Discriminate Modelling of Peak and Off-Peak Motorway Capacity

International Journal of Integrated Engineering - Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 Discriminate Modelling of Peak and Off-Peak Motorway Capacity Hasim Moammed Alassan 1,*, Sundara Parameswaren 2, Ben-Edigbe Jonnie 3 1,2,3 Department of Geotecnics and Transportation Faculty of Civil Engineering, Universiti Teknologi MALAYSIA * Department of Civil Engineering, Bayero University Kano, NIGERIA Abstract: Traffic teory is concerned wit te movement of discrete objects in real time over a finite network in 2 Dimensions. It is compatible wit or dependent on fundamental diagram of traffic. Witout question traffic flow is an essential quantitative parameter tat is used in planning, designs and roadway improvements. Road capacity is significant because it is an important indicator of road performance and can point road managers in te rigt road maintenance and traffic management direction. In tis paper four direct empirical capacity measurement metods ave been considered. To test te efficacy of eac metod, data for peak period, off-peak and transition to peak ave been used. Te eadway and te volume metods lack predictive capability and are suitable only for current assessment of flow rates. Te product limit metod is weak in its predictive capability in view of te arbitrariness in te selection of te capacity value. It is also an extreme value metod; ence not all volume data can be used wit tis metod. Te fundamental diagram metod as good predictive capability and furnises capacity values consistent wit te standard of te facility. Unlike oter metods, it does not rely on bottleneck conditions to deliver te capacity value. Te paper concluded tat eac metod is uniquely suited to prevailing conditions and can be so employed. Keywords: Fundamental Diagrams, Road Capacity, Traffic Flow, Headway, Modelling 1. Introduction Traffic flow measurements are vital for te performance evaluation of existing igway facilities as well as teir design and maintenance. Traffic flow rate, speed and density are usually required to completely describe te state of traffic on any roadway section. However, teir measurements are time consuming and costly but often required for preliminary designs, planning and roadway improvements. Traffic flow models enable quick estimates to be made in addition to describing te relationsips between flow, speed and density. Basically, tree approaces to traffic flow modelling can be pursued depending on te level of detail desired. Microscopic models describe te way individual veicles move on te road. Veicles in close proximity to eac oter may need to overtake (lane canging) or may continue beind te leading veicle (platoon formation). In situations like tese mesoscopic models are more suitable to describe traffic beaviour. At ig flow rates, traffic flow needs to be described at te level of detail of te flow rater tan te individual veicles, macroscopic models are ten used. Traffic flow modelling using te macroscopic approac could be by simulation or direct empirical metods. Tis paper explores direct empirical traffic flow modelling approaces wic yield traffic flow rates as te outcome. Te aim is to determine te efficacy of tese *Corresponding autor: madorayi@gmail.com 2012 UTHM Publiser. All rigt reserved. penerbit.utm.edu.my/ojs/index.pp/ijie models in predicting traffic flow rates particularly at capacity. Models tat utilise veicle eadway data, traffic volume data, traffic volume and speed data and volume, speed and density data sall be pursued. Tese metods also cover flow rates measurements during peak periods, non-peak periods as well as transitions to peak. Te rest of te paper is organised as follows: section 1.2 covers te literature review on te subject; section 1.3 describes te data collection procedure. In section 1.4, we present te flow rate (capacity) modelling tecniques wile te results follow in section 1.5. Finally we draw te conclusions in section 1.6. 2. Concepts on Empirical Roadway Capacity Estimation s Roadway capacity is central to igway traffic analysis. Capacities ave sometimes been derived in many literatures wit extreme values tat ave no resemblance to actual traffic flowrate. Determination of roadway capacity is one of te main outputs in traffic studies and traffic teory analysis. Its value is a key input for facility selection, design and reabilitation. Capacity can be taken as te maximum number of veicles per time period traversing a section or point along a roadway lane under prevailing circumstances. Te definition suggests tat roadway can be at peak, off-peak, and mixed-peak capacity per time unit under prevailing 53

