Alternative Impedances for Shortest Path Network Analysis for Cycling

Alternative Impedances for Shortest Path Network Analysis for Cycling Introduction Traditional impedances used in network analysis are travel time, travel distance and travel cost or some type of cost function. For cycling, these parameters might not be adequate for cyclists who like to consider the degree of difficulty of traveling by bike on roads in terrain with even modest grades. Alternative impedances that estimate the degree of difficulty of a route for a cyclist are the work and power required for the trip. Work is defined as the application of a force over a distance. For example, lifting a weight of 100 pounds vertically 10 ft would result in 100 lbs times 10 ft of work or 1,000 ft lbs. The force is that required to lift the 100 lb weight. The distance is 10 ft. For a cyclist, two forces were considered in calculating the amount of work required for each link on a trip weight and air resistance. In overcoming the force of weight, only the component of weight that is acting against him or her as they travel uphill is considered. (If the cyclist was traveling vertically up a wall, then their total weight would have to be overcome as in the example above). weight times the upgrade decimal slope of the street link approximates the component weight force that needs to be overcome on that upslope. The distance is the length of the street link. Multiplying this force times the length of the street link gives the amount of work in ft lbs required to move your weight uphill. The force in pounds of air resistance is given by the equation: 0.5 * C d * ρ * A * V² (slug ft per sec 2 ) C d is the coefficient of drag (1.2), rho (ρ) is the density of air in slugs per ft 3 (.00237), A is the cross sectional area of the cyclist facing the wind (7 ft 2 ), and V is velocity in ft per sec 2. Multiplying this force times the length of the street link gives the amount of work in ft lbs required to overcome air resistance. One slug ft per sec 2 = 1 lb of force. Multiplying this force times the length of the street link gives the amount of work in ft lbs required to overcome air resistance. However, work is a metric that is independent of time if 100,000 ft lbs of work is required for a bike trip from point A to point B, it does not matter the time required to make that trip. Power on the other hand takes both work and time into consideration. Power is the rate of work, i.e. work divided by time (ft lbs per sec). One horsepower is equivalent to 550 ft lbs per sec. Page 1 of 16

The velocity model used in this study made the following assumptions: (slope is in decimal; i.e. a 1% slope is.01 decimal) Base Velocity = 10 mph or 15 ft per sec Uphill Downhill Velocity = 15 [(+slope) * (125) ] ft per sec Minimum of 3 ft per sec (walking speed) Velocity = 15 [( slope) * (500) ] ft per sec Maximum of 37 ft per sec (25 mph) Data Acquisition and Preparation Two data sources were used in this project: the Portland METRO RLIS street shapefile and the Oregon 10 meter resolution Digital Elevation Model (DEM). Both datasets are found on the Portland State University (PSU) I: drive. In order to calculate velocity, time, work and power, the profile slope of each street link is needed in each direction. End point elevation is therefore needed for each link. The first phase in preparing the RLIS street shapefile for use in Network Analysis was: (1) Convert the shapefile to a coverage. This step creates directional characteristics for each street link by establishing the required FNODE and TNODE fields needed in a network; (2) Second, additional fields had to be created: FNODE_Elevation and TNODE_Elevation; (3) Third, create a point coverage composed of the end nodes of each street link; (4) Fourth, extract elevations from the DEM to these end points; (5) Fifth, add X,Y coordinates to this coverage so that later a feature class of these nodes can be created to be joined back to the street link coverage; and (6) Export the attribute table as a.dbf file so that it could be manipulated in ACCESS and EXCEL. The second phase involved manipulating the end node attribute table in ACCESS and EXCEL in order to produce the following directional (FromNode, ToNode) fields for each record: FNODE_Elev, TNODE_Elev, FT_Slope, TF_Slope, FT_Velocity, TF_Velocity, FT_Time, TF_Time, FT_Work, TF_Work, FT_Power, and TF_Power. In order to determine which RASTERVALU elevation from the DEM belonged to the FNODE_Elev and TNODE_Elev field, the end node attribute table and the street coverage file had to be investigated to determine what pattern or logic existed in the end point coverage. After investigation of this issue, a pattern was discovered in the end point file. There were pairs of records FNODE to TNODE pairs for each corresponding link in the street coverage with a unique identifier present in the attribute tables for each pair of records: LOCALID. The number of records in the end node file was exactly twice the number of records in the street link coverage Page 2 of 16

