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Genetic fuzzy logic traffic signal control with cell
transmission modeling
Yu-Chiun Chioua & Yen-Fei Huanga a
Institute of Traffic and Transportation, National Chiao Tung University, Taipei, Taiwan. Published online: 25 Jul 2013.
To cite this article: Yu-Chiun Chiou & Yen-Fei Huang (2014) Genetic fuzzy logic traffic signal control with cell transmission
modeling, Journal of the Chinese Institute of Engineers, 37:4, 446-460, DOI: 10.1080/02533839.2013.814995 To link to this article: http://dx.doi.org/10.1080/02533839.2013.814995
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Genetic fuzzy logic traf
fic signal control with cell transmission modeling
Yu-Chiun Chiou* and Yen-Fei Huang
Institute of Traffic and Transportation, National Chiao Tung University, Taipei, Taiwan (Received 19 August 2011;final version received 2 October 2012)
This study develops an adaptive traffic signal control model based on an iterative genetic fuzzy logic controller (GFLC). The proposed model considers traffic flow and queue length as state variables and extension of green time as control variable, toward the minimization of total vehicle delays. For the learning efficiency of GFLC and the capability in cap-turing traffic behaviors, cell transmission model is used to replicate the traffic condition. To investigate, the performance of the proposed model in the case of an isolated intersection, comparisons to pretimed signal timing plans determining by Webster and total enumeration methods, and two queue-length based adaptive models are conducted. Results show that our proposed GFLC model performs best. As traffic flows vary more noticeably, the GFLC traffic signal control model performs even better than any timing plans. In the case of sequential intersections with four coordinated signal systems: simultaneous, progressive, alternate, and independent, the experimental example study also show that the pro-posed GFLC model can also perform better than current and pretimed timing plans, suggesting that the propro-posed GFLC signal control model is effective, robust, and adaptable.
Keywords: adaptive signal control; genetic fuzzy logic controller; cell transmission model
1. Introduction
On-line traffic signal control typically feeds real-time traf-fic data, collected by sensors, into a build-in controller to produce timing plans. Thus, it can provide signal-timing plans in response to real-time traffic conditions. Actuated signal control, dynamic signal control, and adaptive signal control are examples of on-line control. Because of its flexibility, adaptability, and optimality, adaptive signal control tends to be the mainstream of signal controls now-adays. The well-known adaptive signal controllers, such as SCOOT, SCATS, and OPAC, employ mathematical equations or models to determine ‘crisp’ threshold values as the cores of control mechanisms; thus, the control per-formance may be negatively affected by the uncertainty of traffic conditions. Since a fuzzy control system has excellent performance in data mapping as well as in treat-ing ambiguous or vague judgment (Teodorovic 1999), many recent studies have employed fuzzy set theory to develop fuzzy logic controllers (FLC). The underlying theory for an FLC system is to use fuzzy logic rules to form a control mechanism to approximate expert percep-tion or judgment under given condipercep-tions (Zadeh 1973). The applications of FLC to signal control are to determine the signal phasing and timing plans, including priority of phases, cycle length and split, by utilizing real-time traffic data, such as vehicle arrivals or arrival rate, occupancy, queue length and speed, collected by detectors.
Most FLC signal control models consider some traf-fic variables as state variables, such as vehicle arrivals, queue length, occupancy, and green elapsed time and use of extension of green time as a control variable (Pappis and Mamdani 1977; Mohamed, Mohamed, and Murali 1999; Niittymäki 2001). Some studies further determine the phase sequence and green times of each phase (Hoyer and Jumar 1994; Murat and Gedizlioglu 2005). However, most of these studies subjectively preset the combination of logic rules and shapes of membership functions, lacking a learning procedure. Thus, the perfor-mance of the models cannot be assured. Adjusting the combination of logic rules and membership functions very often requires tremendous effort, but there is no guarantee of obtaining good control performance. Genetic algorithms (GAs) have been proven suitable for solving both combinatory optimization problem (e.g. selecting the logic rules) and parameter optimization problem (e.g. tuning the membership functions). Employing GAs to construct an FLC system with a learning process from examples, hereafter termed a genetic fuzzy logic controller (GFLC), cannot only avoid the bias caused by subjective settings of logic rules or membership functions but also greatly enhance the control performance. Thus, a considerable number of studies relating to different areas of GFLC have been published in recent years (Herrera, Lozano, and
*Corresponding author. Email: [email protected]
–460, http://dx.doi.org/10.1080/02533839.2013.814995
Ó 2013 The Chinese Institute of Engineers
Verdegay 1995, 1998; Lekova et al. 1998; Wang and Yen 1999; Chiou and Lan 2005). The iterative GFLC model proposed and validated by Chiou and Lan (2005) is adopted in this study.
However, to develop a GFLC-based signal control requires an efficient traffic simulation model to replicate traffic behaviors and determine the performance of the control logic. Many studies use microscopic traffic sim-ulation software to evaluate the performances of signal control models, such as CORSIM (Pham et al. 2011), SimTraffic (Lin, Tung, and Ku 2010), AIMSUN (Fang and Elefteriadou 2010), VISSIM (Xu and Zheng 2009), MITSIMLab (Ben-Akiva et al. 2003), INTEGRATION (Dion, Rakha, and Zhang 2004), and PARAMICS (Wu and Ho 2009); however, it would be too time-consum-ing to use such simulation software for the evolution of genetic generations. Thus, this study employs a cell transmission model (CTM), a cell-based model pro-posed by Daganzo (1994, 1995), to evaluate the perfor-mance of learned logic rules and tuned membership function. CTM is a first-order discrete Godunov approx-imation to the kinematic wave partial differential equa-tion of Lighthill and Whitham (1955) and Richards (1956). The popularity of CTM is due to its very low computation requirements compared with micro-simula-tion models; the ease with which it can be calibrated using routinely available point detector data (Munoz et al. 2004); its extensibility to networks (Waller and Ziliaskopoulos 2001) and urban roads with signalized intersections (Lo 2001; Wong, Wong, and Lo 2010); and the flexibility with which it can be used to pose questions of traffic assignment (Lo and Szeto 2002) and ramp metering (Zhang, Ritchie, and Recker 1996) and freeway speed-limit control (Chiou, Huang, and Lin 2010). Despite their simplicity, field data have sug-gested that they fit measurements well. See for exam-ple, Lin and Ahanotu (1995) and Smilowitz and Daganzo (1999). These two studies validated CTM for freeway and arterial traffic. According to the description above, CTM is a widely used discrete macroscopic model and can simulate, as well, plausible models for signalized urban streets.
