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If the demand for rides out of the urban center (in the morning) were as high as demand for a ride in, then we'd already see equal movement in both directions.
You can algorithmically optimize things all you want, but in the end, it isn't really an improvement unless the car can find a fare relatively quickly and relatively close by. If it has to park downtown for any period of time, it's spending money rather than making money, and it's likely more money than a private commuter would pay for his car since he'd probably have a lease (and the pay structures of most garages make things even more complicated and expensive). If the car drives around to find fares, picks up a fare outside of downtown, or goes to a lot where it can park for free, it's contributing to traffic. Additionally, the optimization is only concerned with losing the least amount of money when the cars don't have passengers. Minimizing traffic doesn't play into the equation.
Consider the following scenario: A downtown area gets 20,000 commuter vehicles per day, and a garage costs $15/day on average. Assuming demand to leave downtown is minimal until later mid-afternoon, the optimal move is to simply have the cars drive around downtown. Perversely, if the cars are electric or hybrid (which they are usually assumed to be), it's in the interest of the car service to create as much traffic as possible. Since the bulk of the energy consumption only occurs when the cars are actually moving, it's best for the companies to ensure that the cars are stopped as much as possible. If people need to leave downtown during the day, then, well, there aren't enough of them to make it worth it to park the cars somewhere.
Deal with the consequences of. Sitting traffic as a passenger isn't exactly much of an improvement when you're trying to get somewhere.
I'll jump off from this point.
Aside from just being an Elon dream, I think a lot of potential Autonomous Cars As A Service (ACaaS) would seriously consider whether EVs are in their interest (at least in areas where the weather is such that EVs make sense at all). Per-mile energy costs tend to be lower, and I haven't kept up with the stats on current models, but there are simplicity reasons to believe that, for a given level of non-propulsive tech in a car, simply swapping out ICE for electric can plausibly reduce maintenance costs (practical numbers would definitely be needed for battery replacement on a car running taxi service every day compared to a comparable ICE). So, there are inherent reasons for the service providers to want to consider EVs.
Anyway, let's get to building a model. I'll start with the premise that we pretty much just model the vast majority of peak traffic as commuter traffic, the timing/quantity of which we hold essentially fixed. I'll also assume for now that the autonomous taxi service supplies almost all of the commuter traffic. Crucially, this model doesn't really say almost anything about the total number of cars owned or the number that are personally owned. People may still keep the same number of cars at home, in their garage all week, ready to use for the weekends, trips to grandma's, etc. I think there's a lot of confusion in the thread that is flipping back and forth between ownership and utilization. In any event, I think this model is somewhat like what you have in mind in your example.
Now, basically all of the service's EVs have presumably magically found a home to cheaply charge all night, presumably somewhere in the suburbs where it's hopefully cheap. Given the current 200-300mi range, they can basically all come online and 'work' through the morning rush hour(s). I think there are a couple crucial questions, one which you've brought up, but another which I think has been missing. First, "How much do they deadhead during rush hour?" Second, "Do they have to charge to make it through the evening rush hour? If so, what method would they prefer?"
For deadheading, the simplest model is to just to assume that there is approximately zero demand to go in the opposite direction of the main commuter traffic. To a first approximation, a small amount of deadheading has the obvious cost of driving back trading off with a reduction in the number of vehicles the ACaaS has to operate. Presumably, an ACaaS startup will have a nerd in the back room doing calculations with a more detailed model of traffic, and I think this would be a key parameter. Obviously, as you point out elsewhere, as that parameter increases, you run the risk of tipping into two-way congestion, which would also increase the cost of deadheading. But something else to note is that, in this simple model, this parameter is pretty directly correlated to the number of cars that potentially end up somewhere in the urban area through midday. That is, for example, if each autonomous commuter taxi makes one deadhead trip to pick up another commuter, and we assumed that each commuter would have brought one vehicle into the urban area, we've reduced the number of vehicles in the urban area during midday by half. It's very directly (inversely, lol) just a 1/(N+1) relation. At what value of N does two-way congestion really start to become a problem? I haven't the foggiest. Hell, it could even be a non-integer less than one; I have truly no idea.
An important challenge of this model to people who are pro-ACaaS is that they really kind of need to say what sort of N they're expecting, would be okay with, and think is plausible. Else, they need to propose something specific that the model is wrong about that can plausibly make their other claims work. If they're not okay with nonzero N, they better have something good, or we might think they're slipping magic into their imprecise model.
