We would then substitute this into the constraints of eq 4-2 and solve the resulting

^

problem linearly in the *d*j values. However, we would still have a significant task in

the solution of the system of equations. For this reason we will abandon the

possibility of achieving a solution by this approach.

We have an infeasible solution from the unconstrained problem. For each of the

consumers whose absolute pressure distribution of Figure 8 exceeds the constraints,

we need to reduce the piping pressure losses by increasing pipe diameters enough

to allow for constraint satisfaction. Since the system constraints all deal with

pressure levels in the network, we need to find a strategy to resolve these constraint

violations. Let's attempt to find a solution by starting with our optimal independent

design and identifying methods to move from this design to one that satisfies all the

constraints. We will attempt to conduct this process of modifying the solution so that

it satisfies the constraints in a manner that will keep us as close as possible to the true

globally optimal design. We have the distinct advantage of knowing that our opti-

mal independent design will form an absolute lower bound on system cost. At any

point we can compare the cost of our feasible design to the cost of the optimal inde-

pendent design and determine if further attempts at improvement are warranted.

Examining Figure 8, we see that to bring excessive pressure differences within the

bounds of the constraints, we will need to reduce the slope of the pressure vs.

distance lines. The slope of these lines is the pressure loss per unit length of pipe.

Equations 5-8 and 5-9 tell us that if we are to reduce the slope we must do so by

increasing the pipe size. We would like to identify a method of determining which

pipe sizes to increase and by how much to satisfy constraints with minimum cost

increase.

At first it might seem that the best procedure would be to start by increasing pipe

sizes at the consumer's end of the system, where the sizes are smallest and the pipes

tend to be shorter. In the smaller pipe sizes, the incremental increases in diameters

are in general less than for the larger pipe sizes. Thus, we could make smaller moves

away from the lower bounding cost. Starting at the consumer appears to be the most

logical way to proceed if the critical consumer is the only consumer who has

exceeded the absolute pressure constraint. In the more general case, however, more

than one consumer will have violated the absolute pressure constraints; thus, we

will examine that case first.

If more than one consumer has violated the absolute pressure constraints, we

could achieve constraint satisfaction by increasing pipe sizes that serve each

consumer individually until all the constraints are satisfied. Alternately, we could

increase pipe diameters in pipes that serve all of the consumers with violated

constraints. Because the pipe sizes are discontinuous (discrete) and the incremental

differences between adjacent diameters are nonuniform, it's not possible for us to

predict a priori which diameters would be the best candidates for increasing. Thus,

we need to identify a method that will guide our search for a feasible and acceptable

solution expediently. In deciding when to stop our search, we always have the

benefit of knowing our lower bounding cost.

If we refer back to Figure 8, we see that satisfaction of the absolute pressure

constraints relies on keeping the pressure in the supply and return lines within the

bounds prescribed by the maximum absolute pressure constraint and the two

minimum absolute pressure constraints. We can adjust pressures at the plant to

achieve a state that satisfies all the constraints, as long as the maximum pressure

difference within the system does not exceed the absolute pressure constraints

discussed above. Since the critical consumer previously identified will be the

consumer who requires the largest pressure differential within the system, we will

examine this consumer's requirements first and attempt to resolve the constraint

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