5. Network Analysis. 5.1 Introduction

Transcription

1 5. Network Analysis 5.1 Introduction With the continued rowth o this country as it enters the next century comes the inevitable increase in the number o vehicles tryin to use the already overtaxed transportation network. Since it is evident that our current practices o traic manaement and the construction o additional lanes o hihway can not hope to relieve the conestion bein caused, another solution is needed. And AHS is that solution. Assume by the middle o the next century an entire AHS Malev network has been built as depicted in Fiure 3-. As the Interstate Hihway System has over the last 0 years, rowth patterns o both existin cities and new population centers will be centered around the AHS and its interchanes. With these chanes will come new traic patterns based around the uideways and the uideway-reeway interchanes. It is thereore o interest to see how the traic will respond to the new transportation network. Traditionally an analysis o a hihway network is completed usin what is known as the Urban Transportation Plannin Process (UTPP). This process consists o our steps trip eneration, trip distribution, modal split, and traic assinment. Trip eneration is concerned with estimatin the number o trip productions and trip attractions each node, or zone, produces. This depends heavily on the socio-economic characteristics o the population utilizin the network, and is thereore outside the scope o this work. Thus trip eneration will not be considered and trips will be loaded onto the network at interchanes usin arbitrary volumes. This should not be a problem as the oal is to et a eel or how the new network will behave rather than to model an actual system. As or modal split, this is actually taken care o by the network itsel. Recall throuh Fiure 3- that separate uideway and reeway lanes are provided or both cars and trucks and buses. By havin the car traic separated rom the truck and bus traic, modal split is actually perormed between the respective roadways. 67

2 This leaves trip distribution and traic assinment. Trip distribution is used to ind out where the trips loaded at each production node are destined. This can then be used to estimate the number o trips on a link i perormed alon a sinle route (i.e. allowin no route choice). Traic assinment is used to decide which route a trip will be made on. Usin one o two available methods user equilibrium and system optimization the link volumes over an entire network can be ound. In the case o an AHS parallel to the existin Interstate Hihway System, the results o trip distribution and traic assinment are o interest. 5. Trip Distribution Here trip distribution will be used to determine what the link volumes between interchanes on the AHS uideway would be. Assume that interchanes alon the AHS are equally spaced and Q r vehicle trips per hour are enerated at each interchane Q r in each direction. Recall that Q r is the ramp capacity o an interchane, ound in Section 4.4. The ramp traic then proceeds to travel downstream and exits at its destination interchane. The number o trips that exit at each interchane is inversely proportional to the number o interchanes between the oriin and destination interchanes (see Fiure 5-1). This can be mathematically expressed as where Q = q + q + q + + q = q r 1 3 n k k= 1 q k n (5.1) q (5.) = 1 x k and x is a measure o the importance o proximity to the traveler and is a unction o the trip purpose. Substitutin Equation (5.) into Equation (5.1) and solvin or q 1 : The link volume, q, between adjacent interchanes can be expressed as q 1 = n Q k = 1 r k x (5.3) 68

3 Q r q 1 q q 3 q q 3 q 3 Trip Distribution Analysis Fiure

4 q = n k= 1 kq k Then by substitutin Equation (5.) into Equation (5.4) n 1 1 q = q k x k= 1 Substitutin Equation (5.3) into Equation (5.5) inally results in q = Q r n k= 1 n k= 1 k k 1 x In Fiure 5-, q/q r is plotted aainst n or various values o x. x (5.4) (5.5) (5.6) It can be seen rom Fiure 5- that the uideway link volume q increases as the lenth o trip n increases. This is especially true or low values o x. As said beore, x is a measure o the importance o proximity to the driver, and is a unction o trip purpose. For example, i the trip is or work, a person is just as willin to drive a lon distance as a short one i.e. the proximity to the workplace is not an issue. Meanwhile, or shoppin a person is much more likely to o to a nearby location than to drive a lon distance to shop. Thereore proximity is an issue when discussin a shoppin trip. To represent these dierences, each trip type is iven values o x which best demonstrate its pattern. Historically, x < 1 has been used or work trips; 1 x or school and shoppin trips; and x > or social and recreational trips. By examinin the equation or q/q r, obtained rom Equation (5.6), the requirement or converence can be seen to be x. Now look at the case where there is a minimum trip lenth o a interchanes on the uideway. Then where n r a a+ 1 a+ n k k= a Q = q + q + q + + q = q (5.7) 70

