Speed adjustment considerations

[Gumboots]

Ok, more fun. So for pulling power of 1, free weight of 0, locomotive weight of 34 tons, and one cargo car of 34 tons, the speeds found earlier were:

0% grade: 48 mph
2% grade: 18 mph
4% grade: 5 mph
6% grade: 1 mph

So I just changed one thing. I set free weight to 34 tons. This resulted in speeds of:

0% grade: 95 mph
2% grade: 35 mph
4% grade: 10 mph
6% grade: 2 mph

Part of this isn't surprising. The full top speed being maintained with one car on 0% grade makes sense, since the free weight is the weight of that one car, so the train does the same top speed it would do with no cars at all.

The surprising part is that all those people who were saying free weight only affected speed on flat terrain were dead wrong. Again. Just like with pulling power.

It's clear from the figures that halving the weight of the train via use of the free weight setting exactly doubles the speed up all grades.

So there ya go. Again.

Next time anyone says that pulling power affects speed on grades, and free weight affects how soon speed on flat terrain will drop as you add additional cars, tell them they are partly right but really have no idea.

Oh yeah, and what are the figures for two cars?

0% grade: 48 mph
2% grade: 11 mph
4% grade: 2 mph
6% grade: 1 mph

So that again makes sense for flat terrain, since the speed there is the same as for 0 free weight and 1 car. Clear enough. Haven't yet run the calculations for grades, but it's bound to be simple once I nail it.

[Gumboots]

Hey I just realised something. Those speed figures for a free weight of 34 and 2 cars are figures I've seen before.

0% grade: 48 mph
2% grade: 11 mph
4% grade: 2 mph
6% grade: 1 mph

They first showed up back here.

Ok, so that appears to be how it goes for one car and zero loco weight. Next thing is to start adding loco weight and see what happens. To keep things simple I added loco weight in multiples of car weight, so went 0, 34, 68, 102. That means total train weight goes 100%, 200%, 300%, 400%.

<snip>

Total train weight of 300% gives:
0% grade: 48 mph
2% grade: 11 mph
4% grade: 2 mph
6% grade: 1 mph

The first time those speed figures showed up I was using a locomotive weight of 68 tons, one car at 34 tons, free weight of 0, and pulling power of 1. Now exactly the same speeds are showing up for a locomotive weight of 34 tons, two cars at 34 tons each, free weight of 34, and pulling power of 1.

So the overall train weight is the same, and all other factors apart from free weight are also the same. What this means is that although the change in free weight has doubled speeds up all grades for a one car consist, it has had absolutely no effect on speeds up grades for a two car consist.

This may be how the myth about free weight only applying to performance on flat terrain got started. When I get time tonight I'll test with higher values of free weight and get more data.

[Gumboots]

Nevermind this post. It turned out to be completely wrong anyway. See the next posts for the real story.

[Gumboots]

Hey this is even simpler than I thought. I decided to get some good data together, so hauled out the long-suffering Schools beta for yet more indignities. I left free weight and locomotive weight at zero, just to get the most basic situation to start with. I also increased top speed to 1,000 mph, not because such speeds will ever be useful in the game but simply to get the best range of figures with the minimum of effects from rounding errors. This has worked out really well.

I've only gathered data for pulling powers of 1, 2 and 3, but the results are so solid that they are sure to extrapolate to any other values. For a start, the conclusion about the pulling power affecting all speeds for all grades and for any number of cars, and doing it by using the inverse of the pulling power value as an exponent, is confirmed. I've attached screenshots of various bits of spreadsheet. If you pick any speed cell, from any of the tables for pulling power = 1, then if you check the same cell in the corresponding table for pulling power = 2 you will find that the speed there is 1000(((Speed for PP=1)/1000)^0.5) and if you check the same cell for pulling power =3 the speed there will be 1000(((Speed for PP=1)/1000)^0.333).

The next bit is that doubling the pulling power exactly doubles the weight you can haul at that speed and on that grade. Tripling the pulling power exactly triples the weight you can haul. This is handy because a/ it's so obvious I can't miss it and b/ it'll help limit the possibilities for the function behind all of this, thereby making that function easier to nail down.

