24 January 2019

Palo Alto: Designing in a Vacuum

Palo Alto is continuing the fraught public process of winnowing down the feasible and acceptable options for grade separating the four remaining rail crossings. Having hired an engineering consultant, the city is busily making plans for railroad land that doesn't belong to it and over which it has no jurisdiction.

The fancy renderings from a recent meeting, envisioning a tunnel, a trench, a hybrid embankment, or a viaduct, invariably show expansive new landscaping when construction is finished. This is reflective of the ample railroad land available through most of Palo Alto. Caltrain's land is typically about 100 feet wide, excepting a few short sections of the corridor near Southgate and Peers Park that are just 60 feet wide. South of those narrow spots, there is plenty of room to accommodate four tracks (about 75 feet required) if needed in the future, no matter what the pot-stirring local press may say.

Palo Alto's planning process thus far seems to have missed these important facts:

  1. Caltrain's nascent business plan envisions ambitious expansions of service in the next two decades, growing far beyond the initial goal of electrification. Service planning thus far strongly suggests (pp. 64-67) that new overtake tracks will be needed approximately from south of Peers Park to the Mountain View border. The additional tracks in south Palo Alto, featured in all remaining options (p. 34), would allow express trains to pass local trains.
     
  2. In other cities to the north and south where Caltrain has become directly involved in the planning process, it has levied a requirement that city-generated grade separation designs preserve the future option of adding overtake tracks, expanding the corridor from two to three or four tracks. Two examples:
     
    • Whipple Ave in Redwood City, where the city recently hired Caltrain to lead the planning effort. On page 138 of the October 1st, 2018 city council meeting agenda, a letter from Caltrain states: "... the Project Study Report must include at least one design option that accommodates the potential overtake. In this context, "accommodate" is understood to have the following minimum threshold of meaning: the grade separation design maximizes the preservation and configuration of existing right of way such that overtake tracks could be built later with no or minimal right of way acquisition; the grade separation design does not force future overtake tracks to be built in a way that substantially increases their cost and complexity."
       
    • Rengstorff Ave in Mountain View, where the city recently hired Caltrain to lead the preliminary engineering and environmental clearance effort. On page 105 of the December 2018 JPB board meeting agenda, we read that "the design will consider and accommodate Caltrain / high-speed rail blended system improvements and be designed to allow for up to four tracks."
In practical terms, this adds a new constraint to Palo Alto's grade separation deliberations. We can reasonably infer that Caltrain will require at least the Charleston / Meadow grade separation to be engineered for four tracks, or at least not to preclude four tracks. The sooner this constraint is incorporated into the city's planning process, the less anguish and recrimination there will be in arriving at an acceptable design.

When planning construction on someone else's land, it helps to know what the owner wants.

16 December 2018

Billions of Seconds Wasted

The latest tweaks to the design of the San Francisco Downtown Extension (DTX) rail alignment can be seen in a March 2018 track plan and profile drawing. Because it largely follows the street grid, it's no secret that the alignment is full of sharp curves, which can only be traversed at slow speed. However, compared to a 2012 drawing, speed limits have dropped in several places from 40 mph to just 30 mph, because train speed evidently isn't a design priority when civil engineers get a blank check.


Back in 2012, the speed profile sort of made sense: starting from the basement of the Transbay Transit Center (left end of the diagram) the train would screech at about 20 mph through the sharp curve towards 2nd Street, speeding up to 35 mph along 2nd and through the curve towards Townsend. On that mostly straight bit along Townsend, speeds could pick up to 40 mph before dropping back briefly to 35 mph through the curve to 7th Street, then exiting along 7th Street at 40 mph (right end of diagram). If only one criticism were allowed, it wasn't clear why that final curve should be limited to 35 mph; there was plenty of space at Townsend and 7th to flatten it out to 40 mph, resulting in a simple and efficient stepped speed profile for the approach to Transbay.
Fast forward to 2018, and things are much worse. There is a new kink in the alignment where it connects to the existing tracks. The new underground 4th and Townsend station, at the city's request, has been shoved into the Townsend Street right of way in the hope of freeing up the existing rail terminal parcels for high rise redevelopment (where the 2012 alignment might have clashed with new building foundations). The rigid requirement for a straight island platform has resulted in a series of 30 mph kinks in the track. Elsewhere, the speed limit along Townsend has dropped by 5 mph.
The designers might argue this is only a few seconds lost, so no big deal, right?
How many seconds are wasted?
A train traversing the DTX will have to observe the speed limits not just for the length of each speed restriction, but for the added length of the train itself, as the limit applies from the moment the head end of the train enters a speed restriction until the tail end leaves the speed restriction. High-speed trains will be up to 400 m long, so this can really add up. We can simulate the time needed for a train to travel from a standing start at the end of a Transbay platform to a 40 mph entry into the existing Tunnel 1, a distance of about 2.2 miles. The results depend on the train type, and whether a stop is made at 4th and Townsend:
  • 2012 alignment, single-length HSR: 4:04
  • 2012 alignment, double-length HSR 4:17
  • 2012 alignment, 8-car Caltrain EMU, no stop at Townsend 4:06
  • 2012 alignment, 8-car Caltrain EMU, 30-second stop at Townsend: 5:04
     
