Category: Good Transit

Little Things That Matter: Interchange Siting

I’ve written a lot about the importance of radial network design for urban metros, for examples here, here, here, here, and here. In short, an urban rail network should look something like the following diagram:

That is, every two radial routes should intersect exactly once, with a transfer. In this post I am going to zoom in on a specific feature of importance: the location of the intersection points. In most cities, the intersection points should be as close as possible to the center, first in order to serve the most intensely developed location by all lines, and second in order to avoid backtracking.

The situation in Berlin

Here is the map of the central parts of Berlin’s U- and S-Bahn network, with my apartment in green and three places I frequently go to in red:

(Larger image can be found here.)

The Ring is severed this month due to construction: trains do not run between Ostkreuz, at its intersection with the Stadtbahn, and Frankfurter Allee, one stop to the north at the intersection with U5. As a result, going to the locations of the two northern red dots requires detours, namely walking longer from Warschauer Strasse to the central dot, and making a complex trip via U7, U8, and U2 to the northern dot.

But even when the Ring is operational, the Ring-to-U2 trip to the northern dot in Prenzlauer Berg is circuitous, and as a result I have not made it as often as I’d have liked; the restaurants in Prenzlauer Berg are much better than in Neukölln, but I can’t go there as often now. The real problem is not just that the Ring is interrupted due to construction, but that the U7-U2 connection is at the wrong place for the city’s current geography: it is too far west.

As with all of my criticism of Berlin’s U-Bahn network layout, there is a method to the madness: most of the route of U7 was built during the Cold War, and if you assumed that Berlin would be divided forever, the alignment would make sense. Today, it does not: U7 comes very close to U2 in Kreuzberg but then turns southwest to connect with the North-South Tunnel, which at the time was part of the Western S-Bahn network, running nonstop in the center underneath Mitte, then part of the East.

On hindsight, a better radial design for U7 would have made it a northwest-southeast line through the center. West of the U6 connection at Mehringdamm it would have connected to the North-South Tunnel at Anhalter Bahnhof and to U2 at Mendelssohn Park, and then continued west toward the Zoo. That area between U1/U2 and Tiergarten Park is densely developed, with its northern part containing the Cold War-era Kulturforum, and in the Cold War the commercial center of West Berlin was the Zoo, well to the east of the route of U7.

Avoiding three-seat rides

If the interchange points between lines are all within city center, then the optimal route between any two points is at worst a two-seat ride. This is important: transfers are pretty onerous, so transit planners should minimize them when it is reasonably practical. Two-seat rides are unavoidable, but three-seat rides aren’t.

The two-seat ride rule should be followed to the spirit, not the letter. If there are two existing lines with a somewhat awkward transfer, and a third line is built that makes a three-seat ride better than connecting between those two lines, then the third line is not by itself a problem, and it should be built if its projected ridership is sufficient. The problem is that the transfer was at the wrong location, or maybe at the right location but with too long a walk between the platforms.

Berlin’s awkward U-Bahn network is such that people say that the travel time between any two points within the Ring is about 30 minutes, no matter what. When I tried pushing back, citing a few 20-minute trips, my interlocutors noted that with walking time to the station, the inevitable wait times, and transfers, my 20-minute trips were exceptional, and most were about 30 or slightly longer.

The value of an untimed transfer rises with frequency. Berlin runs the U-Bahn every 5 minutes during the daytime on weekdays and the S-Bahn mostly every 5 minutes (or slightly better) as well; wait times are shorter in a city like Paris, where much of the Metro runs every 3 minutes off-peak, and only drops to 5 or 6 minutes late in the evening, when Berlin runs trains every 10 minutes. However, Parisian train frequencies are only supportable in huge cities like Paris, London, and Tokyo, all of which have very complex transfers, as the cities are so intensely built that the only good locations for train platforms require long walks between lines.

New York of course has the worst of all worlds: a highly non-radial subway network with dozens of missed connections, disappointing off-peak frequencies, and long transfer corridors in Midtown. In New York, three-seat rides are ubiquitous, which may contribute to weak off-peak ridership. Who wants to take three separate subway lines, each coming every 10 minutes, to go 10 kilometers between some residential Brooklyn neighborhood and a social event in Queens?

Bronx Bus Redesign

New York is engaging in the process of redesigning its urban bus network borough by borough. The first borough is the Bronx, with an in-house redesign; Queens is ongoing, to be followed by Brooklyn, both outsourced to firms that have already done business with the MTA. The Bronx redesign draft is just out, and it has a lot of good and a great deal of bad.

What does the redesign include?

Like my and Eric Goldwyn’s proposal for Brooklyn, the Bronx redesign is not just a redrawing of lines on a map, but also operational treatments to speed up the buses. New York City Transit recognizes that the buses are slow, and is proposing a program for installing bus lanes on the major streets in the Bronx (p. 13). Plans for all-door boarding are already in motion, to be rolled out after the OMNY tap card is fully operational; this is incompetent, as all-door boarding can be implemented with paper tickets, but at this stage this is a delay of just a few years, probably about 4 years from now.

But the core of the document is the network redesign, explained route by route. The map is available on p. 14; I’d embed it, but due to file format issues I cannot render it as a large .png file, so you will have to look yourselves.

The shape of the network in the core of the Bronx – that is, the South Bronx – seems reasonable. I have just one major complaint: the Bx3 and Bx13 keep running on University Avenue and Ogden Avenue respectively and do not interline, but rather divert west along Washington Bridge to Washington Heights. For all of the strong communal ties between University Heights and Washington Heights, this service can be handled with a high-frequency transfer at the foot of the bridge, which has other east-west buses interlining on it. The subway transfer offered at the Washington Heights end is low-quality, consisting of just the 1 train at the GWB bus station; a University-Ogden route could instead offer people in University Heights a transfer to faster subway lines at Yankee Stadium.

Outside the South Bronx, things are murkier. This is not a damn by faint praise: this is an acknowledgement that, while the core of the Bronx has a straightforward redesign since the arterials form a grid, the margins of the Bronx are more complicated. Overall the redesign seems fairly conservative – Riverdale, Wakefield, and Clasons Point seem unchanged, and only the eastern margin, from Coop City down to Throgs Neck, sees big changes.

The issue of speed

Unfortunately, the biggest speed improvement for buses, stop consolidation, is barely pursued. Here is the draft’s take on stop consolidation:

The spacing of bus stops along a route is an important factor in providing faster and more reliable bus service. Every bus stop is a trade-off between convenience of access to the bus and the speed and reliability of service. New York City buses spend 27 percent of their time crawling or stopped with their doors open and have the shortest average stop distance (805 feet/245 m) of any major city. London, which has the second closest stop spacing of peer cities, has an average distance between stops of 1,000 ft/300 m.

Bus stop spacing for local Bronx routes averages approximately 882 feet/269 meters. This is slightly higher than the New York City average, but still very close together. Close stop spacing directly contributes to slow buses and longer travel times for customers. When a bus stops more frequently along a route, exiting, stopping, and re-entering the flow of traffic, it loses speed, increases the chance of being stopped at a red traffic signal, and adversely affects customers’ travel time. By removing closely-spaced and under-utilized stops throughout the Bronx, we will reduce dwell time by allowing buses to keep moving with the flow of traffic and get customers where they need to go faster.

Based on what I have modeled as well as what I’ve seen in the literature, the optimal bus stop spacing for the Bronx, as in Brooklyn, is around 400-500 meters. However, the route-by-route descriptions reveal very little stop consolidation. For example, on the Bx1 locals, 3 out of 93 stops are to be removed, and on the Bx2, 4 out of 99 stops are to be removed.

With so little stop consolidation, NYCT plans to retain the distinction between local and limited buses, which reduces frequency to either service pattern. The Bx1 and Bx2 run mostly along the same alignment on Grand Concourse, with some branching at the ends. In the midday off-peak, the Bx1 runs limited every 10 minutes, with some 12-minute gaps, and the Bx2 runs local every 9-10 minutes; this isn’t very frequent given how short the typical NYCT bus trip is, and were NYCT to eliminate the local/limited distinction, the two routes could be consolidated to a single bus running every 4-5 minutes all day.

How much frequency is there, anyway?

The draft document says that consolidating routes will allow higher frequency. Unfortunately, it makes it difficult to figure out what higher frequency means. There is a table on p. 17 listing which routes get higher frequency, but no indication of what the frequency is – the reader is expected to look at it route by route. As a service to frustrated New Yorkers, here is a single table with all listed frequencies, weekday midday. All figures are in minutes.

Route Headway today Proposed headway
Bx1 10 10
Bx2 9 9
Bx3 8 8
Bx4/4A 10 8
Bx5 10 10
Bx6 local 12 8
Bx6 SBS 12 12
Bx7 10 10
Bx8 12 12
Bx9 8 8
Bx10 10 10
Bx11 10 8
Bx12 local 12 12
Bx12 SBS 6 6
Bx13 10 8
Bx15 local 12 12
Bx15 limited 10 10
Bx16 15 15
Bx17 12 12
Bx18 30 20
Bx19 9 9
Bx20 Peak-only Peak-only
Bx21 10 10
Bx22 12 8
Bx23 30 8
Bx24 30 30
Bx26 15 15
Bx27 12 12
Bx28 17 8
Bx38 (28 variant) 17 discontinued
Bx29 30 30
Bx30 15 15
Bx31 12 12
Bx32 15 15
Bx33 20 20
Bx34 20 20
Bx35 7 7
Bx36 10 10
Bx39 12 12
Bx40 20 8
Bx42 (40 variant) 20 cut to a shuttle, 15
Bx41 local 15 15
Bx41 SBS 10 8
Bx46 30 30

A few cases of improving frequency on a trunk are notable, namely on the Bx28/38 and Bx40/42 pairs, but other problem spots remain, led by the Bx1/2 and the local and limited variants on some routes.

The principle of interchange

A transfer-based bus network can mean one of two things. The first, the one usually sold to the public during route redesigns, is a grid of strong routes. This is Nova Xarxa in Barcelona, as well as the core of this draft. Eric’s and my proposal for Brooklyn consists entirely of such a grid, as Brooklyn simply does not have low-density tails like the Bronx, its southern margin having high population density all the way to the boardwalk.