The next bit is really obvious too, now that I have enough data in a handy format for spotting it. If you look at any row for any grade, in any of the tables, the speeds across that row as you go from 1 to 8 cars can always be found by taking the speed in the first cell, dividing it by the locomotive's top speed, and then using the result of that division to simply multiply the speed in each cell to get the speed for the next cell.

Taking the first screenshot as an example, the B era (mixed) table at upper left has a speed for 1 car and 0% grade of 709 mph. That's 0.709 of the top speed. If you multiply 709 by 0.709, you get 503. Multiply 503 by 0.709 and you get 356. Do it again and you get 253, then 179, then 127, then 90, then 64. So there is some small rounding error (less than 0.5%) on a couple of the values in the table, but nothing to worry about. This same relationship holds for all rows, for all values of pulling power and cargo car weight, on all grades. All you need to know is the top speed, and the speed on that grade with 1 car, and you can work out the other speeds really easily.

So all that's needed now is to nail down the function that generates those speeds for one car. It'll be something fairly simple, that will automatically spit out the number for speed if you feed a weight into it. The effects of pulling power will be applied on top of this, so won't affect the basic function. Since I now have three sheets for three values of pulling power, and each sheet has 12 speeds tabulated for each combination of grade and number of cars, but with varying weights, there shouldn't be many places this function can hide for long (famous last words).

[Gumboots]

Got the speed factors graphed and yep, it's obviously just a function of weight. Open Office's automatic exponential trend lines are very slightly out. The true function would go to a speed multiplier of exactly 1 at a car weight of 0. Anyway it's all pretty clear. The next trick will be to start throwing locomotive weight into the mix and see what happens on grades. We already know that adding locomotive weight will reduce speeds on grades. It's just a matter of nailing down exactly how it does this and to what extent.

[Gumboots]

Ok, got the basic function sorted. It's just an exponential. For each grade there is an initial value of the speed multiplier set for when weight = 1 ton. Values for higher weights just have the weight in tons applied as an exponent on the initial value.

For example, for a 0% grade the initial value is 0.980 for a weight of 1 ton. The value for 10 tons cars is 0.817, which is simply 0.98^10, while for 20 ton cars the value is 0.668, which is simply 0.98^20, and it goes on from there, all the way up to 80 tons cars having a value of 0.98^80. Once you know the initial value for any grade, you can work out the speed multiplier for any car weight. Once you have the speed multiplier for that car weight, you can work out the speeds for any number of cars on that grade.

Once I twigged to how it worked it was easy to get the initial values for 0%, 2%, 4% and 6% grades, just by taking the speed on that grade as a fraction of top speed and then applying the inverse power of the weight in tons. I also plotted some extra points for 1%, 3%, 5%, 7%, 8%, 9% and 10% grades just for the heck of it. This was just done by basic curve fitting by eye. I haven't yet figured out the actual function behind these initial values, but am not too fussed about it. The information will only ever be used for generating basic speed stats on reasonable grades, so just having empirically derived values is good enough. It'll still give us our precious speed stats to an accuracy of 1 mph.

Pic attached.

[Gumboots]

Ok folks, if anyone is interested here's the first beta version of a speed vs grade vs consist calculator. This one doesn't yet deal with free weight, so is of limited practical use, but correcting the formula for that is next on the hit list.

However, just as a proof of concept, it gives accurate results for any loco+tender weight and free weight of 0, with any consist and any value of pulling power, on grades from 0% up to 8% in 1% intervals. The curve fitting for non-standard grades (anything that's not 0, 2, 4 or 6) has been improved too, so those will be slightly more accurate than the graph shown in the previous post.

The fourth sheet shows the effect of adding a locomotive weight of 80 tons, compared to a weight of 0 tons. For pulling power of 1, speed up a 2% grade is about 1/3 of the speed with locomotive weight of 0. This is a constant fraction of speed, despite the locomotive weight not being a constant fraction of total train weight.

Even though the locomotive weight as a proportion of total train weight decreases from 50% of an 8 car freight train's weight in the A era, to 11.1% of an 8 car freight train's weight in the D era, the speed of both trains up a 2% grade is still a constant 1/3 compared to the same train with locomotive weight of 0. This also applies when there are fewer than 8 cars. It's exactly the same for a 1 car consist. The train's speed up a 2% grade with an 80 ton loco will still be about 1/3 of the speed with a 0 ton loco. The corresponding figure for pulling power of 2 is about half the speed, and for pulling power of 3 is about 2/3. This is pretty obviously bonkers coding, but it's what we've got. Welcome to RT3.