H.M. Alassan et. al, Int. J. of Integrated Engineering Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 conditions. Te peak roadway capacity under dry weater condition is often employed as te reference flowrate because it represents te worst traffic stream scenarios. Hypoteses about te reasons for te capacity drop postulated in many studies include among oters: acceleration constraints, drivers beaviour and location bottleneck. Terefore te question of data collection period is pertinent to te issue of roadway capacity drop. As contained in many literatures, capacity estimation metods include estimation wit eadways, estimation wit traffic volumes, estimation wit traffic volumes and speeds, and finally estimation wit traffic volumes, speeds and densities. However, not all empirical estimation metods can be used for peak and off-peak traffic capacity prediction. It can be argued tat only eadway and fundamental diagram of flow can be so employed. Higway capacity can be determined using volume and speed data but te metod remains essentially te same as te extreme value metod. Te speed component of te data is not employed in te evaluation. It is owever, used to assess te qualitative performance of te traffic flow. Capacity can be estimated using te tree macroscopic parameters of volume, speed and density in bivariate relationsips called fundamental diagrams. Since speed and volume data are more readily obtainable from observation sites te density is derived using te fundamental equation of traffic. Flow-Density polynomial functions are ten used for quantitative prediction of te value of capacity because roadway capacity can be estimated using equation 1 below; (1) Were: Q is capacity, u f is free flow speed and k j is jam density. Traffic flow rate measurements employing direct empirical metods require observation and data collection at specific points or sections of te igway and determining te flow rate terefrom. Headway estimation metods assume traffic to be categorised into two: traffic following eac oter closely oterwise known as forced flow and traffic arriving in a free-flowing fasion. Two models are available to fit te eadway data and deriving te capacity (maximum flow rate). Tese are te Branston model [1] and te Buckley model [2]. Te two models are respectively stated in Equations 2 and 3 below. f g e g e d ( ) ( ) ( 1 ) ( ) = θ + θ λ λ λ (2) Te model by Branston results in a generalised queuing model wit te function g() being te probability density function of te time eadway. HA et al (2010) [3] suggest tat te gamma distribution given in Equation 3 provides te best fit for te time eadway. 0 ( ) ( τ) Γ( β) β 1 β α α( τ) g = e x1 { τ}( ) Buckley [2] used a Semi-Poisson process to model te eadway data as given in Equation 4. In bot models λ is te arrival rate in te free-flowing traffic wile θ represents te constrained portion of traffic. ( ) θ ( ) ( 1 ) f g θ λe ( ) ( λ) G (3) λ = + (4) L In eiter case, estimation of te traffic capacity is made using Equation 5 below. 