6. Calibration and Evaluation

This section describes a calibration procedure for the speed-density relationship and preparation of WAF.dat file discussed in the previous section. WAF.dat file contains information about changes in traffic parameters according to visibility and precipitation intensity such that weather impact can be reflected in a simulation process in DYNASMART. The weather and traffic data used for calibration are obtained from Archived Data Management System (ADMS) in Virginia.

6.1 Data for Calibration

The Archived Data Management System (ADMS) Virginia archives both traffic data and weather data for selected locations in Virginia. Weather data consists of visibility (1-10 mile), precipitation (inch/hour) and weather type. Collection times are not at a fixed interval. Usually there are 1-5 observations for one hour. Traffic data consists of vehicle count, occupancy and speed at 5-minute aggregation interval. The study site is the Hampton Roads network, composed of three freeways segments: (1) I-64 between Bay Avenue and the Virginia Beach-Chesapeake City Limits, I-64 forming the outer loop, (2) I-564 between Terminal Boulevard and I-64, I-264 between Broad Creek and Rosemont Road, and (3) entire I-664.

Figure 6-1. The Hampton Roads Network

The densities could be converted from the occupancy data using the following relationship:

where

k = density [veh/mi/ln]

L_v = average vehicle length [feet]

L_s = average sensor length [feet]

occ = occupancy [%]

L_v is assumed to be 5 meters (approximately 16.4 feet); and L_s is set to 2 meters (approximately 6.5 feet).

6.2 Calibration of Speed-Density Function

DYNASMART uses a modified Greenshields model for traffic propagation. In the current version, two types of the modified Greenshields family models are available. Type one is a dual-regime model in which constant free-flow speed is specified for the free-flow conditions (1st regime) and a modified Greenshields model is specified for congested-flow conditions (2nd regime). Dual-regime model is generally applicable to freeways. The reason why a two-regime model is applicable for freeways in particular is that freeways have typically more capacity than arterials and can accommodate dense traffic (up to 2300 pc/hr/ln) at near free-flow speeds. Hence, a slight increase in traffic would not significantly deteriorate prevailing speeds in the 1st regime.

Figure 6-2. A Dual-Regime Modified Greenshields Model

In mathematical terms, the dual-regime modified Greenshields is expressed as follows:

where

v_i = speed on link i

v_f = speed-intercept

u_f = free-flow speed on link i

v_o = minimum speed on link i

k_i = density on link i

k_jam = jam density on link i

α = power term

k_bp = breakpoint density

Type two uses a single regime to model traffic relations for both free- and congested-flow conditions, i.e.

Figure 6-3. A Single-Regime Greenshields Model

In mathematical terms, the type 2 modified Greenshields is expressed as follows:

Parameters for the dual-regime Greenshields model (in DYNASMART-P) can be calibrated for the freeways in the Hampton Roads network, that is, I-64, I-264 and I-564, using the time-dependent traffic data from ADMS Virginia. There are six parameters to be calibrated, namely, breakpoint density (k_bp), free-flow speed (u_f), speed-intercept (v_f), minimum speed (v_o), jam density (k_jam), and the power term (α ).

The speed-density relationship could be approximated by two portions, a straight portion and a curvilinear portion. Hence two equations must be estimated to correctly and adequately represent the freeway traffic model structure. The straight portion of the speed-density relationship is represented by the equation:

For the straight portion of the model, only one parameter needs to be estimated, namely the mean free speed uf, which reflects the true prevailing freeway speed under uncongested conditions. On the other hand, the modified Greenshields' model is used to describe the curvilinear portion (second regime) of the speed-density relation, which is expressed by the following equation.

Linear regression analysis is the major tool for the calibration of the link traffic flow models. This can be achieved by transforming the modified Greenshields' model into a linear form by taking the natural logarithm on both sides:

which is in the form of

and can be estimated directly by conducting a simple linear regression analysis. The parametric analysis procedure is also implemented to help for linear regression analysis.

The data required to calibrate this component includes:

• Time-varying link density
• Average speed for the corresponding time intervals

The MOE for this task include:

• Goodness of fit – R-squared value in the linear regression;
• Root mean squared error for speed.

The procedures of calibrating speed-density function are as follows.

Step 1. Process observation data

Step 1.1. Categorized the traffic data (speed and occupancy), for each location, into five data sets according to the weather condition (i.e., precipitation intensity), namely, normal, light rain (less than 0.1 in./hr), moderate rain (0.1 to 0.3 in./hr), heavy rain (greater than 0.3 in./hr), and light snow (less than 0.1 in./hr). Since there have been not much data available for heavier snow, only one category is used for snow.

Step 1.2. Convert occupancy into density using Equation (6 1).

Step 1.3. For each location and each weather condition, perform Step 2 to 5.

Step 2. Fit the data into a dual-regime model. For initial kbp of 10 vpmpl, do the followings.

Step 2.1. Divide the data set into to subsets based on the initial kbp, that is, the first and second regime observations.

Step 2.2. For the first regime, the free-flow speed, uf, is estimated as the mean of the speeds. Root mean squared error for speeds is also calculated.

Step 2.3. For the second regime, set v0 and kjam based on the observations, that is, the minimum speed observed and maximum density observed.

