Power density



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Pumped storage And there is year another challenge in calculating power densities of water power: how to treat pumped hydroelectric storage (PHES) projects. These stations (the first ones were built in Swiss and Italian Alps during the 1890s) use cheaper off-peak (that is usually night-time) electricity to pump water from a lower-lying reservoir to a higher reservoir built on an adjacent elevated land; they can fill it typically in 5-6 hours, creating hydraulic heads of more than 300 m, and the record difference of nearly 700 m (690 m at Chaira in southwestern Bulgaria). During the hours of peak demand they serve as a rapidly deployable reserve to provide, virtually instantly but for relatively short spells of time, additional supply.

PHES projects use reversible pumps/turbines that can switch from pumping to generating mode in just 6-10 minutes, and that can usually reach full capacity generation in a just a few minutes, while turbines synchronized to the grid and spinning on air can go from standby to full load in almost instantly. For example, the largest British PHES, Dinorwig in northern Wales completed in 1984 with installed capacity of 1.728 GW, can reach maximum power in just 16 seconds (First Hydro Company 2013). Their maximum power is easy to calculate: all we need to know is the mass of released water (usually given as a volume) and the generating head; g is, of course, acceleration of gravity at 9.81 m/s2.

For example, Dinorwig can release 7 Mm3 over a five-hour period to turn its six reversible Francis turbines receiving water from the upper reservoir that is 500 m above the lower storage (First Hydro Company 2013). This translates into theoretical maximum power of about 1.9 GW, and the plant’s actual rated capacity, taking into consideration roughly 90% conversion efficiency by its large turbines, is about 1.7 GW) and its output is 8.5 GWh:
Power of Dinorwig pumped storage plant
P = mgh

m = 7 Mm3/5 x 3,600 = 388.9 m3/s x 1,000 = 388,900 kg

= 388,900 kg x 9.81 x 500 = 1.9 GW

= 1.9 GW x 0.9 = 1.7 GW x 5 = 8.6 GWh


The same equation can be used to calculate the cost of water pumping. Because it takes 6.5 hours to fill the reservoir the pumping flow rate is 299.1 m3/s and due to frictional drag the effective pumping height is 530 m and efficiency is, once again, about 90%: these specifics result in the pumping cost of 11.2 GWh and overall efficiency of 77%. We are willing lose a substantial share of electricity in order to be able generate almost instantly at the time of peak demand:
Energy cost of water pumping
P = mgh

m = 7 Mm3/6.5 x 3,600 = 299.1 m3/s x 1,000 = 299,100 kg

= 299,100 kg x 9.81 x 530 = 1.55 GW

= 1.55 GW x 1.11 = 1.72 GW x 5 = 8.6 GWh 8.6/11.2 = 0.77


PHES stations are also generally expensive to build but they remain the only means to store electricity (by converting it to potential power) on multi-MW to GW scale, and hence the most widespread means of massive energy storages for use during the periods of peak demand. In 2010 the worldwide capacity of pumped storage stations reached 120.7 GW, with Europe having slightly more than a third of the total, Japan a fifth and the US nearly a fifth (USEIA 2013). This means that the worldwide pumped storage capacity was equal to about 13% of all installed hydropower but the comparison is misleading for two major reasons. First, hydrostations and PHES ate two very different modes of generation: in hydrostations we generally aim at producing as much electricity as precipitation and water storage will allow, in pumped storages we are generating according to peak demand. More importantly, due to inevitably higher energy costs of pumping overall energy balance of pumped storages are always negative.

PHES is always a net energy loser, with the most efficient projects consuming about 20% more electricity than they generate. In 2010 the global electricity cost of running PHES amounted to about 22 TWh and in the US pumped storages consumed about 5.5 TWh more electricity than they produced (USEIA 2013). This reality makes it difficult to compare the technique’s performance with other modes of electricity generation. Moreover, because of their intermittent operation and generally low capacity factors, power densities of pumped storages will have very different when calculated for peak flows or as an annual mean. A project with installed capacity of 1 GWe, reservoir area of about 800 ha and load factor of 13% will have annually averaged power density of just 16 W/m2.

