Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling ASSIGNMENT TECHNIQUES FOR NETWORKS WITH JUNCTION MODELLING ABSTRACT The modelling of congested urban road networks has been the subject of ongoing research around the world for some time. A variety of methods have been developed with varying levels of success and accuracy. A fundamental issue has been the trade off between obtaining results that can be interpreted for strategic planning purposes, and the need to represent traffic operations accurately at a local level. Many cities require a flexible modelling tool that is capable of assessing both. Ove Arup & Partners have recently developed and validated an EMME/2 model to assess transport projects for the Auckland City, New Zealand that incorporates detailed representation of junction delay, which draws on previous work undertaken by RO Hill(1). The research and development of the modelling tool highlighted many important modelling issues. This paper discusses the issues related to congested assignment model development and describes an approach to achieve the objectives of city-wide models. Topics discussed include: assignment methods; stability and convergence; appropriate level of detail for citywide models; link and turn penalty functions; and delay functions at signalised junctions. 1 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling ASSIGNMENT TECHNIQUES FOR NETWORKS WITH JUNCTION MODELLING AUTHORS Peter Dunn, B.Eng (Civil), MIE Aust, Ove Arup & Partners New Zealand, Manager Transportation Planning New Zealand Bruce Johnson, B.Eng (Hons), M.Eng.Sc, MIE Aust Ove Arup & Partners, Manager Transportation Planning South-East Australasia; Contact Details Ove Arup & Partners New Zealand Ltd PO Box 2503 Auckland 1 New Zealand Phone +64 9 309 4713 Fax +64 9 307 3917 Email: peter.dunn@arup.com INTRODUCTION The representation of congested road networks for the purposes of evaluating alternative traffic scenarios has been a continuing area of transport modelling research for some time. Increasing desire by traffic authorities to assess the network effects of localised traffic projects at a strategic level has lead to more detailed and sophisticated assessment tools being developed. Assignment convergence is a critical issue to ensure that comparisons between alternative scenarios represent real differences of network supply or travel demand and not varying degrees of convergence that may lead to erroneous cost-benefit analysis. Many different algorithms and assignment methodologies have been developed and used. However for the purposes of this paper we have divided techniques into two broad categories, where travel cost is expressed either as a function of: link flows (Type1); or link flows, and turning volumes or capacity at junctions (Type 2). Traditionally Type 1 models have proven popular, as a unique assignment solution is obtainable using equilibrium algorithms. However when applied to congested urban networks the level of network detail is limited without junction delay being represented. Type 2 models can provide a finer level of detail. However to reach a converged equilibrium solution Type 2 models require turn delay functions that predominantly relate to the assigned volume on that approach link only (Type 2a). Where the function strongly relates to assigned volumes on competing links at a junction (Type 2b) there is no known minimisation technique. Type 2a functions make the realistic representation 2 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling of turning movement delays difficult when the test scenario predicts significantly changed flow patterns on competing links at a junction, whilst Type 2b functions can result in convergence problems. The chosen methodology is very much dependant on the objectives of the model. There has been much work undertaken in Auckland, New Zealand relating to Type 2 models. Arup have recently validated an EMME/2 traffic model for Auckland City Council that required a new procedure to be developed. This required research into different techniques to represent delay at junctions. This paper discusses junctionmodelling issues and outlines a process using EMME/2 that achieves the objectives of a sub-regional traffic model, using the Auckland model as an example. BACKGROUND Auckland Model Hierarchy Auckland has a well-defined model hierarchy. At the highest level there is the Auckland Regional Transport (ART) model, a 4-step EMME/2 model capable of assessing transport strategies at a regional level. There are four city councils in the Auckland region. Each has developed, or is in the process of developing, more detailed subregional traffic models capable of being used to assess the impact of individual projects at a citywide level. Figure 1 illustrates the model hierarchy. ART Model Auckland Regional Council Public Transport Assessment Model Subregional Traffic Models Auckland City (being developed) Manukau City Waitakerie City North Shore City Project Models Specific Projects Auckland Model Hierarchy Figure 1 Whilst the ART model provided the basis for the development of a regional wide strategy, the network and zoning