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Power Density Primer: Understanding the Spatial Dimension of the Unfolding Transition to Renewable Electricity Generation (Part IV – New Renewables Electricity Generation)

By Vaclav Smil -- May 13, 2010
Editor’s note: This is Part IV of a five part series that provides an essential basis for the understanding of energy transitions and use. The previous posts in this series can be seen at:
Part I – Definitions
Part II – Coal- and Wood-Fired Electricity Generation
Part III – Natural Gas-Fired Electricity Generation


Photovoltaic Electricity Generation

Satellite measurements put the solar constant – radiation that reaches area perpendicular to the incoming rays at the top of the atmosphere (and that is actually not constant but varies with season and has negligible daily fluctuations) – at 1,366 W/m2. If there were no atmosphere and if the Earth absorbed all incoming radiation then the average flux at the planet’s surface would be 341.5 W/m2 (a quarter of the solar constant’s value, a sphere having four times the area of a circle with the same radius: 4?r2/?r2). But the atmosphere absorbs about 20% of the incoming radiation and the Earth’s albedo (fraction of radiation reflected to space by clouds and surfaces) is 30% and hence only 50% of the total flux reaches the surface prorating to about 170 W/m2 received at the Earth’s surface, and ranging from less than 100 W/m2 in cloudy northern latitudes to more than 230 W/m2 in sunny desert locations.

For an approximate calculation of electricity that could be generated on large scale by photovoltaic conversion it would suffice to multiply that rate by the average efficiency of modular cells. While the best research cells have efficiencies surpassing 30% (for multijunction concentrators) and about 15% for crystalline silicon and thin films, actual field efficiencies of PV cells that have been recently deployed in the largest commercial parks are around 10%, with the ranges of 6-7% for amorphous silicon and less than 4% for thin films. A realistic assumption of 10% efficiency yields 17 W/m2 as the first estimate of average global PV generation power density, with densities reaching barely 10 W/m2 in cloudy Atlantic Europe and 20-25 W/m2 in subtropical deserts.

PV panels are fixed in an optimal tilted south-facing position and hence receive more radiation than a unit of horizontal surface but the average power densities of solar parks are low. Additional land is needed for spacing the panels for servicing, for access roads, inverter and transformation facilities and for service structures, and only about 85% of a panel’s DC rating will be transmitted from the park to the grid as AC power. Olmedilla de Alarcón, the world’s largest solar park in Spain, has installed capacity of 60 MW of peak power (MWp) but its annual generation of 85 GWh (or 9.7 MW of electricity as an average annual rate) translates to capacity factor of just 16%. Portuguese Moura (46 MWp, 88 GWh or 10 MW of average annual generation) has the capacity factor of nearly 22% and the capacity factor for Germany’s largest solar park (Waldpolenz rated at 40 MWp) is only 11%. Power density of Olmedilla is only 9 W/m2, that of Moura almost 8 W/m2 while Waldpolenz rates just above 4 W/m2.

Olmedilla 85 GWh/year = 9.7 MW 9.7 MW/108 ha = 9 W/m2
Moura 88 GWh/year = 10 MW 10 MW/130 ha = 7.7 W/m2
Waldpolenz 40 GWh/year = 4.56 MW 4.56 MW/110 ha = 4.1 W/m2

The largest solar PV parks thus generate electricity with power densities that is roughly 5-15 times higher than for wood-fired plants but that is at best 1/10 and at worst 1/100 of the power densities of coal-fired electricity generation. Again, if only 10% of all electricity generated in the US in 2009 (395 TWh or about 45 GW) were to be produced by large PV plants, the area required (even with average power density of 8 W/m2) would be about 5,600 km2. No dramatic near-term improvements are expected either in the conversion efficiency of PV cells deployed on MW scale in large commercial solar parks or in the average capacity factors. But even if the efficiencies rose by as much as 50% within a decade this would elevate average power densities of optimally located commercial solar PV parks to no more than 15 W/m2.

Part IV Olmedilla

Olmedilla PV plant with 162,000 panels and 60 MWp generates electricity with average power density less than 9 W/m2 of its total area.

Concentrating Solar Electricity Generation

Concentrating solar power (CSP) projects use tracking parabolic mirrors in order to reflect and concentrate solar radiation on a central receiver placed in a high tower. This technique has several technical advantages compared to PV, above all: higher conversion efficiencies (thanks to a conventional steam-powered generation) and the possibility to augment the solar-heated steam by fuel combustion. Still, power densities of CPS are not all that different from PV generation.

