Diatribes of Jay

This is a blog of essays on public policy. It shuns ideology and applies facts, logic and math to economic, social and political problems. It has a subject-matter index, a list of recent posts, and permalinks at the ends of posts. Comments are moderated and may take time to appear. Note: Profile updated 4/7/12

16 February 2013

Real Solar Energy Cost Parameters


Introduction
Power per land area: one nominal megawatt for three football fields
Linear warranties and effective plant lifetimes
Real cost figures
Conclusion
Note on the time value of money
Note on Maintenance
Encouraging Update (3/25/13)

Introduction

A year ago, I published a post analyzing coal promoters’ claim that wind and solar power are uneconomic. That post had three salient conclusions.

First, no one yet knows what the real cost of wind and solar energy will be. Because they have no fuel, refining, fuel-transportation or pollution-remediation costs, their energy cost per kilowatt-hour depends only on two variables: maintenance expense and amortized capital cost of plant. Of these two, the latter is by far the dominant factor.

We don’t know what amortized capital cost will be because it depends on the plants’ actual working lifetimes, which only experience can tell. No one knows those lifetimes for sure today because (1) no plants have yet been worn out and (2) plant technologies are rapidly moving targets.

The post’s last two conclusions were related. Close analysis of quantitative trends (but without real, accurate numbers) strongly suggests that wind and solar power will beat coal on the cost of energy by considerable margins. That suggestion, in turn, explains why the coal barons are trying to kill wind and solar energy with a PR campaign, rather than by competing in the marketplace.

While nothing I have read or thought since contradicts any of these conclusions, that post was mostly theory. Now, as a result of my own efforts to have a solar array built for my home, I have some real, current commercial numbers.

The calculations in this post are based on LG Solar’s commercially available panels, Part Number LG255S1C-G3. Their technical specifications are available here. This post is intended neither as a review nor an endorsement of these particular solar panels. Their specs are just an example of what a major multinational commercial manufacturer can produce and is willing to warrant today. The point of this post is how these current commercial specs corroborate solar energy’s cost superiority.

Power per land area: one nominal megawatt for three football fields

According to LG’s online spec sheet, the surface area of each panel is 1640 mm x 1000 mm, or 1.64 square meters. The panel produces a nominal power of 255 Watts. (The actual power depends on insolation, which in turn depends on the solar declination and hence on latitude at the array location, and of course on weather.)

So the panel area for a nominal one-megawatt array would be 1.64 x 1,000,000/255 square meters, or 6,431 square meters. That’s an area about 80.2 meters on a side, in US terms 6,431/4,047 = 1.6 acres, or less than three American football fields—hardly a huge area of land. (The actual area might be somewhat larger, because the panels must be tilted to account for solar declination, and each row of panels must stay out of neighboring rows’ shadows. Alternatively, rows can be staggered in height, like seats in a movie theater, without increasing horizontal land area.)

My own solar array will produce 6.24 nominal kilowatts, more than enough (on average) to run a well-equipped house and charge an electric car. If we round to 6 nominal kilowatts per house, our nominal one-megawatt array could serve about 160 homes, with a solar array occupying a land area of less than three football fields. That’s about 100 homes per acre of panel land.

These facts have two big community-planning implications. First, in suburban areas, each new subdivision could, on average, power itself. For example, in a new subdivision of one-half acre lots, reserving one lot for a solar array could power about fifty homes. Second, in sunny rural areas, especially in the Southwest, land is cheap and unpopulated, with nothing but cattle to complain about views. In those areas, a one-gigawatt plant would occupy 1,600 acres.

The first point is the key to rapid adoption of solar energy. Solar arrays are infinitely scalable. You can build one for a single home, a single block of homes, a subdivision, an industrial plant, a whole neighborhood, or a whole city.

Big centralized power plants are no longer necessary as long as either (1) large, regional smart grids tie big areas together and average out local fluctuations in sun and wind, and/or (2) conventional power plants are available to smooth out fluctuations and avoid intermittency. The best plants for (2) today are natural-gas plants, which are scalable, rapidly adjustable in output, and low in carbon output for fossil-fueled power sources.

Linear warranties and effective plant lifetimes

The “Certifications and Warranty” portion of the LG technical specifications reveals some important facts about solar panels. They are solid-state devices with no moving parts. They are not like machinery, whose parts break down at random and which, after too many parts break down, have to be replaced.

