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Discussion Starter · #1 · (Edited)
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Here's my first draft at a formula for working out a BEV's ability to travel a given distance successfully.

Basically we need four facts to work out the chances of success (Ps) in completing a trip.

These are:

Probability of a failed charge point (Pf)
Range of vehicle (r)
Total Distance (d)
Number of points enroute* (n)

* I have assumed equidistant spacing.

Ready for this ? :eek::D

Ps = (1 - Pf / [r/(d/n)]) ^ [d/r]

Plugging some numbers in to test:
250 mile journey in a 300 mile range car = 100% (obviously)
250 mile journey in a 60 mile range car, 5 points, 5% chance of a point being down = 80%
250 mile journey in a 100 mile range car, 5 points, 5% chance of a point being down = 95%
250 mile journey in a 200 mile range car, 5 points, 5% chance of a point being down = 99.8%

So clearly battery range has a big impact on actually reaching your destination (not just time or convenience) all things being equal.

What if we double the number of points?
250 mile journey in a 60 mile range car, 10 points, 5% chance of a point being down = 90%
250 mile journey in a 100 mile range car, 10 points, 5% chance of a point being down = 98%

What if we double the reliability?
250 mile journey in a 60 mile range car, 5 points, 2% chance of a point being down = 92%
250 mile journey in a 100 mile range car, 5 points, 2% chance of a point being down = 98%

This formula by the way could equally be applied to PHEVs or ICE's but of course range, number of points (i.e. petrol stations) and reliability are all MUCH better so it would be pointless ;)

I calculated the 550 mile Aviemore trip at 77% guessing at 10 chargers along the way, and for giggles a Model S if stopping at Superchargers only on the same trip, a 95% chance of making it. (Though I suspect Supercharger up-times are much higher, but not guaranteed as the unfortunate incident at the Hyatt proves)

I have this in an Excel spreadsheet, if anyone wants to scrutinise.. but not sure how to upload it here though as xlsx files aren't allowed..

There maybe errors in my transposition to a formula rather than Excel, and anyone with some better software for formatting it is welcome to post an image (For example I have used [ to represent floor rather than the more modern symbol for example)

Enjoy :)
 

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Interesting idea. Shows how doubling up Rapids massively increases your chance of success, given the ecotricity network is usually about 15% offline...
 

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For me I'd add Temperature in. However there would be two reasons. 1) Battery range loss due to cold, 2) How long I could drive in such cold temps in my Twizy. :eek:
 

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Discussion Starter · #6 ·
Interesting idea. Shows how doubling up Rapids massively increases your chance of success, given the ecotricity network is usually about 15% offline...
I edited my post slightly Edd, I'd copied and pasted incorrectly (the figures for doubling reliability were out):oops:

Doubling reliability gives the same benefit as doubling posts in this example.

Makes sense really because this journey the car can pull in at the first post, see it's not working, and drive to the next. Effectively the same as doubling reliability ;)

I'd argue reliability is more important, because it's effect is linear.. Adding more posts will be result in stepped results.
 

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Discussion Starter · #7 ·
Yeah I think there is no point counting the out leg from home as the car is already (100% likely) charged at home.
lol, I deleted my quoting post of your deleted post... but yes the assumption is the car is full at the start of the journey
 

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Hmm, not only do you have to assume equidistant charging points, but also equal consumption, IMO energy consumption needs some variability in there to factor in the things that normally cause issues, i.e. outside temperature/traffic/weather/terrain/detours...

I guess "availability" of a chargepoint also includes the probability of being ICEd :)
 

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Discussion Starter · #9 ·
Hmm, not only do you have to assume equidistant charging points, but also equal consumption, IMO energy consumption needs some variability in there to factor in the things that normally cause issues, i.e. outside temperature/traffic/weather/terrain/detours...

I guess "availability" of a chargepoint also includes the probability of being ICEd :)

Jimbo, you are quite right, my real intention with this is to move away from the subjective speculation on Vehicle Range vs Reliability vs Point count.

As with all models, some assumptions must be made.

Points not being equidistant, would make the formula much harder to comprehend, but equally achievable, just you'd need a set input into the equation.

