Hedging Strategy and Electricity Contract Engineering - IFOR

u
ø j 1
u
ì ø j 1
ì
u
ø i 1
u
ì ø i 1
ì
140 Power portfolio optimization
and solved by checking the functional value at u i i Ü
u j j 1Ü Ükô 1.
Ü
Ü r
1 and at
1Ü
á?á@á
áaá@á
Ü zÛ Ý
ë
At u i either j such that u i u j u j 1 and u i is given by
û where
given by
u i
u j
, or else u i î
uö
ë
1 Û u i Ý 0 and zÛ u i Ý is consequently
zÛ u i Ý Û Wú ¦KÝ u i Û u i Ý û Û u j Ý Û 1 ûüÝ Û u j
ë 1Ý Ü
zÛ Ý Wú zÛ Û Ý ¦KÝ
á
Ü Ý zÛ
ë
u i u i u i
Similarly, at u j either i such that u j u i u i 1 and u j is given by
û where
by
u j
u i
, or else u j î u r
ë
1 Û u j Ý 0 and zÛ u j Ý is hence given
zÛ u j Ý Û Wú ¦KÝ u j û Û u i Ý Û 1 û}Ý Û u i
ë 1Ý Û u j Ý?Ü
zÛ u j Ý Û Wú ¦bÝ u j Û u j Ý
á
The optimum is consequently found by choosing the minimum among
u zÛ i r 1ãåäæä ä ã Ý 1Ü zÛ u iâ j jâ 1ã ä äæä ã ö Ý
ë á
With the help of the dual we can now sharpen the results of Corollary 6.6.
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 water õ ë1
ë 1
ì õ
1
1.
(ii) In the general case where water is scarce, either one production weight
equals one, ë i. e.
i
1 for 1Ü an Ü i r , and two adjacent pumping
weights õ ìj á@á?á
1 and ìj
for a j 2Ü are non-zero and given by
õ õ ì
Ükô
á@á?á