5 16 x = x = ½ Volume Ratio, q/q r x = 1 4 x = x = Lenth o Trip, n (No. o Interchanes) Volume Ratio versus Lenth o Trip Fiure 5- x = trip purpose coeicient 71

6 q k qa = ( k a+1) x (5.8) Then the link volume, q, between interchanes can be expressed as q = n k= a By substitutin Equation (5.8) into Equation (5.10) q = q n a k= a kq k k ( k a+ 1) Substitutin Equation (5.9) into Equation (5.11) inally ives q = Q r n k= a n k= a k ( k a+ 1) ( k a+ 1) x x x (5.10) (5.11) (5.1) In Fiure 5-3, q/q r is plotted aainst n or various values o a usin x =. While trip distribution was able to ind what the maximum link volumes would be dependin on what the ramp capacity is, there is no way to determine whether the vehicle trips would use the uideway over the reeway, especially or short trips. Thereore traic assinment will be used to determine how vehicle trips will distribute themselves over the uideway and reeway or a trip with the same oriin and destination. 5.3 Traic Assinment While in the previous section it was assumed that the volume o traic enterin the uideway was equivalent to the ramp capacity, the actual volume which will access the uideway will depend on the volume o traic enterin the uideway-reeway system and how these vehicles assin themselves to either the reeway or the uideway to reach their ultimate destination. This assinment decision is based mostly on travel time which is dependent on the characteristics o the transportation network. This section will demonstrate the interactions and equilibrium 7

7 7 a=5 6 a=4 Volume Ratio q/q r a=3 a= a= Lenth o Trip, n (no. o interchanes) Relationship Between Volume Ratio and Lenth o Trip Fiure

8 between the uideway and the reeway or both cars and trucks, and will show the dynamic character o the system that causes adjustments to establish new equilibrium states based on controlled chanes to land-use activity and/or to the uideway-reeway interace. In relation to the proposed uideway-reeway system, traic assinment becomes the splittin o the entrance ramp volumes at each interchane into the uideway entrance ramp volume, Q r, and the reeway entrance ramp volume, Q r, so that Q + Q = Q (5.13) r This route choice is a classic equilibrium problem, since travelers route choice decisions are primarily a unction o route travel times, which are determined by traic low. Traic low is itsel a product o route choice decisions. This relationship between travel time and traic volume, called the hihway supply unction, has many mathematical orms that have evolved over the years. r r Two theoretical approaches to obtainin a mathematical expression o the hihway supply unction are: (1) continuous low phenomenon such as in a one-dimensional compressible luid, and () discrete low usin queuin theory as a oundation. The ormer ives rise to a relationship between speed and low on a conventional roadway that is parabolic in nature, with an increasin reduction in speed as the volume approaches the roadway s capacity. However, when the reciprocal o speed is taken, it has been ound that the travel time ound is siniicantly underestimated. Blunden [] has applied the second approach to the development o the ollowin supply unction: T = T ( j) ρ ρ (5.14) where T is the travel time, T is the ree-low travel time, j is the level o service actor, and ρ is the volume to capacity ratio, q/q. In relation to the proposed system, T and T would be travel times between interchanes. 74

9 In applyin the above model, consider the ollowin scenario. Both the uideway and reeway lanes run parallel thus providin travelers with the choice o two trips. Assume that the minimum lenth o a uideway trip is a = 3. The section bein analyzed will consist o our interchanes with eastbound traic enterin at interchane 0 destined or interchane and westbound traic enterin at interchane 3 destined or interchane 1. A directional split will be assumed, so that the number o vehicles travelin rom interchane 0 to interchane is equal to the number o vehicles travelin rom interchane 3 to interchane 1. The choice or eastbound travelers is either to travel links 0-1 and 1- on the reeway or 0-1, 1-, and -3 on the uideway and then 3- on the reeway. The choice or vehicles enterin at interchane 3 and travelin west is similar, but reversed. Fiure 5-4 sketches out the hypothetical network. O interest in the scenario is to ind the link volumes usin two theories o travel route choice: user equilibrium and system optimization. With user equilibrium a traveler selects his or her route by which route would have the shortest travel time. The aect on the other users o the system is not considered. The ultimate result is that as lon as vehicles are usin all available routes the travel times on all those routes would be equal. Under system optimization the traveler is assined to the path which increases the total system travel the least. In this instance the vehicle would not necessarily be assined to the route with the best travel time or itsel as in user equilibrium. Also, the travel times on dierent routes do not have to be equal like in user equilibrium, even i vehicles are loaded onto all the available routes. The travel times which will be used in the example oriinate rom the hihway supply unction shown in Equation (5.14), and are: T c T T c t = T = T = T ( ) Qc jc qc 1 (5.15) Qc q c ( ) Qc jc qc 1 (5.16) Qc q c ( ) Qt jt qt 1 (5.17) Qt q t 75

