Traffic Flow PDF

CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 Chapter 31 Fundamental relations of traffic flow 31.1 Overview Speed is one of the basic parameters of traffic flow and time mean speed and space mean speed are the two represent...

CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 Chapter 31 Fundamental relations of traffic flow 31.1 Overview Speed is one of the basic parameters of traffic flow and time mean speed and space mean speed are the two representations of speed. Time mean speed and space mean speed and the relationship between them will be discussed in detail in this chapter. The relationship between the fundamental parameters of traffic flow will also be derived. In addition, this relationship can be represented in graphical form resulting in the fundamental diagrams of traffic flow. 31.2 Time mean speed (vt ) As noted earlier, time mean speed is the average of all vehicles passing a point over a duration of time. It is the simple average of spot speed. Time mean speed vt is given by, 1X n vt = vi , (31.1) n i=1 where v is the spot speed of ith vehicle, and n is the number of observations. In many speed studies, speeds are represented in the form of frequency table. Then the time mean speed is given by, Pn qi vi vt = Pi=1 n , (31.2) i=1 qi where qi is the number of vehicles having speed vi , and n is the number of such speed categories. 31.3 Space mean speed (vs ) The space mean speed also averages the spot speed, but spatial weightage is given instead of temporal. This is derived as below. Consider unit length of a road, and let vi is the spot speed of ith vehicle. Let ti is the time the vehicle takes to complete unit distance and is given by v1i. If there are n such vehicles, then the average travel time ts is given by, Σti 1 1 ts = = Σ (31.3) n n vi 1 If tav is the average travel time, then average speed vs = ts. Therefore from the above equation, n vs = Pn 1 (31.4) i=1 vi Introduction to Transportation Engineering 31.1 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 qi No. speed range average speed (vi ) volume of flow (qi ) v i qi vi 1 2-5 3.5 1 3.5 2.29 2 6-9 7.5 4 30.0 0.54 3 10-13 11.5 0 0 0 4 14-17 15.5 7 108.5 0.45 total 12 142 3.28 This is simply the harmonic mean of the spot speed. If the spot speeds are expressed as a frequency table, then, Pn qi P vs = ni=1 qi (31.5) i=1 vi where qi vehicle will have vi speed and ni is the number of such observations. Example 1 If the spot speeds are 50, 40, 60,54 and 45, then find the time mean speed and space mean speed. Solution Time mean speed vt is the average of spot speed. Therefore, vt = Σv n = i 50+40+60+54+45 5 = 249 5 = 49.8 n 5 5 Space mean speed is the harmonic mean of spot speed.Therefore, vs = Σ 1 = 1 + 1 + 1 + 1 + 1 = 0.12 = 48.82 vi 50 40 60 54 45 Example 2 The results of a speed study is given in the form of a frequency distribution table. Find the time mean speed and space mean speed. speed range frequency 2-5 1 6-9 4 10-13 0 14-17 7 Solution The time mean speed and space mean speed can be found out from the frequency table given below. First, the average speed is computed, which is the mean of the speed range. For example, for the first speed range, average speed, vi = 2+5 2 = 3.5 seconds. The volume of flow qi for that speed range is same as the frequency. The terms vi.qi and vqii are also tabulated, and their summations in the last row. Time i vi mean speed can be computed as, vt = Σq 142 Σqi = 12 = 11.83 Similarly, space mean speed can be computed as, Σqi 12 vs = Σ qi = 3.28 = 3.65 vi 31.4 Illustration of mean speeds Inorder to understand the concept of time mean speed and space mean speed, following illustration will help. Let there be a road stretch having two sets of vehicle as in figure 31:1. The first vehicle is traveling at 10m/s with 50 m spacing, and the second set at 20m/s with 100 m spacing. Therefore, the headway of the slow vehicle Introduction to Transportation Engineering 31.2 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 10 m/s 10 m/s 10 m/s 10 m/s 10 m/s 50 50 50 50 20 m/s 20 m/s 20 m/s 100 100 hs = 50/20 = 5sec ns = 60/5 = 12 ks = 1000/50 = 20 hf = 100/20 = 5sec nf = 60/5 = 12 kf = 1000/100 = 10 Figure 31:1: Illustration of relation between time mean speed and space mean speed hs will be 50 m divided by 10 m/s which is 5 sec. Therefore, the number of slow moving vehicles observed at A in one hour ns will be 60/5 = 12 vehicles. The density K is the number of vehicles in 1 km, and is the inverse of spacing. Therefore, Ks = 1000/50 = 20 