Modelling Heterogeneous Traffic Flow on Upgrades of Intercity Roads

The effect of an upgrade and its length is very significant for traffic flow characteristics. Road traffic in developing countries like India is highly heterogeneous comprising vehicles of wide ranging physical dimensions, weight and dynamic characteristics such as engine power, acceleration rate, etc. Due to these variations, the effect of grade on vehicles in heterogeneous traffic may vary significantly among vehicle categories. Variation in the level of the interaction between vehicles on upgrades may result in different sets of traffic flow characteristics. Hence, it is necessary to model traffic flow on upgrades and study, in depth, changes in traffic flow characteristics with alteration in the magnitude of an upgrade and its length. Computer simulation has emerged as an effective technique for modelling traffic flow due to its capability to account for randomness related to traffic. This study is concerned with applying a simulation model of heterogeneous traffic flow, named HETEROSIM, to study the traffic flow characteristics and performance of different vehicle types on upgrades of different magnitudes. Keywords: upgrade, power-to-weight ratio, acceleration rates, heterogeneous traffic, simulation, speed–volume relationships, speed–distance profiles


Introduction
Knowledge of tra c ow characteristics is very important for planning, analyzing, designing and operating roadway systems (Helbing and Greiner 1997;Arasan and Koshy 2005;Kerner and Klenov 2006;Berezhnoy et al. 2007;Bonzani 2007;Burinskienė 2007, 2009;Bogdevičius 2007, 2009;Akgüngör 2008aAkgüngör , 2008bBazaras et al. 2008;Gowri and Sivanandan 2008;Youse and Fathy 2008;Antov et al. 2009;Beljatynskij et al. 2009;Gražulevičienė and Bendokienė 2009). Roadway capacity values and speed-ow relationships used for planning, designing and operating roads, in most of the developed countries, pertain to fairly homogeneous tra c conditions comprising vehicles of more or less uniform static and dynamic characteristics. However, tra c scenario in developing countries like India di ers signi cantly from conditions observed in developed countries. Road tra c in India is highly heterogeneous comprising vehicles of wide ranging physical dimensions, weight and dynamic characteristics.
Di erent types of vehicles of heterogeneous tra c on Indian roads may be broadly grouped into the following categories: 1. buses; 2. trucks; 3. light commercial vehicles comprising large vans and small trucks; 4. cars including jeeps and small vans; 5. motorised three-wheelers, including threewheeled motorised vehicles to carry passengers and three wheeled motorised vehicles to carry small quantities of goods; 6. motorised two-wheelers, including motorcycles, scooters and mopeds; 7. bicycles; 8. tricycles to carry passengers or small quantities of goods; 9. animal drawn vehicles. ese motorised and non-motorised vehicles share the same road space without any physical segregation. To manoeuvre, these vehicles take any lateral position on the roadway based on space availability. When such different types of vehicles with varying static and dynamic characteristics are allowed to mix and move on the same roadway facility, from time to time, a variable set of longitudinal and transverse distributions of vehicles may be noticed. e e ect of grade and its length is very signi cant for tra c ow characteristics. On upgrades, heavy vehi-cles such as trucks, buses, etc. will experience signi cant reduction in their speeds, whereas passenger cars and other smaller vehicles such as motorized-two-wheelers may experience relatively lesser speed reduction. Under heterogeneous tra c conditions, this variation in speed reduction among di erent vehicle categories can be attributed to their wide ranging physical characteristics such as dimensions, weight, etc. as well as to their dynamic characteristics such as engine power, acceleration rate, etc. Variation in the level of the interaction between vehicles on upgrades may result in a di erent set of trafc-ow characteristics when compared with tra c ow characteristics on level roads. Hence, it is necessary to model tra c ow on upgrades and to study changes in tra c ow characteristics with changes in the magnitude of an upgrade and its length. Simulation, from microscopic through macroscopic, is increasingly becoming a popular tra c-ow modeling tool for analyzing various tra c operations and vehicular interactions. Research work reported in this paper is related to the study on the e ect of an upgrade and its length on the performance of di erent vehicle categories under heterogeneous trafc conditions. e methodology adopted for modelling heterogeneous tra c ow on roads having upgrades of di erent magnitudes, the details of analysis and relevant results along with a review of the previous studies are presented in the following sections.

