Pipeline network problems

II. Steady-State Pipeline Network Problems Formulation.127... 

Equations (16) and (17) show the important link between fundamental cycles and cut-sets of a graph. Thus, a spanning tree provides a convenient starting point for formulating a consistent set of governing equations for steady-state pipeline network problems. 

We are now in a position to formulate the steady-state pipeline network problem based on the laws governing the behavior of the network and its elements. As it turns out, there is more than one way of formulating the problem, and since the computational efforts required for the solution are unequal, it behooves us to examine the ramifications of these formulations. 

In our treatment so far we have dwelt on the description, formulation, and specification of steady-state pipeline network problems. As we stated at the... 

Under all but laminar flow conditions, the steady-state pipeline network problems are described by mixed sets of linear and nonlinear equations regardless of the choice of formulations. Since these equations cannot be solved directly, an iterative procedure is usually employed. For ease of reference let us represent the steady-state equations as... 

The Newton-Raphson method has been applied to pipeline network problems since 1954 (Wl). Its performance has been generally very good, although convergence difficulties have been reported (S2), when starting from inappropriate initial guesses. In some cases large oscillations around... 

Unlike the previous method Wolfe s method requires the inversion of an (n + 1) x (n + 1) matrix at each iteration, but a short-cut procedure may be used to take advantage of the fact that at each iteration only one column of this matrix is modified. Another disadvantage of this method is that it cannot use the information from a previous case (e.g., the Jacobian) to obtain a better starting point. To the best of our knowledge there is no demonstrable advantage to recommend the use of this method in pipeline network problems. 

Finally, a special point to look for in comparing iterative methods for pipeline network problems is to use the same problem formulation for both methods otherwise the results may reflect differences in formulations as well as iterative methods. 

In principle, the steady-state pipeline network problems can always be solved by the transient solution methods after allowing sufficient time steps for the solution to reach steady state. This possibility was discussed by Nahavandi and Catanzaro (Nl) who made a comparison of a transient solution method with the Cross method of balancing flows (R4). For the particular 35-node and 45-branch hydraulic network problem tested, the transient solution method took 108 seconds as compared with the 134 seconds required by the Hardy-Cross method. (See also Section V,A,2.)... 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

---