The effect of locomotive weight is just another basic exponential function. It's simply a case of getting the relevant speed from the basic speed function for that grade, then multiplying it by x^(w/p) where x is the initial value of the multiplier for a 1 ton loco on that grade, w is loco+tender weight in tons, and p is pulling power.

So, the only tricksy shenanigans left is free weight, but I'm betting that won't be very complex either.

Edit: Early beta spreadsheet removed.

[Gumboots]

Hey cool. This free weight thingy is a no-brainer. I just graphed up some basic data for it and it's pretty clear what's going on. All it's doing is taking the basic speed curve for 1 to 8 cars on that particular grade, and scaling the whole thing vertically. The attached graph is for pulling power of 1 and locomotive+tender weight of 0. Again, this is just to keep the variables under control while I nail down each factor.

For a 0% grade, the amount of vertical scaling is such that when the free weight is equal to 1 car weight the speed curve will cross the top speed line (1000 in this graph). So if you are running 10 ton cars on flat terrain, a free weight of 10 means you can haul 1 car at top speed, a free weight of 40 means you can haul 4 cars at top speed, and a free weight of 80 means you can haul 8 cars at top speed.

For a 6% grade, a free weight of 40 for a consist of 10 ton cars means you can haul 1 car up that grade at top speed.

I haven't yet reverse engineered the equation behind this for all grades, but in conceptual terms I can see how it works, so getting it sorted won't be too much of a problem. Frankly the trickiest bit is probably going to be figuring out how to make the spreadsheet formulae handle the conditionals I'll need to throw in. It may end up needing nested IF statements, etc to stop it flipping out.

I'm really getting a crash course in advanced spreadsheet usage here, but I suppose it is time I learnt how to use the things so it's all good. I can see how they're a useful invention, although for stuff like this it'd be great if they could work in the old HP Reverse Polish Notation (which totally rocks for complex equations because you don't have to keep track of brackets).

Anyway, pic attached.

And yes, the data I have now totally confirms that the free weight setting does have a significant effect on speed up grades. Don't believe anyone who says otherwise.

[Wolverine@MSU]

And yes, the data I have now totally confirms that the free weight setting does have a significant effect on speed up grades. Don't believe anyone who says otherwise.

You seem to be the expert on this Gumboots so I'll believe what YOU say. I'm impressed with your tenacity on this and the other issues, with an approach that exemplifies the scientific process:

Develop hypothesis --> test hypothesis --> interpret results --> refine hypothesis --> test revised hypothesis --> reevaluate results --> rinse --> repeat.

Spreadsheets are a great tool. I use them all the time for all sorts of things, and the graphing/curve-fitting features are excellent. I'm pretty adept at eeking out as much as I can, so if you have any questions about how to go about doing what you want to do, feel free to ask. One limitation with most of them (I use MS Excel) is that the trendline (curve-fitting) functions are somewhat limited in scope. If you need to do curve-fitting with a complex function, check out the "Solver" feature in Excel. It will take a complex, multi-coefficient function, and do a non-linear least-squares regression to suss out the coefficients.

[Gumboots]

Thanks. I may take you up on that offer. Will see how I go.

The main syntax sticking point is going to be nesting the conditionals, all of which are going to have formulae inside them. The "free weight" thing will need a conditional that returns the top speed if the y value is greater than that. I was also wanting to put in other conditionals.

At the moment, if the speed is calculated to be less than 0.5 the spreadsheet will show a speed of 0. RT3 seems to have a coded-in minimum speed of 1 mph, which I think is to make sure trains are never actually stopped by any grade or consist. It doesn't really matter if the spreadsheet shows 0 mph for some cells, but it's probably not difficult to get it working with a lower limit of 1, and I need to sort the upper limit anyway, so might as well do it.

The third conditional I want is to limit the numbers of cars in the consist, so you can get meaningful results for specific custom consists of less than 8 cars without the remaining cells generating garbage. I've had that one working already, but ditched it while I get the spaghetti for all the exponential functions disentangled.

