Introduction of Cooperative Vehicle-To-Infrastructure Systems to Improve Speed Harmonization

Figure 6. Equation. Speed-based algorithm minimum group speed approach.

Figure 6. Equation. Speed-based algorithm minimum group speed approach. u superscript des subscript i, m times open parenthesis k closed parenthesis equals open bracket five options: (1) u subscript m times open parenthesis k closed parenthesis, if V superscript min subscript i times open parenthesis k closed parenthesis is greater than or equal to 50, (2) 0.3 times V superscript min subscript i times open parenthesis k closed parenthesis plus 0.7 times u subscript m times open parenthesis k closed parenthesis, if V superscript min subscript I times open parenthesis k closed parenthesis is greater than or equal to 40 and less than or equal to 50, (3) 0.5 times V superscript min subscript i times open parenthesis k closed parenthesis plus 0.5 times u subscript m times open parenthesis k closed parenthesis, if V superscript min subscript i times open parenthesis k closed parenthesis is greater than or equal to 30 and less than or equal to 40, (4) 0.7 times V superscript min subscript i times open parenthesis k closed parenthesis plus 0.3 times u subscript m times open parenthesis k closed parenthesis, if V superscript min subscript i times open parenthesis k closed parenthesis is greater than or equal to 15 and less than or equal to 30, and (5) 0.85 times V superscript min subscript i times open parenthesis k closed parenthesis plus 0.15 times u subscript m times open parenthesis k closed parenthesis, if V superscript min subscript i times open parenthesis k closed parenthesis is less than or equal to 15.

Back

Figure 7. Equation. Speed-based algorithm average group speed approach.

Figure 7. Equation. Speed-based algorithm average group speed approach. u superscript des subscript i, m times open parenthesis k closed parenthesis equals open bracket five options: (1) u subscript m times open parenthesis k closed parenthesis, if V superscript ave subscript i times open parenthesis k closed parenthesis is greater than or equal to 50, (2) 0.3 times V superscript ave subscript i times open parenthesis k closed parenthesis plus 0.7 times u subscript m times open parenthesis k closed parenthesis, if V superscript ave subscript i times open parenthesis k closed parenthesis is greater than or equal to 40 and less than or equal to 50, (3) 0.5 times V superscript ave subscript i times open parenthesis k closed parenthesis plus 0.5 times u subscript m times open parenthesis k closed parenthesis, if V superscript ave subscript i times open parenthesis k closed parenthesis is greater than or equal to 30 and less than or equal to 40, (4) 0.7 times V superscript ave subscript i times open parenthesis k closed parenthesis plus 0.3 times u subscript m times open parenthesis k closed parenthesis, if V superscript ave subscript i times open parenthesis k closed parenthesis is greater than or equal to 15 and less than or equal to 30, and (5) 0.85 times V superscript ave subscript i times open parenthesis k closed parenthesis plus 0.15 times u subscript m times open parenthesis k closed parenthesis, if V superscript ave subscript i times open parenthesis k closed parenthesis is less than or equal to 15.

Back

Figure 11. Illustration. Lane-drop bottleneck.

Figure 11. Illustration. Lane-drop bottleneck. This illustration shows a lane-drop bottleneck. The road section is divided into three zones: the speed harmonization zone on the left, the acceleration zone in the middle, and the bottleneck on the right. There are five sets of sensors: one in the speed harmonization zone, one directly upstream of the bottleneck, one directly downstream of the bottleneck, one on the on-ramp into the acceleration zone, and one on the off-ramp out of the acceleration zone. The sensor in the speed harmonization zone collects q superscript s open parenthesis t closed parenthesis, k superscript s open parenthesis t closed parenthesis, and v superscript s open parenthesis t closed parenthesis. The sensor directly upstream of the bottleneck collects q superscript u open parenthesis t closed parenthesis, k superscript u open parenthesis t closed parenthesis, and v superscript u open parenthesis t closed parenthesis. The sensor directly downstream of the bottleneck collects q superscript d open parenthesis t closed parenthesis, k superscript d open parenthesis t closed parenthesis, and v superscript d open parenthesis t closed parenthesis. The sensor on the on-ramp into the acceleration zone collects q superscript in subscript i open parenthesis t closed parenthesis. The sensor on the off-ramp out of the acceleration zone collects q superscript out subscript j open parenthesis t closed parenthesis. This illustration shows a lane-drop bottleneck. The road section is divided into three zones: the speed harmonization zone on the left, the acceleration zone in the middle, and the bottleneck on the right. There are five sets of sensors: one in the speed harmonization zone, one directly upstream of the bottleneck, one directly downstream of the bottleneck, one on the on-ramp into the acceleration zone, and one on the off-ramp out of the acceleration zone. The sensor in the speed harmonization zone collects q superscript s open parenthesis t closed parenthesis, k superscript s open parenthesis t closed parenthesis, and v superscript s open parenthesis t closed parenthesis. The sensor directly upstream of the bottleneck collects q superscript u open parenthesis t closed parenthesis, k superscript u open parenthesis t closed parenthesis, and v superscript u open parenthesis t closed parenthesis. The sensor directly downstream of the bottleneck collects q superscript d open parenthesis t closed parenthesis, k superscript d open parenthesis t closed parenthesis, and v superscript d open parenthesis t closed parenthesis. The sensor on the on-ramp into the acceleration zone collects q superscript in subscript i open parenthesis t closed parenthesis. The sensor on the off-ramp out of the acceleration zone collects q superscript out subscript j open parenthesis t closed parenthesis.