detail was too coarse for the assessment of individual schemes. The Auckland planning authorities saw the need to develop more detailed, local road network assignment models based on the ART model to assess and evaluate 3 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling road and traffic management projects. These became known as sub-regional models. The Auckland Regional Council (ARC) provides the cities with traversals of their vehicle trip matrices from the ART model for use in the subregional models. The subregional models are essentially assignment only models. Sub-regional Traffic Model Development Objectives The objectives of the sub-regional models are to: assess impacts of regional transport strategies on the road network, to a level where appropriate mid-block lane requirements, junction control and signal timing strategies can be identified; estimate design traffic volumes for transport projects; provide input to more detailed junction design programs such as SIDRA to assess junction requirements; define future traffic conditions in road corridors; assess the effect of public transport measures on traffic flow; and provide outputs for economic evaluation. In particular, they aim to: represent intersection delay for each movement; include all roads with a significant through traffic function; and apply a finer zone system to the vehicle trip matrices traversed (extracted) from the ART model. Initial Development Initially a Type 2 model process named ASRTaM (Auckland Sub Regional Transport Assessment Model) was developed within the EMME/2 framework by Waitakere City Council. ASRTaM is a complex assignment process carried out in EMME/2, which uses two Fortran programs (SIGMA and OPFLOW) to estimate optimum timings at signals and opposing flows at priority junctions. The ASRTaM process is documented in a previous paper by Hill described below as it relates to the new process developed here: (1) and is briefly an initial set of junction parameters is calculated based on an assignment assuming no junction delay; ASRTaM then carries out a full equilibrium assignment and recalculates the opposing flows at priority junctions, before re-assigning. This is undertaken for a specified number of iterations and represents the inner loop of the process; once the inner loop is completed, the signalised intersection parameters are recalculated and the next iteration begins. This is the outer loop; and 4 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling this process is repeated until the green splits (outer loop) and opposed flows (inner loop) do not change significantly between equilibrium assignments. This is the criterion for convergence. The model process was initially developed using the Waitakere sub-regional traffic model as a test-bed. Waitakere is located to the west of Auckland City, with a mix of urban and rural environment. The network includes 36 signalised and 193 priority intersections. Six full outer iterations (green split calculations) were necessary to achieve convergence of the base year model. Meanwhile Auckland City Council undertook extensive work in 1996 and 1997 to validate their sub-regional model (called the Isthmus model) using the ASRTaM process. The Isthmus model consists of 366 zones while the road network is represented in more detail than the ART model, particularly around the CBD and inner areas. A total of 423 intersections are modelled, of which 195 are signalised. Traffic conditions within Auckland City are significantly more congested than Waitakere. Figure 2 shows the extent of Auckland City’s model area. ART Model Isthmus Model Auckland City Model Area Figure 2 A validation report was produced in March 1997, however the validation did not meet the requirements of the Council. Arup were appointed in September 1998 to review and validate the Auckland City model. 5 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling ISTHMUS MODEL ISSUES An initial review of the Isthmus model revealed problems which made use of the model difficult and validation of the model impossible using the ASRTaM process. They are summarised below: in order to obtain an acceptable converged assignment 141 (65%) of the 195 signalised intersections required their parameters, such as green splits, saturation flows and cycle time, to be estimated externally and provided as fixed inputs to the process. There are two issues here, firstly it compromised the ASRTaM approach which aims to optimise signal strategies during the assignment and secondly it requires a very significant external analysis of individual junction operations. Importantly this also implies externally estimating parameters for at least 141 junctions in future scenario tests; the process required 25 iterations of the outer loop, and 4 iterations of the inner loop to achieve acceptable stability, requiring a total of 100 equilibrium assignments; due to the above total run times ranged from 10 hours on a Pentium 166MHz to 4.5 hours on a Pentium 266MHz, resulting in slow and costly turn over of tests; no consistent method was documented of estimating basic saturation flows for those junctions with fixed