Operational optimization at signalized metering roundabouts using cuckoo search/local search algorithm

Abstract

A metering roundabout where traffic is controlled by traffic lights with phase times influenced by queue detector occupancy might be the solution to enhance performance when there are unbalanced traffic flows at roundabouts. There have, however, been minimal studies on how the distance of the queue detector from the stop line affects signal phase time durations and the queuing lengths. This research, therefore, seeks to develop a Cuckoo Search/Local search Algorithm using parameters such as arrival volumes, conflicting volumes, detector distance and phase time to investigate the relationship of signal setting, detector location and queuing formulations. Also, some additional statistical tests were performed for the fitness of the data. In order to conduct solid model calibrations and validations, model output data was compared against the AIMSUN model. The results from the analyses demonstrated that the queue detector distance can affect phase time durations and vehicle queuing lengths on the controlling approach as well as queuing lengths on the metered approach. This study showed that, based on the study for the Old Belair Road roundabout, the total queue length (controlling + metered) will be minimized from 689 to 499 m by the proposed methods when the detector is relocated at 210 m from the roundabout stop line, giving longer phase green times and resulting in decreased intersection queuing lengths.

Keywords

Metering roundabout detector location AIMSUN cuckoo search/local search algorithm

Introduction

Motivation

Roundabouts, a type of intersections, can experience congestion problems, particularly during peak hours when there are unbalanced traffic conditions.^1–5 Delays on one or two approaches can be observed because entering vehicles may be interrupted by rotating vehicles. Thus, in order to form balanced traffic flow conditions and obtain enough gaps between rotating vehicles, the signalized roundabout emerged as a popular alternative, often called a metering roundabout.

The concept of metering roundabout⁶ is described in Figure 1. In this roundabout, detectors are installed on the controlling (dominant) approach and the traffic signals on the metered (sub-dominant) approach are differently set up from traditional ones. Stevens,¹ Azhar and Svante,⁷ and Mosslemi⁸ mentioned that a signal roundabout can be grouped into three categories: traffic flow control, operation time and number of approaches controlled as presented in Table 1.

Figure 1.

Concept of metering roundabout.⁶

Table 1.

Categories of signal roundabout.

Classification	Control type	Description
Traffic flow control	Direct control	Entering traffic flow and circulatory traffic flows are controlled by signals
	Indirect control	Only entering traffic flows are controlled by signals
Operation time	Full time control	Signal is operated for 24 h
	Part time control	Signal is activated by time of day or by detectors
Number of approaches controlled	Full control	All approaches are controlled by signals
	Partial control	Approaches are controlled by signals while the remaining approaches operate under give way control

Among the categories of signal roundabouts given in the table above, unbalanced traffic flow conditions usually occur during peak hours, thus, indirect with part-time control is the most appropriate control system.^1,3,9,10 In addition, the simple form of a signal metering roundabout has a valuable benefit-cost ratio because it is easy to operate with low installation cost.¹¹

The operational principle of a metering roundabout is based on queuing length, detector location and signal phase durations. Further, signal phase times can be adjusted when pre-scheduled queue length is detected. However, there are no clear guidelines about how to use traffic signals for roundabouts, and no specific way to determine detector locations.

Literature review

In order to increase intersection or road network capacity, typical artificial intelligent optimization methods are widely used.^12–14 These methods try to find minimum delay time, queuing length or the number of stops through the adjustment of cycle length and signal phase duration.

Sergey et al.¹⁵ analyzed the associated characteristics of traffic signal control and dynamic guidance systems, using intelligent technology and fusion technology to study its intelligent collaboration, and proposed a structural framework for urban traffic signal control and traffic guidance system intelligent collaboration, which is controlled by traffic signals. Yu et al.¹⁶ discussed the problem of traffic demand balance under the condition of limited environmental capacity of road sections using mathematical programing methods to describe the continuous network balance problem of demand changes in the waiting situation. However, the impact of urban traffic signals on the passage capacity of road sections at intersections and how to meet the increasing traffic demand by traffic signal timing was not considered.

Brederode and Verlinden¹⁷ considered how to increase network traffic demand under traffic signal control conditions using a bi-level optimization framework. Lin et al.¹⁸ studied the user balance distribution problem and the system optimal distribution problem based on emissions. Ding et al.¹⁹ discussed the generalized user optimal equilibrium assignment problem and the generalized system optimal equilibrium assignment problem. Fontaine and Minner²⁰ combined the road environment capacity limitation, the optimal traffic signal setting problem and the urban traffic discrete network design problem. However, the construction of new road sections maximizes the traffic demand to meet the growing traffic demand in the city and the maximum demand of the transportation network can meet the requirements of modern urban environmental protection through the restriction of road environmental capacity. In addition, the optimization methods concept such as particle swarm optimization and multi-objective optimization show the successful application in many fields.^21–23 Also, Brits et al.²⁴ and Naves et al.²⁵ conducted studies to find the best traffic signal durations using these methods. However, these studies only focused on the signalized intersections and signal phase time resulting different performance. Moreover, the optimization methods did not apply to the metering roundabouts to find the best detector location.

With regard to the metering roundabout studies, several researchers attempted to make a definition of unbalanced traffic conditions using the origin-destination factor, which is able to enhance metering roundabout capacity.^9,26 According to past studies, the operational performance on a roundabout can be exacerbated by unbalanced traffic conditions. In such cases, an effective solution is the metering roundabout leading to a decrease in queues or delays. In addition, past studies explained that vehicle movements on a roundabout and T-junction are similar and they operate individually. Another studies,^9,27 however, gave a definition with respect to unbalanced conditions with the prerequisite that ingressing vehicles from one approach might affect the vehicle movements on other approaches at roundabouts.

According to Akçelik,^6,10 the distance of the queue detector from the give-way line should be in the range of 50–120 m using the probability of the blockage parameter that is used when a road has limited queue storage. The parameter describes how downstream traffic flow affects upstream congestions. Fortuijn,²⁸ a study of the queue detector distance from the stop line is important because signal phase durations are greatly influenced by detector occupancy rates with respect to the metering roundabout. Fortuijn,²⁸ also tried to find the queue detector distance (demand detector) using the Pollaczek-Khinchine M/M/1 waiting-time formula. However, this study was based on a different type of control metering roundabout (turbo roundabout) which had the queue detector installed on the dominant approach and a delay detector on the sub-dominant approach.

