Multi-Variable Regression and Sensitivity Analysis Factor Selection

Several factors with demonstrated effects on fuel efficiency of passenger rail transportation were selected for this analysis. Each factor and representative range of values was selected to reflect real-world conditions on a mixed-use corridor (Table 5.2). Previous studies and train energy modeling consider grade one of the most significant factors in train fuel consumption (Hay 1953, Sierra Research 2004, Fullerton et al. 2014, Tolliver et al. 2014). Although existing mainline grades can exceed 3.0% for short distances, the maximum grade for freight railroad that most passenger trains operate on is about 1.0% (AREMA 2003).

Table 5.2 Factors investigated in freight and passenger fuel efficiency study

Passenger Experiment Factors Low Medium High

Grade (%) -2.22 0 2.22

Traffic Volume (trains per day) 8 16 24

Percentage Freight Trains 25 50 75

Passenger Speed (miles per hour) 50 79 110

Train Length (number of coaches) 6 9 12

Locomotives - 1 2

Station Spacing (miles) 5 120 240

Traffic volume was included as a proxy for congestion, representing realistic train volumes for a single-track line with passing sidings. The percentage of freight trains represented the heterogeneity of traffic on the line. Lines with greater heterogeneity have been shown to experience more train delay for a given traffic volume (Dingler et al. 2013, Dingler et al. 2014). The extra delay could disproportionately affect freight train fuel consumption with increased passenger traffic (Sogin et al. 2012a).

Speed of freight and passenger trains was also considered due to the increased train resistance associated with higher speeds. Train resistance forces also increase with train length and weight, represented by number of cars and gross railcar load. Train length was also considered, but passenger coach weight was held constant across all simulations. These factors also determined the transportation productivity of each train, as they control the number of seat- miles to be used in the fuel efficiency calculation. The number of locomotives providing tractive effort was also a factor.

Design of Experiments (DOE) methodology was used to develop a partial-factorial experiment matrix that designates unique cases with varying combinations of factors in Table 5.2 to accurately describe the response surface. Using DOE reduced the required computing time and power by using a partial-factorial experiment, rather than full-factorial. The partial-factorial passenger experiment constructed using DOE required 80 cases, compared to 1,458 total cases for a full-factorial experiment.

Multivariate Regression Analysis

Fuel efficiency results from the simulations in RTC were used to construct a multivariate regression model, with the factors in Table 5.2 as input parameters and fuel efficiency as the response output. JMP was used to analyze the results from the RTC simulations and construct the regression model. The model recreated the response surface using the least squares regression technique with the results from the RTC simulations. Fuel consumption (gallons) was predicted by the passenger fuel efficiency regression model. Efficiency (seat-miles per gallon) was then evaluated separately based on the output of the model and the respective number of seats in the train consist. This was done to avoid over-fitting in the software due to the effects of seating capacity on the relationship between the response variable and inputs.

Sensitivity Analysis

After constructing the regression model, the sensitivity of the calculated fuel efficiency to changes in the inputs (Table 5.2) was quantified by the arc elasticity method as described by Allen and Lerner (1934). This approach has been used in similar research investigating the influence of different factors on train delay (Sogin et al. 2013). The arc elasticity method quantified how fuel efficiency responds to normalized unit changes in each factor with respect to

a base case. Normalization of units is desirable due to the bias that could be introduced by varying the units and order-of-magnitude ranges of the analyzed factors.

The base case for the experiment used the combination of the medium factor values (Table 5.2), with a few exceptions. The factor ranges used to construct the regression model include the extreme values of each parameter in order to produce a larger response surface. Therefore, the high and low values for several factors did not present a reasonable range of uncertainty or variability in anticipated service conditions. Thus, some of the base, minimum, and maximum factor values were modified in the sensitivity analysis to reflect more realistic ranges. Each factor was varied between its low and high values, while the other factors were held constant at their respective medium value and recording the output fuel efficiency (Figure 5.3). The arc elasticity of the fuel efficiency response was then calculated (Equations 5.1 and 5.2). The output of each equation was the ratio of the percent change in fuel efficiency to the percent increase or reduction in the factor. Arc elasticity values represented the corresponding percent- change in fuel consumption to a one-percent change in each input factor. Larger magnitudes of arc elasticity indicate that a particular factor has a larger influence on fuel efficiency than factors with lower magnitudes of arc elasticity.

Table 5.3 Base, minimum, and maximum values used in passenger fuel efficiency sensitivity analysis

Passenger Experiment Factors Minimum Base Maximum

Grade (%) -2.22 0 2.22

Traffic Volume (trains per day) 8 16 24

Percentage Freight Trains 25 50 75

Passenger Speed (miles per hour) 50 79 110

Train Length (number of coaches) 6 9 12

Locomotives - 1 2

Where:

𝑒_{ℎ𝑖𝑔ℎ} = arc elasticity with respect to high input variable value 𝑌 = output variable

𝑋_𝑖 = input variable

(𝑋_𝑖0, 𝑌_𝑖0) = base input and corresponding response

Where:

𝑒_𝑙𝑜𝑤 = arc elasticity with respect to low input variable value