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Decision-making-based optimal generation-side Chapter | 11 283
should be noted however that the elements of d up ; d down Aℜ NG may be non-
positive. In the base case scenario where there are no outages, the power
mismatch is negative P m , 0, and some elements of d up are also negative. In
this case the network is congested, hence to relieve it, the generators corre-
sponding to d up negative should provide down-spinning reserves, while the
rest of the units would provide up-spinning reserves.
11.4 Probabilistic security-constrained reserve scheduling
An optimization horizon N t 5 24 with hourly steps 1 is considered, the sub-
script t indicate the value of the quantities for a given time instance t 5 1, 2,
3, .. ., T. For the production cost a quadratic form is considered, and for the
reserves a linear cost is also considered [38]. Let c 1 ; c 2 ; c up ; c down Aℜ NG be
generation and reserve cost vectors and [c 2 ] denotes a diagonal matrix with
vector c 2 on the diagonal
For each step t, the vector of decision variables is defined as follows:
h h i i
ð
x t 5 P G;t ; d i ; d i ; R i ; R i Aℜ NG14NG 11 ϒjjÞ ð11:3Þ
up;t down;t up;t down;t
iAϒ
where R i up;t ; R i down;t are the probabilistically worst-case up down spin-
ning reserves that the system operator needs to purchase for every iAϒ.
Therefore the optimization problem is written as
!
T
X i X T T X T i T i
min P P w Þ 5 c P G;t 1 P ½P G;t 1 c R 1 c down down;t
R
ð
c 2
G;t
1
up up;t
m
N t
χ t fg t51 t51 kAZ w =K i iAZ
ð11:4Þ
11.4.1 Deterministic constraints
These are constraints that correspond to a case where the wind power is
equal to its forecast. Here the reserves are determined based on the
generation-load mismatch that may occur due to an outage.
1 T C G P G;t 1 C W P f 2 C L P L;t 5 0 ð11:5Þ
w;t
i i i f i
2P f # A P P ð11:6Þ
t inj;t w;t # P f
i
i
i
P # P i G;t 1 R P f w;t # P G ð11:7Þ
G
t
i
2R i # R P f # R i ð11:8Þ
down;t t w;t up;t
i i
R ; R $ 0 ð11:9Þ
up;t down;t
