Environmental Engineering Reference
In-Depth Information
Appendix G
KKT optimality conditions
For convex non-linear constrained optimisation problems of the form the time-
coordinated optimal power flow (TCOPF) problem presents, optimality conditions
due to Karush, Kuhnt and Tucker (KKT) apply, provided some circumstances are
satisfied. These conditions are based on the Lagrange function problem, which can
be stated as [121]:
L
(
x
,
λ
)
=
f
(
x
)
+
λ
1
g
(
x
)
+
λ
2
h
(
x
)
(G.1)
⎧
⎨
f
(
x
) is a scalar-valued objective function
g
(
x
) is the vector of equality constraints
h
(
x
) is the vector of inequality constraints
λ
1
is the vector of Lagrange multipliers from equality constraints
λ
2
is the vector of Lagrange multipliers from inequality constraints
where
⎩
Finally, the KKT conditions need to meet the following set of circumstances:
∂L
∂x
=
0
(G.2)
g
(
x
)
=
0
(G.3)
h
(
x
)
≤
0
(G.4)
λ
2
h
(
x
)
=
0
(G.5)
λ
2
≥
0
(G.6)
Once these conditions are satisfied, the results provide relevant information such
as the marginal objectives of the variables.
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