Review of the Previous Studies
In the past, researchers used a force balance equation and vehicle mechanics for developing models predicting truck speeds on upgrades of any percentage and length. e impact of road gradient on speed was analyzed by Yagar and Aerde (1983) who found that the operating speed at a location is expected to decrease by approximately 1.8 km/h for each 1% of grade when going uphill. Gillespie (1985) developed a simpli ed means of predicting truck hill-climbing performance based on the characterization of the available power for accelerating and overcoming grade. e author also found that the ratio of the available power to weight is speed dependent but provides an easy means for calculating truck speed pro les on arbitrary grades. Archilla and Fernandez De Cieza (1996) developed truck speed pro les on di erent grades. e simple force balance equation was tted to the eld data. e model explained about 80% of the observed variation. For developing speed pro les on grades, the weight-to-power ratio of 190 kg/kW was considered. Bester (2000) developed a procedure for calculating speed pro les of trucks by using the assumption that the acceleration of a truck was a linear function of speed.
A Policy on Geometric Design … (2004) provides a single set of speed pro les over distance to characterize the deceleration and acceleration of typical heavy trucks based on the weight-to-power ratio of 120 kg/kW on different grades. Bennett and Greenwood (2001) as a part of Highway Development and Management series, popularly known as HDM-IV manual, has given the governing equation for the forward movement of a road vehicle, when the latter encounters a grade. e manual has also given the values of di erent parameters that can be used for estimating the magnitude of resistance (air resistance, rolling resistance, grade resistance, etc.) involved in the forward movement of vehicle on grades. Lucic (2001) extended the vehicle-dynamics model by introducing the concept of variable power in order to capture the buildup of power as the vehicle engages in gearshi s. e proposed extension has resulted in the signi cant enhancement of the state-of-the-art vehicledynamics model. e author nally concluded that the use of constant transmission e ciency results in an overestimation of vehicle speeds at low speeds and an underestimation at high speeds. Lan and Menendez (2003) developed a well-dened speed pro le for trucks based on dynamic, kinematic and operating characteristics of moving trucks on grades. For developing speed pro les on grades, the weight-to-power ratios of 121.7 kg/kW and 182.5 kg/kW were considered. e exact speed-pro le model was derived through numerical integration. e authors also described an approximation model suiTable for calculation by hand. All these studies, however, are mainly related to the characterization of truck performance on upgrades under homogeneous tra c conditions and hence, the results of these studies are not applicable for Indian conditions.
A review of Indian studies related to the tra c ow characteristics of grades proposed that there had been only one study on the subject matter. Chandra and Goyal (2001) measured a free speed of di erent types of vehicles on two-lane intercity roads having grades with di erent magnitudes. e analysis pertaining to the average change in the free speed of different types of vehicles with a grade indicated that each percent of an upgrade decreases the free speed of a car by 2.96 km/h, that of a bus by 2.43 km/h, motorised twowheelers -by 2.50 km/h, light commercial vehicles -by 1.26 km/h, a tractor with trailer -by 2.06 km/h, a bicycle -by 0.51 km/h, a cycle-rickshaw -by 0.36 km/h and an animal drawn vehicle -by 0.19 km/h. Finally, the authors concluded that the free speed of a vehicle decreases uniformly with an increase in a gradient. e actual amount of reduction depends upon the type of a vehicle and the magnitude of the gradient. e literature survey clearly shows that, at present, there is no ready-to-use reference material available regarding developing speed-distance pro les for di erent vehicles on grades of varying magnitudes under heterogeneous tra c conditions in India. Hence, there is a need to model vehicular movement on grades at micro-level, which may enable researchers to study tra c ow characteristics with variation in the magnitude of a grade and its length and to develop speed-distance pro les of different vehicle categories for upgrades of magnitude varying over a wider range.

