A couple of days ago (January 11, apparently shortly after midnight) on Watts Up With That, Christopher Monckton published a piece that ran under the headline “The Final Nail in the Coffin Of ‘Renewable’ Energy.” The piece contained a short and apparently elegant mathematical proof — which Monckton attributes to a guy named Douglas Pollock — of a proposition that Monckton stated as follows:

In plain English, the maximum possible fraction of total grid generation contributable by unreliables turns out to be equal to the average fraction of the nameplate capacity of those reliables [sic — should be “unreliables”?] that is realistically achievable under real-world conditions.

Monckton (and Pollock) thus seem to be saying that if (for example) a wind turbine system can only generate about 35% of nameplate capacity “realistically achievable under real-world conditions,” then it’s futile to build any more wind turbines once you get to 35% wind penetration into output, because the 35% penetration is a mathematical limit that cannot be exceeded.

My immediate reaction was that that couldn’t possibly be right. I was planning to write a comment over there pointing out what I thought was the flaw. But before I got around to it there were 300 or so comments on the post, which had to a sad degree degenerated into a name-calling match between Monckton and some adversaries. So rather than writing a long comment there that would then be buried at the bottom of all of that, I decided to write a post here, which may or may not then get cross-posted at WUWT (it’s up to them).

This matter illustrates why, when I dabble in math in my posts, I try to stick to simple arithmetic. Not that mathematical proofs aren’t fun — I have done more than a few in my day — but it’s very easy to make an implicit assumption that you don’t recognize and end up with a result that does not really support the conclusion you think it does.

First, here’s what I think is the flaw: Monckton/Pollock, perhaps without realizing it (or maybe, because they think it’s too ridiculous to even consider), have assumed that there would be no “overbuilding” of intermittent generation capacity. By overbuilding, I mean building so many generators that when the wind and sun are at full strength the system produces more electricity than the demand, which electricity then has to be discarded or wasted. (You often see the term “curtailed.”)

But unfortunately, overbuilding is very much on the table as a way to get more wind and solar input into the system and, supposedly, reduce the use of fossil fuels. As an example, in a post on July 30, 2022 I compiled some statistics for the country of Germany that had been put out by the U.S. Energy Information Agency for the year 2020. According to that data, in 2020 Germany had average electricity usage of about 57 GW, and peak usage of about 100 GW. But it had wind turbines with “nameplate capacity” of 62 GW and solar panels with nameplate capacity of about 54 GW, for a total between the two of 116 GW. So when the wind and sun are both producing at full strength and usage is average, Germany has more than twice the electricity it needs just from the wind and sun even if everything else is turned off. They need to “curtail,” or alternatively, as I understand it, wholesale power prices go negative and they have to pay Poland to take the excess power off their hands. And yet, in the effort to get to the imaginary “all renewable” future, Germany continues to build more wind turbines and solar panels. So, as ridiculous as it may seem, overbuilding is actually occurring in the real world, and more of it is coming.

According to the German Umwelt Bundesamt (Federal Environmental Agency), Germany got 41% of its electricity from renewables in 2021. That well exceeds the “average fraction of nameplate capacity” of the wind and solar generators that is “realistically achievable” (however that may be defined), which is around 30% averaged between the two of them. The difference, I believe, is a result of the overbuilding. Thus the case of Germany demonstrates that overbuilding can, and in the real world does, lead to exceeding what Monckton calls the “Pollock limit.”

To illustrate how this works, let me introduce some math. However, in accordance with my practice, I will avoid fancy proofs and stick to simple arithmetic.

Consider an electricity system with a constant 1 GW demand, to be supplied to the extent possible with wind turbines. Assume that the wind turbines operate at 50% of nameplate capacity averaged over the course of the year. In this location, it turns out that the weather is such that the wind blows at full strength 25% of the time, half strength 50% of the time, and not at all the remaining 25% of the time. You build 1 GW of nameplate capacity of wind turbines to exactly match demand when the wind is at full strength. Over the course of the year, you get from the wind turbines all of your demanded electricity 25% of the time, half of it 50% of the time, and none the remaining 25% of the time, which as stated comes to an average of 50% over the course of the year. The penetration of wind power on the grid at 50% is equal to the capacity factor of the wind turbines at 50%, and thus is exactly at the “Pollock limit.”

Can you get more than the 50% grid penetration from wind production, even though the turbines only produce at 50% of nameplate capacity? Yes — by overbuilding. You can double the amount of wind turbines. Then, in the 25% of the year when the wind blows at full strength, you will get double the electricity you need, and will have to discard or “curtail” half of the production. In the 50% of the time when the wind blows at half strength, you will get exactly the amount of electricity you need. And in the remaining 25% of the time when the wind does not blow at all, you get nothing. Averaged over the course of the year, even though the wind turbines only operate at an average of 50% of capacity, you have gotten 75% of your electricity from the wind system, at the cost of doubling the size of the system and throwing away 25% of the electricity produced. And you still have no electricity 25% of the time.

But suppose you want to get nearer to all of your electricity from the wind. Too bad — you have maxed out. If you again double the amount of wind turbines, you will have 4x the amount of electricity you need during the 25% of the time when the wind is at full strength, 2x the amount you need when it is at half strength, and still nothing when the wind is calm. In other words, under these assumptions, with a 2x overbuild, you have maxed out your percent of electricity from wind at 75%. The 75% is well more than the 50% “Pollock limit” of these assumptions, and happens to correspond exactly to the amount of time when there is any usable wind at all. In simple terms, overbuilding can get you above the “Pollock limit,” but no amount of overbuilding can solve the problem of complete calms. For solar, the same principle applies to nights.