10 Q r Q r Q r Q r Q r Q r Q r Q r Q r Q r Q r Q r Q r Q r Guideway Freeway Sample AHS Network Fiure

11 T t = T ( ) Qt jt qt 1 (5.18) Qt q t where T j i = travel time on route j by vehicle type i T j = ree-low travel time on route j Q j i = capacity o route j or vehicle type i q j i = volume o vehicle type i on route j j j i = level o service actor or route j or vehicle type i The indices i and j can be classiied as ollows: vehicle type i can be either car (c) or truck (t), and route j can be either reeway () or uideway (). The irst point to notice about the system is the symmetry o the network (see Fiure 5-4). Because loadin is identical at both interchane 0 and interchane 3, the link volumes throuhout the system will be symmetric. That is, the volume on reeway link 0-1, q 0-1, is equal to the volume on reeway link 3-, q 3-. Also note that the link volumes on the entire uideway in both directions are equal. Because o this and that the minimum trip on a uideway is three interchanes, the uideway links 0-1, 1-, and -3 will be considered as one link, 0-3. Likewise in the other direction the uideway will be called link 3-0. By examinin the symmetry o the network, the ollowin relations can be ound: q = q = Q (5.19) r q = q = Q (5.0) 1 1 r q = q = Q + Q (5.1) q1 0= q 3= 0 (5.) Due to this symmetry, all o the ollowin work will be done with traic travelin in the eastbound direction, namely rom interchane 0 to interchane. The reader should realize that all the results obtained hereater apply in the westbound direction also. r r First the theory o user equilibrium will be explored. As stated beore, the concept is to balance the travel times on the two routes so that they will be equal. Thereore the travel time alon the 77

12 uideway link 0-3 plus the reeway link 3- must be equal to the travel time alon the reeway link 0-1 plus the reeway link 1-. Mathematically, it would be shown as T + T = T + T (5.3) Note rom Equation (5.1) that T 3- = T 0-1 so that we end up with T = T (5.4) or user equilibrium to exist. To model this system vehicles were entered into the network sinly, each one checkin what the existin travel times were at that time with the traic loadin that was present rom those who entered the system beorehand. For example, i a vehicle prepares to enter the system and inds that the uideway route has the shorter travel time, that vehicle is added to the volume on that route. The next vehicle is then prepared to be loaded. When the travel times are checked, this time the reeway route may have the shorter travel time as the vehicle just loaded has increased the uideway volume and thereore the uideway travel time. This model was run or both car and truck systems usin the ollowin values or the reeway and uideway characteristics: or cars T c = 6.0 min., Q c = 400 veh/hr, j c = 0.5, T c = 3.6 min., Q c = veh/hr, and j c = 0.10; or trucks T t = 6.0 min., Q t = 100 veh/hr, j t = 0.50, T t = 5.4 min., Q t = 1000 veh/hr, and j t = 0.0. The results o the model were outputted as plots o the uideway volume Q r and the reeway volume Q r versus the total volume Q r or cars in Fiure 5-5 and trucks in Fiure 5-6. By settin the uideway travel time to ree low travel time (the reeway travel time when there are no vehicles loaded) and solvin or the uideway volume it was ound that Q r Q T 1 T = T 1+ j T (5.5) beore any vehicles bein to choose the reeway as a route. As a note, the model was set up so that i a vehicle was to encounter identical travel times on the two routes, the vehicle would select the route with the shorter distance, namely the reeway route. Usin the above 78