Objective and Scope
e general objective of research work reported here is to study the e ect of upgrades of di erent magnitudes on tra c ow characteristics and to develop speed-distance pro les (performance curves) for different categories of vehicles under heterogeneous tra c conditions. e recently developed micro-simulation model of heterogeneous tra c-ow HETEROSIM (Arasan and Koshy 2005) is used to study tra c ow characteristics at micro-level over a wide range of tra c ow conditions on upgrades.
Field data collected on tra c ow characteristics such as free speed, physical dimensions of di erent vehicle types, lateral clearance between vehicles, etc. are used in the calibration and validation of the simulation model.
Acceleration rates over di erent speed ranges required for the simulation of the heterogeneous tra c ow of grades were estimated for di erent upgrades. e validated simulation model is then used to develop speed-volume relationships for upgrades of di erent magnitudes.
Speed-distance pro les are also developed for different vehicle categories for upgrades of di erent magnitudes using the simulation model.

e Simulation Model
As this study pertains to heterogeneous tra c conditions prevailing in India, the available tra c simulation models based on homogeneous tra c conditions where a clear lane and queue discipline exist are not applicable to study heterogeneous tra c ow characteristics.
Also, the models developed through research attempts made to model heterogeneous tra c ow (Khan and Maini 2000;Kumar and Rao 1996;Ramanayya 1988) are limited in scope and do not address all aspects comprehensively. e recently developed model of heterogeneous tra c ow HETEROSIM (Arasan and Koshy 2005), however, is comprehensive and capable of replicating heterogeneous tra c ow. e modeling framework is explained brie y here to provide the background to the study. For the purpose of simulation, the entire road space is considered as a single unit and vehicles are represented as rectangular blocks on road space, the length and breadth of which represent the overall length and the overall breadth of the vehicles respectively. e front le corner of the rectangular block is taken as the reference point and the position of vehicles on road space is identi ed based on the coordinates of the reference point with respect to origin chosen at a convenient location on the space. e simulation model uses the interval scanning technique with a xed increment in time. For the purpose of simulation, the length of road stretch as well as road width can be varied as per user speci cation. e model was implemented in the C++ programming language with modular so ware design. e simulation process consists of the following major sequential steps related to tra c ow on the midblock sections of roads: 1. vehicle generation; 2. vehicle placement; 3. vehicle movement. e snapshot of the animation of heterogeneous tra c ow obtained using the animation module HETEROSIM is shown in Fig. 1.

Model Validation
When used for studying the e ect of grade, the simulation model HETEROSIM requires data particularly on the free-ow speeds of di erent vehicle types at the start of the sections and acceleration rates of di erent vehicle categories while negotiating a particular gradient, in addition to other relevant data pertaining to the roadway and tra c conditions.
Hence, for the purpose of data collection, study stretches were selected a er conducting a reconnaissance survey to satisfy the following conditions: 1. the stretch should have a uniform gradient for considerable length; 2. the stretch should be fairly straight; 3. the width of the roadway should be uniform; 4. there should not be any direct access from the adjoining land used on both sides. e three study stretches selected for data collection are on National Highway No. 4 near Pune, in Maharashtra (Western part of India) and their speci c locations are: 1. between km 31.2 and km 31.5 with a gradient of 3%; 2. between km 2.6 and km 3.0 with a gradient of 3.78%; 3. between km 5.2 and km 5.9 with a gradient of 5%. e selected stretches are four lane divided roads with a total carriageway width of 8.75 m (including shoulder) for each direction of tra c ow. us, there was a need to estimate acceleration rate for di erent cat- egories of vehicles, for a set of speed ranges and for three observed grades of 3%, 3.78% and 5%. Free speed data on di erent categories of vehicles were measured at the initial positions of the chosen sections of di erent grades. e model was validated for all three grades mentioned above. In this paper, the results pertaining to the validation of 5% upgrade are presented, as an example.
Estimation of Acceleration Rates e thrust required for the forward movement of a road vehicle is determined by the summation of forces acting on the vehicle in the longitudinal direction. e forces acting on the vehicle while negotiating an upgrade of magnitude i % are: 1) rolling resistance; 2) air resistance; 3) grade resistance; 4) inertial forces 5) during acceleration and deceleration, as shown in Fig. 2.
