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U R B A N A RT E R I A L S P E E D F L O W E Q U AT I O N S F O R T R AV E L D E M A N D M O D E L S 111
TABLE 2 Functional Form Candidates for SpeedFlow Curves
Functional Form Example Comments
Linear s ax b Not acceptable. Reaches zero speed at high v/c.
Logarithmic s a ln x b Not acceptable. Has no value at x = 0 (the logarithm of x approaches negative infinity).
Exponential s as0 exp(bx) Has all required traits for equilibrium assignment.
Power s a/xb Not acceptable. It goes to infinity at v/c = x = 0.
Polynomial s ax2 bx c Not acceptable. It reaches zero speed at high v/c.
BPR s s0/(1 a(x)b) Has all required traits for equilibrium assignment.
Akcelik s L/{L/s0 0.25[(x 1)
[(x 1)2 ax ]}
Has all required traits for equilibrium assignment.
Note:
s = predicted speed; a, b, c global parameters for equation; L link length; x volume/capacity ratio; s0 link free-flow speed.
where solution. As a practical matter, the speedflow equations
should never intersect the x-axis (that is, the predicted
L = link length (mi), speed should never reach precisely zero), so that the com-
S = average link travel time (h), puter implementing the travel demand model is never
S0 = free-flow link travel time (h), confronted with a "divide by zero" problem.
x = v/c ratio, Three of the candidate functional forms meet the equi-
T = duration of analysis period (h), librium assignment requirements for a speedflow
c = capacity (vph), and curve--exponential, BPR, and Akcelik.
J = calibration parameter.
MODEL SPEEDFLOW EQUATION
PRELIMINARY SCREENING OF CANDIDATE CALIBRATION--V/C < 1.00
SPEEDFLOW EQUATIONS
The exponential, BPR, and Akcelik equations were fit-
Speedflow equations must meet several behavioral ted through a least-squares error-fitting process to the
requirements to permit capacity-constrained equilibrium observed speedflow data. Figure 2 compares the fit of
assignment to be performed by travel demand models. the standard BPR and the other fitted curves to the data.
The speedflow equations must be monotonically As can be seen, the wide scatter of the observed data
decreasing and continuous functions of the v/c ratio for allows almost any speedflow curve to be drawn
an equilibrium assignment process to arrive at a unique through the cloud of data. All three functional forms
SCAG Arterial Speed Study
60
50
Field Data Points
40
Mean Speed (mph)
Standard BPR
30
20
Fitted BPR
10
Fitted Akcelik
0
0 0.2 0.4 0.6 0.8 1.0 1.2
One-Hour Volume/Capacity Ratio
FIGURE 2 Speedflow equations versus field data for v/c < 1.00.