But then there is the second meaning, deployed on networks where trunk routes split into branches. In this formulation, instead of through-service from the branches to the trunk, the branches should be reduced to shuttles with forced transfers to the trunk. Jarrett Walker’s redesign in Dublin, currently frozen due to political opposition (update: Jarrett explains that no, it’s not really frozen, it’s in revision after public comments), has this characteristic. Here’s a schematic:

The second meaning of the principle of interchange is dicey. In some cases, it is unavoidable – on trains, in particular, it is possible to design timed cross-platform transfers, and sometimes it’s just not worth it to deal with complex junctions or run diesels under the catenary. On buses, there is some room for this principle, but less than on trains, as a bus is a bus, with no division into different train lengths or diesels vs. electrics. Fundamentally, if it’s feasible to time the transfers at the junctions, then it’s equally possible to dispatch branches of a single route to arrive regularly.

New York’s bus network is already replete with the first kind of interchange, and then the question is where to add more of it on the margins. But the Bronx draft includes some of the second, justified on the grounds of breaking long routes to improve reliability. Thus, for example, there is a proposed 125th Street crosstown route called the M125, which breaks apart the Bx15 and M100. Well, the Bx15 is a 10.7 km route, and the M100 is an 11.7 km route. The Bx15 limited takes 1:15-1:30 end to end, and the M100 takes about 1:30; besides the fact that NYCT should be pushing speedup treatments to cut both figures well below an hour, if routes of this length are unreliable, the agency has some fundamental problems that network redesign won’t fix.

In the East Bronx, the same principle of interchange involves isolating a few low-frequency coverage routes, like the Bx24 and Bx29, and then making passengers from them transfer to the rest of the network. The problem is that transferring is less convenient on less frequent buses than on more frequent ones. The principle of interchange only works at very high frequency – every 8 minutes is not the maximum frequency for this but the minimum, and every 4-6 minutes is better. It would be better to cobble together routes to Country Club and other low-density neighborhoods that can act as tails for other trunk lines or at least run to a transfer point every 6-8 minutes.

Is any of this salvageable?

The answer is yes. The South Bronx grid is largely good. The disentanglement of the Bx36 and Bx40 is particularly commendable: today the two routes zigzag and cross each other twice, whereas under any redesign, they should turn into two parallel lines, one on Tremont and one on 180th and Burnside.

But outside the core grid, the draft is showing deep problems. My semi-informed understanding is that there has been political pressure not to cut too many stops; moreover, there is no guarantee that the plans for bus lanes on the major corridors will come to fruition, and I don’t think the redesign’s service hours budget takes this into account. Without the extra speed provided by stop consolidation or bus lanes, there is not much room to increase frequency to levels that make transfers attractive.

Positive and Negative Interactions

This is a theoretical post about a practical matter that arises whenever multiple variables interact. Two variables x and y, both correlated positively a dependent variable z, are said to positively interact if when x is larger, the effect of y on z gets larger and vice versa, and to negatively interact if when x is larger, the effect of y on z gets smaller. If z is transit ridership, let alone any of the direct benefits of good transit (good job access, environmental protection, public health, etc.), then it is affected by a slew of variables concerning service provision, infrastructure, and urban design, and they interact in complex ways.

I have not found literature on this interaction, which does not mean that this literature does not exist. The papers I’ve seen about correlates of bus ridership look at it one variable at a time, and yet they are suggestive of positive as well as negative interactions. More broadly, there are interactions between different types of service.

Positive interactions tend to involve network effects. These include the interaction between transit and transit-oriented development, as well as that between different aspects of rail modernization. Whenever there is positive interaction between variables, half-measures tend to flop; some are a reverse 80/20 situation, i.e. 80% of the cost yields 20% of the benefits. In some cases, compromises are impossible without making service useless. In others, some starter service is still viable, but in its presence, the case for expansion becomes especially strong, which can lead to a natural virtuous cycle.

Negative interactions occur when different improvements substitute for one another. One straightforward example is bus stops and frequency: frequency and the quality of bus shelter both impact bus ridership, but have a negative interaction, in that at higher frequency, the inconvenience coming from not having bus shelter is less important. In some cases, negative interactions can even lead to either/or logic, in which, in the presence of one improvement, another may no longer be worth the economic or political cost. In others it’s still useful to pursue multiple improvements, but the negative interaction implies the benefits are not as great as one might assume in isolation, and transit planners and advocates must keep this in mind and not overpromise.

Door-to-door trip times

The door-to-door trip time includes walking distance to and from the station, waiting time, transferring time, and in-vehicle time. Each of these components affects ridership in that longer trips reduce people’s propensity to choose public transport.

There is strong positive interaction between variables affecting the trip time. This is not directly attested in the literature that I know of, but it is a consequence of any ridership model that lumps the different components of trip time into one. If public transportation runs faster, that is if the in-vehicle time is reduced, then the share of the other components of the trip time rises, which means that the importance of frequency for reducing wait time is increased. Thus, speed and frequency have a positive interaction.

However, at the same time, there is a subtle negative interaction between speed and service provision on buses. The reason is that bus operating expenses are largely a linear function of overall service-hours, since costs are dominated by driver wages, and even maintenance is in practice a function of service-hours and not just service-km, since low speeds come from engine-stressing stop-and-go traffic conditions. In this case, increasing the speed of buses automatically means increasing their frequency, as the same resources are plugged into more service-km. In that case, the impact of a further increase in service is actually decreased: by speeding up the buses, the transit agency has reduced the share of the door-to-door trip time that is either in-vehicle or waiting at a stop, and thus further reductions in wait time are less valuable.

In the literature, the fact that investing in one portion of the trip makes its share of the overall trip length smaller and thus reduces the impact of further investments is seen in research into ridership-frequency elasticity. My standard references on this – Lago-Mayworm-McEnroe and Totten-Levinson – cite lit reviews in which the elasticity is far higher when frequency is low than when it is high, about 1 in the lowest-frequency cases and 0.3 in the highest-frequency ones. When frequency is very low, for example hourly, the elasticity is so high that adding service increases ridership proportionally; when frequency is a bus every few minutes, the impact of service increase on ridership is much smaller.

I’ve focused on in-vehicle time and waiting time, but the other two components are sometimes within the control of the transit agency as well, especially on rapid transit. Station design can reduce transfer time by providing clear, short passageways between platforms; it can also reduce access time by including more exits, for example at both ends of the platform rather than just at one end or in the middle. As such design positively interacts with other improvements to speed, it makes sense to bundle investments into more exits and better transfers with programs that add train service and speed up the trains.

Network effects

There is positive interaction between different transit services that work together in a network. In the presence of a north-south line through a city, the case for east-west transportation strengthens, and vice versa. This is not a new insight – Metcalfe’s law predicts usage patterns of communications technologies and social networks. The same effect equally holds for fixed infrastructure such as rail, and explains historical growth patterns. The first intercity steam railway opened in 1830, but the fastest phase of growth of the British rail network, the Railway Mania, occurred in the late 1840s, after main lines such as the London and Birmingham had already been established. 150 years later, the first TGV would start running in 1981, but the network’s biggest spurt of growth in terms of both route-km and passenger numbers occurred in the 1990s.

Using a primitive model in which high-speed rail ridership is proportional to the product of city populations, and insensitive to trip length, the United States’ strongest potential line is naturally the Northeast Corridor, between Boston and Washington. However, direct extensions of the line toward Virginia and points south are extremely strong per the same model and, depending on construction costs, may have even higher return on investment than the initial line, as 180 km of Washington-Richmond construction produce 540 km of New York-Richmond passenger revenue. In some places, the extra link may make all the difference, such as extending New York-Buffalo high-speed rail to Toronto; what looks like a basic starter system may be cost-ineffective without the extra link.

Network effects produce positive interactions not just between different high-speed rail lines, but also between transit services at lower levels. Rail service to a particular suburb has positive interaction with connecting bus service, for which the train station acts as an anchor; in some cases, such as the Zurich model for suburban transit planning, these are so intertwined that they are planned together, with timed transfers.

Network effects do not go on forever. There are diminishing returns – in the case of rail, once the biggest cities have been connected, new lines duplicate service or connect to more marginal nodes. However, this effect points out to a growth curve in which the first application has a long lead time, but the next few additions are much easier to justify. This is frustrating since the initial service is hard to chop into small manageable low-risk pieces and may be canceled entirely, as has happened repeatedly to American high-speed rail lines. And yet, getting over the initial hurdle is necessary as well as worth it once subsequent investments pan out.

Either-or improvements

In the introduction, I gave the example of negative interaction between bus shelter amenities and frequency: it’s good to have shelter as well as shorter waits, but if waits are shorter, the impact of shelter is lessened. There are a number of other negative interactions in transit. While it is good to both increase bus frequency and install shelter at every stop, some negative interactions lead to either-or logic, in which once one improvement is made, others are no longer so useful.

Fare payment systems exhibit negative interactions between various positive features. The way fare payment works in Germany and Switzerland – paper tickets, incentives for monthly passes to reduce transaction costs, proof of payment – is efficient. But the same can be said about the smartcard system in Singapore, EZ-Link. EZ-Link works so rapidly that passengers can board buses fast, which reduces (but does not eliminate) the advantage of proof-of-payment on buses. It also drives transaction costs down to the point of not making a monthly pass imperative, so Singapore has no season passes, and it too works.

Interior circulation displays negative interactions as well. There are different aspects of rolling stock design that optimize for fast boarding and disembarking of passengers, which is of critical importance on the busiest rail lines, even more than interior capacity. Trains so designed have a single level, many doors (four pairs per 20-meter car in Tokyo), interiors designed for ample standing space, and level boarding. Each of these factors interacts negatively with the others, and in cities other than Tokyo, regional trains like this are overkill, so instead designers balance circulation with seated capacity. Berlin has three door pairs per car and seats facing front and back, Zurich has double-deckers with two pairs of triple-wide doors and has been quite tardy in adopting level boarding, Paris has single-level cars with four door pairs and crammed seats obstructing passageways (on the RER B) and bespoke double-deckers with three pairs of triple-wide doors (on the RER A).

Finally, speed treatments on scheduled regional and intercity trains may have negative interactions. The Swiss principle of running trains as fast as necessary implies that once various upgrades have cut a route’s trip time to that required for vigorous network connections – for example, one hour or just a few minutes less between two nodes with timed transfers – further improvements in speed are less valuable. Turning a 1:02 connection into a 56-minute one is far more useful than further turning a 56-minute service into a 50-minute trip. This means that the various programs required to boost speed have negative interactions when straddling the boundary of an even clockface interval, such as just less than an hour, and therefore only the cheapest ones required to make the connections should receive investment.