Propulsive e ort (drive force) is derived from the engine to overcome these forces. Any reserve in drive force, if available, may be used either to accelerate the vehicle or to overcome the drag arising from upgrades. When encountering a grade requiring thrust greater than the available drive force, de ciency is made up by the deceleration of the vehicle. e speed at which the available drive force is zero represents the steady-state or terminal or 'crawl' speed of the vehicle. is steady-state speed that a vehicle can maintain on an upgrade occurs when forces are in balance. e following (Equation 1) is the fundamental equation of motion for a road vehicle (Bennett and Greenwood 2001): where: a -acceleration, m/s 2 ; P d -driving power delivered to wheels, kW; M -vehicle mass; EMRAT -effective mass ratio; F a -aerodynamic drag resistance, N; F r -rolling resistance, N; F g -gradient resistance, N; v -vehicle speed, m/s. e equation used for calculating aerodynamic drag resistance is as follows: where: F a -aerodynamic force opposing the motion of the vehicle; ρ -mass density of the air, kg/m 3 (1.2 kg/m 3 ); CD -aerodynamic drag coe cient; CDmult -aerodynamic drag coe cient multiplier; AF -the projected frontal area of the vehicle, m 2 ; v -the speed of the vehicle, m/s. e equation used for calculating rolling resistance is as follows: where: F r -rolling resistance, N; M -the mass of the vehicle, kg; f -the coe cient of rolling resistance (0.01). e value of the coe cient of rolling resistance (f) is approximately constant up to the speed of 50 km/h (Kadiyali et al. 1982). At higher speeds (more than 50 km/h), the values can be approximately calculated as: where: f -the coe cient of rolling resistance at speed v; v -speed in km/h; f o -the coe cient of rolling resistance assumed constant up to a speed of 50 km/h. e equation used for calculating grade resistance is given as follows: where: F g -gradient resistance, N; M -the mass of the vehicle, kg; i -the magnitude of the gradient in percentage.
Equation (1) indicates that the ability of a vehicle to accelerate on an upgrade is dependent on the used power-to-weight ratio, the mass and various forces opposing motion. Vehicles with low power-to-weight ratios such as trucks, while negotiating up grades, may decelerate to a crawl or terminal speed where forces are in balance and acceleration is equal to zero. is steady-state terminal speed can, thus, be obtained by solving Equation (1) for the velocity term. Hence, the data pertaining to maximum power (kW) and gross vehicle weight (kg) of di erent makes, for each vehicle category, was collected from the manufacturers' websites, brochures, etc. Finally, the average representative values of power and weight for each vehicle category were arrived at for calculating acceleration rates at various speed ranges. e values pertaining to the coe cient of drag (CD), CD multiplier (CDmult), and the projected frontal area (AF), for each vehicle category, were taken from the manual of Highway Development and Management (HDM-4) (Bennett and Greenwood 2001). CD is a function of the direction of vehicle movement relative to wind direction. e apparent direction of the wind, which is the vector resultant of vehicle-movement direction and wind direction, is termed the yaw angle (Ψ). e values of CD reported in literature are usually from wind tunnel tests conducted with a zero degree yaw (i. e. front on to the wind) and accordingly, are the minimum for the vehicle. Hence, to obtain typical values of CD that would be found on roads, it is necessary to adjust to the variation in wind direction which can be achieved using a 'typical' wind angle increasing zero degree-yaw CD leading to wind av-Dr iv e Fo rc e Weight = mg F r F a F g A ir R e si st a n ce R o ll in g R e si st a n ce G ra de R es is ta n ce F g G ra de R es is ta n ce F g G ra de R es is ta n ce F g G ra de R es is ta n ce i Fig. 2. Forces acting on a moving vehicle on upgrades eraged CD or CD (Ψ). e ratio of CD (Ψ) divided by CD is termed CD multiplier (CDmult). e E ective Mass RATio (EMRAT) for each vehicle category is also calculated at various speeds as per the guidelines of HDM-4 manual (Bennett and Greenwood 2001). e coe cient of rolling resistance (f) is taken as 0.01 (Kadilyali et al. 1982) as the pavement surface of the study stretch is made of asphaltic concrete. For calculating used driving power delivered to the wheels (P d ), the variable 'power factor' is used. e calculation of