We have two possible constraints for each ∆*P*cv,i

(∆*P*cm ,i )

(4-2)

v

j

i

(∆*P*c- ,i ) .

(4-5)

v

The inequality constraint *g*1 is monotonically decreasing in ∆*P*cv,i so it bounds ∆*P*cv,i

in the proper sense. The equality constraint *h*1 could also be "directed" (see

Papalambros and Wilde [1988] for procedure for directing equality constraints) such

that it would bound ∆*P*cv,i in the proper sense. At this point it is not clear which of

these two constraints would bound each of the ∆*P*cv,i variables in the proper sense.

In fact it's entirely possible that the active constraint may vary depending on the

particular consumer in question. If no other decision variables appeared in these two

constraints, they would form a conditionally critical set for each ∆*P*cv,i (see

Papalambros and Wilde [1988] for definition of conditional criticality).

However, we see that pressure increase across the pump at the heating plant ∆*P*hp

has not yet been fixed. ∆*P*hp is monotonically increasing in constraint *h*1; thus, it

becomes a monotonic nonobjective variable in our problem. The Second Monotonic-

ity Principle, MP2 (Papalambros and Wilde 1988) tells us that either ∆*P*hp is

irrelevant and can be deleted from the problem together with all the constraints in

which it appears, or it is relevant and bounded by two active constraints, one

bounding it from above and one bounding it from below. If for just one consumer

constraint is an active constraint whose deletion would cause the problem to become

unbounded. This active constraint would have the following monotonicities when

directed to bound ∆*P*cv,i in the proper sense

(∆*P*c-v,i , ∆*P*h+p ) .

i

j

If one constraint is critical for more than one variable in the problem, it is said to

be "multiply critical" (Papalambros and Wilde 1988) and this would be the case for

should be eliminated from the problem whenever possible. Here, we have that

option since we can combine the *h*1 constraints for any two consumers and eliminate

∆*P*hp from the problem. Before we do so, let's consider briefly what is physically

happening in our problem.

First, we note that the pressure increase required across the pump at the heating

plant ∆*P*hp appears in eq 4-2 for each consumer. Since, physically, we know that ∆*P*hp

can only assume one value, it must be the greatest value that results from consider-

ation of all the consumers. For the remaining consumers, ∆*P*cv,i must be greater than

the minimum value ∆*P*cvm,i. The consumer who requires the greatest ∆*P*hp will be

called the "critical" consumer. Notice that the equality constraint of eq 4-2 can be

satisfied for the remaining consumers by letting ∆*P*cv,i > ∆*P*cvm,i as allowed by eq 4-

5. This is, in fact, how it is done in practice in most cases; the ultimate balancing of

the pipe network flows is done by the consumer's control valves. For the case of the

critical consumer, eq 4-5 will be satisfied as a strict equality, i.e., ∆*P*cv,i > ∆*P*cvm,i.

While these arguments of constraint activity would appear to be completely

intuitive, since we would not want to supply any more pumping energy than

necessary, they can also be shown analytically as follows.

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