Conclusion

Good transit advocates should always keep the complexities that affect transportation in mind. Negative interactions between different investments have important implications for activism as well as management, and the same is true for positive interactions.

When variables interact negatively, it is often useful to put a service in the good enough basket and move on. In some cases, further improvements are even cost-ineffective, or require unduly compromising other priorities. Even when such improvements remain useful, the fact that they hit diminishing returns means advocates and planners should be careful not to overpromise. Cutting a two-hour intercity rail trip to an hour is great; cutting a 40-minute trip to a 20-minute one may seem like a game changer, but really isn’t given the importance of access and egress times, so it’s usually better to redeploy resources elsewhere.

Conversely, when variables interact positively, transit service finds itself in an 80% of the cost for 20% of the benefits situation. In such case, compromises are almost always bad, and advocates have to be insistent on getting everything exactly right, or else the system will fail. Sometimes a phased approach can still work, but then subsequent phases become extremely valuable, and it is useful to plan for them in advance; other times, no reasonable intermediate phase exists, and it is on activists to convince governments to spend large quantities of upfront money.

Transportation is a world of tradeoffs, in which benefits are balanced against not just financial costs but also costs in political capital, inconvenience during construction, and even activist energy. Positive and negative interactions have different implications to how people who want to see better public transport should allocate resources; one case encourages insisting on grand plans, another encourages compromise.

Optimization

This post may be of general interest to people looking at optimization as a concept; it’s something I wish I’d understood when I taught calculus for economics. The transportation context is network optimization – there is a contrast between the sort of continuous optimization of stop spacing and the discrete optimization of integrated timed transfers.

Minimum and maximum problems: short background

One of the most fundamental results students learn in first-semester calculus is that minimum and maximum points for a function occur when the derivative is zero – that is, when the graph of the function is flat. In the graph below, compare the three horizontal tangent lines in red with the two non-horizontal ones:

A nonzero derivative – that is, a tangent line slanting up or down – implies that the point is neither a local minimum nor a local maximum, because on one side of the point the value of the function is higher and on the other it is lower. Only when the derivative is zero and the tangent line is flat can we get a local extreme point.

Of course, a local extreme point does not have to be a global one. In the graph above, there are three local extreme points, two local maxima and one local minimum, but only the local maximum on the left is also a global maximum since it is higher than the local maximum on the right, and the local minimum is not a global minimum because the very left edge of the graph dips lower. In real-world optimization problems, the global optimum is one of the local ones, rather than an edge case like the global minimum of the above graph.

First-semester calculus classes love giving simplified min/max problems. This class of problems is really one of two or three serious calc 1 exercises; the other class is graphing a function, and the potential third is some integrals, at universities that teach the basics of integration in calc 1 (like Columbia and unlike UBC, which does so in calc 2). There’s a wealth of functions that are both interesting from a real-world perspective and doable by a first-semester calc student, for example maximizing the volume of some shape with prescribed surface area.

My formulas for stop spacing come from one of these functions. The overall travel time is a function of walking time, which increases as stops get farther apart, and in-vehicle time, which decreases as stops get farther apart. A certain stop spacing produces the minimum overall trip time; this is precisely the global minimum of the travel time function, which is ultimately of the form f(x) = ax + b/x where a and b are empirical parameters depending on walking speed and other relevant variables.

Continuous optimization

The fundamental fact of continuous optimization, one I wish I’d learned in time to teach it to students, is that at the optimum the derivative is zero, and therefore making a small mistake in the value of the optimum is not a big problem.

What does “mistake” mean in this context? It does not mean literally getting the computation wrong. There is no excuse for that. Rather, it means choosing a value that’s slightly suboptimal for ancillary reasons – perhaps small discontinuities in the shape of the network, perhaps political considerations.

Paul Krugman brings this concept up in the context of wages. The theory of efficiency wages asserts that firms often pay workers above the bare minimum required to get any workers at all, in order to get higher-quality workers and incentivize them to work harder. In this theory, the wage level is set to maximize employer productivity net of wages. At the employer’s optimum the derivative of profit is by definition zero, so a small change in wages has little impact to the employer. However, to the workers, any wage increase is good, as their objective function is literally their wage rather than profits. They may engage in industrial action to raise wages, or push for favorable regulations like a high minimum wage, and these will have a limited effect on profits.

In the context of transit, this has the obvious implication to wages – it’s fine to set them somewhat above market rate since the agency will get better workers this way. But there are additional implications to other continuous variables.

With stop spacing specifically, the street network isn’t perfectly continuous. There are more important and less important streets. Getting transit stops to align with major streets is important, even if it forces the stop spacing to be somewhat different from the optimum. The same is true of ensuring that whenever two transit lines intersect, there is a transfer between them. This is the reason my bus redesign for Brooklyn together with Eric Goldwyn involved drawing the map before optimizing route spacing – the difference between 400 and 600 meters between bus stops is not that important. For the same reason, my prescription for Chicago, and generally other American cities with half-mile grids of arterial roads, is a bus stop every 400 meters, to align with the grid distance while still hewing close to the optimum, which is about 500.

When I talked about stop consolidation with a planner at New York City Transit who worked on the Staten Island express bus redesign, the planner explained the philosophy to me: “get rid of every other stop.” In the context of redesigning a single route, this is an excellent idea as well: the process of adding and removing bus stops in New York is not easy, so minimizing the net change by deleting stops at regular intervals so as to space the remaining stops close to the optimum is a good idea.

The world of public transit is full of these tradeoffs with continuous variables. It’s not just wages and interstations. Fares are another continuous variable, involving particular tensions as different political factions have different objective functions, such as revenue, social rate of return, and social rate of return for the working class alone. Frequency is a continuous variable too in isolation. Top speed for a regional train is in effect a continuous variable. All of these have different optimization processes, and in all cases, it’s fine to slightly deviate from the strict optimum to fulfill a different goal.

Discrete optimization

Whereas continuous optimization deals with flat tangent lines, discrete optimization may deal with delicate situations in which small changes have catastrophic consequences. These include connections between different lines, clockface scheduling, and issues of integration between different services in general.

An example that I discussed in the early days of this blog, and again in a position paper I just wrote to some New Hampshire politicians, is the Lowell Line, connecting Boston with Lowell, a distance of 41 km. The line is quite straight, and were it electrified and maintained better, trains could run at 160 km/h between stops with few slowdowns. The current stop spacing is such that the one-way trip time would be just less than half an hour. The issue is that it matters a great deal whether the trip time is 25 or 27 minutes. A 25-minute trip allows a 5-minute turnaround, so that half-hourly service requires just two trainsets. A 27-minute trip with half-hourly service requires three trainsets, each spending 27 minutes carrying passengers and 18 minutes depreciating at the terminal.

A small deterioration in trip time can literally raise costs by 50%. It gets to the point that extending the line another 50 kilometers north to Manchester, New Hampshire improves operations, because the Lowell-Manchester trip time is around 27-28 minutes, so the extension can turn a low-efficiency 27-minute trip into a high-efficiency 55-minute trip, providing half-hourly service with four trainsets.

In theory, frequency is a continuous variable. However, in the range relevant to regional rail, it is discrete, in fractions of an hour. Passengers can memorize a half-hourly schedule: “the inbound train leaves my stop at :10 and :40.” They cannot and will not memorize a schedule with 32-minute frequency, and needing to constantly consult a trip planner will degrade their travel experience significantly. Not even smartphone apps can square this circle. It’s telling that the smartphone revolution of the last decade has not been accompanied with rapid increase in ridership on transit lines without clockface schedules, such as those of the United States – if anything, ridership has grown faster in the clockface world, such as Germany and Switzerland.

Transit networks involving timed connections are another case of discrete optimization in which all parts of the network must work together, and small changes can make the network fall apart. If a train is late by a few minutes and its passengers miss their connection, the short delay turns into a long one for them. As a result, conscientious schedule planners make sure to write timetables with some contingency time to recover from delays; in Switzerland this is 7%, so in practice, out of every 15 minutes, one minute is contingency, typically spent waiting at a major station.

But this gets even more delicate, because different aspects of the transit network impact how reliable the schedule is. If it’s a bus, it matters how much traffic there is on the line. Buses in traffic not reliable enough for tight connections, so optimizing the network means giving buses dedicated lanes wherever there may be traffic congestion. Even though it’s a form of optimization, and even though there’s a measure of difficulty coming from political opposition by drivers, it is necessary to overrule the opposition, unlike in continuous cases such as wages and fares.

Infrastructure planning for rail has the same issues of discrete optimization. It is necessary to design complex junctions to minimize the ability of one late train to delay other trains. This can take the form of flying junctions or reducing interlining; in Switzerland there are also examples of pocket tracks at flat junctions where trains can wait without delaying other trains behind them. Then, the decision of how much to upgrade track speed, and even how many intermediate stations to allow on a line, has to come from the schedule, in similar vein to the Lowell Line’s borderline trip time.

Continuous and discrete optimization

Many variables relevant to transit are in theory continuous, such as trip time, frequency, stop spacing, wages, and fares. However, some of these have discontinuities in practice. Stop spacing on a real-world city street network must respect the hierarchy of more and less important destinations. Frequency and trip times are discrete variables except at the highest intensity of service, perhaps every 7.5 minutes or better; 11-minute frequency is worse to the passenger who has to memorize a difficult schedule than either 10- or 12-minute frequency.

New York supplies a great example showcasing how bad it can be to slavishly hew to some optimal interstation and not consider the street network. The Lexington Avenue Line has a stop every 9 blocks from 33rd Street to 96th, offset with just 8 blocks between 51st and 59th and 10 between 86th and 96th. In particular, on the Upper East Side it skips the 72nd and 79th Street arterials and serves the less important 68th and 77th Streets instead. As a result, east-west buses on the two arterials cross Lexington without a transfer.

Just east of Lex, there is also a great example of optimization on Second Avenue Subway. The stops on Second Avenue are at 72nd, 86th, and 96th, skipping 79th. It turns out that skipping 79th is correct – the optimum for the subway is to the meter the planned stop spacing for the line between 125th and Houston Streets, so it’s okay to have slightly non-uniform stop spacing to make sure to hit the important east-west streets.

Frequency and trip times are subject to the Swiss maxim, run trains as fast as necessary, not as fast as possible. Hitting trip times equal to an integer or half-integer number of hours minus a turnaround time has great value, but small further speedups do not. Passengers still benefit from the speedup, but the other benefits of higher speed to the network, such as better connections and lower crew costs, are no longer present.