the variable 'power factor' involves nding speed at which vehicle power reaches its maximum. For this purpose, a steady state or terminal speed for each vehicle category, at which the vehicle power reaches its maximum, is calculated considering acceleration, a = 0 in Equation (1). en, the ratio of vehicle speed under consideration and maximum speed at which vehicle-power attains its maximum is considered as the variable 'power factor' . e used driving power (P d ), then, can be estimated by multiplying the maximum power value by the variable power factor. us, using all these parameters acceleration rates at various speed ranges with respect to each upgrade considered for the study, are calculated for di erent vehicle categories. Acceleration rates, thus estimated over di erent speed ranges, used as the input to the simulation model, for model validation on a grade of 5% are given in Table 1. Table 1 shows that acceleration rates of such types of vehicles as buses, trucks, light commercial vehicles and motorised three-wheelers, in the speed range of more than 40 km/h, are negative. e reason behind these negative values can be explained as follows. Force balance Equation (1) consists of two components; namely, drive force derived based on the engine power of the vehicle and types of resistance (air, rolling and grade) to be overcome by a moving vehicle. e propulsive e ort (drive force) is derived from the engine (based on the power of the vehicle) to overcome these forces. Any reserve in drive force, if available, may be used either to accelerate the vehicle or to overcome the drag arising from upgrades. When encountering a grade of certain magnitude requiring thrust that is greater than the available drive force (a magnitude of resistance is higher than drive force generated from the engine of the vehicle), deciency is made up by the deceleration of the vehicle. On upgrades, tra c ow characteristics (the speed of vehicles) will change as its length increases. So, in the case of upgrades, for the purpose of model validation, it is important to compare simulation-output values with the corresponding observed values at di erent sections on the stretch of roads with a gradient. erefore, it is necessary to collect tra c-ow data for a particular upgrade at di erent intervals of the length of an upgrade selected for model validation. Data on tra c required for the study were collected using video recording of tra c ow at the selected locations. Tra c ow was recorded for one hour using a video camera mounted on an adjoining elevated ground, which enabled recording all tra c ow characteristics. Tra c ow was captured at each of the chosen locations (on 5% upgrade at: km 5.200, km 5.500, km 5.700 and km 5.900). e video captured tra c data on all locations and then transferred it to a Work Station (computer) for detailed analysis. e inputs required for the model to simulate heterogeneous tra c ow include road geometry, tra c volume and composition, vehicle dimensions, minimum and maximum lateral spacing between vehicles, minimum longitudinal spacing between vehicles, free speeds of di erent types of vehicles, acceleration and deceleration characteristics of vehicles, the type of headway distribution and the simulation period. e required input tra c data was obtained by running the video of trafc ow at a slower speed (⅛ th of the actual speed) to enable one person to record data observing the details displayed on the monitor of the computer. Tra c composition, for the measured total tra c volume of 598 vehicles per hour, at the starting point of the study stretch with 5% upgrade, was noted. It was found that Animal drawn vehicles, bicycles and tricycles which might be present in small numbers in certain intercity roads, were not present at any of the study stretches (as these vehicles may not be able to negotiate such steep grades). Similarly, using the same data extraction technique, the observed tra c volume and composition at all other locations for each of the grades were also recorded. It was observed that for each of the grades there was no signi cant di erence in tra c volume and composition obtained from tra c ows passing through various locations of a particular grade. e overall dimensions of all categories of vehicles and the minimum and maximum values of lateral-clearance share (corresponding to, respectively, zero speed and free speed conditions of respective vehicles) were adopted from an earlier study (Arasan and Koshy 2005).