parameters. Errors were found in the estimation of the shared lane effect as applied to those intersections with fixed parameters; a common volume delay function was applied to all links with minimal capacity restraint. Whilst this is often not critical in congested urban networks, the Isthmus network includes several motorway and arterial road links where the link capacity does constrain traffic flows and speed. Further, only roads with a significant through traffic function are modelled, and therefore there is a need to distinguish between roads with different access and activity characteristics; the use of external Fortran programs to estimate signal timings and opposed flows placed constraints on alterations to the model procedures and validation of the algorithms employed; the database entry facility was inflexible; and comparison of future year scenario tests revealed inconsistent results after 25 iterations of the outer loop, particularly on parallel routes and there were large differences between alternative equilibrium iterations. A review of previous iterations indicated that the model was not converging to a unique solution. It was clear that the ASRTaM process in its current form could not practically or reliably be applied to the Isthmus Model, and therefore an alternative model process was developed to meet the objectives of the model. 6 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling MODEL DEVELOPMENT Methodology We conducted a literature review of a small number of papers and studies that drew on Arup’s international experience to better understand the attributes of various model approaches. The issues we were particularly interested in were: is this an appropriate use of EMME/2? If not what are the options?; what assignment procedure should be used, and what options are available?; and what level of precision is appropriate? Type 1 models have employed delay functions that relate to characteristics of that link. Some of the models incorporate link based functions that reflect to some degree intersection approach capacity and delay characteristics. Type 1 models result in a diagonalised function for which there are known mathematical minimalisation algorithms. Many urban models have been satisfactorily calibrated using link-based functions to estimate junction delays as well as link delay. Type 2 models that have turn delay functions relating to characteristics of another link at an intersection have a non-diagonal relationship. If this relationship is strong, researchers have not been able to develop a mathematically proven minimisation method. Researchers have therefore explored other ways of estimating delays at junctions. For example SATURN runs through an assignment and junction simulation loop. However it lessens the impact of the non-diagonal effect by employing a factor that reduces flows though the junction above saturation. This has a flow on effect through the network. Even so a stable solution is still difficult to achieve in congested large citywide networks. TRIPS on the other hand allows alternative assignment techniques, such as volume averaging, which smoothes the variations between iterations. Again a converged solution can be difficult to obtain. There have been several attempts to model junction delay in EMME/2. All attempts we know of doing this other than by functions that relate strongly to the traffic flow along the link have resulted in unstable results on congested networks. It is clear however that the structured nature of EMME/2 the software provides capability to develop a process at least to the level of say TRIPS. The review revealed that there is no clear consensus on the junction modelling issue, except that a wide variety of techniques are used worldwide, which result in continuing research and debate. Relating this to the needs of the Isthmus model, it was concluded: given the objectives of the model, it would be beneficial to model turn delays that are estimated by a non-diagonal relationship. In such cases the uniqueness of the process must be proven; 7 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling it is possible to produce a model in EMME/2 that will have the same capability as other strategic model packages with junction modelling capability, as say TRIPS; assignment methods such as updating the intersection parameters after a smaller number of equilibrium iterations and continuing the assignment, and damping processes such as volume averaging should be explored to assist convergence; and linkages between EMME/2 and SATURN have been tried (5), but to our knowledge were found to be unsatisfactory, and were not explored further in this study. Linkages between a strategic model and a detailed junction model such as SIDRA have been mooted, but not tried. SIDRA outputs have been used as inputs to congested assignment models, but this has been achieved by manual methods. There is some suggestion that this could be explored further. MODEL PROCEDURE Based on the review a new model procedure was developed. The elements of the process are described in this section and a flowchart illustrating the process is displayed