Page 484 - A Comprehensive Guide to Solar Energy Systems

Chapter 25 • Optimal Renewable Energy Systems 497

in the winter. on the other hand, hydropower and wind energy costs are positively correlated,
with hydro and wind energy being potential substitutes. Price spikes on individual days (due
to low ambient energy availability) are not necessarily a large concern, since energy from
previous, less expensive days can be stored for such days. The objective is to balance differ-
ent energy sources and storage costs to arrive at the minimum total cost for the year.
a
optimization software can be used to identify solutions for constrained optimization
problems such as the one described in section 25.3. For this example, the optimization
software is set to pick combinations of energy production capacity from solar PV, wind,
hydropower, and biomass energy, and well as energy storage capacity and storage quan-
tity. Capacity for each intermittent source results in production estimates by source for
each of the 365 days in a year, given ambient conditions for each day. if biomass invest-
ments have been chosen and demand exceeds ambient supply on a particular day, bio-
mass energy may be used. similarly, including storage investments allows any excess
energy to be stored and consumed later, if demand should exceed supply.
Production capacity choices result in capital expenditures, which are amortized at an
interest rate of 8%. Annual operating expenses are added to arrive at total annual expense.
The optimization software attempts to minimize total annual expense, subject to the con-
straint that energy supplied is greater than or equal to demand on each of the 365 days in
the year. Biomass use is also constrained to the total annual biomass availability. The opti-
mization software iteratively picks combinations of renewable energy sources and storage,
until total cost cannot be further reduced.
results for this example are shown in Table 25.2. Given the assumptions earlier, and the
weather patterns observed for single representative years (which vary by energy source),
the minimum-cost solution includes 42% of Vermont’s electricity from solar PV, 45% wind
power, 13% hydropower, and no biomass, plus an energy storage plant of approximately
41% of the size of the northfield Mountain case-study site.
While this solution reflects the equimarginal principle, this result is not obvious from
looking at any particular day, where marginal costs may differ. Though on a day when
marginal costs differ, supply could be shifted from higher to lower marginal cost sources
to reduce costs, this may result in failing to meet the supply constraint on other critical or
near-critical days. Thus all critical days must be considered as a group. For example, con-
sidering for simplicity only the nondispatchable sources (scenario 4 in Table 25.3), the five
days where supply is most limited have less than 6% excess capacity. For the total energy
produced on these five days, there is less than a 5% difference between the lowest and
highest marginal costs. The equimarginal principle thus holds, approximately, when all
the critical and near-critical days are considered together.

a Finding a global minimum cost from among the many combinations of different energy sources is a
nontrivial mathematical problem. For this example, Microsoft Excel’s Solver add-in is used. note that results
may be sensitive to Solver settings, for example, the length of time in which solver is allowed to search for
lower-cost combinations. other software including Mathematica, Matlab, GAMS, and Stata can perform similar
optimization routines. An optimization result is not necessarily unique, and is not guaranteed to be a global
minimum—results should always be checked for consistency and reasonableness.