Aggregation of Traffic Networks Using Sensitivity Analysis

AGGREGATION OF TRAFFIC NETWORKS USING SENSITIVITY ANALYSIS

Dr Richard Connors Institute for Transport Studies, University of Leeds Prof David Watling Institute for Transport Studies, University of Leeds

1. ABSTRACT

Transport modelling encompasses many scales of analysis, from junction design and signal timing to urban planning and infrastructure provision for predicted land use across whole regions. The transport network is modelled using different levels of detail at each scale of analysis, and yet it is not well understood what effect the level of aggregation (with respect to data and/or model structure) has on the predictions of the network model, and hence on the overall analysis. These two issues, the level of aggregation used in the network model and the fidelity of data used in calibration, can be considered separately though of course their combined impact is ultimately of interest. While network aggregation is a well recognised problem in transport analysis, no systematic aggregation methodology exists, with only empirical evidence reported in the literature. In this paper we consider the particular problem of network aggregation: specifically the question of how a detailed network model can be aggregated. We propose a systematic method of network aggregation based on sensitivity analysis. The aim of this method is to represent the aggregate dependence of origin-destination travel times on network-wide demands. We illustrate this technique with an example motivated by real-life problems in which it is desired to represent the impacts of a smaller urban area within a larger-scale network.

2. INTRODUCTION

Transport analysis must span many scales. For example, at one extreme its objective may be the design of traffic signals at an urban intersection for bus priority, or the shock-wave analysis on traffic flow of a motorway incident. At a wider geographical level, transport planning over a city network may be required to examine the impact on route choice and travel demand of measures such as road user charging or new retail developments. At a still wider level, the analysis may need to address problems of regional or national impact, such as the introduction of a national road user charging system. An element in all such analyses is that as geographical scope of the problem changes, so does the scale of the models used, in terms of the fidelity of network representation, the size of the zones over which trip demands are modelled, and the level of disaggregation of traveller responses by, say, socio-economic group. The question then naturally arises: are these models all consistent in some sense, across the different aggregation scales? Such questions regarding aggregation are not new. In the landmark review of network equilibrium approaches given by Friesz (1985), he notes an ‘interest

in network aggregation [that] stems primarily from the high computational cost of solving large network equilibrium and design models’. Purely from a viewpoint of computational speed, it is interesting to note that such a motivation is still potentially valid today, even given ongoing increases in computer power: this is particularly the case for network design problems, where one may wish to find network equilibrium flows a great many times as part of an algorithm to evaluate, for example, the efficacy of alternative road pricing cordon designs (Sumalee, 2004). Yet even in cases where computational speed is not an obstacleas witnessed by the increasing number of large-scale regional and national network models in operation containing tens of thousands of linksthere still exist basis questions of consistency of scale. For example, even if we could do so, would it be desirable to run a national model each time we wished to adjust the traffic signals at one junction? Clearly this would ensure that the urban, regional and national scales are consistent, but would we believe impacts at this level, and how would we deal with validation of the model? In any case, we cannot neglect that fact that in practice there are a wide variety of transport modelling software packages in use, which have different capabilities directed at different kinds of problem. Many packages that are able to deal with land- use/transport interactions or demand modelling in a sophisticated way are not able to deal with network modelling: and interfacing such software programs is both time-consuming and generates problems of its own in ensuring convergence to self-consistent solutions. We would argue, therefore, that there is a justifiable need to examine and understand the problem of aggregation scale. In transport we use the term aggregation to refer to the level of detail included in network and behavioural models, it is also used to refer to methods used in order to summarise characteristics of detailed models for larger scale analyses. Looking to the transport literature, it is possible to identify several aggregation themes that have attracted attention. Firstly, there is decision aggregation (dating back to Theil, 1957). Aggregation here is with respect to the distribution of (socio-economic) attributes across the population of decision makers: with a continuous distribution of different individuals, determining the number of people choosing each alternative involves an integral. This introduces a computational burden into the equilibrium assignment of transport network analysis; aggregation in this context has been investigated for both logit (Sheffi, 1984) and probit (Bouthelier and Daganzo, 1979) models. However, these approaches to the aggregation of decision makers do not shed much light on the issue of network aggregation. Secondly, there is traffic aggregation, which covers a large spectrum of approaches (indeed it could be argued that such a class also incorporates the network aggregation methods described below, though we have chosen to separate these classes). At one extreme, this class contains an analysis of traffic flow dynamics on a single link, leading to theory on the aggregation of individual car-following models to give fluid flow PDEs (Lighthill and Whitham, 1955). Recent work in this category: following (Herman and Prigogine, 1979) and (Ardekani and Herman, 1987), Daganzo (2007) employs the two-fluid model to aggregate dynamic urban flow into zones, resulting in a multiple-