Contribution and organization

At present, many researches on the optimization and control of urban traffic are mostly based on the optimization of traffic flow distribution in the urban transportation network, or the optimal control based on the shortest time between the traveler’s starting points, or the largest urban capacity. Optimal control of traffic flow distribution optimizes for the signal period or signal time interval mostly using conventional optimization methods. Indeed, there have been minimal studies on the determination of detector location affecting queuing length at metering roundabouts. Consequently, when a metering roundabout is planned, no guideline for the determination of optimal detector location is provided and majority of studies relies on the use of existing analysis software. There is no research to reveal a relationship between the detector location and signal phase duration at metering roundabouts. Furthermore, there are still relatively little research to investigate detector locations, which is an important parameter affecting the entire performance of the roundabout.

Therefore, this study investigates not only the optimal detector position but also a relationship between the detector locations and signal phase duration at metering roundabouts using the Cuckoo Search/Local search Algorithm. The detail of detector locations and the queuing length on major (controlling) approach and minor (metered) approach presented in detail. The proposed method simulated using Matlab software and compared with microscopic simulation model AIMSUN. To achieve this purpose, drone footage data will be used in the process of CS/LS algorithm, and AIMSUN model development, calibration and validation. Furthermore, Geoff E. Havers (GEH) tests will be performed to assess the fitness of the model in AIMSUN.

Research background

The metering roundabouts model

It is the case that the metering signal phase lengths can be determined by the position of the approach detector. An et al.²⁹ applied the concept and formulated the queuing length models at metering roundabouts in order to estimate the queuing length based on detector locations and signal phase times. In their study numerical models were formulated with six main parameters that significantly affect the formation of queuing length for each approach. Three variables (i.e. arrival volume, conflict volume and vehicle space) commonly have a positive relationship to queuing length. However, detector location on the controlling and metered approaches, vehicle presence time on the controlling and metered approaches, and signal phase green and red time duration can be differently applied. Therefore, the queuing length estimation model in controlling and metered approaches can be expressed in equations (1) and (2).²⁹

Q_{con} = \frac{(\frac{P_{gre}}{T} \times \frac{\frac{V_{i}}{NL}}{T} \times \frac{\frac{V C_{i}}{NL}}{T} \times D L_{C} \times P T_{C} \times VS)}{D L_{M} \times P T_{M}} \times 6700

(1)

Q_{met} = \frac{(\frac{P_{red}}{T} \times \frac{\frac{V_{i}}{NL}}{T} \times \frac{\frac{V C_{i}}{NL}}{T} \times D L_{M} \times P T_{M} \times VS)}{D L_{C} \times P T_{C}} \times 15000

(2)

The phase green and red times can vary theoretically by up to 300 s (5 min). To explain in detail, in a cycle of 300 s, the green and red time may take a share of the cycle time. Thus a relationship between Pred and Pgre is presented in equation (3).

P_{red} = 300 - P_{gre}

(3)

Where, Qcon is the queuing length of the controlling approach (m), Qmet is the queuing length of the metered approach (m), Pgre is the green time (s), Pred is the red time (s), NL is the number of the lane, Vi is the arrival volume of subject i approach (vehs), Vci is the conflict volume against subject i approach, DL_C is the detector location on the controlling approach (km), DL_M is the detector location on the metered approach (km), VS is the vehicle space (m), PT_C is the vehicle presence time on the Detector C (s), PT_M is the vehicle presence time on the Detector M (s), and T is time.

The Old Belair Road roundabout has no detector on the metered approach, thus, constant “1” was applied for the cases (i.e. DLM = 1 and PTM = 1).

The cuckoo algorithm

The cuckoo search (CS) algorithm is an emerging algorithm that combines the common cuckoo breeding mechanism with the local search (LS) method.^30–32 The CS algorithm has a good initial search ability, but poor optimization ability in the later periods, low search accuracy and slow convergence speed, so it needs to be improved for multi-objective optimization.

For a 1-dimensional optimization problem, d variables are needed:

x = [x_{1}, x_{2}, \dots . ., x_{d}]

(4)

The Levy Flight location update formula is:

\begin{matrix} x_{i}^{t + 1} = x_{i}^{t} + α \oplus L (λ) & i = (1, 2, \dots n) \end{matrix}

(5)

L (λ) ~ u = t^{- λ}, \begin{matrix} 1 < λ \leq 3 \end{matrix}

(6)

Where: $x_{i}^{t + 1}$ is the updated bird’s nest position; $x_{i}^{t}$ t is the current bird’s nest position; ⊕ is the point-to-point multiplication; $α$ is the step size; $L (λ)$ is the search path; $u$ is the standard normal variable and $u$ follows the normal distribution of $N (0, σ_{u}^{2}) .$

If the parasitic nest’s communication threshold is δ, the current optimal parasitic nest position is $x_{b}^{t}$ and the probability of parasitic nest i to communicate is $p_{i}$ . If $p_{i} < δ$ , then the communication in equation (7) is performed.

\begin{matrix} x_{i}^{t} = r \cdot x_{i}^{t} + (1 - r) \cdot x_{b}^{t}, & i = 1, 2, \dots, n \end{matrix}

(7)

In the formula: $x_{i}^{t}$ is the bird’s nest position after communication; r is a random number between (0,1). When the CS algorithm parameters are set, it is found that the probability $p_{a}$ and the step size α are fixed, which will cause the algorithm’s convergence and search speed to deteriorate. When setting the two, if $p_{a}$ is small and α is large, it will increase the number of iterations. If $p_{a}$ is large and α is small, it will improve the convergence speed, but the optimization ability is poor. Therefore, setting the two as dynamic quantities causes change with the number of iterations, which can both improve the convergence speed of the algorithm and increase its search accuracy. The specific improvement formula is as follows:

p_{a} (t) = p_{a, \max} - \frac{t}{g} (p_{a, \max} - p_{a, \min})

