Tuesday, March 19, 2013

154> Fractals!



One thing concerns me about the typical vision of a PRT system is the idea that should consist entirely of a network of one-way loops that would be expanded by adding new loops contiguously.  This seems to me to be an oversimplification, and generally not the most efficient design in terms of system cost or travel time.  Yet even bi-directional routing can be thought of as a loop, of sorts – just one that has been squeezed along its length. So how could the paradigm of one-way loops be lacking?

First of all, let’s take an idealized example where the density of potential riders is completely even and the track forms a perfect grid of squares. (round the block loops)  Clearly there will be a bias towards higher ridership on routes crossing the middle than at the outer edges.  So what do we do?  Is the middle subject to wait times or are the outer routes paid for, but underutilized?  Add to this the fact that the middle will probably have more potential riders anyway, and you can see that the problem is just that much worse.

Beyond that, the assumption of an ever-expanding network of contiguous loops tends to ignore the uneven distribution of important destinations.  Although cities have a central business district that needs to be the heart of a PRT system, after that there are seemingly no rules for what should come next.  A spider web design is better than the grid, since the converging radial routes enable a somewhat denser concentration toward the middle.  But cities don’t grow evenly and the transportation bottlenecks can be far from downtown, an inconvenient fact that would-be PRT providers will surely downplay.  After all, there is plenty to be done downtown, so why worry that far ahead?  I say, “Inconvenient” because really dealing with it increases the complexity of a system immensely.  In my opinion the solution will have to include some combination of multiple speeds, multiple vehicles sizes, and multiple, parallel tracks- like there are multiple lanes on a freeway.  Anything less, and the system will be one that can only solve some of the problem in some of the situations, and that fact may be baked-in by the design and therefore forever unfixable.  

If there is only one speed, all track must be routed without any sharp turns if the system is to be fast enough to be useful for anything more than very short trips.  This means acquiring right-of-way on nearly every corner or bypassing needed, but troublesome routes.  Then there is the matter of multiple vehicle sizes.  Here I am referring to GRT.  Certain routes, say to an airport , stadium or “park&ride” lot  might benefit from a larger shuttle-type service rather than only individual smaller vehicles.  This would help with the rush hour demand for routes that many passengers share in common, without requiring extra track or vehicles just for a few hours each day.  The third option, multiple parallel tracks, is also for this situation.  One of the great things about PRT is that it can be designed to have very inexpensive track, which makes such possibilities more palatable.  I think following the familiar highway model, where there is both a high-speed express and a more local feeder aspect might be particularly advantageous.  Such an investment paves the way for future loops along the way, and could often be done using existing highway easements.     

In either of these last two cases it is assumed that there are enough destinations along the way (or at either end) to justify a PRT-compatible track rather than say, an ordinary bus.  Having GRT sharing the track reduces what would otherwise be a requirement for more parallel tracks, but parallel tracks, if spread apart by a few blocks, would provide enhanced access all along the way.  Like I said, there should be some combination, if at all possible, of these three methods.  This may complicate the system design, but it makes the system more versatile and therefore ultimately a more compelling value.    

So how are we to determine the relative merits of the various systems and routing schemes?  I think there are probably some mathematical formulas that describe the general problem and give some ballpark ratios and other guidelines that might be useful as a starting point.  Such formulas seem to follow the general principles of fractals, something that occurred to me when Nathan Koren ttp://www.podcar.org/blogs/nathan-koren/, in this excellent two-part (similarly themed) post, used a leaf as an example of a transportation system.

Imagine a growing town building new roads outward into surrounding countryside.  Along each new road are natural “sweet spots” to develop housing, warehousing, retail, etc.  Any closer, and land is too expensive.  Any farther out, and the commute is too far. Maximizing the usefulness of the new road and access to these areas can be accomplished with simple branching.  Now you have twice as many sites with the same travel time without needing separate roads.  Branch the branches and now you have four. Branch once more and you have eight.  Indeed, radially emanating roads must diverge if they are to access any reasonable portion of the ever widening land mass anyway.

 
This follows the rules of a type of fractal geometry known as the Lindenmayer system, (L-system) which is seen thoughout biology and is efficient for movement of blood, plant nutrients, etc.  The math for the above example is simple.  Go a distance, split, go half as far, split, go half as far, split, etc. (I cut the "trunk" to save space)  Add a bit of randomness and you can create forms which look like photographs from a botany book.


Want to live on a cul de sac?  This “H tree” (left) gives everyone the piece and quiet of their own dead end, equidistant from the main road.  To the right is what is called a quadric cross, which represents a three-way split at right angles. 


