Improving convergence of quasi dynamic assignment models

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1 Imroving convergence of quasi dynamic assignment mels Luuk Bredere,*,,2, Adam Pel, Luc Wismans 2,, Erik de Romh Deartment of Transort & Planning, Delft University of Technology 2 DAT.mobility a Goudael Comany Centre for Transort Studies, Faculty of Engineering Technology, University of Twente * Corresonding author: Extended abstract submitted for resentation at the 6 th International Symosium on Dynamic Traffic Assignment (Sydney, 28-0 June, 206) Intruction For decades congestion levels around the world are rising. To roerly incororate the effects of congestion into strategic transort mels, a shift from static caacity restrained towards caacity constrained and dynamic traffic assignment mels has occurred. In this aer we focus on quasi dynamic assignment mels (more secific: static-caacity and storage constrained mels by the definitions in Bliemer et al (205)). These mels exlicitly cature the flow metering and sillback effects of congestion, but assume stationary demand during a single time eri (e.g. a whole eak hour) and are therefore more scalable and mathematically tractable, both imortant roerties for strategic transort mels. Although comutational caabilities of current hardware allow for large scale alication of such mels, the incororation of caacity constraints causes route cost functions to be much more sensitive and to be insearable over sace (the latter occurs when routes share bottleneck nes). Furthermore, the incororation of storage constraints further increases insearability (which occurs when queues sill back onto ustream links) and causes cost functions to become imlicit. As such quasi dynamic mels do not fully contain the favorable mathematical roerties that are exloited in many algorithms to solve their caacity restrained counterarts and in fact do not necessarily comly with the requirements for existence and/or uniqueness of the user equilibrium (theorems.4 and.8 in Nagurney (99)). Although in reality these unfavorable roerties exist, a substantial by of research suggests that their (satial) occurrence is limited and as such have minimal ractical temoral and satial consequences (Peeta and Zilliaskooulos (200)). However, several large scale alications using the quasi dynamic assignment mel STAQ (first described in Bredere et al (200)) have shown that esecially the addition of storage constraints causes oor or non-convergence in real world alications. Further investigations in this aer will show that also the caacity constraints on their own can cause serious convergence issues. Contributions in this aer are (i) to give an overview of meths in literature and logical extensions to those meths that could imrove convergence of quasi dynamic assignment mels, (ii) to reveal and illustrate mechanisms that cause the convergence issues using examles on theoretical networks and (iii) to investigate to what extent enhancements to existing algorithms can be used to (artly) get around the convergence issues encountered. Ultimately, this research should lead to a meth that generically solves quasi dynamic assignment mels. 2 Methology The scoe of the research is narrowed down by assuming that (i) a ath based mel is used and that (ii) travelers have ercetion errors on route travel times, leading to the stochastic user equilibrium (SUE). We choose to use the multinomial logit (MNL) mel to calculate route choice robabilities, such that route demand is defined by: f = ex( µ c ) / ex( µ c ) D P, () where is the route cost on route, is the scale arameter describing the degree of travelers ercetion errors on route travel times (where erfect knowledge is assumed when aroaches infinity) and is the travel demand for OD air. Here (and in most real world alications)