  • 2018 alignment, single-length HSR: 4:25 (+21 sec)
  • 2018 alignment, double-length HSR 4:42 (+25 sec)
  • 2018 alignment, 8-car Caltrain EMU, no stop at Townsend 4:27 (+21 sec)
  • 2018 alignment, 8-car Caltrain EMU, 30-second stop at Townsend 5:19 (+15 sec)
To summarize and simplify, we can assume that every Caltrain will stop at Townsend, so the performance loss is 15 seconds per Caltrain movement, and roughly 20 seconds per HSR movement. That doesn't sound like much, but consider that trains are carrying hundreds of passengers, each of whom are individually delayed. The collective waste of time can be measured by multiplying the train delay by the expected ridership.
Today Caltrain has about 15,000 weekday boardings in SF, a number that Caltrain says could eventually quadruple. Let's say it only triples, and that 35,000 of those weekday boardings occur at Transbay and 10,000 at 4th and Townsend (which we won't count) making for 70,000 trips through the DTX approach. That's 70,000 trips x 15 seconds/trip = a million seconds wasted every weekday, or about 3 person-years of productive labor time per month of DTX operation. Over a year, about a quarter billion seconds would be wasted!
HSR eventually expects 18 million annual trips originating in the Bay Area, of which maybe half might involve Transbay. Combine that with a similar number of HSR trips terminating at SF, and you get 18 million annual HSR trips through the DTX approach. That would be a waste of another third of a billion seconds.
Every year then, about half a billion seconds would be wasted due to careless DTX alignment design.
How do we fix it?
Fixing it involves realizing that
  1. every second matters, a lot
  2. the marginal cost of the next second saved is more expensive than the last
  3. saving seconds is most efficiently and cheaply done in the slow parts of a system
Making up 20 seconds through minor fixes to the DTX track alignment design, before any concrete is poured, is far cheaper and easier and more productive than trying to make up 20 seconds somewhere faster, for example in the Central Valley by running trains at 220 mph instead of 215 mph.
What ought to still be possible is an alignment that starts at 20 mph through the screecher to 2nd Street, rises to 35 mph along 2nd Street, then rises to 40 mph along Townsend continuing without slowing around the curve to 7th Street. With this improved speed profile, train run times from Transbay to Tunnel 1 (relative to the 2018 alignment plans) would be:
  • Single-length HSR: 4:02 (23 seconds faster)
  • Double-length HSR 4:14 (28 seconds faster)
  • 2018 alignment, 8-car Caltrain EMU, no stop at Townsend 4:04 (23 seconds faster)
  • 2018 alignment, 8-car Caltrain EMU, 30-second stop at Townsend 5:02 (17 seconds faster)
The combined annual time savings would exceed half a billion seconds per year. As we watch the cost of the DTX project reach ever more dizzying heights, we should at the very least expect to get more transportation value out of the project. Careless and inexcusable engineering of a rail alignment that wastes so much of everyone's time only adds insult to the injury.

08 December 2018

Grade Crossing Trouble Ahead

Grade crossing in Denver (photo: RTD)
Denver's RTD has been operating a new 25 kV electrified commuter railroad since 2016. There's a big problem with it: the grade crossings gates are down for too long, which the FRA and Colorado PUC consider hazardous because impatient motorists frustrated by a longer-than-expected wait may drive around the gates just as the train finally shows up. The problem has festered, with  millions spent on human flaggers to supervise traffic at each grade crossing, contractual acrimony leading to lawsuits, and in recent days a threat by the FRA to shut down the entire railroad until the issue is resolved.

