Where p (x) denotes the probability density of peak flow x;
c (x) is the corresponding social cost (‘flood damage’); and
x^1 denotes the moderated outflow corresponding to the flood discharge
level x.
In the case of a reservoir storage system of flood control, let x1 = ?(x) represent the
reservoir operation policy. Then, sub-stituting this equivalence into the first equation
gives
B = ƒx p (x){c(x)-c(?(x)^1 )]}dx
Assuming a stationary probability distribution p (x) over the relevant period, the present
value of an annuity of B per period over the life of the project at the appropriate rate of
discount gives the gross benefit of the project. By subtracting from this figure the
relevant investment and operation costs similarly ‘discounted’ to the base year; we can
get the appropriate measure of the net benefit of the project.
If c (x^1 ) = 0, we have B= ƒx p (x). c(x).c.dx which representsGross benefit of complete protection. The corresponding present value gives the
‘expected present social value’ of complete protection. A comparison of this value with
the social cost involved can in principle determine whether complete protection is ‘worth
while’.
It is likely that the reservoir operation policies as represented by x^1 =?(x) will vary with
the peak flow, x. Suppose they vary as follows:
If x > x1, then x^1 =? 1 (x) ;
If x 1 = x < x2, then x1 = ?2 (x)
......................................
And so on.
The gross benefit of flood control becomes
B = ƒx p (x){c(x)dx [ƒ^8 x1 p(x)c{? 1 (x)}dx+ ƒ^1 x2 p(x)c{? 1 (x)}dx.......]
Given the operating policy, the probability distribution p(x) and the flood damage
function c(x), B can be readily calculated.
- AN APPLICATION OF THE ALTERNATIVE APPROACH
In this section, the principles described in section 10.4 are applied in order to evaluate
flood-control policies in the Damodar Valley Project. There are two main problems; (a)
the estimation of the probability distribution of the peak flow; and (b) the estimation of
the peak flow/flood damage relationship. These problems are taken up in turn.