The most general rule here is really that continuous optimization tolerates small errors, whereas discrete optimization does not. Therefore, it’s useful to do both kinds of optimization in isolation, and then modify the continuous variable somewhat based on the needs of the discrete one. If you calculate and find that the optimal frequency for your bus or train is once every 16 minutes, you should round it to 15, based on the discrete optimization rule that the frequency should be a divisor of the hour to allow for clockface timetable. If you calculate and find that the optimal bus stop spacing is 45% of the distance between two successive arterial streets, you should round it to 50% so that every arterial gets a bus stop.

Getting continuous optimization right remains important. If the optimal stop spacing is 500 meters and the current one is 200 meters, the network is so far from the local maximum of passenger utility that the derivative is large and stop consolidation has strong enough positive effects to justify overruling any political opposition. However, it is subsequently fine to veer from the optimum based on discrete considerations, including political ones if removing every 1.7th bus stop is harder than removing every other stop. Close to the local maximum or minimum, small changes really are not that important.

Construction Costs in the Nordic Countries

I write a lot about stereotypes in the context of construction costs. Countries with a reputation for corruption, such as Spain, South Korea, Greece, and Italy, often build subways very cheaply. Germany, for all its stereotype of efficiency, has high costs and some dysfunctional decisionmaking in what to build. Singapore, the self-styled most efficient government, pays its transport minister more than a million dollars per year to make excuses for why it has such high construction costs.

In the Nordic countries, the stereotype is correct: those countries have transparent, clean governments, and also build infrastructure cheaply.

Subway tunnels

All four mainland Nordic capitals have recent or ongoing metro expansion projects:

Stockholm just opened Citybanan, a regional rail connection including 6 km of tunnel with two deep stations in Central Stockholm and a 1.4 km bridge. The total cost was 16.8 billion SEK in 2007 terms, which in today’s PPP terms is about $330 million per km. It’s expensive for a suburban subway but not for regional rail.

Copenhagen is currently wrapping up construction on the fully underground, driverless City Circle Line. It is a circular but not circumferential line through city center. With repeated schedule slips, the budget is now 24.8 billion DKK, or $3.4 billion in PPP terms, which is $220 million per km.

Stockholm is expanding its metro in three directions. The fully underground extensions are together 19 km and 22.4 billion SEK, which in PPP terms is $130 million per km.

Helsinki has just opened an expansion of its metro westward to Espoo. This is a 13.5 km, 8-station fully underground line with a water crossing. After cost overruns, the current cost estimate is 1,186 million, which is in PPP terms $115 million per km.

Oslo recently opened a short connection, called Lørenbanen. It’s 1.6 km long and includes a single new station, for a total of NOK 1.33 billion, including 150 million for modernization of an existing connecting line. In PPP terms this is just $90 million per km in today’s money.

Other rail infrastructure

Sweden is investing heavily in mainline rail modernization. This includes a planned high-speed rail network connecting the country’s three biggest cities, which are spaced far apart and not on a line, requiring the total system to be 740 km long. The cost projection as of 2015 is 125 billion SEK, which in PPP terms is $14 million per km; I do not know if it is in 2015 prices or expected year of construction prices. This cost figure is comparable to that of Madrid-Barcelona and about half the at-grade norm for Europe.

Sweden is simultaneously investing in its mainline network, rather than neglecting it in favor of just HSR the way France is. A document from 2009 lists some of these on p. 38 based on the national plan of 2010-21, which did not include HSR. Of note, two full double-track projects are coming it at about $10 million per km or slightly more. In contrast, in Berlin, suburban S-Bahn double-tracking is around twice as expensive per the list on PDF-pp. 73-77 of the official wishlist.

In Denmark, a recent double-tracking project cost 675 million DKK for 20 km, or $4.6 million per km, even cheaper than in Sweden. The project includes not just double track but also an upgrade to 160 km/h.

Denmark is also investing heavily in electrification – see here for a list of projects, without costs. Costs for some of these projects are provided by Railway Gazette. The Fredericia-Aalborg line is 249 km and 4.7 billion DKK, the Roskilde-Kalundborg line is 56 km and 1.2 billion DKK, and the Esbjerg-Lunderskov line is 57 km and 1.19 billion DKK; all three lines are double-track. The longer line is $2.6 million per km, the shorter two are $2.9 million. This is much cheaper than in the core Anglosphere but more expensive than projects for which I have data in France, Israel, and New Zealand.

It’s cheap, but do people ride it?

Absolutely. Low construction costs can occur for projects that nobody has any reason to build, they’re so low-ridership, while some high-cost projects remain cost-effective if they have extremely high ridership, like Second Avenue Subway Phase 1.

In the case of the Nordic capitals, the recent extensions are well-patronized. The ridership prognosis for the City Circle Line is 289,000 per weekday, which means its cost is $11,800 per rider. The link above for the Stockholm T-bana extension projects 170,000 riders per day, which I believe means weekday rather than literal day; in that case, the projected cost per rider is $14,500. Løren’s ridership is 8,000 per day, which one former resident says is just boardings without alightings, which means total ridership is actually 16,000, making the cost of the line just shy of $9,000 per rider. And Helsinki’s West Metro is projected to get 100,000 daily riders, which means its cost is about $15,500 per rider.

Moreover, Stockholm’s overall use of public transportation is very healthy. The first 6 pages of this PDF comprise a report on modal split in Stockholm, out of all trips, not just work trips. In 2015, 32% of all trips in Stockholm County were by public transport, 38% were by car, 9% were by bike, and 16% were on foot. There had been a notable shift from cars to the other modes since 2004.

Converting this statistic to work trip mode share, the most stable metric and the one reported for the US, Canada, UK, and France, requires some additional work. However, where both statistics are available, they do provide some insight: in Hamburg in 2008, the overall car mode shares for all trips and for just work trips were similar (48% for work trips vs. 42% for all trips in the city, 65% vs. 63% in the suburbs); work trips alone exhibit much higher transit mode share (33% vs. 18% in the city, 16% vs. 8% in the suburbs), at the expense of non-motorized trips, which are disproportionately for short errands. It is very likely that the work trip public transport mode share in Stockholm County is comparable to Ile-de-France’s 43%, in a metro area one fifth the size.

Transit ridership in the other Nordic capitals is weaker, though still impressive for their size. Copenhagen lags in transit but has a strong bike network. Oslo had 118 million metro riders in 2017 (source, PDF-p. 31 – per same link you can also see the operating costs per car-km work out to just short of PPP$4, compared with a typical first-world range of $4-7), plus some additional commuter rail ridership (65 million nationwide, not just around Oslo). Helsinki had 63 million annual metro passengers in 2015, before the extension opened, and somewhat fewer additional commuter rail passengers, for a total ridership of perhaps 120 million. Both of the smaller cities have about the same metro area rail ridership per capita as New York, which is about fifteen times their size.

What does this mean?

Scandinavia has a reputation for efficient government at home as well as abroad. Right-wing pundits are far more likely to look for aspects of its governance that play to their desire for privatization, such as Sweden’s school voucher system or the contracting out of urban rail, than to assert that Scandinavia is a socialist failure. Unlike autocracies that have cultivated such reputation, the Nordic countries fully deserve this praise when it comes to building infrastructure cost-effectively. Sweden appears to consistently build rail for half the per-unit cost of Germany.

And yet, I don’t see that much praise for Nordic infrastructure. There are people in the English-speaking world making grandiose claims about how democratic countries need to be more like China and about how authoritarianism is just more efficient. I don’t know of any making that claim about how Nordic social democracy is more efficient, with its depoliticized state apparatus, multiparty elections, high levels of transparency, bureaucratic legalism, and near-universal collective bargaining.

Across all levels of public transportation investment, from high-speed rail down to routine track upgrades, we see inexpensive, efficient projects in the Nordic countries. They achieve high levels of rail usage without megacities in which only masochists drive, and keep expanding their networks in order to complete the green transition. Public transit managers in not just the laggard that is the US but also Germany and other relatively solid countries should make sure to study how things work in Scandinavia and how they can import Nordic success.

Circles

Rail services can be lines or circles. The vast majority are lines, but circles exist, and in cities that have them they play an important niche. Owing to an overreaction, they are simultaneously overused and underused in different parts of the world. However, that some places overuse circles does not mean that circles are bad, nor does it mean that specific operational problems in certain cities are universal.

In particular, what I think of as the ideal urban rapid transit network should feature circles once the network reaches a certain scale, as in the following diagram that I use as my Patreon avatar:

Circles and circumferentials

Circles are transit lines that run in a loop without having a definitive start or end. Circumferentials are lines that go around city center, connecting different branches without passing through the most congested part of the city. In the ideal diagram above, the purple line is both a circle and a circumferential. However, lines can be one without being the other, and in fact examples of lines that are only one of the two outnumber examples of lines that are both.

For example, here is the Paris Metro:

Paris has a circle consisting of Metro Lines 2 and 6, which are operationally lines; people wishing to travel on the arcs through the meeting points at Nation and Etoile must transfer. Farther out, there is an incomplete circle consisting of Tramway Line 3, where the forced transfer between 3a and 3b is Porte de Vincennes. Even farther out there is an under-construction line not depicted on the map, Line 15 of Grand Paris Express, which has a pinch point at its southeast end rather than continuous circular service. All three systems are great example of circumferential lines with very high ridership that are not operationally circles.

Another rich source of circumferential lines that are not circles is cities near bodies of water. In those cities, a circumferential line is likely to be a semicircle rather than a circle. This is responsible for the current state of the Singapore Circle Line, although in the future it will be closed to form a full circle. The G train in New York is a single-sided circumferential line to the east of Manhattan, not linking with anything to the west of Manhattan because of the combination of wide rivers and the political boundaries between New York and New Jersey.

In the opposite direction – circles that are not circumferentials – there are circular lines that don’t neatly orbit city center. The Yamanote Line in Tokyo is one such example: its eastern end is at city center, so it combines the functions of a north-south radial line with those of a north-south circumferential line connecting secondary centers west of Central Tokyo. London’s Circle Line is no longer operationally a circle but was one for generations, and yet it was never a circumferential – it combined the central legs of two east-west radial mainlines, the Metropolitan and District lines.

