Equation 6 · Power Density and What It Permits
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol P_d
is part of the quantity the equation computes from the expression on the right.
Symbol E
E occurs above the fraction bar. The numerator is divided by the entire denominator below it.
Symbol A
A occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol t
t occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
Denominator: At
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
The second confusion is with capacity factor . Power density is an area-normalised output rate; capacity factor is a time-normalised one. They multiply rather than substitute. Writing energy generated as E over a period t across area A , with nameplate capacity C and capacity factor f : . The first factor, capacity density , is a siting and engineering question; the second is a resource and dispatch question. Miller and Keith’s US measurements make the separation visible: they report 2016 mean capacity factors of 32.9 per cent for wind and 22.1 per cent for solar, with 90th-percentile values of 43 and 27.5 per cent, alongside power densities that are not in the same…
Read the full surrounding passage
The second confusion is with capacity factor . Power density is an area-normalised output rate; capacity factor is a time-normalised one. They multiply rather than substitute. Writing energy generated as E over a period t across area A , with nameplate capacity C and capacity factor f : . The first factor, capacity density , is a siting and engineering question; the second is a resource and dispatch question. Miller and Keith’s US measurements make the separation visible: they report 2016 mean capacity factors of 32.9 per cent for wind and 22.1 per cent for solar, with 90th-percentile values of 43 and 27.5 per cent, alongside power densities that are not in the same ratio at all [ 4 ] . A technology can improve its capacity factor for years without moving its power density, and the reverse. Van Zalk and Behrens found solar the only one of nine energy types with a statistically significant trend in power density over time, rising an estimated 0.42 watts per square metre per year; wind’s 0.17 per year did not reach significance [ 3 ] .
Sources cited in the surrounding passage
- [4] Observation-based solar and wind power capacity factors and power densities ↗
- [3] The spatial extent of renewable and non-renewable power generation: A review and meta-analysis of power densities and their application in the U.S. ↗
These citations give research context. Read each source to check which claims it supports.
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