How much storage is needed, part 4

Image credit: Unknown

My previous post in this series related some of the drawbacks of my simplistic analysis, the objectives I want to achieve, and a sketch of the methodology I've employed. Briefly, I've used a Monte Carlo simulation (yes, I linked to something other than Wikipedia, you're welcome Doctor Steve) to determine a likely outcome for generation of energy by a single hypothetical wind turbine in Dalhart, TX.

I ran 1000 simulations of 8760 data points of wind speed from what turned out to be a mixture distribution combining a normal and a Gamma distribution. This generated a total of 8,760,000 wind speeds. Each was delivered to an interpolating function generated from a digitized power curve of a 3MW nameplate capacity wind turbine. This resulted in 8,760,000 data points, each representing the average power delivered by the turbine for a hypothetical hour.

From there, finding the average power delivered was a simple process, and the result of this simulation was a mean power of 788 kW. This equates to a capacity factor of ~788*100/3000=26.3\%~. This is a surprisingly low number for the location and turbine chosen, given published figures at sites such as this that yield an implied capacity factor of 33.1%.  Further, the model estimate has no allowance for planned and unplanned maintenance outages. And, of course, the 33.1% number is ostensibly from measured data, so, as Dr. Steve might say, "who ya gonna believe, me or your lyin' eyes?"

All that said, perhaps Dalhart isn't the ideal location, perhaps the wind gradient is steeper than the model I used, perhaps they've used more highly optimized equipment, perhaps the measured year had, for some reason, particularly strong (but not too strong) winds. I'm going to proceed with my analysis based on the model data.

So, the next step is to determine the storage required for the ability to deliver a given power at, say, 99.99% reliability. That is, the system should be able to supply the specified power for all but ~8760/10000=0.876~ hours/year. This is actually less than the SAIDI*SAIFI (system average interruption duration index, measured as the average duration of outages*system average interruption frequency index, measured as the average number of outages per customer per year) and so sounds quite reasonable if not overly conservative.

One assumption will be that, when the turbine is delivering more than the power under consideration and the storage facility is "topped off," we can send the power to the grid. Another will be that, for the level of power being considered, the storage system is capable of delivering power at that level. As I've discussed in previous posts, there are two primary characteristics of an energy storage installation: the quantity of energy that the system can store; and the rate at which it can deliver that energy.

Of note, approximately 4.0% of the time, the wind is below the cut in speed of the turbine and thus all energy delivered by the system must come from storage. The modeled wind exceeded the cut out speed of the turbine a negligibly small 0.0004% of the time. But there are no black swan events in the distribution (think tornadoes).

It took me a little time to decide on an effective way to proceed, but ultimately I decided to start with a guess of storage and loop through each increment (i.e., each hour's worth) of power (since the power is in kilowatts and the increments are hours, no conversion is necessary). If the storage plus the increment minus the steady use exceeded the maximum available storage, the excess was discarded and the maximum was kept for the next iteration. If the sum was less, that was kept for the next iteration. Upon completion, determine the number of iterations at which storage was zero or less, adjust maximum storage if and as necessary and try again. Using the mean power from all of the trials, no amount of storage sufficed, but reducing it to 725kW gave me what I wanted.

And finally, the result: If our 3MW turbine plus storage system is committed to delivering 725 kilowatts and we can provide 40MWh* of storage, there's effectively zero chance of not having the committed power available. Of course, the system can deliver greater power than that when the wind blows and/or when plenty of energy is stored but committing to greater power than 725kW or installing less storage than 40MWh means that there will be times when the system cannot deliver. Obviously, installing it in an integrated grid system can offset this, but the goal here was to determine what storage will enable what level of reliable base load power for a single turbine so the result is likely to be conservative. This is a virtue in the world of engineering. Below is a chart showing the first 100,000 increments with increment number on the x-axis and energy stored on the y-axis.

