Hedging Strategy and Electricity Contract Engineering - IFOR

g
b4i d G9:7j b4
j
G 1 9j7j H b4
j 7 1H WK d
(ii) In the general case where water is scarce, either one production weight
197
equals one, 4 i. e.
i
1 for an
8J88 6 i 1 6 r , and two adjacent 9 pumping
weights 7j 1 and j
for a j 2 6 88J8 6 < are non-zero and given by
7 9 9 7
9:7j
d d b4i j
WK b4
1 7
b4
j j 1H 6 9:7j 1
1 9:7j 7 8
b4 G 7 d
Or else one pumping weight equals one, i. 9 e. 7j
1 for an j 1
8J88 6 < 6
i 1
and 94
i
for an i 2 6 88J8 6 r
7
and two adjacent production 94 weights
are non-zero and given by
4
i
9
d j
b4i
d
1
WK
9 4 4
i 1
9 4 1
i
8
b4i
6
b4i 1 7 4
Proof Assume u j minimizes the dual unique i 1
8J88 6 6 r such that
u j u i u i4 1 . Then by applying the complementarity condition
6
and by observing that b4k u j a4k
0 k i, it directly follows that
4
i
9
1. Further by again applying the complementarity condition
7 q G d b4q u j 9
d
H = 8J88 6 < 6 6
and observing that b4q u j 6 g h d i 6
9 9 7 6 9 7 6
7 9
a4q 0 q 1
a4q 0 q j 1 j , it follows that
q 0 q j 1 j and hence that 7j 1
7j
1. Observe that the
special case, where u 1 u 1 0 minimizes the dual, corresponds to abundant
water and 941 1
1 follows from the complementarity conditions. Else
observe that, since the water constraint is binding
we can directly solve for G9 7j 7 16 9 7j H .
Else u i minimizes the dual unique j 1
8J88 6 < 6 such that
u i u j u j4 1 . A similar procedure now applies and by the complementarity
conditions it follows that 9
6
7j
1 and the production weights can