FLOOD01647

• County: Statewide
• Title: Optimal Sizing of Urban Flood Control Systems
• Date: 3/1/1974
• Prepared For: US
• Prepared By: COE
• Floodplain - Doc Type: Educational/Technical/Reference Information

"

project the trend to determine improved component sizes is the Newton-Raphson
convergence procedure. The optimization methodology proceeds as follows:

I. Trial sizes of all system components are nominated and the entire system
is simulated in all of its hydrologic, cost, and economic detail to calculate the
value of the objective function, which for unconstrained optimization is the
sum of the equivalent annual cost and expected annual damage.
Z. The size of one component is decreased by a small selected amount (1%)
and the simulation is repeated for the entire system to compute a new value
of the objective function. This is repeated again resulting in three unique values
of the objective function for small changes in the size of one component.
3. From these three values, an estimate is made of the component size that
would result in the minimum value of the objective function. The computation
of the adjustment is shown in Fig, 4 and proceeds as follows:

r ( Xo - t..ZX) = tan e = f' ( Xo - t..ZX) [ ( Xo - t..ZX) - X* rl . . . . . . (4)

or X*=Xo-[f'(Xo- t..;)][r(xo'- t..;)rl _ t..;....," .(5)

in which f' (Xo - t..ZX) = U(Xo) - f(Xo - t..X)] (t..X) -I . . . . , . . . . (6)

r (x - t..X) = [f(X - Zt..X) - Zf(X - t..X) + f(Xo)] (t..X) -2 . . . , . (7)
o 2 0 0

and t..X = incremental change in X; X = size of variable being optimized;
Xo = present size of component X; and X* = projected "new" size for X.
4, After adjustment of the size of the system component, the entire system
is simulated again in detail to compute the new value of the objective function
and, provided the objective function has decreased, the procedure then moves
to the second system component whose scale is to be optimized.
5, The foregoing procedure is repeated for the second and all subsequent
components to be optimized.
6, A single adjustment has now been made for each component for one complete
search of the system component sizes. The procedure is then repeated for two
more complete system s~arches.
7. The component whose change contributed the most to decreasing the
objective function is adjusted next before another complete system search is
performed.
8. The procedure is terminated when either no more improvement in the
objective function can be made (within a tolerance) for the component making
the greatest contribution to decreasing the objective function, or the complete
search cycle is completed.

I'

The efficiency of the search procedure and the degree of success in determining
the optimum sizes for the components is a function of the behavior of the
objective function and the starting values, If the objective function varies
erratically with small adjustments in the component scales, chances of finding