13 x 104 Guideway Traic Volume, q (vph) Total Traic Volume, q (vph) x Freeway Traic Volume, q (vph) x 10 4 Total Traic Volume, q (vph) Traic Assinment o Cars Under User Equilibrium Conditions Fiure

14 Guideway Traic Volume, q (vph) Total Traic Volume, q (vph) 800 Freeway Traic Volume, q (vph) Total Traic Volume, q (vph) Traic Assinment o Trucks Under User Equilibrium Conditions Fiure

15 parameters, it was ound that this required uideway loadin was Q r = 6087 veh/hr or cars and Q r = 486 veh/hr or trucks. Once the loadin on the uideway reaches that point, the trips bein to be distributed amon both routes as can be seen in Fiures 5-5 and 5-6. Also by examinin the raphs one will notice that once loadin has beun on both the routes, the pattern appears to ollow a quadratic orm. It would be o interest to ind the unction or which this loadin ollows to help ain an understandin o the relationships between the uideway and reeway ramp volumes, Q r and Q r, and the total ramp volume, Q r. Three orms o equations were itted: q a bq j = + (5.6) q j = a+ bq+ cq (5.7) q j = a+ bq (5.8) where j indicates the route type. These equations were it usin the least squares method with the results bein shown in Appendix A in Fiures A-1 throuh A-6 or cars and Fiures A-7 throuh A-1 or trucks. The reression curves are plotted as a dashed line while the actual data curve is shown as a solid line. Upon examinin the curves one will reconize that the itted curves created by Equation (5.7), in Fiures A-, A-5, A-8, and A-11, best estimate the curves created by the model. Statistical analysis o the reression curves conirmed that the orm o Equation (5.7) best explained the relationship between the uideway and reeway volumes and the total volumes. The our equations or this network are q = q q (5.9) c q = q q (5.30) c q = q q (5.31) t q = q cq (5.3) t where Equations (5.9) and (5.30) are or car volumes on the reeway and uideway, respectively, and Equations (5.31) and (5.3) are or truck volumes on the reeway and uideway, respectively. It is interestin to note how once loadin beins on both routes, the reeway appears to et more loadin relative to its capacity than does the uideway. But as 81

16 loadin continues to increase the volumes closer to capacity, the uideway aain takes most o the traic loadin. While as stated above the hiher order models shown in Equations (5.9) throuh (5.3) are statistically more siniicant, they don t provide a practical relationship or use. It is thereore useul to have the linear relationships available or use as they are also much more appealin conceptually. The linear orm or cars is converted to the orm q c ( q ) = (5.33) and is plotted in Fiure 5-7. The linear orm or trucks is converted to the orm q t c ( q ) = (5.34) t and is plotted is Fiure 5-8. The equations suest that traic beins to divert rom the two uideways when traic volumes on the uideways reach 5700 cars per hour and 4800 trucks per hour. While user equilibrium is an interestin case to examine in act necessary to examine since it is in exactly this manner that today s roadways are loaded the introduction o automation to the transportation system provides the opportunity to actually put system optimal operations into aect. System optimal operations actually attempt to achieve the lowest possible travel time or the transportation system as a whole. The problem with implementin this in the past has been the requirement that some drivers actually access the route with the loner travel time, somethin that is hard, i not impossible, to et done. With the automated vehicles it is possible to do just that and ultimately achieve a system optimal operation. When modelin the user equilibrium scenario the decision process upon enterin the system was straiht-orward the vehicle was assined the route with the lowest travel time. With system optimal operations the decision process becomes more complicated. Each vehicle must analyze the system and decide which route would increase the total travel time by the least amount i it was to join it. To develop the unctions needed to allow a comparison to be made, start with the minimization unction: 8

17 Freeway Traic Volume, q (vph) x 10 4 Total Traic Volume, q (vph) Freeway Traic Volume as a Function o the Total Traic Volume For Cars Under User Equilibrium Conditions Fiure

18 Freeway Traic Volume, q (vph) Total Traic Volume, q (vph) Freeway Traic Volume as a Function o the Total Traic Volume For Trucks Under User Equilibrium Conditions Fiure

