Roundabout Capacity Estimation Using Weaving – based on a design approach

Regarding weaving capacity, there are those who postulate that traffic volume per weaving section per time factor, is the determinant of practical capacity, while others prefer to use the number of lane change operations performed within the given weaving section. One thing is clear though, weaving can cause traffic stream disturbance that may lead to a bottleneck. According to HCM (1985), the weaving section will operate satisfactorily, only if traffic on the approach road is well below the practical capacities of these approaches and the weaving section has one more lane than would normally be required for the combined traffic from both approaches. When a merge area is closely followed by a diverge area, weaving segments are formed. Weaving segments require intense lane-changing manoeuvres because drivers jockey to access lanes appropriate to their desired exit points. The most critical aspect of a weaving segment is lane changing. Hence, the practical capacity of a roundabout can be estimated with a weaving-based equation 2.1(CSS, 1972)

𝑄𝑝= 280𝑤(1+𝑒 𝑤⁄ )(1−𝑝 3 ⁄ )

1+ (𝑤 𝑙⁄ ) [2.1]

Where; Qp = practical capacity;

19 e = average entry width;

w = weaving width and Ɩ = weaving length.

Figure 2.3: Typical yield rule movement at roundabouts

Assuming no U-turn, where A denotes ahead, L denotes left turning vehicles, and R denotes right turning vehicles, it can be seen from figure 2.3 that:

Entry flow rate rate per arm, QE = qL + qA + qR [2.2]

Circulating flow rate rate per arm, Qc = qA + 2qR [2.3]

From the definition in HCM (2010), some parameters are described as follows:

𝑞_𝐾𝐷𝑒 _{denotes flow rate rate of turning vehicles at approach D}

q L : _q A : _q R W E N

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D∈ {South, East, North, West} represents either the south, east, north, or west approach K ∈{A, L, R} represents through, left-turning, or right-turning vehicles, respectively  qDc _{denotes flow rate rate of the conflict stream of approach D}

𝑞𝐾𝑒 denotes total flow rate rate of turning vehicles on all approaches at the roundabout

qe _{denotes total flow rate rate of all approaches at the roundabout}

Entry flow rate of roundabouts can be divided into three directions (straight, left and right). However, assuming the probabilities of through vehicles entering the inner and outer circulatory lanes are 𝑝𝑖𝑛𝐷𝑒and 𝑝𝑜𝑢𝑡𝐷𝑒 respectively; 𝑝𝑖𝑛𝐷𝑒+ 𝑝𝑜𝑢𝑡𝐷𝑒 = 1 . For the entry flow rate rate 𝑞𝐷𝑒, there will

be 𝑞_𝐿𝐷𝑒+ 𝑞_𝐴𝐷𝑒. 𝑝_𝑖𝑛𝐷𝑒 entering the inner lane, crossing the two streams. Meanwhile 𝑞_𝑅𝑒+ 𝑞_𝐴𝑒. 𝑝𝑜𝑢𝑡𝐷𝑒

select the outer lane and only need to pass through one stream, the relationship can be written as (provided U-turn is not allowed):

𝑞𝑖𝑛𝑆𝑐= 𝑞𝐿𝑁𝑒+ 𝑞𝐿𝑊𝑒+ 𝑞𝐴𝑊𝑒. 𝑝𝑖𝑛𝑊𝑒 [2.4] 𝑞𝑜𝑢𝑡𝑆𝑐 = 𝑞𝐴𝑊𝑒. 𝑝𝑜𝑢𝑡𝑊𝑒 [2.5] 𝑞_𝑖𝑛𝐸𝑐 = 𝑞_𝐿𝑊𝑒+ 𝑞_𝐿𝑆𝑒+ 𝑞_𝐴𝑆𝑒. 𝑝_𝑖𝑛𝑆𝑒 [2.6] 𝑞_𝑜𝑢𝑡𝐸𝑐 _{= 𝑞} 𝐴𝑆𝑒. 𝑝𝑜𝑢𝑡𝑆𝑒 [2.7] 𝑞𝑖𝑛𝑁𝑐 = 𝑞𝐿𝑆𝑒+ 𝑞𝐿𝐸𝑒+ 𝑞𝐴𝐸𝑒. 𝑝𝑖𝑛𝐸𝑒 [2.8] 𝑞𝑜𝑢𝑡𝑁𝑐 = 𝑞𝐴𝐸𝑒. 𝑝𝑜𝑢𝑡𝐸𝑒 [2.9] 𝑞_𝑖𝑛𝑊𝑐 = 𝑞𝐿𝐸𝑒+ 𝑞𝐿𝑁𝑒+ 𝑞𝐴𝑁𝑒. 𝑝𝑖𝑛𝑁𝑒 [2.10] 𝑞_𝑜𝑢𝑡𝑊𝑐 _{= 𝑞} 𝐴𝑁𝑒. 𝑝𝑜𝑢𝑡𝑁𝑒 [2.11]