Based on this, this study aims to develop an adaptive signal control model for both isolated and sequential inter-sections based on the iterative GFLC with a cell transmis-sion modeling approach. The study is organized as follows. Section 2 states the rationales for signal control with an iterative GFLC model and CTM. Section 3 utilizes experimental cases to validate the effectiveness, robust-ness, and applicability of the proposed iterative GFLC model in controlling the signal at isolated intersections. Section 4 further validates the effectiveness and applicabil-ity of the iterative GFLC model in controlling the signal of sequential intersections. Finally, the concluding remarks and suggestions for future research follow.
2. Methods
2.1. The GFLC model
To develop a self-learning GFLC-based signal control model, the iterative GFLC model, proposed by Chiou and Lan (2005), is adopted in this study. The encoding meth-ods, genetic operators, and iterative evolution algorithm for the iterative GFLC model are briefly described as follows. 2.1.1. Encoding method for logic rules
Each logic rule is represented by one gene and its lin-guistic degree of control variable is indicated by the value of the corresponding gene. Taking two state vari-ables and one control variable as an example, if each variable has five linguistic degrees (NL: negative large, NS: negative small, ZE: zero, PS: positive small, PL: positive large), then the chromosome length is 25. Genes take the integers from zero to five, where zero represents the exclusion of the rules; other numbers indicate the inclusion of the rules and the linguistic degrees of control variable. This encoding method is depicted in Figure 1. A chromosome with gene sequence of 0002040010000001000030000, for example, will representfive logic rules being selected:
Rule 1: IF x1= NL and x2= PS THEN y = NS
Rule 2: IF x1= NS and x2= NL THEN y = PS
Rule 3: IF x1= NS and x2= PS THEN y = NL
Rule 4: IF x1= PS and x2= NL THEN y = NL
Rule 5: IF x1= PL and x2= NL THEN y = ZE
Figure 1. Encoding method for logic rules (Chiou and Lan 2005).
2.1.2. Encoding method for membership function Consider a triangle fuzzy number and let parameters cr
k,
cc
k, and clk, respectively represent the coordinates of right anchor, cortex, and left anchor of kth linguistic degree. Then 15 parameters need to be calibrated for a variable with five linguistic degrees. Furthermore, it is assumed that the first and last degrees of fuzzy numbers are left-and right-skewed triangles, respectively, left-and that the others are isosceles triangles as shown in Figure 2. Therefore, a variable with five linguistic degrees has eight parameters to be calibrated and their orders are
cmax¼ cc5¼ c r 5P c r 4P cl 5 cr 3 P cl4 cr 2 P cl3 cr 1 P cl 2P c c 1 ¼ cl 1¼ cmin; ð1Þ cck¼ ðcr kþ clkÞ 2 ; k ¼ 2; 3; 4; ð2Þ where cmaxand cminare the maximum and minimum
val-ues of the variable, respectively. The orders between cl 5 and cr3, cl4 and cr2 and, cl3 and cr1 are indeterminate. In order to tune these eight parameters, nine position variables r1,…, r9are designed as follows:
cl2¼ cminþ r1 h; ð3Þ cr1¼ cl2þ r2 h; ð4Þ cl3¼ c l 2þ r3 h; ð5Þ cr 2¼ maxfc r 1; c l 3g þ r4 h; ð6Þ cl 4¼ maxfc r 1; c l 3g þ r5 h; ð7Þ cr 3¼ maxfc r 2; c l 4g þ r6 h; ð8Þ cl 5¼ maxfc r 2; c l 4g þ r7 h; ð9Þ cr4¼ maxfcr3; cl5g þ r8 h; ð10Þ
whereh ¼ðcPmaxcminÞ 9 i¼1ri
.
To achieve two significant digits, each position vari-able is represented by four real-coding genes also depicted in Figure 2. The maximum value of the position variables is 99.99 and the minimum value is 0. Thus, in the example of two state variables and one control vari-able (each withfive linguistic degrees), the chromosome is composed of 108 genes.
Figure 2. Encoding method for membership functions (Chiou and Lan 2005).
2.1.3. Genetic operators
The max-min-arithmetical crossover proposed by Herrera, Lozano, and Verdegay (1995), and the nonuniform mutation proposed by Michalewicz (1992) are employed. In the max-min-arithmetical crossover, let Gtw¼ fgt
w1; . . .; gtwk; . . .; gtwKg and Gtv¼ fgtv1; . . .; gtvk; . . .; gtvKg be two chromosomes selected for crossover, the follow-ing four offsprfollow-ings will be generated (Herrera, Lozano, and Verdegay 1995): Gtþ11 ¼ aGtwþ ð1 aÞGtv; ð11Þ Gtþ12 ¼ aG t vþ ð1 aÞG t w; ð12Þ Gtþ13 with gtþ13k ¼ minfgt wk; g t vkg; ð13Þ Gtþ14 with g4ktþ1¼ maxfgwkt ; gvkt g; ð14Þ where a is a parameter (0 < a < 1) and t is the number of generations. In the nonuniform mutation, let Gt
j¼
fgt
j1; . . .; gtjk; . . .; gjKt g be a chromosome and the gene gjkt be selected for mutation (the domain of gt
jk is½gjkl; gujk), the value of gtþ1k after mutation can be computed as follows,
gtþ1jk ¼ g t jkþ Dðt; gjku gtjkÞ if b ¼ 0; gt jk Dðt; gjkt gljkÞ if b ¼ 1; ð15Þ where b randomly takes a binary value of 0 or 1. The function Dðt; zÞ returns a value in the range of [0, z] such that the probability of Dðt; zÞ approaches 0 as t increases:
Dðt; zÞ ¼ zð1 rð1t=TÞh
Þ; ð16Þ
where r is a random number in the interval [0, 1], T is the maximum number of generations, and h is a given constant. In Equation (16), the value returned by Dðt; zÞ will gradually decrease as the evolution progresses.