(8)

α (t) = α_{\max} \exp (c . t)

(9)

c = \frac{1}{g} \ln (\frac{α_{\min}}{α_{\max}})

(10)

Where: $p_{a} (t)$ is the probability function of discovery; $α (t)$ is the step function; c is the intermediate variable; $p_{a, \max}, p_{a, \min}$ are the control parameters of $p_{a}$ ; g is the number of iterations; t is the current evolution algebra; $α_{\max}, α_{\min}$ are the control parameters of the step size, respectively. For the minimization of multi-objective problems, if the solution vector, $x_{1}$ , $x_{2}$ satisfy equation (11), then $x_{1}$ dominates $x_{2}$ :

\forall_{i} \in {1, 2, \dots ., M} : f_{i} (x_{1}) \leq f_{i} (x_{2})

(11)

\exists_{i} \in {1, 2, \dots ., M} : f_{i} (x_{1}) \leq f_{i} (x_{2})

(12)

Where, M is the number of solution vectors; $f_{i} (x_{1}), f_{j} (x_{1})$ are the minimum objective function values; $f_{i} (x_{2}), f_{j} (x_{2})$ are the maximum objective function values. After the optimal solution is generated, the optimal satisfactory solution is calculated using the CS algorithm which provides the optimal detector location and signal solution for the roundabout. The satisfaction of the objective function is in equation (13).

μ_{i} = {\begin{matrix} 1 & f_{i} \leq f_{i, \min} \\ \frac{f_{i, \max} - f_{i}}{f_{i, \max} - f_{i, \min}} & f_{i, \min} < f_{i} < f_{i, \max} \\ 0 & f_{i} \geq f_{i, \max} \end{matrix}

(13)

Where, $μ^{k}$ represents the satisfaction of the i-th optimal solution; $f_{i}$ is the i-th objective function value; $f_{i, \min}$ , $f_{i, \max}$ are the minimum and maximum objective function values, respectively.

The local search

In order to improve the solution quality, a LS scheme was carried out, in which it explored the area less congested in the current archive which may obtain more non-dominated solutions.^33,34 The general operation of the scheme is described in the following steps.

Step 1: It began with a randomly selected point $(x_{m} \in R_{n}) \in E^{t}$ , and the set step lengths $Δ x_{i}$ in all of the organized directions $u_{i}, i = 1, . . ., n$ . Setting $m$ = 0, and that $m$ was the size of E^t.

Step 2: Setting the beginning of the counter m =m + 1, and $h$ = 1 where h is the number of trials ( $h = 1, . . ., h_{\max}$ ) to obtain a more preferred solution than $x_{m}$ .

Step 3: The variable x_i was perturbed around the present temporary base point $x_{m}$ to get the new provisional base point $x_{m}^{'}$ as:

x_{m^{'}} = 〈 \begin{matrix} x_{m} - Δ x_{i} u_{i} & i f f^{-} (*) > (f (*) \cap f^{+} (*)) \\ x_{m} & i f f (*) > (f^{-} (*) \cap f^{+} (*)) \\ x_{m} + Δ x_{i} u_{i} & E l s e i f f^{+} (*) > f (*) \end{matrix}

(14)

\forall_{0} i = 1, 2, . . . . . ., n

Where, $f (*) = f (X_{m}), f^{+} (*) = f (X_{m} + Δ x_{i} u_{i}), and f^{-} (*) = f (X_{m} - Δ x_{i} u_{i}$ ). Besides, $f (*)$ was assumed as the valuation of the objective functions at a point.

Step 4: The condition of point X_m was unchanged.

Although the number of trials $h$ was not satisfied, the step length Δxi was decreased by using the next dynamic equation (15),

Δ x_{i} = Δ x_{i} (1 - (r)^{h / h_{\max}}

(15)

Where, $r$ is a random number, and $r \in [0, 1]$ , then, step 3 was repeated.

Step 5: Or else, if $x_{m}'$ was preferred than, $x_{m}$ , $(i . e$ ., f( $x_{m}'$ ) $> f$ ( $x_{m}$ ) then, the new base point would be $x_{m}'$ , hence, step 6 must be repeated.

Step 6: Also, it can be aided by the base points $x_{m}$ and $x_{m}'$ , in establishing a pattern direction S, as follows:

S = x_{m}' - x_{m}

(16)

Point $x_{m} ″$ had to be identified.

x_{m} ″ = x_{m}' + λ S

(17)

Where, λ is the step length, which could be taken as λ = 1.

Step 7: If $f (x_{m} ″) \leq f (x_{m}')$ set $x_{m}$ = $x_{m}'$ , then step 4 had to be repeated.

Step 8: Or else, if f( $x_{m} ″) > f$ ( $x_{m}'$ ), set $x_{m}$ = $x_{m}'$ , $x_{m}'$ = $x_{m} ″$ , then, step 6 had to be repeated.

Step 9: If the $x_{m}$ had been the best possible points, they must be compared with predefined parameters to get to the end.

Data collection and proposed methods

Data collection

This research selected Old Belair Roundabout in Adelaide, South Australia, for data collection. The roundabout is operated by one detector (115 m prior to the stop line) on the southern approach and traffic signal is located on the northern approach during morning peak hours. One hour during peak period was carried out as below:

– 1st survey: October 7, 2015 between 07:45 and 08:45 (Wednesday)

– 2nd survey: October 8, 2015 between 07:25 and 08:25 (Thursday)

– 3rd survey: November 17, 2015 between 07:50 and 08:50 (Tuesday)

There are six major parameters for calculating queuing length on each approach, and data were collected from three sources: field survey using drones, Sydney Coordinated Adaptive Traffic System (SCATS) and South Australian Department of Planning, Transport and Infrastructure (DPTI) documents, as presented in Table in Annex. For traffic volumes and signal phase time data, SCATS data were used. The SCATS, developed in Australia, is now used internationally as an adaptive traffic signal control.³⁵ SCATS obtains traffic information from detectors located behind the stop line on each lane. Traffic data information from SCATS includes two types, traffic count data (SCATS VS) and traffic signal operation data (SCATS SM). All these counts are detector based, further, this data is stored in a NEXUS database managed by DPTI at the Barbara Hardy Institute, University of South Australia, and the use of historical data is also available.³⁶ SCATS VS data can be extracted from a selected period of a minimum of 5 min using Traffic Reporter.

