The Physics-Economics of Solar Panels

Whenever there’s more heat than light on an issue, you can bet there’s a gap in public understanding.

Take immigration, for example. The GOP’s business wing relies on illegal immigrants’ cheap labor to make money. Its political wing fosters fear and loathing of illegal immigrants to get votes. That schizophrenia is why we have had no immigration reform since 1986. The stalemate is likely to continue, as the GOP’s backers continue to seek both money and votes, without seeming to notice the contradiction.

Solar energy is similar. What drives the “debate” is public misunderstanding of what solar panels are and how they work.

In several earlier essays (1, 2 and 3), I tried to shed light on this subject by doing the engineering-accounting-economic calculations the right way. But as my wife points out, my writing is sometimes verbose and complex. This essay is the Cliff Notes version for our Twitter age.

Solar panels are a unique way to generate electricity. Our species has never used anything remotely like them before. So the attitude of many people toward them (unfortunately including more than a few engineers) is a bit like savages who just discovered fire contemplating a gas turbine. “How,” they might say, “can we do anything useful with something that just goes round and round in circles?”

Solar energy’s detractors are a lot like those savages. They don’t understand what solar panels are and how they work. But they do understand that their output is “intermittent,” i.e., it decreases on cloudy days and stops at night. So since they’re used to coal and natural-gas generators, which aren’t intermittent, they just can’t get their minds around the other unique features of solar panels, which are decisive advantages.

Solar panels use no fuel and create no effluent while operating. Most people know that. They know it, but they don’t really consider the importance of having no greenhouse-gas effluent whatsoever, and therefore no acceleration of global warming, or the absence of acid rain, mercury pollution of oceans and lakes, and asthma-causing particulate smog—things that coal power plants produce every day in great profusion.

Yet this is just the beginning of solar panels’ advantages. What many people don’t know is that they are not mechanical devices. They don’t have moving parts, so they don’t wear out or break down. They are solid-state devices whose entire “works” are hidden inside their molecular structure. They generate power at the subatomic level, through a solid-state-physics process called the “photoelectric effect” that Albert Einstein got the Nobel Prize for explaining some 108 years ago.

So solar panels don’t move, turn, revolve, whiz or buzz. They just sit there, reliably (and quietly) turning the Sun’s radiant energy into useful electricity, day after day, without fuss, rotary motion, smoke or noise.

But nothing is perfect. Instead of breaking down or wearing out, solar panels slowly and predictably degrade. Their power output decreases according to a linear formula, over very long periods of time. This is not economics, politics, or ideology. This is physics and engineering.

Don’t take my word for it. Take the manufacturers’. The industry-standard warranty now uses this formula. An example appears on LG’s online solar-panel spec sheets [path: Technical Specifications > Certifications and Warranty].

The formula is high-school algebra. Power output decreases by 3% the first year, and by 0.7% per year thereafter. That means the panel continues to produce useful power for nearly 140 years. (Manufacturers only guarantee this formula for the first 25 years, but lawyers and accountants can’t change physics.)

If you replace the panel when its power output drops by 67%—to 33% of initial power—it will have run for about a century. By then it will have generated the equivalent of 66 years of full-power output.

You can use these basic facts to calculate the generating cost of solar-panel energy per kilowatt-hour. That cost is quite different from the buying price of the panels, per Watt of nominal power-output capacity (an industry-standard figure explained below).

To do the calculation correctly, all you need is three other numbers, as follows:

1. The Manufacturer's Per-Watt Panel Price M. The manufacturer’s wholesale price M for solar panels, in dollars per Watt of (nominal) power output, or capacity. This is an industry-standard parameter. Only a few years ago, it was over a dollar per Watt. Today it is close to 50 cents per Watt, with First Solar the industry price leader (using non-silicon technology).

2. The “Turnkey Factor” T. The ratio of the total cost of building a working solar array, including buying the panels, to the price of the panels (of which there are many, all the same). Analysts express this cost as a “turnkey factor” T. It’s the cost of the ready-to-run solar array, per Watt of nominal output capacity, divided by the per-Watt panel price.

3. The Solar Irradiance Factor. This number accounts for the facts that (1) the Sun does not shine at night, (2) the Sun shines less brightly on cloudy days, and (3) the Sun’s irradiance differs at different locations on the Earth’s surface, depending mostly on latitude. Solar-panel installers use calculated solar irradiance tables for the last element (3). But we can make a good approximation on all three elements by assuming that, in sunny climes like our American Southwest, and at that latitude, the sun shines eight hours per day on about two out of three days.

