstream section to the stop line at the speed limit; and n c is the current cycle number.
Hence, the original constraint ( 8) on the number of vehicles must be modifed to

Simulation Experiments
An idealized arterial, consisting of three four-leg intersections, was simulated using Vissim 6.00-19.Two experiments were conducted to compare the MMPC with benchmark control techniques in terms of control objectives and operational performance.Te benchmark control techniques include fxed-time control (FTC), semi-actuated control (SAC), and base model predictive control (BMPC).Te control techniques in this paper were implemented using Python and the Vissim COM interface.For a more detailed description of the simulation experiment confguration, see reference [36].
8.1.Road Geometry.In Figure 1, intersections 1-3 are closely spaced, from west to east.Except for these intersections, there are no entrances and exits along the arterial.Te posted speed limit remains at 50 km/h, while the desired free-fow speed follows a uniform distribution ranging from 48 to 58 km/h.
Te intersections, as shown in Figure 4, share identical geometric designs and trafc control devices.Radar sensors are installed on the roadside to collect raw data (position and speed) from the stop line up to 130 m upstream [37][38][39].Variables such as the number of queued vehicles and the number of vehicles in the approach lanes can be derived from the raw data.In addition, virtual loop detectors can be created from the raw data, positioned 40 m upstream of the stop lines for SAC [40].

Trafc Demand.
To replicate the demand patterns during the heavy load scenarios of the day, the total simulation period (11700 s) was divided into three demand loading periods: 0-2700, 2700-9900, and 9900-11700 s.For every simulation run, a random sample of the trafc demand was taken at the start of the demand loading period.Te sampling ranges are shown in Tables 4 and 5. Vehicle inputs contained only passenger cars.

Signal Timing
Values.Te cycle length was calculated using the Webster model, and the green time of phases for FTC was calculated based on the critical fow ratio of phases, as presented in Table 6.Te result was also used as the background green time for the SAC and as the initial green time for BMPC and MMPC.
Te other shared parameters are as follows: ( Te lag-lag left-turn sequence was employed on the minor streets for all of the control techniques.Unless specifed otherwise, the major street at intersection 1 set phases K2 and K5 as leading phases, intersection 2 set phases K1 and K5, and intersection 3 set phases K1 and K6.Two benchmark control techniques were employed using BMPC, referred to as BMPC-A and BMPC-B.BMPC-A defnes the leading phases as described above, while BMPC-B modifes the leading phases of all of the major streets to phases K2 and K5.BMPC-B was exclusively employed for experiment 2 to free the transition by modifying the phase sequence. Ofsets were chosen as the subject to optimize for both FTC and SAC using the signal timing tool Vistro 2020 [41].For BMPC and MMPC, the ofset was calculated based on a time-space diagram that considers the distance and speed between adjacent intersections, as well as the queue service time of the downstream intersection (QS 2,K2 = 3 s and QS 3,K2 = 5 s).A gap time of 3 s was utilized for SAC.
(1) BMPC: Te objective function was defned as ∀i,Kj (X i,Kj (n)/SC i,Kj ) 2 , and constraints (1)-( 13) were applied.Given that the results were real numbers, the fnal green time was determined by rounding.Furthermore, r i,Kj was calculated as the median value of the sampling ranges of the percentage of turning vehicles.
(2) MMPC: Te objective function was defned as ∀i,Kj X i,Kj (n), and constraints (2), ( 4), (7), and ( 9)-( 23) were applied.Te remaining steps were the same as for BMPC, the only diference being the use of ri,Kj in the calculation instead of r i,Kj .8.5.Simulation Modeling.Te parameters of the Wiedemann 74 model, which allow the HCM2010 method [44] to calculate a base saturated fow rate of 1900 veh/h/lane, were chosen.For every control technique, 50 simulation runs were conducted, each consisting of a 900 s warm-up period followed by a 10800 s data analysis period.