Back

Figure 14. Illustration. Fundamental diagram of traffic flow.

Figure 14. Illustration. Fundamental diagram of traffic flow. This illustration shows two arcs plotting capacity on the y-axis and density on the x-axis. Both arcs begin at the origin of the two axes. The following points are on the higher arc:

• q superscript d subscript c (downstream capacity of the bottleneck) and k subscript 0 (target density) (as the arc ascends).
• q superscript u subscript c (capacity directly upstream of the bottleneck) and k superscript u subscript c (density-at-capacity directly upstream of the bottleneck) (at the apex of the arc).
• q superscript d subscript c (downstream capacity of the bottleneck) and v subscript 0 (advisory speed recommendation) (as the arc descends).
• 0 (where the descending arc intersects the x-axis) and k superscript u subscript j (jam density directly upstream of the bottleneck).

The following points are on the lower arc:

• q superscript d subscript c (downstream capacity of the bottleneck) and k superscript d subscript c (target density) (at the apex of the arc).
• 0 (where the descending arc intersects the x-axis) and k superscript d subscript (jam density directly upstream of the bottleneck).

Back

Figure 19. Flowchart. Density-based algorithm logic and advisory speed recommendations.

Figure 19. Flowchart. Density-based algorithm logic and advisory speed recommendations. This flowchart illustrates the density-based algorithm with several phases as follows. Step 1. When t is less than or equal to t subscript 0, assign the speed harmonization zone advisory speed recommendation as v tilde subscript 0 times open parenthesis t closed parenthesis equals v subscript f. Optimal flow rate in the speed harmonization zone is set as q tilde subscript 0 times open parenthesis t closed parenthesis equals q superscript d subscript c plus q subscript r times open parenthesis t closed parenthesis.

Step 2. At each time step t, check the two conditions open bracket q superscript s times open parenthesis t closed parenthesis is less than q superscript d subscript c plus q subscript r times open parenthesis t closed parenthesis and k superscript u times open parenthesis t plus 1 closed parenthesis is less than or equal to k superscript u subscript c where l equals L divided by v subscript l. If both conditions are satisfied, set the advisory speed recommendation as v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis equals v subscript f. If either one of the conditions is violated, first compute a target flow rate in the speed harmonization zone as q hat subscript 0 times open parenthesis t plus 1 closed parenthesis equals beta times q superscript d subscript c plus q subscript r times open parenthesis t closed parenthesis. If k superscript u times open parenthesis t plus 1 closed parenthesis greater than k superscript u subscript c and absolute value of v tilde times open parenthesis t closed parenthesis minus v tilde times open parenthesis t minus 1 closed parenthesis is less than delta v subscript thr, set beta equal to beta subscript 0. If beta subscript 0 is less than 1, then beta times q superscript d subscript c is less than the maximum bottleneck discharge flow rate when capacity drops happen. Otherwise, let beta equal 1. Target flow rate in the next time step is q tilde times open parenthesis t plus 1 closed parenthesis equals a q hat subscript zero times open parenthesis t plus 1 closed parenthesis plus open parenthesis 1 minus a closed parenthesis q tilde subscript 0 times open parenthesis t closed parenthesis. Advisory speed recommendation at t plus 1 is v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis equals V times open parenthesis Q superscript -1 times open parenthesis q tilde subscript 0 times open parenthesis t plus 1 closed parenthesis closed parenthesis closed parenthesis.

Step 3. If delta v times open parenthesis t closed parenthesis equals absolute value of v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis minus v tilde 0 times open parenthesis t closed parenthesis is greater than delta v subscript thr, then v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis equals open bracket v tilde subscript 0 times open parenthesis t closed parenthesis plus delta v subscript thr, if v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis is greater than v tilde subscript 0 times open parenthesis t closed parenthesis. v tilde subscript 0 times open parenthesis t closed parenthesis minus delta v subscript thr, if v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis is less than or equal to v tilde subscript 0 times open parenthesis t closed parenthesis. Let v subscript min less than or equal to v hat subscript 0 times open parenthesis t plus 1 closed parenthesis less than or equal to v subscript f, for example as v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis equals max times open bracket min times open bracket v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis, v subscript f closed bracket, v subscript min closed bracket. Also set q tilde subscript 0 times open parenthesis t plus 1 closed parenthesis equals Q times open parenthesis V superscript -1 times open parenthesis v tilde subscript 0 times open parenthesis t plus 1 closed parenthesis closed parenthesis closed parenthesis.

Step 4. If t is less than T, t equals t plus 1 and go back to step 2. Otherwise, stop iterations.

Back