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J. Cent. South Univ. (2013) 20: 2897−2904 DOI: 10.1007/s11771­013­1811­5 Modified stochastic user­equilibrium assignment algorithm for urban rail transit under network operation ZHU Wei(朱炜) 1, 2, HU Hao(胡昊) 1, XU Rui­hua(徐瑞华) 2, HONG Ling(洪玲) 2 1. Center of Transportation Research, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China; 2. School of Transportation Engineering, Tongji University, Shanghai 201804, China © Central South University Press and Springer­Verlag Berlin Heidelberg 2013 Abstract: Based on the framework of method of successive averages (MSA), a modified stochastic user­equilibrium assignment algorithm was proposed, which can be used to calculate the passenger flow distribution of urban rail transit (URT) under network operation. In order to describe the congestion’s impact to passengers’ route choices, a generalized cost function with in­vehicle congestion was set up. Building on the k­th shortest path algorithm, a method for generating choice set with time constraint was embedded, considering the characteristics of network operation. A simple but efficient route choice model, which was derived from travel surveys for URT passengers in China, was introduced to perform the stochastic network loading at each iteration in the algorithm. Initial tests on the URT network in Shanghai City show that the methodology, with rational calculation time, promises to compute more precisely the passenger flow distribution of URT under network operation, compared with those practical algorithms used in today’s China. Key words: urban rail transit; stochastic user equilibrium; assignment algorithm; method of successive averages; network operation too obvious to be neglected in influencing passengers’ 1 Introduction behaviors, and passengers may choose different routes for the same origin−destination (O−D) pair. Passenger flow is the foundation of making and 2) Under the condition of network operation, coordinating operation plans for urban rail transit (URT) whether a route is chosen by a URT passenger is systems. Models to solve passenger flow assignment subjected to not only the network’s topology and problems can be classified according to whether different travel costs but also the rail lines’ service time. Wardrop’s principle is followed. One model is the In other words, during some certain period of time, a non­equilibrium assignment, and the other is the route cannot be taken part in the passenger flow equilibrium assignment model which is sounder in theory assignment and thus will be excluded in the route choice and can be applied into a large­scale network with rapid set in case of some rail lines of it beyond their service development of computer technology. time. Currently, with the fast development of economy, Moreover, it is assumed that passengers’ choice the URT systems of big cities in China, such as Beijing, processes in reality have more or less random Shanghai and Guangzhou, have entered into the new characteristics because of imperfect knowledge of travel stage of network operation with technology for time, individual differences, measurement errors, and so one­ticket transfer, whose characteristics with regard to on [1−3]. Therefore, confronting with today’s URT passenger flow assignment are concluded as follows: systems in China, the result from passengers’ route 1) The dramatic expansion of those cities’ URT choices can be described more appropriately by the networks has enabled more flexible and complex travels stochastic user­equilibrium (SUE) with time constraint, within the URT systems because more and more transfer which is also proved by some simulation experiments stations have provided better connecting services. On the [4−5] and full­scale case tests [6]. other hand, most of the URT lines have been overloaded The SUE problem has been studied for a long time. since travel demand increases rapidly in these Thorough reviews were presented in part of the fast­expanding cities. Hereby, the factor of crowding is literature [7−10]. It can be solved either in the space of Foundation item: Project(2007AA11Z236) supported by the National High Technology Research and Development Program of China; Project (2012M5209O1) supported by China Postdoctoral Science Foundation Received date: 2012−04−23; Accepted date: 2013−03−25 Corresponding author: ZHU Wei, PhD; Tel: +86−21−62933091; E­mail: [email protected] 2898 J. Cent. South Univ. (2013) 20: 2897−2904 rs link flows or in the space of path flows [11−12]. Among min Z ( X ) = - q rs S rs [ c ( X )] + x å various solution algorithms, the well­known method of rs x successive averages (MSA) developed by SHEFFI and x t ( x ) - a t (w ) d w (6) å a a a åò 0 a POWELL [13] is the first algorithm applied to solve the a a SUE problem and can be applied with any stochastic s.t. Eqs. (3)−(5). network loading method. Especially, the MSA is widely where ta(ω) is the unit generalized travel cost