Page 480 - A Comprehensive Guide to Solar Energy Systems

Chapter 25 • Optimal Renewable Energy Systems 493

Though this proof is simplified, it can be expanded to include any number of energy
sources and energy storage, and to include any number of time periods, for example, the
365 days in a year or 8760 hours in a year. The results are always the same: at critical times
when demand constraints bind, the marginal costs of all energy options are equal, and the
equimarginal principle holds.
Though Eqs. (25.2) through (25.8) demonstrate how the equimarginal principle applies
to designing a renewable energy system, the lagrangian approach does not necessarily
represent an easy empirical way to identify critical days or their associated marginal costs.
And the Kuhn-Tucker conditions are necessary (they must hold at the optimum solution)
but not sufficient (they may also hold in a solution that is not the global minimum). The
following section turns to an empirical example where optimization software is used to
identify the optimum portfolio of investments in a renewable energy system. As we will
see, the equimarginal principle holds for the critical days.

25.4 Economics of Renewable Energy Intermittency:
Empirical Example from Vermont

Vermont is small state in the northeastern, new England region of the United states, with
a 2016 population of 625 000. The state is mostly rural, with its largest metropolitan area,
Burlington, having 216 000 residents [11]. Vermont contains both small and large-scale
examples of most renewable energy sources, including solar PV, wind energy, hydroelec-
tric energy, and electricity generation from woody biomass. As of 2017, the state had a
goal of using 90% renewable energy by 2050, a goal endorsed by recent administrations
from both major political parties [12]. For illustration, a 100% renewable electricity goal is
considered, with all electricity sourced from within the state (though in reality, some out-
of-state sources are likely). daily electricity demand data are taken from the new England
grid that serves Vermont [13], and scaled to Vermont based on population.
The Vermont example is based on case studies of actual solar PV, wind power, hydro-
power, and biomass plants in Vermont, in addition to a pumped-hydro energy storage
facility in nearby Massachusetts. Energy data from actual generating facilities are used
to illustrate real possibilities for integration of renewable energy sources that are already
proven on a limited scale. But as specific project costs vary greatly, and in some cases
costs have declined sharply in recent years, current cost data are used to represent Us
national averages.
The solar PV component of the example is based on a 2016 installation of 2.7 MW capacity,
in Williston, Vermont. The project is owned by Vermont’s largest electric utility, Green Moun-
−1
tain Power. it produces an estimated 8186 MW h a (where a refers to year) (19.9% capacity
factor) from 21 417 solar PV panels on land area of approximately 11 ha [14]. daily produc-
tion is estimated from 2005 solar insolation data at the nearby Burlington airport [15]. The
−1
estimated cost of $1375 (kW AC ) is based on the midpoint of the lazard [16] range, yielding
6
6
an expected cost for a new project of $6.5 × 10 (actual cost was approximately $12 × 10 ) [17].