In contrast, here are the specifics for a few top projects that show peak power densities up to about 50 times higher. Bath County project in Virginia (completed in 1985) remains the world’s largest pumped storage with 3 GW of installed capacity (Dominion 2013), followed by two 2.4 GW stations in China. As already noted, Bath pumped storage has elevation difference of 385 m, upper reservoir covers 1.07 km2, lower reservoir has 2.25 km2 (Dominion 2013). This gives it peak power density of just over 900 W/m2, an impressively high rate that could be sustained only for a few hours a day and hence it is not comparable with power densities of conventional hydrostations that have much higher capacity factors. China’s largest pumped storage in Guangdong, completed in 2000, has the head of 535 m, upper reservoir of 1.2 km2 upper and lower reservoir 1.6 km2 (Chincold 2013). The station uses night-time electricity from the Dayawan nuclear station to provide peak power to Hong Kong and Guangzhou with maximum power density of nearly 860 W/m2.

China’s second 2.4 GW station, Huizhou, is also in Guangdong and in 2012 the country had 24 PHES with the total capacity of 16.95 GW. Japan’s Okutataragi (1.93 GW) in southern Honshū and Ludington pumped storage in Michigan (1.872 GW delivered within 30 minutes after start-up) make up the world’s top five pumped storages. Germany, which would benefit from a higher PHES capacity to accommodate its rising solar and wind electricity generation, had 7. GW installed in 2012 (about 5% of total capacity) and despite a common belief that the country’s PHES potential is largely exhausted Steffen (2012) shows that another 4.7 GW could be realized in the coming years. Similarly, more projects are planned for several other EU countries (including not only Switzerland and Austria but also Portugal, Spain and Slovenia), in the US and in Japan (Deane, Gallachóir and McKeogh 2010).



Phytomass for traditional and modern uses

In Europe, North America and Japan traditional phytomass fuels -– dominated by wood and charcoal but in many agricultural also including large amounts of crop residues and in some societies also dried dung of domestic and wold animals –- were the most important source of thermal energy until the late 19th century, throughout the rest of the world (including populous China, India, Indonesia and Brazil) until different periods of the 20th century. There were only two great exceptions to the dependence on wood in the early modern (1500-1800) world, England and Wales, and the Netherlands. England was the first country that accomplished its transition to coal, while the Dutch relied heavily on peat. Data compiled by Warde (2007) suggest that the most likely time when energy from coal combustion surpassed that from the burning of wood was around 1620, and that by 1650 coal provided two-thirds of all primary energy. Dutch Golden Age of the 17th century was fueled by peat: its per capita consumption was higher than India’s average energy supply in the year 2000 (de Zeeuw, 1978; Smil 2010a).

In the US coal (and still relatively small flow of crude oil) began to supply more than half of all primary energy by the mid-1880s, in Japan the tipping point came a generation later, in the USSR it was delayed until the early 1930s, in China until the mid-1960s (Smil 2010a). At the beginning of the 20th century traditional phytomass fuels provided half of the total primary energy supply (TPES); by its end their global harvest had actually doubled as hundreds of millions of poor villagers and many people in smaller towns throughout Asia, Africa and Latin America continued to rely on wood, charcoal and straw for cooking and heating, but in relative terms phytomass fuels became a marginal part of the TPES, providing no more than 12% by 2010 (Smil 2010a).

In the poorest parts of the world household fuel is still gathered mostly by families, much of it in the way unchanged for millennia, not by cutting trees but by collecting fallen and dry branches of small trees and shrubs. This often requires lengthy walks to the nearest sources of woody phytomass, making it a time-consuming chore that is usually done by women and children. At the same time, in many tropical countries a surprisingly large share of woody phytomass does not come from forests but from roadside and backyard trees and from small groves. And while an increasing availability of kerosene, LPG and electricity reduced the dependence on crop residues, burning of cereal straws is still a common practice in many rural areas, particularly in Asia.