We can collect this distinction into a table:

Circle, not circumferential Circumferential, not a circle Circumferential circle
Yamanote Line
Osaka Loop Line
Seoul Metro Line 2
London Circle line (until 2009)
Madrid Metro Line 12
Paris M2/6, T1, T2, T3, future M15
Copenhagen F train
New York G train, proposed Triboro
London Overground services
Chicago proposed Circle Line
Singapore Circle line (today)
Moscow Circle Line, Central Circle
Berlin S41/S42
Beijing Subway Line 2, Line 10
Shanghai Metro Line 4
Madrid Metro Line 6

Operational concerns: the steam era

In the 19th century, it was very common to build circular lines in London. In the steam era, reversing a train’s direction was difficult, so railways preferred to build circles. This was the impetus for joining the Metropolitan and District lines to form the Circle line. Mainline regional rail services often ran in loops as well: these were as a rule never or almost never complete circles, but instead involved trains leaving one London terminus and then looping around to another terminus.

Another city with a legacy inherited from steam-era train operations is Chicago. The Loop was built to easily reverse the direction of trains heading into city center. At the outer ends they would need to reverse direction the traditional way, but there was no shortage of land for yards there, unlike in the Chicago CBD since named after the Loop.

As soon as multiple-unit control was invented in the 1890s, this advantage of circles evaporated. Subsequently rapid transit lines mostly stopped running as circles unless they were circumferential. London’s Central line, originally pitched as two long east-west lines forming a circle, became a single east-west line, on which trains would reverse direction.

Operational concerns: the modern era

Today, it is routine to reverse the direction of a rapid transit train. The vast majority of rapid transit routes run as lines rather than circles.

If anything, there have been complaints that circles are harder to run service on than lines. However, I believe these concerns are all specific to London, which changed its Circle line from a continuous loop to a spiral in 2009. I have heard concerns about the operations of the Ringbahn here, but as far as I can tell the people who express them are doing so in analogy with what happened in London, and are not basing them on the situation on the ground here. Moreover, there are no plans to make the Yamanote Line run as anything other than the continuous loop it is today.

The situation in London is that the Circle line has always shared tracks with both the Metropolitan and District lines. There has always been extensive branching, in which a delay on one train propagates to the entire network formed by these two mainlines. To this day, Transport for London does not expect the lines in the subsurface network to have the same capacity as the isolated deep tube lines: with moving block signaling it expects 32 trains per hour, compared with 36 on isolated lines.

What’s more, the junctions in London are generally flat. Trains running in opposite directions can conflict at such junctions, which makes the schedules more fragile. Until 2009, London ran the Circle line trains every 7 minutes, which was bound to create conflicts with other lines.

The importance of this London-specific background is that the argument against circles is that they make schedules more fragile. If there is no point on the line where trains are regularly taken out of service, then it is hard to recover from timetable slips, and delays compound throughout the day. However, this is relevant mainly in the context of an extensively-branching system like London’s. Berlin has some of that branching as well, but much less so; one of the sources of reverse-branching on the S-Bahn is a line that should get its own cross-city route anyway, and another is a Cold War relic swerving around West Berlin (S8/85).

The benefits of complete circles

The complete circle of the Yamanote Line or the Ringbahn can be compared with incomplete circles, such as the Oedo Line or the various circumferentials in Paris. From passengers’ perspective, it’s better to have a complete circle, because then they can undertake more trips.

Circumferential lines broadly have two purposes:

  1. They offer service on strong corridors that are orthogonal to the direction of city center, such as the various boulevards hosting the M2/6 ring as well as the Boulevards des Maréchaux hosting T3.
  2. They offer connections between two radial lines that may not connect in city center, or may connect so far from the route of the circumferential that transferring via the circumferential is faster.

Both purposes are enhanced when the route is continuous. In the case of Paris, a north-south trip east of Nation is difficult to undertake, as it requires a transfer at Porte de Vincennes. Passengers connecting from just south, on M8 or even on M7, may not save as much time traveling to lines just north, such as M9 or M3, and might end up transferring at the more central stations of Republique or Opera, adding to congestion there.

In contrast, in Berlin the continuous nature of the Ring makes trips across the main transfer points more feasible. Just today I traveled from my new apartment to a gaming event on the Ringbahn across Ostkreuz. At Ostkreuz the trains dwelled longer than the usual, perhaps 2 minutes rather than the usual 30 seconds, which I imagine is a way to keep the schedule. That delay was, all things considered, minor. Had I had to transfer to a new train, I would have almost certainly taken a different combination of trains altogether; the extra waiting time adds up.

Why are circles so uncommon?

The operational concerns of London aside, it’s still uncommon to see complete circles on rapid transit networks. They are the ideal for cities that grow beyond the scale of three or four radial trunks, but there are only a handful of examples. Why is that?

The answer is always some sort of special local concern. If city center is offset to one side of the built-up area, such as in a coastal city, then circumferential lines will be semicircles and not full circles. If there is some dominant transfer point that requires a pinch, then cities prefer to build a pinch into the system, as is the case for Porte de Vincennes on T3 or for some of the lines cobbled together to form the London Overground.

This is similar to the question of missed connections. Public transportation networks must work hard to ensure that whenever two lines meet, they will have a transfer. Nonetheless, missed connections exist in virtually all large rapid transit networks. Some of those are a matter of pure incompetence, but in many, rail networks that developed over generations may end up having one subway line that happens to intersect another far from any station on the older line, and there is little that can be done.

Likewise, it is useful to ensure that circumferential lines be complete circles whenever the city is symmetric enough to warrant circles. Paris, like other big cities with strong transit networks, is good but not perfect, and it is important to call it on the mistakes it makes, in this case building M15 to have a jughandle rather than running as a complete circle.

Stop Spacing and Route Spacing

Six months ago I blogged a model for optimal stop spacing on an urban transit route. These models exist in the published literature, but they assume that the speed benefit of stop consolidation reduces operating costs, which requires introducing new variables for the value of time. My model assumes the higher speed of stop consolidation is plugged into higher frequency, which means only five variables are needed, and only two of them vary substantially between different cities and their networks. The formula is a square root.

In this post, I’m going to extend this formula to optimizing route spacing on a grid.

I’m using mode-neutral language like “vehicle,” but this is really just about buses, because to a good approximation, urban rail networks are never grids. I’m sorry, Mexico City, I know your Metro network does its best to pretend you have an isotropic city, but your three core radial lines are just far busier than the tangential ones.

Optimal stop spacing: a recap

My previous post uses words rather than symbolic language, since there are only five relevant parameters. Here I’m going to use symbols for the variables to make the calculation even somewhat tractable. All units I’m using are base SI units, so speed is expressed in meters per second rather than kilometers per hour, but the dimensional analysis works out so that it’s not necessary to pick units in advance.

  • s: stop spacing
  • v: walk speed
  • p: stop penalty
  • d: average distance traveled
  • w: walk/wait penalty, expressed as a ratio of perceived walk or wait time to in-vehicle time
  • λ: average distance between successive vehicles, or in other words headway in units of distance, not time

The variables v and p are fairly consistent from place to place. The variable w is as well, but may well differ by circumstance, e.g. people with luggage may have a higher walk penalty and a lower wait penalty, and people who are more familiar with the system usually have lower w. The parameter λ is a function of how much service runs on the line, as we will see when we expand to cover route spacing.

A key assumption in this model is that d does not change based on the network. This is a simplification: if s is too low then it will drag down d with it, as people who are discouraged by the slow in-vehicle speed avoid long trips or choose other modes of travel, whereas if s is too high then it will drag d up, as people who have to walk too long to the stop may just walk all the way to their destination if it’s nearby. In Carlos Daganzo’s textbook this situation is resolved by replacing an empirically determined d with the size of the city, assuming travel is isotropic, but the effect is essentially the same as just setting d to be half the length of a square city.

The formula for perceived travel time is

\frac{sw}{2v} + \frac{dp}{s} + \frac{\lambda wp}{2s}

if travel along the line is isotropic, or

\frac{sw}{4v} + \frac{dp}{s} + \frac{\lambda wp}{2s}

if one end of the travel (e.g. the residential end) is isotropic and the other is at a fixed node (e.g. a subway transfer). In either case, in-vehicle time excluding stops is omitted, as it is constant.

The minimum travel time occurs at

s = \sqrt{2\cdot \frac{v}{w}\cdot p\cdot(d + \frac{\lambda w}{2})}

if travel is isotropic and

s = \sqrt{4\cdot \frac{v}{w}\cdot p\cdot(d + \frac{\lambda w}{2})}

if there is a distinguished node at one end of the trip.

Observe that there is negative interaction between stop consolidation and other aspects of bus modernization. First, higher frequency, as expressed in concentrating service on strong routes, reduces the value of λ and therefore slightly reduces the optimal stop spacing. Second, the model assumes the same penalty w for walking and waiting, but sometimes these two activities have distinct penalties, and then the walk penalty is responsible for the occurrence of w in the denominator in the formula whereas the wait penalty supplies the appearance of w in the numerator. Improving bus stop facilities reduces the wait penalty, pushing the optimal s farther down, even though at the same time it’s cheaper to improve bus stops if there are fewer of them.

The empirically determined values of the five variables in the formula are as follows:

  • v is 1.45 m/s in Forde-Daniel, 1.3-1.4 m/s in Bohannon, and 1.38 in TRB Part 4, PDF-p. 16; I take v = 4/3
  • p is 25 seconds based on examining the differences in schedules between local and limited buses in New York and Vancouver
  • d is 3,360 meters per unlinked trip per the NTD
  • w is around 2 for waiting in Fan-Guthrie-Levinson, 2 in general for buses in Teulings-Ossokina-de Groot, PDF-p. 25, 1.75 in the New York MTA’s internal model, 2.25 in the MBTA’s (as mentioned in one of Reinhard Clever’s papers), and a range of 2-3 in Lago-Mayworm-McEnroe; I take w = 2
  • λ is single-lane network length (that is, twice the route-length, modulo one-way loops) divided by fleet size in actual use, which is 1,830 meters in Brooklyn today and 1,160 based on what Eric Goldwyn and I recommend

This leads to optimal stop spacing equal to

s = \sqrt{2\cdot \frac{4/3}{2}\cdot 25\cdot(3360 + \frac{1160\cdot 2}{2})} = 388 \mbox{ meters}

if travel is isotropic and

s = \sqrt{4\cdot \frac{4/3}{2}\cdot 25\cdot(3360 + \frac{1160\cdot 2}{2})} = 549 \mbox{ meters}

if there is a distinguished node. The numbers are slightly lower than in my older post since I’m using a slightly lower walk speed, 1.33 m/s rather than 1.5.