I'll check out the solver if I get stuck. OO's solver only handles linear equations by default but there are extensions which claim to handle non-linear stuff. I've already found the limits of the default trend line fitting too. Often I can clearly see the type of function required and what tweaks the default line would need (even a simple 180 degree rotation at times) but there doesn't appear to be any convenient way of getting them done. OTOH, sometimes it works perfectly, with the graph in my previous post being a good example.

Edit: Deleted incorrect hypothesis. That means wrong stuff. Right stuff can be found here.

[Gumboots]

Anyway, I have some general conclusions about locomotive stats balancing, based on what has been found so far. You might also be able to get some ideas for strategy out of this.

For a start, the game's coding makes no attempt to generate real train physics. This is definitely a train game rather than a train simulator. The mathematics behind it just ensures that our trains go fast down hills and slow up hills, and assumes that's enough to keep us entertained. Seems to work.

Pulling power is well-named. It does what it says on the tin. "Free weight" is badly named, because it's not really that at all. I can see why it might have been given that name, because its effects can make it appear that you are being given "free cars" in your consist on 0% grade, but it's still not a very good name. OTOH, "exponential curve arbitrary modifying factor #3" is a bit of a mouthful.

Although its effects are probably most obvious on 0% grades, at least for later locomotives with high pulling power, "free weight" definitely has an effect on all grades and for all locomotives. The lower the pulling power, the greater the effect "free weight" will have up grades. Similarly, pulling power also has an effect on all grades, even on 0%, and this is interwoven with the effect of "free weight".

Long consists are heavily penalised in terms of speed, especially when the free weight setting is low. If FW = 0, adding an eighth car to a seven car consist will cause the same percentage speed reduction as adding a second car to a one car consist, and this will apply to all grades. This is obviously nowhere near realistic, but it's how RT3 coding works. For higher values of FW this will only strictly apply to a 0% grade, and will only apply if FW is not so high that any consist at all can be hauled at full top speed. I haven't yet worked out exactly how the speed varies for non-zero grades in these cases, but it probably still penalises long consists more than it should.

Locomotive weight and tender weight are very heavily penalised in terms of speed up grades. If you want your choofer to go up hills like a mountain goat, making it weigh next to nothing is a really good way of doing that. A very light locomotive will, for given settings of pulling power and free weight, haul an 8 car consist faster up grades than a heavy locomotive. This is obviously nowhere near realistic either. In reality, hauling a heavy consist up hills requires a lot of weight in your loco. RT3 is the direct opposite of that.

The way it works is that, for the basic case where pulling power = 1 and "free weight" = 0, every 1 ton added to locomotive or tender weight will always cause the same percentage reduction in speed up a given grade. For example, a locomotive weight of 1 ton will reduce speed up a 4% grade by 3.25%, and will do this for any number of cars in the consist, and regardless of the weight of each car. Increasing the weight of a 320 ton locomotive to 321 tons will also reduce speed up a 4% grade by 3.25%, and will do this for any number of cars in the consist, and regardless of the weight of each car.

At this point your reaction is probably somewhere between and "Bovine droppings!" but really, that's how RT3 works.

Pulling power > 1 modifies this. For example, if pulling power = 3 and "free weight" = 0, then every 1 ton added to loco weight will cost you 1.1% in speed up a 4% grade, but this will still apply regardless of the number of cars in the consist, and regardless of the weight of each car, and regardless of how heavy the locomotive was before you added that extra 1 ton.

Increasing the "free weight" to anything more than 0 can change this relationship, but only if the "free weight" is high enough to allow hauling that number of cars at full top speed on that grade. If the "free weight" is not high enough to allow hauling that number of cars at full top speed on a 4% grade, then if pulling power = 3 every 1 ton added to locomotive weight will still cost you 1.1% in speed.

[Gumboots]

Got it!

Ok, "free weight" works like this. Remember how there was a basic speed reduction factor for 0% grade, and that this just kept getting used to multiply speed for the last number of cars to get speed for the next number of cars? That's easy to turn into an equation.

Call the speed reduction factor SRF. FW is "free weight". CW is car weight. N is the number of cars.