3600 q = / n (5) p Capacity estimation tat employ observed volume metods also utilise two broad sub-models. Te submodels are te observed extreme and te expected extreme metods. Observed extreme metods may temselves be furter categorised as bimodal metod and te selected maxima. [4]. In te observed extreme metod, probability distributions are used to fit te data similar to te eadway metod in wic traffic is categorised into constrained and unconstrained flows. In tis case, constrained flow corresponds to flows at capacity wile unconstrained flows are te below capacity flows. Te selected maxima metod involves collecting data over a period of time and identifying te maximum flows over te period. Te capacity is ten assumed to be te average of te maximum flows over te period as given below. q = q / n (6) c i i Were q c is te capacity value (veicles per our), q i is te maximum flow rate observed over period i, n is te number of cycles and i is te lengt of cycle period over wic a maximum flow rate is determined. Estimation using extreme value metods essentially use extreme value teory particularly, te Weibull distribution to fit te volume data. Te parameters of te distribution are determined and cumulative distribution function is ten used to obtain te capacity value. [4]. 3. Data Collection Data was obtained from a principal road in Joor Baru 23km from Universiti Teknologi Malaysia. Te J5 is a two-lane two way facility tat runs along te west coast of peninsula Malaysia from te soutern state of Joor to te nortern state of Keda. Two automatic traffic counters were installed. One on eac lane and data was collected for two monts during te monsoon period in 2010. Detailed veicular information logged by te counters were retrieved and processed into macroscopic parameters. Te eadway data (also obtained from te counters) were processed separately for te eadway estimation metods. Only day ligt traffic data ave been g 54

H.M. Alassan et. al, Int. J. of Integrated Engineering Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 used in tis paper. Peak period, non-peak and transition to peak data ave been used appropriately for te metods used in tis paper. All traffic flows were converted to Passenger Car Equivalent (PCE) units prior to analysis using te standard Malaysia PCE values. 4. Analysis Peak our flows obtained during te observation period are sown in Table 1. For te eadway metod, te respective average eadways were determined and were ten used to compute overall average for te period of observation. Peak Volume PCE/r Table 1 Peak Flow Mean Headway Results Speed km/r Headway seconds Peak Volume PCE/r Speed km/r Headway seconds 1260 65.49 2.8571429 1393 69.45 2.5843503 1335 66.99 2.6966292 1271 69.60 2.8324154 1245 66.48 2.8915663 1313 68.55 2.7418126 1255 65.72 2.8685259 1238 66.28 2.9079160 1332 65.60 2.7027027 1390 71.12 2.5899281 1251 66.39 2.8776978 1345 69.30 2.6765799 1270 65.90 2.8346457 1266 70.36 2.8436019 1284 65.78 2.8037383 1312 68.75 2.7439024 1272 65.76 2.8301887 1232 71.65 2.9220779 1383 64.83 2.6030369 1363 71.42 2.6412326 1226 66.67 2.9363785 1313 72.70 2.7418126 1399 64.95 2.5732666 1304 72.34 2.7607362 1263 63.63 2.8503563 1267 74.31 2.8413575 1262 64.88 2.8526149 1383 64.58 2.6030369 1185 62.79 3.0379747 1344 63.57 2.6785714 1288 66.40 2.7950311 1302 65.48 2.7649770 1236 73.92 2.9126214 1171 65.65 3.0742955 1377 70.81 2.6143791 1323 64.03 2.7210884 1385 73.23 2.5992780 1415 62.98 2.5441696 1286 74.61 2.7993779 1257 65.24 2.8639618 1288 75.96 2.7950311 1475 64.94 2.4406780 1358 66.63 2.6509573 1253 68.63 2.8731045 1293 67.47 2.7842227 1288 67.10 2.7950311 Te mean eadway from te data in table 1 is 2.78 seconds. Te capacity was determined using Equation (5). Fig. 1 Flow/Time Profile Te product limit metod utilises te volumes and speed data to determine te capacity value. Te metod also involves classifying te data into two. Flows below te capacity value and flows at capacity and beyond. Given tat G (q) is te probability tat te capacity value q c is greater tan a flow rate q. Te capacity cumulative distributive function is ten given by F (q) = 1-G (q) (6) Te product limit function resulting is given as: K i 1 q G( q) =Π i, qi { C} (7) q K i q Were G(q) = Prob (q c > q) Equation (6) is used to compute te probabilities of capacity q c being greater tan any flow rate q. Te capacity cumulative distribution function is ten determined using equation (6) and sown in column 7 of Table 2. Using te median volume as te basis for defining te capacity te flow rates were divided into two: values tat represent flow rates below capacity and values for flow rates at or above capacity. Te ensuing plot of te cumulative distribution function is sown in Figure 2. Te capacity value is taken to be te 90 t percentile value of te cumulative distribution function. To estimate te capacity using te volumes data, te flow profiles were plotted as sown in Figure 1. Te maximum flows obtained from te profiles are ten used as in equation 6 to obtain te capacity value. Fig. 2 Cumulative Distributions from Product Limit 55