Maths isn't my strong suit, I could easily code such an algorithm, but posting up some C#, VBA, C++, Java
would be a bit daft, and get us into endless debates over coding styles and language wars :(

You are 100% correct one of the "failure modes" of a Rapid is it being ICE'd (as is not having the right RFID card ;) )

I should also include an overall Pm, which is the probability of being a Muppet, and leaving your cable at home :)
 

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What about if there's no chargers within range in the direction you want to travel and all the available ones are 180 degrees the other way? Which is pretty much the case for many places outside suburbia. I guess we could call it the 'Round the houses factor' ;)
 

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Ps = (1 - Pf / [r/(d/n)]) ^ [d/n]

Plugging some numbers in to test:
250 mile journey in a 60 mile range car, 5 points, 5% chance of a point being down = 80%
OK - what am I doing wrong!

Using these figures, the inner term of d/n = 250 / 5 = 50, so r/(d/n) = 60 / 50 = 1.2, and Pf/(r/(d/n)) = 0.05/1.2 = 0.041667. Then 1 - that is 0.958333, then raise to d/n (which was 50) gives 0.119 or an 11.9% chance of success.
 

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Discussion Starter · #12 ·
OK - what am I doing wrong!

Using these figures, the inner term of d/n = 250 / 5 = 50, so r/(d/n) = 60 / 50 = 1.2, and Pf/(r/(d/n)) = 0.05/1.2 = 0.041667. Then 1 - that is 0.958333, then raise to d/n (which was 50) gives 0.119 or an 11.9% chance of success.
Good spot Matt. it should read [d/r] ... I will correct.

The square parentheses indicate rounding down to whole integer btw.
 

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Discussion Starter · #13 ·
I spotted a few issues, in my formula (specifically it didn't give 100% for an exact match).
Haven't got time to put it back into notation form, so I'll do what I said I wouldn't :)

Function ReliabilityCalc(reliability As Double, range As Double, pointCount As Double, distance As Integer) As Double
Dim distanceBetweenPoints As Double
Dim skippablePoints As Integer
Dim chargesRequired As Integer


distanceBetweenPoints = distance / pointCount
skippablePoints = Int(range / distanceBetweenPoints)
chargesRequired = Int(distance / (range + 1))

ReliabilityCalc = (1 - reliability / skippablePoints) ^ chargesRequired



End Function


Anyway, plotting it out I get this:

Graph.PNG

It steps as I would expect, and I get 100% for the vehicle range.

I think there might still be a small problem because it makes no sense a 230 mile journey is harder than a 240 mile journey, in the 100 mile range car. It's a boundary condition somewhere I'm sure.
 

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I spotted a few issues, in my formula (specifically it didn't give 100% for an exact match).
Haven't got time to put it back into notation form, so I'll do what I said I wouldn't :)

Function ReliabilityCalc(reliability As Double, range As Double, pointCount As Double, distance As Integer) As Double
Dim distanceBetweenPoints As Double
Dim skippablePoints As Integer
Dim chargesRequired As Integer


distanceBetweenPoints = distance / pointCount
skippablePoints = Int(range / distanceBetweenPoints)
chargesRequired = Int(distance / (range + 1))

ReliabilityCalc = (1 - reliability / skippablePoints) ^ chargesRequired



End Function


Anyway, plotting it out I get this:

View attachment 2479

It steps as I would expect, and I get 100% for the vehicle range.

I think there might still be a small problem because it makes no sense a 230 mile journey is harder than a 240 mile journey, in the 100 mile range car. It's a boundary condition somewhere I'm sure.
Smashing graph, actually illustrates the issues quite well !
 

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Discussion Starter · #15 ·
Smashing graph, actually illustrates the issues quite well !
:oops: Thanks..

I have fixed my formula (I think... still waiting peer review ;) ) But this is the outcome of 50 mile spacings of charge points.
upload_2014-11-25_14-16-4.png


It clearly illustrates that because the 100 mile car can use either the 50 or 100 mile charging station on it's journey to 110 miles it is twice as likely to make it than the 60 mile car which can only use the 50 mile charge point.

I am worried the points after this are out, and my generalisation is incorrect, so really would appreciate any maths whizzes giving it the once over.
 

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Discussion Starter · #17 ·
Needs adjustment for the fact that the final rapid charger is not needed as you have reached your destination.
Assuming the destination isn't a scrapyard, you'd still need to charge it up though wouldn't you?
Sure you've arrived, but the flat bed will still have to come and take you home? (I could be being dense of course, it's been a long day!)

No points on this list need to be a rapid, and ideally the final one would be a destination charge at a hotel or car park, a relatives, or indeed back home.


To use this as a proper planning tool, you'd need something like a list to be representing distances of the real world route e.g: {10,44,102,155,180,233} and probably each one assigned a probability. E.g. A charge master unit might be twice as reliable as a Source London one.