For the basic case where "free weight" = 0 you have: Speed = (Top speed) x (SRF)^N

For any non-zero value of "free weight", which also works for zero anyway, the equation is: Speed = (Top speed) x (SRF)^(N-(FW/CW)).

Which is not as scary as it looks. All it does is give you your extra "free car" to haul at top speed every time "free weight" goes up by an amount equal to car weight, while still allowing speeds to be calculated for intermediate values of "free weight".

So that's a done deal. The party bit is that I just figured out how "free weight" boosts speed for grades, which was the last bit that was holding this nifty new calculator up. In the great tradition of RT3 coding, this one is really simple too. It's the same scaling for all grades, and only depends on FW/CW.

So call this grade boost factor GBF.

For FW = 0, GBF =1. In other words, the speed is just what it would have been anyway.

For FW > 0 it's GBF = 1.2239^(FW/10) so: Speed= 1.2239^(FW/10)(Top speed) x (SRF)^N

That's all it is. Works for all non-zero grades and all car weights. Why 1.2239? Don't know, and don't care. It is, so I'll use it.

Now this is all for pulling power = 1, so I'll have to do some more checking with other values of pulling power just to make sure which one gets applied first. Once I have that figured out, writing the complete equation will be easy.

[Gumboots]

Edit: Warning. Everything in this post turned out to be wrong. I must have made a basic error when testing, due to being tired.

Just thought of something else to test. I was idly wondering if loco and tender weight were treated the same. IOW, just added together to get a total weight, and that got used as "loco weight" in the formula. I was hoping the answer was yes, because that would be the simplest case.

OTOH, I did wonder why they had bothered including a tender weight if it made no real difference. If tender weight and loco weight were just added together to get a total figure there would be no real need for tender weight. You could just increase the loco weight by the same amount and save the game engine having to read some extra code, and save yourself the trouble of having to edit tender files during development. Since I knew the RT3 devs were as lazy as I am, I thought they might have deprecated the tender weight in the same way that they deprecated the weights in the .cgo files.

No such luck.

Turns out that loco weight and tender weight are treated differently. I ran a quick test with the Big Boy, since I knew that by default it has a loco weight of 560 tons and a tender weight of 60 tons. A quick and obvious test would be to reverse those figures and see what happened, so that's what I did. The result is that a 60 ton Big Boy with a 560 ton tender will haul 8 monster freight cars up a 6% grade about twice as fast as a 560 ton Big Boy with a 60 ton tender.

So now I have to figure that bit out too.

Pic attached, showing the variation in speed for the two cases, with two different cargo cars weights for each case.

[RulerofRails]

Well done on finding the formulas!

I haven't had enough time to run lots of numbers in the spreadsheet, but it's looking great.

I doubt that I understood everything you were saying when I was reading through, however I have a question mark over your latest post regarding tender weight being treated differently to engine weight. If that were true, I can't see why the workaround principle in some of my latest posts of changing combined engine + tender weight, free weight, pulling power, and car weight using the same modifier would end up with identical stats but at a new car weight. I tested that multiple times on different engines. I treated engine and tender weight as a whole and would only make the change to one value.

Just now I tried to replicate your results with the Big Boy and failed to do so. I switched around engine and tender weights with no change in stats. From your pic, the number "1" and number "3" sets respectively appeared no matter which way I switched the weights. I even tried to put all 620 tons in the tender with 0 engine weight. But, behold, the stats seemeth not to have changed.

[Gumboots]

Eh? That's weird, because I took those figures straight from RT3 when I swapped the weights around. I'll test it again just to make sure.

If both weights are just added into one lump that would be ideal, because it'll be simpler to deal with (I already have everything else figured out) but then I wonder how RT3 threw out different sets of figures in the last test.

[Gumboots]

Hey question: how are you doing that test? Are you exiting back to the main menu, then editing the hex and saving it, then going back into the sandbox to check the stats again? Or are you just editing the hex while still in the sandbox?

I did it the first way. I always exit back to the main menu when testing hex changes, just to make sure the file is reloaded by the game before being called for display.

[RulerofRails]

Depending on what I am testing, but the most common thing with the engine stats is to reload a saved game/sandbox after I have changed and then saved the hex. All previous engine changes I have made show up immediately once the game loads afresh. However, before questioning your analysis, I did completely restart the game between tests and even tried starting new scenarios.

