A Dynamic Shortest Path Algorithm Using Multi-Step Ahead Link Travel Time Prediction

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1 Journal ofadvanced Transportation, Vol. 39, No. I, pp www, advanced-transport. corn A Dynamic Shortest Path Algorithm Using Multi-Step Ahead Link Travel Time Prediction Young-Ihn Lee Seungiae Lee Shinhae Lee Jeunggyu Chon In ths paper, a multi-step ahead prediction algorithm of link travel speeds has been developed using a Kalman filtering technique in order to calculate a dynamic shortest path. The one-step and the multi-step ahead link travel time prediction models for the calculation of the dynamic shortest path have been applied to the directed test network that is composed of 16 nodes: 3 entrance nodes, 2 exit nodes and 11 internal nodes. Time-varying traffic conditions such as flows and travel time data for the test network have been generated using the CORSIM model. The results show that the multi-step ahead algorithm is compared more favorably for searchmg the dynamic shortest time path than the other algorithm. Key Words: Dynamic Shortest Path Algorithm, Multi-step ahead Shortterm Prediction Introduction Route guidance systems provide motorists with step-by-step instructions on how to get fiom any origin to any destination in a network. The systems determine the best route from a user-supplied origin to destination, based on minimum link travel times. As a result of Young-Ihn Lee, Associate Professor, Graduate School of Environmental Studies, Seoul National University, South Korea Seungiae Lee, Associate Professor, Department of Transport Engineering, University of Seoul, Seoul, Korea, Shinhae Lee, Research Fellow, Department of Transportation Studies, Seoul Development Institute, Seoul, Korea Jeunggyu Chon, Researcher, Korea National Housing Corporation

2 6 Y-I. Lee, S. Lee, S. Lee, J. Chon this, existing studies are mainly focusing on the short-term prediction of link travel times or speeds in order to predict the travel times dynamically. Okutani (1987) proposed a model employing the Kalman filtering theory for predicting short-term traffic volumes. Prediction parameters are updated using the most recent prediction error in order to avoid error propagation in time. The method was tested using data collected from a street network in Nagoya. And it showed that the method can be easily adapted into a method of predicting future traffic volume from observed data. Sisiopiku and Rouphail (1993) investigated the correlation between traffic flow and occupancy obtained from detectors for predicting the link travel time. They showed that the correlation between travel time, traffic flow or occupancy is independent in the low traffic demand, but is becoming dependent as occupancy increases. Dailey (1995) presented an algorithm for estimating mean traffic speeds using volume and occupancy data from a single inductive loop. The algorithm is based on the statistics of the measurements obtained from management systems. The algorithm produced reasonable speed estimates in the reliability test. Iwasaki and Shirao (1996) showed that a short-term prediction scheme of travel time can be developed on a long section of motorways using the pseudo-traffic patterns. The autoregressive method was introduced as a prediction model in which the parameters were identified by adapting an extended Kalman filtering method. The predictions errors were almost less than lo%, which was significantly smaller than the results of the existing studies. Chen and Dougherty (1997) suggested a process for finding a suitable location of detectors using a neural network. Cohen (1 997) tested the efficiency of on-line travel time information on urban arterials in Paris. Lee et. al. (2004) tested the performance of the ARIMA, an artificial neural network, an exponential smoothing, and Kalman filtering techniques. The results show that the Kalman filtering technique is more suitable for predicting short-term travel speeds than the other techniques. Most algorithms on the route guidance applications have been developed based on only one-step ahead prediction of the link travel times, speeds or traffic volumes. The multi-step ahead prediction process should be considered in order to represent realistically the time-varying traffic conditions of the upstream links on the dynamic time interval basis. The multi-step ahead prediction of link travel time, and then searching the dynamic shortest time path based on these multi-step ahead prediction between OD pairs are key elements in the route guidance applications. In this paper, a multi-step ahead prediction algorithm of

