1.3. Techniques for the Optimization
of a Water Resources System
23
Fig. 1.9 Three-reservoir system.
Fig. 1.9. Using units for the notation as given in Appendix C, let Sj t be the
storage volume of water in reservoir j at the beginning of period t 9 Dj t the
amount of water supplied for irrigation from reservoir j in period t } and
Q mt the flow in stream m during period t—all in acre-feet. The mass balances
on the water flow are
accumulation
input
output
Su
= Qot -Qu -D u
Sn
= Qu -Qu -
D u
Szt = Qu -- Qzt
and the initial conditions are Sjq. Inequality constraints come into play
because releases to the streams between the reservoirs must always exceed
certain minimum levels for downstream usage but must not exceed flood
levels:
QT
< Qmt < « x
m = 1, 2, 3,
t = 1,. . ., rmax
Similar upper and lower bound exist on the capacity of the irrigation
canals:
Df
n
< D jt < Z)jT
j = 1, 2, 3, t = 1,..., 5T max
A third set of constraints pertains to the storage capacity and minimum
level in each reservoir:
< Sjt ^ S™**
j = 1, 2, 2, 2 = 1,..., Tmax
Keep in mind that the upper and lower bounds are known constants. To
put the above bounds in the form of inequality (1.1b), the lower bounds for
Qmt are QZ
n
- Q m t < 0 and the upper bounds are Q mt - Q£f
x
< 0; the
other bounds can be similarly rearranged.