Innovations in Travel Demand Modeling, Volume 2: Papers (2008)

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where L = link length (mi), S = average link travel time (h), S0 = free-flow link travel time (h), x = v/c ratio, T = duration of analysis period (h), c = capacity (vph), and J = calibration parameter. PRELIMINARY SCREENING OF CANDIDATE SPEED–FLOW EQUATIONS Speed–flow equations must meet several behavioral requirements to permit capacity-constrained equilibrium assignment to be performed by travel demand models. The speed–flow equations must be monotonically decreasing and continuous functions of the v/c ratio for an equilibrium assignment process to arrive at a unique solution. As a practical matter, the speed–flow equations should never intersect the x-axis (that is, the predicted speed should never reach precisely zero), so that the com- puter implementing the travel demand model is never confronted with a “divide by zero” problem. Three of the candidate functional forms meet the equi- librium assignment requirements for a speed–flow curve—exponential, BPR, and Akcelik. MODEL SPEED–FLOW EQUATION CALIBRATION—V/C < 1.00 The exponential, BPR, and Akcelik equations were fit- ted through a least-squares error-fitting process to the observed speed–flow data. Figure 2 compares the fit of the standard BPR and the other fitted curves to the data. As can be seen, the wide scatter of the observed data allows almost any speed–flow curve to be drawn through the cloud of data. All three functional forms 111URBAN ARTERIAL SPEED–FLOW EQUATIONS FOR TRAVEL DEMAND MODELS TABLE 2 Functional Form Candidates for Speed–Flow 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. SCAG Arterial Speed Study M ea n Sp ee d (m ph ) 60 50 40 30 20 10 0 One-Hour Volume/Capacity Ratio 0 0.2 0.4 0.6 Field Data Points Standard BPR Fitted BPR Fitted Akcelik 0.8 1.0 1.2 FIGURE 2 Speed–flow equations versus field data for v/c < 1.00.