What does any of this have to do with Caltrain? The peninsula corridor electrification project uses the same electrification technology installed by the same contractor (Balfour Beatty), uses the same positive train control technology installed by the same contractor (Wabtec), must contend with more than three times as many grade crossings, and therefore, faces the same looming grade crossing problem. For months, the issue has topped the list of risks that threaten the project, and the search for a viable solution is causing the electrification contractor to fall significantly behind schedule.

How grade crossings are supposed to work

The simplest way to activate a grade crossing is for the train to shunt a track circuit at some set distance before the crossing. This is known as a conventional track circuit warning system, and doesn't work well if different trains arrive at different speeds. The point where the crossing activates must be set far enough ahead to give the required warning time before the fastest train arrives at the crossing; this makes the gates stay down too long for slower trains.

The usual solution to this problem is a Constant Warning Time (CWT) system, which uses electrical signals sent through the track to sense the distance and speed of the approaching train. The grade crossing controller can then predict when to activate the crossing such that the warning time is approximately constant regardless of train speed. This is the type of warning system installed today on the many grade crossings of the peninsula rail corridor.

The FRA provides a nice overview discussion of how various types of grade crossings work. The applicable federal regulations are under 49 CFR Part 234.

What happened in Denver

Because the Denver system is electrified, there are large 60 Hz AC traction return currents (at safe low voltage!) commonly present in the rails when a train is nearby. These currents interfere with and prevent the use of a traditional Constant Warning Time system.

The contractor came up with a "smart" solution: the crossings have a traditional track circuit warning system overlaid with a wireless crossing activation system (WCAS) that interfaces with the positive train control system. Software sends wireless messages back and forth between the train computer and the crossing controller. The train and crossing enter into a contract: the train predicts when it will arrive at the crossing and promises not to get there any sooner, and the crossing commits to activate at some fixed time interval before the appointed arrival, staying closed until the train passes. Depending on the circumstance, the train may arrive at the crossing later than anticipated when the contract was entered into, resulting in extended gate down time. When WCAS is inoperative, the old-school track circuit takes over, also resulting in extended gate down time when a train is operating at less than maximum speed.

In early 2016, before the Denver train opened for revenue service, FRA and PUC inspectors found that the crossings activation times were inconsistent, with frequent occurrence of long gate down times and erosion of what is known as "credibility" of the warning system. Things went gradually downhill from there:
  • So as not to delay the much anticipated start of revenue service, the regulatory agencies granted a temporary waiver to allow RTD to begin operating without WCAS, on the condition that human flaggers supervise traffic at each affected crossing, at the expense of the contractor.
  • The contractor tried to tweak the WCAS software to make warning times more consistent. A fudge factor known as the "Approach Condition Adjustment Factor" (ACAF, so known because every fudge factor needs an acronym to sound legitimate) was applied based on the observed statistical distribution of warning times at each crossing.
  • In September 2017, the FRA gave RTD relief in its interpretation of the consistency required for gate downtime, relaxing its unofficial consistency criterion from +/-5 seconds or +/-10% of programmed warning time to +15/-5 seconds for RTD's system.
  • Performance of WCAS failed to satisfy the increasingly picky regulatory agencies. RTD began to penalize the contractor for failing to deliver a working grade crossing solution. FRA inspectors kept writing up excessive downtime violations.
  • The FRA forbade the start of revenue service on a newer rail line that has since been completed. The original plan to create quiet zones, where train horns are not used at grade crossings, was delayed indefinitely to the continuing aggravation of neighboring residents.
  • In September 2018, the contractor decided that the regulatory agencies had invented and enforced new consistency requirements that were not in the official regulations, and sued RTD claiming "force majeure" of a regulatory change. The complaint makes a fascinating read.
  • In October 2018, the FRA provided the latest inspection report (of many) showing continuing non-compliance with the -5/+15 second consistency tolerance.
  • On November 15th, 2018, the FRA fired off a letter indicating that it was fed up with the continuing grade crossing non-compliance, among other things, and threatened to shut down the entire commuter rail system by revoking the 2016 waiver.
  • RTD is lawyering up against the FRA, and submitted a strongly worded legal memorandum with numerous exhibits effectively claiming that the grade crossing problem exists solely in the imagination of the regulators. RTD provided evidence that other railroads (including Caltrain!) commonly experienced long gate down times in violation of the criteria imposed on RTD.
Whatever happens next is sure to be dramatic. The entire saga can be reviewed under docket FRA-2016-0028, which organizes all the documents exchanged between RTD and the FRA relating to the temporary operating waiver.

