6.1. Free-flow travel time

The free-flow travel time usually corresponds to the time for a vehicle to pass through a road link when no other vehicles are present, such as the travel time experienced in the early morning. In research studies and engineering practice, it is usually taken as the time to go through a link when traveling at the designated speed limit, sometimes multiplied by a slowing-down factor of roughly 1.3 to reflect the minor delays caused by stopping at intersections and other factors in an urban environment (Çolak et al., Reference Çolak, Lima and González2016). For example, the DfT defines the free-flow journey time using time delay coefficients for different road types, speed limits, link lengths, widths, gradients, and traffic junctions, which is obtained for our study sites and plotted as the horizontal axis value in Figure 7a (DfT, 2002).

Figure 7. Comparison between the theoretical traffic parameter value and observed value at 24 ATC sites. (a) Free-flow speed; (b) capacity or saturation flow.

In this study, it is recognized that using speed limit to calculate the free-flow travel time based on road type frequently ignores the impact of localized, irregular factors, for instance, a curve or pothole on the road. Rather than making assumptions about the free-flow time, we estimated it from the Google travel time data collected over the period of the study. For each site, the 5th percentile (fastest) travel time among all observation points is used to represent the observed free-flow travel time of that particular site. Free-flow speeds for each direction of the same road are estimated separately. Figure 7a shows the contrast between the speed limit and the observed maximum speed calculated from the observed free-flow travel time. Each point represents a specific site in the dataset. On average, the observed maximum speed of a road link is 20% lower than the specified speed limit. However, the former tends to be higher (exceeding the speed limit) on roads with the low posted speed limit. A 45-degree line is also plotted in Figure 7a for reference to compare the speed limit against the observed maximum speed.

6.2. Road capacity

Compared to the free-flow travel time, road link capacity is an ambiguous quantity to define. For example, the UK Design Manual for Roads and Bridges (DMRB) defines capacity vaguely as “the maximum sustainable flow of traffic passing in 1 hr, under favourable road and traffic conditions” (The UK Highways Agency, 1999). The DMRB also provides lookup tables that feature traffic capacities for a range of road types, road widths, and number of lanes. A manual survey of satellite imagery was carried out to assess the lane count, estimate the road width for each of the ATC locations and thus provide an estimated capacity by this DMRB definition. However, such a method was deemed unacceptable due to the uncertainty in what constitutes as favorable road and traffic conditions.

Instead, a more nuanced definition from Spiess (Spiess, Reference Spiess1990) is adopted, which defines capacity as the volume at which congested speed is half the free-flow speed. In the combined Google and ATC data, since there is hardly any observation when the speed is exactly half of the free-flow speed, a more relaxed definition of capacity is subsequently utilized. For each site, the capacity is taken as the highest 95th percentile hourly traffic volume among all observations where the travel time falls between 1.8 and 2.2 times the free-flow speed. Capacities for each direction of the same road are estimated independently, since each direction may have different contexts, such as the presence of bus stop, street parking, or lane width. As a result, each individual road is given a capacity attribution. This capacity attribution is observed to be consistently lower than the DMRB recommended value, as can be seen in Figure 7b. On average, the observed capacities are only 34% of those recommended by the DMRB, reflecting factors such as signalized intersections, bus stops, and street parking that cannot be perfectly captured using simple lookup tables. The 45-degree line is plotted in Figure 7b for reference. Also, it can be seen that there are only a few values of the DMRB road capacity, clustered according to the road types and lane counts.

6.3. Volume–delay relationship

Based on observations of the road link travel time and hourly traffic counts presented above, the context-specific volume–delay relationship can be constructed. The BPR formulation of the volume–delay relationship is chosen due to its mathematical advantages (differentiable and convex) and widespread use (Equation (1)). In particular, for the baseline case, we used $ \alpha =1.0 $ and $ \beta =2.0 $ as per TfL recommendation, while $ {t}_0 $ and $ {v}_c $ are based on the speed limit and DMRB recommended capacity as computed in Section 6.1 and Section 6.2. It should be noted that when the BPR curve is used in practice, $ {t}_0 $ is often corrected by a delay factor to reflect the city-specific conditions (Çolak et al., Reference Çolak, Lima and González2016) while $ {v}_c $ is often corrected by factors such as road geometry, gradient, and signal status (Kurth et al., Reference Kurth, Van den Hout and Ives1996). However, these corrections usually require labor-intensive site inspections and the correction factors are often empirical or generalized for the whole city. The aim of this analysis is to propose an efficient way to obtain alternative, site-specific function parameters from real observations. The purpose of this study is to also contrast the performance of the data-informed model parameters with the base curve in terms of their ability in calculating the time-delay at different congestion levels.

