in all of the constraints of eq 4-2 at any instance in time, so we can write this constraint

set in the form

∑ (∆*P*s

+ ∆*P*r) +∆*P*cv +∆*P*he ∑ (∆*P*s + ∆*P*r) +∆*P*cv + ∆*P*he = 0

(5-4)

j

k j

i

where *i *≠ *k*.

If we have *n *consumers, there will be (*n * 1) such equality constraints containing

∆*P*cv,i that apparently could be "directed" such that they would bound ∆*P*cv,i from

below as required. However, these constraints are not all independent. Since we

started with *n *independent equations and then eliminated ∆*P*hp, 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 ∆*P*cv,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 ∆*P*cv,i term for the consumer arbitrarily chosen to be

consumer "1." Now, ∆*P*cv,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 ∆*P*cv,1. Now we have

(*n * 2) constraints remaining that can bound ∆*P*cv,2 in the proper sense, since the

equality constraint with consumer 1 has been directed such that it would bound

∆*P*cv,2 in the improper sense

Similarly, let ∆*P*cv,2 be bounded by properly directing its equality constraint with

consumer "3." Now, at first it would appear that ∆*P*cv,3 could be bounded by (*n * 2)

constraints as well, since we have only directed the constraint involving ∆*P*cv,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 ∆*P*cv,1 and ∆*P*cv,2 such

that it bounded ∆*P*cv,1 below, we are not free to direct the constraint between ∆*P*cv,1

and ∆*P*cv,3 as needed; in fact, it must be directed in the opposite sense of that

required, that is, if

(

)

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he -

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he ≡ < 0 ∆*P*cv,1 , ∆*P*cv,2

+

j

j

2

1

and

(

)

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he -

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he ≡ < 0 ∆*P*cv,2 , ∆*P*cv,3

+

j

3

j

2

then

(

)

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he -

∑ (∆*P*s + ∆*P*r )j + ∆*P*cv + ∆*P*he ≡ < 0 ∆*P*cv,1 , ∆*P*cv,3 .

+

j

3

j

1

{(∆*P*

), (∆*P*cv,3 , ∆*P*c+v,5),L(∆*P*cv,3 , ∆*P*c+v,n)}

Thus, ∆*P*cv,3 has only (*n * 3)

+

cv,3 , ∆*P*cv,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

46

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