reservoir model. This representation approaches aspects of the area speed- flow and continuum models mentioned below. Thirdly, zonal aggregation considers the method by which continuous space is partitioned into discrete zones; in transport modelling this is typically for the representation of trip demand patterns. Simplification of trip origins and destinations into representative (centroids of) traffic agglomeration zones (TAZs) is a particular instance of the modifiable areal unit problem, which is the subject of many papers outside of transport network analysis journals (see Martinez et al., 2005). The effect of zonal aggregation on parameter estimates and goodness-of-fit in spatial interaction models (without a network) has been investigated (Batty and Sikdar, 1982a, Batty and Sikdar, 1982b, Batty and Sikdar, 1982c, Batty and Sikdar, 1982d, Openshaw, 1977). Openshaw considers optimal design of a zoning system: the combinatorial problem of aggregating N zones into M

of the large detailed network). Hearn introduces a set of artificial links to connect the two decomposed parts of the original network, and these provide the OD trip matrix for the focus area network. While the decomposed problems are individually easier to solve, it is not clear that solving them in a combined iterative fashion is more efficient than solving the original network assignment problem. However, Hearn establishes analytical properties of the decomposed components that may allow efficient approximate solutions to be found (also see comments in Friesz, 1985). From this review of existing transport network literature, it can be observed that (with the notable exception of the work of Hearn 1984) research to date on network aggregation mostly comprises empirical reports (for example Bovy and Jansen, 1983) simply demonstrating that the level of aggregation alters model prediction, or that good model prediction does not necessarily rely on the same scale of aggregation for the network links and the TAZs (Chang et al., 2002). For the great majority of problems faced, the state of methodology is no further advanced than that described by Friesz in 1985, when commenting that ‘aggregation has been practiced in an essentially ad hoc fashion since the advent of widespread use of network transportation planning models’. Looking to related fields where more systematic advances have been made, work in the geography and planning literature (e.g. Xu and Sui, 2007), is not directly applicable to the analysis of transport networks as only constant cost networks are considered, with no concept of capacity or congestion. A recent review of such measures and their relevance to transport is provided by Xie and Levinson (2007). Likewise, while several network aggregation studies exist in the operational research literature, they focus on uncongestible, constant link cost problems: see the review of Rogers et al. (1991), or the work of (Zipkin, 1980) who simplifies a linear minimum-cost network-flow optimisation problem by replacing groups of nodes with aggregate nodes, and derives bounds for the solution to the aggregated problem. With these comments in mind, the purpose of the present paper is to present preliminary investigations into developing a systematic procedure for aggregation in network models, focusing at this stage on network aggregation. Specifically, with an aim to achieve some generality, the focus will be on aggregation of the stochastic user equilibrium (SUE) model, which is well known to approximate the more widely used User Equilibrium (UE) model to an arbitrary accuracy by suitable choice of perception error variances. The structure of the paper is as follows: section 2 sets out our notation for network analysis and section 3 briefly presents the sensitivity analysis of SUE flows, section 4 then describes the aggregation methodology via an example and conclusions are given in section 5.

