Driverless cars and freeway traffic flow

  • 21-Sep-2017 06:23 EDT
Fuhs LEAD traffic.jpg

The five-lane I-80 freeway looking south near Berkeley, CA, a scene repeated many times over within the U.S. (WikiCommons image).

The integration of driverless cars onto multilane freeways brings challenges for vehicle and infrastructure planners, not the least of which is ever increasing congestion. The threshold for serious congestion is a population of 300,000. The U.S. currently has 51 traffic-congested cities with population of 380,000 or more.

Can driverless cars with full vehicle-to-infrastructure (V2I) connectivity favorably affect congestion? The answer is yes. This means construction of new freeway lanes may not be needed except in exceptional cases.

Demand vs. mitigation

Freeway traffic demand depends on the local ambient vehicle population, the number of citizens owning cars, and the mobility companies owning driverless cars. Driving habits, especially the time the car is parked, play an important role. In addition, demand depends on three subsets: 1) fraction of ambient car population which is driverless, 2) fraction of ambient population which is parked compared to rolling, and 3) fraction of driverless, f45, which are at SAE Level 4 or 5. For subset 3), see the top of Figure 2. In this article, f45 = 1.0 is assumed.

The balance between demand and mitigation will be different for each of the cities with population over 300,000. Also, the balance is time dependent—which side is winning may shift in time.

To provide insight, a speculative and qualitative graph (Fig. 2) provides a focus for the discussion of possible future trends. The gray area is for conventional cars while the white areas (lower part of graph) represent driverless cars. In the lower drawing, the fraction of driverless cars in motion is cross hatched.

According to this graphical portrait, the population of conventional cars, C-cars, will decrease markedly. The growth of the number of driverless cars, D-cars, will initially be steep, followed by slow steady growth. The total number of cars, C-cars plus D-cars, decays slowly.

A dominant factor, at least early in the driverless-car era, is the relative time spent parked. Driving habits indicate a typical conventional car spends 95% of its life parked. In contrast, most of the early driverless cars will be owned and operated by mobility companies. To earn a profit the driverless vehicle must be moving 75% of the time. The 5% matched to the 75% greatly magnifies the relative number of driverless cars moving on the freeway.

Even though only 1 out of 16 cars in the total-car pool is driverless, one-half of the cars on the freeway are driverless. As another example, for a driverless fraction of the total vehicle population equal to a tiny 1/32, the driverless vehicle fraction for the mixture on the freeway is 1/3.

Mitigation: Freeway platooning

The maximum gain in traffic flow due to platooning, based on geometry, is 400% to 500%. However, this is quite optimistic because of practical considerations, such as allowing a car in an inner lane to move across lanes and to exit at the next off ramp. Realistic gain may be one-half or 200% to 250 %—like adding four new lanes to an existing four-lane freeway. Think of doubling the traffic flow on the 405 freeway in L.A.!

The big question for demand versus mitigation is when will mitigation become significant? To state a specific year is pure speculation. A valid and accurate statement for “when” is based on the fraction of driverless cars. For mixed conventional-driverless traffic flow on a multilane freeway, the surprising fact is that the lane assignments have eigenvalues.

Figure 3 not only shows the lane assignments but also introduces several discoveries made during the course this study. The four lanes are numbered at the top of each column. The sequence of lane numbers is consistent with the DoT method of numbering lanes. Letters (either C or D) appear in a column below each lane number; these letters are the lane assignment. A column is a lane. The assignment is either C-Lane or D-Lane. A set of lane assignments form a “state.” For example, (D, D, C, C) is the state 23. The eigenvalues are not evenly spaced.

The word “state” was selected since the definition matches that used in quantum mechanics. Think of the QM vibrational states for a harmonic oscillator. The analogy between the QM oscillator and the mixed traffic flow is imperfect but share enough features to note the similarity. The states of the traffic flow are discrete and are given by eigenfunctions with specific and discrete QN. Lane management is fundamentally an example of discrete (not smoothly varying) mathematics.

