Journal of Multidisciplinary Engineering Science Studies (JMESS) ISSN: 2912-1309 Vol. 2 Issue 2, February - 2016 Fundamental Diagram For Spatial Analysis Of Urban Traffic Flow: A Case Of Kenya’s Nyeri Municipality Lekariap Edwin Mararo Prof. Abiero Gariy Prof. Mwatelah Josphat Department of Civil, Department of Civil, Department of Geomatic Construction and Environmental Construction and Environmental Engineering and Geospatial Engineering, Jomo Kenyatta Engineering, Jomo Kenyatta Information Systems, Jomo University of Agriculture and University of Agriculture and Kenyatta University of Technology, Nairobi, Kenya Technology, Nairobi, Kenya Agriculture and Technology, [email protected] zagariy@eng,jkuat.ac.ke Nairobi, Kenya [email protected] Abstract—Nyeri town’s historic background and 1. INTRODUCTION strategic position it occupies in Kenyan politics has contributed to its infrastructural Analysis of traffic flow and modeling of vehicular development. This has resulted in unprecedented congestion has mainly relied on fundamental laws, traffic congestion. This congestion, besides inspired from physics using analogies with fluid leading to a number of traffic-related accidents, mechanics, many particles systems and the like. One has derailed the economic development due to main difference of physical systems and vehicular the time wasted in traffic. It is against this traffic is that humans make choices in terms of backdrop that this study sought to establish routes, destinations and driving behavior, which fundamental traffic flow models and derive the creates additional complexity to the system. While flow characteristics associated with traffic most of the traffic science theories make a clear operations in the town. The study was guided by distinction between free-flow and congested traffic a number of important models which included states, empirical analysis of spatio-temporal speed density model, speed flow model, and congestion patterns has revealed additional flow-density model. The study relied on complexity of traffic states and non-steady state macroscopic data based on traffic characteristics conditions (Munoz & Daganzo, 2003; Helbing et al., at an aggregated level. The data collection 2009). Thus, the known fundamental diagram, initially methods addressed traffic flow, average speed, observed for a stretch of highway to provide a steady- free-flow speed, traffic composition and state relationship between speed, density and flow, is directional distribution. The collected data was not sufficient to describe the additional complexity of analyzed using both classification analysis and traffic systems; it also contains significant event count analysis. The study found that MFD experimental errors in the congested regime for a could be employed to enhance accessibility in highway stretch (Kerner and Rehborn, 1996) or for a Nyeri town. The study deduced that there were city street (Geroliminis, 2008). Nevertheless, it was certain road links that performed better than recently observed from empirical data in Downtown others which was concluded to be very Yokohama that by aggregating the highly scattered significant to urban roads planners and plots of flow vs. density from individual loop engineers. The study recommended that the detectors, the scatter almost disappeared and a well designed capacities of the road segments should defined Macroscopic Fundamental Diagram exists be provided so that further analyses that pertain between space-mean flow and density (Geroliminis, to comparing the performance levels of different 2008). The determined MDF is important in the study links can be undertaken. of the performance of the road networks. Keywords—Microscopic fundamental diagram, Nyeri town, spatial analysis, traffic flow, traffic Performance of road networks over the years is flow models, traffic operations. typically measured at link and intersection levels and this makes it quite difficult when assessing the www.jmess.org JMESSP13420064 251 Journal of Multidisciplinary Engineering Science Studies (JMESS) ISSN: 2912-1309 Vol. 2 Issue 2, February - 2016 performance of an entire network. To be able to put characteristics associated with traffic operations in effective traffic and transport management practices Nyeri town? in place for an entire town, it was ideal to consider measuring performance of roads at higher levels. 