2. DEFINITIONS AND NOTATION

We represent the road network by a directed graph consisting of N links labelled a = 1,2,..., N ; a demand vector q, with entries q r [r = 1,..., R], representing the travel demand on the rth origin-destination (OD) movement, which is served by a set of paths K r . An assignment of flows to all paths is

r th denoted by f, with elements fk ≥ 0 giving the flow on the k path connecting the rth OD. The assignment is feasible for demand vector q if and only if r r r ∑ fk = q ∀r and fk ≥ 0 ∀k,r . (1) k∈K r The (closed, convex) set of feasible path flows thus defined is denoted F. The link travel times are single-valued, monotonic, continuous functions, ta (x), of the vector of link flows, x=∆⋅f, where the link-path incidence matrix, ∆, has elements denoting the links a that are part of path k, ordered by OD movement r. The mapping between flows and costs arises from the standard link-additive model: cf()= ∆∆T t( ⋅f). (2) The foundation text for deterministic network equilibrium is Wardrop (1952), who proposed that travellers will change route in order to reduce their travel cost, c(.). The consequence is that, at equilibrium, all used routes have equal minimum cost for each OD movement; this condition is called User Equilibrium (UE). The path flow vector f% is a solution to the UE if it satisfies (1) and T cf()%%(f−≥∀f) 0 f∈F . (3) If we begin with a less constrained representation of individual choice behaviour, we can derive a generalization of the UE paradigm. Adopting a random utility model, the perceived utility of using path k is rrr Uck=− k+ε k, (4) where the random variable ε is drawn from a zero-mean probability distribution. The proportion of the r-th OD demand choosing to use path k are those individuals who perceive this to be the least cost (maximum utility) path: rrr PUkk=>Pr(Uj∀j). (5) P is used to denote the vector of all such route-choice proportions; from (4), P is a function of route costs and hence of route/link network flows. The stochastic user equilibrium (SUE) is then a feasible path flow vector such that fq%=⋅P(c()f%). (6) From here on we use x%=∆⋅f% to denote the SUE link/path flow vectors. At SUE, all paths are used even though they do not have equal cost; there is not a unique “OD travel cost” as is the case with UE. The satisfaction provides an analogous quantity. For the r-th OD the satisfaction is defined to be r rr ∂S r SE= max{Uk } , whence r = −Pk . (7) k  ∂ck Note (Sheffi, 1985) that if the ε-variances are all zero, then the perceived OD travel time equals the minimum (actual/measured) OD travel time, and we recover the common OD travel cost of UE. Since not all travellers use the (actual) minimum cost path at SUE, the average experienced SUE travel time for the OD pair is always in excess of the minimum deterministic OD travel time. These inequalities give: SErr=≤max U min cr≤Prcr (8) { kk} { } ∑ kk k

In the limit as variances tend to zero we recover UE and all terms become equal.

3. SENSITIVITY ANALYSIS AND THE SATISFACTION

We wish to approximate the aggregate changes in costs across the network as OD demands change. Our method stems from the sensitivity analysis derived for probit SUE link flows in Connors et al (2007); this provides an approximation to the variation in SUE link flows under perturbations to network parameters. Consider the OD demands, q, as functions of s: qq= (s). In fact in this paper we will only explicitly consider the simple case where qsrr()=qr+sr. Following the analysis in Connors et al (2007), gives xq%%()+=s x(q)+A⋅s, (9) where the required Jacobian matrices of link costs and path choice probabilities have been combined to give the constant matrix, A. The sensitivity analysis result (9) therefore allows us to write the SUE link flows as a function of the OD demand perturbations. While this is a linear approximation, the dependence of link flows on demands is typically much closer to a linear relationship than (for example) the function relating link costs to OD demands, and so we can have some optimism that the approximation may be acceptable over a useful range. Using (9) we can compute approximate link flows and hence link costs, sum them accordingly to get path costs and hence calculate the satisfaction for each OD as an explicit function of the OD demands: SSrr=≈()cs() Sr(∆T⋅t(x%(q+s))) . (10) For logit SUE, where the random terms in (4) are drawn from Gumbel distributions, the satisfaction function can be written in closed form. We can therefore write (an approximation to) the satisfaction for each OD explicitly as a function of the OD demand perturbations: rrrr% Sc=log ∑exp(−θ k (x)) so that Sc()sx≈−log ∑exp( θ k ( (q+s))) . k k This expression is trivial to evaluate for any new OD demand profile q+ s; the predicted satisfactions include the ‘network effect’ where changes in demand on one OD movement cause congestion and hence re-routing of traffic on other OD movements. For probit SUE the satisfaction cannot be written in closed form, though the same approach leads to an explicit and efficient expression for S r (s).