Proposed methods

In a metering roundabouts system model, the typical transfer function can be considered as shown in Figure 2. The optimal detector location and traffic signal of the roundabout model is based on ensuring the minimum queuing length load, while satisfying the dual-objective with minimum queues on controlling and metered approaches. In the roundabout model, the constraint conditions of the objective function are as follows.

(1) The controlling approach objective:

\min f_{1} = Q_{Con} (x_{1}, x_{2})

(18)

Figure 2.

Block diagram of optimize queuing length.

Where, $Q_{Con}$ is controlling approach in equation (1); $x_{1}, x_{2}$ are the $DLc$ detector location and $Pgre$ green time; $f_{1}$ is the objective function of controlling queuing length.

(2) The metering approach objective:

\min f_{2} = Q_{met} (x_{1}, x_{2})

(19)

Where, $Q_{met}$ is metering approach in equation (2); $f_{2}$ is the objective function of metering queuing length.

(3) The total queuing goals:

Integrating the two objective functions, the objective function used to identify the function of metering roundabouts system can be calculated as:

minY (x) = \min [f_{1} (P_{1}), f_{2} (P_{2})]

(20)

$Y (x)$ is the objective function after integration.

The optimization process is to extract the optimal parameters Pgre, Pred, NL, Vi, Vci , Vs, and DL_C, which makes for minimum queue (see Appendix Table A1). Where, the searching area of the coefficients is set according to the experience, Pgre= 160–258 s, Pred = 43–150 s, VN = 35–71 vehs, VS = 45–116 vehs, VcN = 4–9 vehs, VCS = 28–59 vehs, Vs = 7 m, DL_C = 50–250 m, PT_C = 4 s, and PT_m = 3 s, NLcon = 2 lanes, NLmet = 1 lane.

Modeling, calibration, and validation

This section presents model verification using AIMSUN7 software to compare the optimum detector locations between cuckoo search/local search algorithm and AIMSUN results. In order for metering roundabout modeling, three procedures including modeling, calibration and validation were implemented. Moreover, in order to fit AIMSUN model output to the field data, the GEH statistical test was performed. The GEH statistic, as shown in equation (21), is a type of chi-squared statistic and used for goodness-of-fit in transport engineering.^37–39 If the GEH value is less than five, the model results fit with the observed data well.^40,41

GEH = \sqrt{\frac{{(V - E)}^{2}}{{(E + V)}^{/ 2}}}

(21)

where, $E$ is the estimated output by model, and $V$ is the observation.

Modeling of Old Belair Road metering roundabout

For the layout modeling, the Old Belair Road roundabout was extracted using the function “template create” in AIMSUN 7 as presented in Figure 3. The current detector was installed on the southern approach at a distance of 115 m with a traffic signal used on the northern approach for the peak periods.

Figure 3.

Old Belair Road metering roundabout in AIMSUN 7.

Queuing lengths on each approach were collected using drones in 5-min intervals in this study and traffic volumes in Table in Annex on each approach (Northern approach: VN, Southern approach: VS, Western approach: VW, and Eastern approach: VE) was input for performance evaluation. In order for more precise analysis, 20 min of warm-up and cool-off periods were applied before and after the 1-h modeling durations.

The metering system was set with control strategies under the “traffic management” function in AIMSUN 7, which creates a connection between the installed traffic light and detectors for operating the metering system according to queuing length on the controlling (northern) approach. Start and stop triggers were specified for monitoring the occupancy time in the queue detector zone.

In addition, the occupancy value can be defined as the “percentage of cycle time that the detector detected presence in the last cycle,” as shown in equation (22).

\begin{matrix} Occupancy = \\ \frac{\sum_{i = 1}^{Total number of intervals} t_{final} (i, cycle) - t_{init} (i, cycle)}{Cycle} \times 100 \end{matrix}

(22)

Where, $t_{init} (i, cycle)$ is an initial time, $t_{final} (i, cycle)$ is a final time, and Cycle is 120 s.

This study input 55 occupancy value as starting detection and a value of 20 was specified for detection end.

In order to duplicate the real-life signal phase timing and double cycle, two control plans were applied as follows:

– Control Plan 1: Queuing length is shorter than detector location

– Control Plan 2: Queuing length is longer than detector location

Control Plan 1 is for normal conditions which is that vehicle queues did not reach the detector on the controlling approach. In this case, maximum green time (90 s: 45 + 45 s) and minimum green time (42 s: 21 + 21 s) were input. Control Plan 2 is used during metering system activation and if the detector detects the queues on the controlling approach for 3 s, AIMSUN 7 will switch to Control plan 2. When the detector occupancy is cleared, Control plan 1 would be activated automatically.

Model calibration and validation

After modeling, parameter calibration is necessary for matching the queuing length from the drone data. This study compared 2 days queuing length data for the model calibration on October 7th and 8th, and November 17th data was compared for the model validation. Maximum speed (approaching approach and exiting approach), visibility distance, yellow-box speed, maximum desired speed (vehicle) were adjusted as shown in Table 2.

Table 2.

Calibrated parameters.

Parameters	Default value	Adjusted value
Maximum speed (approaching)	60 km/h	50 km/h
Maximum speed (exiting)	60 km/h	36 km/h
Visibility distance	10 m	6 m (north), 12 m (south)
Yellow-box speed	10 km/h	4 km/h (north), 5 km/h (south)
Maximum desired vehicle speed	120 km/h	60 km/h

With the adjusted values in AIMSUN, the queuing lengths on the controlling and metered approaches were compared in Table 3. It can show that the queuing length of AIMSUN outputs are closely matched to the field data on the north and south approaches during the peak hour.

Table 3.

The queuing length comparison between drone data and AIMSUN.