Optimal route spacing: stops at intersection points

Studying route spacing has to incorporate stop spacing for a simple reason: there should be a stop at every intersection between routes, and therefore the route spacing should be an integer multiple of the stop spacing. There are three modifications required to the above formula, of which the first is easy, the second requires defining more parameters but is mathematically still easy, and the third is very hard:

  1. Passengers need to walk not just along the route to their stop but also from their origin to the route, which increases walk time
  2. The value of λ may change, since fewer routes imply more vehicles per route and thus denser vehicle spacing, and in particular wait time depends not just on how many stops are on the way but also on the speed net of stops
  3. Increasing the route and stop spacing in tandem reduces the number of stops involved in waiting for the bus (this is λ again) twice, that is quadratically

The first modification means that instead of traveling an average distance of s/4 to the stop at each end, assuming isotropy, people have to travel a distance of s/4 along the route and also s/4 to the route itself. In the travel time formula, we replace sw/2v with just sw/v with isotropic travel.

To deal with the second modification, we define the following variables, in addition to the ones from the section above on stop spacing:

  • f: fleet size in independent vehicles in actual revenue operation (buses or trains, not train cars)
  • a: area of the network to be covered by the grid, e.g. a city, metro area, or borough
  • u: speed assuming there are no stops along the route

If the area is a, then we can approximate it as a square of side \sqrt{a}, which has \sqrt{a}/s north-south and \sqrt{a}/s east-west routes, each of length \sqrt{a}, and thus the total two-way network length is 2a/s. Since the value of λ is the one-way length divided by fleet size, we write

\lambda = \frac{4a}{sf}

Moreover, people wait an additional λw/2u; in the previous section this wait existed as well but was ignored in the formula as it did not depend on s, but here it does, and thus we need to add this wait factor.

We deal with the third modification by replacing λ with 4a/sf in the formula for wait time. If people travel isotropically and do not transfer, the travel time formula is now

\frac{sw}{v} + \frac{dp}{s} + \frac{d}{u} + \frac{2aw}{sfu} + \frac{2awp}{fs^{2}}

The summand d/u is constant but is included for completeness here, in analogy with the no-longer-constant summand 2aw/sfu.

But it’s the last summand that gives the most problems: it turns the optimization problem from extracting a square root to solving a cubic. This is technically possible, but the formula is opaque and does not really help showcase how the parameters affect the final outcome. We need to solve for s:

\frac{w}{v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0

We can plug in the above values of w, v, d, and p, as well as the following values of the new variables, and use any cubic solver:

  • f = 612 buses in Brooklyn, excluding vehicles in turnaround, non-revenue service, etc. (it’s actually slightly lower today, around 600, but our network is a bit more efficient with depot moves)
  • a = 180,000,000 m^2 for Brooklyn
  • u = 5.3 m/s net of stops, assuming our other proposals, such as bus lanes, are implemented

The cubic formula turns into

1.5s^{3} - 305976s - 58823529 = 0

for which the positive solution is s = 528 meters.

We can complicate this formula in two ways.

First, we can let go of the assumption of isotropy. If there is a distinguished node at one end, then walk time is halved, as in the formula for stop spacing on a given route. The overall travel time is equal to

\frac{sw}{2v} + \frac{dp}{s} + \frac{d}{u} + \frac{2aw}{sfu} + \frac{2awp}{fs^{2}}

and this is optimized when

\frac{w}{2v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0.

Plugging the usual values of the parameters, we get

0.75s^{3} - 305976s - 58823529 = 0,

for which the positive solution is s = 719 meters. The ratio between the results with isotropy and a distinguished node is 1.36, close to the square root of 2 that we get in the formula for stop spacing on a predetermined route; the reason is that in the cubic formula the linear term is much larger than the constant term near the root, so the effect of changing the cubic term is much closer to the square root than to the cube root.

The second complication is introducing transfers. Transfers do not change the walk time – the walking time between platforms or curbside waiting areas is small and constant – but introduce additional wait time, which means we need to double both terms that include waits. But if we have transfers we need to restore the assumption of isotropic travel, since for the most part the distinguished nodes for Brooklyn buses involve subway transfers.

In that case, the travel time formula is

\frac{sw}{v} + \frac{dp}{s} + \frac{d}{u} + \frac{4aw}{sfu} + \frac{4awp}{fs^{2}}

which is minimized at the positive root of the cubic

\frac{w}{v}s^{3} - (dp + \frac{4aw}{fu})s - \frac{8apw}{f} = 0.

We need to figure out the value of d, which is difficult in this case – the New York bus network discourages bus-to-bus transfers through low frequency and poor bus stop amenities. That the formulas I’m using do not allow for how the shape of the network influences d is a real drawback here. But if we let d be the usual 3,360 meters that it is for unlinked trips, and plug the usual values of the other parameters, we get,

1.5s^{3} - 527951s - 117647059 = 0

to which the solution is s = 683 meters.

Optimal route spacing: the general case

The above section makes a critical assumption about route spacing and stop spacing: they must be equal, making every stop a transfer. However, this assumption is not strictly necessary. Indeed, if we assume isotropy, and let the route spacing be 860 meters, then it’s better for passengers to double the density of stops to one every 430 meters just from looking at the formula for stop spacing.

In this section, we look at the optimal formulas assuming route spacing is twice or thrice the stop spacing. Then in the next section we will compare everything together.

We keep all the variable names from before, and set s to be the stop spacing, not the route spacing. Instead, we will find formulas for route spacing equal to 2s and 3s and compare their optima with that for the special case in which stop and route spacing are equal.

We need to modify the formula in the previous section in two ways. First, walk time is, in the isotropic case, half the stop spacing plus half the route spacing. And second, the dependence of λ on the shape of the network comes from route spacing rather than stop spacing. If route spacing is 2s, the formula for travel time is

\frac{3sw}{2v} + \frac{dp}{s} + \frac{d}{u} + \frac{aw}{sfu} + \frac{awp}{fs^{2}}

and its minimum is at the positive solution to

\frac{3w}{2v}s^{3} - (dp + \frac{aw}{fu})s - \frac{2apw}{f} = 0.

We retain the New York- and Brooklyn-oriented variables from the above sections and obtain

2.25s^{3} - 194989s - 29411765 = 0.

The solution is s = 352 meters, i.e. routes are to be spaced 704 meters apart, with one intermediate station on each route between each pair of successive crossing routes.

If we have three interstation segments between two successive routes, then we need to solve the cubic

\frac{2w}{v}s^{3} - (dp + \frac{2aw}{3fu})s - \frac{4apw}{3f} = 0

or

3s^{3} - 157992s - 19607843 = 0

to which the solution is s = 276 meters.

In the above section we also looked at two potential complications: introducing transfers, and introducing non-isotropy. Non-isotropy, expressed as an isotropic origin and a distinguished destination, halves the cubic term; transfers double the wait times and thus double the constant term and the larger of the two summands adding up to the linear term.

If the route spacing is exactly twice the stop spacing, then the non-isotropic formula is

\frac{3w}{4v}s^{3} - (dp + \frac{aw}{fu})s - \frac{2apw}{f} = 0

or, using the same parameters as always,

1.125s^{3} - 194989s - 29411765 = 0.

The solution is s = 420 meters, with routes spaced 840 meters apart.

The isotropic cubic with transfers is

\frac{3w}{2v}s^{3} - (dp + \frac{2aw}{fu})s - \frac{4apw}{f} = 0

and with the usual parameters, again sticking with d = 3,360 even though in practice it is likely to be higher, this is

2.25s^{3} - 305976s - 58823529 = 0

and then the root is s = 442 meters, with routes spaced 884 meters apart.

We conclude this section with the same formulas assuming the route spacing is not 2s but 3s. The non-isotropic, one-seat ride formula is

\frac{w}{v}s^{3} - (dp + \frac{2aw}{3fu})s - \frac{4apw}{3f} = 0

or with the usual parameters

1.5s^{3} - 157992s - 19607843 = 0,

of which the positive root is s = 374 meters, with routes spaced 1,123 meters apart,

The transfer-based isotropic formula is,

\frac{2w}{v}s^{3} - (dp + \frac{4aw}{3fu})s - \frac{8apw}{3f} = 0

or

3s^{3} - 231984s - 39215686 = 0.

The positive root is s = 340 meters, with routes spaced 1,021 meters apart.

What’s the best route spacing?

We have optimums based on assumptions about the interaction between stop and route spacing, but so far we have not compared these assumptions with each other. In this section, we do. For each scenario – isotropic, transfer-free travel; a distinguished node along transfer-free travel; and isotropic travel with a transfer – we look at the optimal values of route spacing equal to one, two, or three times the stop spacing.

In the table below, the walk and wait times are without penalty; but the penalty is applied to them when summed with in-vehicle time.

Scenario Component Route spacing = s Route spacing = 2s Route spacing = 3s
Isotropy; 1-seat ride Optimal s 528 352 276
Walk time 396 396 414
Wait time 262.954 216.997 198.394
In-vehicle time 793.053 872.599 938.31
Total time 2110.962 2098.593 2163.097
Distinguished node; 1-seat ride Optimal s 719 420 374
Walk time 269.625 236.25 280.5
Wait time 182.811 173.812 133.965
In-vehicle time 750.791 833.962 858.561
Total time 1655.663 1654.086 1687.49
Isotropy; 2-seat ride Optimal s 683 442 340
Walk time 512.25 497.25 510
Wait time 388.05 326.378 302.432
In-vehicle time 756.949 824.008 881.021
Total time 2557.549 2471.263 2505.885

 

The table implies that in all scenarios it’s optimal to have two interstations between parallel routes, though if there’s a distinguished node the difference with having just one interstation between parallel routes is very small. The three-interstation option is never optimal, but is also never far from the optimum, only half a minute to a minute worse.

But please interpret the table with caution, especially the two-seat ride section. The total time for a 3.36-kilometer trip without applying the walk or wait penalty is about 28 minutes regardless of whether the route to stop spacing ratio is 1, 2, or 3. This is still faster than walking, but not by much, and riders may well be so discouraged as to walk the entire way. If the trip is much shorter than 3.36 kilometers or the rider’s particular disutility of walking is much lower than 2 then transit will not be competitive with walking. In turn, a network set up with the stop spacing implied by the above formulas will only get transfer trips if they’re much longer, which should raise the optimal interstation somewhat. If d = 6,000 then in the transfer scenario the optimum if stop and route spacing are equal is 711 meters and that if route spacing is twice as high as stop spacing is 470 meters, and the latter option is noticeable faster.

How does our bus redesign compare with the theory?