in Figure 3. Assignment Process Several alternatives were investigated. The adopted method involved carrying out one full equilibrium assignment using both link and turn delay functions and then updating the turn delay parameters during the assignment at specified intervals. An issue was at what interval would the model produce a relatively stable result, to enable the final portion of the equilibrium to be completed. Previous experience suggested that turn delay parameter updates after a small number of iterations were likely to produce better results. We tested several alternatives, and concluded that optimal results for this model were obtained with junction parameters updated every five equilibrium iterations until the 50th iteration, then concluding the assignment without updates to meet the equilibrium stopping criteria. The process allows the user flexibility to specify different update intervals and stopping criteria. However it is important that the update intervals are consistent for alternative scenario tests. Figure 4 shows the model convergence of a typical Isthmus model run. 8 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling Model Process Figure 3 9 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling R e la tiv e G a p 140 120 % 100 80 60 40 20 0 0 10 20 30 40 50 60 70 80 60 70 80 I te r a ti o n N u m b e r N o rm a lis e d G a p 45 40 m in/trip 35 30 25 20 15 10 5 0 0 10 20 30 40 50 I te r a ti o n N u m b e r Figure 4 The effect of the parameter updates can be seen in both graphs, noting that the relative gap cannot be calculated in the first iteration after an update. A comparison of the second iteration after each update generally shows a progressively decreasing relative gap. The assignment takes 73 iterations to converge to a relative gap of 0.5% and a normalised gap of 0.1 after the final junction parameter update. Importantly tests conducted on the Isthmus network have shown that different starting assumptions for assignments produce exactly the same assignment. While the mathematical evidence does not guarantee uniqueness of solutions for Type 2 models, the experience with application of this model shows that in practice a unique solution is obtainable. The adoption of this assignment procedure was a major factor in reducing run times from 4.5 hours to 10 minutes. 10 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling Turn Delay Functions Several different turn delay functions were assessed. The ASRTaM process had used a conic function that aggregated the various junction parameters into the standard conic form. This was quite an involved procedure using module 2.41 to aggregate the parameters into the three available user turn attributes, and resulted in 32 different turn penalty functions. The turn delay function applied to signalised junctions in the Isthmus model is based on the SIDRA (2) delay functions. As the SIDRA functions use conditional statements curve fitting was necessary to define a continuous function in a form that could be read by EMME/2. The function is shown graphically in Figure 5 for three green splits. Signal Delay Functions Comparison for s=1800 vph, C=120 sec, T = 2h 2.00 Delay (min) Model u=0.25 SIDRA u=0.25 Model u=0.50 SIDRA u=0.50 Model u=0.75 SIDRA u=0.75 1.00 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Degree of Saturation Figure 5 The benefits of this approach are: only one signal delay function is necessary for a given progression condition; it is consistent with the methodology used to assess junction operations in Australasia; and the overflow component of the SIDRA delay function is already in a conic form, which aids convergence. Volume Delay Functions The process uses the link component of the volume delay functions specified in the Auckland Regional Transport model, which are based on recent surveys carried out in New Zealand (8). A further two volume delay functions were defined to represent flow conditions on roads performing a more local function. In all, 15 volume delay functions were defined. 11 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling Junction Parameter Input Process The process requires junction parameters to be read into extra attributes using module 2.41. Table 1 indicates the turn parameters and format required. In addition to the turn parameters there are several node parameters required including junction control type and cycle time. Table 1: Extra Turn Attributes Extra Turn Attribute @tpf Description Column Default Code Options 1 - 12 1 – 20 21 –32 defines turn at junction 4 0 @t critical gap (sec) 5 4 @ml @sat @pha number of lanes saturation flow (vph) Phase A 6 7 8 1 1850 0 0,1 @phb Phase B 9 0 0,1 @phc Phase C 10 0 0,1 @phd Phase D 11 0 0,1 @phe Phase E 12 0 0,1 @phf Phase F 13 0 0,1 @phg Phase G 14 0 0,1 @u @gmin green split Minimum Green Time (sec) 16 17 0 6 0 –1 0 – 120 @pr Progression Flag 18 1 1,2,3 @I @l @flt Intergreen Time (sec) Lost Time (sec) Free Left Turn Flag 19 20 21 5 4 0 0 – 120 0 – 120 1 @opprt Opposed Right Turn Flag 22 0 0-8 @short @share Short Lane Length (m) Shared Lane Flag 23 24 0 0 0 – 200 1 @ped Pedestrian Phase 25 0 2 @phi Proportion