in all of the constraints of eq 4-2 at any instance in time, so we can write this constraint
set in the form
∑ (∆Ps
+ ∆Pr) +∆Pcv +∆Phe ∑ (∆Ps + ∆Pr) +∆Pcv + ∆Phe = 0
(5-4)
j
k j
i
where i ≠ k.
If we have n consumers, there will be (n 1) such equality constraints containing
∆Pcv,i that apparently could be "directed" such that they would bound ∆Pcv,i from
below as required. However, these constraints are not all independent. Since we
started with n independent equations and then eliminated ∆Php, we will have at
most (n 1) independent equations remaining. Below, we will show that these
(n 1) independent equations may bound at most (n 1) of the ∆Pcv,i objective
variables in the proper sense. The arguments made will be for an instant in time but
must hold for any time during the yearly cycle.
We begin by examining the ∆Pcv,i term for the consumer arbitrarily chosen to be
consumer "1." Now, ∆Pcv,1 for consumer 1 can be bounded from below as required
by any one of the (n 1) constraints in which it appears with another consumer.
Suppose we let the constraint with consumer "2" bound ∆Pcv,1. Now we have
(n 2) constraints remaining that can bound ∆Pcv,2 in the proper sense, since the
equality constraint with consumer 1 has been directed such that it would bound
∆Pcv,2 in the improper sense
Similarly, let ∆Pcv,2 be bounded by properly directing its equality constraint with
consumer "3." Now, at first it would appear that ∆Pcv,3 could be bounded by (n 2)
constraints as well, since we have only directed the constraint involving ∆Pcv,2 in the
improper sense and any one of the remaining (n 2) constraints can be directed as
needed. However, since we directed the constraint between ∆Pcv,1 and ∆Pcv,2 such
that it bounded ∆Pcv,1 below, we are not free to direct the constraint between ∆Pcv,1
and ∆Pcv,3 as needed; in fact, it must be directed in the opposite sense of that
required, that is, if
(
)
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe -
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe ≡ < 0 ∆Pcv,1 , ∆Pcv,2
+
j
j
2
1
and
(
)
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe -
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe ≡ < 0 ∆Pcv,2 , ∆Pcv,3
+
j
3
j
2
then
(
)
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe -
∑ (∆Ps + ∆Pr )j + ∆Pcv + ∆Phe ≡ < 0 ∆Pcv,1 , ∆Pcv,3 .
+
j
3
j
1
{(∆P
), (∆Pcv,3 , ∆Pc+v,5),L(∆Pcv,3 , ∆Pc+v,n)}
Thus, ∆Pcv,3 has only (n 3)
+
cv,3 , ∆Pcv,4
constraints that could be directed to bound it in the proper sense.
If we continue to follow this line of reasoning, we find that when we reach
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