2.1.4. Iterative evolution algorithm
The iterative evolution algorithm for selecting the logic rules and tuning the membership functions is similar to bi-level mathematical programming. The upper level is to solve the composition of logic rules using the membership functions tuned by the lower level. The lower level is to determine the shape of membership functions using the logic rules learned from the upper level. Consider an FLC with n state variables x1, x2,…,
xn and one control variable y, each with d1, d2.,…, dn
and dn+1 linguistic degrees. Assume the membership
functions of all linguistic degrees to be triangle-shaped. The iterative evolution algorithm is structured as follows:
Step 0: Initialization: s = 1. Step 1: Selecting logic rules.
Step 1–1: Generating an initial population with p chromosomes. Each chromosome has Qn
i¼1digenes, and each gene randomly takes one integer from [0, dn+1].
Step 1–2: Calculating the fitness values of all chromosomes based on incumbent shapes of membership functions.
Step 1–3: Selection. Step 1–4: Crossover. Step 1–5: Mutation.
Step 1–6: Testing the stop condition. The stop condition is set based on whether the mature rate (the proportion of same chromosome in a population) has reached a given constantη. If so, proceed to Step 2; otherwise go to Step 1–3.
Step 2: Tuning membership functions.
Step 2–1: Generating an initial population with p chromosomes. Each chromosome has 36(n + 1) genes and each gene randomly takes one integer from [0, 9].
Step 2–2: Calculating the fitness values of all chromosomes based on the incumbent combination of logic rules.
Step 2–3: Selection. Step 2–4: Crossover. Step 2–5: Mutation.
Step 2–6: Testing the stop condition. Let fsbe the
largestfitness among the population for the sth evolution epoch. The stop condition is set based on whether the mature rate has reached a given constantη. If so, proceed to Step 3 and let s = s + 1; otherwise go to Step 2–3. Step 3: Testing the stop condition. If (fs+1 fs)5 ɛ, where
ɛ is an arbitrary small number, then stop. Incumbent combination of logic rules and shapes of membership functions are the optimal learning results. Otherwise, go to Step 1.
2.2. The signal control 2.2.1. Fitness value
The performance of signal control for an isolated inter-section or sequential interinter-sections is frequently measured in terms of total number of stopped vehicles, proportion of stopped vehicles, average vehicle delays, total vehicle delays (TVD), maximal green band, etc. This study will aim to minimize the TVD and thus defines the fitness function of GAs as follows,
f ¼ 1
TVD: ð17Þ
2.2.2. Variables
Following most of the previous literature, this study chooses average traffic flows in green phase (TF) and queue length in red phase (QL) as two state variables.
The control variable is extension of green time (EGT), to determine the timing of phase change. For the case of sequential intersections, TF is the summation of traffic flows at all approaches in green phase; while QL is the summation of queue length at all approaches in red phase. Assume that these variables are five lin-guistic degrees and represented by triangle membership functions. This makes a total of 125 potential logic rules. With one gene for each rule, there would be 25 genes in a chromosome; thus, a total of 36 position parameters are required to calibrate the tuning of membership functions. With four genes for each parameter, there would be a total of 108 genes in a chromosome.
2.2.3. Activation points
In consideration of pedestrian safe crossing, a minimum green time (Gmin) in each green phase is preset. At
the end of Gmin, the proposed iterative GFLC model will
be activated automatically to conclude an EGT. If EGTP EGTmin(a preset value), current green phase will
be extended by EGT seconds. If EGT < EGTmin, current
green phase will be terminated. The GFLC model will not be activated again until the end of this extension time. If total green time exceeds the preset maximum green time (Gmax), current green phase is forced to terminate. A short
all-red period is designed in each signal change interval. The activation points are also depicted in Figure 3.
2.2.4. Models comparison
For validation, the proposed GFLC model is compared with three pretimed plans – Optimal single, Optimal multiple, and Webster, as well as two adaptive timing plans – Vanished queue length (VQL) and Maximum queue length (MQL). The Optimal single timing plan is determined by total enumeration method to search for an optimal cycle length and green time during the study
period. The Optimal multiple timing plans comprise some optimal single timing plans which depend on traf-fic flow patterns. The determination of cycle length and green time of Webster timing plan can be referred to Equations (18–20) and any textbook in traffic control (May 1990; McShane and Roess 1990).
C¼ 1:5L þ 5 1 y1 y2 . . .yn
¼1:5L þ 5
1 Y ; ð18Þ where C: cycle length; L: total lost time; yi: the ratio of
maximum flow rate and saturation flow rate in phase i; Y ¼Pyi.
GE¼ C L; ð19Þ
GEi¼
yi
YGE; ð20Þ
where GE: effective green time; GEi: green split in
phase i.
The VQL based model adopted in Lin and Lo (2008) is an adaptive control system. It switches traffic signal to serve the other approach, whenever the queue on the cur-rent approach vanishes. In contrast, the MQL-based model turns the traffic signal into green, whenever the queue length of the approach reaches a preset maximum value which is optimized through trial and error. The activation points of VQL and MQL are depicted in Figures 4 and 5.