Just now, I tried this in 1.05 just in case it was something happening in 1.06. I also tried to put the Big Boy stats with reversed engine and tender weights into the Class 01. It's all the same for me.

There must be some explanation. Try these Class 01 files with the Big Boy stats except for the weight swap. What do you see?

[Gumboots]

I'm about to crash for the night. I'll double check everything over the weekend.

[Gumboots]

You are dead right about pulling power. I noticed that too. There is no way to make a loco better on grades. All we can do is:
Increase the free weight to make it better on the flat ground with a tiny boost to graded performance.
Increase the engine or tender weight which will make the loco worse on grades with no effect on flat-ground performance.
Increase pulling power which increases all values with flat-ground performance obviously benefiting more from higher values.

I was just thinking about this again. If we want to make a loco better on grades, the most effective way is to simply reduce the loco weight. Even changes of 1 ton have a direct and even multiplying effect on any speed up any grade. We were already thinking of artificially reducing weights for heavy locos to get their fuel consumption down to a useful level. This will automatically give them a boost on grades, and a pretty strong one too.

I think loco weight is going to end up being one of most powerful modifiers for adjusting stats. Given that the exponential formula behind it is now known, it shouldn't be hard to use that to keep relative performance of heavy locos the same while drastically reducing their weights.

Also, it turns out that increases in pulling power actually benefit speed up grades more than speed on the flat, because pulling power applies the same exponent on all grades (including 0) but non-zero grades have a steeper speed drop with every added car, which means they get a proportionally larger bonus once the pulling power exponent is applied.

I had a thought, hey, some people have examined speed and pulling power a fair bit in the past. So I went looking at some of the old spreadsheets that people have made. Of interest is Low_grade's one from the tips/tutorials download section. He used a simple scale of 1 to 10 to relate most of the levels i.e. Good to Very Good for example. He made pretty complicated formulas, but I wonder if we could update the scales to nearer their true relationships. What do you think?

I had a look at Low_grade's sheet too. I'm not sure how much use it is. For example, he appears to just take the PopTop fuel ratings at face value, without any consideration for how actual fuel cost is affected by loco weight. We'd have to find and dissect every assumption in his sheet before we'd know what it all meant, and whether the assumptions were good ones.

[Gumboots]

I doubt that I understood everything you were saying when I was reading through, however I have a question mark over your latest post regarding tender weight being treated differently to engine weight. If that were true, I can't see why the workaround principle in some of my latest posts of changing combined engine + tender weight, free weight, pulling power, and car weight using the same modifier would end up with identical stats but at a new car weight. I tested that multiple times on different engines. I treated engine and tender weight as a whole and would only make the change to one value.

Just now I tried to replicate your results with the Big Boy and failed to do so. I switched around engine and tender weights with no change in stats. From your pic, the number "1" and number "3" sets respectively appeared no matter which way I switched the weights. I even tried to put all 620 tons in the tender with 0 engine weight. But, behold, the stats seemeth not to have changed.

I just tested it again and you're right. I must have clicked the wrong buttons or something when I tested it before. I was pretty tired at the time so I'm putting those results down to basic human error.

This is good, because it means loco+tender weight is a simple proposition and I can just get on with compiling the final version of the spreadsheet.

I also think that when we get around to revising stats, it'd make sense to just set all tender weights to zero and do all the required adjustments on the loco weight only. It'll give the same result, and mean fewer things to keep track of.

Oh and with the spreadsheet, the assumption I made is that the game calculates the car-weight-related speed reduction coefficients on the basis of average car weight. This is the simplest way of doing it, so is probably what they did. I can't really tell at the moment though because the stats pop-up only gives speeds for consists that have one car weight for all cars. The only way of checking this assumption would be to check train speeds while they're running in the game.

What I'm thinking is that if the car-weight-related speed reduction coefficients for each grade and each car in the consist are calculated on the basis of individual car weights, with each individually-calculated coefficient being applied to that stage of the consist before the next is calculated, then this should give noticeably different speeds (compared to using an average weight) as each car is added if the consist alternates between heaviest and lightest cars. Should be easy enough to check, since the results can be predicted for both cases.