For the purpose of model validation, it was decided to compare the observed and simulated speeds of di erent categories of vehicles at xed intervals when the vehicles negotiated for a particular upgrade. Road length on 5% upgrade was 700 m. e total width of the roadway available for one direction of movement on the divided highway was 8.75 m. For the purpose of validation, for a grade of 5%, simulation was run with three random number seeds and the averages of the three runs were taken as the nal output of the model. e observed roadway condition, measured free speed param-eters at the initial position, observed tra c volume and composition and acceleration rates (estimated at various speed ranges) were given as the input to the simulation model. e inter-arrival time (headway) of vehicles was found to t into negative exponential distribution and free speeds of di erent categories of vehicles based on the results of an earlier study (Kadiyali et al. 1981) were assumed to follow normal distribution. ese distributions, then, formed the basis for an input of two parameters for the purpose of simulation. e speeds maintained by di erent categories of vehicles at the selected sections on the study stretches were obtained as simulation output. A comparison of the observed and simulated speeds maintained by di erent types of vehicles at three selected sections on 5% upgrade (km 5200 to km 5500 (300 m), km 5200 to km 5700 (500 m) and km 5200 to km 5900 (700 m)), namely, sections I, II and III are shown in Table 2 indicating that the simulated speed values signi cantly match with the eld observed speeds for all vehicle types at all selected sections. A statistical validation of the model based on the observed and simulated speeds of di erent categories of vehicles was also done through the paired t-test for 5 degrees of freedom at the level of signi cance of 0.05. e calculated t-statistic values for sections I, II and III on 5% upgrade are 1.23, 0.343 and 0.862 respectively, against the Table value of 2.57. It can be seen that the value of t statistic calculated based on the observed data is less than the corresponding Table value. is implies that there is no signi cant di erence between the simulated and observed mean speeds at all sections indicating the validity of the simulation model.

Model Application
e HETEROSIM model can be applied to study various heterogeneous tra c scenarios under varying trafc and roadway conditions. Here, the application of the model is to develop relationship between tra c volume and speed under varying roadway conditions (four-lane divided roads with upgrades varying from 2% to 6%). Further, the validated simulation model is also used to develop speed-distance pro les on upgrades for di erent vehicle categories under heterogeneous tra c conditions.

Development of Speed-Flow Relationships
Since research work reported here is aimed at studying the e ect of an upgrade and its length on tra c ow characteristics and also on the performance of di erent categories of vehicles in heterogeneous tra c, a four-lane divided road with 8.75 m of road space (main roadway plus shoulder) available for each direction of movement with a magnitude of upgrades of 2%, 3%, 4%, 5% and 6%, were considered for model application. For this purpose, a representative tra c composition was considered for simulating tra c ow on the upgrades. is representative tra c composition commonly seen on divided intercity roads in India consists of buses (21%), trucks (35%), light commercial vehicles (11%), cars (17%), motorised three-wheelers (3%) and motorised two-wheelers (13%). e input data on free speed, at the initial position required for simulating tra c ow on grades, was collected on the stretch between km 77.2 and km 77.4 of National Highway No. 45 near Chennai (Southern part of India) which is a four-lane divided road (total width for one direction makes 8.75 m). e stretch is straight and level with no side road connections. Also, tra c ow on the study stretch was unhindered by road side land uses. By giving free speeds as input at the entry position of road stretch considered for simulating tra c ow on upgrades, it was possible to model the movement of vehicles approaching an upgrade of the given magnitude from level roads. Acceleration rates over di erent speed ranges required for the simulation of heterogeneous tra c ow on grades were estimated for di erent upgrades of a magnitude varying from 2% to 6% using equation (1) as explained earlier. e length of road stretch (used for noting outputs) considered for simulation was 1600 m.
