Constraint activity for consumer control valve pressure losses
We have two possible constraints for each ∆Pcv,i
h1 = ∆Php - ∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe = 0
(∆Pcm ,i )
(∆Pc- ,i ) .
g1 = ∆Pcvm,i ∆Pcv,i ≤ 0
The inequality constraint g1 is monotonically decreasing in ∆Pcv,i so it bounds ∆Pcv,i
in the proper sense. The equality constraint h1 could also be "directed" (see
Papalambros and Wilde  for procedure for directing equality constraints) such
that it would bound ∆Pcv,i in the proper sense. At this point it is not clear which of
these two constraints would bound each of the ∆Pcv,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 ∆Pcv,i (see
Papalambros and Wilde  for definition of conditional criticality).
However, we see that pressure increase across the pump at the heating plant ∆Php
has not yet been fixed. ∆Php is monotonically increasing in constraint h1; thus, it
becomes a monotonic nonobjective variable in our problem. The Second Monotonic-
ity Principle, MP2 (Papalambros and Wilde 1988) tells us that either ∆Php 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
i the constraint h1 is critical for ∆Pcv,i, then ∆Php becomes relevant. A critical
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 ∆Pcv,i in the proper sense
(∆Pc-v,i , ∆Ph+p ) .
h1 = ∆Php - ∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe ≡ < 0
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
h1 above. Papalambros and Wilde (1988) warn that multiply critical constraints
should be eliminated from the problem whenever possible. Here, we have that
option since we can combine the h1 constraints for any two consumers and eliminate
∆Php 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 ∆Php appears in eq 4-2 for each consumer. Since, physically, we know that ∆Php
can only assume one value, it must be the greatest value that results from consider-
ation of all the consumers. For the remaining consumers, ∆Pcv,i must be greater than
the minimum value ∆Pcvm,i. The consumer who requires the greatest ∆Php will be
called the "critical" consumer. Notice that the equality constraint of eq 4-2 can be
satisfied for the remaining consumers by letting ∆Pcv,i > ∆Pcvm,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., ∆Pcv,i > ∆Pcvm,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.