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420 Decision Making Applications in Modern Power Systems
d X
P 5 pUη c ð16:6Þ
d
nAN p tðÞ
d
b d agg Up # PtðÞ # b d agg UP ; ’tA 0; T ð16:7Þ
½
1=2
The aggregate energy boundaries, that is, E d ðtÞ, are obtained by summing
1
up e ðtÞ, which is the accumulated energy from the as-fast-as-possible charg-
n;d
2
ing pattern, and e ðtÞ, which is from the as-late-as-possible charging pattern.
n;d
In addition, the total power consumption value should be lower than the aggre-
gated power from all available vehicles at time t. Note that the discontinuity of
the aggregated power is also modeled, similar to that in Eq. (16.1). The optimal
power consumption profiles for day-ahead operations can be used as the refer-
ence for EVs to follow during the real-time operations in distributed and asyn-
chronous fashions, which are, however, not the focus of this chapter.
16.3.3 Time-of-use tariff structure
For commercial sites in California TOU markets, two categories of costs are
generally applied to customers’ bills, that is, energy charge and demand
charge. Energy charges are calculated by the product of amount of electricity,
measured in kilowatt-hours (kWh) used per time period, and the per-kWh rate
for those respective time periods. Demand charge is calculated by using the
maximum load measurement in each demand period, multiplied by the corre-
sponding demand charge rate, in $/kW. Thus the total monthly cost of energy
charge is modeled by Eq. (16.8), where the cost of energy consumption in dif-
ferent time periods is also included. Eq. (16.9) models the total monthly
demand charges, where I denotes the set of the demand charge periods. In the
case of the E-19 tariff in the PG&E territory, there are three demand charge
periods for summer months, that is, peak, part-peak, and anytime max periods,
while two periods in the winter, that is, part-peak and anytime max periods.
d
X X X N p ðtÞ
d
d
C EC 5 ðL tðÞ 1 P ðtÞÞUΔtUλðtÞ ð16:8Þ
dAD tAT n51
d
X X N p ðtÞ
d
d
C DC 5 max L tðÞ 1 P tðÞ Uω i ð16:9Þ
n
T i AfT p ;T pp ;T M g tAT i
dAD
Thus to minimize the monthly energy bills by considering only the
energy charge and demand charges, a deterministic optimization problem is
formulated as follows:
Problem 1—TOU charges (energy charge 1 demand charge)
Objective minimizeðC EC 1 C DC Þ
Subject to (16.1) (16.9)
