Storage Situation is Worse than Mearns and Rogers Calculated

The following graphs were prepared using data from the California Independent System Operator with one hour resolution from 1 January 2011 until 30 November 2020, and five minute resolution thereafter. They show the net energy content (or deficit) that would have been in storage, assuming all supply came from renewable sources. At first, the analyses assume a system with unlimited but empty storage capacity at the beginning of the study period. Analyses are repeated with bounded storage capacity.

In early sections, quantities in storage were calculated by assuming average renewable capacity is equal to average demand. In later sections, the effect of average renewable supply being larger than average demand is analyzed. Mearns and Rogers compared instantaneous renewable supply to average renewable output. The method of calculation used here is explained below.

Daily average for solar and wind

The top left graph shows that solar + wind output decreases at the time when demand increases — people come home from work, turn on the air conditioner, turn on the television, and cook dinner.

The bottom left graph shows the average daily trend for solar and wind output, as a fraction of average total demand during the period of analysis.

The top right graph shows what fraction of total demand would be satisfied by solar and wind, if they were the only sources and their average output were magnified to equal average demand.

The bottom right graph shows the daily variation of the amount of energy that would be in storage if solar and wind were the only sources. The vertical axis is watt hours in storage per watt of average solar + wind capacity. This rather rosy average-day picture is the basis for claims that only small amounts of storage are necessary. But look carefully and you'll notice that the average daily deficit is four watt hours per watt, while the average surplus is three watt hours per watt: Storage is being continuously depleted.

Energy in storage 2020-2022

When a time range longer than one day is considered, it is clear that the daily average is not an adequate description.

Some days are better than average, and some are worse. It is necessary to consider the cumulative effect of good and bad days, especially the cumulative effect of consecutive good and bad days.

The units of the vertical axis are watt hours in storage per watt of average demand, compared to the amount in storage on 1 April 2020.

The "Weighted Increase" (purple) line is computed by magnifying renewables in proportion to their average production change rates during the period of analysis:

Unnormalized Weights
Geothermal Biomass Biofuel Small Hydro Wind Solar PV Solar Thermal Big Hydro
-0.04% -0.21% -0.95% -10.12% +0.19% -13.29% +1.23% +0.94%

The storage capacity necessary to avoid blackouts is the difference between the maximum surplus and the deepest deficit. The maximum surplus was 276 watt hours per watt on 9 August 2020. The depest deficit was 1446 watt hours per watt on 6 March 2022. To avoid blackouts, a storage system would need to have a capacity of 1,722 watt hours per watt of average demand, to avoid dumping power when storage is charged to full capacity, and to have been precharged to 1446 watt hours per watt of average demand on 1 April 2020 to avoid outages on 6 March 2022. The effect of precharging would be to shift the graphs upward by 1446 watt hours, and the "weighted increase" line would nowhere have been negative.

With unlimited storage capacity, not precharged, when the amount was negative and its trend was negative, i.e., 59.3% of the time, there would have been outages (see How the Graphs are Computed below).

The "Equal Increase" (green) line increases all renewables by the same factor. It is unlikely that hydro can grow much. Environmentalists want to remove dams, and they complain that fracking for geothermal causes earthquakes.

How the graphs were computed

To compute the amount of energy accumulated into (or discharged from) storage at any particular instant, using historical data, compute the difference δ(t) between instantaneous power production in watts, per watt of average demand, and instantaneous demand in watts, per watt of average demand.

The amount of energy in storage at time t since the beginning of the analysis, in watt hours per watt of average demand, is then the integral of that instantaneous surplus (or deficit):

$S\left(t\right)=\underset{0}{\overset{t}{\int }}\delta \left(t\right)d\tau \approx \sum _{i=1}^{N}\delta \left(t\right)$n Δ τn

where δ(t) has units of watts of surplus (or deficit) per watt of average demand, N is the number of measurement instants, Δτn has units of hours, and S(t) has units of watt hours per watt of average demand.

Rectangular quadrature is justified by the fine resolution of measurements (one hour from 1 January 2011 until 30 November 2020, and five minutes thereafter).