We drew our redesigned map with full knowledge of how to optimize stop spacing on a single route, but we didn’t look at route spacing optimization. Of course, the assumption of regular route spacing is less realistic than that of regular stop spacing, as some areas have higher demand, or more distinguished arterials. But we can still discuss the average route spacing in our plan, by comparing our proposed route-length with Brooklyn’s land area.

With a 356-kilometer network in a borough of 180 km^2, effective route spacing is 1,010 meters. This is a little longer than I expected; in Southern Brooklyn the north-south and east-west routes we propose are spaced around 800-850 meters apart, and in Bed-Stuy the east-west routes tighten to 600 meters as they’re all radial toward Downtown Brooklyn and quite busy. The reason the answer is 1,010 meters is that there are margins of the borough with no service (like Floyd Bennett Field) or grid interruptions due to parks (such as Prospect Park) or already-good subway service (South Brooklyn).

The stop spacing we use is 480 meters, excluding nonstop freeway segments in the Brooklyn-Battery Tunnel and toward JFK. In the Southern Brooklyn grid, we’re pretty close to a regular spacing of two interstations between parallel routes. In the Bed-Stuy grid, the north-south routes have a stop per crossing route since the east-west routes are so densely placed, and the east-west routes have one, two, or three interstations between crossing routes, but the average is two.

To the extent the optimization formulas tell us anything, it’s that we should consider adding a few more routes. Target additions include another north-south Bed-Stuy route, an east-west route in South Brooklyn restoring the discontinued B71, and a north-south route through Southern Brooklyn on 16th Avenue. Altogether this would add around 20 km to our network. Beyond that, additional routes would duplicate subway routes, which my analysis above excludes even when they form a coherent grid with the buses.

Rules of thumb for your city

If your city has streets that form a coherent grid, then you can design a bus grid without too many constraints. By constraints I mean street networks that interrupt the grid so often so as to force you to use particular streets at particular spacing, for example the Bronx or Queens. Constraints in a way make planning easier, by reducing the search space; I contend Brooklyn is the hardest of the four main boroughs to redesign precisely because it has the fewest constraints in its grids and yet its grid is just interrupted enough that it cannot be treated as tabula rasa.

In general, you probably want buses spaced around 800 meters to a kilometer apart. While the value of d will differ between cities, the optimum route spacing isn’t that sensitive to it. If d rises to as high as 10,000, the optimal s in the scenario with transfers is 753 meters if route spacing equals stop spacing and 511 meters if it equals twice stop spacing, compared with 683 and 442 meters respectively with d = 3,360; the one-interstation-per-parallel-route scenario becomes better than the two-interstation scenario, but the difference is half a minute, compared with a minute and a half in favor of two interstations with d = 3,360.

In practice I don’t know of any city whose grid is so unconstrained and so isotropic that you can seriously debate 700, 800, 900, 1,000, etc. meters between routes. At that resolution you’re always constrained by arterial spacing, which in American cities tends to be 800 because it’s half a mile and in Canada is irregular (de facto close to a mile) due to constant grid interruptions on intermediate would-be arterials in both Toronto and Vancouver. In this range of arterial spacing, you want exactly two interstations between parallel routes; if you want more or fewer then you should have a very good reason, such as a major destination such as a hospital located at an awkward offset.

Something that does matter very much is fleet size relative to the area served – the quantity a/f. If you aren’t running much service, then you need wider route spacing just to avoid reducing frequency to unusable levels. If instead of f = 612 we use f = 200, then the optimum with one interstation per parallel routes with the transfer scenario is s = 1087, with two it’s s = 676, with three it’s s = 508, and with four it’s s = 414, and among these three is best and even four is a few seconds faster than two. In that case route spacing of about a kilometer and a half, which may be a mile in American arterials, is fully justified.

Conversely, if buses are faster, that is if u is higher, then the optimal interstations fall in all cases. This is because the impact of u comes from its effect on wait times, so faster buses mean that it’s less important to reduce λ.

The effects of a/f and u relate again to the negative interactions between various components of bus reform. Running more service means it’s justifiable to have more closely-spaced routes, since pruning routes to increase frequency from 10 to 5 minutes is much less valuable than pruning them to increase frequency from 30 to 15 minutes. Likewise, running faster service means wait times fall, again reducing the need to prune routes.

If you’re tasked with designing bus routes, then make sure to use correct values for a, f, u, and d for your city, as they are likely to be very different from those of New York. The formulas are more intricate when optimizing route spacing and it’s useful to play with them until you get comfortable with them on an intuitive level, but ultimately they do give reasonable answers for how to design a bus network.

Little Things That Matter: Bus Shelter

Many years ago, probably even before I started this blog, I visited family in Hamden, a suburb of New Haven. I took the bus from Union Station. When it was time to go back to New York, I timed myself to get to the bus that would make my train, but it rained really hard and there was no shelter. The time passed and as the bus didn’t come, I sought refuge from the rain under a ceiling overhang at a store just behind the bus stop, in full view of the road. A few minutes later, the bus went through the station at full speed, not even slowing down to see if anyone wanted to get on, and to get to my train I had to hitchhike, getting a ride from people who saw that I was a carless New Yorker.

Fast forward to 2018. My Brooklyn bus redesign plan with Eric Goldwyn calls for installing shelter everywhere, which I gather is a long-term plan for New York but one that the city outsourced to a private advertising firm, with little public oversight over how fast the process is to take. When I asked about the possibility of reducing costs by consolidating stops I was told there is no money for shelter, period. It was not a big priority for us in the plan so we didn’t have costs off-hand, but afterward I went to check and found just how cheap this is.

Streetsblog lists some costs in peripheral American cities, finding a range of $6,000-12,000 per stop for shelter. Here‘s an example from Florida for $10,000 including a bench. In Providence I asked and was told “$10,000-20,000.” In Southern California a recent installation cost $33,000 apiece. I can’t find European costs for new installation, but in London replacing an existing shelter with a new one is £5,700, or $8,000.

So let’s say the costs are even somewhat on the high American side, $15,000. What are the benefits?

I’ve found one paper on the subject, by Yingling Fan, Andrew Guthrie, and David Levinson, entitled Perception of Waiting Time at Transit Stops and Stations. The key graph is reproduced below:

The gender breakdown comes from the fact that in unsafe neighborhoods, women perceive waits as even longer than the usual penalty, whereas in safe ones there is no difference between women and men.

The upshot is that if the wait time is 10 minutes, then passengers at a stop with a bench and shelter perceive the wait as 15 minutes, and if there’s also real-time information then this shrinks to 11 minutes. If there are no amenities, then passengers perceive a 15-minute wait when they’ve waited just 6.5 minutes and an 11-minute wait when they’ve waited just 4. In other words, to estimate the impact of shelter we can look at the impact of reducing waits from 10 minutes to 6.5, and if there’s also real-time info then it’s like reducing waits to 4 minutes.

If the wait is 5 minutes then the impact is similar. With bench and shelter the perceived wait is 8.5 minutes, equivalent to a 3-minute wait without any amenities; with real-time information, the perceived wait is 6.5 minutes, equivalent to a 2-minute wait without amenities. There is some scale-dependence, but not too much, so we can model the impact of shelter as equivalent to that of increasing frequency from every 10 minutes to every 6.5 minutes (without real-time displays) or every 4 minutes (with real-time displays).

I have some lit review of ridership-frequency elasticity here. On frequent buses it is about 0.4, but this is based on the assumption that frequency is 7.5-12 minutes, not 4-6 minutes. At the low end this is perhaps just 0.3, the lowest found in the literature I’ve seen. To avoid too much extrapolation, let’s take the elasticity to be 0.3. Fan-Guthrie-Levinson suggests shelter alone is equivalent to a 50-66% increase in frequency, say 60%; thus, it should raise ridership by 15%. With real-time info, make this increase 30%.

What I think of as the upper limit to acceptable cost of capital construction for rail is $40,000-50,000 per weekday rider; this is based on what makes activists in Paris groan and not on first principles. But we can try to derive an equivalent figure for buses. On the one hand, we should not accept such high costs for bus projects, since buses have higher operating expenses than rail. But this is not relevant to shelter, since it doesn’t increase bus expenses (which are mostly driver labor) and can fund its ongoing maintenance from ads. On the other hand, a $40,000/rider rail project costs somewhat more per new rider – there’s usually some cannibalization from buses and other trains.

But taking $40,000/rider as a given, it follows that a bus stop should be provided with shelter if it has at least ($15,000/$40,000)/0.15 = 2.5 weekday boardings. If the shelter installation includes real-time info then the denominator grows to 0.3 and the result falls to 1.25 weekday boardings.

In New York, there are 13,000 bus stops, so on average there are around 180 boardings per stop. Even in Rhode Island, where apparently the standard is that a bus stop gets shelter at 50 boardings (and thus there is very little shelter because apparently it’s more important to brand a downtown trunk as a frequent bus), there are 45,000 weekday riders and 3,000 stops, so at 15 riders per stop it should be fine too put up shelter everywhere.

The only type of stop where I can see an exception to this rule is alighting-only stops. If a route is only used in a peak direction, for example toward city center or away from city center, then the outbound stops may be consistently less used to the point of not justifying shelter. But even that notion is suspicious, as American cities with low transit usage tend to have weak centers and a lot of job and retail sprawl. It’s likely that a large majority of bus stops in Rhode Island and all stops within Providence proper pass the 2.5 boardings rule, and it’s almost guaranteed that all pass the 1.25 boardings rule. And that’s even before consolidating stops, which should be done to improve bus speed either way.

At least based on the estimates I’ve found, installing bus shelter everywhere is a low-hanging fruit in cities where this is not already done. In the situation of New York, this is equivalent to spending around $550 per new weekday rider on transit – maybe somewhat more if the busier stops already have shelter, but not too much more (and actually less if there’s stop consolidation, which there should be). Even in that of Providence, the spending is equivalent to about $6,600 per rider without stop consolidation, or maybe $3,000 with, which is much better than anything the state will be able to come up with through the usual channels of capital expansion.

If it’s not done, the only reason for it is that transit agencies just don’t care. They think of buses as a mode of transportation of last resort, with a punishing user experience. Cities, states, and transit agencies can to a large extent decide what they have money for, and letting people sit and not get drenched is just not a high priority, hence the “we don’t have money” excuse. The bosses don’t use the buses they’re managing and think of shelter as a luxury they can’t afford, never mind what published transportation research on this question says.