One widely discussed concept in energy generation is "capacity value," a very different concept (and number) than capacity factor. Basically, this number represents how much other generating capacity can be avoided with the installation of a generator and, for wind in particular, it is typically much lower than the capacity factor. Since there are times when no wind is blowing and demand does not abate, for an unaided turbine, sufficient generating capacity must be available to meet the demand, even though it may only be used sporadically. The goal of adding storage in this analysis is to bring the capacity value of the wind turbine close to the capacity factor.

As I noted in my previous post (on another topic), most utilities are not looking for days of storage (my analysis above determined that 48 hours of storage at 24.2% of the turbine's nameplate capacity would provide that power continuously and reliably), they're looking for a few hours. And, of course, the myriad complexities of transmission constraints, demand side variability, planned and unplanned generator outages, etc. have not been considered. Others have taken some of these into account using a similar methodology (i.e., Monte Carlo simulation). None that I've found, however, incorporate storage into the analysis. If I were a professor at a research institution or an NREL researcher or, perhaps, if I worked for a turbine manufacturer or a storage technology firm, I'd implement a much more sophisticated model incorporating the above factors as well as a wind farm as opposed to a single turbine.

Next in this series (which, as readers may have noted, may be interrupted by posts on other topics) will be an analysis of the economics of such a system, or at least the beginning of such an analysis. I anticipate that the cost will be prohibitive without pricing the externalities of fossil fuel generation (i.e., without implementing a carbon tax).




*In several trials, 35MW would have sufficed with no increments less than 0, but this run had a particularly calm stretch and, even with 40MW, had 0.0088% of the increments less than 0. However, this met the criteria of 99.99% reliability at 99.9912%.

Drag and weight as parameters of fuel economy in passenger cars

Image credit: www.modified.com

I've published previously that, for my car of that time that, below about 50 m.p.h., rolling resistance is the greater contributor to my need to burn fuel and above, it's aerodynamic drag. That vehicle was a Land Rover LR3 HSE, a much larger, heavier, draggier vehicle than my current chariot (a Lexus CT200h). Going through the same calculations, I find the crossover point to be about 38 m.p.h. That is (at steady speeds), below 38 m.p.h, rolling resistance provides the greater force to be overcome by burning fuel (or running electrons from high potential to low), above 38 m.p.h., it's aerodynamic drag. Below is a graphic taking these fractions from 0 to 40 m/s (about 89 m.p.h., far above my maximum). It should be noted that, in all of this, I only consider the external forces being overcome.

At my highway speed of 55 m.p.h., about 68% of my fuel is burned to overcome aerodynamic drag. And, since something like 70% of the miles I drive are on the freeway at my typical freeway speed, it's clear that drag represents a large portion of my fuel expenditures.

So let's take a look at how fuel economy in miles per gallon varies with the coefficient of aerodynamic drag (C_d). Below is a graphic showing an estimate of fuel economy as a function of C_d at 60 m.p.h. for a Toyota Camry-like vehicle (note that axes are not zero scaled). While the curve in this range looks to be close to linear, over larger ranges it's not, since fuel economy is inversely proportional to C_d and thus the graph is that of a hyperbola.

So what can be accomplished by reducing C_d from, say, 0.32 to 0.29? At 60 m.p.h. (and using my very simple model), this would result (for the Camry-like vehicle) in an increase from about 47.5 m.p.g. to 50.7 m.p.g. In a typical 12,000 mile year with 50% of the miles driven at highway speed, this would save some 8 gallons of fuel that might cost $32. Meh.


I attended a conference sponsored by the American Physical Society entitled "Physics of Sustainable Energy" (this was the third triennial such conference, I attended the second as well) at UC Berkeley. Amory Lovins of the Rocky Mountain Institute was the banquet speaker and made a presentation during the course sequence as well. Mr. (though Lovins has several honorary doctorates, I'm not sure that the "Dr." honorific is appropriate) Lovins has a huge portfolio of concepts that he claims, if implemented, would result in massive reductions in energy use in buildings (industrial, commercial, institutional, residential), transportation, and manufacturing. As time allows, I'll look into some of these.