Unless destroyed by accident, tornadoes, hurricanes, lightning or hail, solar panels continue producing power for long periods of time. But because of internal aging and other poorly understood phenomena (probably including microscopic manufacturing imperfections), the power they produce slowly decreases as they age.

Accordingly, LG offers a so-called “linear” warranty on the power output of its panels, which reads verbatim as follows:

“Linear warranty

Output warranty of Pmax (measurement Tolerance ± 3%)
1) 1st year: 97%, 2) After 2nd year: 0.7% annual degradation, 3) 80.2% for 25 years”

Thus LG warrants its panels’ power output to 80% of initial power for 25 years. But, unlike mechanical devies, solar arrays won’t just stop working some time shortly after the warranty expires. They’ll keep producing power, but at lower and slowly declining rates, for a long, long time.

How long? No one knows, because no one has yet used them at commercial scale for even 25 years.

It’s impossible to overemphasize the importance of these points. Solar photovoltaic arrays use a new power technology, with characteristics that differ from those of all previously used power sources.

One characteristic—intermittency—has been much discussed in the popular press. The sun doesn’t shine at night, and solar panels produce less power when clouds decrease solar irradiance reaching the ground. This makes them harder to use than conventional power sources, but by no means impossible. There are many ways to handle intermittency; as one German put it, intermittency is not a “problem.” Accommodating it is just “a task.”

In any event, another characteristic of solar arrays is decidedly positive, as compared to all mechanical means of generating electricity. Unless physically destroyed, photovoltaic solar panels don’t break down. Like old soldiers, they just slowly fade away.

LG’s so-called “linear warranty,” which reflects this fact, will undoubtedly become an industry standard. It tells how solar panels actually perform. With greater experience in plant longevity, the present 25-year limit on commercial guarantees will no doubt rise. And the 0.7% annual power degradation rate will likely come down with advances in semiconductor and fabrication-process technology.

Let’s analyze how all this affects energy pricing. First, let’s look at effective plant lifetimes. If a panel, on average, loses 3% of its power the first year and undergoes a 0.7% annual power degradation thereafter, its power output will not fade away entirely until 1 + 97%/0.7% = 139.6 years.

Thus, century-long photovoltaic power-plant lifetimes are consistent with LG’s commercial warranties today (although not, of course, guaranteed.) If LG’s linear formula is accurate in the long term, its panels will still produce over a third of their initial power a century after manufacture. By that time, improvements in conservation and efficiency might allow the same power plant to serve the same customers, with proportionally reduced total power needs.

But by far the most vital point is cost. The cost of solar energy, like the cost of wind energy, depends primarily on how long the initial, fixed capital investment in plant continues to produce useful power. As far as the cost of energy is concerned, the annual degradation is irrelevant. What matters is the total energy produced over the plant’s useful life.

Let’s suppose, just for discussion, that the user makes an arbitrary decision to discard the plant (or the panels) when their output falls to 34.7% of nominal. That will occur in about a century. The total energy that the plant produces over this useful lifetime is just the area under power-output curve, which we can calculate using integral calculus, simple geometry, or (in this case) simple averaging.

Over the course of the first year, the power output drops from 100% to 97%, for an average of 98.5%. Over the course of the plant’s remaining useful life, the power output drops from 97% to 34.7%, for an average of 65.85%. So the “effective” working life of the panel, at 100% of nominal power, is 0.985 + 0.6585 x 99 = 66.17 years. That is, over its useful lifetime, until its “retirement” at 34.7% of nominal power, the plant would produce as much energy as if it ran at full power for about 66 years.

Real cost figures

With that figure, we can now make some pretty accurate estimates of what photovoltaic solar energy really costs.

We start with the routinely achieved industry standard of one dollar per Watt as the manufacturing cost of solar cells. Then we multiply by the the ratio of the cost (per Watt capacity) of a deliverable, working plant, with infrastructure, wires, control gear, and grid attachment, to the cost of manufacturing a one-Watt cell.

After the plant lifetime, this multiplier is the key numerical factor for analyzing solar photovoltaic power pricing. Since it reflects the cost of building a “turnkey” plant, I call it the “turnkey factor.”

My own commercial array vendor offers a turnkey factor a bit south of six. I have read in the business press that commercial builders of large-scale commercial solar arrays are now achieving turnkey factors as low as three.

With the turnkey factor and the plant lifetime now understood and estimated, we can estimate the real cost of solar photovoltaic energy, not from theory, but from commercial fact. We need only one additional figure: the number of hours of useful power that solar panels generate per year.

