Power density



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Radiant flux
F = T4

σ (Stefan-Boltzman constant) = 5.67 x 10-8 W/m2/K4

T = 5,779 K,

F = 5.67 x 10-8 x (5,779)4

= 63.24 MW/m2


This means that the Sun’s isotropic radiation is 63.2 MW for every m2 of its photosphere.

This granulated layer radiates along a wide spectrum whose wavelengths range from less than 0.1 nm (γ rays) to more than 1 m (infrared radiation). The peak radiation –- dictated by Wien’s displacement law λmax= 0.002898/T (K) -– is at about 500 nm, close to the lower limit of green light (491 nm) and about two-fifths of all energy is radiated in visible wavelengths between 400 (deep violet) and 700 nm (dark red). Ultraviolet radiation carries about 8% of all solar energy, infrared the rest (53%). Visible wavelengths energize photosynthesis (it proceeds mainly by the means of blue and red light, with green reflected) and are sensed by organisms ranging from to humans: our vision is most sensitive to green (491-575 nm) and yellow (576-585 nm) light with the maximum visibility at 556 nm. But the heating of the biosphere is done mostly by IR radiation by wavelengths shorter than 2 μm.

Once the radiation leaves the Sun’s photosphere it travels virtually unimpeded through space and its power density at the top of the Earth’s atmosphere is easily calculated by dividing the star’s total energy flux (3.845 x 1026 W) by the area of the sphere whose radius is equal to the planet’s mean orbital distance of 149.6 Gm. This rate -– 1,367 W/m2 –- is known as the solar constant although a quarter century of satellite observations (unhindered by the radiation’s travel through the Earth’s atmosphere) clearly documented both short-term and longer-term deviations (Foukal 1990; de Toma et al. 2004). The mean value of satellite observations is 1,366 W/m2 with brief declines of 0.2-0.3 W/m2 (caused by the passage of large sunspots across the Sun) and periodic undulations (caused by the 11-year solar cycle) with the peaks as high as 1366.9 W/m2.

Although the Earth is a rotational ellipsoid rather than a perfect sphere, dividing the solar constant by four (the difference between the area of a circle and the sphere of the same radius) yields a fairly accurate mean of solar radiation (341.5 W/m2) that would reach the rotating planet if it were a perfect absorber (black body) and had no atmosphere. In reality, the Earth’s atmosphere absorbs about 16% and hence even without any clouds the annual mean irradiance would be no more than 287 W/m2. Clouds absorb another 3%, reducing the mean to about 278 W/m2. But the atmosphere is not only an absorber, it is also a reflector or incoming radiation; long-term satellite measurements confirmed that the Earth’s albedo (the share of the reflected radiation) averages almost exactly 30%, with obvious seasonal fluctuations between the northern and southern hemispheres.

Atmosphere reflects 6% of the incoming radiation and clouds 20% (continental surfaces and water accounts for the remaining 4%). This means that about 55% of incoming radiation could be absorbed by an average square meter of a perfectly non-reflecting horizontal ground surface, the rate that translates to about 188 W/m2 on annual basis. This global annual mean is a useful number for assessing solar radiation as the energizer of photosynthesis or as energy resource to be converted into heat and electricity but actual irradiance is subject to daily and seasonal variations that can be perfectly predicted for any site on the Earth rotating on a tilted axis, to much less predictable meso-scale influences of cloud-bearing cyclonic systems, and to highly unpredictable local cloudiness. Values of the total annual global irradiance are, obviously, function of latitude and cloudiness and I will cite just a few representative numbers averaged from daily measurements between 1981 and 2010 (Ineichen 2011).

Annual irradiance The rate for Berlin, representative of populated northern hemisphere mid-latitudes influenced by regular cyclonic flows, is just 116 W/m2 (annual energy total of 1.016 MWh/m2) in average year; within the same zone Dublin is only about 100 W/m2, London is very close to Berlin (110 W/m2) wile Paris gets around 125 W/m2 (NASA 2013). Murcia in Spain, representative of sunny Mediterranean locations, receives 196 W/m2, similar to Athens and a bit more than Rome (at about 175 W/m2); the southern states of America’s Great Plains experience the same range of average irradiance (Tulsa in Oklahoma at about 180 W/m2, San Antonio in Texas about 200 W/m2).