Time	Oct 7th				Time	Oct 8th
	North		South			North		South
	D (m)	A (m)	D (m)	A (m)		D (m)	A (m)	D (m)	A (m)
07:45-07:50	600	520	80	70	07:10-07:15	550	540	70	75
07:50-07:55	800	740	60	70	07:15-07:20	600	620	60	70
07:55-08:00	500	530	150	125	07:20-07:25	700	760	90	100
08:00-08:05	910	950	130	140	07:25-07:30	850	820	140	150
08:05-08:10	620	630	135	140	07:30-07:35	920	960	150	135
08:10-08:15	400	450	155	150	07:35-07:40	690	730	160	150
08:15-08:20	665	700	150	160	07:40-07:45	700	710	155	160
08:20-08:25	490	510	125	130	07:45-07:50	550	520	140	160
08:25-08:30	535	565	135	130	07:50-07:55	600	620	150	170
08:30-08:35	440	395	70	80	07:55-08:00	520	535	150	145
08:35-08:40	420	390	50	70	08:00-08:05	380	390	80	95
08:40-08:45	220	250	50	70	08:05-08:10	270	250	60	65
*D = Drone data, A = AIMSUN output

In addition, the GEH test was conducted for model calibration between the drone survey and AIMSUN output. Twodays’ data (October 7th and 8th) were used as presented in Table 4 and the GEH values were less than five in all periods. Thus, it can be seen that AIMSUN results fit with drone data well and model calibration was successfully conducted.

Table 4.

GEH results for model calibration.

Statistic	Oct 7th			Oct 8th
	Time	North	South	Time	North	South
GEH	07:45-07:50	3.38	1.15	07:10-07:15	0.42	0.59
	07:50-07:55	2.16	1.24	07:15-07:20	0.80	1.24
	07:55-08:00	1.32	2.13	07:20-07:25	2.22	1.03
	08:00-08:05	1.31	0.86	07:25-07:30	1.03	0.83
	08:05-08:10	0.40	0.42	07:30-07:35	1.30	1.25
	08:10-08:15	2.42	0.40	07:35-07:40	1.50	0.80
	08:15-08:20	1.33	0.80	07:40-07:45	0.37	0.40
	08:20-08:25	0.89	0.44	07:45-07:50	1.29	1.63
	08:25-08:30	1.27	0.43	07:50-07:55	0.81	1.58
	08:30-08:35	2.20	1.15	07:55-08:00	0.65	0.41
	08:35-08:40	1.49	2.58	08:00-08:05	0.50	1.60
	08:40-08:45	1.95	2.58	08:05-08:10	1.24	0.63
Average GEH	07:45-08:45	1.67	1.18	07:10-08:10	1.02	1.00

For model validation, November 17th data was used. Table 5 shows the queuing length of drone and AIMSUN on the north and south approaches in 5-min periods, and the GEH values. Similar to model calibration, the queuing length on the north and south approaches match well and the GEH values are all less than five. Thus, it can be seen that the AIMSUN outputs are reliable.

Table 5.

GEH results of model validation.

Time	Nov 17th
	North		South		Statistic	North	South
	D (m)	A (m)	D (m)	A (m)
07:50-07:55	680	640	65	70	GEH	1.55	0.61
07:55-08:00	770	750	65	80		0.72	1.76
08:00-08:05	700	650	55	70		1.92	1.90
08:05-08:10	905	880	65	60		0.84	0.63
08:10-08:15	790	800	60	70		0.35	1.24
08:15-08:20	1005	910	155	160		3.07	0.39
08:20-08:25	835	725	170	180		3.94	0.75
08:25-08:30	505	585	130	145		3.43	1.28
08:30-08:35	650	550	135	160		4.08	2.05
08:35-08:40	60	85	160	170		2.93	0.78
08:40-08:45	45	70	130	110		3.30	1.83
08:45-08:50	95	110	45	60		1.48	2.07
					Average GEH	2.30	1.27

Result and discussion

Optimal detector location in AIMSUN

In order to find the optimum detector location on November 17th, the detector on the controlling approach was moved in 25-m increments between 50 and 225 m in AIMSUN. Then, the total queuing length (north + south) from 10 replications for peak hour is presented in Table 6.

Table 6.

Queuing length on the north and south in accordance with detector moves.

Time	DL = 50 m		DL = 75 m		DL = 100 m		DL = 125 m
	N (m)	S (m)	N (m)	S (m)	N (m)	S (m)	N (m)	S (m)
07:50-07:55	1320	25	820	45	795	60	720	95
07:55-08:00	1300	30	900	50	850	75	700	115
08:00-08:05	1250	30	920	50	860	70	820	120
08:05-08:10	1630	40	920	40	830	80	770	110
08:10-08:15	2465	40	1330	50	910	100	810	135
08:15-08:20	2745	45	1680	70	1255	125	860	165
08:20-08:25	2550	30	1740	80	920	145	700	190
08:25-08:30	2010	40	1305	50	780	120	570	150
08:30-08:35	1860	40	1425	60	750	130	555	170
08:35-08:40	960	30	485	75	180	135	80	175
08:40-08:45	925	30	425	70	140	100	75	125
08:45-08:50	900	35	325	50	115	100	75	130
Average (N + S)	1694		1080		802		701

Time	DL = 150 m		DL = 175 m		DL = 200 m		DL = 225 m
	N (m)	S (m)	N (m)	S (m)	N (m)	S (m)	N (m)	S (m)
07:50-07:55	560	105	430	110	290	135	330	165
07:55-08:00	580	105	450	125	340	150	380	170
08:00-08:05	650	110	490	140	460	160	450	190
08:05-08:10	680	135	510	165	470	185	460	190
08:10-08:15	700	150	520	180	430	210	500	220
08:15-08:20	825	190	555	200	480	240	520	255
08:20-08:25	655	195	660	215	560	225	550	250
08:25-08:30	520	160	530	185	520	205	500	235
08:30-08:35	525	160	550	180	520	185	510	220
08:35-08:40	80	170	80	200	40	230	40	240
08:40-08:45	80	130	75	150	45	170	45	200
08:45-08:50	65	130	70	135	45	155	40	195
Average (N + S)	638		575		538		571

The optimum detector location reflects the minimum queuing length considering both (controlling and metered) approaches. Figure 4 illustrates the queuing length results in AIMSUN based on detector location changes.

