As noted in the Chapter 4, all district heating networks, with the exception of pure

transmission systems, will have multiple consumers and pipes. If each pipe were

independent of the others, it would be possible to apply the procedure developed

earlier in Chapters 2 and 3 to each pipe independently and develop a complete

design in that way. Unfortunately, as the constraints introduced in Chapter 4 show,

each pipe segment does not operate independently of the others. Thus, the system

can not be designed completely in that way for all but the most trivial cases.

However, the "optimal independent design," as we will call it, is very useful, even

though we can not guarantee that it would be feasible, for we can use it to form a

lower bound on any other designs that we might propose. We know that it is not

possible for us to achieve a lower cost design for any one of the pipe segments than

the one we have determined independently. Thus, it follows that we also are assured

that no design for the entire system of pipe segments can be lower in cost than the

sum of the costs for the optimal independent designs. While this may appear to be

of little significance to the designer who is subject to system constraints, it's actually

a very useful result. It will serve two very important functions for us. First, it will give

us a lower bound on total system cost to which we can compare other designs to see

if they are sufficiently close to render additional effort at achieving better designs

impractical or unnecessary. The second function of this "optimal independent

design" will be as a starting point for a solution strategy that will move towards an

optimal solution that satisfies all the system constraints. Both of these attributes of

the optimal independent design will be exploited in this chapter.

Our objective here will be to develop methodologies that will help us find the

optimal discrete pipe diameters for systems with multiple pipes and consumers,

while minimizing the computational effort necessary, such that large networks

often encountered in practice may be treated with an acceptable degree of effort.

Because of excessive computational effort, many of the methods that have been

previously applied to problems of this type are felt to be unsuitable. Several of the

more common approaches are discussed very briefly below.

The classical approach to a constrained optimization problem like this one is to

include all the constraints in the problem solution and find a solution that satisfies

all the constraints. Here, we have many constraints that would need to be included.

For our problem from eq 4-2 and 4-5, we would have two constraints for every

consumer. Equations 4-11 and 4-22 would result in two constraints for every node

in the supply pipe. Equation 4-23 gives us one constraint for each node in the return

pipe, and at the heating plant there are two additional constraints (eq 4-24 and 4-25).

If, for example, we considered a moderate sized system, with 100 consumers and 125

nodal points, we would have 577 constraints. Several of the more common methods

for handling such problems are discussed below.

The method of linear programming (Wilde and Beightler 1967) is very efficient at

solving optimization problems with large numbers of constraints. However, as the

name of the method implies, the objective and constraints must be linear. Here, we

have a highly nonlinear problem because of the pressure losses being proportional

to the pipe diameters raised to approximately the negative five power. Methods

have been devised (Reklaitis et al. 1983) to use linear programming algorithms on

nonlinear problems by making linear approximations about a point using a Taylor

series expansion about that point. The computational effort involved in the use of

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