file. And the FNODE TNODE record pairs were in order: the first record being the FNODE and the second record being the TNODE. So, beginning with record one in the node file, this record corresponded to an FNODE and the RASTERVALU elevation from the DEM on that record was assigned to the FNODE_Elev field. Record two was the corresponding TNODE to that FNODE. The RASTERVALU elevation from the DEM on record two was assigned to the TNODE_Elev in record one. Then, record two was deleted. This process was repeated for the entire end node attribute table, cutting the number of records in half. Once a FNODE_Elev and TNODE_Elev were established on a given record, the remaining fields listed above could be calculated. This revised end node table was then joined back to the street coverage using the unique identifier present in the attribute tables for both files: LOCALID. The third phase involved a moderate amount of checking the revised street coverage with the new directional impedance fields joined to it. This phase consisted of checking FNODE and TNODE elevations in the attribute table with elevations from the DEM. Also, in the joined file, FNODE to TNODE values were compared from the original street coverage fields to the joined end node table fields. Analytical Method With the updated street coverage attribute table, shortest path network analysis was conducted for several origins and destinations within the Portland metro area. Two shortest path routes were calculated using Network Analysis for each OD pair, one minimizing time and one minimizing work. Origins and destinations were chosen to be positioned on either side of hilly or mountainous terrain in Portland in order to see if the shortest path that was minimizing work would try to avoid steep grades. Likewise, the analysis wanted to determine if the shortest path that minimized time would be less sensitive to steep grades when selecting its path. Power was calculated for each of the two routes above after they were determined by dividing the total work for that route (ft lbs) by the total time (sec) and converting this to horsepower. Power could not be summed as an impedance for each link, as was done for time and work, because power is a rate work per time. It would be analogous to trying to sum velocities (distance per time) for each link on a route. See the flow chart at end of document which summarizes the study methodology. Page 3 of 16

Findings and Conclusions Five origin/destination pairs were evaluated for routes that minimized time, work and power. The following figures and tables summarize the results for each OD pair. The study results indicate that there is a trade off between the amount of time and the amount of work and power required to make a trip via bike over hilly terrain. The case study origin destination pairs in this study showed a reduction in the amount of power required of about 25% to 50% when taking the minimum work path versus taking the minimum time path. Similarly, there was a corresponding increase in travel time by about 25%. The velocity model used in this study was based on subjective judgment and personal experience. A different velocity model might yield different results. For example, it was assumed that velocity would decrease by 1.25 ft/sec per 1% upgrade. With a base speed of 15 ft/sec, the velocity would be reduced to a walking speed of 3 ft/sec on an 8% upgrade. If the velocity were decreased at a faster rate on an upgrade then the shortest paths selected based on minimizing time might have been more likely to avoid steeper upgrades. Page 4 of 16

Case Study Origin Destination Pairs Origin/Destination Pair #1 Portland Art Museum to East of Mt. Tabor The first OD pair was from the Portland Art Museum to east of Mt. Tabor. As seen in the graph below, the route that minimized work avoided climbing part of Mt. Tabor, unlike the minimum time route. Portland Art Museum to Mt. Tabor Shortest Paths Minimize Time O D Minimize Work 1 The following table summarizes the results: Portland Art Museum to Mt. Tabor Route Summary Statistics Shortest Path Type Minutes Miles Work (ft lbs) Avg Speed (mph) Average Horsepower Time 33 5.2 90,501 9.4 0.082 Work 42 6.9 67,153 9.9 0.049 37 Page 5 of 16