For that parameter, we assume that the sun shines eight hours per day, that two days out of three (on the average) have useful sun, and that a year has 365 days (ignoring leap years). With those assumptions, each year provides

8 x 2/3 x 365 hours = 1,947 hours


of useful energy output. Since these numbers are approximate, and to make calculation easier, let’s just call it 2,000 hours.

With these preliminaries, the actual cost of solar photovoltaic energy, in cents per kilowatt-hour, is as follows:

C = MT / 2L


where M is the cost per Watt capacity of manufacturing the solar panel (in cents, not dollars), T is the turnkey factor discussed above, and L is the effective lifetime of the solar panel, in years, at 100% of nominal-power equivalent. (We reduce the solar-hours factor from 2,000 to 2 because we are calculating energy in kilowatt-hours, not Watt-hours.)

At present we are assuming a turnkey factor T of 3 for commercial installations and 6 for home installations. We have just estimated an effective 100%-power lifetime of the LG solar panels as 66 years. The following table shows the cost of energy from LG’s solar panels using this formula and the estimates we have made:

Cost of Solar Photovoltaic Energy, Excluding Plant Maintenance

Mfg Cost M
of 1 W cell
Turnkey Factor TCost of Energy C
(cents)(dimensionless)(cents per kWh)
10064.8
10032.4
5062.4
5031.2

The results in this table are both surprising and encouraging. Over the last several years, the retail price of electric energy (derived 87% from coal) I have paid has been between 11 and 12 cents per kilowatt-hour. Let’s just call it 11.5 cents.

According to our Energy Information Administration, the national-average ratio of electricity-production expenses to all expenses of electric utilities for 2011 was 122,520 / 247,118 = 45%. So the total proportion of non-production costs was 55%. If we add a 10% profit margin for a regulated utility, the total proportion of non-production costs plus profit was 65%, or a production cost of 35% of selling price. Applied to the prices I actually pay, namely 11.5 cents per kilowatt-hour, that implies an internal energy cost to my power company of 0.35 x 11.5 cents, or 4 cents per kilowatt-hour.

Every figure in our table but the first is lower than that. Accordingly, unless plant maintenance (which the table neglects) adds 10% or more to the price of solar photovoltaic energy, the cost of solar energy from all but small home solar arrays like mine beats that of the biggest (and presumably most efficient) coal-fired power plant in the nation—the Four Corners power plant from which I get 87% of my power.

Even my little home solar array comes within 20% of the big coal plant’s internal cost of power. When you add in all the distribution costs, overhead and profit, I’ll be making a killing producing my own electrical power.

Conclusion

The numbers in this post are not theoretical. They come from commercial spec sheets of a major solar-panel producer. Although I am not at liberty to disclose a private bid, I have a real proposal from a real business to install a home solar array at an M of 100 cents and a T of 6.

There are only two aspects of this analysis that are not hard numbers. The first is the plant lifetime L, which is a reasonable deduction from LG panel warranties and the properties of solar cells. The second is the assumption that maintenance of solar arrays, which have no moving parts, will not add more than about 10% to the price of energy. (For further discussion of this second assumption, click here.)

The table above suggests that solar panels are capable of delivering energy near a penny a kilowatt hour, under only two conditions. First, the industry must get solar-cell manufacturing costs down to 50 cents per Watt capacity. Some participants, such as First Solar, are close to that goal already. Second, commercial installers must get the Turnkey Factor for building large-scale commercial solar arrays down to three, if it is not already there. There is also a third implicit assumption: that any new cell technology that decreases the manufacturing cost per Watt capacity won’t also decrease the cell’s effective working lifetime L.

Note on the time value of money

Business readers may object that the foregoing analysis ignores the time value of money, that is, interest on the capital investment in plant (if it is borrowed) or a lost opportunity to receive interest on the same money (if it is not).

There are three answers to this objection. First, this analysis estimates the cost to the producer of producing electrical energy. If the producer has the money, he can invest in a solar array and, over its lifetime, produce energy at a lower effective cost than with coal.

Over the long lifetime of a coal power plant, the costs of the coal, its transportation, smokestack scrubbers and other regulatory/pollution-remediation measures will go up with inflation, which will roughly track the cost of money. But there will be no comparable variable costs for solar energy. Both investments will require maintenance, but that cost will be lower for solar photovoltaic energy. The cost of money comes into play only if you compare generating and selling electricity with other ways of making money.