But with respect to the topic of this post, Mr. Lovins stated that "two thirds of the energy used in a personal car is mass dependent." It seemed high when I heard it, let's consider. Energy is used in a car to accelerate (very much mass dependent but, in a hybrid, some of the kinetic energy imparted by accelerating a car's mass can be recovered by regenerative braking during deceleration), climb hills (very much mass dependent but descending hills can recover some, and in the case of hybrids with regenerative braking, much of the energy used in climbing), overcoming rolling resistance (mass dependent), overcoming aerodynamic drag (not mass dependent), and overcoming drive line friction and inertia moments of the rotating masses (both indirectly mass dependent in that lighter cars will need smaller, less powerful engines and, hence, lighter drive line components).


So, a lot of the energy is mass dependent. Is two-thirds a reasonable estimation? This is a complex question and will vary by car and by driver, but surely we can approach it. I'll assume that the car is not a hybrid. In addition to the usual coefficient of rolling resistance (C_rr) assumptions, a number of others are required, among them: fraction of city vs. highway miles (I assumed 0.4 and 0.6); stops and starts per mile for both city and highway (I assumed 4 highway accelerations per 25 miles and 4 per mile in the city), engine efficiency (I assumed 22%). I ignored hill climbing (this would sway the fraction we're seeking higher). Should my readership clamor for it, I can elaborate on the process I used to calculate. In the end though, my estimate of the fraction of energy used in mass dependent aspects of fuel economy in this personal vehicle is 37%.

It's actually more complex than this since, at very low speeds, a large proportion of the energy used is devoted to keeping the engine going. In the extreme, stopped at a light, all of it is (though in my hybrid, the engine shuts off at a stop and I used to turn off the engine in my LR3 to eliminate this). And the amount of energy devoted to keeping the engine turning is dependent on the size of the engine and, thus, on the mass of the car. This argues for increasing the estimate of the mass dependent portion of energy used. But for the car I'm considering, it's hard for me to imagine that that portion exceeds half.

It's clear though that changes in the assumptions will have a large effect on this calculation. For example, reducing the highway portion would increase mass dependent energy; decreasing the average stop/accelerate cycles per mile in city driving would decrease it. In any event, it's clear that mass reduction in a vehicle will significantly enhance fuel economy. This post is already pretty long, so I'll elaborate in a future post.

More on fuel saved by regenerative braking

I published a post regarding how much energy is captured in the regenerative braking system in my Lexus CT200h hybrid. After some discussion with commenter Gabriel Grosskopf, I estimated that about 59% of the energy available (after subtracting the energy used to overcome aerodynamic drag, rolling resistance, and internal friction) was recaptured and used to charge the battery.

Since I (and others) have represented that the regenerative braking system is among the key reasons that hybrids achieve superior fuel economy, I decided to check the actual impact.

My round trip commute, generally downhill in the morning and uphill in the evening, is 62.46 miles and, for the last 10 fill ups, my average m.p.g. has been 52.47. So, to make my commute, I use, on average 62.46/52.47=1.190 gallons of gasoline. My display showed me today that my regenerative braking system added 700 watt hours or 2,520,000 joules to my battery that I could use for accelerating, hill climbing, etc. If I assume my electric motor is 90% efficient, I put 2,268,000 of these joules to work.

A gallon of gasoline (reformulated blend in this case) has an energy upon oxidation of 111,836 btu or 117,993,000 joules. I estimate that my internal combustion engine is about 25% efficient, so I put about 29,498,000 of these joules to work. My 1.19 gallons thus provide 35,103,000 joules that propel my vehicle (the remainder being lost as waste heat in myriad ways).