2.3. The CTM
Previous studies have often employed traffic simulation software to evaluate the performance of signal control models. However, it would be too time-consuming to use simulation software for the evolution of genetic gen-erations. To facilitate the learning process of the pro-posed model, an efficient traffic simulator is necessary to evaluate the performance of selected logic rules and
Gmin EGT1EGT2 AR
R AR Gmin
R
EGT1EGT2EGT3 AR AR Time Time East-west directions North-south directions TF1 TF2 TF3 TF1 TF2 TF3 TF4 QL1 QL2 QL3 QL4 QL1 QL2 QL3
Legend : Activation point : Red phase : Green phase : All red Figure 3. GFLC activation points for an isolated intersection.
tuned membership functions in a short period. Thus, a cell-based traffic simulator is considered. CTM, proposed by Daganzo (1994, 1995) for simulating traffic hydrody-namic behavior in a cell-based manner, uses several sim-ple equations to govern traffic movements along the roadway which is represented by a series of equal-length cells. These equations are expressed as follows depend-ing on normal connection, divergdepend-ing connection, or merging connection (as shown in Figure 6):
2.3.1. Normal connection SBkðtÞ ¼ minfQBkðtÞ; nBkðtÞg; ð21Þ REkðtÞ ¼ minfQEkðtÞ; ðw=vfÞðNEkðtÞ nEkðtÞÞg; ð22Þ qEkðtÞ ¼ minfSBkðtÞ; REkðtÞg; ð23Þ nBkðt þ 1Þ ¼ nBkðtÞ qEkðtÞ þ qjkðtÞ; ð24Þ nCkðt þ 1Þ ¼ nCkðtÞ þ qEkðtÞ qlkðtÞ; ð25Þ
where SBk(t) represents the potential moving vehicles in
cell Bk at time t. QBk(t) represents the maximum number
of vehicles entering into cell Bk at time t. nBk(t) represents
the number of vehicles in cell Bk at time t. REk(t)
repre-sents the potential vehicles moving into cell Ek at time t. NEk(t) represents the maximum number of vehicles stored
in cell Ek at time t. v and w are the free-flow and shock-wave speeds, respectively. qEk(t) represents the number of
vehicles flowing into cell Ek from cell Bk at time t. The q k fundamental diagram can be depicted as Figure 7. 2.3.2. Diverge connection
In deriving boundary conditions for divergences, it should be recognized that the left- and right-turning ratios generally depend on the mix of vehicle destina-tions presenting in cells upstream of the junction. Thus, the cell transmission equations can be expressed as follows: SBkðtÞ ¼ minfQBkðtÞ; nBkðtÞg; ð26Þ G(serve QLNS) AR R AR R AR AR Time Time East-west directions North-south directions QLNS QLEW
Legend : Activation point : Red phase : Green phase : All red G(serve QLEW)
Figure 4. VQL activation points for an isolated intersection.
G AR R AR R AR AR Time Time East-west directions North-south directions QLNS-max QLEW-max
Legend : Activation point : Red phase : Green phase : All red G
Figure 5. MQL activation points for an isolated intersection.
REkðtÞ ¼ minfQEkðtÞ; ðw=vfÞðNEkðtÞ nEkðtÞÞg; ð27Þ RCkðtÞ ¼ minfQCkðtÞ; ðw=vfÞðNCkðtÞ nEkðtÞÞg; ð28Þ qBkðtÞ ¼ minfSBk; REk=bEk; RCk=bCkg; ð29Þ qEkðtÞ ¼ bEkqBk; ð30Þ qCkðtÞ ¼ bCkqBk; ð31Þ nBkðt þ 1Þ ¼ nBkðtÞ qBkðtÞ þ qjkðtÞ; ð32Þ nEkðt þ 1Þ ¼ nEkðtÞ þ qEkðtÞ qlk1ðtÞ; ð33Þ nCkðt þ 1Þ ¼ nCkðtÞ þ qCkðtÞ qlk1ðtÞ; ð34Þ
where bEk and bCk are left- and right-turning ratios, respectively.
2.3.3. Merge connection
A merge can present in one of three following cases: Case 1: Receiving is more than sending ðREkP SBkþ SCkÞ SBkðtÞ ¼ minfQBkðtÞ; nBkðtÞg; ð35Þ SCkðtÞ ¼ minfQCkðtÞ; nCkðtÞg; ð36Þ REkðtÞ ¼ minfQEkðtÞ; ðw=vfÞðNEkðtÞ nEkðtÞÞg; ð37Þ qBkðtÞ ¼ SBk; ð38Þ qCkðtÞ ¼ SCk; ð39Þ qEkðtÞ ¼ REk; ð40Þ nBkðt þ 1Þ ¼ nBkðtÞ qBkðtÞ þ qjk1ðtÞ; ð41Þ nCkðt þ 1Þ ¼ nCkðtÞ qCkðtÞ þ qjk2ðtÞ; ð42Þ nEkðt þ 1Þ ¼ nEkðtÞ þ qEkðtÞ qlkðtÞ; ð43Þ
Case 2: Receiving is less than sending ðSBk[REkpk^
SCk[REkpCkÞ
If the condition in Case 1 is not satisfied, the model assumes that the maximum number of vehicles, REkðtÞ; advance into cell Ek. As long as the supply of vehicles from both approach SBkðtÞ and SCkðtÞ; is not exhausted, assume that a fraction (pEk) of vehicles comes from
cell Bk and the remainder (pCk) from cell Ck, where
pEK+ pCK= 1. Thus, Equations (38) and (39) can be
modified as follows:
qEkðtÞ ¼ pEkREk; ð44Þ
qCkðtÞ ¼ pCkREk; ð45Þ
Case 3: Sending of one of two cells is limited by receiving
Which is less common, arises when an approach with priority crowds out traffic on its complementary approach. Thus, Case 3 can be expressed by the follow-ing two conditions:
(1) SBk\REkpk^ SCk[REkpCk; Equations (38) and (39) can be modified as Bk Ek Bk Ek Ck Ek Bk Ck Ek q jk q qlk jk q qEk Ck q Bk q 1 lk q 2 lk q lk q 1 jk q 2 jk q Bk q Ck q Ek q (a) (b) (c)
Figure 6. Representation of three connections of CTM. (a) normal, (b) diverge, and (c) merge.