3 A Dynamic Shortest Path... 7 link travel speeds has been developed using a Kalman filtering technique in order to calculate a dynamic shortest path. The one-step and the multistep ahead link travel time prediction models for the calculation of the dynamic shortest path have been applied to the directed test network that is composed of 16 nodes: 3 entrance nodes, 2 exit nodes and 11 internal nodes. Time-varying traffic conditions such as flows and travel time data for the test network have been generated using the CORSIM model. The results show that the multi-step ahead algorithm is compared more favorably for searching the dynamic shortest time path than the other algorithm. In section 2, the dynamic route guidance algorithm, which are composed of the multi-step ahead prediction model using the Kalman filtering technique, and the dynamic shortest path algorithm, are presented. In section 3, a numerical analysis of the developed algorithm is presented using a small example network. In section 4, some conclusions and hrther studies are remarked. Dynamic Shortest Path Algorithm Multi-ster, Ahead Link Travel Time Prediction Algorithm Kalman filtering is a recursive method, which estimates a temporary condition of linear dynamic system disturbed by Gaussian White noise. And it is a prediction process applied to dynamic systems that predict the optimal condition. This method is characterized as state average and covariance. In order to make this algorithm, the state equation and observation equation should be made as follow: X, +, = <D, X + w, (State equation) (1) Z, = HkXk + vk (Observation equation) (2) where, xk :(nx 1) state vector at time tk a, :(nxn) transfer vector fiom time k to time k+l, :(mx 1) observation vector at time tk Wk :(nx 1) white sequence vector with known covariance that element s average is 0 and correlation with no other parameter Hk :(mxm) connection vector between state vector and observation vector at time tk

4 8 Y-I. Lee, S. Lee, S. Lee, J. Chon vk :(mxl) observation vector with known covariance and uncorrelated with wk A multi-step ahead prediction algorithm is developed using the Kalman filtering technique that is described in the previous paragraph. In the travel speed prediction process, the travel time of each link is to be predicted at a time step when the individual vehicle passes through the respective link. The travel time from the origin to the destination, then, is predicted by adding travel times of downstream links based on those are predicted by respective-step ahead. The required time steps are varied depending on the travel speeds of upstream links of the respective links. The conceptual difference between one step and mult-step ahead algorithms is described in the figure 1. t- 1 t t+ 1 t+2 tt3 't k-i (a)one step ahead prediction method t- 1 t t+ 1 t+ 2 tt3 ft ahead prediction method pigure 1. Conceptual Difference of One Step and Multi-Step Ahead Methods The multi-step ahead prediction algorithm is as follows: Step 0: Initialization. Estimate travel speeds of links at current time step t = 0. Step 1: Set time step t = 0 and link number k = 0. Step 2: Update time step t = t + r (r: the time step required to traversing link k with the speed estimated). Predict the speed and the travel time of link k until total travel time from origin to link k reaches to the time step t +r using the Kalman filtering technique. Step 3: Update link number k = k + 1. Predict the travel time of link k until total travel time from origin to the link k reaches to the time step t +r using the Kalman filtering technique. Step 4: Continue step 2 and 3 for all k.

5 A Dynamic Shortest Path... 9 Dynamic Shortest Path Algorithm Finding a path from origin to destination, which dynamically minimizes the total link travel time, is important from an applied standpoint on the route guidance systems. The Dijkstra's algorithm was enhanced to find the dynamic shortest time path based on the predicted link travel times using the Kalman filtering technique. Step 1: (Initialization) Cij(t)=, Ft,[li,: Ij Cii(t)=O, r:t,!,i C, (t): link(ij) travel time of time step t update C,(t) for the downstream link of link (ij) Step 2: (Get next vertex) Search node m with minimum DG, for Uj save node i to path(i,m) Step 3: (Update least-cost paths) if t D,<t+l, then use cost table C, (t) Update D, = Cim(t) + Cmj(t), if D,>Cim(t) + C,(t) for nj goto step 2, if m is not the sink node. Step 4: (Find minimum time path) stop if node i is the source node of path(i,m) Application Test Network and Data Generation The proposed algorithm was applied to the test network to investigate the algorithm performance. The test network in this paper is a directed freeway corridor that is composed of 16 nodes: 3 entrance nodes, 2 exit nodes, and 11 internal nodes. Figure 2 and Table 1 show the configuration of the test network. Tables 2 and 3 represent the geometry of the test network and flow rates of the entrance nodes. There are four routes from origin node 1 to destination node 9: 1) the first route: link 2-link 3 -link 4 -link 5-link 6-link 7 -link 8, 2) the second route: link 2-link 15-link 16-link 4-link 5-link 6 -link 7 - link 8,