Of course, in the simplest model, we don't super care about congestion in the other direction except to the extent it increases the cost of deadheading. That is, the simplest model is that it was a wide open, completely empty freeway heading back to the suburb, and nobody cares if a bunch of deadheading empty ACaaS are clogging it. One would need more complications in the model to capture anything else.
N is not just limited by cost and two-way congestion considerations; it's also limited by time constraints. If an average round trip takes an hour, for example, you can't make more than a couple within the peak hours. This also leads us to the second question about charging. If you're driving back and forth for N trips for a few hours in the morning, do you need to charge to make it through the rest of your day?
EV owners typically prefer slower charging, as it's cheaper and better for the battery, reducing their lifetime costs (ACaaS operators may also have incentives to just abuse the hell out of their batteries; typical taxis certainly have incentives to abuse the hell out of their cars). Of course, if it's sitting there charging more slowly, it's not making any fares. But if it's lollygagging around in traffic trying not to do anything so as to conserve energy, it's not making any fares either. In this model, there's not many fares to be had at this time, so it's kinda dead time anyway. I think I see three options: 1) Not pay for a spot to park half of the afternoon, just eat the cost of fast charging, then hold up traffic conserving energy, 2) Pay to take up a spot for a while, but get cheaper slower charging, 3) Just drive back out of town to get cheaper slower charging without paying the spot fee.
To flesh out a hopeful possibility for (2), though, as the morning rush tapers off, they could start to duty cycle off for charging. Math would need to be done, but if you need your duty cycle to be Y% through the day, you'll have Y% of your peak capacity available. Hopefully, those nerds in the back will figure out how to get that percentage right so that you have enough charge left in enough tanks to get through the evening peak.
But then, I think the math conclusion is that, during the day, between rush hours, Y% of the autonomous taxis will be roaming for possibly cheap fares (maybe still doing bad energy conservation stuff), whereas (100-Y)% of them will be charging somewhere. So, we won't have peak rush hour quantities of cars on the road all the time throughout the day. Also, even assuming that the entire (100-Y)% of chargers are finding their charge homes in the urban area, that's plausibly still a lot fewer charging parking spaces than would normally be housing the full 100% of peak traffic all day that we currently have. Ya know, if the limit on N will allow it. The hope and promise here over individually-owned vehicles would be that you save on parking, can recoup some costs with deadheading, and even getting some fares on a duty cycle in the afternoon is worth more than having it sit in the parking garage all day while you're working.
Obviously, there are a ton of detailed cost comparisons that would have to be made. But I think that EVs are potentially different from ICEs in that essentially the only sensible model of 'charging' the latter is 'fast charging', at one given price point. I'm sure someone out there will make some models/approximations where they say, "Assume our average charging voltage through the afternoon is V," then proceed to compute duty cycles, cost of charging, impacts on maintenance, expected fares in the slow periods, cost of however many parking/charging spaces they need at that duty cycle, etc.
We'd kinda need to search out the space of at least (N,Y,V). I don't know if there will be any ranges of parameters that work for any models with real-world costs/traffic patterns/etc., and they may all end up with perverse incentives like, "Poke around and hold up traffic all afternoon." But I also kinda don't think I'm comfortable concluding that there is no range of parameters at all where it might make sense. I kinda just think we'd need a more sophisticated model. I also think the most likely conclusion would be, "It does not make sense for ACaaS to grow all the way to the point of providing all of the commuter traffic," and the model would get more complicated still, as one would have to vary the quantity of commuter traffic they think they could capture.
As a note, ChatGPT pulled a number out of its giant inscrutable matrix and estimated that traffic volumes during peak hours are often 30-50% higher than midday, rising above 100% in certain cities like NY/LA/London. I honestly have no other frame of reference in my mind to know if that's a complete fabrication or is in the right order-of-magnitude, but it could give a modeler some start in estimating plausible duty cycles (I mean, not exactly giving us Y, but maybe something like it?). Probably the simplest starting assumption would be that all of that midday traffic is entirely in the urban area, but obviously that's not "correct", and again, one would need a more complicated traffic model.
One final note is that nothing in this simple model has any dynamics. There's no, "Well, A got cheaper, so more people decided to B, so..." It's a purely static model, in line with your scenario.
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