A summary of the three models used for comparison is given in Table 3. The base curve uses BPR coefficient $ \alpha =1 $ and $ \beta =2 $. In particular, the free-flow travel time $ {t}_0 $ is calculated from the speed limit, while the capacity $ {v}_c $ follows the DMRB recommended value. In traffic modeling, the transport agencies may apply some adjustment coefficients to consider external influencing factors on $ {t}_0 $ and $ {v}_c $. However, in most of the cases, these factors are predetermined and may not be up-to-date or localized to each road segment. The alternative formulations are developed to address this issue: for the first data-informed alternative formulation (DD1), the base value of $ {t}_0 $ and $ {v}_c $ is replaced by observed values obtained in Section 6.1 and Section 6.2. For the second data-informed alternative (DD2), not only $ {t}_0 $ and $ {v}_c $ are replaced by the observed values, but also $ \alpha $ and $ \beta $ are calibrated using regression models on the real-world data. Once the data collection and processing pipeline have been built, it is relatively easy to obtain the coefficients used in the data-informed volume–delay curves, even updating weekly or monthly to take into account any temporal changes in the road characteristics (e.g., roadworks or weather conditions).

Table 3. Summary of volume–delay relationships tested.

It is evident from Figure 6 that the observed relationships between the volume and delay are site-specific, nonlinear, and heteroskedastic (variance of the data increases with the independent variable). All these characteristics are expected from the traffic perspective. In particular, the nonlinearity means that the travel time increases more dramatically as the hourly traffic volume approaches the capacity. The heteroskedasticity implies more variations in vehicle speed when the road is congested due to interactions with other vehicles. Usually, to deal with such data, a nonlinear model needs to be used (which is already the case for the BPR curve). Heteroskedasticity can be accounted for by using techniques that include data transformation or using weighted least square method (Vynck, Reference Vynck2017). However, it is not a requirement if the goal is to estimate regression parameters, as the ordinary least square (OLS) method also produces unbiased, though inefficient, estimation of the coefficients. Standard errors of the estimation parameters are not used, so the OLS method is deemed sufficient for obtaining the $ \alpha $ and $ \beta $ parameters used in the DD2.

Figure 8 shows the fitting results of all candidate functions for three representative sites, each from a different road class. It can be seen that the base curves with the default speed and capacity values fit the poorest to the data across all three examples. In particular, the base curves underestimate the free-flow time for Site 9 S (Figure 8a, Trunk road) and 11 N (Figure 8b, Principal road), while overestimating the free-flow time for Site 35 S (Figure 8c, B road). The base curves overstate the capacity in all three examples, which has already been shown in Figure 7b. By simply substituting the free-flow time $ {t}_0 $ and capacity $ {v}_c $ in the base curve with the observed values, the fitting of the DD1 improves significantly based on a visual inspection. Quantitative evaluations of the fitting performance will be given next in Section 6.4. The DD2 (green curve in Figure 8) fits the data more closely than the DD1. However, in certain cases, the DD2 may result in unrealistic $ \beta $ values of less than one, due to a large number of observations at medium volume but large delay such as the hyper-congestive conditions, thereby making the volume–delay curve unreasonably concave. Hence, the DD1 is a good approach in general to improve the performance of volume–delay curves.

Figure 8. The fitting three different models in the observed data at three sites.

By dividing the length of the road link with the link travel time shown in Figure 8, the speed–flow relationship can be obtained. The DfT TAG uses linear or piece-wise linear models to predict the reduction in speed with increasing flow. Depending on the road class (e.g., whether it is urban, suburban, rural), different formulae are used. For each class, the speed–flow formulae again depend on a variety of factors (e.g., percentage of frontage development, numbers of major and minor intersections). It should be noted that, although a speed–flow formula is provided for urban roads, the preferred way of doing traffic assignment is still to model junctions and low cruise speed in congested urban areas explicitly, rather than relying on the speed–flow curves. If to be used, it should be recognized that the speed–flow curves are developed based on network-average conditions, which limits their applicability to individual links independently. Fully recognizing these limitations, a comparison plot is still made for the observed data points and the speed–flow curves in the TAG document for illustration purposes (Figure 9). The blue dots are the speed–flow observation data. The black and green curves are the same as the ones in Figure 8, only showing in the speed–flow plane. The purple lines are the speed–flow relationship made according to the TAG model. Parameters of the purple curves are given in Table 4 and these values are obtained by examining Google Street View of each site presented in Figure 9. It can be seen that the purple curves do not agree well with the local data observations. Part of it is because link observations (e.g., the percentage of frontage development for this particular street) are used in place of network-average values. While it is also true that the curve parameters, even if to be collected at the network level, are somewhat subjective/elastic and may not represent all context-specific information (e.g., bus stops, subtle geometry effects, or temporary closures). In comparison, the proposed crowd-sourced data collection approach can then provide a more localized, context-specific estimation of the volume–delay curves, or even used to calibrate parameter values of the TAG models.