Simulation Results
8.6.1.Experiment 1.In this experiment, the control objectives of MMPC in Sections 5-7 were verifed using the mean absolute error (MAE) of the number of vehicles and the actual ofset deviation.Tese performance measures were calculated based on the actual number of vehicles and ofset for every cycle.BMPC-A, which shared the same phase sequence as MMPC, was chosen as the benchmark control technique.
Te absolute error in the number of vehicles was calculated by comparing the actual and estimated values.Te MAE values were averaged for every cycle in the arterial, as presented in Table 7, which demonstrates the efectiveness of MMPC in reducing the MAE compared with BMPC-A, which can be attributed to the introduction of a real-time vehicle distribution ratio and the percent arrival before the end of green.Te accurate vehicle distribution ratio facilitated the precise estimation of the number of vehicles for the current cycle, while the constraint modifed by the percent arrival before the end of green enhanced the model's consistency with the realistic trafc environment.
Te ofset of intersection 1 is set to 0. Figure 5 shows the deviation of the actual ofset from the initial ofset for intersections 2 and 3. Due to the absence of transition, BMPC-A exhibited varying degrees of early return to the green problem there.Te transition-free structure in MMPC played a crucial role in maintaining a constant actual ofset throughout the control process.
MMPC demonstrated exceptional capabilities in accurately estimating and predicting the trafc state while effectively maintaining the ofset at every intersection.Consequently, it was justifable to anticipate that these distinct advantages of MMPC would yield signifcant performance enhancements.
Journal of Advanced Transportation 8.6.2.Experiment 2. Te performance measures employed in this experiment comprised: (1) high-priority path travel speed; (2) average vehicle delay; (3) total travel time (TTT); and (4) average number of stops.With the exception of the high-priority path travel speed, all of the performance measures for a given period were averaged across 50 simulation runs.
Te travel speed was measured from the stop lines of Phase K2 at intersection 1 to those at intersection 3, following the coordinated direction.
Figure 6 shows the summary statistics and probability density of the travel speed in a high-priority path.Te MPC techniques (BMPC-A, BMPC-B, and MMPC) outperformed the conventional control techniques (FTC and SAC) in terms of the mean travel speed.For the conventional control techniques, the majority of vehicles traveled at speeds less than 30 km/h.MMPC performed better than BMPC-A and BMPC-B in terms of the mean, median, and 15 th and 85 th percentiles of the travel speeds.Moreover, MMPC had nearly half of the vehicles traveling at speeds greater than 40 km/h.Te measurement zone, which extended from a signifcant distance away from the stop lines to the beginning of the intersection exit, was used to measure the number of stops and vehicle delay for every vehicle movement.TTT was calculated as the cumulative travel time of all of the vehicles in the arterial.
Tables 8 and 9 demonstrate that the MMPC-enabled arterial outperforms the arterial utilizing other benchmark techniques in terms of system average vehicle delay, TTT, and system average number of stops.Tis is attributed to MMPC's ability to maintain the performance of major streets at a suboptimal level, while simultaneously optimizing the performance of minor streets to nearly optimal levels.
Te computation efciency of the control techniques was evaluated using the runtime per cycle.In this experiment, BMPC-A, BMPC-B, and MMPC have average runtimes per cycle of 0.25, 0.25, and 0.35 s, with maximum runtimes per cycle reaching 1.87, 0.91, and 1.22 s, respectively.Compared to BMPC, the average runtime for MMPC increased by 0.1 s.However, it is more important, from the perspective of implementation, whether the maximum runtimes exceed the  Journal of Advanced Transportation threshold value.In trafc signal control, the green time of the phase needs to satisfy the minimum green.Terefore, the signal controller does not need to determine the timing plan at the beginning of the cycle; instead, it should be determined at the latest when the minimum green is frst satisfed.According to the defnition of the minimum green, this threshold is at least 5 s.Tis ensures that the signal controller employing the mentioned techniques can obtain the timing plan in a timely manner while satisfying all the basic conditions for implementation.

Conclusions
Tis paper proposes modifed model predictive control (MMPC) for coordinated signals along an arterial, with the aim of approximating the realistic trafc environment.Te modifcations proposed in this paper, namely, the transition-free ring-barrier structure, vehicle distribution ratio, and percent arrival before the end of green; collectively contribute to the distinctive characteristics of MMPC.Te simulation results demonstrate that MMPC efectively maintains coordination without the need for transition and accurately estimates current and future trafc states by employing modifed constraints.Tese fndings highlight MMPC's ability to closely approximate real-world trafc environments.MMPC outperforms the benchmark techniques in terms of vehicle progression for coordinated movement, resulting in signifcant system-wide improvements in delays, number of stops, and TTT, with only a minor and acceptable increase in runtime.
Certainly, there is still signifcant potential for improving the performance of MMPC.Future research should focus on integrating real-time percent arrivals before the end of green into the optimization problem.Strictly speaking, MMPC only accomplishes a part of the signal control process, i.e., splits, because the ofset and cycle length are still to be determined using other techniques.Hence, future research should also explore the utilization of reinforcement learning to generate signal cycle lengths, where MMPC can contribute as prior knowledge to expedite the training process.Signifcantly, this research transcends the confnes of typical assumptions and large-scale trafc networks, exploring a new avenue to drive the implementation of MPC in signal control.

Table 2 :
Assumptions and diferences from typical assumptions.ID Assumptions Diferences 1 All streets are two-way, and trafc channelization is taken into account Streets changed from one-way to two-way, and consideration of trafc channelization introduced 2 Trafc demand simultaneously takes into account both the vehicle demand and pedestrian crossing demand Inclusion of pedestrian crossing demand in addition to vehicle demand 3 A dual-ring-barrier, eight-phase signal controller, as specifed in the National Electric Manufacturers Association (NEMA) standards, is installed at every intersection Transition from simpler signal controllers to more common types (NEMA standards) that support more phases and functions 4 Te ofset of every intersection is determined using alternative methods, taking into account the ofset transition Change in the determination of ofset, now considering alternative methods and ofset transition 5 Te turning ratio randomly fuctuates within a certain range, and turning vehicles are not uniformly distributed over the link Introduction of variability in turning ratios and non-uniform distribution of turning vehicles 6 Some of the vehicles entering the link cannot pass the stop line during the current cycle Recognition that not all vehicles can pass the stop line during the current cycle, aligning with the practical reality that some vehicles must wait until the next cycle 4 Journal of Advanced Transportation

Figure 4 :
Figure 4: Layout of the test-bed intersection.

Table 3 :
Variables and their descriptions.Percent arrival before the end of green of the n th cycle for phase Kj at the i th intersection C(n)

Table 4 :
Sampling ranges of the trafc demand.

Table 5 :
Sampling ranges of the percentage of turning vehicles (%).

Table 6 :
Green time of phases for FTC (s).

Table 7 :
Mean absolute error of number of vehicles (veh).

Table 8 :
Average vehicle delay and total travel time.Te bold values represent the minimum values for each column, which are also the optimal values.

Table 9 :
Average number of stops.