With these numbers, we can calculate the cost of using solar panels to generate electricity. (See also, 1 and 2.) In these calculations, the panel’s effective working lifetime is a key parameter, because solar panels have no fuel or fuel transportation cost, no pollution-remediation cost, and very low maintenance (occasionally cleaning dust, snow or leaves off the panels). Therefore, by far the dominant determinant of the cost of solar-panel power is the amortized cost of building the solar power plant.

The results are 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)
400	3.2
300	2.4
200	1.6
150	1.2
100	0.8

Three conclusions leap out from this table. First, when properly calculated, the cost of electric energy produced by solar panels is very low.

Industry experts now tout MTs of $4 per Watt and $2 for large-scale commercial arrays. With the lower number, the long-term generating cost of solar energy is already at 1.6 cents a kilowatt-hour. That is less than one-seventh of the national-average residential retail price of electric energy for 2010, namely, 11.54 cents per kilowatt-hour.

No other means of generating electric energy now known can match, let alone beat, that cost. For example, I have calculated the generating cost of the gigantic Four Corners coal power plant, from which I (and most of Northern New Mexico) get 87% of my electricity, at about four cents per kilowatt-hour, 2.5 times as high. This plant is the biggest, and presumably the most efficient, coal-fired power plant in our nation.

The second thing to note in the table is that energy cost depends strongly on the parameters M and T. These are the numbers to watch as the industry progresses. M has dropped from over a dollar to a bit over fifty cents in just the last three years. T factors range from 11-12—the apparent turnkey-to-panel cost ratio reported for a recently powered-up solar array in Las Vegas—to solar industry reports of 2.

Third, the calculation underlying the table depends entirely on the observed rate of slow decline in solar panel output. If advances in semiconductor technology decrease this rate of decline, the generated cost of solar-panel energy will decrease accordingly. This industry is just entering its exponential growth phase, and there is every reason to expect such advances.

An Example in Las Vegas

The comment pages of online media were all agog recently with news of a new solar array in Las Vegas. It’s a medium-scale three-megawatt array designed to help power a city sewage-treatment plant. According to the local news report, “The $20 million project will generate 6 million kilowatt hours of electricity annually.”

With those simple facts—cost, nominal power output, and annual energy output—we can apply the foregoing analysis and test our calculations.

The news report doesn’t include the panel price M, but we can calculate the product MT from these figures. The array’s nominal power output is three megawatts, and its turnkey price was $20,000,000. So the turned-on-plant price per nominal Watt capacity is 20 million dollars over 3 million Watts, or about $6.67 per Watt. That’s the product MT in our analysis above.

The first thing to note is that Las Vegas didn’t get a very good deal. The state of the art is an M near fifty cents per Watt, and a T for commercial-scale arrays near 2, or at least below 3. So the product MT should be no higher than $1.50 per Watt, over four times lower than Las Vegas’.

Their are many possible explanations for this discrepancy, some innocent, some not so much. City projects involve a lot of red tape and delay; the city may have contracted for this array several years ago, when industry cost parameters were higher. Since the project was only medium-sized, contractors may have required the city to pay closer to home retail prices. Since Las Vegas is known for gambling and entertainment, not technology innovation, it may have had to import talent at additional expense. And this being Las Vegas, we cannot rule out sweetheart deals with contractors, or even outright corruption.

For whatever reason, Las Vegas paid over four times more than today’s state of the art can do. But did the city get a bad deal in the long term? Let’s analyze.

The array will generate 6 million kilowatt-hours of energy every year. At the 2010 national-average residential retail electricity price [Click on “All Tables,” then download Table 4 into Excel] of 11.54 cents per kilowatt-hour, the 6 gigawatt-hours represent a $692,400 annual savings to the city. On the $20 million investment, that’s a 3.46% rate of return.

If the city qualified for lower commercial electricity prices, 10.19 cents, that annual savings would be $611,400, for a 3.06% rate of return. (Such a small array would not likely qualify for the much lower industrial electricity pricing. Anyway, industrial power is interruptible—not a good thing for a sewage treatment plant.)

A rate of return around 3% is hardly stellar, although not bad in today’s low-interest-rate environment.

But the immediate rate of return is not the point. That’s short-term thinking. That’s the take-the-money-and-run philosophy that got us in such trouble in the Crash of 2008, from which we are still recovering.