If I assume that I used all of the energy my brakes provided, then 35,103,000 + 2,268,000 = 37,371,000 joules of work were done to propel my car. Then, dividing by 0.25, I can estimate that 149,484,000 joules of oxidized gasoline would have been necessary to do this work. This is the energy in 1.267 gallons. Dividing this into 62.46, I find that the fuel economy without the regenerative braking would have been about 49.30 m.p.g. The regenerative braking thus upped my m.p.g. by 3.17.

As I've often said, it's much more intuitively informative to discuss gallons per mile, or gallons per 100 miles. So, the regenerative braking took me from 2.03 gallons per 100 miles to 1.91 gallons per 100 miles. So it takes me 5.9% less fuel to go a given distance, ceteris parabus.

There's no question that I'm carrying a lot more significant figures (apologies to John Denker) than are warranted by the precision of my data, but I think that the figure I've determined is probably in the ballpark.

Wasted energy

Nope, I'm not talking about the 75% of the energy in the gasoline in your car going out the exhaust pipe as waste heat, or the cooled/heated air escaping your house because it's not as "tight" as it could be. I'm not talking about anything to do with the inefficiency of the use of primary energy. I'm (against my usual nature) using a more colloquial meaning of "energy" here. I'm talking about foolish expenditures of "mental energy" on schemes that either will produce no useful energetic results or such a trivial amount of energy that it's a waste of time to bother.

I've previously written of the foolishness of humans as generators via walking and revolving doors, the silliness of one of the Discovery Channel's "Project Earth" programs (they were all fairly silly but I focused on a specific one). I've got two more examples.

The first is something akin to the "human powered generator," that is, it would probably work from a technical point of view but be useless in any practical way. I was pointed to it by a post in Tom Sanwson's Swans on Tea blog entitled "This Claim Won't Fly." The post referred to an article on CNN's site about Airbus working with UNESCO to challenge engineering students in a "Fly Your Ideas" competition.

Of the ideas described in the article, one sounds far out (shape shifting engines to reduce noise footprint), one marginally practical (powering jets with a supercooled mixture of biomethane and LNG), and one ... let's just say silly. That last one is to use seats upholstered in a thermoelectric fabric to use passengers' body heat to generate electricity. University Putra Malaysia team leader Tan Kai Jun envisions generating "100 nanowatts of voltage." Like Tom Swanson, I'll ignore the fact that watts is not a unit of voltage, but rather a unit of power.

"It's a small amount, but imagine this collected from 550 seats throughout 10 hours of flight. A plane has a lifespan of a few hundred flights -- over time that's a big reduction," Mr. Jun Tan says."

Tom ran some numbers, but I'll do the same. The "age" of an airliner is measured in pressurization cycles, and a "typical" airliner may have 51,000 flight hours and 75,000 pressurization cycles in its useful life. Let's consider an airliner with 400 seats (say, a B777) and 51,000 hours. We'll assume the plane flies with an average load capacity of 95%. So we have 400*51000*.95 = 19,380,000 seat hours. Using the the Google Chrome extension "Cloudy Calculator" and multiplying hours times 100 nanowatts (the calculator does all conversions) we find that we've generated 1.9 watt hours over the life of the aircraft. I pay about $0.12 (12 cents) per kilowatt hour at my house, so this almost two watt hours is worth a smidgen (one of my favorite units - slightly smaller than a skosh) over a fiftieth of a penny.

Looked at another way, Jet A fuel (used in airliners) has an energy density of 35.3 megajoules per liter. 1.9 watt hours is about 7,000 joules, or 0.007 megajoules. So, over the lifetime of the airplane, the passengers would generate electrical energy equal to the chemical potential energy in 0.2 milliliters of jet fuel. Of course, a heat engine such as a turbofan might operate at, say, 45% efficiency so we'd actually need to burn a bit under half a milliliter to generate those 1.9 watt hours. I suspect that this development will not revolutionize flying.

The next post will cover a different type of silliness - one that's either a fraud or pie in the sky.