In modern countries virtually all phytomass is now harvested by modern means and its share available for energy conversions comes from three major sources. The first one is woody phytomass explicitly destined for energy conversions (combustion, gasification), some of it originating in harvests of natural growths, an increasing share coming from tree plantations of fast-growing species. The second one is a diverse group of residual phytomass that includes those tree parts that do not become merchantable timber or pulp for making paper (bark, chips, wood shavings, saw dust) and crop residues (a category dominated by cereal straws and sugar cane bagasse). The third one, the latest addition to the two well-established sources, is the cultivation of annual or perennial field crops that are converted to liquid biofuels (mostly to ethanol and biodiesel) or are gasified.

But no matter what is the harvested species, power density of phytomass energy use is always relatively very low, an inevitable consequence of inherently poor efficiency of the conversion chain that starts with solar radiation and ends with actually harvested phytomass. Photosynthetic conversion of select wavelengths of solar radiation (the process proceeds by the mans of blue and red light) to chemical energy of new phytomass (first to carbohydrates, eventually also to proteins and lipids) is a remarkable transformation, the foundation of all heterotrophic life and hence also of all human societies and civilizations (Smil 2013), but one with inherently low efficiency because only a small part of solar energy that is initially converted to new chemical bonds in those plant tissues that ends up as harvestable phytomass.



From solar radiation to phytomass Cannell (1989) offered a representative sequence of efficiencies and conversion losses leading from the insolation to actual wood yield and I have updated some of his multipliers and recalculated all the steps in terms of power densities for a location with 115 W/m2 of annual irradiance (typical for the forests in southern Sweden) and with all progressive rates expressed in W/m2 (rounded to the nearest 0.1).
Power densities of a photosynthetic progression
Insolation 115 W/m2

Photosynthetically active radiation

(blue and red light) x 0.43 = 49.5

Leaf reflectance and transmittance x 0.85 = 42.0

Quantum efficiency of photosynthesis

(theoretical carbohydrate output) x 0.25 = 10.5

Limited diffusion of CO2 to chloroplasts x 0.4 = 4.2

Limits due to light interception

at favorable temperature x 0.3 = 1.2

Photosynthate reduced by respiration x 0.5 = 0.6


Conversion of the power density of 0.6 W/m2 back to annual phytomass yields (assuming energy density of 19 GJ/t and specific density of 500 kg/m3) roughly 10 t/ ha in terms of dry matter. This would be a fairly representative mean for a growing (non-climax) forest on fairly good soils and receiving adequate precipitation. Indeed, Luyssaert et al. (2009) found that the best available model of the net primary productivity of European forests (for EU-25) produces an annual mean of 520 ± 75 g C/m2, that is (assuming that wood is 50% C) 10.4 ± 0.75 t/ha. Another set of estimates put the mean annual NPP of EU-25 cropland at 646-846 g C/m2, that is roughly 13-17 t/ha (Ciais et al. 2005). Most of the world’s terrestrial plants will have yields, and hence production power densities, of the same order of magnitude. Net primary productivity (NPP, with all values in dry matter) of tropical ecosystems is on the order of 20 t/ha, means for temperate and boreal forests and woodlands will be around 10 t/ha, and a very similar average applies to the world’s cultivated land.

Global terrestrial NPP is between 110-115 Gt/year (Zhao et al. 2005; Ito 2011); assuming (conservatively) 18 GJ/t this is at least 2 ZJ or nearly 63 TW. Prorated over the Earth’s ice-free surfaces the average power density of terrestrial photosynthesis would be just 0.4 W/m2 but a more accurate rate results from also leaving out both hot and cold deserts: this raises the mean power density to about 0.6 W/m2 (11 t/ha). During the periods of the most rapid growth NPP additions could amount to 200 kg/ha a day for the inherently more efficient C4 plants and up to 150 kg/day for C3 species. But NPP is not a measure of actual phytomass harvest, it is a modelled construct of the gross productivity of a plant (or a community or an ecosystem) adjusted for its respiration: hence it does not include heterotrophic consumption –- what we would call in simple terms pre-harvest losses to bacterial and fungal infestations, to insects, birds, rodents and other mammals –- as well losses due to inclement weather (lodging of stems and other wind damage, flooding, drought).