Giving relevant data as input, tra c ow on di erent upgrades was simulated for volume levels ranging from a very low level to the maximum possible value (capacity) and speeds corresponding to each of the volume levels were obtained as output. In this regard, it may be noted that when simulation runs are made with successive increments in the tra c volume (input), there will be a commensurate increase in the exit volume at the end of simulation stretch. When the volume reaches capacity level, the increments in the input tra c volumes may not result in the same amount of an increase in the exit volume resulting in a decrease in the rate of tra c ow. Some successive decrease in the exit volume (in spite of an increase in input) indicates that the roadway has reached its capacity. e plots made relating speed and ow for all upgrades (2% to 6%) on the same set of axes are shown in Fig. 3. It can be seen that the developed speed-ow relationships follow the well established trend establishing the validity of the model to simulate heterogeneous tra c ow on di erent upgrades of magnitudes varying from 2% to 6%. e capacity values of 8.75 m wide roads having upgrades of a magnitude of 2%, 3%, 4%, 5% and 6% at a negotiated distance of 1600 m for the representative tra c composition considered are found to be 1360, 1210, 1120, 1050 and 1010 vehicles/h respectively.

Speed-Distance Pro les for Di erent Vehicle Types
Speed-distance relationships for di erent types of vehicles on grades can be developed by simulating heterogeneous tra c ow on the chosen roadway with a speci ed gradient and noting down a change in the speed of vehicles at regular intervals over space while the vehicles negotiate the upgrade. For the purpose of simulation, the volume level corresponding to the Volume-to-Capacity (V/C) ratio of 0.5 (normally used as design service volume in India) was considered for developing speed-distance curves for di erent vehicle categories along di erent grades. Free speed values for di erent vehicle categories, tra c composition and estimated acceleration rates for di erent vehicle categories at di erent speed ranges were given as input to the simulation model for simulating tra c ow along di erent grades ranging from 2% to 6%. e total length of road stretch, for simulation purpose, was taken as 3400 m. e middle 3000 m length of the simulation stretch was used to collect data on the simulated average speed for each vehicle category at every 200 m interval. e simulation model was run with three random number seeds and the average of the three runs was taken as the nal output of the model. Speed-distance relationships were thus developed for di erent vehicle categories for each di erent grade. e relationships thus, obtained in re-spect of buses, trucks, light commercial vehicles, cars, motorised three-wheelers and motorized two-wheelers are depicted in Figs 4-9, respectively. e gures display that vehicle performance (speed reduction) pertaining to various grades is di erent across di erent vehicle categories. In case of heavy vehicles such as buses, trucks and light commercial vehicles having lower power-toweight ratio, there is a signi cant reduction in speed, whereas for other smaller vehicles such as cars, motorised three-wheelers and motorized two-wheelers having higher power-to-weight ratio, there is a relatively lesser reduction in speed. It can also be noted that there is a signi cant speed reduction in all upgrades up to a distance of 1600 m beyond which the speed-distance curve is relatively atter and indicates that there is no signicant speed reduction beyond that point. 1. e validation results of the simulation model of heterogeneous tra c ow indicate that the model is capable of replicating heterogeneous tra c ow on intercity roads having upgrades of different magnitudes to a highly satisfactory extent. 2. From the speed-volume curves developed using the simulation model, it has been found out that for the representative tra c composition, the capacities of four-lane divided (8.75 m wide) roads having upgrades of the magnitudes of 2%, 3%, 4%, 5% and 6% for one direction of the tra c ow of the section at 1600 m from the start of the grade, are found to be 1360, 1210, 1120, 1050 and 1010 vehicles/h respectively.
3. From the speed-distance pro les developed for di erent vehicle types, it has been discovered that the rate of speed reduction in the vehicles, on upgrades, will vary based on the type of a vehicle and it is a function of a magnitude of an upgrade and the power-to-weight ratio of the vehicle. 4. Based on the simulation experiments conducted for developing speed-distance pro les for each type of the vehicle, it was established that the e ect of an upgrade on vehicle performance (speed) may not be signi cant beyond a length of 1600 m. 5. e developed speed-volume and speed-distance relationships provide the basis for estimating Passenger-Car-Unit (PCU) values of vehicles with due consideration to upgrades and their lengths. us, through this study, the problem of measuring the volume of heterogeneous trafc on upgrades can be addressed by converting di erent types of vehicles into equivalent passenger cars and expressing the volume in terms of PCU/h. 3% 4% 5% 6% Fig. 9. Speed-distance pro les for motorised two-wheelers on upgrades