Battery-Electric Buses: New Flyer

Two months ago, after my article about battery-electric buses appeared in CityLab, New Flyer reached out to me for an interview. Already in one of the interviews I’d done for the article, I heard second-hand that New Flyer was more reasonable than Proterra and BYD and was aware of the problem of battery drain in cold weather. I spoke to the company’s director of sustainable transportation, the mechanical engineer David Warren, and this confirmed what I’d been told.

Most incredibly, I learned at the interview that the headline figures used in the US for electric bus performance explicitly exclude heating needs. The tests are done at the Altoona site and only look at electricity consumption for propulsion, not heating. New Flyer says that it is aware of this issue and has tried not to overpromise, but evidently Proterra and BYD both overpromise, and regardless of what any vendor says, American cities have bought into the hype. In Duluth this was only resolved with fuel-fired heaters; the buses only use electricity for propulsion, which is not the majority of their energy consumption in winter.

Warren and I discussed New York specifically, as it has a trial there on the M42. The heater there puts out 22 kW of energy at the peak, but on the day we discussed, January 29th, when the air temperature was about -7*, actual consumption was on average about 10 kW. Electricity consumption split as 40% heating, 20% propulsion, and 40% other things, such as the kneeling system for easier boarding.

The battery can last many roundtrips on the M42, specifically a very slow route. Electric vehicles tend to do much better then fuel-powered ones at low speed in city traffic, because of regenerative braking and higher efficiency. When I discussed the Proterra trial with MVTA, I was told specifically that the buses did really well on days when the temperature was above freezing, since the battery barely drained while the bus was sitting in rush hour Downtown Minneapolis traffic. This pattern is really a more extreme version of one that may be familiar to people who have compared fuel economy ratings for hybrid and conventional cars: hybrids are more fuel-efficient in city driving than on the highway, the opposite of a non-hybrid, because their electric acceleration and deceleration cycles allow them some of that regeneration.

The current system is called OppCharge (“opportunity charging”), and currently requires the bus to spend 6 minutes out of every hour idling for recharge; the Xcelsior presentation shows a bus with a raised pantograph at a charging station, and I wonder whether it can be extended to an appropriate length of wire to enable in-motion charging.

The New Flyer examples I have seen are in large cities – New York and Vancouver. New York’s system for opportunity charging does not require an attendant; Vancouver’s may or may not, but either way the charging is at a bus depot, where the logistics are simpler. In contrast, in Albuquerque the need for midday charging was a deal breaker. When I talked to someone who knew the situation of Albuquerque’s BRT line, ART, I was told that the BYD midday charge system would require an attendant as well as room for a charging depot. Perhaps an alternative system could get rid of the attendant, but the land for a bus that at the end of the day isn’t that busy has nontrivial cost even in Albuquerque.

Even with opportunity charging, batteries remain hefty. Warren said that they weigh nearly 4 tons per standard-length bus; the XE40 weighs 14 metric tons, compared with 11.3 for the older diesel XD40 platform. Specifically on a short, high-ridership density like the M42 and many other New York buses, there is likely to be a case for installing trolleywire and using in-motion charging. In-motion charging doesn’t work well with grids, since it is ideally suited to when several branches interline to a long trunk route that can be electrified, but ultimately it’s a bus network with ridership density comparable to that of some big American light rail networks like Portland’s.

*In case it’s unclear to irregular readers, I exclusively use metric units unless I mention otherwise, so this is -7 Celsius and not -7 Fahrenheit; the latter temperature would presumably drain the battery a lot faster.

Park-and-Rides (Hoisted from Comments)

My post about the boundary zone between the transit-oriented city and its auto-oriented suburbs led to a lot of interesting discussions in comments, including my favorite thing to hear: “what you said describes my city too.” The city in question is Philadelphia, and the commenter, Charles Krueger, asked specifically about park-and-ride commuter rail stations. My post had mentioned Southeast on the Harlem Line as an interface between commuter rail and the Westchester motorway network, and the natural followup question is whether this is true in general.

The answer is that it’s complicated, because like the general concept of the cars/transit boundary zone, park-and-rides have to be rare enough. If they’re too common, the entire rail system is oriented around them and is not really a boundary but just an extension of the road network. This is the situation on every American commuter rail system today – even lines that mostly serve traditional town centers, like the New Haven Line, focus more on having a lot of parking at the station and less on transit-oriented development. Even some suburban rapid transit lines, such as the Washington Metro, BART, and the recent Boston subway extensions, overuse park-and-rides.

However, that American suburban rail systems overuse such stations does not mean that such stations must never be built. There are appropriate locations for them, provided they are used in moderation. Those locations should be near major highways, in suburbs where there is a wide swath of low-density housing located too far from the rail line for biking, and ideally close to a major urban station for maximum efficiency. The point is to use suburban rail to extend the transit city outward rather than the auto-oriented suburban zone inward, so the bulk of the system should not be car-oriented, but at specific points park-and-rides are acceptable, to catch drivers in suburbs that can’t otherwise be served or redeveloped.

Peakiness and park-and-rides

I’ve harped on the importance of off-peak service. The expensive part of rail service is fixed costs, including the infrastructure and rolling stock; even crew labor has higher marginal costs at the peak than off-peak, since a high peak-to-base ratio requires split shifts. This means that it’s best to design rail services that can get ridership at all times of day and in both directions.

The need for design that stimulates off-peak service involves supportive service, development, and infrastructure. Of these, service is the easiest: there should be bidirectional clockface schedule, ideally with as little variation between peak and off-peak as is practical. Development is politically harder, but thankfully in the main example case, the Northeastern United States, commuter rail agencies already have zoning preemption powers and can therefore redevelop parking lots as high-intensity residential and commercial buildings with walkable retail.

Infrastructure is the most subtle aspect of design for all-day service. Park-and-ride infrastructure tends to be peaky. Whereas the (peakier, more suburban) SNCF-run RER and Transilien lines have about 46% of their suburban boardings at rush hour, the LIRR has 67%, Metro-North 69%, and the MBTA 79%. My linked post explains this difference as coming from a combination of better off-peak service on the RER and more walkable development, but we can compare these two situations with the Washington Metro, where development is mostly low-density suburban but off-peak frequency is not terrible for regional rail. Per data from October 2014, this proportion is 56%, about midway between Transilien and the LIRR.

This goes beyond parking. For one, railyards should be sited at suburban ends of lines, where land is cheap, rather than in city center, where land is expensive and there is no need to park trains midday if they keep circulating. But this is mostly about what to put next to the train stations: walkable development generating a habit of riding transit all day, and not parking lots.

Where parking is nonetheless useful

In response to Charles’ comment, I named a few cases of park-and-rides that I think work well around New York, focusing on North White Plains and Jersey Avenue. There, the parking-oriented layout is defensible, on the following grounds:

  1. They are located in suburban sections where the reach of the highway network is considerable, as there is a large blob of low density, without much of the structure created by a single commuter line.
  2. They are near freeways, rather than arterials where timed connecting buses are plausible.
  3. They are immediately behind major stations in town centers with bidirectional service, namely White Plains and New Brunswick, respectively.

The importance of proximity is partly about TOD potential and partly about train operating efficiency. If the park-and-rides are well beyond the outer end of bidirectional demand, then the trains serving them will be inefficient, as they will get relatively few off-peak riders. A situation like that of Ronkonkoma, which is located just beyond low-ridership, low-intensity suburbs and tens of kilometers beyond Hicksville, encourages inefficient development. Thus, they should ideally be just beyond the outer end, or anywhere between the city and the outer end.

However, if they are far from the outer end, then they become attractive TOD locations. For example, every station between New York and White Plains is a potential TOD site. It’s only near White Plains that the desirability of TOD diminishes, as White Plains itself makes for a better site.

On rapid transit in American suburbia, one example of this principle is the Quincy Adams garage on the Red Line just outside Boston. While the station itself can and should be made pedestrian-friendlier, for one by reopening a gate from the station to a nearby residential neighborhood, there’s no denying the main access to the station will remain by car. Any TOD efforts in the area are better spent on Quincy Center and Braintree, which also have commuter rail service.

Where parking should urgently be replaced by TOD

American suburban rail lines overuse park-and-rides, but there are specific sites where this type of development is especially bad. Often these are very large park-and-ride structures built in the postwar era for the explicit purpose of encouraging suburban drivers to use mainline rail for commuter and intercity trips. With our modern knowledge of the importance of all-day demand, we can see that this thinking is wrong for regional trips – it encourages people to take rail where it is the most expensive to provide and discourages ridership where it is free revenue.

The most important mistake is Metropark. The station looks well-developed from the train, but this is parking structures, not TOD. Worse, the area is located in the biggest edge city in the Northeast, possibly in the United States, possibly in the world. Middlesex County has 393,000 jobs and 367,000 employed residents, and moreover these jobs are often high-end, so that what the Bureau of Economic Analysis calls adjustment for residence, that is total money earned by county residents minus total money earned in the county, is negative (Manhattan has by far the largest negative adjustment in the US, while the outer boroughs have the largest positive one). The immediate area around Metropark and Woodbridge has 46,000 jobs, including some frustratingly close to the station and yet not oriented toward it; it’s a huge missed opportunity for commercial TOD.

In general, edge cities and edgeless cities should be prime locations for sprawl repair and TOD whenever a suburban rail line passes nearby. Tysons, Virginia is currently undertaking this process, using the Silver Line extension of the Metro. However, preexisting lines do not do so: Newton is not making an effort at TOD on the existing Green Line infrastructure, it’s only considering doing so in a part of town to be served by a potential branch toward Needham; and the less said about commuter rail, the better. Mineola and Garden City on Long Island, Tarrytown in Westchester, and every MBTA station intersecting Route 128 are prime locations for redevelopment.

Commuter rail for whomst?

I believe it’s Ant6n who first came up with the distinction between commuter rail extending the transit city into the suburbs and commuter rail extending the suburbs into the city. If the trains are frequent and the stations well-developed, then people from the city can use them for trips into suburbia without a car, and their world becomes larger. If they are not, then they merely exist to ferry suburban drivers into city center at rush hour, the one use case that cars are absolutely infeasible for, and they hem car-less city residents while extending the world of motorists.

Park-and-rides do have a role to play, in moderation. Small parking lots at many stations are acceptable, provided the station itself faces retail, housing, and offices. Larger parking structures are acceptable in a handful of specific circumstances where there is genuinely no alternative to driving, even if the rest of the rail service interfaces with walkable town centers. What is not acceptable is having little development except parking at the majority of suburban train stations.