Going downward in a column increases the driverless car fraction, f. An eigenvalue occurs at the top and bottom horizontal edges of a state. As is apparent, the state is identified by using the eigenvalue subscripts. Once the value of the fraction, f, crosses an eigenvalue, the lane assignment must be changed. At an eigenvalue, all four lanes of the freeway are running full with k = k* and K = K*. The fact that the freeway is running full forces the lane reassignments.

States 12, 23, and 34 have an eigenfunction. The eigenfunction is the traffic density for each lane of the freeway; dimensions are cars/km-lane. The eigenfunction also has a discrete quantum number, QN. Inserting the QN into the eigenfunction yields the traffic density for that lane within the state. For the problem at hand, the QN are C = 30, 60, and 90.

Density of mixed C-D traffic flow

Jump ahead to Figure 4. State 01 is called Stealth Driverless because, even though D-cars are in the State, the traffic density is unaffected. The States 12, 23, and 34 are labeled as Quasi-Quantum since these states have the behavior of eigenfunctions giving the Traffic Density.

The ending State 45 on the upper-right side tagged NK* Conundrum. For the typical values used here, NK* has a value of 480 cars/km. The traffic density flattens at 390 cars/km beginning at the eigenvalue 4. As f increases to f = 1.0, the traffic density remains unchanged at 390 cars/km. The apparent discrepancy of 90 cars/km is the NK* Conundrum.

The equation for the eigenfunction for States 12, 23, and 34 is simply:

[INSERT EQUATION 1 HERE]

The values for C to use as QN’s are C = 90, 60, & 30; as C decreases, the value of f increases. The number of vehicles is C and D.

Look at State 01 in Figure 4. The driverless car fraction is increasing as f increases. However, the all-lane traffic density remains fixed. For the typical values used in Figure 4, at f = 0, there are 120 C-cars and zero D-cars. At f = 0.25 there are 90 C-cars and 30 D-cars. Note f = 1/N = 0.25 is the first eigenvalue. At f = 0.25, the threshold k* = 30 is attained. Until f = 0.25 is reached, D-cars cannot provide platooning improvement in traffic flow.

When an HOV lane is converted to a D-Lane, conventional cars are excluded from the lane, and total traffic density decreases. The change of a HOV to D-Lane too soon causes the all-lane traffic to decrease, not increase.

Platooning and wave damping

The familiar Greenshield Model offers an excellent method to illustrate the impact of driverless cars on traffic flow. Figure 5 shows the results. Platooning is possible with driverless vehicles. Platooning allows very large traffic density while safely maintaining high vehicle speed. Instead of decreasing speed with increasing density, as is the case with human-driven cars, driverless cars allow a high fixed speed even at high traffic density. This fact is shown in the left hand graph of Figure 5 where the platooning speed is V* = VP.

Since the platooning speed is constant at VP, the traffic flow, q, increases linearly with increasing traffic density, k. This behavior for q is shown in right hand graph in Figure 5. The gain in traffic flow compared to human-driven cars is illustrated with the vertical arrow. The gain at k equal approximately to 1.65 k* is as large as the q* for human driven cars. At k = kG the traffic flow is twice as large or 2q*. In fact, for driverless cars gridlock never occurs! (At least not at k = kG.)

Then there are waves in traffic flow. Whether created spontaneously or by lane blockage or bottleneck, waves make the throttle and V fluctuate like a yo-yo toy on a string. The fluctuating V yields an average V which reduces traffic flow compared to smooth V. A yo-yo throttle also increases fuel/energy consumption.

Experimental data show with f = 0.05, waves can be damped. When a driverless car is immersed in a traffic lane, with a suitable control law one driverless car can dampen the instability of the 19 other conventional cars. Damping of traffic waves offers up to 25% gain in traffic flow; this is like adding a new lane to an existing four-lane freeway. It can be effective in mixed-flow lanes where platooning is impossible.

Mechanical engineer Prof. Allen Fuhs, Ph.D, is an SAE Fellow and recipient of the society’s Ralph Teetor Award. He is author or editor of a dozen books. He encourages feedback at allen.fuhs@sbcglobal.net.

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