The rate that adjusts for all heterotrophic claims is net ecosystemic production (NEP) and if there were no weather-induced pre-harvest losses and if the entire above-ground growth is harvested (such as with whole-tree utilization for wood chips or with a forage or silage crop) then the NEP should be identical to actual harvest. But more often our harvests do not include all of the above-ground phytomass. When trees are grown for stemwood (to be cut into sawnwood or processed into pulp) the actually harvested net stem growth is only about 20% of NPP and 40% of NEP (Pretzsch 2009). And for cereal crops (by far the largest category of agricultural products) grain is typically no more than 40-45% of the above-ground phytomass at the harvest time (the rest being cut straw and the remaining stubble). Power density of actual phytomass harvests has changed with time due to changes in harvesting methods as well is in the degree of phytomass utilization.

For millennia, collecting woody phytomass for household use has been an activity with by far the lowest power densities, but even modern commercial operations –- relying on mechanized harvesting (chainsaws, trucks) and converting wood to charcoal in large kiln operations (most notably in Brazil to feed the country’s blast furnaces using charcoal to smelt iron) –- rank below other conversions of renewable energy sources. Maximum power densities for collecting phytomass fuels can be approximated by using data on litter fall, phytomass that is shed by plants in many smaller shapes and sizes (leaves, needles, seeds, cones, nuts, blooms) and larger woody parts (strips of bark, twigs, branches, pieces of logs).



Harvests, yields and power densities Studies in tropical and subtropical forests show annual production rates of as little as 5-6 t/ha and as much as 12-15 t/ha, in temperate forests the rates are mostly less than 5 t/ha (Bala et al. 2010; Odiwe and Muoghalu 2003; Liu et al. 2003; Clark et al. 2002). Converted by using an average of 18 MJ/kg these rates would translate to 90-270 GJ/ha in the tropics and subtropics and mostly to less than 80 GJ/ha elsewhere, or between roughly 0.25 and 0.85 W/m2. Many litter fall studies (including the cited ones) also show the composition of the fallen phytomass: leaves (or needles) typically account for 65-75% of the total mass, reproductive parts (flowers, seeds, nuts) for 5-10% and coarse woody debris is only 15-25% of the total, or as little as 750 kg and as much as 3.75 t/ha a year. Only the latter part of litter fall is usually collected and annual power densities of collectable woody phytomass would be no higher than 0.04-0.20 W/m2.

But in some extreme circumstances virtually the entire litter fall was harvested in regions with severe fuel shortages, particularly in long-ago deforested areas of North China where peasants used to sweep every piece of plant litter, including leaves, needles and small twigs, shed by remaining wood groves and carry them in baskets on their backs to their house in order stir-fry their meager meals: this practice could be seen in North China even during the 1980s! But because litter yield in those low-density, low-productivity groves was more than a 3-4 t/ha even its complete harvest (an unrealistic assumptions as some bits are too small to carry and as decomposition is constantly reducing the litter layer) would prorate to less than 0.2 W/m2. The lowest woodfuel collection rates have been recorded in the poorest arid parts of Africa; for example, in Eritrea’s scrublands and wooded grasslands the annual yield is as low as 30-50 kg/ha of air-dry (15 GJ/t) wood (Arayal 1999); even the latter rate is just 5 g of woody phytomass per square meter, just a tiny twig and a minuscule power density of 0.0024 W/m2.

Once wood harvests began on a large, truly commercial, scale to supply towns and growing cities, cutting down of mature virgin forests or older succession growths tapped much richer stores of phytomass but it did not proceed with higher much higher power densities. Cutting down a rich temperate primary forest could yield 400 t/ha but as such a harvest could not be repeated (even after 80 years a secondary growth would yield at least 20% less) a properly prorated annual power density mean was no more than about 0.25 W/m2. Those large cities that were located in temperate regions and needed winter heating –- China’s northern capitals Xi’an and Beijing, London and Paris –- had considerable demand for wood, a function not only of their size but also of wasteful combustion in open fireplaces (diffusion of more efficient enclosed stoves was surprisingly slow even in the richest parts of Europe), and similarly wasteful conversion of wood into charcoal, the preferred (smokeless) fuel for heating.