H.M. Alassan et. al, Int. J. Of Integrated Engineering Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 Table 2 Computed Capacity Probabilities using Product Limit 1 2 3 4 5 6 7 Interval i q i Set Order j K q, q i ε{c} G(q) F(q) 7.00-8.00 1260 Q 13-0.700 0.300 7.00-8.00 1335 C 31 1 0.300 0.700 7.00-8.00 1245 Q 7-0.825 0.175 7.00-8.00 1255 Q 10-0.750 0.250 7.00-8.00 1332 C 30 1 0.325 0.675 7.00-8.00 1251 Q 8-0.815 0.185 7.00-8.00 1272 Q 20-0.570 0.430 7.00-8.00 1383 C 37 1 0.110 0.890 7.00-8.00 1226 Q 3-0.982 0.018 7.00-8.00 1399 C 41 1 0.050 0.950 7.00-8.00 1185 Q 2-0.920 0.080 7.00-8.00 1288 Q 23-0.452 0.548 7.00-8.00 1236 Q 5-0.850 0.150 7.00-8.00 1377 C 36 1 0.188 0.812 7.00-8.00 1385 C 38 1 0.050 0.950 7.00-8.00 1286 Q 22-0.450 0.550 7.00-8.00 1393 C 40 1 0.050 0.950 7.00-8.00 1271 Q 19-0.573 0.427 7.00-8.00 1313 C 28 1 0.330 0.670 7.00-8.00 1238 Q 6-0.850 0.150 7.00-8.00 1390 C 39 1 0.050 0.950 7.00-8.00 1345 C 33 1 0.238 0.762 7.00-8.00 1266 Q 16-0.475 0.425 7.00-8.00 1312 C 27 1 0.338 0.662 7.00-8.00 1232 Q 4-0.238 0.762 7.00-8.00 1363 C 35 1 0.188 0.812 7.00-8.00 1313 C 28 1 0.188 0.812 7.00-8.00 1304 Q 26-0.400 0.600 7.00-8.00 1267 Q 17-0.650 0.350 7.00-8.00 1383 C 37 1 0.163 0.837 7.00-8.00 1344 C 32 1 0.225 0.775 7.00-8.00 1302 Q 25-0.400 0.600 7.00-8.00 1171 Q 1-0.991 0.019 7.00-8.00 1323 C 29 1 0.328 0.672 7.00-8.00 1415 C 42 1 0.025 0.975 7.00-8.00 1257 Q 12-0.750 0.250 7.00-8.00 1475 C 43 1 0.000 1.000 7.00-8.00 1253 Q 11-0.750 0.250 7.00-8.00 1288 Q 23-0.450 0.550 Average Total I =46 Volume I in Q = 27 1305 I in C = 19 56

H.M. Alassan et. al, Int. J. Of Integrated Engineering Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 Capacity estimation using te fundamental diagram approac utilises traffic flow parameters of volume, speed and density. To proceed, te flow-density plot is fitted to a quadratic function and te speed-density plot is fitted to linear function. Tese two are sufficient to derive te traffic state parameters including te capacity. Te capacity value is obtained by finding te derivative of te quadratic function and determining te critical density and maximum flow rate. Te plots for te two functions are sown in Figure 3. Te results of te analysis for peak, non-peak and transition to peak are presented in te next section. Fig. 3 Flow/Density Relationsip (off-peak) q = - 0.5255x 2 + 64.395x 1.3709 R 2 = 0.97 (8) Te model coefficients in equations in te paper ave te expected signs and te coefficients of determinations (R 2 ) for bot road directions A are muc greater tan 0.85, tus, it can be suggested tat a strong relationsip between flows and densities exists and te model could be used to estimate roadway capacity for te igway sections. Te F observed statistics at 10 degree of freedom is muc greater tan F critical value of 4.94 suggesting tat te relationsip did not occur by cance. Also te t observed statistic at 10 degree of freedom tested at 5 % significance level is muc greater tan 2 tus suggesting tat density is an important variable wen estimating flow. For maximum flowrate; q / x = 1.051k + 64.395 = 0 k c = 61.27veicles/km Estimated capacity (Q) = 1972pcu/r 5. Results and Discussion Te results of te remainder capacity evaluations for peak, off-peak and transition to peak are sown in Table 3. Te eadway, volume and product limit metods gave similar results for te peak period data. Te fundamental diagram approac returned a value iger tan te oter metods. Te eadway metod produced te least capacity value. Tis may be due to te large average eadways obtained. Te eadway of 2.78 seconds is above te perception-reaction time of drivers normally stated as 2.50 seconds. Te implication is tat drivers travelled wit less indrance on te facility and consequently a low capacity value would result. Te average eadways for te off-peak and