4. AGGREGATION

The process for network aggregation follows from the sensitivity analysis described above. For a given network, we compute the SUE link flows and SA Jacobians, giving us the A matrix in (9). Using the SA predicted equilibrium link flows, we can then calculate the satisfaction for each OD as a function of perturbations to the OD demands without re-computing SUE. The aim is to derive a simplified network representation such that the prediction of how changes to OD demands affects OD costs is made without recourse to

information about network connectivity or link cost functions – this network information is implicit in the sensitivity expression (9) that is included in (10).

4.1 Example Probit Network Aggregation

The fixed demand, single user class Headingley network has 73 nodes, with 240 OD movements using 188 road links. The path set is generated up front (as described in Connors et al., 2007), resulting in 1545 paths through the road network. The probit variance for each link is set to be a fixed value, a multiple of the free flow time on that link. Clearly, any other variance structures could be adopted with this model. The test network is shown in Figure 1 below. In the road network, triangles represent origins and destinations for the intra-urban traffic, with dashed lines being centroid connectors, and the solid lines designating two way road links.

Figure 1a: Detailed network Figure 1b: Simplified network Figure 1

We consider a ‘wide area’ analysis of commuters travelling from a satellite commuter town into the city centre; these commuters can choose to drive through the city road network, or drive to a nearby rail station and then catch a train into the city centre. The commuters who travel by car are distributed between several routes through the urban road network. We wish to focus on how well we can approximate the road network with a single surrogate link under changes to OD demand. In the initial investigation presented here, we make several simplifying assumptions, though our approach is easily extended to a more general analysis.

• The additional cost at the city centre destination arising from parking and walking to workplace is the same as that for access from rail station to work place • The alternative specific constants for rail/road are zero • The road leading from the commuters’ origin to the station is congestible • The generalised cost for rail travel from the commuter station to city centre is constant, and includes a fixed average waiting time, constant travel time and fare. Note that as our model does not reflect aspects of overcrowding on public transport, which would need an estimate of person-trips by mode, we shall assume the OD demands are in the form of vehicle-trips (i.e. so that vehicle occupancy is implicit in the flows defined). We therefore represent the generalised cost of travelling via the rail link (the dot-dash line) as 5 xTRAIN gcTRAIN ( xTRAIN ) =+cTRAIN ( xTRAIN ) τ = 100 ++200600 1240 where xTRAIN is the rail flow (i.e. number of vehicle trips on the rail link), where driving from the out-of-town origin to the commuter station along the road link has flow-dependent cost cxTRAIN ( TRAIN ) and where the total generalised cost of the train journey is denoted by the fixed value τ. The specific parameter values used in the numerical example (below) are shown. We begin by computing the SUE flows for the entire detailed network under a demand of 800 on the commuter OD movement. Note that there are other flows originating at the ‘satellite’ origin, but whose destinations are within the detailed urban network rather than in the city centre; in this model these travellers do not take the train. With this base scenario the split of commuter flows is shown in Table 1.

Commuter OD Demand = 800 Flow into Rail Link 194.50 (605.50 go via Road Network) Total Flow onto Road Network from 1557.48 Commuter Origin Table 1: Modal split in base scenario

We now consider the road network only – without the rail link – under the base scenario inflow; this means a total inflow of 1557.48 from the satellite origin, of which 605.50 is on the commuters’ OD movement. Equivalently: we delete the rail link and set the commuter OD to be 605.50, leaving all other aspects of the network and OD matrix unchanged. We now calculate the sensitivity analysis Jacobians for the road network (under fixed demand) and hence can compute the SA predicted network flows under different demand scenarios. In particular, we calculate the approximate predicted flows resulting from perturbing the level of commuter demand. Our interest is in characterising the effective cost faced by commuters passing through the road network under different levels of demand. To this end we use the SA predicted network flows to calculate the commuter OD satisfaction for several levels of commuter OD demand. In Figure 2 the SA-based OD

