consumer *i = n*, we have no constraints remaining that can be directed in the proper

sense to bound the ∆*P*cv,n term in the objective. Note that because the assignment of

the numerical value for the consumer index *i *is arbitrary, its assignment will have

no impact on what we have shown here and that this result would hold for any set

of consumer indices.

Since we have *n *variables ∆*P*cv,i in the objective, one for each of the *n *consumers,

and the objective is monotonically increasing in each of these variables, we must

have *n *constraints bounding the ∆*P*cv,i from below. Above we have shown that at

most (*n * 1) of these constraints could result from eq 4-2. The only remaining

constraints on the ∆*P*cv,i are the set formed by eq 4-5. Let's assume for the moment

that all consumers have the same minimum pressure differential requirement for

their control valves, i.e., that ∆*P*cvm,i is the same for all *i*. Now we see that it must be

the consumer with the minimum value of ∆*P*cv,i whose constraint from eq 4-5 is

active. This is true since a consumer with any greater value would cause at least one

of the other constraints from the set of eq 4-5 to be violated, i.e., if consumer *i *has the

minimum control value pressure loss

for all *k *≠ *i*.

∆*P*cv,k > ∆*P*cv,i

And if for all *i *and *k *the minimum allowable control valve pressure losses are equal

∆*P*cvm,i = ∆*P*cvm,k .

Now, if the constraint for consumer *i *is active

∆*P*cv,i > ∆*P*cvm,i

then the constraint for consumer *k *can not be active

∆*P*cv,k > ∆*P*cvm,k .

We have already shown that (*n * 1) of the constraints from the set of eq 4-2

are active and can thus be treated as equalities. These (*n * 1) constraints

force the pressure loss summations to be equal for all *n *values of the consumer's

index *i*

∑ (

∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he =

∑ (

∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he

for all *i *≠ *k.*

j

j

i

k

This means that we identify the consumer whose value of ∆*P*cv,i is at its mini-

mum allowed ∆*P*cvm,i by finding the consumer with the maximum value for the

quantity

∑ (

∆*P*s + ∆*P*r )j + ∆*P*cvm,i + ∆*P*he .

j

i

Once the pipe sizes are known, this quantity is easily calculated. This consumer who

has the minimum value of ∆*P*cv,i is our so called "critical" consumer.

It follows then that there are only two cases where more than one of the

constraints from the set of eq 4-5 may be active:

1. In the case where the ∆*P*cvm,i values are identical for all consumers, if the

maximum value of

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