2 a global scale arameter is normalized over ODairs by = / min, where is the free flow cost on route. This normalization ensures that the relative effect of ercetion errors is the same on all ODairs (regardless of their average route travel time). As measure of convergence we use the ga function derived in Bliemer et al (20) that will reach zero uon convergence when using MNL: G ( o, d ) P = f c f ( + µ ln ψ ) ( o, d ) D ψ, (2) where = min[ + ln ] reresents the minimum stochastic ath cost. Note that by omitting the summation over OD airs in both enumerator and denominator, the ga value for a single OD can be obtained, useful when investigating which ODairs cause convergence issues. In order to enforce and seed u convergence, stochastic route based traffic assignment mels tyically average route demands over iterations using some averaging scheme that alies iteration secific ste sizes. To ensure that the averaging scheme itself does not cause divergence, the conditions = and = must hold (Blum (954)) meaning that the ste sizes must decrease in every iteration. The simlest averaging scheme tested is the meth of successive averages (MSA) that uses =/. A well-known roblem of MSA is that convergence tends to slow down as the number of iterations increases. Tested variations on MSA that give more emhasis on later iterations using redetermined ste sizes include raising to some ower <, to reset every other iterations while maintaining current route costs (MSA-reset), to use constant ste sizes or to use = /.. (MwSA, Liu et al (2007)) where is a constant. More intelligent averaging schemes tested use information of revious iteration(s) to determine are the Self Regulating Average (SRA, Liu et al (2007)) and SRA with dynamic ste size (Taale and Pel (205)). 2. Test Cases For all test cases (an udated version of) the quasi dynamic assignment mel STAQ (Bredere et al (200)) was used. STAQ uses a static mel with caacity constraints to determine flow metering effects and combines this with the dynamic link transmission mel (LTM) to determine average sillback effects. Both submels use the ne mel described in Tamere et al (20). Its low comutational cost allows for quickly running a lot of convergence schemes. Because the mel basically is a static version of the LTM, results described in the remainder of this aer are likely to be also alicable on fully dynamic ath based assignment mels with stochastic route choice. To investigate the mechanisms that cause convergence issues a generic test network dislayed in Figure is constructed. In this network the effects on convergence of adding caacity and storage constraints can be isolated because the two asects of interest (sensitivity and extent of insearability of the imlicit cost functions, see section ) can be controlled by adjusting link lengths. Assuming a demand of 2000 veh on both OD airs and free flow link seeds and caacities defined on the right side of Figure, the ustream nes of links 5 and 7 form otential bottleneck locations. For both ODairs two routes exists: a constrained route using a otential bottleneck ne and an unconstrained (direct) route. 2

3 2 Link Length (l) Free seed (v) Caacity (C) # [km] [km/h] [veh/h] O D l l l -l L D l -l Figure : generic network geometry (left) and link attributes (right) The lengths of links, 4 and 6 are variable, while the definitions of link lengths 5 and 7 ensure that in all secific networks derived from this generic network, the constrained routes have lower free flow travel time than the constrained route, meaning that in all secific networks, both bottlenecks are activated under SUE conditions. Three secific networks are considered: The Indeendent network, in which l =0, l 4>0 and l 6>0. In this network all routes are indeendent The Deendent network, in which l >0, l 4=0 and l 6=0. In this network link is shared by two routes. The Sillback network, in which l >0, l 4>0 and l 6=0. In this network l 4 is chosen such that sillback from link 4 onto its ustream link () occurs under SUE conditions. The influence of insearable route costs due to caacity constraints (routes sharing a bottleneck) can be examined by comaring assignment results on the deendent with the indeendent network, whereas the influence of satial insearability of the cost function due to storage caacity constraints (sillback on ustream links) can be examined by comaring assignment results on the sillback network with the deendent network. It is exected that increased insearability worsens convergence and thus that the indeendent network will converge best, the deendent network will converge worse and the deendent network with sillback will converge worst. The influence of sensitivity of the imlicit cost functions has been examined by scaling all link lengths, thereby controlling the ratio between free flow travel time and travel time in congestion, while maintaining the same solution under free flow conditions. The extent to which results are transferable into real size networks has already artly been examined by alication on a network of the city of Den Bosch (the Netherlands) consisting of 50 centroids, 8500 links, 7000 nes and routes. Because this is still work in rogress and due to sace constraints, both analyses will not be discussed here. Preliminary results and conclusions (existing meths) All averaging schemes mentioned in section 2 where imlemented and tested using recommended arameter values rovided in literature. When ranked by the number of iterations needed to achieve true SUE conditions (ga value to machine recision) SRA roved to be the best meth on all networks. Results are dislayed on the left hand side of figure 2, MSA results where added as a reference. Considering the left art of figure 2, the ga values show oscillations in the first 9-22 iterations until a value around E-02 is reached. Further investigation showed that these oscillations are formed due to iterations in which the averaging scheme overshoots, causing the bottleneck on (one of) the constrained routes to deactivate, and thus become inconsistent with the state under SUE conditions. Only after this unstable hase the correct state is maintained and convergence accelerates and smoothens. Because MSA takes less iterations to stabilize into the correct state, it outerforms SRA until about iteration 24-0, after which SRA clearly takes over. Aarently, SRA suffers from using the information from iterations in the unstable hase, causing the ste sizes to be decreased too much. After reaching the stable state, SRA maintains relatively large ste sizes whereas MSA s continue to decline slowing down convergence.