Figure 9. Comparison with the DfT TAG speed–flow curves for three sites.

Table 4. Parameter values for curves in Figure 9.

6.4. Variation quantification

As it can be seen in Figure 8, the real-world traffic data are very noisy to be represented by a single curve. There have been second-order models that include a family of curves, which can reproduce the variations in travel time around the same traffic flow conditions to a certain degree (Fan, Reference Fan2020). However, well-calibrated first-order models that only include the speed term (no time derivatives of it), such as the BPR curve are still beneficial for specific tasks, including analysing the impact of building new infrastructures and conducting regional-wide static and semi-static traffic simulations. After fitting the three candidate models to the data, their fitting performances are analysed in this section.

The fitting performance is evaluated with the mean absolute error (MAE, Equation 2) because the MAE metric has the same unit as the dependent variable (time in seconds or speed in km/hr) and is less sensitive to outliers compared to the frequently used root mean squared error metric. As the variations in the observed travel time are uneven and more substantial when the hourly traffic volume approaches the capacity, MAE is calculated for different subsets of the data, divided based on the quartile of the hourly traffic volume. Figure 10 shows the MAE in terms of travel time (a)–(c) and speed (d)–(f) estimations of all roads, grouped by the volume-to-capacity ratio of the observed data. For example, Q1 refers to the subset of the data where the hourly traffic volume are within 0–25% of the estimated capacity. Q5 refers to the subset of the points where the hourly traffic volume exceeds the estimated saturation flow. The Q5 partition exists because capacity in this study is defined as the 95th percentile hourly traffic volume for observations with travel times between 1.8 and 2.2 times the free-flow time (Section 6.2). It can be seen that, in terms of travel time or speed, the base curve leads to the largest errors in estimating the time delay given the traffic volume (Figure 10a,d). While for the two data-informed models, the errors of the DD2 appear to be less, but not much lower than the DD1. As mentioned before, engineers and planners would normally apply a series of factors on $ {t}_0 $ and $ {v}_c $ for the base curve to reflect city or street characteristics (e.g., signal green time ratio or road curvature). We tried to apply these factors to scale $ {t}_0 $ and $ {v}_c $ so that the base value matches more closely with the observed value in Figure 7, but it still produces a larger MAE compared to the two data-informed curves due to its lack of ability to adapt to individual street situations.

(2)$$ {\displaystyle \begin{array}{c}\mathrm{For}\quad \mathrm{travel}\quad \mathrm{time}:{MAE}_t=\frac{\sum_{i=1}^n\mid {t}_{i,\mathrm{model}}-{t}_{i,\mathrm{obs}}\mid }{n},\\ {}\mathrm{For}\quad \mathrm{speed}:{MAE}_u=\frac{\sum_{i=1}^n\mid {u}_{i,\mathrm{model}}-{u}_{i,\mathrm{obs}}\mid }{n},\end{array}} $$

where $ {t}_{i,\mathrm{model}} $ and $ {u}_{i,\mathrm{model}} $ are estimated travel time and speed from the model; $ {t}_{i,\mathrm{obs}} $ and $ {u}_{i,\mathrm{obs}} $ are individual travel time and speed observations; $ n $ is the total number of observations.

Figure 10. Boxplots of the overall fitting MAE for all study sites for three model types, grouped by the traffic volume level. (a–c) MAE for the estimated travel time. (d–f) MAE for the estimated link speed.