Second, the effective lifetimes of solar arrays (66 years, in our estimate) are far longer than the terms of commercial loans, especially for power-plant construction. It is not customary to consider the time value of money in calculating power producers’ profit after loans have been paid off. That’s why nuclear energy is considered so “cheap” today, because the plants have long ago been paid off, and many are obsolete. Just so, solar arrays will produce decades of “free” energy (to the producer, except for maintenance) long after any plant-construction loans have been paid off. (This “free” energy will come without the risks of meltdowns and radioactive releases posed by nuclear power plants.)

Third, as a conceptual and accounting matter, it is not clear that the time value of money is relevant. There is no way to accurately estimate, let alone calculate, the effect of changing interest rates and inflation over periods as long as fifty or more years. Any attempt to do so would likely produce garbage in hindsight. (Just think of someone in the high-interest, high-inflation late 1970s trying to imagine what interest rates might be today.) The revenue received for energy might vary with inflation and interest rates, but the cost of the plant is a sunk cost, best conceived as a single, fixed expense.

Footnote: This quotation is from Hans-Joseph Fell, the legal architect of Germany’s energiewende. See Osha Gray Davidson, Clean Break: The Story of Germany’s Energy Transformation and What Americans Can Learn from It, Kindle e-book (99 cents), Chapter 1 (Kindle at 10%).

Note on Maintenance

Analysis of existing data strongly suggests that solar panels satisfy the above post’s assumption of maintenance adding less than 10% to energy cost.

Our Energy Information Administration publishes online figures for electric utility expenses of various kinds. For 2011, maintenance's fraction of total production cost was as follows:

Maintenance/production cost = 15,772/122,520 = 12.8%


This figure is a nationwide average over all utilities and all sources of electric power. Yet as I explained last year, solar photovoltaic arrays are highly likely to incur the lowest maintenance expense of any means of producing electric power.

Solar panels and their inverters have no moving parts. They operate at ambient temperature, not the blast-furnace temperatures of coal or natural-gas burners, or even the high-pressure steam temperatures of turbines or reciprocating engines in coal, natural-gas or nuclear power plants. And of course they suffer only solar radiation, not the nuclear radiation that is rampant in the most critical parts of nuclear power plants, and that is almost as hard on materials as it is on living things.

Therefore solar photovoltaic arrays require only two foreseeable types of maintenance. First, someone has to inspect them periodically for damage to or destruction of solar panels (mostly after destructive weather events), to replace inoperable parts, and to see that they are still operating properly. Second, if snow, dust, leaves or other obstructions block solar radiation, someone has to clean them off.

That sort of maintenance is highly unlikely to cost more than the regularly scheduled inspections, diagnostic tests, routine part replacements, and occasional maintenance shutdowns of far more complicated plants with many moving parts. Coal plants, in particular, have another huge maintenance expense that solar plants don’t have: periodic replacement of expensive active elements in the stack “scrubbers” that try to remove from plant effluent (1) sulfur dioxide (the cause of acid rain), (2) mercury (which makes tuna dangerous for pregnant women and children to eat), and (3) particulates (which cause asthma and other respiratory diseases).

Based on this comparison, it is highly unlikely that maintenance cost for solar arrays would add more than 10% to their cost of energy, when the industry average is 12.8% And even if the cost for solar arrays hit that industry average, it wouldn’t change any conclusion in the post above.

Erratum: An earlier version of this post miscalculated my power company’s internal cost of producing a kilowatt-hour of electrical energy. The error, which affected only the last three paragraphs before the Conclusion, has been corrected.

Encouraging Update (3/25/13): A recent news article in Bloomberg.com suggests that commercial-scale solar arrays are already approaching long-term generation costs of a penny per kilowatt-hour. Based on data provided by a Sharp executive, it quotes turnkey costs of solar power (the product of M and T, or MT in the table above) of $4 per watt capacity for residential arrays, and $2 for large-scale commercial arrays.

If these figures are right, the calculated cost of solar energy for the life of the array, as figured above, is as follows:

Cost of Solar Photovoltaic Energy, Excluding Plant Maintenance

Turnkey Cost MT
of 1 W cell
Cost of Energy C
(cents)(cents per kWh)
4003.2
2001.6

permalink

1 Comments:

Post a Comment

Links to this post:

Create a Link

<< Home