Page 323 - Decision Making Applications in Modern Power Systems

284 Decision Making Applications in Modern Power Systems

T i
T i
1 d up;t 5 1; 1 d down;t 5 1 ð11:10Þ

11.4.2 Probabilistic constraints
These are constraints that deal with the uncertainty of the wind power fore-
cast. The reserves are now characterized by both the generation-load mis-
match that may occur due to the wind power forecast error and the outages.
We thus have that for all t 5 1, 2, 3, ..., N t ,

i i i i
PðP w;t Aℝ 2 P # A P P w;t # P

f inj;t f
i
i

P # P i 1 R P w;t # P i ð11:11Þ
G G;t t G

i
2 R i # R P w;t # R i
down;t t up;t ; for all ðiAZÞ $ 1 2 ε t
The probability is meant with respect to the probability distribution of the
w
wind power vector P w Aℜ . The last constraint in (11.11) is included to
determine the reserves R i ; R i as the worst case, in a probabilistic
up;t down;t
i
sense, R is the power correction term. The reserves that the system operator
t
will need to purchase are then determined as
R up;t 5 max R i up;t ð11:12Þ
iAZ
R down;t 5 max R i down;t ð11:13Þ
iAZ
which denote the worst-case values, given all the outages, of R i ; R i ,
up;t down;t
respectively. In (11.11) the same probability level is considered for each
time step t 5 1,..., N t ; different probability levels per stage or a joint chance
constraint for all stages can be captured by the proposed framework as well.
In line with the formulation an additional AGC/LFC functionality is pro-
posed. The system operator must monitor both the production of the tripped
plant and the deviation of the wind power from its forecast. Thereafter, by
using (11.1) as a look-up table the appropriate distribution vector, among
those computed in the optimization problem, is selected as shown in
Fig. 11.2.
The resulting problem given by (11.10) and (11.11) is a chance-
constrained bilinear program whose stages are only coupled due to the tem-
poral correlation of the wind power. Further coupling among the stages can
be obtained if a unit-commitment problem was included or ramping con-
straints of the generating units and minimum up and down times were mod-
eled [35]. The two major challenges faced when attempting to solve problem
(11.3) (11.11) are as follows: the first is due to the presence of bilinear
terms that are a result of the products of d i up;t ; d i down;t , and P G,t for iAϒ G , the
second is owing to the presence of the chance constraint.