To use historical data to compute what δ(t) would have been if all sources were renewable sources, it is necessary to increase measured renewables' average production to match average demand. Let $\stackrel{}{R}$ be current average renewables' production, and $M\stackrel{}{W}$ be the additional average renewables' production needed to match average total demand $\stackrel{}{T}\left(t\right),$ where $\stackrel{}{W}$ is a weighted average of renewables' production (proportions might be different from current proportions), and M is a magnification factor. Then

$\stackrel{}{R}+M\stackrel{}{W}=\stackrel{}{T}$, or $M=\frac{\stackrel{}{T}-\stackrel{}{R}}{\stackrel{}{W}}.$

To compute the relationship of S(t) to average total demand, that is, how much storage capacity is needed per watt of average demand, we need

$\delta \left(t\right) =\frac{R\left(t\right) +\mathrm{G M}W\left(t\right)}{\stackrel{}{T}}-\frac{T\left(t\right)}{\stackrel{}{T}}=\frac{R\left(t\right)}{\stackrel{}{T}\left(t\right)}+G\frac{W\left(t\right)}{\stackrel{}{W}\left(t\right)}\left(\left(,1,-,\frac{\stackrel{}{R}\left(t\right)}{\stackrel{}{T}\left(t\right)},\right)\right)-\frac{T\left(t\right)}{\stackrel{}{T}}$

where G is a general growth factor that allows to increase the weighted average of renewables' production above average demand,

$R\left(t\right)=\sum _{i=1}^{N}\mathrm{Ri}\left(t\right),$ $W\left(t\right)=\sum _{i=1}^{N}\mathrm{gi}\mathrm{Ri}\left(t\right),$ and $\sum _{i=1}^{N}\mathrm{gi}= 1,$ If all gi were equal, this method would assume that all renewable sources can be magnified by the same amount $\left(\stackrel{}{T}-\stackrel{}{R}\right)/\stackrel{}{W}$ so as to increase their total average output to total average demand. This is not going to happen. For example, environmentalists want to remove dams, not build more of them. In the initial analyses we assume G = 1. Later, we examine the effect of larger G.

Because neither average demand nor average renewables' production are constant, compute "instantaneous" average demand and production using least-squares fits to straight lines, $\stackrel{}{R}i\left(t\right)=\mathrm{mi}t+\mathrm{bi}$ and $\stackrel{}{T}\left(t\right)=\mathrm{m0}t+\mathrm{b0}.$ This assumes linear change in the averages.

How to read the graphs

Where S(t) is increasing, δ(t) > 0, more electricity was produced than demand, and energy would be flowing into storage. Where S(t) is decreasing, δ(t) < 0, less electricity was produced than demand, and energy would be withdrawn from storage. Where S(t) > 0, renewable sources plus stored electricity produced sufficient power to satisfy demand. Where S(t) < 0, renewable sources plus stored electricity did not produce sufficient power to satisfy demand, and blackouts would occur where S(t) is decreasing, for example, between November and March. This shows the necessity for non-renewable sources — coal, gas, and nuclear — or significant storage, in California's electricity systems.

Observe that in mid 2020, total energy that would be in storage as a result of all renewables being increased equally, and renewables having produced more than demand, was about 400 watt hours per average watt of capacity. When the amount in storage is negative, for example between November 2020 and June 2021, any time that demand exceeds supply, i.e., δ(t) < 0, there would be blackouts.

If an all-renewable generating system had been in place on 1 April 2020, with a storage system having capacity less than about 1200 watt hours per watt of average demand, and had not been precharged to 800 watt hours per watt of average demand, there would have been prolonged blackouts throughout 2022.

Including data from earlier dates

Solar thermal output grew more between 2015 and 2020 than between 2020 and 2022, so different weights were used (hydro decrease was actually -6.8%, but that would have made the situation look very much more dire):

Unnormalized Weights
Geothermal Biomass Biofuel Small Hydro Wind Solar PV Solar Thermal Big Hydro
-0.05 0.12 0.05 -0.05 1.33 4.82 12.5 -0.05

Between 1 April 2015 and 31 December 2022, the greatest surplus was 1375 watt hours per watt of average demand on 25 July 2022, and the deepest deficit was 1822 watt hours per watt of average demand on 28 February 2017. To avoid blackouts, a capacity of 1375 + 1822 = 3197 watt hours per watt of average demand would be necessary, precharged to 1822 watt hours on 1 April 2015.

The recent situation looks better for the "weighted increase" line in this graph than in the previous graph because solar PV is magnified by 4.82 instead of being decreased by 13.29. The magnification factors were computed from generation growth, which depends upon climate variation, not from capacity growth, because the California Department of Energy does not provide projections for capacity growth, neither in toto nor by generation technology. They only forecast total demand. Average demand is assumed to grow linearly. The reference for average demand is computed using a least-squares fit to a straight line, which is slightly different for the two periods. For 4 April 2020 until 12 January 2023, the line is 24,591 + 0.169 t MWe, where t is hours since midnight 4 April 2020. For 4 April 2015 until 12 January 2023, the line is 25,778 + 0.00778 t MWe, where t is hours since midnight 4 April 2015.