Figure 4.

Queuing length (North + South).

When the detector location is at 50 m, queuing length of two approaches was at the maximum of 1694 m for 1 h. Thus, the detector location of 50 m is the worst distance. When detector was located between 50 and 100 m from the stop line the queuing length reduced significantly. Detector location at 100–200 m decreased the queuing length slightly. When the detector location is at 200 m, the sum of both approaches was on average 538 m. In this case, queuing length can be decreased by around 1156 m for peak hour.

Best detector location in cuckoo search/local search algorithm

In order to verify the proposed optimization algorithm to determine the detector location in accordance with arrival volume and conflicting volume, the two signal timing schemes are obtained by using equation (3), which are respectively recorded as optimize equations (1) and (2). Furthermore, the tuning parameters for the CS/LS proposed algorithm, Levy exponent (β) = 0.25, number of iterations = 50, number of nests = 10 were set in order to get accurate results.

Table 7 shows that each iteration indicates different queuing lengths on the controlling and metered approach in accordance with a relationship between detector locations and phase green time. Total queuing length (controlling + metered approaches) looks steady after 40th iteration, which means 210 m of detector location is the best detector location.

Table 7.

Effect of detector location and signal time.

No. iteration	Pgre (s)	DLC (m)	Qcon (m)	Qmet (m)	Qtotal (m)	No. iteration	Pgre (s)	DLC (m)	Qcon (m)	Qmet (m)	Qtotal (m)
1	180.8	57.5	46.3	1206.6	1253.0	26	190.6	196.6	191.4	349.9	541.3
2	203.6	55.8	50.6	1005.5	1056.2	27	210.9	147.7	163.6	376.5	540.1
3	178.4	73.4	58.3	964.2	1022.5	28	222.2	123.4	147.2	391.7	538.9
4	161.5	86.9	62.5	927.3	989.8	29	176.9	245.9	218.7	316.5	535.2
5	182.8	77.3	62.9	883.2	946.1	30	252.9	67.1	100.6	433.5	534.1
6	179.5	102.3	81.8	685.7	767.5	31	190.1	227.0	217.1	306.9	524.0
7	189.6	102.4	86.5	627.5	714.0	32	211.1	165.7	180.7	337.5	518.2
8	183.5	112.2	91.7	604.7	696.4	33	203.6	194.2	201.0	314.1	515.1
9	182.6	117.0	95.2	584.1	679.2	34	195.3	234.6	229.0	284.8	513.8
10	196.8	109.3	95.8	549.5	645.3	35	237.6	102.5	133.4	379.7	513.1
11	168.1	151.5	113.4	506.8	620.2	36	214.7	168.9	186.5	319.0	505.5
12	213.3	96.3	91.4	524.4	615.8	37	224.9	137.7	162.9	342.4	505.3
13	233.2	72.2	75.0	538.9	613.9	38	241.8	97.4	129.9	373.0	502.9
14	195.1	147.7	128.3	413.5	541.9	39	204.7	235.7	239.8	260.5	500.3
15	244.4	69.7	75.9	464.1	539.9	40	208.7	210.2	175.1	323.9	499.0
16	205.1	131.7	145.3	444.6	589.9	41	208.7	210.2	175.1	323.9	499.0
17	228.7	94.3	121.0	465.3	586.3	42	208.7	210.2	175.1	323.9	499.0
18	224.8	101.6	126.7	456.2	582.9	43	208.7	210.2	175.1	323.9	499.0
19	177.1	185.9	171.6	410.0	581.6	44	208.7	210.2	175.1	323.9	499.0
20	178.9	192.2	178.1	391.9	570.0	45	208.7	210.2	175.1	323.9	499.0
21	197.9	159.3	165.4	398.3	563.7	46	208.7	210.2	175.1	323.9	499.0
22	164.3	230.9	193.9	367.2	561.1	47	208.7	210.2	175.1	323.9	499.0
23	222.6	113.7	137.7	421.5	559.2	48	208.7	210.2	175.1	323.9	499.0
24	215.0	129.1	148.6	408.4	557.0	49	208.7	210.2	175.1	323.9	499.0
25	183.0	209.2	195.5	350.5	546.0	50	208.7	210.2	175.1	323.9	499.0

Figure 5 shows the total queuing length for optimization arrived at for different detector locations and signal times. Optimizations from equations (1) and (2) are calculated based on collected traffic flow. From the 1st to 13th iteration, the queuing length on metered approach is decreased dramatically by 55.3% (from 1206 to 539 m). The queuing length on the controlling approach, however, is slightly increased by 30 m. Thus, the total queuing (controlling + metered) length is 613.9 m for 1 h (51% decrease). On the other hand, there is not much difference between the 14th and 40th iterations and the gap in the total queuing length is only 42.9 m an hour. It can be seen that as the iteration is repeated, the queuing length on the controlling approach is increased while the queuing length on the metered approach is decreased. The total queuing length cannot be changed after the 40th iteration by CS/LS algorithm, which means detector location at 210 m is the optimal queuing length on the controlling and metered approaches.

Figure 5.

Optimizing the location and green time signal of total queuing length.

Figure 6 illustrates a relationship between detector location and signal green time with iteration increases in CS/LS algorithm (left-side y axis is detector location and right-side y axis is signal green time). Each iteration finds the best phase green time to minimize the queuing length on the controlling and metered approaches. The 30th iteration indicates that the maximum green time, detector location and total queuing length are 252 s, 67 m and 534 m respectively. In addition, 209 s of phase green time, 210 m of detector location indicate the optimal roundabout operation.

Figure 6.

Relationship between detector location and signal green time.