Some Observations
Measured distribution of 38255 grade
crossing activation times in Denver.
  1. Denver solved the wrong problem. They tried to invent a better mousetrap, something more sophisticated than a constant warning time grade crossing predictor. All they needed to do was to provide the same simple function with a substitute detection method that didn't rely on traditional audio-frequency AC circuits, which are incompatible with electrification. Instead, they decided to invent a better mousetrap involving lots of software, GPS, and wireless messaging, which naturally attracted regulatory scrutiny.
     
  2. Complexity is bad. Multiplying the number of interfaces and creating dependencies between elements of the system leads to expensive aerospace avionics-like hardware and software that is cumbersome to deploy, test and maintain. System complexity leads to a proliferation of strange and unanticipated corner cases and failure modes.
     
  3. Software can anticipate when to activate a crossing and prevent a train from showing up too soon, but there is no software in the world that can make a train show up on time.
     
  4.  Grade crossing activation times naturally follow a statistical distribution that arises from random environmental factors beyond the control of the warning system. The low end of the distribution must never be shorter than the mandated 20 seconds, but the long end of the distribution will inevitably have some outliers. The diagram above shows the measured distribution of 38255 crossing activation times on RTD. Notice the long tail.
     
  5. Even traditional "constant" warning time systems have this statistical tail. If the FRA inspectors applied the same regulatory zeal to Caltrain as they did to RTD, Caltrain would certainly be found in non-compliance. This isn't idle speculation: RTD gathered the data to prove it.
     
  6. The criteria for non-compliance, namely a "significant difference" from the prescribed warning time, are subjective. Guidance from the FRA acknowledges as much: "Thus, prudent judgment must be exercised when reviewing the results of warning time testing to determine whether the actual warning time provided during testing was compliant with the standard."
     
  7. The regulators painted themselves into a corner. They imposed a strict -5/+15 second criterion, which is easy to verify for an inspector with a stop watch and a clip board, but makes the long tail of the activation time distribution an automatic violation that is almost impossible to avoid. In recognition of the environmental factors beyond the control of the warning system, the regulators should have used controlled test conditions or applied a different criterion, such as X% of activations within Y% of programmed warning time. This is harder to verify for an inspector with a clipboard, but the grade crossing controller ought to be able to maintain these statistical records across a very large number of crossing activations.
     
  8. While electrification is relatively rare in the US, there are numerous railroads abroad that have solved the constant warning time problem in electrified territory. This probably isn't rocket science. The mousetrap already exists.
Lessons for Caltrain
With the grade crossing warning system already at the top of the Caltrain electrification project's risk list and the contractor falling behind, this problem is already getting a lot of attention. The people involved hopefully already realize:

Keep it simple - the job is to come up with a grade crossing predictor that works in the presence of traction return currents. It will be tempting to come up with a more sophisticated custom solution that uses lots of software, but we learned from the CBOSS project, and Denver's travails, that complexity usually leads straight to disaster. The dumber the better.
Document existing conditions - a large database of activation time statistics should be assembled for each crossing as it exists today, to head off a conflict over the subjective nature of the FRA warning time consistency criteria. In the event of a Denver-like disagreement with FRA or CPUC, Caltrain would be in a position to quantify precisely how much more (and hopefully not less) consistent the new warning solution will be, regardless of the selected criterion. Caltrain enjoys the advantage that it isn't building new crossings like Denver, so there is an existing system performance baseline that is already accepted by regulators. That baseline will only be useful if it is thoroughly documented.
Plant the goal posts firmly - Work with FRA towards mutually agreed verification criteria that don't repeat the mistakes made in Denver of specifying a rigid range and then testing in the uncontrolled conditions of revenue service. The activation time distribution will always have a statistical tail. If the consistency criterion can't be met by today's existing grade crossing system, then it's probably a bad criterion.
Make sure we aren't paying for Denver - the contractor needs to be held accountable for the extent to which Caltrain electrification funds (and schedule delays!) are accruing to the Denver project's benefit, if the same grade crossing solution is ultimately pursued in both projects.