Instead, let’s think long term. At a 3% rate of return, the city would have paid for its investment in a little over 33 years. After that—on our assumption of replacing the panels at 33 percent output—the city will have over 67 years of virtually free power, albeit with a steadily declining output.

Should we forego these impressive cost and environmental advantages because the power output is intermittent and declining? Or should we figure out how to use this cheap, clean power to our economic and health advantage?

Before you answer, consider the common home mortgage. Many families work and invest for thirty years to pay off their thirty-year mortgages, so that their children can inherit their home free and clear.

That’s just what Las Vegas will do. After a payback period roughly equivalent to a thirty-year mortgage’s, it will have another three-and-one-half generations’ worth of electric power for free, with a very small maintenance charge, no pollution and no contribution to global warming or positive global-warming feedback.

That’s not quite as good as a home mortgage, whose payoff yields fee simple title forever. But it’s a hell of a lot better than high-maintenance, high-fuel cost and high-pollution alternatives, which provide no fuel-cost protection, let alone free power for over three generations.

So Las Vegas may have paid too much, but it did the right thing. Advances in technology may reduce the sewage plant’s power requirements as time goes on. Even if not, and even if the city has to add new panels periodically to maintain the solar array’s rated output, the replacement-panel costs will be lower, likely much lower, due to interim advances in the industry. Las Vegas made a wise long-term investment in its future, something our American culture once did routinely, but has trouble doing today.

Note: the time value of money

Business-savvy readers will note that the foregoing plant-payback analysis (except for the last paragraph) neglects the time value of money. If Las Vegas had to borrow the money to build the array, the interest on the loan would decrease the array’s net rate of return and prolong the payback period. The foregoing analysis assumes that Las Vegas has the money in the bank and is self-financing the array.

The are good reasons for this approach, some of which I’ve noted previously. But in this particular case, there is an even better reason. If Las Vegas had not built the array, it would have had to continue paying the power company the $611,400 per year it previously paid for power (at commercial rates) indefinitely. At very least, it would have to pay that sum for the 66 years of effective full-power operation of the array, for a total cost of $40.3 million, double the plant’s cost.

The present value of that payment stream would of course be less. But it’s hard to know how to discount the payment stream for coal power back to present value, especially over such a long period. If we use a convenient online annuity PV calculator over 66 years, we get a present value of $ 22.3 million at an interest rate of 2%, but only $17.5 million at 3% and about $10 million at 6%. So the present-value of the coal-power payments could be more or less than the cost of the solar array.

But who wants to gamble on interest rates (especially for cities) being closer to 6% than 2% over 66 years, while also gambling on the stability of coal-produced electricity rates? Gambling is how Las Vegas fleeces tourists, not how it runs itself. It knows the House always wins.

Insofar as this array is concerned, Las Vegas has insured its future with an investment now. Even if it had to borrow money for the array, a fixed-interest-rate loan makes its costs known; and its power costs will be zero (except for maintenance) for a long time after the loan is paid off. If solar panels have to be replaced, their replacement prices will likely go down over time, with improvements in technology and manufacturing, while the price of coal-produced electricity will likely go up, due to both increasing fuel scarcity and increased shifting of pollution costs onto polluters.

The clincher reason for ignoring the cost of money is that interest rates and commodity prices usually rise together. So as interest rates triple from 2% to 6%, taking the accumulated cost of coal power (after discounting to present value) down to half the price of the solar array, the cost of coal (and therefore coal-produced power) is likely to rise accordingly. So the $10 million cost of coal power (after discounting to present value), will become more like $30 million, due to commodity price inflation. The $20 million solar array is still the better buy.

Finally, just for fun, let’s do the numbers for the same facility, assuming a current state-of-the-art MT product, for commercial-scale arrays, of $2 per Watt. Then the 3 megawatt plant would have cost $6 million to build, not $20 million.

In that case, the annual $611,400 savings in unused coal energy would provide a 10.2% annual rate of return. That’s a stellar return for any investor today, let alone a city!

If the city borrowed money to build the plant, even at 5% interest, those savings could pay off the loan in less than fourteen years. Thereafter, the city would enjoy over 86 years of free power (except for maintenance) before the array’s output dropped to one-third its original value. If we considered probable increases in coal-power pricing—whether due to commodity price inflation or costly pollution-remediation measures—the rate of return would be even higher, and the payback even faster.

Footnote: Some solar arrays have very sluggish motors that rotate the panels to face the sun as it appears to move across the sky during the day. But economics disfavors this unnecessary embellishment. Most solar arrays today use fixed panels, which are cheaper to build and maintain.

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