Online Companion for
Robust Unit Commitment with Dispatchable Wind:
An LP Reformulation of the Second-stage
Germán Morales-España
a,†, Michael Davidson
b,
Laura Ramírez-Elizondo
a, and Mathijs M. de Weerdt
ca
Department of Electrical Sustainable Energy, Delft University of Technology, The Netherlands
b
Engineering Systems Division, Massachusetts Institute of Technology, USA.
c
Department of Software and Computer Technology, Delft University of Technology, The Netherlands
†
E-mail: g.a.moralesespana@tudelft.nl
This document is an online companion1for [1].
1
LP Reformulation of the Second-stage Robust UC
1.1
The Second Stage Problem for uncertain Wind or Solar
This report shows the step-by-step procedure of eliminating some variables of the problem presented in [1] through the Fourier Motzkin procedure.
As stated in [1], the the second-stage max-min problem of the robust unit commitment with dis-patchable wind, which leads to a bilinear problem, is equivalent to the following LP formulation:
min p,w,υ+,υ− c⊤p+ d⊤w s.t. Hp+ Jw ≤ h (1) Bp+ Cw ≤ ˜g (2) w+ W⊤υ+− W⊤υ−≤0 (3) υ−− υ+≤ 1 (4) p, w, υ+, υ−≥0 (5)
where W and W are diagonal matrices containing the vectors w and w in the diagonal, respectively. Since W and W are diagonal symmetric matrices, then W = W⊤ and W = W⊤.
By applying the Fourier Motzkin elimination, we obtain the following equivalent formulation where 1Last update 2015-07-24
variables v+ and v−no longer appear: min p,w c⊤p+ d⊤w s.t. Hp + Jw ≤ h (6) Bp+ Cw ≤ ˜g (7) w≤ w (8) p, w ≥ 0. (9)
The following subsections show the step-by-step Fourier Motzkin procedure of eliminating variables v+ and v−.
1.2
Eliminating υ
+To start, constraints (3)-(5) can be rewritten as
υ+ ≤ W−1 W υ−− w (10)
−1+ υ−≤ υ+ (11)
0 ≤ υ+ (12)
w, υ−≥0 (13)
where (10) is equivalent to (3) after multiplying (3) by W−1. Notice that W−1 is a positive matrix, then the inequality does not change its sign (≤).
The Fourier Motzkin elimination consists in combining all the upper bounds of υ+ (10) with their
lower bounds (11)-(12), hence the following equivalent set of inequalities can be obtained:
−1+ υ− ≤ W−1 W υ−− w (14)
0 ≤ W−1 W υ−− w (15)
w, υ−≥0 (16)
by multiplying both sides of (14)-(15) by W, we obtain:
−w+ Wυ−≤ W υ−− w (17) 0 ≤ W υ−− w (18) w, υ−≥0 (19) Now, reorganizing: W− W υ− ≤ w − w (20) W υ−≥ w (21) w, υ−≥0 (22)
and multiplying (20) and (21) by W − W−12
and W−1, respectively: υ−≤ W − W−1 (w − w) (23) W−1w≤ υ− (24) 0 ≤ υ− (25) w≥0 (26)
2Notice that W − W−1≥ 0 since W ≥ W by definition.
1.3
Eliminating υ
−Again, by applying the Fourier Motzkin elimination, we combine all the upper bounds of υ−(23) with
the lower bounds (24)-(25). Hence, the following set of constraints are equivalent to (23)-(26) where variables υ−no longer appear:
W−1w≤ W − W−1
(w − w) (27)
0 ≤ W − W−1
(w − w) (28)
w≥0 (29)
and multiplying (27) and (28) by W − W
W− W W−1w ≤ w − w (30)
w ≤ w (31)
w ≥0 (32)
which by reorganizing becomes
WW−1− I+ I w ≤ w (33)
w≤ w (34)
w≥0 (35)
where I is the identity matrix.
Now, by multiplying both sides of (33) by WW−1−1
, which is equivalent to W⊤W⊤−1 =
WW−1:
w≤ W W−1w (36)
w≤ w (37)
w≥0 (38)
Since W is a diagonal matrix with elements in its diagonal wii 6= 0 , the inverse W −1
is also a diagonal matrix with elements in its diagonal1/w
ii. Then the matrix operation W −1
w is a column vector of ones (1). Therefore, (36)-(38) become
w≤ w (39)
w≤ w (40)
w≥0 (41)
where (40) is redundant because it is dominated by (39), which imposes a tighter upper bound to w. Finally, after Fourier Motzkin elimination, constraints (3)-(5) are equivalent to
w≤ w (42)
w≥0 (43)
therefore, the formulation (1)-(5) is equivalent to min p,w c⊤p+ d⊤w s.t. Hp + Jw ≤ h (44) Bp+ Cw ≤ ˜g (45) w≤ w (46) p, w ≥ 0. (47) 3
References
[1] G. Morales-Espana, D. Michael, L. Ramírez-Elizondo, and W. Mathijs M. de, “Robust Unit Com-mitment with Dispatchable Wind: An LP Reformulation of the Second-stage,” Delft University of Technology, Technical Report, 2015.