v -w
q
km qm
kj Figure 7. q-k diagrams obtained from the CTM.
qBkðtÞ ¼ SBk; ð46Þ
qCkðtÞ ¼ REk SBk; ð47Þ
(2) SBk[REkpk^ SCk\REkpCk, Equations (38) and (39) can be modified as
qBkðtÞ ¼ REk SCk; ð48Þ
qCkðtÞ ¼ SCk; ð49Þ
For the cases of isolated and sequential intersections, TF is the average traffic flows at all green-phase approaches. While QL is the summation of queue length at all red-phase approaches. Vehicle delay at each time tick can be calculated by multiplying queue length (QL) at all red-phase approaches with the time tick. The TVD is then calculated by summing up the vehicle delay within the whole evaluation horizon.
3. Isolated intersection
To investigate the applicability and performance of the proposed signal control model, comparisons to other signal control models, including Webster, Optimal single, Optimal multiple, and two queue-length based models, are conducted in experiments.
3.1. An experimental case 3.1.1. Data
To validate the effectiveness and robustness of the proposed iterative GFLC signal control model, an experimental example for an isolated four-leg intersec-tion (Figure 8) is demonstrated. To simplify the analysis, the study neglects the turning traffic. The parameters of the CTM model are set as: free-flow speed = 50 km/h, time step = 2 s, kj= 130 veh/km/lane. Assume that the
intersection has two lanes (Ni(t) = 3.6 veh/cell for all i
and t) in each approach with saturation flow of 1800 pcu/hr/lane (qmi(t) = 2.00veh/time step for all i and t).
The flow patterns of five-minute flow rates in different approaches are given in Figure 9. A noticeable peak and off-peak traffic patterns are assumed in east and west directions; while rather flat traffic patterns are assumed in north and south directions. The parameters of the iter-ative GFLC model are set as the same as population size = 100, crossover rate = 0.9, a = 0.3, h = 0.5, η = 80%, ɛ = 0.05. The center of gravity method is employed for defuzzification. The parameters of signal control are: Gmax= 100 s, Gmin= 20 s, all red + lost time = 6 s,
EGTmax= 20 s, and EGTmin= 4 s. Left- and right-turning
ratios are both set as 20%.
3.1.2. Model training
The training results of the iterative GFLC signal control model for various mutation rates are reported in Table 1.
1 2 3 10 11 12 13 14 20 21 22 23 46 45 44 38 37 36 35 34 27 26 25 24 60 67 68 58 48 57 59 47 69 81 91 83 92 80 72 71 82 70 N S E W
Figure 8. Configuration of the experimental isolated intersection.
As shown in Table 1, the GFLC performs best at the mutation rate of 0.05 with corresponding TVD of 12.13. The values of TVD achieved by the GFLC model under various mutation rates do not significantly differ, but the number of generations tends to rapidly grow as the mutation rate increases. Figure 10 further depicts the learning process of the GFLC at the mutation rate of 0.05. Note that the GFLC converges after three iterative evolutions with 117 generations. The value of TVD decreases from 27.73 to 12.31.
Table 2 presents the selected logic rules and Figure 11 shows the tuned membership functions for traffic flow, queue length, and EGT by GFLC. Note from Table 1 that seven and five out of fifteen logic rules have con-cluded that EGT are ‘NL’ and ‘NS’, and two and only one logic rule has concluded that EGT are ‘ZE’ and ‘PS’. Thus, those fifteen logic rules being selected were represented as follows:
Note also from Figure 11(c) that the membership function of ‘NL’ degree ranging from 0 to 9 s implies
that the rules of concluding ‘NL’ tend to terminate the current green phase if EGT is less than 4 s or to extend the green phase if EGT is between 4 and 9 s. The mem-bership functions of ‘NS’, ‘ZE’, and ‘PS’, respectively ranging from 5 to 10, 9 to 13 and 10 to 18 s suggest that the rule of concluding ‘NS’, ‘ZE’, and ‘PS’ tends to extend the current green phase.
3.1.3. Model validation and comparisons
To validate the effectiveness, the control performance of iterative GFLC model is compared with three pretimed plans – Optimal single, Optimal multiple, Webster, and two adaptive timing plans– VQL and MQL. The Optimal single timing plan is determined by total enumeration method to search for an optimal cycle length and green time during the study period. The Optimal multiple timing plan comprises eight optimal single timing plans which depend on traffic flow patterns as shown in Figure 9. Since the Optimal multiple model designs the optimal sig-nal timings for each traffic flow rate, its control perfor-mance is optimal if traffic pattern remains unchanged. The determination of cycle length and green time for Webster timing plan can be referred to Equations (18–20). The VQL based model adopted in Lin and Lo (2008) is an adaptive control system switching traffic signals to serve the other approach, whenever the queue on the cur-rent approach vanishes. In contrast, the MQL based model turns traffic signal green whenever the queue length of the approach reaches a preset maximum value which is opti-mized through a trial and error.
