Hedging Strategy and Electricity Contract Engineering - IFOR

4i 6
a4i
and b4i 7 16
a4i 7
1 8
_
196 Proofs
Proof The trivial case where we have abundance of water is obvious, and in the
general case the average price of produced electricity is given by
G 1
H i
a4i 1
9 4
i 7
i H b4i 7
1
94
a4i
i
b4i
8
Hence we are comparing the slope of the line connecting the points
G 1 9 4
94
0 6 G 0 9 4 and H 1
i
b4i
9 4
1 i 6 G b4i
9 4 1 H i
a4i 1
9 4
i 7 7
a4i
with the line between
Observe that a r and b r will go to zero when the threshold S4r goes to infinity,
hence following from the concavity of a in b, it is obvious that the latter slope
cannot exceed the former slope.
Corollary 6.6
In the solution of (6.26)
(i) If b41 then 1
1 and 9 7
1
1 is optimal
4 WK 9
(ii) If WK d X
b4r d b4
e then the problem is infeasible
(iii) Else at most two adjacent 9 4
production weights
two adjacent 9 pumping weights 7j 1 and j
7 9 7
i
and at most
will be non-zero.
i 7
1 and 9 4
Proof The results again follows from Theorem 6.1. In case (i) we optimally
produce and pump at as low prices as possible, which again follows from
a being monotone increasing in b. Case (ii) is obvious and case (iii) again
follows directly from the concavity.
_
Proposition 6.7
In the solution of (6.26) at the most three weights are non-zero and given by
(i) In the trivial case, where there is abundant 9 water 9:7 41 1
1.