6 10 Y-I. Lee, S. Lee, S. Lee, J. Chon 3) the third route: link 2-link 3 -link 4-link 5-link 6-link 17-link 18-link 8, 4) and the fourth route: link 2-link 15-link 1 6-link 4-link 6-link 17-link 18-link 8. The first route is consisted of all the freeway links whereas the other routes include detour links from freeway to arterials on some sections. Figure 2. Test Network Table 1. Node Characteristics Nodes Node numbers Characteristics Entrance nodes 1, 10,13 Traffic production Exit nodes 9, 12 Traffic attraction Internal nodes others NETSIM nodes: 15,16,17,18 FRESIM nodes: others 1 I I I

7 A Dynamic Shortest Path Table 2. Network Geometry Geometry Number of lanes Freeway 1 Detour routes Lane widths 12ft 12ft 1 Linklengths or 6600ft or 1400ft 1 Max. speed 65mileh 40mileh Table 3 represents flow rates of the entrance nodes: node 1, 10 and 12. As the entrance flows are increased as time goes on in Table 3, the congestion is growing and the shortest path from node 1 to node 9 is changed. The CORSIM model was used to obtain time-varying link travel time data on one-minute interval basis. Table 4 shows the link travel time data for the test network that were obtained by the CORSIM model, which are assumed the observed (or true) link travel time. Table 3. Flow Rates of Entrance Nodes (vph) 30:OO -45:OO 45:OO 60:OO 75:OO 90:OO :OO I I 6300 I

8 12 Y-I. Lee, S. Lee, S. Lee, J. Chon Table 4. Simulated Link Travel Time of the Test Network (1 minute interval basis, sec) WO I 533 I l* bwelbine bavelhe bavelhe $eveltine baveltine baveim baveitine I S s $ dl d d 8 d 9 b B # Performance of the Multi-Step Ahead Algorithm The proposed algorithm was applied to the test network. The process of the proposed algorithm is demonstrated in Figure 3. The figure shows the conceptual estimation process by which these variables are measured or estimated. The process of finding the dynamic shortest time path is conducted on the multi-step ahead link travel time estimation on oneminute time interval basis. The results of the one-step ahead and the multi-step ahead algorithms were compared using the simulated actual travel time obtained from the CORSIM model. These lead to determine whether the multi-step ahead algorithm results are more consistent with the observed travel times given by the CORSIM model for finding the true dynamic shortest time path. Table 5 and Figure 4 show the predicted and observed shortest path travel times respectively from node 2 to node 8 according to each starting time from node 2. The result shows that the dynamic shortest path algorithm based on the multi-step ahead algorithm produces more accurate results than the static shortest path algorithm and

9 A Dynamic Shortest Path the dynamic shortest path algorithm based on the one-step ahead algorithms. The one-minute ahead and five-minute ahead time intervals have been used in the one-step ahead algorithm for the comparison of the performance. The five-minute interval estimates give rise to more accurate results than the one-minute ahead time interval. ESUnrdcdy*iJRd timeat-sbep t 0b.aFvsdbltbawl Urn at tm rrp t Figure 3. Short-Term Prediction Process of the Multi-Step Ahead Algorithm

10 14 Y-I. Lee, S. Lee, S. Lee, J. Chon Table 5. The Shortest Path Travel Time Estimates from Node 2 to Node 8 According to Starting Times Start time step min) static Simulated method (Observe) (based Multitravel time on step bythe current ahead CORSIM travel method (sec) time) (set) * One- step ahead ahead ion ahead ion Start time ' Simulated (Observe) travel time by the CORSIM (set) on Current travel time) (set) One step ahead ahead step 1 min. 5 min. method ahead ahead predict- predict ion ion I I I I I I I 448 I 466 I 61 I 581 I 615 I 602 I 585 I I 466 I 62 I I I I I I 544 I 525 I I 521 I 74 I 578 I 591 I 566 I I 529 I 75 I 553 I 595 I I 549 I 76 I I 561 I I 79 I 557 I 610 I I I 579 I 80 I I 604 I 598 I I 571 I 81 I 565 I 657 I 619 I 613 I I 581 I 82 I 599 I 654 I I I 559 I I 601 I 542 I I 575 I 87 I 647 I 670 I 646 I 639 I I I I