Origin/Destination Pair #2 Portland Art Museum to Portland Golf Club The second OD pair was from the Portland Art Museum west to the Portland Golf Club. As seen in the graph below, the route that minimized work avoided climbing what appears to be the steeper grades west of downtown Portland and waited for as long as possible to turn west and head in the direction of the destination. Portland Art Museum to Portland Golf Club Shortest Paths Minimize Time O D Minimize Work 38 The following table summarizes the results: Portland Art Museum to Portland Golf Club Route Summary Statistics Shortest Path Type Minutes Miles Work (ft lbs) Avg Speed (mph) Average Horsepower Time 49 7.0 193,428 8.5 0.119 Work 66 10.8 134,893 9.8 0.062 Page 6 of 16

Origin/Destination Pair #3 PCC Cascade to PCC Sylvania The third OD pair was from PCC Cascade to PCC Sylvania. These routes do not differ as much as the previous two OD pairs. The destination, PCC Sylvania, is located on top of a peak and there is really not too much of an option for either route to take. We can still see that the minimal work route waited until the end before climbing the steeper grades to the west. Also, this route appears to make a switch back type of maneuver upstream of the destination on top of the peak. PCC Cascade to PCC Sylvania Shortest Paths O Minimize Time Possible Switch Back Maneuver Minimize Work D 40 The following table summarizes the results: PCC Cascade to PCC Sylvania Route Summary Statistics Shortest Path Type Minutes Miles Work (ft lbs) Avg Speed (mph) Average Horsepower Time 71 11.0 212,954 9.2 0.091 Work 82 12.7 188,376 9.2 0.069 Page 7 of 16

Origin/Destination Pair #4 PCC Cascade to Tigard The fourth OD Pair went from PCC Cascade to Tigard. Here we see the minimal work path avoiding the steep terrain to the west altogether. PCC Cascade to Tigard Shortest Paths O Minimize Time Minimize Work D 42 The following figure and table summarize the results: PCC Cascade to Tigard Route Profile 43 Page 8 of 16

Origin/Destination Pair #4 PCC Cascade to Tigard (cont) PCC Cascade to Tigard Route Summary Statistics Shortest Path Type Minutes Miles Work (ft lbs) Avg Speed (mph) Average Horsepower Time 86 14.6 223,943 10.2 0.079 Work 112 19.5 160,752 10.5 0.044 Page 9 of 16

Origin/Destination Pair #5 PCC Cascade to PCC Rock Creek The fifth OD Pair went from PCC Cascade to PCC Rock Creek. We can speculate from the 2.5D figure that the grades in the terrain directly west of downtown are steeper and thus avoided by the minimal work route, even though the additional distance and time is considerable as summarized in the table below. PCC Cascade to PCC Rock Creek Shortest Paths D O Minimize Time Minimize Work PCC Cascade to PCC Rock Creek Route Summary Statistics Shortest Path Type Minutes Miles Work (ft lbs) Avg Speed (mph) Average Horsepower Time 91 13.7 300,129 9.0 0.100 Work 120 21.3 194,112 10.7 0.049 Page 10 of 16

Flow Chart Prepare RLIS Street Shapefile for Use in Network Analysis Convert Shapefile to Coverage Result: Directional attributes are established; From NODE (FNODE) & To NODE (TNODE) fields are created. Create additional fields Create a point coverage from the street coverge Create fields Feature Vertices to Points (Both Ends) Result: fields to hold FNODE_Elev and TNODE_Elev are created; Also: FT_Slope, TF_Slope, FT_Velocity, TF_Velocity, FT_Time, TF_Time, FT_Work, TF_Work, FT_Power, and TF_Power Result: point coverage of street link end points where DEM elevations can be assigned. Get elevations of end node point coverage from DEM Extract Values to Points Result: a RASTERVALU elevation field is generated for each end node record in the node attribute table Create X,Y coordinates of the end nodes so that later, a feature class of the end node table can be created Add XY Coordinates Result: X,Y coordinate fields are generated for each node record A Page 11 of 16