Step 2.4. Transform the second regime data, speed and density, as follows:

Step 2.5. Perform linear regression of the function Y = αX + b to estimate α and b.

Step 2.6. Recover v_f from the estimated b, that is, v_f = e^b + v₀.

Step 2.7. Calculate R-squared value for the second regime.

Step 2.8. Calculate difference in estimated speeds at the joint of two regimes by comparing u_f in the first regime and the modeled speed value at k_bp in the second regime.

Step 3. Increase k_bp by 1 vpmpl and repeat Step 2.1 to 2.8 until k_bp becomes 30 vpmpl.

Step 4. Find the optimal value of k_bp based on MOEs of the fitted models for each regime and joint fit observations for the entire models.

Step 5. Choose the function that fits best to the data set for each weather condition.

The calibration result for one freeway section of I-64 is presented in Figure 6-4. It shows the weather effect on speed-density relation and flow-density relation due to reductions in speed for both 1st and 2nd regime under rain and snow events. Detailed calibration results for all study sections are presented in Appendix A.

Figure 6-4. Calibrated Speed-Density and Flow-Density Curves for a Freeway Section I-64
(Data source: The Archived Data Management System Virginia)

6.3 Calibration of Weather Adjustment Factors

Once speed-density functions for different weather conditions (i.e., normal, light rain, moderate rain, and light snow) are obtained for each location, another linear regression is conducted to estimate weather adjustment factor coefficients in Equation (5 1) for the WAF.dat file.

The procedures for preparing WAF.dat are as follows.

Step 1. Calculate the WAFs for each parameter using the relation , where u and u' represent the parameters under the normal condition and the rain (or snow) condition respectively.

Step 2. Conduct linear regression for each parameter using the WAF for a dependent variable and visibility and categorized precipitation intensity of each data point for independent variables.

Step 3. Prepare WAF.dat file using calibrated coefficients for each parameter from Step 2.

Note that not all of the parameters listed in Table 5-3 can be calibrated using the observation data. Some parameters could be inferred from other calibrated parameters.

(1) Traffic flow model related parameters, that is, speed-intercept (v_f), minimum speed (v₀), density break point(k_bp), jam density(k_jam), shape term alpha (α) and maximum service flow rate (f_max) can be calibrated from the traffic data. However, as minimum speed, jam density and shape term alpha turn out to be insensitive to weather conditions from the calibration results, WAF for those parameters are assumed as 1, which indicates these are not affected by weather conditions.

(2) Link characteristics: saturation flow rate, and posted speed limit adjustment could be inferred from the calibrated traffic flow model.

(3) Signal control: the adjustments in cycle length, offset, green, amber, maximum green, and minimum green could be inferred from the saturation flow rate.

(4) Left turn/stop sign/yield sign capacities could be calibrated using the traffic data, for example, maximum observed flow rate could be used as a surrogate of capacity.

The detailed calibration results of WAF are provided in Table 6-1.

Table 6-1. Coefficients of Weather Adjustment Factor

Input data	Traffic properties	β0	β1	β2	β3	β4	β5
Traffic flow model	1. Speed-intercept, (mph)	0.91	0.009	-0.404	-1.455	0	0
Traffic flow model	2. Minimal speed, (mph)	1	0	0	0	0	0
Traffic flow model	3. Density break point, (pcpmpl)	0.83	0.017	-0.555	-3.785	0	0
Traffic flow model	4. Jam density, (pcpmpl)	1	0	0	0	0	0
Traffic flow model	5. Shape term alpha	1	0	0	0	0	0
Link	6. Maximum service flow rate, (pcphpl or vphpl)	0.85	0.015	-0.505	-3.932	0	0
Link	7. Saturation flow rate, (vphpl)	0.91	0.009	-0.404	-1.455	0	0
Link	8. Posted speed limit adjustment margin, (mph)	0.91	0.009	-0.404	-1.455	0	0
Left-turn capacity	9. g/c ratio	0.91	0.009	-0.404	-1.455	0	0
2-way stop sign capacity	10. Saturation flow rate for left-turn vehicles	0.91	0.009	-0.404	-1.455	0	0
2-way stop sign capacity	11. Saturation flow rate for through vehicles	0.91	0.009	-0.404	-1.455	0	0
2-way stop sign capacity	12. Saturation flow rate for right-turn vehicles	0.91	0.009	-0.404	-1.455	0	0
4-way stop sign capacity	13. Discharge rate for left-turn vehicles	0.91	0.009	-0.404	-1.455	0	0
4-way stop sign capacity	14. Discharge rate for through vehicles	0.91	0.009	-0.404	-1.455	0	0
4-way stop sign capacity	15. Discharge rate for right-turn vehicles	0.91	0.009	-0.404	-1.455	0	0
Yield sign capacity	16. Saturation flow rate for left-turn vehicles	0.91	0.009	-0.404	-1.455	0	0
Yield sign capacity	17. Saturation flow rate for through vehicles	0.91	0.009	-0.404	-1.455	0	0
Yield sign capacity	18. Saturation flow rate for right-turn vehicles	0.91	0.009	-0.404	-1.455	0	0

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