Page 478 - A Comprehensive Guide to Solar Energy Systems

Chapter 25 • Optimal Renewable Energy Systems 491

For renewable energy systems, the equimarginal principle can be modified as follows:
for an energy system that can meet energy demand at each moment in time, marginal
costs should be equal for energy sources plus storage options at critical times of limited
energy availability. For example, on a still night with no wind or solar energy available, the
marginal cost of obtaining more hydropower should equal the marginal cost of wind or
solar energy plus required storage.
Formally, the objective is to minimize total cost TC for renewable energy. For simplicity,
here we consider only solar energy s and wind energy w on days 1 and 2, but the proof can
be extended to any number of energy sources and energy storage options operating over
any number of time periods. The function f describes the capital and operating costs of
obtaining energy quantities s and w. The marginal cost of obtaining energy depends on the
renewable source (s or w) and the day (1 or 2), given daily fluctuations in ambient energy.
The objective function is thus:
,,
(,
Minimize TC = fs ws w ) Minimize TC=f(s 1 ,w 1 ,s 2 ,w 2 )
1
1
2
2
subject to constraints that total energy production exceeds demand d on each day:
s 1 + w 1 ≥ d 1
s 2 + w 2 ≥ d 2
s 1 +w 1 ≥d 1 s 2 +w 2 ≥d 2
and to nonnegativity constraints, as energy quantities cannot be negative:
sw sw 2 ≥ 0 (25.2) s 1 ,w 1 ,s 2 ,w 2 ≥0
,
,
1
1
, 2
The lagrangian method can be used to describe the optimum solution, where the
Greek letter lambda (λ) is the lagrangian multiplier [10]:
+
L = fs ws w ) λ 1 d ( 1 − s 1 − w ) λ 2 d ( 2 − s 2 − w ) (25.3) l=f(s 1 ,w 1 ,s 2 ,w 2 )+λ 1 (d 1 −s 1 −w 1 )+λ 2 (
+
,
,
2
1
1
, 2
( 1
2
d 2 −s 2 −w 2 )
since the objective function (Eq. (25.2)) includes inequality constraints, the Kuhn-
Tucker conditions (Eqs. (25.4)–(25.7)) are necessary for the least-cost solution ([10];
[eq. 13.17, p. 410]):
2 λλ
sw sw , , 2 ≥ 0 (25.4) s 1 ,w 1 ,s 2 ,w 2 ,λ 1 ,λ 2 ≥0
,
,
1
1
1
, 2
This condition reiterates the nonnegativity requirement for the energy quantities, and
extends it to the lagrangian multipliers. in this type of optimization problem, λ has a spe-
cific interpretation: it is the marginal cost of slightly tightening the associated constraint
[10], a cost which cannot be negative.
L 1 ≤ 0 → d 1 − s 1 − w 1 ≤ 0 → d 1 ≤ s 1 + w 1
λ
L 2 ≤ 0 → d 2 − s 2 − w 2 ≤ 0 → d 2 ≤ s 2 + w 2 (25.5) lλ1≤0→d 1 −s 1 −w 1 ≤0→d 1 ≤s 1 +w 1 lλ2≤
λ
0→d 2 −s 2 −w 2 ≤0→d 2 ≤s 2 +w 2