21 October 2018

Thinking Big in Redwood City

The architecture of Amsterdam Bijlmer
(photo by tataAnne) could represent
the future Redwood City station.
In a seamless transportation network that runs on a regular clockface schedule with timed, well-coordinated transfers, connecting nodes play a key role. Redwood City has natural potential as a connecting node, being located approximately at the midpoint of the peninsula rail corridor, serving as a logical transfer point between local and express trains, serving as the entry point to the peninsula from the future Dumbarton rail corridor, and being in of itself a significant destination with extensive connecting bus service and a willingness to grow.

With Redwood City currently renewing its interest in grade separations, it's important to think big and to re-imagine the station as a key node in the Bay Area's transportation network.

Start with a good timetable

Using our handy service pattern generator, let's see what we could do if we organized a blended system that made Redwood City a key transfer node. When you make a business plan, the first thing to be crystal clear about is: what is your product? In Caltrain's case, the timetable is the product, and all these stations and tracks should only be built as long as they contribute directly to delivering a quantifiably better timetable for the ordinary rider. Building a major new station in Redwood City isn't about trite superlatives like "Grand Central of the West," but simply about efficient and seamless coordination of timely and reliable ways to get from point A to point B.

Let's set some ground rules for our timetable:
  • Caltrain expresses will operate every 10 minutes on a regular clockface schedule. A base 'takt' of 10 minutes reduces gracefully to 20 minutes or 30 minutes in the off-peak.
  • In Silicon Valley, there will be no skip-stop service because the population and jobs are evenly sprawled. Every station in Silicon Valley needs to be served frequently, doing away with the ridership distortions induced by the Baby Bullet effect.
  • In San Mateo county, where stop spacing is closer, slower local trains will operate every 20 minutes. These local trains will meet the express at Redwood City, before turning back north.
  • Dumbarton service will operate every 20 minutes, meeting the express at Redwood City with little or no wait to transfer to trains on the peninsula corridor, before turning back towards the East Bay.
  • Because the overall pattern repeats every 20 minutes, HSR will operate 3 trains per hour rather than the planned 4. Otherwise, there is a harmonic mismatch between the HSR frequency and the Caltrain frequency. 4 HSR trains per hour in a clockface timetable forces the base 'takt' to increase to 15 minutes, which is not desired.
  • If we're going to make Redwood City a major node, it certainly rates HSR service, so we will create a new mid-peninsula stop for HSR.
This is the resulting timetable (see also additional data on service pattern), shown here for one hour in the southbound direction only (the northbound side is symmetrical). Colors denote the 10-minute Caltrain express, the San Mateo local, Dumbarton service, and HSR.

Notice the express arriving at Redwood City at 7:43 meets the Dumbarton train departing at 7:44, and the local arriving at Redwood City at 7:52 meets the next express at 7:53. Every ten minutes there is a cross-platform transfer, alternating between express-to-Dumbarton and local-to-express. Counting both directions, a cross-platform transfer occurs at Redwood City every five minutes!

Implicit in this timetable are a number of other capital improvements besides a new Redwood City station, such as overtake tracks in various locations along the corridor (highlighted in yellow in this view of the timetable... and while we're here, look how much less yellow is needed if HSR uses the Dumbarton corridor via Altamont Pass). It's important to remember that there is no formulation of the blended system that avoids the need for overtake tracks, unless one is willing to push slower trains into station sidings to sit for at least five minutes while a faster train catches up and pulls ahead. If you are a Caltrain rider, you should be wary of the cheapskates at the HSR authority who want to do this to your commute.

Deriving the functional requirements for the Redwood City node

To enable this timetable, we need the Redwood City station to have the following attributes:
  1. Four platform tracks serving two 400-meter long island platforms to facilitate both northbound and southbound cross-platform transfers of very long, high-capacity trains.
     
  2. Platforms centered on the best cross-town corridor, namely Broadway, for convenient access to and from local destinations on foot, by bike or scooter, by bus, or using the planned Broadway Streetcar.
     
  3. A turnback track that enables certain Dumbarton corridor trains to originate and terminate in Redwood City, without fouling other train traffic, long enough for an EMU-8 train.
     
  4. A turnback track that enables the San Mateo local to turn back in Redwood City, without fouling other train traffic, long enough for an EMU-8 train.
     