The above results not only indicate the performance of the models, but also give engineers quantitative evidence on what amounts of variations to be expected when using the simple volume–delay function curves in their estimation. For example, Figure 10a–c shows the variations between the observed link traversal time and the estimated travel time by each individual model. For the roads selected as the study sites, the link travel time predictions can be 1–2 min off using the base curve, while the errors are usually below half a minute using the data driven models (DD1 and DD2), except in Q5, where the traffic volume is relatively high (>observed capacity) and the errors increase to 0.5–1 min. The model prediction errors are better discussed by comparing the predicted speed against the observed speed at different traffic levels, as in Figure 10d–e since the speed measure can be compared across roads. As seen in Figure 10d–e, without the site-specific knowledge, the speed estimations using the default volume–delay curve can vary by 10–25 km/hr (Figure 10d). Nevertheless, merely fixing the speed and capacity estimations (DD1), the variation in the speed estimation can be reduced to less than 5 km/hr for low to medium traffic volume cases, and less than 10 km/hr when the volume is close to the capacity (Figure 10d). If using calibrated coefficients $ \alpha $ and $ \beta $ as in the DD2, the variation in the speed estimations are almost always below 5 km/hr from the observations (Figure 10e). However, it should be noted that calibrating $ \alpha $ and $ \beta $ without constraints may produce concave volume–delay curves in the case of the DD2.

Table 5 shows the MAE of speed (km/hr) by model, road class, and volume level, which is an extension of the information in Figure 10d–f disaggregated by the road facility type. As can be seen in the table, the Principal roads generate the highest number of observations, while the Unclassified roads have the least. Among the observations from Principal roads, the largest group of observations have the volume-to-capacity ratio around 75–100%, indicating a high demand. At the same time, less important roads (e.g., C or Unclassified) have the majority of the observations in the low traffic category (less than 75%), which means less congestion in general on these types of roads. The general trend on the MAE for the estimated speed is not much different across different road classes. For all road classes, the MAE of the estimated speed is significantly higher when using the base curve and lower when using the data-informed models. Also, the MAE for speed, in general, grows with the volume level: when the traffic volume-to-capacity ratio is higher, the MAE for speed goes up as well, which is in agreement with the visual judgements made in Figure 10.

Table 5. Speed MAE (km/hr) by model form, road type, and traffic volume level.

7.1. Example scenario and possible factors

Figure 11a presents the observed volume and delay scatter plot for ATC location 66 in both directions. It is obvious that the southbound direction consistently experiences higher delay than the northbound direction, even at same volumes of traffic. In an attempt to better understand and possibly explain this distinct directional behavior, satellite photography, and street-level photography of the road are assessed. Figure 12 shows a satellite image of ATC location 66 obtained from Google Maps (Google Inc., Reference Inc2020). The red marker gives the location of the ATC counter itself, and the two blue markers illustrate the origin/destination location used in the Google Directions API request. From this satellite imagery and street view imagery, possible explanatory factors can be identified:

1. 1. The southbound lane features road parking.

2. 2. The southbound lane features a large bus stop and taxi lay-by (for Leytonstone Station) and a smaller bus stop lay-by.

Figure 11. Challenging scenarios: volume and delay observations at ATC site 66 (a) and 67 (b).

Figure 12. Satellite and street view images of ATC 66 (Google Inc., Reference Inc2020).

The larger bus stop and taxi lay-by serving Leytonstone is a significant geometric feature that is likely to have a significant impact on the southbound traffic. The second smaller bus stop lay-by and on-road parking may also have an impact, albeit to a smaller extent. Such factors may explain the exhibited differences from the location-device-informed journey data and thus permit their inclusion in an informal way.

7.2. Challenging scenarios

Inspections of the volume–delay scatter plots reveal a subset of roads that are not suitable to be described with the BPR-style monotone function. For example, as previously explained in Section 5, 15 sites are eventually removed from the analysis due to invalid data. Among the remaining 24 sites, many have outliers even after the data cleaning process (e.g., Figure 6). Figure 11b shows another “abnormal” example, particularly with the northbound direction: the delay has a considerable variation when the hourly volume is around 200 vehicles, far from its capacity of about 400 vehicles per hour. This is probably resulted from the hyper-congested stage of the traffic, where the density has exceeded the optimum value and flow decreases despite continuous increase in the delay. Such phenomena cannot be correctly modeled by the monotone volume–delay function and should be dealt with using dynamic models if the hyper-congested stage is crucial in the analysis (which is usually the case for short-term predictions, such as evacuations). This is also reflected in the modeling guidance from the DfT, where the preferred choice of modeling congested urban roads in traffic assignment is to explicitly represent the junction delay and low cruise speed (DfT, 2020). The proposed data-driven models are useful in certain scenarios, such as traffic assignment at the city-scale, where simplicity of the volume–delay curves is crucial to ensure the computational tractability of the model. However, its limitation, namely allowing unrealistic high-traffic flow to be assigned to certain links, should be aware of. We plan to compare the performance of the data-driven method with the more detailed models involving junctions in our future work. The errors of using the monotonic volume–delay function in face of hyper-congested real-world observations are reflected in the MAE quantification as shown in Figure 10. Apart from these reasons, external factors could also affect the sensibility of the data, for example, a traffic collision or flooding may result in a high level of delay in a short period of time.