5. THEORETICAL FRAMEWORK This will also help in determining the state of the network. The performance of a town’s road network 5.1 Speed - Density Model which can be studied at a macro level could be attributed to the planning of the town. A well planned The development of a traffic flow model begins with a town has, as part of its network characteristics, good relation between speed and density (van accessibility and mobility, thus, less congestion. Maarseveen et al., 2005). The relationship between these two parameters provides essential information Nyeri town, to which the study was delimited, is for planning purposes, which is one reason why the served by a reasonably well-maintained tarmac road MFD is being investigated for Nyeri. In real life network connecting it to Nairobi, Nakuru, Nanyuki, situations, a typical example of this relation is shown Othaya and other surrounding towns. Most in Figure 2-2. The graph shows that when a single transportation of cargo to and from Nyeri is by road, road user uses the roadway he/she could drive at although the town has a largely underutilized railway any desirable speed because density (k) is low; this station at Kiganjo (about six kilometres out of town is the free-flow speed (uo) since the choice of speed towards Nanyuki) on the branch line of the railway is not limited by other road users. When more and from Nairobi to Nanyuki, and an airstrip at Mweiga more drivers begin using the roadway, density (about 15 Kilometres out of Town towards Nakuru) increases and the speed decreases significantly and another airstrip at Nyaribo (off the Nanyuki - (because of the many interactions amongst vehicles) Naromoru highway). The traffic congestion evident in till the road capacity is reached. When the product of the town was a reason good enough to necessitate a density and speed results in the maximum flow, the study on fundamental traffic models. capacity is reached and the speed at this point is referred to as optimum speed uc, often called critical 2. STATEMENT OF THE PROBLEM speed, (Highway Capacity Manual, 2000). Nyeri town is not only the capital of Nyeri County but At a point in time, density becomes so high such that it has till 2013 been the administrative headquarters all vehicles stop and speed is now zero; the density of the previous Central Province that comprised of at this point is the jam density, kj. Information such as the current Nyandarua, Kiambu, Nyeri, Kirinyaga, the measure of these parameters from this diagram and Murang’a Counties. Its historic background and (Figure 1) will aid the town authorities in planning the strategic position it occupies in Kenyan politics has town by knowing how various land uses contribute to contributed to its infrastructural development. This this u-k diagram as a result of land use transport has resulted in unprecedented traffic congestion. interactions. Also, the town’s free flow and critical This congestion, besides leading to a number of speeds are essential for planning, traffic traffic-related accidents, has derailed the economic management and safety purposes. Knowledge about development due to the time wasted in traffic. It is the town jam density helps to put in place effective against this backdrop that it was necessary to measures to control traffic flow in the city so that the formulate measures which would potentially address city’s flows do not get to such high levels of density. this challenge. Therefore, it was important to carry out a study on fundamental traffic models and derive the flow characteristics associated with traffic operations in this town. 3. STUDY OBJECTIVE To establish fundamental traffic flow models and derive the flow characteristics associated with traffic operations in Nyeri town 4. RESEARCH QUESTION What are the fundamental traffic flow models and flow www.jmess.org JMESSP13420064 252 Journal of Multidisciplinary Engineering Science Studies (JMESS) ISSN: 2912-1309 Vol. 2 Issue 2, February - 2016 Speed uo uc u (km/hr) k (vh/m) kj k c Density Figure 1: Speed- density relation for the city of Yokohama, Japan (Geroliminis & Daganzo, 2008) This kind of relation can be shown theoretically as: if the town has the overall traffic characteristics as such. Density (k) can be derived from equation 1 u= uo ………………………………… (1) as: k=ks(1-u/uo) ……………………………(2) Where: Substituting equation 2 the flow can be obtained as k = density shown in equation 3 2 kc = critical density q=ks(u–u /uo) ……………...………......... (3) kj = jam density u = speed uo = free speed uc = critical speed 5.2 Speed- Flow Model An aggregation of the average speed (u) and flow (q) of vehicles on the road network in the Nyeri town could establish a relation such as in Figure 2, www.jmess.org JMESSP13420064 253 Journal of Multidisciplinary Engineering Science Studies (JMESS) ISSN: 2912-1309 Vol. 2 Issue 2, February - 2016 uo uc qc Figure 2: Speed-Volume Relationship observed on Northbound I-405 (Los Angeles), three Mondays in September-October 2007 (Federal Highway Administration, 2009) 2 Through the study of the relationship between these q = uo(k - k /ks) (4) parameters, one is able to interpret and know the kind of relation between land use and transport that Where; contributes to this type of graph. Initially with no vehicles on the roadway, speed and flow are both q = flow uo = free speed zero. As the number of vehicles increases, the flow increases and speed decreases due to the vehicles k = density ks = jam density interactions, until a capacity is reached; then flow decreases with decreasing speed. The capacity of Some characteristics of an ideal flow-density the road is reached at this point. Also at this point, relationship which serve as very useful information for planning purposes are that when there are no the maximum flow (qc) and critical speed (uc) can be known.