Page 479 - A Comprehensive Guide to Solar Energy Systems

492 A CoMPrEhEnsiVE GUidE To solAr EnErGy sysTEMs

subscripts denote partial first derivatives of the lagrangian equation (e.g., l λ1 = ∂l/∂λ 1 ).
These conditions reflect the demand constraints, that total energy production on each
day must be equal to or greater than demand.
≥
L s1 ≥ 0 → f s1 λ ≥ 0 → f s1 λ 1
−
1
−
L w1 ≥ 0 → f w1 λ ≥ 0 → f w1 λ 1
≥
1
ls1≥0→fs1−λ 1 ≥0→fs1≥λ 1 lw1≥0→ L s2 ≥ 0 → f s2 λ ≥ 0 → f s2 λ 2 (25.6)
≥
−
2
fw1−λ 1 ≥0→fw1≥λ 1 ls2≥0→fs2−λ 2 ≥0 L w2 ≥ 0 → f w2 λ ≥ 0 → f w2 λ
−
≥
→fs2≥λ 2 lw2≥0→fw2−λ 2 ≥0→fw2≥λ 2 2 2
These are similar to the classic first-order conditions for a lagrangian equation with
equality constraints, and reflect the marginal costs of energy on each day. For example,
f s1 is the marginal cost of solar energy on day 1. By themselves, these conditions do not
require that marginal costs be equal.
=
1 = 0 → ( 1 λ− ) 0
s L s1 s f s1 1
1 = 0 → 1 λ ) 0
=
−
w L w1 w f ( w1 1
=
−
s 1 ls1=0→s 1 (fs1−λ 1 )=0w 1 lw1=0→w 1 (fw1−λ 1 s L s2 2 = 0 → s f ( s2 2 λ 2 ) 0 (25.7)
=
−
)=0s 2 ls2=0→s 2 (fs2−λ 2 )=0w 2 lw2=0→w 2 (fw2 w L w2 2 = 0 → w f ( w2 2 λ 2 ) 0
−λ 2 )=0
This is the first of two so-called complementary slackness conditions. For solar energy
on day 1, for example, either s 1 = 0 (no solar energy is used in the solution), or (f s1 − λ 1 ) = 0
(which requires f s1 = λ 1 , making the value of the second term zero). note that if a λ i is zero,
these conditions require that either the respective s i or w i is zero (no solar or wind is used),
or that f si or f wi is zero (that marginal cost is zero). For a feasible solution, this cannot hold
in all four equations, that is, some energy with a nonzero cost must be provided. Either λ 1
or λ 2 must take a positive value in the solution.
=
1 λ L 1 = 0 → 1 λ d ( 1 − s 1 − w ) 0
λ
1
λ 1 lλ1=0→λ 1 (d 1 −s 1 −w 1 )=0λ 2 lλ2=0→ 2 λ L 2 = 0 → 2 λ d ( 2 − s 2 − w ) 0 (25.8)
2 =
λ
λ 2 (d 2 −s 2 −w 2 )=0
The final conditions are again for complementary slackness: on day 1, either λ 1 = 0 or
d 1 = s 1 + w 1 (so that supply exactly equals demand, and the second term is zero). This condi-
tion is key to demonstrating equality of marginal costs. For example, when d 1 − s 1 − w 1 = 0,
or when the demand constraint is binding on a critical day, λ 1 can take a value greater than
zero, that is, if just enough solar panels have been deployed to meet demand on this day.
in Eq. (25.7) for solar energy on day 1, a positive value of λ 1 allows s 1 to take a positive
value (the optimum solution uses solar energy on this day) and the marginal cost of solar
to have a nonzero value (f s1 > 0), but only if f s1 = λ 1 . similarly, a positive value of λ 1 allows w 1
to be positive in the second line of Eq. (25.7), and f w1 = λ 1 . And if both f s1 and f w1 are equal to
λ 1 , then f s1 and f w1 are equal to each other (λ 1 = f s1 = f w1 ): the marginal costs of solar and wind
energy are the same. When demand constraints bind, that is, on critical days when the total
energy supply equals demand without exceeding it, the marginal cost of obtaining renew-
able energy is equal for each renewable energy source and for stored energy. if this were not
the case, that is, if marginal energy costs were different on critical days, it would be possible
to reduce total cost with a solution that more fully exploited the lower-cost resource.