19 min T = tq n i= 1 i i (5.35) where i is the route and n is the number o routes in the network, in this case two. The hihway supply unctions rom Equations (5.15) and (5.16) are inputted or t, thus ivin T = T ( 1 ) ( 1 ) Q j q Q q q + T Q j q Q q Takin the partial derivatives o Equation (5.36) in terms o q and q and simpliyin produces T q T q = T = T ( ) ( ) + q ( Q q) q ( ) Q j q q Q j q 1 1 Q Q ( ) ( j ) q ( ) + q ( Q q) 1 q Q 1 j q Q (5.36) (5.37) (5.38) These two equations become the decision unctions or the system optimal loadin. Equations (5.37) and (5.38) determine the chane in T in terms o q and q. The route with the smallest chane is the one which will receive the next vehicle enterin the system. As with user equilibrium, the loads on both the uideway and reeway routes are plotted aainst the total traic volume, in Fiure 5-9 or cars and Fiure 5-10 or trucks. The values used in the user equilibrium example are also used here. As expected, the traic initially enters onto the uideway and continues to do so until the system travel time chane is equal or both routes. To ind this point Equation (5.37) was set equal to six (the value o Equation (5.38) with q = 0). Upon solvin the resultin quadratic traic volumes o q = veh/hr or cars and q = 3779 veh/hr or trucks were ound to satisy the conditions. Aain the volume must reach lare values beore any loadin will even bein on the reeway. Once loadin does bein on both routes simultaneously, the load curves aain resemble a parabolic curve. Due to the similarities between the nature o the curves ound under user equilibrium and system optimization the same basic orms, Equations (5.6), (5.7), and (5.8), were used to it these curves. Aain the least squares method o reression was applied with the results bein shown in Fiures A-13 throuh A-18 or cars and Fiures A-19 throuh A-4 or trucks. Upon examinin the curves, and 85

20 .6 x Guideway Traic Volume, q (vph) Total Traic Volume, q (vph) x Freeway Traic Volume, q (vph) Total Traic Volume, q (vph) x 10 4 Traic Assinment o Cars Under System Optimal Conditions Fiure

21 Guideway Traic Volume, q (vph) Total Traic Volume, q (vph) Freeway Traic Volume, q (vph) Total Traic Volume, q (vph) Traic Assinment o Trucks Under System Optimal Conditions Fiure

22 throuh statistical analysis, the reression curves portrayed in Fiures A-14, A-17, A-0, and A- 3 did the best job o explainin the relationship between the reeway and uideway volumes, q and q, and the total traic volume, q. The resultin equations are q = q q (5.39) c q = q q (5.40) c q = q q (5.41) t q = q q (5.4) t where Equations (5.39) and (5.40) are or car volumes on the reeway and uideway, respectively, and Equations (5.41) and (5.4) are or truck volumes on the reeway and uideway, respectively. One will notice that aain the reeway takes on most o its loadin in the beinnin and beins to level o at the end, while the uideway continuously increases the rowth in traic volume. Aain, while the hiher order models are more statistically siniicant, the linear relationships have more conceptual appeal and are more practical or application and use. The linear relationship or cars is converted to the orm q c ( q ) = (5.43) and is plotted in Fiure The linear orm or trucks is converted to the orm q t c ( q ) = (5.44) t and is plotted in Fiure 5-1. The equations suest that traic beins to divere rom the uideway to the reeway when traic volumes on the uideway reach cars per hour and 800 trucks per hour under system optimal conditions. Applyin the two dierent theories o traic assinment to the network brouht one clear issue to the oreront the volume needed to utilize both routes. The ability to obtain such an accurate curve when compared to the actual loadin curve created by the models is a bit surprisin but not unounded. The model is ultimately based on a ew equations which when compared with one another decided which route a iven vehicle would be loaded onto. Because o this there is a stron relationship between the decision unctions and the inal output curve. 88

23 Freeway Traic Volume, q (vph) x 10 4 Total Traic Volume, q (vph) Freeway Traic Volume as a Function o the Total Traic Volume For Cars Under System Optimal Conditions Fiure

24 Freeway Traic Volume, q (vph) Total Traic Volume, q (vph) Freeway Traic Volume as a Function o the Total Traic Volume For Trucks Under System Optimal Conditions Fiure

25 Since the curve was ultimately created by equations, it is reasonable to expect that there exists a unction which would match its behavior extremely closely. 91