All pre-industrial averages of per capita fuelwood consumption are just approximations but they suffice to illustrate annual urban claims. My estimate for the early Roman Empire is no more than 600 kg/capita (Smil 2010b), in London of the early 14th century it was 1.5 t (Galloway,Keene and Murphy 1996), during the 18th typical German mean was close to 3.5 t/capita (Sieferle 2001), in 1830 it was about 4.5 t/capita in Austria (Krausmann and Haberl 2002), in the forest-rich US the mean was nearly 6 t/capita (roughly split between household and industrial uses) during the 1850s (Schurr and Netschert 1960) but in Paris it declined from nearly 1 t/capita in 1815 to less than 300 kg/capita by 1850 (Clout 1983). Even if the average consumption were only 1 t/capita a pre-industrial city of million people would have needed every year would harvested from about 250,000 ha, an equivalent of a square with sides of nearly 50 km.

And the prevailing power densities were further reduced by the conversion of wood to cleaner-burning but inefficiently produced charcoal. As already described in the first chapter, English charcoal-making during the 18th century -– when a typical charcoal:wood ratio was 1:5 by mass and (assuming 29 GJ/t of charcoal and 19 GJ/t of wood) about 1:3.3 in energy terms –- operated with power density of less than 0.1 (just 0.07) W/m2. Ironmaking reduced the transportation costs of this low power density fuel by locating in forested regions (resulting, inevitably, in extensive local deforestation), but cities had to import charcoal from increasingly distant sources and yet the easily crushable fuel is not suited for long-distance transport.

An 18th-century city consuming roughly half of its fuel demand as fuelwood and half as charcoal would draw on energy supply produced with power densities less than 0.15 W/m2, and for every 100,000 of its inhabitants its annual fuel supple would have required wood harvests from about 40,000 ha of forested land. Obviously, before the advent of coal (and except for the places where inexpensive water-borne transport could bring fuel from distant forested regions) the combination of expanding rings of deforestation around large cities and high cost of land transport was an important factor in limiting the size of preindustrial cities that had to supplied by wood delivered by heavy carts (pulled by oxen or horses), on the backs of donkeys or by camel trains (common practice in pre-modern Beijing) from increasing distances.

Harvests of crop residues had much lower power densities than those of woody forest phytomass, in temperate regions almost always less than 0.05 W/m2 (< 1 t/ha). But the two kinds of biofuels are not in the same category and hence the comparison is in many ways inappropriate: wood is the only targeted harvest, while crop residues are just by-products of harvesting cereal, leguminous, oil, sugar or fiber crops. Moreover, straw and stalks and leaves have always had many more valuable uses than to be burned inefficiently in small household stoves: those uses range from animal feed to substrate for mushroom cultivation, and the best choice might be often to leave most of the residues in the field to be incorporated into soils to replenish their organic matter, retain moisture and help to prevent erosion (Smil 2013).

Some modern harvests of wood for energy (with conversion options including combustion for heat, combustion for electricity generation, gasification and production of liquid biofuels) still come from natural forests (mostly from secondary and tertiary growth) but the clear trend has been to cultivate the trees. Planting of fast-growing tree species grown in short rotations began during the 1960s, mostly with poplar cones (Dickmann 2006). These plantings are dense (some with more than 50,000 stems/ha, to be thinned later) and are harvested after 4-6, and rarely more than 10, years and their harvest is followed either by coppiced growth or by replanting (West 2013). Recent interest in biomass energy has led to a wave of experiments with fast-growing tree species and to reports of some extraordinarily high yields, some in excess of 50 t/ha. Such claims are often based on a simplistic extrapolation of optimally tended small experimental plots. Care must be also taken in properly converting volumetric yield reports (a common practice in forestry) to mass and energy equivalents.