of bunched traffic for priority intersections 26 0 0.05 - 1 Comments 4 leg Priority/Roundabout 5 leg Priority 4 leg Signals For priority and roundabouts Turn Lanes (incl shared lanes) User specified saturation flow Flag 1 if movement is allowed in Phase A, otherwise 0 Flag 1 if movement is allowed in Phase B, otherwise 0 Flag 1 if movement is allowed in Phase C, otherwise 0 Flag 1 if movement is allowed in Phase D, otherwise 0 Flag 1 if movement is allowed in Phase E, otherwise 0 Flag 1 if movement is allowed in Phase F, otherwise 0 Flag 1 if movement is allowed in Phase G, otherwise 0 User specified green split proportion Minimum Green is 15 secs with parallel ped phase 1 = Average Progression 2 = Good Progression 3 = Bad Progression 1 = Free Left Turn (All intersection types) 0 = No Opposed Turn 1 - 7 = phase no (A - G) 8 = 2 opposed phases 0 = No short lane 0 = No Shared Lane 1 = Movement shares lane 0 = No Parallel Pedestrian Phase 1 = Parallel Pedestrian Phase 2 = All red pedestrian phase As per ASRTaM manual Data entry has been simplified by the development of an Excel spreadsheet to produce the standard input files required by the process. 12 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling Signalised Intersection Module This comprises three components: cycle time selection; saturation flow estimation; and green time estimation. Full use of the capabilities of the network calculator module is made here, particularly the hierarchical tree related to attributes in an expression evaluation. Cycle time selection The cycle time is input as a node attribute for each signalised intersection. The choice of cycle times needs to be considered by the user as a strategic network management issue and should reflect considerations such as practical cycle times for signal linking and pedestrian requirements. The process allows the user the option to specify a maximum cycle time, with the program estimating the optimum cycle time. Saturation Flow Estimation This macro undertakes the following steps: it assumes a user-specified base saturation flow for different speed environments as follows: Conditions VDF Functions Basic Saturation Flow (vph) 12 - 13 1750 Good 1-5 1850 Very Good 6-8 2000 Poor for shared lanes, it apportions saturation flow on the basis of the current assignment for the shared turns; for movements where the right turn is opposed, a user-specified factor is applied to the base saturation flow. ARR123 recommends this factor should be 1/3 for approximate analysis. This method avoids the need to use opposed flow directly to influence the capacity of the turn and so is desirable from the viewpoint of assignment convergence. There is potential to include the more detailed and technically correct method which relates to the green time allocation amongst other parameters, but we believe the model's precision does not warrant this; 13 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling where short lane effects occur a factor is applied to the base saturation flow. This again is based on the ARR123 method. The queue length is estimated, and a factor applied to the basic saturation flow if the queue exceeds the lane length; and an option is available that allows the user to specify the saturation flow as input and ignore the shared, opposed right turn and short lane effects. This would allow specific input from SIDRA for complicated combinations of lane sharing, short lanes or phase arrangements. Green Split Estimation This is a separate macro that follows as closely as practical the equal degrees of saturation approach documented in ARR123 which is the theoretical basis of SIDRA that has been accepted as a standard throughout most of Australasia. In summary, features of the process are that: it allows the user to specify the cycle time; it estimates the required green time for each movement at a junction; the user specifies the minimum green time allocated to each movement. This is an important constraint and limits the range of resultant green splits, hence aiding convergence; the user specifies the practical degree of saturation; it calculates the required cycle time by extracting the maximum green time estimated for each phase and intergreen time. As this is carried out on a phase by phase basis, movements that occur in more than one phase have their time apportioned equally to each phase. Although this is a simplification, it is considered an appropriate approximation for a network model; it estimates new green splits based on the above, but checks that the sum of the minimum phase times is not greater than the cycle time, in which case the new cycle time is used; free left turns are allocated 100% green split; if the required cycle time is greater than the maximum cycle time then green splits are reallocated considering the minimum green time requirements; pedestrian phases are allowed for; and the user can choose to specify a green split. Priority Intersection Module This is similar to the ASRTaM process, except that the opposed flows are calculated within EMME/2 using module 2.41 and the registers. Generally the priority function performed satisfactorily and wasn’t a critical element of the model. It is intended that both the priority and roundabout function be reviewed for future model revisions. Output and Results Module This macro produces standard reports and plots of the results of the assignment to enable the model results to be easily assessed. 