Table 3 summarizes the comparison results. The Optimal multiple timing plan, composed of eight optimal single timing plans, each of which lasts for 15 min, cor-responding to various traffic conditions, is the optimal control under the given traffic conditions. Notice that only 0.13 vehicle-hours or 1.08% additional delays are incurred by iterative GFLC model in comparison with the optimal multiple timing plan. In other words, the GFLC model has achieved almost optimal control. Also notice that the GFLC model performs better than Webster, Optimal single, VQL, and MQL models by respectively curtailing 6.47, 4.40, 1.88, and 1.36 vehicle-hours (or 34.78, 26.62, 13.42, and 10.08%) of TVD. The results demonstrate the effectiveness of our proposed iterative GFLC model. The performance comparison between the Optimal multiple timing plan and the pro-posed GFLC model shows that our propro-posed GFLC model can achieve almost optimal control.
Note: RB-# represents the #th evolution of selecting logic rules. DB-# represents the #thevolution of tuning membership functions.
10 12 14 16 18 20 22 24 26 28 30 0 20 40 60 80 100 Generations TVD RB-1 DB-1 RB-2 DB-2 RB-3 DB-3
Figure 10. Learning process of iterative GFLC at the isolated experimental intersection. 0 20 40 60 80 100 120 140 160 0 5 10 15 20 25
Time (every 5 min)
Traffic flows (pcu/5min/approach)
East (West) North (South)
Figure 9. Five-minute flow rates at the experimental isolated intersection.
Table 1. The results of iterative GFLC with various mutation rates (Pm).
Pm 0.01 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50
No. of generations 233 173 117 319 195 365 262 1054 1331
TVD 12.35 12.25 12.13 12.41 12.89 12.91 12.61 12.30 12.20
The eastwest cycle length and green splits in each cycle by GFLC are illustrated in Figure 12and in which a total of 102 cycles are progressed with cycle lengths ranging from 56 to 110 s. The distribution of northsouth green splits in Figure 12(b) has approximately reflected the traffic patterns of the same directions in Figure 9. This indicates that our iterative GFLC model can control the traffic signal adaptively.
To further examine the robustness of the iterative GFLC model, we randomly vary the traffic flows by 10–50% as shown in Figure 13. The timing plans of
pretimed signal control models (i.e. Webster, Optimal single, and Optimal multiple) remain unchanged. The GFLC timing plans corresponding to various traffic flows are generated by the same logic rules and mem-bership functions, which are learned from the original traffic patterns as given in Subsection 3.1.2. The results are summarized in Table 4. Note that the GFLC outper-forms the three pretimed and two adaptive control tim-ing plans. Moreover, the GFLC can do much better than any other models as the traffic flows vary more conspicuously, indicating the robustness of the GFLC model.
The sensitivity analysis of different percentages of turningflow is shown in Table 5. The timing plans of all models also remain unchanged. Note that the GFLC has outperformed other timing plans at each level of turning flow rates, except the training case (PLT= 0.2, PRT= 0.2).
Moreover, the GFLC can do much better than any other models as the turningflows increase.
0 0.2 0.4 0.6 0.8 1 TF (pce / 5 mins) µ 0 0.2 0.4 0.6 0.8 1 0 20 40 60 QL (pce) µ 0 0.2 0.4 0.6 0.8 1 0 0 50 100 150 200 250 300 5 10 15 20 EGT (sec.) µ (a) (b) (c)
Figure 11. Tuned membership functions by iterative GFLC. (a) traffic flow (TF), (b) queue length (QL), and (c) extension of green time (EGT).
Table 2. Selected logic rules by iterative GFLC.
Y (EGT) X1(TF) NL NS ZE PS PL X2(QL) NL NS ZE NS NS NL ZE NS ZE ZE NS NL PS NS PS NS NS PS PL NS PL PS PS NL
Rule 1: IF TF = NL and QL = NL THEN EGT = NS. Rule 2: IF TF = NL and QL = NS THEN EGT = NL. Rule 3: IF TF = NL and QL = ZE THEN EGT = ZE. Rule 4: IF TF = NL and QL = PS THEN EGT = NS. Rule 5: IF TF = NL and QL = PL THEN EGT = NS. Rule 6: IF TF = NS and QL = NL THEN EGT = ZE. Rule 7: IF TF = NS and QL = NS THEN EGT = ZE. Rule 8: IF TF = NS and QL = ZE THEN EGT = NS. Rule 9: IF TF = NS and QL = PS THEN EGT = PS. Rule 10: IF TF = NS and QL = PL THEN EGT = PL. Rule 11: IF TF = ZE and QL = NL THEN EGT = NS. Rule 12: IF TF = ZE and QL = NS THEN EGT = NS. Rule 13: IF TF = ZE and QL = ZE THEN EGT = NL. Rule 14: IF TF = ZE and QL = PS THEN EGT = NS. Rule 15: IF TF = ZE and QL = PL THEN EGT = PS. Rule 16: IF TF = PS and QL = PS THEN EGT = NS. Rule 17: IF TF = PS and QL = PL THEN EGT = PS. Rule 18: IF TF = PL and QL = PS THEN EGT = PS. Rule 19: IF TF = PL and QL = PL THEN EGT = NL.
Table 3. Comparison of control performance at the
experimental isolated intersection.
Timing plan TVD (vehicle-hours)
ΔTVD compared with GFLC Vehicle-hours % GFLC 12.13 – – Webster 18.60 6.47 34.78 Optimal single 16.53 4.40 26.62 Optimal multiple (8) 12.00 0.13 1.08 MQL 14.01 1.88 13.42 VQL 13.49 1.36 10.08
Note: Optimal multiple (8) represents a total of sub-periods. The TVD of each sub-period is determined by the optimal single timing plan.
4. Coordinated intersections
This study further extends the proposed iterative GFLC model to the signal control of consecutive intersections. To synchronize the signal control for sequential intersec-tions, three coordinated signal systems including simulta-neous, alternate, and progressive systems are considered. The simultaneous system implements exactly the same signal timing plans simultaneously in sequential intersec-tions without offset (time lag). The progressive system implements these plans with offset. The alternative sys-tem implements two timing plans with inverse green and red times. In addition, an independent operation which implements the timing plans at the sequential intersec-tions without any coordination is also compared. The tim-ing plans of these four signal systems are determined by
the GFLC model and Optimal multiple model, respec-tively. In other words, a total of eight timing plans are to be generated and compared.