4 Comaring the different networks, both MSA and SRA runs show unexected results: using MSA the deendent network shows better convergence than the indeendent network, whereas using SRA, the deendent network with sillback shows better convergence than the deendent network. Additional runs in which demand was increased to ensure that bottlenecks are never deactivated (and thus no unstable hase occurs) did show exected results and much better convergence for all networks and averaging schemes, suggesting that the extent to which the unstable hase occurs determines the seed of convergence in later iterations. 4 Preliminary Results and conclusions (enhanced meths) Based on test results (artly) discussed in section two enhancements to SRA are roosed and tested. Analysis of ga values er OD showed that the least converging ODair contributes the most to oor ga values. Therefore we roose a new averaging scheme called SRA-ODsecific, which determines OD air secific ste sizes, giving SRA a higher degree of freedom during ste size otimization. Furthermore, realizing that the normalization of the scale arameter based on free flow instead of congested cost yields too large scale arameters and thus too high sensitivity of the route choice mel on congested ODairs, we roose to normalize the scale arameter in each iteration based on the current maximum route cost for the considered ODair. We choose to use the maximum (not the weighted average) route cost because this will stabilize the most sensitive ODairs the most, since differences between minimum and maximum route costs are the largest on those ODairs. After the instable hase, differences between minimum and maximum route cost will decline and the overcomensation of the normalized scale arameter automatically diminishes. Additional test runs confirmed that using the maximum route cost clearly outerforms using the (weighed) averaged or minimum route cost for normalization of the scale arameter. Figure 2: Convergence on test networks (left: MSA vs SRA, right: enhanced meths) Results of the enhanced meths are dislayed on the right hand side of figure 2. Comared to SRA, SRA-secific accelerates convergence on the indeendent network after iteration 50, reaching machine recision in just over 20 iterations. This big imrovement makes sense when realizing that on the indeendent network, routecost is searable over both ODairs, which is exactly what SRA- ODsecific imlicitly assumes. On the other networks, routecost is insearable over ODairs, which exlains why SRA outerforms SRA-secific on these networks. Normalization of in each iteration in combination with SRA-ODsecific shortens the stabilization hase on the Indeendent and Sillback networks, thereby greatly imroving convergence. On the Deendent network, the negative effect of SRA-secific aarently outweighs any otential ositive effect of normalization, indicated by the slightly worse erformance comared to SRA-secific. Therefore, for this network a run with normal SRA and normalized was run (dotted blue line in right hand side grah) which shows that also for this network, normalizing can indeed imrove results (comare to SRA). 4

5 5 Overall conclusions and further research directions In this abstract meths in literature and logical extensions to those meths that could imrove convergence of quasi dynamic assignment mels where investigated. Two imortant mechanisms causing convergence issues where identified (existence of an instable hase and satial insearability of route cost functions) using examles on theoretical networks and two enhancements to existing meths where roosed and successfully demonstrated. Based on this research, novel meths that will be tested in the full study include SRA alied to ODairs clustered by usage of bottlenecks, two hybrid forms of MSA and SRA, diagonalization and/or damening of sillback effects, caing routecost for routes where the ratio between time in congestion and total travel time is too high and discarding diverging iterations instead of only lowering their ste size (as SRA does). References Bliemer, M.C.J, Raadsen, M., de Romh, E., Smits, E.-S. (20) Requirements for Traffic Assignment Mels for Strategic Transort Planning: A Critical Assessment, in Australasian transort research forum 20 roceedings, 2-4 october 20, Brisbane, Australia Bliemer, M., Raadsen, M. Bredere, L, Bell, M., Wismans, L. (205) genetics of traffic assignment mels for strategic transort lanning. In Australasian Transort Research Forum 205 Proceedings Bredere, L.J.N., M.C.J. Bliemer, L.J.J. (200) STAQ Static Traffic Assignment with Queuing - Proceedings of the Euroean Transort Conference 200 Liu, He and Bingsheng 2007 meth of succesive wheighted averages MSWA and self regulated averaging schemes for solving stochastic user equilibrium roblem Nagurney, A. (99) Network economics: a variational inequality aroach. Kluwer Academice Publishers, Boston, USA. Peeta, S. and Ziliaskooulos A.K. (200) Foundations of Dynamic Traffic Assignment:The Past, the Present and the Future. Networks and Satial Economics, 200 (), Taale, H. and Pel, A. (205) Better Convergence for Dynamic Traffic Assignment Meths, resented at the 8 th euro working grou on transortation, EWGT 205, 4-6 july 205 Tamere. C.M.J., Corthout R., Cattrysse D., Immers, L.H. (20). A Generic Class of First Order Ne Mels for Dynamic Macroscoic Simulation of Traffic Flows. Transortation Research Part B: methological. Volume 45B issue, 20,

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