The other thing you could do is assign a time to each charge, this is only about actually getting there, not how long it takes. In certain circumstances a 1 hours delay might as well be a flat bed, e.g. you are going to catch a plane.
 

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Just got in! Icy this morning so ran out of charge on the way home ... think that's irony :|

I think this is great, it's fantastic to have some objective projections rather than subjective pondering.

I do have a niggling doubt about the probability (not your model which I think is fine) but I can't work out how to express it into the formula.

If P(r)x is the probability of the driver stopping to recharge at a given point and F(r)x is the probability of a failed point , then having 30 mile points (x = 1, 2, 3 etc.) would mean a 60 mile range car would have an algorithm where (P(r)2 | F(r)1) = 100% (i.e. Given point 1 failed, the driver must stop at point 2). But most drivers will skip point 1 and so upon arrival at point 2 will be stranded. Hmm.

I don't think we can model human behaviour though!

The numbers for the 60 mile car are pretty poor, I've done quite a few long distance trips in mine so have clearly beaten the odds!.

My suspicion is that 200 miles is the sweet spot primarily because at an average speed of 60 mph, most people can do 2-3 hours before needing to stop anyway :)
 

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Discussion Starter · #19 ·
Just got in! Icy this morning so ran out of charge on the way home ... think that's irony :|

I think this is great, it's fantastic to have some objective projections rather than subjective pondering.

I do have a niggling doubt about the probability (not your model which I think is fine) but I can't work out how to express it into the formula.

If P(r)x is the probability of the driver stopping to recharge at a given point and F(r)x is the probability of a failed point , then having 30 mile points (x = 1, 2, 3 etc.) would mean a 60 mile range car would have an algorithm where (P(r)2 | F(r)1) = 100% (i.e. Given point 1 failed, the driver must stop at point 2). But most drivers will skip point 1 and so upon arrival at point 2 will be stranded. Hmm.

I don't think we can model human behaviour though!

The numbers for the 60 mile car are pretty poor, I've done quite a few long distance trips in mine so have clearly beaten the odds!.

My suspicion is that 200 miles is the sweet spot primarily because at an average speed of 60 mph, most people can do 2-3 hours before needing to stop anyway :)
Sorry to hear you ran out, I hope I haven't jinxed things!

I am completely onside with your suggestion. For maximum reliability you have to stop at every point enroute, until you get enough range to get to your destination. Again a mantra in the EV community, but one driven by blind faith or instinct, not hard numbers.

Shouting we need more rapids, or better reliability (not something I disagree with BTW, but not for this exercise) on prejudices clouded by the plain fact the ability to cover distance is more effected by vehicle range than the other two factors. I hope I've proven this. I agree 200 miles is a really good spot to be in. The only slight issue is the only cars that can claim this right now are PHEVs/REXs/ICEs.

Even the Tesla Model S can't because it's Superchargers are so interspersed, and the only other option currently available is 22kW, and you might as well have broken down as you wait for it to recharge :(

TBH the figures for the 60 mile range car are indicative vs. the others. What is likely out is the probability of a single failure (the 5% I've used). Alternatively maybe you should go for a trip to Vegas :D

P.S. How did your guys get on at Future Decoded?
 

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It was my own fault forgetting to put the charge timer on last night, all the battery was hoovered up trying warm up and to see out the windscreen:(

On the rapids, there is also the slight difficulty that having a rapid every 20 miles won't totally solve the failure problem as even if your time is worth nothing and you are pessimistic enough to charge at every opportunity, you can't recharge the battery every 20 minutes anyway without it overheating. I only kept the battery cool on my 300-mile trip by interspersing "fast" and "rapid" charges. What extra rapids would do, though, is give a fighting change to the driver who arrives at his 60-mile mark to find a failed rapid. It might be possible to crawl to the next one along in that case whereas now you would be flat-bedded.

Of course, there is a simpler technical solution, for now, which should cost peanuts. Let the rapids ping their status and let the EV (heck, or your phone) pick that up. A few points do that already. That way you can plan your stops more accurately and have confidence on a working rapid when you need it, without needing to simply throw extra units at the problem.

I think the probability of a single failure is up around the 15% mark based on the Ecotricity list of failed rapids, so I have been very lucky. I think I'll try the roulette wheel first :cool:

Thanks for asking - my guys made it back quite happy and excited about Visual Studio 2015 and ASP.Net vNext. Although they missed their trains home, it took them 3 hours to get back to Peterborough - the Model S slow drive not looking so bad after all!
 
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