Here is a different kind of fractal geometry at work:  Let’s consider, for a moment, the merchant.  Here we have the same desire for cheap property, but it is coupled with the need for exposure to customers.  Clearly the intersections formed by branching roads are particularly advantagous in this regard.  But another plus would be the presence of other businesses, to help draw customers.  This too, can be described in mathematical terms with more fractal geometry, this time with what is called a “Diffusion-limited aggregation.” (DLA)  Here, particles (businesses) randomly migrate from a source, but not too far, only to plant themselves on an edge (2D) or surface. (3D)  This is not unlike coral growth, and can be seen in satellite views of cities, especially aspects like pavement coverage vs. green areas.  In three dimensions, constrained by city blocks, an effect much like crystal formation is seen in the growth of groups of multistory buildings.  Again, these similarities are not merely coincidental. They are the result of similar natural, measurable forces.  
What is interesting about the combination of DLA and L-system effects in city growth is that together they generate satellite communities.  This formation is easy to observe by anyone who takes a farm road out of town.  It usually starts with a gas station/convenience store at a rural intersection.  Soon an eatery or an auto repair garage follows.  As more businesses join the group, land values rise, creating a climate for land speculation and further development.  Much, much later the resultant communities create a traffic nightmare for the host city by interfering with the radial flow of vehicles during rush hour.  In the typical spider web roadway configuration the radial strands that serve the central area are inherently at odds with the concentric routing that serves traffic between neighboring outlying communities.  This creates pockets of traffic congestion that are far from downtown but are still sorely in need of something like PRT.  The classic remedy has been to build a freeway with overpasses over the main crossing roads, cutting the outlying communities in half.
These fractals only go so far in describing the problem facing people tasked with designing PRT routing, because of the subject that I brought up first, which is loops.  After all, notably absent in the earlier discussion about a “sweet spot” was the obvious way to get the most bang for the buck. That is to have the branches loop back upon themselves.  Fractal forms can include loops as well, as in the leaf below. Note the classic fractal multi-scale self-similarity in the tendency towards branching at right angles.
Below is a fractal of loops representing growth along an east/west corridor.  I have included some secondary development, (shown in red) representing the value of shortcuts.  The next step would be to connect the outlying areas directly to form an outer loop.
The “rules” outlined above do more than help explain the uneven geographical distribution of potential PRT traffic.  They also illustrate the different capacity requirements for the track itself.  I drew the L-system tree with separate lines on the “trunk” to illustrate the simple fact that there simply cannot be equal traffic between it and the branches.  The quadric cross example also shows the relative traffic increases (line thickness) toward the center.  Either the branches are at a fraction of capacity or the trunk is overburdened.  As I stated earlier, the idea of simply handling peak loads with massively parallel loops is iffy at best.  But multiple lane trunk lines or GRT have drawbacks as well.
The conventional thinking used to be that starter downtown circulator loops could be added to until eventually there is a network that fulfills the total needs of the covered areas.  There is still some truth to this, but it certainly isn’t the whole story.  Cutbacks in government transportation spending have forced us to examine any and all inefficiencies, including any underutilized track or vehicles.  In the end we will probably end up using some combination of “all-of-the-above.”  I am not sure how useful fractal modeling can actually be in practice, but the subject certainly seems worth pondering.  Surely there is a fractal form whose shape is the result of a mathematical modeling of various forces that shape our cities, and therefore our transportation needs.  One thing is for sure. We shouldn’t be surprised when we find that the most cost effective and expandable PRT solutions mimic nature more than a checkerboard, or use many of the same techniques that have proved effective in moving people via our current network of roads.    

15 comments:

Asko K. said...

Good post as always, Dan.

One fundamental assumption you are taking is that a city would want/need to be fully covered with PRT. They don't. At least here we have fully functioning arterial public transport (subways, local trains). These offer good, existing entry to city centers.

By taking this assumption away, and asking instead: how can PRT do the most to improve the coverage / service level / utilization of existing (arterial) public transport, we have a tremendously smaller challenge at hand. It's also very testable, since we can vary the approaches (i.e. technology or track layouts) taken at individual stations, to find out what works in practice.

This is my take on PRT. Small scale, dynamic, learn by doing. This is where I think our visions differ.

Certainly, one can also apply the fractal thinking to the transport hub (where PRT meets artery) being the trunk.

Cheers, Asko

Andrew F said...

Asko, I disagree that PRT should be view through the lens of improving utilization of existing public transport. PRT should be looked at from a NPV standpoint. It could provide positive NPV by interfacing with existing public transport, but we should not ignore possible deployments that do not utilize existing public transport, or even render it redundant.