At 266 W/m2 (2.328 MWh/m2 in a year) Tamanrasset in southern Algeria indicates the maxima receivable in the planet’s sunniest climates of the arid subtropical belt; the only larger city with a slightly higher long-term rate is Nouakchott, the capital of Mauritania, with 273 W/m2; Saudi Riyadh is ‘’only’’ 251 W/m2. The largest relatively heavily populated regions with average rates above 200 W/m2 are the US Southwest (with 225 W/m2 in Los Angeles and Phoenix) and Egypt’s Delta (Cairo at 237 W/m2). In contrast, frequent cloudiness keeps the rate for such tropical megacities as Singapore, Bangkok well below 200 W/m2.

Differences in monthly insolation averages depend on latitude and cloudiness: in Oslo the difference between January and June is 16-fold, in Riyadh only twofold. These differences are reflected by actual PV electricity generation: in 2012 German output was 4 TWh in May and just 0.35 TWh in January, an order of magnitude disparity (BSW Solar 2013). Predictably, annual variability of irradiance is significantly larger in cloudy climates and in mountainous regions: fluctuations of total irradiance are as high as 7% in Berlin and Zurich, 2% in Murcia and Tamanrasset. Naturally, monthly differences can be much larger: for example, in January 2013 the Czech Republic averaged 50% less sunlight than in January 2012 (Novinky.cz 2013). During cloud-free days the highest daily global irradiance is a perfectly predictable function of a calendar day and latitude; naturally, daily minima in the northern hemisphere are during the winter solstice, maxima six months later. Theoretical expectation of noon-time maxima would be as high as 1,065 W/m2, that is solar constant minus atmospheric absorption and reflection averaging 22% of the solar constant.

Indeed, the highest recorded maxima in subtropical cloud-free deserts come within a small fraction of 1% of that value. For example, the noon-time May and June hourly maxima from Saudi Arabia are 1,059 W/m2 for Dhahran on the Gulf coast and 1,056 W/m2 for Riyadh in the interior (Stewart, Dudel and Levitt 1993). Irradiance delimits the range of power densities that can be harnessed to heat water or air or to be converted by PV cells to electricity: there is an order of magnitude difference between annual averages of less than 100 W/m2 in cloudy temperate mid-latitudes and the just noted maxima over 1,000 W/m2 available for one-three hours a day during the sunniest spells in the great subtropical desert belt that extends from the Atlantic coast of Mauritania to North China and that has its (much more circumscribed) northern hemisphere counterparts in the US Southwest and northwestern Mexico, and throughout large parts of Australia and in a narrow strip along the coast of South America in the southern hemisphere.



Conversions of solar radiation Life’s evolution and the biosphere’s dynamics are determined by the levels and variations of irradiance: it energizes photosynthesis, warms the atmosphere, waters, rocks and soils (creating pressure differences and hence powering the global air circulation, and evaporating moisture and hence powering the global water cycle) as well as bodies of organisms (critical to keep optimal temperatures for enzymatic reactions) and surfaces of buildings (every above-ground structure that shelters is solar-heated). Direct solar radiation has been also used for millennia to make products (to evaporate salt along seashores, to dry crops, to sun-bake clay bricks), to dry clothes. In cold climates modern architectural design has strived to maximize solar gain in passive solar buildings –- and, taking some ancient lessons, to minimize it in hot environments (Athienitis 2002; Mehani and Settou 2012).

Not long after modern indoor plumbing became available solar radiation began to be used to heat water in simple roof-top collectors and this practice has become increasingly more efficient and quite common in many sunny countries (Mauthner and Weiss 2013). Improvements in plant productivity aside, the most important active step in harnessing irradiance has been the development of solar electricity generation, primarily based on the photovoltaic (PV) effect but also using concentrated solar power to run steam turbines. These conversions have power densities higher than any other means of harnessing renewable energy flows; moreover, PV efficiencies have been gradually improving and there is a clear potential for further considerable gains.

Power densities of solar conversions have several unique attributes: they belong to two distinct categories, calculations for nearly all of them are done for tilted rather than horizontal surfaces, and some of them refer to active (adjustable) areas rather than to fixed level surfaces. Most tilted panels on roofs on large solar farms are fixed in one position: optimal angles from horizontal are easily calculated (Boxwell 2012). The best full-year angle for my latitude (Winnipeg is 500N, the same as London or Prague) is 41.10 and that position will capture about 70% of radiation compared to a tracker; for Tōkyō (350N, the same as ) it is 29.70 and, obviously, fixed panels should remain horizontal at the equator. Some panels are adjusted according to seasons (twice or four times a year) while full tracking has been, so far, reserved for some commercial installation.