Europe’s first commercial solar tower, PS (Planta Solar) 10, completed by Abengoa Solar in Sanlúcar la Mayor in 2007, is rated at 11 MWp. With annual generation of 24.3 GWh (87.5 TJ, 2.77 MW), its capacity factor is 25%. Its heliostats occupy 74,880 m2 (624 x 120 m2), and the entire site claims about 65ha; the facility’s power density is thus about 37 W/m2 factoring in the area taken up by the heliostats alone, and a bit more than 4 W/m2 if the entire area is considered. PS20 (completed in 2009) is nearly twice the size (20 MWp; 48.6 GWh or 175 TJ/year at average power of 5.55 MW and capacity factor of nearly 28%). Its mirrors occupy 150,600 m2 and hence the project’s heliostat power density is, at 36.85 W/m2, identical to that of PS10 but, with its entire site covering about 90 ha, its overall power density is higher at about 6 W/m2.

Part IV Abengoa

Abengoa’s two large CSP plants near Sanlúcar la Mayor (Sevilla), PS10 in operation and PS20, in the foreground, shown still under construction.

Bright Source Energy’s proposed Ivanpah CSP in San Bernardino, CA should have an eventual rating of 1.3 GWp and it is expected to generate 1.08 TWh (3.88 PJ) a year and deliver on the average 123.3 MW with a capacity factor of just 9.5%. Heliostat area should be 229.6 ha and the entire site claim is 1645 ha. This implies power densities of 53.75 W/m2 for the heliostats and 7.5 W/m2 for the entire site. Again, no stunning improvements of these rates are expected any time soon and hence it is safe to conclude that optimally located CSP plants will operate with power densities of 35-55 W/m2 of their large heliostat fields and with rates no higher than 10 W/m2 of their entire site area.

Wind-Powered Electricity Generation

Wind turbines have fairly high power densities when the rate measures the flux of wind’s kinetic energy moving through the working surface (the area swept by blades) of this now so popular energy converter. In the windiest, mid-continental regions of America this power density is commonly above 400 W/m2 – but power density expressed as electricity generated per m2 of the area occupied by a large wind farm is a small fraction of that high rate. This is primarily due to necessarily generous spacing of wind turbines (no less than five and up to ten rotor diameters) that is required in order to minimize wake interference. As a result, even a wind farm composed of large 3 MW Vestas turbines with a rotor diameter of 112 m and spacing of six diameters apart will have peak power density of 6.6 W/m2 and even a relatively high average capacity factor of 30% would bring that down to only about 2 W/m2.

Actual power densities vary with average wind speeds and turbine sizes. Altamont, America’s pioneering large wind farm in California, rates only 0.6 W/m2, Puget Sound Energy’s Wild Horse (with a high capacity factor of 32%) has power density of 2 W/m2. The world’s largest offshore wind installation, London Array in the outer Thames estuary – designed to have a capacity of 1 GWp, annual generation of 31 TWh (354 MW) and an area of 245 km2 – will have power density of just 1.44 W/m2. A good approximation of expected power densities for large scale wind generation (year-round average, not the peak power) should not be thus higher than 2 W/m2. If 10% of the US electricity generated in 2009 (395 TWh or 45 GW) were to be produced by large wind farms their area would have to cover at least 22,500 km2, roughly the size of New Hampshire.

Part IV Wind Turbines

Spacing of wind turbines and access roads at the Altamont Pass wind farm in California.


  1. artemis  

    I’m not clear on one of the last points. Is 10% of our current generation 45GW, or are you saying that that it’s ten percent of our current generation at 45GW. So is our current TOTAL generation 45GW or 450GW?


  2. Jon Boone  

    Fascinating piece. The relative power densities of solar and wind are, in modern terms, incredibly dilute. Wind especially. However, beyond the rather simple statistical averaging, using solely the capacity factor, I’d like to see an accounting for wind (and solar) variability in the equation. How does the continuous flux of wind generation, ranging from zero to the installed capacity, affect functional considerations of power density?


  3. Steve C.  

    Never let the science facts get in the way of science fantasy.

    What is it about wind power that incites so much support? Any way you measure it, wind is a marginal (or boutique) solution at best. Solar, at least holds some future promise if we can figure out how to increase the output and reduce the cost. Unfortunately, people seem to think that Moore’s Law applies to solar power because its “high tech”. I think it’s more likely to follow the cost and performance curves of internal combustion engines. In other words, improving the efficiency to something approaching practicality will take another 50 years.