That said, this is a very good post, Dan. I think you raise a very good point regarding planning ahead. A failure to plan is a plan to fail. And failing to plan for how a system might scale could lead to very costly retrofits in the future to accommodate changes in network topography as system scale grows. These would also tend to be in the locations where cost of such retrofits would be greatest in dollars and time/inconvenience: downtown areas, etc.

Dan said...

Dan the Blogger Responds!

Thanks for reminding me to not be so US-centric, Asko… I do not see full coverage of a city as any kind of a short or even medium term goal. Rather, when faced with a collection of interdependent mini-cities spaced a few miles apart, with no public transportation between them except buses, I see a system of a few loops here connected to a few loops there as being an extremely beneficial start, and much superior to, say, light rail, which would not have the coverage of those loops, would cost more, and use much more real estate. (Although I’m for any light rail system that can break even financially.) I see more potential in a lanky, skeletal PRT system that thickens outwardly over time more than a central one that expands radially. (Ironically though, coverage that goes out too far only invites people to move farther away.) But if that skeleton already exists in the form of a different transit means, well, that’s a whole different thing – especially if housing, office and retail space have all been drawn, over time, to this feature. There is a bad “chicken or the egg” problem with new mass transit, since there are rarely enough destinations along a single route to justify its capacity – at least not for many years.

Yours sounds like the DLA example, but where the existing transit has already created the skeleton from which further development emanates. I understand that most places in the world do not share the US’s large proportion of suburban, single family dwellings. If most people live “close in,” in multifamily buildings, and just need a short lift to existing mass transit, then obviously speed is not an issue. That certainly favors lighter, less expensive track, such as your BubbleMotion concept, and perhaps loops are all that is needed.

Hi Andrew, thanks for stopping by… I don’t think there is any right or wrong way to look at this thing. The benefits are diverse and so, then, are the motivations. Maybe that’s why I find it interesting. Personally I think an investment in the environment or the efficiency of the workforce is a good one, even if the numbers don’t add up on day one, but that’s just me. Obviously I agree about planning ahead! By the way – do us a favor and spell out the meanings of your acronyms, OK? Half of my readers don’t speak English as their first language and half of the other half are dumb like me!

Asko K. said...

Thanks for the understansing, Dan!

If I'm right, BRT (bus rapid transit) is making its way also in the States. This is interesting, since it basically offers similar benefits to i.e. light rail, without the cost of the infrastructure. Especially in areas where there's already over-wide roads this might make sense.

Traditionally, there's been this confrontation between the PRT folks (that would be 'us'? :)) and the light rail "industry". I still don't understand why this is - and I'm not seeing it outside of the US. Worlds truly are different.

To anyone who has a chance, I recommend taking a bus in Geneva or Z├╝rich. The Swiss public transport system works like a clock work, and is amazingly clean as well. Hoping to eventually see such service anywhere on the Earth I would travel.

(no need to reply to this - we should be way more busy doing the stuff)

Dan said...

I enjoy responding to comments, Asko. I often get inspirations or insights. I do get busy or away from the internet, but sometimes I delay just because I like to choose my words carefully. Today I have time…

I was thinking the same thing, about the (US) light rail/PRT animosity. Actually it occurred to me, as I was writing about the “chicken and the egg” problem for light rail, that perhaps PRT is the way to “condition” a route for eventual rail. Once the PRT gets overburdened, it’s time for the much bigger investment. Maybe PRT is what light rail needs - not what it should be afraid of. And who better to pitch PRT than the outfit with the more expensive option? I think I’ll try to explore the idea in a future post.

Also, about buses. (BRT) I really don’t see how this scheme gets around holding up traffic for someone, which is at the heart of the gridlock problem in the first place. In Houston, we have HOV (High Occupancy Vehicle) lanes which have now been turned into toll lanes for non-rush hour times. Buses use these, but I often see traffic congestion that would go away if the lanes were just open for everyone.

Last observation… Our beloved “Vermont Transit” (a regional bus line) was purchased by Greyhound, (a national outfit) and went completely to hell. Then Greyhound, itself, was bought by some British company, and now it’s even worse. What I hate most is when they are 45 minutes behind, yet still blow 20 minutes going out of their way to go to Dartmouth College, where there will be nobody getting off or waiting to board… even though everyone will be missing flights, connecting buses, etc… These scheduled stops are from a bygone era, when communication between drivers and passengers could only occur by actually showing up, in person, at a predetermined time. They drive me crazy every time they drive me to the airport!

Andrew F said...

Dan, if anything like your system ever came to fruition, LRT would be more or less obsolete. The LRT fans are rightly concerned about PRT, insofar as they have an irrational love for LRT and not motivated by creating good transport outcomes (they care about their solution, not just having a good solution).