The two distinct categories are land-based systems (large-scale PV plants and concentrated solar power harnessing irradiance with a field of heliostats) and rooftop installations (both for solar heating and PV electricity generation). In the future a third category might become important as vertical surfaces –- mostly suitably oriented building walls –- can be either clad in PV panels or be glazed with PV glass unit windows (Pythagoras Solar 2013). With land-based systems solar power densities will belong to a similar, but not identical, conceptual category as those of wind-powered electricity generation. Similar because PV arrays, much like wind turbines and their associated access roads and structure, will not completely cover the land claimed by the project, but not identical because the degree of the coverage will be greater and, in most cases, the arrays will be fenced. But in both cases the spacing (of turbines and arrays) will be determined by inherent properties of energy flows they aim to harness and by efficiencies with which they convert those fluxes to electricity.

In contrast, power densities of rooftop installations (be they for heat or electricity) are in a special category as no other energy converters are routinely placed on tops of buildings and hence are not claiming any new land surface. The same will be, of course, true about wall-based PV or PV-integrated windows. Moreover, during the periods when rooftop PV modules generate more electricity than can be used by buildings on whose roofs they are situated they are sending the surpluses to the grid –- but this does not mean that this actually always helps to reduce overall land claims of a system in which these installations are embedded. Roof-top thermal and PV solar conversions are thus doubly land-sparing and as they already have power densities (even if land-based) higher than the harnessing of any other renewable energy flux they deserve to be widely promoted and adopted although the intermittency of the flux continues to pose non-trivial challenges to any system that would raise the contribution of such conversions to a relatively high level (this is already, as I will explain later, Germany’s great technical and managerial challenge).

Solar heating Water heating using roof top collectors is not only a highly efficient way of harnessing solar radiation but it is also affordable. With highly selective coatings latest designs of flat plate collectors (typical panels circulating water cover about 2.5 m2 and are less than 8 cm thick) have absorptivity of 95% (Bosch Thermotechnology 2013; Stiebel Eltron 2013). Evacuated glass tubes, developed first in China during the 1980s, have similarly high absorptivity. They have vacuum surrounding a heat pipe fused to an absorber plate and they are more efficient, particularly in colder climates and in winter in sunny climates (Silicon Solar 2008).

Their efficiencies (when compared for insolation at 1,000 W/m2) differ with the desired temperature: for pools (water temperature up to 250C) they have efficiencies 60-80%, for domestic water heating (45-600C) this falls to a range of 40-75% (Thermomax Industries 2010). Australian studies show that in summer evacuated tubes are about 50% more efficient than flat plates, and that they are 80-130% more efficient in winter: for example, when heating water from ambient temperature to 750C in Brisbane (Queensland) flat plate efficiency was about 50% while evacuated tube efficiency was just above 80% (Hills Solar 2008).

During mid-day hours in such sunny climates as California or the Mediterranean high absorptivities of flat plates and evacuated tubes translate into power densities of heat collection in excess of 900 W/m2 , and during those times water heating may proceeds with power densities higher than 700 W/m2: such rates are unequalled by any other commercial renewable energy conversion. Annual power densities of water heating in sunny climates are obviously much lower but, again, much higher than any commercial renewable energy conversion: they can be as high as 110 Wt/m2 in Israel or Arizona, in temperate climate they will be generally no higher than 40-50 W/m2. For example, detailed German data show that by the end of 2012 the country had 16.5 Mm2 of solar heating surfaces whose output was about 685 MWt (BSW 2013) and that implies average power density of 41.5 W/m2.

Obviously, larger storages extend hot water availability but come at a higher cost. Roof placement is usually available but shading (total or partial) by nearby trees, buildings and structures cannot be always avoided. Combined systems can capture irradiance for both space and water heating. Worldwide total of solar heat collectors (including the dominant small rooftop heaters and much larger industrial arrays) is in tens of millions. Mauthner and Weiss (2013) estimated that by the end of 2012 solar thermal collectors had aggregate area of 383 Mm2 (compared to 125 Mm2 in 2005), total capacity of 268 GWt and annual output of 225 TWh: that implies average power density of 67 W/m2.