For the inner flow rates,

𝑞_𝑖𝑛𝑆𝑐+ 𝑞_𝑖𝑛𝐸𝑐+ 𝑞_𝑖𝑛𝑁𝑐+ 𝑞_𝑖𝑛𝑊𝑐 = 2𝑞𝐿𝑒+ 𝑞𝐴𝑊𝑒. 𝑃𝑖𝑛𝑊𝑒+ 𝑞𝐴𝑆𝑒. 𝑃𝑖𝑛𝑆𝑒+ 𝑞𝑇𝐸𝑒. 𝑝𝑖𝑛𝐸𝑒+ 𝑞𝐴𝑁𝑒. 𝑝𝑖𝑛𝑁𝑒 [2.12]

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𝑞𝑜𝑢𝑡𝑆𝑐 + 𝑞𝑜𝑢𝑡𝐸𝑐 + 𝑞𝑜𝑢𝑡𝑁𝑐 + 𝑞𝑜𝑢𝑡𝑊𝑐 = 𝑞𝐴𝑊𝑒. 𝑃𝑜𝑢𝑡𝑊𝑒+ 𝑞𝐴𝑆𝑒. 𝑃𝑜𝑢𝑡𝑆𝑒 + 𝑞𝐴𝐸𝑒. 𝑝𝑜𝑢𝑡𝐸𝑒 + 𝑞𝐴𝑁𝑒. 𝑝𝑜𝑢𝑡𝑁𝑒 [2.13]

If 𝑝𝑖𝑛𝑊𝑒= 𝑝𝑖𝑛𝑆𝑒 = 𝑝𝑖𝑛𝐸𝑒 = 𝑝𝑖𝑛𝑁𝑒= 𝑝𝑖𝑛𝑒 then equation 2.12 becomes

𝑞_𝑖𝑛𝑆𝑐+ 𝑞_𝑖𝑛𝐸𝑐+ 𝑞_𝑖𝑛𝑁𝑐+ 𝑞_𝑖𝑛𝑊𝑐 = 2𝑞𝐿𝑒+ 𝑞𝐴𝑒. 𝑃𝑖𝑛𝑒 [2.14]

If 𝑝𝑜𝑢𝑡𝑊𝑒 = 𝑝𝑜𝑢𝑡𝑆𝑒 = 𝑝𝑜𝑢𝑡𝐸𝑒 = 𝑝𝑜𝑢𝑡𝑁𝑒 = 𝑝𝑜𝑢𝑡𝑒 then equation 2.13 becomes

𝑞𝑜𝑢𝑡𝑆𝑐 + 𝑞𝑜𝑢𝑡𝐸𝑐 + 𝑞𝑜𝑢𝑡𝑁𝑐 + 𝑞𝑜𝑢𝑡𝑊𝑐 = 𝑞𝐴𝑒. 𝑃𝑜𝑢𝑡𝑒 [2.15]

In addition, combining equations 2.11, 2.12 and 2.13, one can obtain:

𝑞𝑆𝑐_{+ 𝑞}𝐸𝑐_{+ 𝑞}𝑁𝑐_{+ 𝑞}𝑊𝑐_{= 2. 𝑞}

𝐿𝑒+ 𝑞𝐴𝑊𝑒. 𝑝𝑖𝑛𝑊𝑒+ 𝑞𝐴𝑆𝑒. 𝑝𝑖𝑛𝑆𝑒+ 𝑞𝐴𝐸𝑒. 𝑞𝑖𝑛𝐸𝑒+ 𝑝𝐴𝑁𝑒. 𝑞𝑖𝑛𝑁𝑒+ 𝑞𝐴𝑊𝑒. 𝑝𝑜𝑢𝑡𝑊𝑒

+𝑞𝐴𝑆𝑒. 𝑝𝑜𝑢𝑡𝑆𝑒 + 𝑞𝐴𝐸𝑒. 𝑝𝑜𝑢𝑡𝐸𝑒 + 𝑞𝐴𝑁𝑒. 𝑝𝑜𝑢𝑡𝑁𝑒 = 2𝑞𝐿𝑒+ 𝑞𝐴𝑒 [2.16]

