A framework for joint management of regional water-energy ... - Orbit

The change in the objective function value at stage t with respect to a change
in inflow during stage t – 1 is obtained from:
*, k
*, k k
t
F Q
t
tm ,
= ⋅
k
t-1, m t, m t-1,
m
L
k l,
k l st
( n)
= ptm ,
+ ltm ,
g t+ 1 ⋅ç
ç rt-
1, t
ç è
l = 1 ÷ ø
ç st
-1( n)
k
= gtm
,
( n)
F
Q Q Q
æ ö æ ö
( n) ( n) ( n)
å ÷ ç ÷
(21)
ç è ø
*,k
k
Note that the partial derivative of F
t
with respect to Q
tm ,
is obtained by adding
the impact of inflow changes on the current and the future operation of
the system. The partial derivative of current inflow with respect to previous
inflow is obtained from the inflow model in (13). The vector of expected
slopes with respect to inflows is given by:
g
K
1 k
tm ,
n gtm
,
K
k=
1
The constant scalar d is calculated as:
tm ,
( ) = å ( n)
(22)
1 K *, k T *
tm ,
= Ft + jtm ,
Etm
,
K
k = 1
d
å (23)
A set of M cut parameters d tm ,
, j , and
tm ,
g are generated at every stage t of
tm ,
each backward iteration, and added to the set of L parameters (calculated in
previous iterations) that approximate the FCF at stage t − 1. The number of
cuts L grows by M in each iteration, gradually improving the approximation
until convergence.
The creation of the backward openings and the cuts is illustrated in Figure 11.
k
The openings are created by adding the inflow noise vector Q
tm ,
to the corresponding
storage E obtained from the mth forward simulation (Figure 11a).
tm ,
*
The K cuts created for each m forward scenario (Figure 11b) are averaged to
construct M cuts (Figure 11c). The maximum over the M cuts is the linear
approximation to the FCF (Figure 11d).
Note that the M resulting cuts will be added as constraints in all M scenarios
at stage t – 1. In order to “share” the cuts among scenarios, the inflow scenario
tree used in the forward simulation must have common samples, i.e. the
distribution of the inflow noise vector must be the same for all scenarios
within a given stage t and independent of the inflow history. In this case, the
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