As I will show, power densities between 40-100 W/m2 are superior to anything any commonly deployed competing renewable energy conversion can offer; moreover, properly scaled distributed units are a perfect choice for household (and also institutional) water heating because in sunny climates they can cover all or most of the daily need even without any voluminous hot water storage and even in less sunny climates they can make significant contributions. And, no less importantly, modern water heating systems have no extraordinary material requirements, are widely affordable and can be installed to operate reliably for long periods of time. Without any doubt, rational system-wide approach to energy use would promote widespread adoption of distributed rooftop solar water heaters for households and smaller-size commercial and industrial buildings in any suitable climate. But, also without any doubt, promotion and expansion of these high power density distributed converters has become overshadowed by rapid (and often incredibly irrational) growth of photovoltaic (PV) electricity generation.

Photovoltaics Conversion of radiation to electricity was discovered by Edmund Becquerel in 1839, the first experimental PV cells were made in 1877, but commercialization of the process began only in 1954 with the production of first silicon solar cells at the Bell Laboratories. In 1962 Telstar, the first commercial telecommunications satellite, opened the way to PV-powered space vehicles but terrestrial applications took off only during the 1990s (Smil 2006). They were driven by increasing interest in carbon-free renewable electricity generation, but rapid diffusion of PV systems began only with the adoption of costly subsidies (in the form of guaranteed long-term feed-in tariffs) in some countries.

PV modules and arrays are now mass-produced both for household rooftop installations and for large-scale industrial projects, with modules designed for nominal irradiance, either at 1,000 W/m2 or 800 W/m2 and ambient temperature of 200 or 250 C. Increasing irradiance raises their current and power output but it has a much smaller effect on the voltage; increasing cell temperature brings significant decline in voltage and it also reduces cell output, efficiency and expected duration. By deploying PV cells with conversion efficiency of at least 10%, peak power densities of PV modules would be then 80-100 W/m2 during a few mid-day hours, with 15% efficiency the rate would rise to 120-150 W/m2. Consequently, approximate calculations should not assume less than 100 Wp/m2 for newly installed PV arrays.

Average annual power densities of PV are rather easily calculated by multiplying measured irradiance by the mean efficiency of modular cells adjusted for their specific performance ratio (difference between their actual and theoretical output). As already noted, the first number is essentially fixed for a given location (with a relatively small inter-annual variation) while the other two parameters have been steadily increasing through innovation. In 2013 the best research-cell efficiencies were as follows: emerging techniques (organic and dye-sensitized cells) 10-14%; thin films 20%; crystalline silicon cells 25%; multijunction cells 44% (NREL 2013). Inevitably, actual field efficiencies of PV cells that have been recently deployed around the world are much lower. Single crystalline modules, the oldest but still the most efficient PV conversion techniques, average 10-12%; cheaper polycrystalline silicon cells is now close behind with 10-11%; string ribbon polycrystalline silicon delivers 7-8%; and amorphous silicon (vaporized Si deposited on glass or on stainless steel) will convert no more than 5-7% of irradiance into electricity.

Assuming, again, average 10% efficiency would result in a fairly representative range of average power densities for the plants operating at the beginning of the second decade of the 21st century. Those densities would range from less than 10 W/m2 in cloudy mid-latitudes (Atlantic Europe, Pacific Northwest) to more than 15 W/m2 in sunnier climates and they would peak at around 25 W/m2 in cloud-free subtropical deserts. These rates are applicable for small modules installed on roofs on the ground, but power densities will have to be lower for large ground-based installations; in these solar parks additional land will be required between long tilted PV arrays in order to avoid shading and to provide access for servicing the modules (including washing and structural repairs), for roads, for inverter and transformation facilities needed to access the grid and for service and storage buildings.

Installations with tracking assemblies will require even more additional land per PV module to avoid shading. Consequently, anywhere between 25-75% of a solar park area will be actually covered by the modules while for the PV field alone the cell assemblies will cover typically 75-80% of land. But it should be noted that many large solar parks often acquire or lease much larger areas intended to accommodate possible future expansion and those areas should not be counted when calculating power densities of actually operating projects. Other projects claim more land outside of their PV array fields in order to create environmental buffers or to provide corridors for wildlife: inclusion of those areas in land denominator is, as with many similar land claims made by other energy installations, arguable.