2.4.2 Gap-acceptance roundabout capacity estimation method

Gap acceptance method is the theoretical approach of estimating roundabout capacity. It operates on two main principles which are; the availability of gaps within the circulating or opposing traffic streams, and the usefulness of the gap by entry traffic. It depends on the driver's reaction and response time, the acceleration of the vehicle and the vehicle length (SANRAL, 2011). This approach is a probabilistic approach that takes headway, follow-up time, critical gaps and the traffic flow rate into consideration, but does not consider the geometry (AL-MADANI and Pratelli, 2014). The gap acceptance capacity estimation considers, first, the critical gap which is identified as the minimum headway between successive vehicles in the circulating approach that entering vehicles can accept to enter the circulating approach. Secondly, the follow-up time headway which is the difference in time between a departure vehicle and the immediate following vehicle at the roundabout entry if the two vehicles accept the same gap in the circulating stream under queuing conditions, and thirdly, the distribution of gaps in the circulating traffic flow rates, which depend on the Poissonian bunched vehicles or random arrivals. The follow-up time and headway change with geometry but are highly influenced by the drivers’ behaviour and traffic composition. The gap acceptance models have not been able to address the inconsistency in the form of the real traffic gap acceptance because the gap acceptance of different vehicles varies compared with the fixed critical gap and follow-up headway stipulated for use in the models (Akçelik, 2003). Drivers reject the large gap and accept smaller gaps in some cases which was not addressed in these models. The vehicle in the circulating roadway gives right of way to entry vehicles, while the entry

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vehicles force their way into the circulating road way at the saturation period (Mark Lenters PE, 2010, AL-MADANI and Pratelli, 2014, Russell and Rys, 2000, Ersoy and Çelikoğlu, 2014, Hagring, 1998), this makes the evaluation of critical headway a difficult task (NCHRP, 2006). Ersoy and Çelikoğlu (2014) discovered that entry capacity has a very sharp change when the accepted follow-up headway is small, this shows that the stipulated critical headway value used for the estimation of capacity may not give the accurate entry capacity in some situations. Moreover, most of the gap acceptance models are exponential models which were discovered not to describe the platooning, they predict short headways which are unrealistic and become more distorted with an increase in flow rate rate, and cannot deal realistically with a high traffic flow rate rate (Vasconcelos et al., 2012). Tanner (1967) developed, and (Troutbeck, 1986, Troutbeck, 1988) refined a roundabout entry capacity as shown below in equation 2.17.

𝑞𝑒=

𝑞_𝑐(1−∆𝑞𝑐)𝑒𝑞𝑐(𝑡𝑎−∆)

1−𝑒−𝑞𝑐𝑡𝑓 [2.17]

Where: 𝑞𝑒 = Enter capacity (veh/s); and 𝑞𝑐= Circulating flow rate (veh/s)

ta = Critical gap (s); 𝑡_𝑓 = Follow-up time (s);

∆ = Minimum headway in the circulating streams (1s for multilane and 2s for single lane)

Tanner’s equation for the capacity of priority intersection forms the fundamental basis for the development of the gap acceptance method. Tanner’s equation was adjusted by Troubeck to relate the equation to the observed field data and adopted with modifications in Australia. All the gap acceptance models are based on the distribution of gaps in the circulating flow rate and acceptance of the gap by the entering traffic (Vasconcelos et al., 2012). This method relies on parameters that have different approach measurements and as to be expected these different ways do not give the same result.

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2.4.3 HCM roundabout capacity estimation method

The highway capacity manual, HCM (2010) introduced exponential regression which is a mixture of empirical and theoretical methods. HCM estimates the entry capacity based on three main parameters; critical gap, follow-up time and the circulating flow rate. Critical gap which is the minimum gap within the circulating traffic that is safe for an entry vehicle to be willing to accept for merging with circulating traffic, the follow-up headway, and the circulating flow rate. The HCM (2010) capacity model was developed as an exponential regression model with parameter estimates based on gap acceptance theory with inherent weaknesses. For example, choosing a negative exponential equation to define the capacity of a roundabout entry, particularly one that is gap-acceptance based. The equation becomes nearly asymptotic to the x-axis making it unreliable to model small entry traffic flow rate when circulating traffic volume is high. It is easier to record the direct measurement of entry and circulating flow rates and more difficult to collect gap data at a roundabout. The absence of a Y-intercept means that the geometric influence of a roundabout is unexplained. A significant advantage of empirical model is sensitivity to roundabout geometric design. Without the capability to predict different capacities for a variety of configurations or number-of-lane-based designs, the designer runs the risk of overdesigning, decreasing safety, and increasing cost (Mark Lenters PE, 2010). Consequently, geometrically-sensitive design methods are sought after by clients, agencies, and owners to achieve required capacity targets while minimising right-of-way impacts, avoiding high construction costs, and balancing the safety of all users (Mark Lenters PE, 2010). Nevertheless, the HCM 2010 model equation is shown below;

𝑄𝑇 = 𝐴𝑒(−𝛽𝑞𝑐) [2.18]

Where; QT denotes theoretical capacity and qc = circulating flow rate 𝐴 = 3600_𝑡

𝑓 ; 𝛽 =

𝑡_𝑐−0.5𝑡_𝑓

3600 ; tf = 3.19s and tc = 4.11s

It can be rewritten as (multilane roundabouts);