So how do you determine what is the maximum percentage limit of wind generation on a grid when overbuilding is allowed? Think about this a little, and maybe do a couple of more simple examples in your head, and you will realize that the following is true: with overbuilding and curtailment allowed without limit, the maximum penetration of renewables on a grid is 1 minus the percent of time when the wind does not blow and/or sun does not shine sufficiently to generate any electricity at all. Where all the renewables are wind, if the wind is completely calm (or at least so light that the wind turbines don’t turn) 10% of the time, then the maximum penetration of wind onto the grid is 90% (i.e., 1 minus 10%). As long as there is even slight generation from the wind, a theoretical massive overbuild could turn that into enough supply to meet demand. Suppose that another 10% of the time the wind only blows sufficiently to generate 1% of nameplate capacity (with all other times generating higher percentages). Then you can get still to the 90% theoretical limit with a 100x overbuild. Even if the wind generates only 0.1% of nameplate capacity for a substantial amount of time, you can still hit the 90% theoretical maximum with a 1000x overbuild. But you can never cover that last 10% when the wind is completely calm, because any number, no matter how large, times zero, equals zero.

So that’s my contribution to the math of this matter. Now a few thoughts on what has occurred over at WUWT. I’m going to quote in full the words in which Monckton states the Pollock proof, highlighting where I think the flaw lies:

Let H be the mean hourly demand met by a given electricity grid, in MWh/h. Let R be the average fraction of nameplate capacity actually generated by renewables – their mean capacity factor. Then the minimum installed nameplate capacity C of renewables that would be required to meet the hourly demand H is equal to H/ R.

It follows that the minimum installed nameplate capacity N < C of renewables required to generate the fraction f of total grid generation actually contributed by renewables – the renewables fraction – is equal to f C, which is also f H / R ex-ante.

Now here comes the magic. The renewables fraction f, of course, reaches its maximum fmax where hourly demand H is equal to N. In that event, N is equal to H ex hypothesi and also to fmax H/ R ex-ante, whereupon H is equal to fmax H/ R.

Since dividing both sides by H shows fmax / R is equal to 1, fmax is necessarily equal to R.

I think that in the bolded phrase “the minimum installed nameplate capacity N < C of renewables” Monckton has assumed that no overbuilding is allowed. It is far from 100% clear, and I have difficulty parsing the sentence. I would agree that if no overbuilding is allowed then the conclusion follows that the maximum possible fraction of grid penetration by the renewables would equal the average fraction of nameplate capacity at which the renewables produce averaged over the year. The maximum would occur, at least as one example, when average electricity demand is constant over the year and the nameplate capacity of the renewables was equal to that constant level of demand. If, on the other hand, demand fluctuated over the year, then there would be times of peak demand when even full nameplate capacity of production could not fulfill demand, and therefore the level of grid penetration by the renewables would fall below their average capacity factor.

Unfortunately, Monckton did not state in his conclusion (quoted in italics way back at the beginning of this post) that the conclusion only applied in a case where there was an assumption of no overbuilding. Several commenters at WUWT (e.g., chadb, Joe Born, “it does not add up”) weighed in to provide examples from places like Texas and the UK where grid penetration could go above Monckton’s “Pollock limit” with overbuilding. Instead of simply recognizing that a small modification to his conclusion was in order, Monckton then launched into a sad round of name-calling. For example, from a comment time-stamped January 11 at 2:40 PM, replying to Joe Born, Monckton calls Born “incompetent,” “idiotic,” “stupid,” a “nitwit,” says he used a “half-witted word salad,” and should “get his kindergarten mistress to read to him.”

Over the course of many, many comments replying to others, I think that Monckton ultimately concedes that his result only applies to a situation where overbuilding is not allowed. He calls such overbuilding “wasteful” and “foolish,” with which I would certainly agree. However, many governments are currently heading down that path. Germany is already there, and proceeding farther and farther by the day. The UK is already there as well, or at least very close. California and New York are not far behind. So I don’t think that we can just dismiss overbuilding cases as so foolish that no one would ever do it.

For any readers who are interested in a deep dive on this subject, I highly recommend Ken Gregory’s definitive August 2022 study titled “The Cost of Net Zero Electrification of the U.S.A.” Gregory explicitly considers paths to “net zero” in the face of the random intermittency of the renewables. Options considered by Gregory include batteries, overbuilding, and carbon capture and storage. For what it’s worth, Gregory finds that within certain ranges and at certain assumed prices, overbuilding is a superior alternative to batteries for increasing grid penetration of renewables — which is not saying a lot, but is saying that overbuilding, while it may be insane, is less insane than other options that seemingly everyone is talking about as if they make sense.

I should say that I have reviewed Mr. Gregory’s study extensively, and I have not found a flaw. That does not mean that there aren’t any. The same goes for my own simple arithmetic above in this post, which could contain flaws as well. If any reader discovers any such, I encourage you to point them out, and I hope that I will accept the criticism with a good spirit, and make any corrections that are appropriate.

Meanwhile, I have long followed Lord Monckton’s work, and respected much of it, and I am saddened to see him go somewhat over the edge on this one. To the extent that my comments here may appear critical, they are offered in the spirit of trying to get the right answer, and hopefully of friendship and cooperation. Unfortunately, I don’t think that the “Pollock limit” is the “final nail in the coffin” that Lord Monckton was hoping for. But it is one more piece of the demonstration that trying to make intermittent renewables the main basis of electricity generation will never be workable or affordable. As to the coffin, it’s going to take a few more nails.