  5. Elevated grade separation of all downtown Redwood City crossings, enabling free flow of pedestrians, bikes and vehicles under the rail corridor and including the re-connection of streets currently cut off by the existing configuration (e.g. Hopkins and James).
     
  6. Bus facilities placed directly under the train platforms for seamless connections without the need for an umbrella. Same for an eventual Broadway Streetcar.
     
  7. No mezzanine level. Mezzanines needlessly drive up the size and cost of stations, and impede and complicate vertical circulation. Street level can fulfill all the functions of a mezzanine, including ticket sales, wayfinding, waiting, retail, and dining.
     
  8. The shortest and fastest possible vertical circulation (stairs, escalators, ramps, and elevators) using a U-shape viaduct cross section to avoid deep and vertical-space-wasting bridge structure. This helps with transferring quickly between the two island platforms, as would be needed for example to continue from the Dumbarton corridor south to Silicon Valley.
The footprint of such a station is not small. However, Redwood City has plentiful available railroad and transit district land, and the street level interface of such a station can be integrated into the city's street grid, opening up cross-corridor access and avoiding a wall effect. The aging Sequoia Station shopping center, with its wasteful surface parking, can be demolished and redeveloped to make room for an expanded station. Station parking can be moved underneath the approach structures, protected from the elements.

One possible station layout
An optimal station layout has four tracks, with the outer tracks for HSR and express commuter trains. The middle tracks are for commuter trains, and allow both northbound (Dumbarton) trains and southbound (San Mateo local) trains the opportunity to turn at Redwood City without impeding the flow of express traffic. The width of the structure is about 130 feet, as shown in the cross section below:
The northbound express track (Track 3) is tangent. The northbound island platform is 400 x 10 m. The center commuter tracks (Tracks 1 and 2) have curves that are not laid out in detail; this detail does not matter since any train that uses these tracks would slow and stop at Redwood City, using standard trackwork and turnouts. The southbound express track (Track 4) is the tricky one: it wows around the station, passing the southbound island platform on a 7500 m radius curve with approximately 1.5 inches of superelevation (not enough to matter for platform lateral tolerances). This track consists of a double reverse curve with six spiral transitions (tangent, spiral, curve, spiral, tangent, spiral, platform curve, spiral, tangent, spiral, curve, spiral, tangent). The curve is necessary to fit a pair of 400-meter island platforms (long enough to berth a double-length high-speed train) without bulldozing too much real estate.

Here is how this all fits (admittedly just barely) in downtown Redwood City:


The sacrificial victim is the Sequoia Station shopping center and associated surface parking crater, which can be redeveloped as part of the station complex with direct access from El Camino Real. Access for high-rise fire apparatus around the viaduct structure might also be a concern for the new condo buildings to the south, although this can be mitigated.

The station includes two pocket sidings to turn commuter trains. The siding south of the station can turn Caltrain locals at Redwood City, while the siding north of the station can turn Dumbarton service. Each siding is sized to store an eight-car EMU. Track center spacing is 15 feet throughout, and platform setback is 6 feet from track center. All viaducts are made from low-profile U-shaped sections that minimize the required height of the tracks and also double as sound walls, reducing the noise of up to 30 trains that would serve the station every peak hour.

Redwood City's slogan, "climate best by government test" would also become "transfer best" with timed, well-coordinated transfers to a variety of destinations. The impending start of designs for grade separations in Redwood City needs to factor in this future, and the city ought to think big.

27 September 2018

Growing Caltrain into an 8-Lane Freeway

Caltrain can and should become an eight-lane freeway. Not like an ugly concrete scar tearing loudly through the landscape, but in terms of throughput capacity in people per hour. Today, Caltrain already carries the equivalent of nearly 3 freeway lanes, and more than doubling the system's capacity is hardly a moonshot. For perspective, BART's Transbay Tube carries up to 27000 people per hour, almost double the entire capacity of the Bay Bridge with its ten freeway lanes.

More than doubling Caltrain's capacity has been proposed before and is now being studied by the agency itself, after a decade of not thinking much past electrification.

Capacity calculations can be controversial and rely on many details and assumptions, so the suggested path to expand Caltrain ridership from 3 to 8 lanes of freeway-equivalent is provided in the form of a spreadsheet, embedded below. You can dig into all the numbers and assumptions for each capacity increase and see the underlying formulas for yourself, down to the detailed number of seats in each train car, to understand how it all adds up.

This is a living document, and feedback is appreciated!