4.1. An experimental case 4.1.1. Data
An experimental example with two consecutive four-leg intersections (Figure 14) is demonstrated. Assume that the intersections have two lanes in each approach with saturation flow of 1800 pcu/hr/lane. The distance between intersections is 222 meters with 8 cells. The five-minute flow rates in different approaches are shown in Figure 15. In this experimental example, a noticeable peak and off-peak traffic patterns are assumed in east and west directions; while rather flat traffic patterns are assumed in north and south direc-tions. The offset of progressive coordinated system is 16 s, since the speed limit between intersections is set as 50 km/hr.
4.1.2. Model comparisons
To validate effectiveness, the control performance of GFLC is compared with Optimal multiple pretimed models with two sub-periods. All signal timing plans of GFLC and Optimal multiple models under various coor-dinated systems are determined separately. To avoid lengthy discussion, the learning results of GFLC are not reported. The control performances of these eight signal control models are reported and compared with Table 6. Obviously, the performances under a progres-sive coordinated system are significantly superior to other systems. Progressive GFLC is the best among these four models with a total delay of 37.35 vehicle hours, followed by progressive optimal multiple model with a total delay of 42.96 vehicle hours. The signal control models under alternate coordinated system per-form relatively poorly. Also notice that all GFLC signal control models perform better than the optimal multiple
0 20 40 60 80 100 120 Seconds
Cycles progressed (number of cycles)
Red
All red and yellow Green 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 1 9 17 25 33 41 49 57 65 73 81 89 97 1 21 41 61 81 101
Cycles progressed (number of cycles)
Green splits
(a)
(b)
Figure 12. Eastwest cycle length and green splits by the GFLC at the experimental isolated intersection. (a) Cycle length, (b) Green splits.
0 50 100 150 200 0 5 10 15 20 25
Time (every 5 min)
Traffic flows (pcu/5min)
East(West) East(West)-10% East(West)-20% East(West)-30% East(West)-40% East(West)-50%
Figure 13. Variedfive-minute flow rates at the experimental isolated intersection.
Table 5. Comparison of control performance with increased turningflow rates. Models 0.2 0.4 0.6 TVD ΔTVD% TVD ΔTVD% TVD ΔTVD% PLT(PRT= 0.2 PS= 1 PLT PRT) GFLC 12.13 – 15.32 – 19.32 – Webster 18.6 34.78 27.98 45.25 36.14 46.54 Optimal single 16.53 26.62 25.24 39.30 33.65 42.59 Optimal multiple 12.00 1.08 18.07 15.22 23.67 18.38 MQL 14.01 13.42 17.81 13.98 22.6 14.51 VQL 13.49 10.08 18.44 16.92 23.54 17.93 PRT(PLT= 0.2 PS= 1 PLT PRT) GFLC 12.13 – 14.99 – 17.90 – Webster 18.6 34.78 24.64 39.16 30.88 42.03 Optimal single 16.53 26.62 20.85 28.11 25.64 30.19 Optimal multiple 12.00 1.08 17.19 12.80 21.13 15.29 MQL 14.01 13.42 17.38 13.75 21.09 15.13 VQL 13.49 10.08 17.69 15.26 22.31 19.77
Table 4. Comparison of control performance with randomly variedflow rates.
Timing plan 10% 20% 30% 40% 50% TVD ΔTVD % TVD ΔTVD % TVD ΔTVD % TVD ΔTVD % TVD ΔTVD % GFLC 12.98 – 13.47 – 14.41 – 15.02 – 15.54 – Webster 20.14 35.55 20.97 35.77 22.57 36.15 24.54 38.79 26.65 41.68 Optimal single 17.75 26.87 19.03 29.22 21.54 33.10 22.91 34.44 23.78 34.65 Optimal multiple 13.45 3.49 13.98 3.65 15.09 4.51 16.55 9.24 18.12 14.24 MQL 15.32 15.27 15.97 15.65 17.41 17.23 18.31 17.97 19.12 18.72 VQL 14.51 10.54 15.09 10.74 16.21 11.10 16.97 11.49 17.57 11.55 1 2 11 12 13 20 22 32 78 86 76 66 75 77 65 87 99 109 101 110 98 89 100 88 N S E W 21 31 33 54 52 45 44 43 34 64 63 53 122 112 121 111 155 145 156 146 124 132 123 133 145 144 135 134
Figure 14. Configuration of the experimental sequential intersections.
models. Compared with the optimal multiple model under the same coordinated system, progressive GFLC can curtail the total delay by the largest amount (13.06%), followed by simultaneous GFLC (11.82%) and by independent GFLC (10.48%). The results show the effectiveness of the proposed iterative GFLC mod-els in controlling sequential intersections.
Six scenario analyses, varying the flow rates in east-west directions and holding the northsouth flows unchanged were conducted. Three levels of flow rates are assumed: the eastboundflow rate is considered as the high level, the westbound flow rate is the low level, and the average of the eastbound and westbound flow rate is defined as the medium level. The control performances of these six scenarios are reported in Table 7 and in which Scenario 6 is the original case. Note that in the cases of same traffic flow level in eastwest direction (scenarios 1, 2 and 3), the rates of TVD reduction by GFLC become more significant as the east west traffic flows get higher. Compared with the optimal multiple models, GFLC can curtail TVD by 17.56% in scenario 1 with progressive coordinated system. However, in the cases of different traffic flow levels in eastwest direction (scenarios 4, 5 and 6), noticeable reduction in TVD by GFLC can be found only for scenario 6 with progressive systems (13.06%). 0 50 100 150 200 250 300 350 0 5 10 15 20 25
Time (every 5 min)
Traffic flows (pcu/5min/approach)
East West North-1 South-1 North-2 South-2
Note: north-# and south-# represent the traffic flows in north and south directions, respectively, at intersection #.