  4. Bruce  

    Fascinating piece. The relative power densities of solar and wind are, in modern terms, incredibly dilute. Wind especially. However, beyond the rather simple statistical averaging, using solely the capacity factor, I’d like to see an accounting for wind (and solar) variability in the equation. How does the continuous flux of wind generation, ranging from zero to the installed capacity, affect functional considerations of power density?


  5. Bill Chaffee  

    Federal wind energy maps don’t take Betz law into consideration do they?


    • KHawkins  

      You may be on the wrong track here. Any wind energy maps I have seen show windspeeds not energy. Betz law relates to the theoretical maximum possible energy content in wind that can be derived using wind turbines, or similar devices.

      Energy output from a wind turbine based on wind speed is described by its power curve, (see http://guidedtour.windpower.org/en/tour/wres/pwr.htm ), which over the range of normal operations, that is while it is climbing, is related to the cube of wind speed.


  6. Scott Brooks  

    Wind and solar’s attractiveness started with the ENRON PR game promoted by Al Gore. Various governors latch on to that till the public saw it as the Green thing to do. Pepole are vary technically naive when it comes to technology, they just buy what they percieve as good.

    Since Nixon established the EPA in response to environmental concern anything a popular president promotes is seen as sanctimonious. They see wind and solar as renewable in regards to the source and not the economics of the final product. They get scammed by all the hype and political fandangaling.

    And then there are liberal educators who are not well informed about the real technology. And the media exaggerate the capabilities of the technology like nuclear one did about the old perceptions of the technology, “too cheap to meter”.

    Ever read Popular Mechanics or Science? They have these colorful and imaginative illustrations that make you go wow! Then they talk about the stuff like people did back in the 60’s and space travel in the near future. They don’t perceive the economics behind the technology and the problems associated with it. They are Disneyfied. (my new age term for mesmerized)

    Of course the computer revolution has also raised their expectations of energy and what not plus things like Star Trek. Oh Mr political beam us up they all sigh. They are taken in like people who go to the magic shows, it’s all slight of the tongue and smoking of the brain, if not outright lying.


  7. Dr Jeffery Lee Johnson  

    Its best that you review your mathematics. The equations and calculation for solar flux (solar insolation) are off by a factor of 6 to 8. http://en.wikipedia.org/wiki/Insolation
    The global standard conditions for qualifying the performance of solar panels is 1000 W/m2. In northern Mexico and southern United States, we routinely experience peak solar incidence between 1150 W7m2 and 1300 W/m2. The issue is that majority of large scale PV plants are in Spain and Germany, where the solar incidence is equal to that of Vancouver Canada. You are making an irregular mistake to assume that solar is equally distributed around the world. The solar conditions for the US and Mexico are significantly different that the calculation you assume here by equating different areas of the world. Do your research before posting number! see global radiation amp and North America map

    For more information contact NAtional Renewable Energy Labs or Sandia National Labs at http://www.NREL.gov


  8. rbradley  

    Vaclav Smil responds to JL Johnson:

    I do not know what that guy is referring to but the numbers are clearly explained at the beginning of the 4th section of my power densities set you posted some time ago, namely:

    Solar constant (radiation at the top of the atmosphere) is 1366 W/m2, distributing this over the planet’s spherical area (that is dividing by four as a sphere’s area is 4?r2 that is four times the circle area of ?r2) yields 341.5 W/m2, subtracting 20% for absorption in the atmosphere and 30% for the planet’s albedo halves the total to about 170 W/m2. That is the global mean of insolation (radiation flux at the Earth’s surface). Cloudy places get less than 100 W/m2, the sunniest deserts above 230 W/m2.

    Taking just about the sunniest US place, Phoenix (see the attached table) has 5.38 kWh/m2/day, which works to 1964 kWh per year or an equivalent of 7GJ, divided by 31.5 Ms in a year this yield average power density of 222 W/m2 or 30% above the global mean.

    The numbers the guy is referring are not annual mean densities, but the peak densities available for 1-3 hours around the noon during the longest summer days, and these are commonly around 1,000 W/m2 during the peak cloudless summer days over much of the northern hemisphere — but these are not power fluxes available on the annual, or daily, average, including the nights: those rates cannot be ever higher than about 250 W/m2: see the attached map!!!

    Of course, when he divides the means into a peak (1300/170 he gets a nearly 8-fold difference) — but he is dealing with two different variables, peak and annual flux.