I think you'd have a really hard time convincing users to switch away from PRT that's faster and more convenient to LRT, even on an overburdened line. And why not just twin the PRT track rather than splash out $50 - $100 million per mile for LRT?

Asko K. said...

Hi, Andrew.

I can see a couple of places here in Finland where your logic does apply. It really comes down to the peak loads. Here that means "morning peaks" when people come to work - they are sharper than the evening peaks.

With high enough peak loads, PRT simply doesn't work. LRT does. If the peaks are below the threshold, your reasoning is valid.

Andrew F said...

Asko, some PRT-compatible GRT solution would seem to be more economical and capable of handling peak loads than LRT.

The problem with LRT is that it is just too costly for the capacity it provides. Heavy rail (subways) will still have a place in very dense areas where there just isn't enough room for the jungle of PRT lines to serve effectively peak loads. LRT is much less compelling. It would also have a more compelling customer experience than being crammed into a streetcar stopping every few hundred meters in mixed traffic.

To put a finer point on it: you can put in a lot of poorly utilized PRT infrastructure for what it costs to build and operate LRT.

Asko K. said...

Andrew, you're probably right.

To me, LRT means "tram" - am I right? They seem to vary a lot - in Sweden they're faster then bus, in Finland they are nicer to ride but generally slower than buses.

One tram infrastructure cost (for putting the rails down, I guess) I saw was 4.5Meur. That's not awfully lot. If the wagons themselves are too expensive, that's simply a market detail, isn't it? Maybe someone should get into the business of making cheap tram systems. :)

Andrew F said...

Asko: anything with low enough demand to be served by a LRT/tram in mixed traffic, could be served by a mature PRT system pretty readily. If an LRT needs grade separation, platforms with prepaid fares, etc. We're talking more like EUR 50 million per km.

I live in Toronto, Canada. It has one of the most extensive LRT networks in North America, and it is being extended at great expense. There is one tunneled portion that is going to cost just about $200 million/km. Many cities manage to build full-blown heavy rail subways for that price.

Asko K. said...

Andrew, thanks for the last paragraph on your surroundings. 200M/km sounds awful - now I understand you.

I genuinly thought BRT had taken over this kind of projects in the Americas (because of its lower cost).

Andrew F said...

Asko, the cult of LRT is still strong in North America. Two smaller cities in Ontario (of which Toronto is the capital) are engaged in billion dollar LRT projects--Kitchener-Waterloo (population of ~300k) and Ottawa (population 1.3 million). The latter already has an extensive BRT system. Building LRT there is madness.

cmf-seattle said...

"As you travel out from the centre less capacity is
required, so trains can be split. For example a N bound train can split into a NW train and a NE train and thereafter stations can be half the size and
cost. Further out the trains can split again, until finally the last stations can be little more than bus stops at grade."


Offline station sidings could double as branching points for loops off of trunk lines (Peter Muller has suggested that breaking up platoons should happen within stations, so as to avoid BWS complications, which was a major stumbling point for the Aramis "virtual coupling" design).

I think a freight system should be built first; reliability and safety data can be collected, in order to show investors and/or insurers that it works. A chassis designed for 2 pallets could be fitted with a passenger cabin.

Andrew F said...

What kind of application would require the kind of freight capacity a modest sized PRT system would need to be economical?

Dan said...

Andrew, as per your comment a couple of days ago – Yes, I do think a fully evolved “SMART” PRT system would pretty much eliminate the need for light rail. I also think, though, that the vehicles, in particular, would be prohibitively expensive at present and probably for decades to come. I am really designing it to act as a guide or benchmark, and expect that early iterations would have more limited performance, probably more in line with other systems. This raises the question of whether a slower “conventional” PRT system with greater headways, more gradual speed changes, etc., still beats light rail in all instances. I would have to vote no, although generally their prices seem so high as to practically warrant criminal investigation, IMHO. I think they spend a lot of their profit on lobbyists and sales people. I say “no” because of the case of a linear configuration with lots of people and only a few stops, particularly after a city has grown around those stops.

And thanks, cmf, for pulling up the Flyda link… That system has always been in inspiration for me. If you want to do something a bit heavier, maybe a bit more GRT based, that Flyda concept is pretty hard to beat. Replacing current light rail with a system like that makes all kinds of sense… And thanks for the insight about breaking up platoons or trains as they branch. You are quite right… The train that subdivides can fit a branching system perfectly – by proportionally incorporating matching fractal geometry itself! (i.e., on a two-way split, the train would divide by two – on a three-way split, the train would divide by 3…) Seems like all (multi-car) light rail ought to do that! Most interesting! It also dovetails nicely with a distributed “active wheel” propulsion system, as opposed to a centralized “engine” that pulls passive cars. I have come to believe that the former has general advantages over the latter.