satisfaction is shown as a dotted line, with the ‘true’ satisfaction calculated using re-computed SUE flows displayed as a solid line. The two lines are difficult to distinguish because the approximation is very close to the actual values, justifying our proposed use of the SA approach. The demand in the base scenario (605.50) is marked. Computation of the probit satisfaction is done using the Clark approximation (Clark, 1961), which approximates the maximum of several normally distributed random variates with a single normally distributed random variate. This calculation of the probit satisfaction also provides its approximate variance. Using the SA link flows under different levels of commuter demand we therefore have the approximate cost of travelling through the road network (the satisfaction) and the approximate probit variance. We use this information to replace the entire road network with a single surrogate link, whose cost function is shown in Figure 2. This leaves us with an aggregate representation of the entire network that has only two links, as in Figure 1b. Importantly, note that the aggregate road network link cost function has flow-dependent probit variance, even though the original network links have flow-independent variances.

Figure 2: Satisfaction for Commuters’ OD Without Rail Link

We now calculate the probit SUE flows for the two link network under several levels of commuter OD demand. The link cost function for the rail link was given above, the other aggregated-network link has the cost function shown in Figure 2. While e.g. a BPR function could be fitted to these data to create a somewhat standardised network model, calculating the satisfaction directly (using SA) is sufficiently simple and quick that another level of approximation seems unnecessary. In Figure 3 we compare the results with the ‘true’ flows resulting from re- solving SUE on the entire detailed network under the same range of demand levels.

Figure 3: Aggregate vs Detailed Model Predictions of Commuter Flows

The commuter flows predicted by the two-link aggregated network agree well with those of the detailed network model. On the aggregate network, the road flows are systematically lower, and the rail flows higher than for the detailed network. Increased probit variances on the aggregate links would cause this behaviour (as travellers move closer to an even split between available alternatives), though we have not determined if this is a valid explanation here.

5. CONCLUSIONS

This paper has reported the first steps of an exploratory study of network aggregation. At this stage of the work, the aim has not been to produce a single method, but rather to take care in setting up systematic procedures whereby alternative elements of network simplification may be made, with a clear awareness of which elements have been agglomerated, approximated or neglected. With a particular case-study of an urban network, one potential “aggregation” process has been illustrated. The first step in this process is to analyse the impact on mean perceived OD travel costs (formally, the ‘satisfactions’) of changes to the OD travel demand matrix, with the process of re-equilibration at each stage implicitly embedded in the relationship. Sensitivity analysis is used for this step to avoid the need to re-solve equilibrium at many points, yielding an explicit functional relationship between OD flows and OD satisfactions (‘explicit’ meaning that an equilibrium problem need not be re-solved each time the OD flows are changed). This has removed the network/equilibrium component, but the resulting problem is not necessarily of lower dimension than the original. There are several ways in which this information may then be subsequently exploited in the overall aggregation process. One way, as illustrated in the example, is to focus on a particular OD movement of interest, and agglomerate all other OD movements as some kind of overall ‘traffic

intensity’, which reflects that as demands on other movements grow they may delay the travel of our OD movement of interest. The resulting problem is an aggregation/simplification of the original problem, consisting of only two dimensions of ‘flow’ and two dimensions of ‘travel cost’, compared with the many OD movements, links and paths in the original network. While this analysis has concentrated on a one-dimensional characterisation of the road network cost as a function of commuter OD demand, this approach is inherently multi-dimensional and allows the interactions of all ODs to be captured, albeit limited by the SA linear approximation to SUE link flows. In this sense the full detail available in the aggregation approach presented here may be thought of as replacing the road network with single links joining each OD, and each such link having flow (and cost) depending on all OD demands – a fully connected and non-separable network graph. One representation for this type of aggregated network comprises two matrices: the original [R x R] OD matrix, and an [R x R] cost-function array wherein each element is the satisfaction function for that OD, a function of the entire OD matrix. A structural analysis of these two matrices may yield analytical methods for further understanding of the aggregation process. Clearly there are still many issues to address in terms of devising appropriate relationships and levels of fidelity, that preserve as much of the underlying network information as is possible.

ACKNOWLEDGEMENTS

This research was funded by EPSRC Platform Grant GR/S46291/01 ‘Towards a unified theoretical framework for transport network and choice modelling’, and grew from ideas that were developed by the first-named author’s visit to the Centre for Collaborative Research at the University of Tokyo.

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