on link a, it considered as a kind of efficient algorithm for two is the function of the flow (ω) on link a. c rs( X) is the reasons [1]. First, the algorithm will converge if the vector of perceived generalized travel cost for the O−D search direction is a descent vector only on the average. rs pair (rs). Srs[c (X)] is the expected perceived generalized Second, the algorithm is based on a predetermined travel cost for the O−D pair (rs). sequence of move sizes along the descent direction so The above description is a SUE model and there are that the difficult calculation of objective function can be several features needed to be noticed when it is applied avoided. Even though the algorithm has been applied in into URT networks: the urban transit analyses extensively in the past, few 1) The generalized travel cost is affected studies have analyzed it with regards to a URT network considerably by in­vehicle congestion rather than [6, 14]. The objective of this work is to propose a vehicle­to­vehicle congestion; modified stochastic user­equilibrium assignment 2) The route­based assignment algorithm is algorithm based on the framework of MSA, which can easier to be implemented due to both of less complex solve passenger flow assignment problem more network compared to road network and time constraint appropriately and precisely for the URT systems under under network operation; network operation especially in today’s China. 3) The transferring time at the transfer station influences the route choices of rail passengers 2 SUE assignment model for URT network considerably. The SUE model is a kind of assignment model 3 Framework of modified algorithm based based on the principle of stochastic user equilibrium, on MSA which is described as: in a stochastic user equilibrium network no user believes he/she can improve his/her The basic algorithm step of the SUE problem above travel cost by unilaterally changing routes. is to search for the descent direction and move size, both The conditions of the SUE are formulated as of which are difficult to be found. However, MSA can be f rs (C rs - C rs ) = 0 (1) used to solve the SUE problem despite the difficulties k k above. The descent direction and move size of the MSA rs rs C k - C ≥ 0 (2) can be written as follows: f rs ≥ 0 (3) 1 k l n = (7) rs n where C k is the perceived generalized travel cost on rs n n n route k for the O−D pair (rs) in equilibrium. C k can be d =y −x (8) expressed as C rs = c rs + e rs , where c rs is the actual k k k k y n = q × P rsn × d rs (9) travel cost and e rs is the random component. C rs is the a åå rs k a , k k rs k perceived generalized travel cost for the O­D pair(rs) in n equilibrium and is the expectation of { C rs }. f rs is the where n is the number of iterations; y is the auxiliary k k n number of trips on route k for the O−D pair (rs). link flows vector at the n­th iteration; y a is the auxiliary rsn Moreover, there are two important relationships can be flow on link a at the n­th iteration; Pk is the choice respectively shown as probability of route k for the O−D pair (rs) at the n­th iteration; d rs =1 if link a is on route k, otherwise f rs = q (4) a, k å k rs d rs =0; xn is the link flows vector at the n­th iteration; λn k a, k is the move size; and dn is a descent direction vector rs rs n x a = ååå f k × d a , k (5) computed at x . r s k Based on the conventional MSA, the modified where qrs is the number of trips for the O−D pair (rs). xa algorithm’s framework is summarized by Algorithm 1. rs is the flow volume on link a. d a ,k =1, if link a is on rs route k for the O−D pair (rs); otherwise, d a ,k =0. Algorithm 1: Modified algorithm for SUE model based The equations above can be solved by indirect on MSA solution where a minimization program is expressed as 1 Input: URT network, rail­use O−D matrix, and all J. Cent. South Univ. (2013) 20: 2897−2904 2899 parameters expressed as 2 Output: link flows vector x n in stochastic user­ T rs = a ×(t rs + t rs ) (11) equilibrium 2,k,m 1,k,m, pq 2,k, q n rs 3 Initialize n←0, t a ←0 where t is the walking time for transferring from n 1,k,m, pq 4 Generate the choice set {k } line p to line q at station m of the k­th route for the O−D 5 Load O−D matrix to network using a route choice pair (rs), and it can be induced from the walking distance model and obtain x 1, let n=1 for transferring and the passenger’s walk speed; t rs 6 Update the set of travel cost on links { t n } with the 2,k, q a is the waiting time at station m, which can be half of the set of current flows x n train’s headway; α is the amplification factor for 7 Load O−D matrix to network again with { t n }, and a transferring time, which is greater than 1 since the obtain a set of auxiliary flows { y n } a perceived transferring time is greater than the rail­ride 8 Calculate the current flows as: xn+1 = x n + 1/ a a travel time in terms of the same actual value [18].

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