  9. David Bergeron  

    Dr Jeffery Lee Johnson, you are being thrown-off by Mr, Smil, somewhat unusual, but valid approach of looking at average daily surface flux level, rather than instantaneous peak levels. The industry normally discusses solar resource in the later terms, but Mr. Smil’s approach is equally valid and, in a way, refreshingly different.

    If you reread his post I’m sure you will conclude his approach is fair. The proof is that his capacity factors are correct. (Another term not typically used in the solar field)

    The bottom line is the capacity factor numbers are right. We have a large plant here in sunny AZ with a capacity factor of 19%.


  10. Mark Heslep  

    The method for calculating solar isolation described by Vaclav Smil, flux distributed over a sphere, is no doubt valid for radiation received by the Earth’s spheroid, but appears to have led to an unwarranted worst case assumption for actual solar collectors, which are seldom deployed tangent to the Earth’s surface. Whenever possible, PV panels are deployed facing due south, tilted to the angle of latitude. Less frequently one or two axis trackers may be used. A square meter of PV panel so deployed maps to more than a sectional square meter of the Earth’s spheroid over the day. With that in mind, let’s reexamine the measured solar collection in (say) Phoenix, Az.
    As it happens we have _measured_ solar radiation data, collected year round at numerous surface sites in the US, by NOAA, often using pyranometers, and for various solar collector types [1]. For a south facing tilted-to-latitude flat plate collector in Phoenix, Az, the measured radiation received is 6.5 kWh/M^2/day, for a power average of 270 W/M^2 over the 24 hour day, with an error margin typically of several percentage points. Similarly, a two-axis flat-plate tracker is measured to receive 8.9 kWh/M^2/day, averaging 370 W/M^2.
    From there we can proceed to PV conversion efficiency. I have no access to PV efficiency data for panels actually deployed in the large farms which are cited by Smil, at a surprisingly low 10% (silicon crystalline?). We do have public access to test data for PV samples submitted to NREL. At the high end, as of 2009 we see (admittedly expensive) experimental triple junction cells (i.e wide solar spectrum), such as those commonly used on spacecraft including the Mars Rovers, tested at 41.6 percent conversion efficiency [2][3]. I have little idea about performance over temperature or expected rate of decline of such a PV cell, but it does give us some idea of what is feasible in the future.
    Currently in the US, the data sheet for SunPower’s commercially available E20 series lists a _new_ panel conversion efficiency of 20.1%, decreasing/increasing at -0.38%/K above/below 25C[4]. I have no experience with that vendor or that product, but again NREL reports test data for other single crystalline silicon cells at 25% efficiency. At 20% PV conversion, and then a 10% loss for AC/DC conversion and assorted connection losses [5], we then have 48 W/M^2 average DC electric power in Phoenix with a tilted plat PV panel, and similarly 66 W/M^2 with a two axis tracker.
    Mark Heslep, P.E.
    Mclean, Virginia

    [1] NREL/NOAA US city solar radiation data http://rredc.nrel.gov/solar/old_data/nsrdb/redbook/sum2/23183.txt
    [2] NREL Solar PV data chart. http://upload.wikimedia.org/wikipedia/commons/e/ed/PVeff%28rev110408U%29.jpg
    [3] http://boeing.mediaroom.com/index.php?s=43&item=810
    [4] SunPower E20 PV panel http://us.sunpowercorp.com/cs/BlobServer?blobkey=id&blobwhere=1300258525337&blobheadername2=Content-Disposition&blobheadername1=Content-Type&blobheadervalue2=inline%3B+filename%3De20_327_ds_en_ltr_w.pdf&blobheadervalue1=application%2Fpdf&blobcol=urldata&blobtable=MungoBlobs
    [5] See, e.g. 98.7% efficient AC/DC MW scale inverter. http://www.solarpowerengineering.com/2011/06/1-4-mw-solar-inverter-for-large-commercial-and-utility-scale/


  11. Mark Heslep  

    Also see David MacKay, Professor in the Dept of Physics at Cambridge University, in his online book Sustainable Energy Without the Hot Air, where he uses 110 W/M^2 average received solar radiation on a south facing flat surface with England’s northerly latitude and climate, and from their using solar thermal heat directly, or whatever PV conversion efficiency is affordable.

    See McKay’s sources starting here:
    especially average W/M^2 received radiation at various locations around the world here
    http://www.inference.phy.cam.ac.uk/withouthotair/c6/page_46.shtml, e.g.
    Rome: 176
    Houston: 197
    Miami: 219
    Los Angeles: 225
    Honolulu: 248


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