The May 2020 price for Tesla PowerWall 2 was \$0.543 per watt hour of capacity, including associated electronics but not including installation. Individual installation costs are \$0.142 to \$0.214 per watt hour of capacity. Industrial scale systems might get price breaks.

Activists insist that an all-electric United States energy economy would have average demand of about 1.7 TWe. Assume California average generating conditions from 2015 through 2022 apply to the entire nation, and therefore 3197 watt hours of storage per watt of average demand is adequate (this is optimistic). The total cost for Tesla PowerWall 2 storage units, not including installation, with $\mathrm{3197}×\mathrm{1.7}×{\mathrm{10}}^{\mathrm{12}}=\mathrm{5.43}×{\mathrm{10}}^{\mathrm{15}}$ watt hours' capacity would be $\mathrm{5.43}×{\mathrm{10}}^{\mathrm{15}}×\mathrm{0.543}=\mathrm{2.95}$ quadrillion, or about 148 times total US 2018 GDP (about \$20 trillion). Assuming batteries last ten years (the Tesla warranty period), the cost per year would be 14.8 times total US 2018 GDP. The cost for each of America's 128 million households would be about \$2,306,000 per month. This analysis assumes 100% battery charge and discharge efficiency. They're closer to 90% (81% round-trip), so the necessary capacity and cost would be about 25% more.

Elon Musk would have more money than God.

Effect of increasing average capacity above average demand

Data were analyzed again with average renewables' capacity increased to G = 1.25 times average demand, with the same relative output magnifications as in the previous graph, and a 100 Wh/W storage capacity. The "flat line" bounding the maximum storage amount means that excess generation would be dumped. Solar thermal and wind output can be adjusted somewhat but solar PV output cannot be adjusted if the panels have fixed mountings. Articulated mountings would be very expensive.

Early large deficits cannot be explained by lesser penetration of the electricity system by renewable sources, because the data were analyzed under the assumption that renewable sources were the only sources, and demand is independent of source. There were better conditions for generation from renewable sources in later years.

If average renewables' generating capacity were to have been increased to G = 3 times average demand, and 12 hours' storage were provided, as is claimed to be sufficient by many environmentalists, power would have been available 97.5% of the time, i.e., blackouts 2.5% of the time. The industry definition for "firm power" is 99.97% availability, or about two hours and forty minutes per year without power. 2.32 gigawatt hours of output — 134% of total demand or 42% of average capacity — would have been dumped.

The cost for only twelve hours' storage, for an all-electric 1.7 TWe American energy economy, would be \$11 trillion, or about \$1.1 trillion per year. The cost per American household would be about \$721 per month (for batteries alone). And electricity would still be available only 97.5% of the time.

Renewables provided 36.6% of California electricity between 2020 April 1 and 2022 December 15. Electricity satisfies about one third of total California energy demand. To provide all California energy from renewable electricity sources whose average generating capacity is three times average demand would require a capacity increase of 2460% above the capacity to satisfy all current California electricity demand. Increasing hydro at all, or increasing biogas, biomass and geothermal by 2460%, is unlikely. With solar and wind alone, blackouts are frequent.

Problem with Increasing Capacity

The problem with renewable energy in general, and increasing capacity in particular, is materials. Professor Simon Michaux at Geologian Tutkimuskeskus — Geological Research Center or Geological Survey of Finland — has quantified the problem. For copper alone, if production were to continue at the 2019 rate, 189 years would be required to build the "technology units" demanded by the IEA. The amount required is almost six times the total amount that humans have so far extracted from the Earth. If all known reserves were completely used, 19% of the units could be built. See the main article for details.

Conclusion

This discussion assumes that the period analyzed includes the deepest deficit that will ever occur — which is, of course, false. When Mount Tambora on the island of Sumbawa in Indonesia erupts again and produces another "year without a summer" such as in 1816 — and it definitely will, the only question is when — there will be no times when δ(t) > 0. The trend of storage content will be everywhere downward. The deepest deficit will be far deeper than any shown here. No physically feasible or economically viable amount of storage could suffice. Renewable generation capacity and storage capacity could not be increased sufficiently rapidly. There would be energy available for only a small fraction of demand. Politicians' homes, and (maybe) hospitals, would have first priority. Civilization would collapse.

References

Typos? Mistakes? Quibble with the analysis? Want the software and data I used?

van dot snyder at sbcglobal dot net.