Comparisons

The optimum detector under a variety of traffic conditions at a metering roundabout is calculated and simulated by CS/LS algorithm and AIMSUN software based on November 17th data. Table 8 shows that the result of the proposed algorithm obtained by adopting the best detector location makes the total queuing length decrease by 6.8% compared to AIMSUN. The best detector location attained from the CS/LS algorithm is 210 m and the total queuing length is 499 m, whereas, AIMSUN simulates that 200 m of detector location can minimize the total queuing length by 538 m. In addition, phase green time, average queuing length on the controlling approach, average queuing length on the metered approach match 90.3%, 91.7%, and 96.1% respectively between the two models.

Table 8.

Comparison of proposed method.

	AIMSUN software	CS/LS Algorithm	Ratio (%)
Pgre	217 s	209 s	96.3
DLC	200 m	210 m	95.2
Qcon	187 m	175 m	91.7
Qmet	350 m	324 m	96.1
Qtotal	538 m	499 m	93.2

The research result indicates that there is not much difference in detector location between CS/LS algorithm and AIMSUN software.

Conclusion

Due to the increasing number of roundabouts in urban areas, there has been growing interest in their operational performance evaluation, in particular in the case of unbalanced traffic flow conditions. This interest has motivated studies on metering roundabouts. With no clear guidelines to determine detector location, research to date has used software to quantify metering roundabout performance.

This paper, therefore, attempts to investigate the optimal detector location at the Old Belair Road metering roundabout using CS/LS algorithm in order to minimize the queuing length not only on the controlling, but also on the metered approaches. The outputs were compared with the microscopic simulation model AIMSUN as algorithm verification. The CS/LS algorithm indicated that detector location at 210 m optimizes the queuing length on the controlling approach (175 m) and metered approach (324 m). Moreover, 209 s of average phase green time was calculated for the 1-h morning peak. It is quite similar to the AIMSUN model output and the differences are just 10 m in detector location and 8 s in phase green time. Thus, it can be expected that the CS/LS algorithm model used in this paper would save time for determining the best detector location at metering roundabouts. Furthermore, the proposed model would serve as a platform for studying the relationship between detector locations and signal phase time at metering roundabouts.

Some limitations in the research are recognized, however, and could be enhanced by more comprehensive research. First, a short lane was considered as one lane and the constant “1” was applied to the queuing models, but a specific constant needs to be defined in accordance with the length of the short lane. Second, in order for a more accurate model output, more varied conditions need to be considered such, as different geometry and traffic volume. Lastly, in order to provide proper detector locations and signal phase durations, more intelligent methods need to be compared.

Footnotes

Appendix

Table A1.

Collected data (volume, conflicting volume, queuing length, and phase time).

October 7th
Time	North (metered) approach			South (controlling) approach			Phase time (s)
	V_CN (veh)	V_N (veh)	Queue (m)	V_CS (veh)	VS (veh)	Queue (m)	Pgre	Pred
07:45-07:50	5	52	600	42	69	80	258	42
07:50-07:55	4	50	800	48	49	60	243	57
07:55-08:00	5	67	500	44	83	150	170	130
08:00-08:05	8	57	910	48	80	130	150	150
08:05-08:10	7	60	620	37	74	135	150	150
08:10-08:15	4	44	400	42	72	155	182	118
08:15-08:20	7	53	665	41	63	150	175	125
08:20-08:25	5	65	490	50	86	125	160	140
08:25-08:30	6	64	535	44	67	135	160	140
08:30-08:35	5	57	440	44	57	70	160	140
08:35-08:40	6	57	420	53	45	50	160	140
08:40-08:45	6	71	220	39	55	50	160	140
October 8th
07:10-07:15	6	37	620	29	77	65	218	83
07:15-07:20	5	43	650	35	88	70	218	83
07:20-07:25	6	40	650	31	91	150	206	94
07:25-07:30	6	57	720	44	94	160	195	105
07:30-07:35	7	35	580	28	109	160	121	179
07:35-07:40	6	58	950	43	118	175	115	185
07:40-07:45	7	60	920	48	127	180	99	201
07:45-07:50	5	48	830	38	116	180	80	220
07:50-07:55	6	54	860	46	112	185	61	239
07:55-08:00	5	62	910	52	102	165	52	248
08:00-08:05	5	65	1000	52	88	115	56	244
08:05-08:10	6	56	1100	42	103	160	58	242
November 17th
07:50-07:55	7	64	680	49	74	70	155	145
07:55-08:00	7	75	768	59	73	70	165	135
08:00-08:05	8	53	700	42	90	50	152	148
08:05-08:10	9	52	905	40	97	70	128	172
08:10-08:15	7	51	790	40	95	60	110	190
08:15-08:20	8	63	1000	55	73	150	125	175
08:20-08:25	7	70	835	52	78	170	148	152
08:25-08:30	5	76	505	55	49	130	170	130
08:30-08:35	7	71	650	48	64	135	193	107
08:35-08:40	5	63	60	41	79	160	215	85
08:40-08:45	4	59	45	36	66	130	237	63
08:45-08:50	6	43	95	32	66	45	250	50

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hong Ki An

References

Stevens

CR.

Signals and meters at roundabouts. In: Proceedings of the 2005 mid-continent transportation research symposium, Ames, IA, 18–19 August 2005.

Akçelik

Roundabout metering signals: capacity, performance and timing. Proc Soc Behav Sci 2011; 16: 686–696.

Hummer

Milazzo

Joseph

The potential for metering to help roundabouts manage peak period demands in the US. In: Transportation Research Board 93rd Annual Meeting, Washington, DC, 12–16 January 2014.

Yue

Stazic

Dual signals roundabout evaluation in Adelaide using SIDRA and AIMSUN Modelling software. Road Transp Res 2017; 26(2): 36–49.

Krivda

Petru

Macha

, et al. An analysis of traffic conflicts as a tool for sustainable road transport. Sustainability 2020; 12(17): 7198.

Akçelik

Capacity and performance analysis of roundabout metering signals. In: TRB national roundabout conference, Vail, CO, 22–25 May 2005, pp.22–25. Washington, DC: Transportation Research Board.

Azhar

Svante

Signal control of roundabouts. Proc Soc Behav Sci 2011; 16: 729–738.