General procedure for calculating power densities of ground-based PV installations is thus quite straightforward: average irradiation (I) is converted from Wh to W, and the result is multiplied by conversion efficiency (η) and an appropriate performance factor (p). For the average US insolation of 1,800 kWh/m2, cell efficiency of 10%, performance factor of 0.85 and 50% of the ground covered by modules the result is almost 9 W/m2, while for cell efficiency of 15% this would rise to 13.1 W/m2:


Power density of a PV module

I = 1,800 kWh/m2

1.8 MWh/m2 x 3,600 = 6.48 GJ/3.1536 x 107 s = 205.5 W/m2

205.5 W/m2 x 0.10 x 0.85 = 17.5 W/m2 x 0.5 = 8.7 W/m2


Power densities of large ground-based projects The most important correction in calculating actual solar power densities is the adjustment for their relatively low capacity factors. Capacities of new solar projects are listed, invariably, in terms of rated peak power (MWp), performance achievable only by perfect conversion during the time of the highest irradiance. As expected, average capacity factors correlate with total irradiance: in places where it is less than 150 W/m2 they will be below 12%, for the insolation between 150-200 W/m2 they will range up to 20%, and in the sunniest locations with irradiance in excess of 200 W/m2 they will be up to 25%. Actual performance data show that even in such sunny locations as Spain most plants have capacity factors of less than 20%, and in cloudy temperate climates that indicator will dip below 10%. In addition, only about 85% of a PV panel’s DC rating will be transmitted to the grid as AC power: these performance ratios vary but in the best systems they should be always above 80% and should be approaching 90%. Several notable examples of large PV plants illustrate actual power densities.

In 2008 Spain’s Olmedilla de Alarcón (Cuenca, Castile-La Mancha) became temporarily the world’s largest solar park with installed capacity of 60 MWp. Olmedilla’s total area of 283 ha of fixed panels and annual generation of 85 GWh (or average power of 9.7 MW) translates to power density of about 3.4 W/m2 and average capacity factor of just 16%. Another plant completed in 2008, Portuguese Moura (46 MWp, 88 GWh or 10 MW of average power) has the capacity factor of nearly 22% and with the area of 130 ha (both fixed and single-ax tracking panels) its power density is 7.7 W/m2. When it was finished in 2011 Sarnia (Ontario) was the world’s largest PV plant; its installed capacity is 97 MWp and annual generation of 120 GWh from 1.3 million panels covering 96.6 ha while the plant’s entire area claims 445.2 ha (Clean Energy 2013). These specification prorate to average annual power density of 14.2 W/m2 of modules and 3 W/m2 of the total land claim. With irradiance of about 180 W/m2 this puts Sarnia’s average conversion efficiency at less than 8% and the capacity factor at 14%.

Another large project completed in 2011, Germany’s Waldpolenz –- about 20 km east of Leipzig on the site of a former Soviet East German air base –- has peak capacity of 52 MW, annual generation of 52 GWh ( 5.94 MW), total panel area of about 110 ha and the total site of 220 ha (Juwi Solar 2008). These specifications yield power density of 5.4 W/m2 for the module field and 2.7 W/m2 for the entire plant area, and annual capacity factor of just 11.4%. Ukraine’s Perovo (located in the western part of sunny Crimean peninsula), has peak power of 100 MW, it generates 132.5 GWh (averaging 15.1 MW) from 200 ha of panels (Clean Energy 2013a), resulting in power density of about 7.6 W/m2 and capacity factor of 15%. Agua Caliente, in 2013 the largest project in North America on 960 ha in Arizona, has one of the world’s highest average annual irradiation rates (2.45 MWh/m2), 290 MWp capacity and it generates 626.2 GWh/year (Clean Energy 2013a). This yields a high capacity factor (24.6%) and power density of 7.45 W/m2. Copper Mountain Solar I (Boulder City, Nevada, 58 MWp, 124 GWh/year, 180 ha) has a nearly identical load factor and a slightly higher power density of 7.86 W/m2.

Consequently, the largest PV projects now operate with power densities of roughly 3-8 We/m2. Because smaller projects use similar or identical PV cells it is not at all surprising that their power density range is pretty much the same. McKay’s (2013) listing of such projects in Italy (with installed capacities between 1 and 10 MW and load factor about 16%) shows the range of 4-9 W/m2, for Spain (projects rated at 7-23 MW with load factors between 16-23%) it is 4-11 W/m2, for the UK (projects averaging about 5 MW with average load factor of about 11%) it is just between 4-5 W/m2 and for the ground-based US installations the range is from just 3.8 W/m2 for a two-axis 2.1-MW tracker in Vermont to 11.43 W/m2 for a fixed 250-kW installation in Florida. In order to minimize their capital costs, most of the large PV projects use relatively inefficient less costly thin-film Cd-Te cells.





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