7.3. Influencing factors

In Section 7.1 the existence of public transport infrastructure and on-road parking was identified as being potentially explanations of the distinct southbound northbound behavior on that road. A range of other possible factors have been identified, which may help explain the unexplained variations:

1. 1. Vehicle type

   In this study, all vehicles are considered to be the same, despite clear distinction in the interactions between two cars compared to an interaction of a car and a lorry (Vap et al., Reference Vap and Sun2007). Further work is planned to assess the impact of the vehicle type. A wide variety of vehicles are observed on different road types, and such differences should be considered. The use of statistical vehicle mix sampled by road type (DfT, 2012b) was discounted for this study due to the small sample size of such statistics compared to the resolution of the data used here. The use of traffic cameras with number plate recognition and sufficient privileges to the Driver and Vehicle Licensing Agency database would enable the disaggregation of vehicle type at a similar spatial and temporal resolution to the Google Directions and DfT ATC data presented here.

2. 2. Weather

   Weather events may impact the journey times by altering the functionality of the vehicle, the performance of the road, and/or the performance of the driver. Ongoing research focuses on combining a large dataset of Google Directions journey times with data from the UK’s Met Office NIMROD precipitation dataset (The UK Met Office, 2003) to assess the relationship between these variables. At a microscopic level, it is known that an increase in precipitation increases journey times as a result of increased risk and the resulting decrease in vehicle speeds to compensate for this factor (Mashros et al., Reference Mashros, Ben-Edigbe, Hassan, Hassan and Yunus2014).

3. 3. Road incidents

   Road works and road traffic collisions can lead to decreased or even zero capacity on a road link, resulting in increased saturation which impacts the journey time. Depending on the warning before such an event, the vehicle traffic may have the ability to adjust to this information, resulting in a greater distribution of traffic thus reducing the mediated travel times. Alternatively, an accident may occur without any warning to other road users committed to their route choice, resulting in longer journey time and perhaps gridlocks in the extreme cases. A method incorporating different accident and roadworks databases with Google Directions data is currently being investigated.

4. 4. Road geometry, type, and land use

   Different road layouts may increase the complexity of vehicle interactions. For example, the curvature of a corner and the road surface quality will impact the speed of a vehicle. The surrounding land use also has an influence on the vehicle speeds (e.g., drivers take safety precautions near a school or leisure center). The inclusion of such factors poses many challenges, including the size and complexity of the data plus the uncertainty and variability in how drivers react to the data. In Figure 12, a series of geometric factors are displayed as an attempt to explain the different behaviors on the same road in different directions. The factors shown in Figure 12 such as on-road parking and bus lay-bys could be quantitatively captured using computer vision and data sources such as Google Street View.

7.4. Generalization and application

This study proposes an efficient data collection and processing pipeline for calibrating the macroscopic volume–delay relationships for urban roads. The methodology considers the context-specific factors that might affect such relationships but are hard to quantify using conventional approaches. Apart from the challenges of using simple volume–delay curves in capturing the complex traffic behavior, another bottleneck for the proposed methodology is the availability of the data. This limitation comes in two parts. First of all, the volume data could only be obtained for a small number of fixed locations, where the ATCs are installed and can provide continuous traffic counts. The DfT also conducts manual traffic surveys at a larger number of locations. However, these manual count data usually come at a lower frequency (e.g., annual data) and may again miss the temporal network changes that have an impact on traffic (e.g., roadworks). This bottleneck could be overcome in the near future by employing alternative data sources of traffic flow information, such as the information extracted from traffic cameras using computer vision techniques (Drake, Reference Drake2018). Secondly, the traffic speed data from Google (or other mobility data providers) are not free, which may require the cost of acquiring data to be budgeted in for large-scale studies. However, this method is expected to be cheaper and more scalable than collecting the speed/delay data from conventional methods, such as the floating car data method.

Nonetheless, the proposed methodology could still be beneficial in understanding the performance of key network locations, such as bridges or major routes, when only point-based traffic measurements are available. In addition, if more types of traffic sensors are installed in the future (e.g., radar or camera), the proposed methodology could then be applied on a larger scale.