vehicles/km. Therefore, by definition, time mean speed vt is given by vt = 12×10+12×20 24 = 15 m/s. Similarly, by definition, space mean speed is the mean of vehicle speeds over time. Therefore, vs = 20×10+10×20 30 = 13.3 m/s This is same as the harmonic mean of spot speeds obtained at 24 location A; ie vs = 12× 1 +12× 1 = 13.3 m/s. It may be noted that since harmonic mean is always lower than 10 20 the arithmetic mean, and also as observed , space mean speed is always lower than the time mean speed. In other words, space mean speed weights slower vehicles more heavily as they occupy the road stretch for longer duration of time. For this reason, in many fundamental traffic equations, space mean speed is preferred over time mean speed. 31.5 Relation between time mean speed and space mean speed The relation between time mean speed and space mean speed can be derived as below. Consider a stream of vehicles with a set of substream flow q1 ,q2 ,... qi ,... qn having speed v1 ,v2 ,... vi ,... vn. The fundamental relation between flow(q), density(k) and mean speed vs is, q = k × vs (31.6) Therefore for any substream qi , the following relationship will be valid. qi = k i × v i (31.7) The summation of all substream flows will give the total flow q. Σqi = q (31.8) Similarly the summation of all substream density will give the total density k. Σki = k (31.9) Let fi denote the proportion of substream density ki to the total density k, ki fi = (31.10) k Introduction to Transportation Engineering 31.3 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 Space mean speed averages the speed over space. Therefore, if ki vehicles has vi speed, then space mean speed is given by, Σki vi vs = (31.11) k Time mean speed averages the speed over time.Therefore, Σqi vi vt = (31.12) q Substituting in 31.7 vt can be written as, Σki vi 2 vt = (31.13) q Rewriting the above equation and substituting 31.11, and then substituting 31.6, we get, ki 2 vt = kΣ v k i kΣfi vi 2 = q Σfi vi 2 = vs By adding and subtracting vs and doing algebraic manipulations, vt can be written as, Σfi (vs + (vi − vs ))2 vt = (31.14) vs 2 Σfi (vs )2 + (vi − vs ) + 2.vs.(vi − vs ) = (31.15) vs 2 Σfi vs 2 Σfi (vi − vs ) 2.vs.Σfi (vi − vs ) = + + (31.16) vs vs vs The third term of the equation will be zero because Σfi (vi − vs ) will be zero, since vs is the mean speed of vi. The numerator of the second term gives the standard deviation of vi. Σfi by definition is 1.Therefore, σ2 = vs Σfi + +0 (31.17) vs σ2 vt = v s + (31.18) vs Hence, time mean speed is space mean speed plus standard deviation of the spot speed divided by the space mean speed. Time mean speed will be always greater than space mean speed since standard deviation cannot be negative. If all the speed of the vehicles are the same, then spot speed, time mean speed and space mean speed will also be same. 31.6 Fundamental relations of traffic flow The relationship between the fundamental variables of traffic flow, namely speed, volume, and density is called the fundamental relations of traffic flow. This can be derived by a simple concept. Let there be a road with length v km, and assume all the vehicles are moving with v km/hr.(Fig 31:2). Let the number of vehicles Introduction to Transportation Engineering 31.4 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 v km B A 8 7 6 5 4 3 2 1 Figure 31:2: Illustration of relation between fundamental parameters of traffic flow counted by an observer at A for one hour be n1. By definition, the number of vehicles counted in one hour is flow(q). Therefore, n1 = q (31.19) Similarly, by definition, density is the number of vehicles in unit distance. Therefore number of vehicles n 2 in a road stretch of distance v1 will be density × distance.Therefore, n2 = k × v (31.20) Since all the vehicles have speed v, the number of vehicles counted in 1 hour and the number of vehicles in the stretch of distance v will also be same.(ie n1 = n2 ). Therefore, q =k×v (31.21) This is the fundamental equation of traffic flow. Please note that, v in the above equation refers to the space mean speed. 31.7 Fundamental diagrams of traffic flow The relation between flow and density, density and speed, speed and flow, can be represented with the help of some curves. They are referred to as the fundamental diagrams of traffic flow. They will be explained in detail one by one below. 