Specific density of commonly exploited species ranges from less than less than 350 kg/m3 for some firs to 500-550 kg/m3 for maples, ashes, elms and oaks. Densities change with age, and they vary even among closely related species: for example, young fast-growing plantation poplars in Washington had density of 370 kg/m3 during the first three years of growth but averaged 450 kg/m3 six years later (DeBell et al. 2002), while diverse families of loblolly pine have densities ranging between 440 and nearly 510 kg/m3 (Belonger, McKeand and Jett 1997). FAO uses generalized density of 500 kg/m3 for conifers and 600 kg/m3 for leafy trees. Energy density (for absolutely dry matter) of commonly harvested trees is mostly between 16-20 MJ/kg.

Leaving many dubious extreme claims aside, the majority of properly conducted experiments –- such as those reported (among many others) by Rao, Joseph and Sreemannarayana (2000), Hytönen (2008) and Sarlls and Oladosu (2010), and many recent studies of hybrid poplars (Klasnja et al. 2003; Paris et al. 2011; Truax et al. 2012; Di Matteo et al. 2012) –- has only confirmed the longstanding knowledge of typical yields in tree plantations (Cannell 1989; Mead 2005; Dickmann 2006). In most temperate settings with natural growth or with moderate inputs (some fertilization, perhaps some supplemental irrigation) fast-growing large-scale plantings of pines, acacias, poplars or willows will yield (depending on climate, soils, cultivars and inputs) between 5-15 t/ha. In the subtropics and tropics commonly cultivated trees (different species of Eucalyptus and Acacia, Leucaena leucocephala, Dalbergia sisoo) will yield 20-25 t/ha. Assuming 19 GJ/t the temperate rates imply power densities of 0.3-0.9 W/m2, the tropical ones range between 1.2-1.5 W/m2.

Mechanical harvesters (wheeled or tracked, first developed in Scandinavia during the 1970s and 1980s) can perform the entire felling/sawing sequence from cutting down a tree near the ground, removing branches and bucking it. Stumps, branches and tree tops remain on the site to recycle nutrients. But there is also an option to harvest entire trees (leaving only stumps) by using a sequence of a feller-buncher, chipper and truck loader; obviously, that practice recycles only nutrient in stumps. Wood chips are transported to their conversion place: their direct combustion (sometimes after preliminary drying) is used to produce heat or steam for electricity generation, or both in co-generation plants.

Combustion of woody phytomass is done most efficiently (with thermal efficiencies close to 90%) in circulating or bubbling fluidized bed boilers (Khan et al. 2009). Gasification (in low- or high-pressure gasifiers) converts as much as 80-85% of energy feed into gas whose composition is dominated by CO and H2 and whose energy density is, depending on the procedure used, as low as 5.4 MJ/m3 or as high as 17 MJ/m3 (Worley and Yale 2012). When using phytomass-derived gas in gas engines or gas turbines to generate electricity the overall efficiency of the sequence could be as high as 35-40%. Converting wood chips to methanol (CH3OH) can be done with efficiencies of up to 70% (Methanol Institute 2013).

Calculating power densities of these fuel outputs is a simple matter of multiplying the power densities of wood yields by appropriate efficiency fractions: for a highly productive tropical wood plantation yielding 20 t/ha (1.2 W/m2) the maximum rates will be then close to 1.1 W/m2 for heat generation, about 1 W/m2 for gasification, around 0.8 W/m2 for methanol production, and less than 0.5 W/m2 for gas used for electricity generation; power densities of methanol production and electricity generation based on woody phytomass harvested in tree plantations in temperate climates would be, in most cases, no more than half of the above values.

A specific example shows what a good temperate-climate harvest of 10 t/ha would imply for generating electricity by burning wood chips in a large power plant. Even if the fuel’s average energy density were 19 GJ/t the plantation would yield no more than 190 GJ/ha, resulting in harvest power density of 0.6 W/m2:




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