1.3. Techniques for the Optimization
of a Water Resources
System
27
vector comprising j = 1, .... m stages, φι the η-dimensional functional,
and xo a constant initial-condition vector. In addition there is an objective
function
Tmax
/(x,y) = Σ
fr(*i,yi,0
(1.8)
and some inequality constraints that define the feasible region for search,
xf
n
< *t < x?
ax
y?
in
< yt < yT
x
(1.9)
The problem is to determine yo,. . . , yr max so that/(x) is maximized subject
to relations (1.7) and (1.9). At any stage (k + 1) the optimum for that
stage is the optimal return from all the downstream (in time or space)
stages plus the return for the (k + l)st stage:
Tmax
/(x, y) = max {/*+i(x, y, t) + max Σ /*(», y> 01
(1.10)
y*+i
t=k+2
As a very simple numerical example, consider the one-period optimization problem of the operation of three reservoirs in series as shown in Fig.
1.9. Storage and release are for one time period t. Minimum and maximum
levels corresponding to expressions (L9) are specified for the river flows
Q jh j=
1,2,3,
4 < Qu < 12
4 < Q 2t < 12
4 < Q zt < 12
and the water for irrigation Dy
0 < D u < 12
0 < D 2t < 12
0 < D zt < 12
The system equations corresponding to Eq. (1.7) are linear equality constraints that represent the material balances on the water flows for one
period At if there are no changes in reservoir levels, i.e., £/ f t+i — Sjt = 0
because the problem is for one period only:
Qu = Qot — Du
Qu = Qu — D 2t
Qzt = Qzt — D zt
Note that the irrigation water releases might become delayed returns to
the next reservoir, so that the irrigation water released during the period
At might be Dj t leaving the jth reservoir but yDj tt -k entering the (j + l)st
reservoir, where k represents a delay of one or more periods and y is the
fraction of water retained.
One of the advantages of dynamic programming is that the objective
function does not have to be an analytical objective function nor contain
continuous variables. Suppose we let the objective function be a table of
data giving the values in thousands of dollars of the irrigation water with-
of a Water Resources
System
27
vector comprising j = 1, .... m stages, φι the η-dimensional functional,
and xo a constant initial-condition vector. In addition there is an objective
function
Tmax
/(x,y) = Σ
fr(*i,yi,0
(1.8)
and some inequality constraints that define the feasible region for search,
xf
n
< *t < x?
ax
y?
in
< yt < yT
x
(1.9)
The problem is to determine yo,. . . , yr max so that/(x) is maximized subject
to relations (1.7) and (1.9). At any stage (k + 1) the optimum for that
stage is the optimal return from all the downstream (in time or space)
stages plus the return for the (k + l)st stage:
Tmax
/(x, y) = max {/*+i(x, y, t) + max Σ /*(», y> 01
(1.10)
y*+i
t=k+2
As a very simple numerical example, consider the one-period optimization problem of the operation of three reservoirs in series as shown in Fig.
1.9. Storage and release are for one time period t. Minimum and maximum
levels corresponding to expressions (L9) are specified for the river flows
Q jh j=
1,2,3,
4 < Qu < 12
4 < Q 2t < 12
4 < Q zt < 12
and the water for irrigation Dy
0 < D u < 12
0 < D 2t < 12
0 < D zt < 12
The system equations corresponding to Eq. (1.7) are linear equality constraints that represent the material balances on the water flows for one
period At if there are no changes in reservoir levels, i.e., £/ f t+i — Sjt = 0
because the problem is for one period only:
Qu = Qot — Du
Qu = Qu — D 2t
Qzt = Qzt — D zt
Note that the irrigation water releases might become delayed returns to
the next reservoir, so that the irrigation water released during the period
At might be Dj t leaving the jth reservoir but yDj tt -k entering the (j + l)st
reservoir, where k represents a delay of one or more periods and y is the
fraction of water retained.
One of the advantages of dynamic programming is that the objective
function does not have to be an analytical objective function nor contain
continuous variables. Suppose we let the objective function be a table of
data giving the values in thousands of dollars of the irrigation water with-