11 A Dynamic Shortest Path Travel hie from tiode 2 to node 8 W d88 MoE RMSE (set> Starting time step at node 2 Figure 4. Link Travel Time for Minimum Time Paths from Node 2 to Node 8 Table 6 represents the performances of the one-step ahead and the multi-step ahead algorithms in terms of the shortest path travel times from node 2 to node 8 according to starting times from node 2. In the RMSE value between the observation and prediction, the multi-step ahead algorithm is superior to the one-step ahead algorithm. In particular, we can see the static shortest path algorithm based on current travel time produces the biggest RMSE value. The evaluation of this study, therefore, shows that the multi-step ahead algorithm is more accurate for searching the dynamic shortest paths than the one-step ahead algorithm. Table 6. RMSE Value of Travel Time from Node 2 to Node 8 I I I 1 Static method One step ahead algorithm (based on Multi-step ahead current travel algorithm 5 minute 1 min. prediction time) prediction The multi-step ahead algorithm also finds more frequently realistic dynamic shortest time paths than the one-step ahead method. Table 7 shows the frequency of finding the true dynamic shortest time paths according to each stating time. In terms of the frequency of finding the true dynamic shortest time paths, the multi-step ahead model hits 97 percent of correct dynamic shortest time paths, whereas the hit ratio of

12 16 Y-I. Lee, S, Lee, S, Lee, J. Chon the one-step ahead algorithm is between 85 and 92 percent ranges. There are four possible routes from node 2 to node 8. We can see the dynamic shortest time paths have been changed from the first route to the other routes as the traffic congestion is occurred in Figure 5. Table 7. Hit Ratio of the True Shortest Path Searches from Node 2 to Node 8 Classification Frequency of the finding true shortest time paths Frequency of finding false shortest time paths accuracy(%) static method (based on :urrent travel time) 52 I 58 I Multi-step ahead model l 2 I "'I One-step ahead model 1 min. predicted minute predicted a4 j3 5 2 :! 20 Minimum time path3 (starting times step 1-> time step 90) Stmtmg time step at node 2 Figure 5. The Routes Used for the Dynamic Shortest Time Paths According to Starting Times

13 A Dynamic Shortest Path Conclusions This paper addresses the novel dynamic shortest path algorithm based on the multi-step ahead short-term prediction of link travel times, which can be applied to the advanced traveler's information systems (ATIS). The multi-step ahead prediction algorithm of link travel speeds was developed using the Kalman filtering technique. The one-step and the multi-step ahead link travel time prediction models for the calculation of the dynamic shortest path have been applied to the directed test network that is composed of 16 nodes: 3 entrance nodes, 2 exit nodes and 11 internal nodes. Time-varying traffic conditions such as flows and travel time data for the test network have been generated using the CORSIM model. The results show that the multi-step ahead algorithm is compared more favorably for searching the dynamic shortest time path than the other algorithm. It is mainly because the multi-step ahead prediction represents realistically the time-varying traffic conditions of the upstream links on the dynamic time interval basis. In turn, it is important to consider that the multi-step ahead prediction of link travel time, and then searching the dynamic shortest time path based on these multi-step ahead prediction between OD pairs are key elements in the route guidance applications. One of the important elements in the hture studies is to the development of the short-term prediction model based on paths rather than links. The path based prediction model enables to consider the causal travel time relationship from the down-stream and up-stream links more logically.

14 18 Y-I. Lee, S. Lee, S. Lee, J. Chon References Okutani, I. (1987), The Kalman Filtering..pproaches In Some Transportation and Traffic Problem, Transportation & Trafpc Theory. Dailey, D.J. (1995), A Statistical Algorithm For Estimation Speed From Single Loop Volume And Occupancy Measurements, The Federal Highway Administration & Univ. of Washington. Virginia, P., Sisiopiku and Nagui, Rouphail, M. (1993), Exploratory Analysis of the Correlation Between Arterial Link Travel Times and Detector Data from Simulation and Field Studies, ADVANCE Working Paper,No. 25, Oct. Iwasah, Shirao( 1996), A Short-term Prediction of Fluctuations using Pseudo-traffic Patterns, ITS World. Congress Chen, H., Dougherty, M. (1997), An Investigation of Detector Spacing and Forecasting Performance using Neural Network, ITS World. Congress I99 7 Cohen, S. (INRET-2, avenue Malleret-Joinville)( 1997), Travel Times on Urban Controlled Links: A Neural Network Approach, ITS World Congress I 997. Lee, S.J., and Y. Lee (2004), Short-term Speed Prediction Models for Time Dependent Shortest Path Algorithms in Car Navigation Systems, Behavior in Networks (BiNs) 2004