A Export the node attribute table so that it can be manipulated (or learn PYTHON and ArcGIS geoprocessing scripting and manipulate attribute table inside ArcGIS) Export table Result:.dbf file that has twice the number of records as the street coverage open.dbf file in EXCEL and/or ACCESS Every two records are a FNODE TNODE Pair Go to first record; assign RASTERVALU in first record to FNODE_Elev field in first record; assign the RASTERVALU in the second record to the TNODE_Elev field in the first record; Go to third record; assign RASTERVALU in third record to FNODE_Elev field in third record; assign the RASTERVALU in the fourth record to the TNODE_Elev field in the third record; Go to the fifth record;... Go to the seventh record;... repeat this process for entire table Delete second, fourth, sixth, etc... records Result: a table with the same number of records as the street coverage and with the appropriate elevations in the FNODE_Elev and TNODE_Elev fields B Page 12 of 16

B Calculate values of impedance fields: FT_Slope, TF_Slope, FT_Velocity, TF_Velocity, FT_Time, TF_Time, FT_Work, TF_Work, FT_Power, and TF_Power Save as.dbf file Convert.dbf file with X,Y coordinates to a feature class Join feature class with the street coverage with LOCALID unique identifier Create Network with the new street coverage Conduct Network Analysis Page 13 of 16

1) Air Resistance Force References Force of Drag =.5 * Cd * ρ * A * V² (lbs) (1 slug ft per sec 2 = 1 lb of force) http://www.ac.wwu.edu/~vawter/physicsnet/topics/dynamics/forces/dragforce.html Coefficient of Drag Cd = 1.2 (conservative estimate: flat plate) http://www.grc.nasa.gov/www/k 12/airplane/shaped.html Density of air ρ = 0.00237 slug per cubic foot http://www.engineeringtoolbox.com/air desity specific weight d_600.html Cross Sectional Area of Cyclist A = 7 ft² (after applying a factor of 2 to be conservative) http://books.google.com/books?id=7mzu1ztzl_kc&pg=pa103&lpg=pa103&dq=frontal+area+cycling&source=bl&ots=_ukhcm vkaq&sig=ob2lxpfdsggeuygmfghzj3xswco&hl=en&sa=x&oi=book_result&resnum=2&ct=result#ppa105,m Page 14 of 16

2) Force Of Pushing A Weight Up An Incline Basic Equation: Force of weight = W * sin (lbs) sin : sine of degree of slope Fundamentals of Physics; Haliday; Resnick; 1974 (p.98) My own calculations: W: Weight of Person + Cycle : 200 lbs (estimate) sin : sine of degree of slope; approximately equal to decimal slope (ft/ft) So, Force (lbs) approximately = W * W in lbs; in decimal (ft per ft) 3) Work Work = Force * Distance (ft lbs) Fundamentals of Physics; Haliday; Resnick; 1974 4) Power Power = Force / Time (ft lbs/sec) Fundamentals of Physics; Haliday; Resnick; 1974 1 Horsepower (HP) = 550 ft lbs/sec Fundamentals of Physics; Haliday; Resnick; 1974 Page 15 of 16

5) Velocity Model Assumptions Base Velocity = 10 mph or 15 ft per sec slope (ft/ft) Uphill Downhill Velocity = 15 [(+slope) * (125) ] ft per sec Minimum of 3 ft per sec (walking speed) Velocity = 15 [( slope) * (500) ] ft per sec Maximum of 37 ft per sec (25 mph) Create a point coverage from the street coverge consisting Page of end 16 points of 16 Feature Vertices to Points Result: point coverage of street link end points where DEM elevations can be assigned.