transition to peak data are respectively 5.86 and 5.88 sec. Te volume metod capacity value was 7.38% iger tan te value returned using te eadways metod. Te selected maxima metod simply requires identification of te maximum flow rates average over te observation period. In addition, te metod requires a bottleneck location to observe te maximum flows. However, te traffic on te facility was free flowing and bottlenecks did not occur. Te maximum flow rates terefore coincided wit te peak periods and te capacity value returned depicts te operating conditions on te facility. Type of Data Headway Volume Product Limit Fundamental Diagram Peak 1297 1400 1373 1972 Non-Peak 614 1033-1869 Transition to Peak 611 1045-2089 Te product limit metod and te fundamental diagram bot require indirect data andling. Unlike te two earlier metods, tese ones are modelled and projected to te capacity level. Te implication is tat te values returned by tese metods are futuristic values wic were not attained on te facility. In te case of te product limit metod, te cumulative capacity distribution function is used to scale off te capacity value. Te point to use is debatable as no consensus as yet been reaced by researcers. In tis paper te 90 t percentile value was used wic was 5.56% iger tan te eadways metod and 1.93% lower tan te volume metod. No results were returned for te off-peak and transition to peak metods using te product limit metod. Tis is understandable because off-peak and transition to peak data are not extreme values as is required before te metod can be used. In te fundamental diagram metod, a value of 1972.73 PCE per.our was obtained wic is 26.06% iger tan te eadway metod. Tis value is more consistent wit igways of principal road standard obtained elsewere[5]. Te fundamental diagram metod as furter advantages over te oter metods; it gives te state of te traffic wic oter metods cannot provide. In all, te metods used eac ave teir individual merits and te operating conditions on a facility sould dictate wic metod to employ in capacity evaluations. In sum, four empirical capacity estimation metods employed in tis paper ave teir strengts and weaknesses. Te eadway and volume metods do not ave predictive capability. Hence, tey are suitable for on te spot assessments and sceduled maintenance purposes. Te product limit metod is weak in its predictive capability because te arbitrariness in te selection of te capacity values from te cumulative distribution function brings about inconsistent results. Furtermore as te metod specifies, only extreme value data can be modelled using te product limit metod. Te fundamental diagram approac is suited for all operating conditions of a roadway. 57

H.M. Alassan et. al, Int. J. of Integrated Engineering Special Issue on ICONCEES Vol. 4 No. 3 (2012) p. 53-58 6. Conclusions Te eadway and te volumes metods lack predictive capability and are suitable only for current assessment of flow rates. Te product limit metod is weak in its predictive capability in view of te arbitrariness in te selection of te capacity value. It is also an extreme value metod, ence not all volume data can be used wit tis metod. Te fundamental diagram metod as good predictive capability and furnises capacity values consistent wit te standard of te facility. Unlike oter metods, it does not rely on bottleneck conditions to deliver te capacity value. Te paper concluded tat eac metod is uniquely suited to prevailing conditions and can be so employed. References [1]. Branston, D., Models of Single Lane Time Headway Distributions. Transportation Science, 1976. 10(2). [2]. Buckley, D., A Semi-Poisson Model of Traffic Flow. Transportation Science, 1968. 2: p. 107-133. [3]. HA, D.-H., M. Aron, and S. Coen, Time Headway Variable and Probabilistic Modelling. Te Frenc National Institute for Transport Network and Safety Researc, 2010. [4]. Minderoud, M.M., H. Botma, and P.H.L. Bovy, Assessment of Roadway Capacity Estimation s Transportation Researc Record 1996. 1572: p. 59-67. [5]. Alassan, H.M. and J. Ben-Edigbe, Higway Capacity Prediction in Adverse Weater. Journal of Applied Sciences, 2011c. 11(12): p. 2193-2199. 58