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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.