Mosslemi

Using metering signals at roundabouts with unbalanced flow patterns to improve the traffic condition. Master’s Thesis, Linköping University, Linköping, 2008.

Krogscheepers

Roebuck

Unbalanced traffic volumes at roundabouts. In: Proceedings of the fourth international symposium on highway capacity (Transportation Research Circular E-C018), Maui, HI, 27 June–1 July 2000, pp.446–458. Washington, DC: Transportation Research Board.

10.

Akçelik

Roundabouts with unbalanced flow patterns. In: ITE 2004 annual meeting, Lake Buena Vista, FL, 1–4 August 2004.

11.

Akçelik

August. Analysis of roundabout metering signals. In: AITPM national conference, Melbourne, Australia, 3–4 August 2006.

12.

Sofronova

Belyakov

Khamadiyarov

DB.

Optimal control for traffic flows in the urban road networks and its solution by variational genetic algorithm. Procedia Comput Sci 2019; 150: 302–308.

13.

Celtek

Durdu

Alı

MEM

. Real-time traffic signal control with swarm optimization methods. Measurement 2020; 166: 108206.

14.

Abdalla

AN.

Prediction of queuing length at metering roundabout using adaptive neuro fuzzy inference system. Meas Control 2019; 52(5-6): 432–440.

15.

Lyapin

Rizaeva

Kadasev

, et al. Methods to analyze traffic demand formation in intelligent transportation and Logistic Regional Network. Transp Res Procedia 2020; 45: 522–529.

16.

Zhang

HM.

Optimal traffic signal control under dynamic user equilibrium and link constraints in a general network. Transp Res Part B Methodol 2018; 110: 302–325.

17.

Brederode

Verlinden

Travel demand matrix estimation methods integrating the full richness of observed traffic flow data from congested networks. Transp Res Procedia 2019; 42: 19–31.

18.

Lin

Pan

Lam

, et al. Stochastic link flow model for signalized traffic networks with uncertainty in demand. IFAC-PapersOnLine 2018; 51(9): 458–463.

19.

Ding

Dai

Chen

, et al. Simulating on-demand ride services in a Manhattan-like urban network considering traffic dynamics. Phys A Stat Mech Appl 2020; 545: 123621.

20.

Fontaine

Minner

A dynamic discrete network design problem for maintenance planning in traffic networks. Ann Oper Res 2017; 253(2): 757–772.

21.

Baghaee

Mirsalim

Gharehpetian

GB.

Multi-objective optimal power management and sizing of a reliable wind/PV microgrid with hydrogen energy storage using MOPSO. J Intell Fuzzy Syst 2017; 32(3): 1753–1773.

22.

Baghaee

Mirsalim

Gharehpetian

, et al. Reliability/cost-based multi-objective Pareto optimal design of stand-alone wind/PV/FC generation microgrid system. Energy 2016; 115: 1022–1041.

23.

Baghaee

Mirsalim

Gharehpetian

, et al. Security/cost-based optimal allocation of multi-type FACTS devices using multi-objective particle swarm optimization. Simulation 2012; 88(8): 999–1010.

24.

Brits

Engelbrecht

van den Bergh

Locating multiple optima using particle swarm optimization. Appl Math Comput 2007; 189(2): 1859–1883.

25.

Naves

de Paula

Balestrassi

, et al. Multivariate normal boundary intersection based on rotated factor scores: A multiobjective optimization method for methyl orange treatment. J Clean Prod 2017; 143: 413–439.

26.

Hagring

Effects of OD flows on roundabout entry capacity. In: Proceedings of the fourth international symposium on highway capacity, Maui, HI, 27 June–1 July 2000.

27.

Akcelik

Chung

Besley

Performance of roundabouts under heavy demand conditions. Road Transp Res 1996; 5: 36–57.

28.

Fortuijn

Robustness of roundabout metering systems (RMS). In: 4th international roundabout conference, Seattle, WA, 16–18 April 2014.

29.

Yue

Stazic

Estimation of vehicle queuing lengths at metering roundabouts. J Traffic Transp Eng 2017; 4(6): 545–554.

30.

Sun

Cui

Dai

, et al. DV-hop localization algorithm with cuckoo search. Sens Lett 2014; 12(2): 444–447.

31.

Zhang

Wang

Cui

, et al. Hybrid multi-objective cuckoo search with dynamical local search. Memetic Comput 2018; 10(2): 199–208.

32.

Cao

Zhang

, et al. Comprehensive learning particle swarm optimization algorithm with local search for multimodal functions. IEEE Trans Evol Comput 2019; 23(4): 718–731.

33.

A simulated annealing based genetic local search algorithm for multi-objective multicast routing problems. Ann Oper Res 2013; 206(1): 527–555.

34.

Yang

Wang

Yang

, et al. Optimization of fixture locating layout for sheet metal part by cuckoo search algorithm combined with finite element analysis. Adv Mech Eng 2017; 9(6): 1687814017704836.

35.

Slavin

Feng

Figliozzi

, et al. Statistical study of the impact of adaptive traffic signal control on traffic and transit performance. Transp Res Rec 2013; 2356(1): 117–126.

36.

Suksri

Analysis and application of traffic signal control data. PhD Thesis, University of South Australia, Department of Transport System Engineering, Adelaide, South Australia 2012.

37.

Dewees

DN.

Estimating the time costs of highway congestion. Econometrica 1979; 47: 1499–1512.

38.

Paz

Molano

Gaviria

Calibration of corsim models considering all model parameters simultaneously. In: Proceedings of the 15th international IEEE conference on intelligent transportation systems, IEEE, Anchorage, AK, 16–22 September 2012.

39.

Vasconcelos

Silva

Seco

, et al. Estimating the parameters of Cowan’s M3 headway distribution for roundabout capacity analyses. Balt J Road Bridge Eng 2012; 7(4): 261–268.

40.

Feldman

The GEH measure and quality of the highway assignment models. Association for European Transport and Contributors 2012; 1–18.

41.

Vuk

Hansen

CO.

Validating the passenger traffic model for Copenhagen. Transportation 2006; 33(4): 371–392.