Figure 15. Five-minute flow rates at the experimental sequential intersections.
Table 6. Comparison of control performance at experimental sequential intersections. Signal coordinated system TVD (vehicle-hours) Rate ofΔTVD reduced by GFLC (%) GFLC Optimal multiple (2) Simultaneous 43.23 49.02 11.82 Progressive 37.35 42.96 13.06 Alternate 50.19 55.17 9.02 Independent 46.00 51.38 10.48
Table 7. Comparison of control performance with varied east west traffic flow scenarios.
Scenarios
Traffic flow
Coordinated system
TVD (vehicle-hours)
Rate ofΔTVD reduced by GFLC (%)
Eastbound Westbound GFLC Optimal multiple (2)
1 High High Simultaneous 67.09 81.23 17.40
Progressive 63.99 77.62 17.56
Alternate 85.27 100.49 15.15
Independent 72.47 86.34 16.06
2 Medium Medium Simultaneous 42.44 47.84 11.28
Progressive 32.23 36.68 12.12
Alternate 51.67 55.69 7.23
Independent 44.69 47.84 6.59
3 Low Low Simultaneous 26.68 28.83 7.44
Progressive 16.10 17.99 10.48
Alternate 46.32 49.70 6.80
Independent 33.83 35.76 5.39
4 High Medium Simultaneous 58.54 62.71 6.65
Progressive 54.84 60.18 8.87
Alternate 78.37 82.68 5.21
Independent 68.25 72.13 5.38
5 Medium Low Simultaneous 37.38 39.54 5.45
Progressive 20.86 22.59 7.67
Alternate 49.52 51.78 4.35
Independent 36.53 38.34 4.74
6 High Low Simultaneous 43.23 49.02 11.82
Progressive 37.35 42.96 13.06
Alternate 50.19 55.17 9.02
Independent 46.00 51.38 10.48
5. Concluding remarks
Based on the iterative GFLC model proposed by Chiou and Lan (2005), this study further develops an adaptive signal control model for both isolated and sequential intersections. We choose average traffic flow and queue length as state variables, EGT as control variable, and TVD as performance measurement. In order to evaluate control performance accurately, the CTM is used to replicate traffic behaviors. For the case of an isolated intersection, the experimental exam-ple has shown that the control performance of GFLC is almost the same as the optimal multiple timing plan and superior to the optimal single, Webster, VQL, and MQL-based timing plans. In the case of sequential intersections, the experimental example has also shown that GFLC performs better than the optimal multiple model, no matter which coordinated signal system is operated. These results demonstrate that the proposed GFLC model is effective, robust, and applicable to real-time signal control.
To further improve the control performance, more effective and efficient encoding methods in selecting the logic rules or tuning the membership functions or both deserve to be explored. It would be interesting to examine whether the learning results of GFLC, the composition of logic rules and the shapes of tuned membership functions are interpretable or not. If so, GFLC can explain an expert’s judgment or decision; otherwise it just works like a black box. For sequen-tial intersections, the performance is measured by TVD in this study; it can also be replaced by other mea-surement such as green band or stopping ratio along the arterial. Because of the computational efficiency of the proposed model, applications to a large-scale net-work deserve further examination. However, it should be noted that the control performance would be greatly degraded as the number of coordinated intersections increases. Thus, to combine with an intersection clus-tering algorithm, the proposed model is able to not only conduct adaptive signal control but also to deter-mine which intersections have to be coordinated. Last but not least, to better account for traffic behavior on many Asian urban streets, mixed traffic including cars, motorcycles, and buses should be considered in signal control model.
Acknowledgments
The authors are indebted to two anonymous reviewers for their insightful comments and constructive suggestions, which helped clarify several points made in the original manuscript. This study was financially sponsored by the ROC National Science Council (NSC 100-2221-E-009-121).
Nomenclature
NL, NS, ZE, PS, PL five linguistic degrees cmax the maximum value of the
corresponding variable cmin the minimum value of the
corresponding variable cr
k the coordinate of right anchor of the kth linguistic degree
cc
k the coordinate of cortex of the kth linguistic degree
cl
k the coordinate of left anchor of the kth linguistic degree
r1,…, r9 the position parameters
t number of generations r random number
T the maximum number of generations Gt
j the jth chromosome in the tth generation
gt
jk the kth gene of the jth chromosome in the tth generation
gl
jk lower bound of g t
jk for all genera-tions
gu
jk upper bound of g t
jkfor all generations a a preset parameter (0 < a < 1) b binary number which randomly takes
a value of 0 or 1
fs the largestfitness value among the
population TVD total vehicle delays
TF average traffic flows in green phase QL queue length in red phase
EGT extension of green time Gmin minimum green time
EGTmax the maximum value of EGT
EGTmin the minimum value of EGT
C cycle length L total lost time
yi the ratio of maximumflow rate and
saturationflow rate in phase GE effective green time
GEi green split in phase i
SBk(t) the potential moving vehicles in cell
Bk at time t
QBk(t) the maximum number of vehicles
entering into cell Bk at time t nBk(t) the number of vehicles in cell Bk at
time t
REk(t) the potential vehicles moving into
cell Ek at time t
NEk(t) the maximum number of vehicles
stored in cell Ek at time t
qEk(t) the number of vehiclesflowing into
cell Ek from cell Bk at time t v free-flow speed
w shockwave speed βEk left-turning ratio
βCk right-turning ratio
pEk a fraction of vehicles comes from
cell Bk to cell Ek
pCk a fraction of vehicles comes from
cell Bk to cell Ck
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