14 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling The following reports are produced in ASCII format: aggregate measures of network performance and screenline comparisons; and junction parameters where the maximum delay of any of the turns at a junction is estimated to be greater than 2 minutes. The following measures are calculated for viewing as plots in module 2.13: maximum delay at a junction; average delay at a junction; degree of saturation at a junction; and average delay on a junction approach. VALIDATION The model was shown to be well validated to 1991 measured travel times and traffic flows. In summary: assigned flows were within 10% of measured traffic volumes across all seven screenlines. Figure 6 shows a comparison of assigned flows against 270 measured flows within the model area. The scattergram shows that a good correlation has been achieved between assigned and measure link flows, resulting in an R2 value of 0.98, and a standard deviation of 360 veh/2h. Link Volume Scattergram Validated 1991 Morning Peak 14000 12000 Assigned Flows (veh/2 hours) 10000 y = 0.9798x + 18.926 R2 = 0.9768 8000 6000 4000 2000 0 0 2000 4000 6000 8000 10000 12000 14000 Measured Flow s (veh/2 hours) Figure 6 15 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling 92% of links had a GEH of less than 12% (GEH is a form of the chi-square statistic which is designed to be tolerant of large errors in low flows); significant outliers were identified for further investigation; and model estimated travel times along 18 corridors are within 2% of measured times. In addition the model's response to increased demand and network changes is logical and justifiable. Figure 7 demonstrates such an example for the 1996 demand case with the addition of a recently constructed arterial road. The assigned flows were checked against counts undertaken within the area of the scheme and were found to closely match measured flows with only two of the 25 locations having a GEH greater than 10. Example of Model Response Figure 7 CONCLUSIONS The project has demonstrated the practicality of applying a Type 2 model using EMME/2. The model process was shown to meet the requirements of Auckland City and there is potential to apply this process to other sub-regional models. We have endeavoured to make the model as flexible as possible by allowing the user to specify inputs at a basic or detailed level. 16 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling In conclusion: the detailed nature of the ASRTaM methodology provided a basis for the development of this process. However in its current form it could not be practically or reliably applied to a congested network scenario; we have improved on this process by: incorporating all modules within EMME/2; improving run times from 4.5 hours to 10 minutes; producing a stable and converged assignment; incorporating flexible input procedures; providing for more detailed junction input if required, consistent with ARR123; and providing standard outputs to make the results easily accessible; further improvements we would like to make to the current process include: simplification of short lane effect methodology; revision of priority and roundabout junctions; application of a priority function for left turn slip lanes; and allowance for data entry using a GIS. the limitations of the method are: there is no allowance for queuing back effects from downstream junctions; the same update intervals should be used when testing alternative scenarios; and there is no allowance for spreading demand to other time periods when high delays are predicted. Overall Arup would appreciate feedback from others who have tried similar approaches and would welcome opportunities to apply this technique elsewhere. REFERENCES 1. RO Hill (1998) An Application of EMME/2 Auto Assignment with Detailed Junction Modelling of Activity at Nodes 2. SIDRA 5 User Guide ARRB Transport Research, January 1998 3. Traffic Signals – Capacity and Timing Analysis, ARR123, 5th reprint 1993 4. Transfund New Zealand Limited, 1998, Project Evaluation Manual 5. Junction Modelling within EMME/2: Ian Taylor and Peter Willoughby, 1998 European EMME2 Users Conference 6. Review of Models Combining Traffic Assignment and Signal Control: Claudio Meneguzzer, Journal of Transportation Engineering – March/April 1997 17 Peter Dunn & Bruce Johnson Assignment Techniques for Networks with Junction Modelling 7. Effect of Intersection Delay Modelling on the Performance of Traffic Equilibrium Models: Koutsopoulos and Habbal 1993 8. ART Model: Road Speed Flow Relationships Working Paper 3, 1994, Auckland Regional Council ACKNOWLEDGMENTS The authors acknowledge Auckland City Transport Planning for whom the project was undertaken, and Dave Thompson and Simon Camm (Ove Arup & Partners London) for their advice and review of the paper. 18