31.7.1 Flow-density curve The flow and density varies with time and location. The relation between the density and the corresponding flow on a given stretch of road is referred to as one of the fundamental diagram of traffic flow. Some characteristics of an ideal flow-density relationship is listed below: 1. When the density is zero, flow will also be zero,since there is no vehicles on the road. 2. When the number of vehicles gradually increases the density as well as flow increases. 3. When more and more vehicles are added, it reaches a situation where vehicles can’t move. This is referred to as the jam density or the maximum density. At jam density, flow will be zero because the vehicles are not moving. 4. There will be some density between zero density and jam density, when the flow is maximum. The relationship is normally represented by a parabolic curve as shown in figure 31:3 Introduction to Transportation Engineering 31.5 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 B qmax A q D E flow(q) C O kjam k0 k1 kmax k2 density (k) Figure 31:3: Flow density curve uf speed u k0 density (k) kj am Figure 31:4: Speed-density diagram The point O refers to the case with zero density and zero flow. The point B refers to the maximum flow and the corresponding density is kmax. The point C refers to the maximum density kjam and the corresponding flow is zero. OA is the tangent drawn to the parabola at O, and the slope of the line OA gives the mean free flow speed, ie the speed with which a vehicle can travel when there is no flow. It can also be noted that points D and E correspond to same flow but has two different densities. Further, the slope of the line OD gives the mean speed at density k1 and slope of the line OE will give mean speed at density k2. Clearly the speed at density k1 will be higher since there are less number of vehicles on the road. 31.7.2 Speed-density diagram Similar to the flow-density relationship, speed will be maximum, referred to as the free flow speed, and when the density is maximum, the speed will be zero. The most simple assumption is that this variation of speed with density is linear as shown by the solid line in figure 31:4. Corresponding to the zero density, vehicles will be flowing with their desire speed, or free flow speed. When the density is jam density, the speed of the vehicles becomes zero. It is also possible to have non-linear relationships as shown by the dotted lines. These will be discussed later. Introduction to Transportation Engineering 31.6 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 uf u2 speed u u u1 u0 q Qmax flow q Figure 31:5: Speed-flow diagram speed u speed u density k flow q qmax flow q density k Figure 31:6: Fundamental diagram of traffic flow 31.7.3 Speed flow relation The relationship between the speed and flow can be postulated as follows. The flow is zero either because there is no vehicles or there are too many vehicles so that they cannot move. At maximum flow, the speed will be in between zero and free flow speed. This relationship is shown in figure 31:5. The maximum flow q max occurs at speed u. It is possible to have two different speeds for a given flow. 31.7.4 Combined diagrams The diagrams shown in the relationship between speed-flow, speed-density, and flow-density are called the fundamental diagrams of traffic flow. These are as shown in figure 31:6 Introduction to Transportation Engineering 31.7 Tom V. Mathew and K V Krishna Rao CHAPTER 31. FUNDAMENTAL RELATIONS OF TRAFFIC FLOW NPTEL May 3, 2007 31.8 Summary Time mean speed and space mean speed are two important measures of speed. It is possible to have a relation between them and was derived in this chapter. Also, time mean speed will be always greater than or equal to space mean speed. The fundamental diagrams of traffic flow are vital tools which enables analysis of fundamental relationships. There are three diagrams - speed-density, speed-flow and flow-density. They can be together combined in a single diagram as discussed in the last section of the chapter. 31.9 Problems 1. Space mean speed is (a) the harmonic mean of spot speeds (b) the sum of spot speeds (c) the arithmetic mean of spot speeds (d) the sum of journey speeds 2. Which among the following is the fundamental equation of traffic flow? k (a) q = v (b) q = k × v (c) v = q × k (d) q = k 2 × v 31.10 Solutions 1. Space mean speed is √ (a) the harmonic mean of spot speeds (b) the sum of spot speeds (c) the arithmetic mean of spot speeds (d) the sum of journey speeds 2. Which among the following is the fundamental equation of traffic flow? k (a) q = v √ (b) q = k × v (c) v = q × k (d) q = k 2 × v Introduction to Transportation Engineering 31.8 Tom V. Mathew and K V Krishna Rao