![](bg6.png)
ates and a continuous degree of freedom appears in
the solution. This degree of freedom corresponds to
the disbalance of the flows in the compressors, that is,
the addition of ±δQ to the flows canceling each other
in the total flow through the station. In this situation,
shown in Fig.3b, there is a local Qundef-conflict be-
tween the compressors.
The ambiguity is not necessarily limited to the
compressor station, it can also go beyond it. Fig.3c
shows an extended Pundef-conflict. Here two SM-
compressors are connected by an intermediate sub-
graph R, which in a particular case can be a pipeline.
Since there are no equations in the subgraph that
could fix the pressure, it turns out to be undefined
along the pipeline. In a more general case, one can
choose for R any connected subgraph that satisfies
the generalized resistivity conditions and does not in-
clude supplies with a given pressure Pset. Let there
be a solution in which both compressors are stably lo-
cated inside their SM-faces. In this case, they can be
replaced by supplies or exits with given flows Qset.
The solution exists only if these Qsets are balanced
with the Qsets in the subgraph R, that is, their to-
tal sum is zero. Further, by choosing an arbitrary
node of the subgraph R, one can discard the Kirch-
hoff condition written in it, since it follows from the
conservation conditions and is fulfilled automatically.
This condition can be replaced by the P = Pset con-
dition, with the pressure value taken from the chosen
solution. The result of these manipulations is a con-
nected generalized resistive graph with a given Pset,
for which the system has a unique solution. Now Pset
can be shifted by an arbitrary small value δP, and the
solution will also exist and be unique. In this case, by
continuity, the shift of the solution will be arbitrarily
small and still be inside the SM-faces, that is, it will be
a solution to the originally posed problem. As a result,
we have demonstrated the presence of continuous ar-
bitrariness in the solution associated with a change in
pressure, Pundef-conflict. It is also noteworthy that
this consideration can be applied to any node of the R
subgraph, including those with Qset = 0. Therefore,
the arbitrariness affects the pressure at all nodes of the
subgraph.
Similarly, one can get an extended Qundef-
conflict, see Fig.3d. Here, two SPO compressors con-
trol the pressure in one node. The equations repeat,
the system is degenerate, and the solution has a con-
tinuous degree of freedom associated with the ±δQ
disbalance of flows through the compressors. Now
this disbalance extends to the outer subgraphs R
1,2
,
which in a particular case can be pipelines with Psets
at the free ends, and for subgraph R
3
– a pipeline with
Qset at the free end. In a more general case, one can
choose for R
1−3
connected generalized resistive sub-
graphs, for R
1,2
– with Psets, and for R
3
– with Pset or
without it. Next, consider a solution where the com-
pressors are stably inside their SPO-faces. The net-
work can be dissected at compressors into three dis-
connected parts, while for R
1,2
the compressors can
be replaced by Qsets, and for R
3
by Pset = SPO. The
solution exists and is unique in each of these three
subgraphs, while in R
3
the solution does not change
and in R
1,2
it is deformed when ±δQ-flows change.
Thus, we have shown the presence of Qundef-conflict
in the considered scenario.
While the problems of a local type can be noted
and ignored, for extended conflicts, entire network re-
gions may have undefined characteristics, which re-
quires additional analysis. At first glance, the de-
scribed conflicts can be recognized and eliminated,
by an automatic algorithm or manually, for example,
not allowing SPO conditions to collide in the same
node. However, the conflict may arise not between
the main target faces, but between the auxiliary ones,
that is, PIMIN, QMAX and all other conditions in the
equation can enter into conflict. Some of these con-
flicts do not occur directly, but persist through sub-
graphs. It is impossible to predict in advance which
faces will be activated, and consideration of all poten-
tially emerging possibilities for large networks turns
out to be combinatorially unacceptable. Conflict can
also arise at intermediate iterations, on the path from
the starting point to the solution. This will lead to the
degeneration of the Jacobi matrix and to the immedi-
ate divergence of the solution algorithm.
Compressors also have a refined, so-called ad-
vanced model (Clees et al., 2018b; Baldin et al.,
2021) that includes additional conditions in the con-
trol equation. These conditions do not fix the prob-
lem, as they only add new faces that are curved and
with the correct signature, but the old faces still re-
main in the equation and continue to generate con-
flicts. In practice, if the compressor can fulfill its
target conditions, then it is right on those potentially
conflicting faces.
In gas transport problems, not only the compres-
sors bring the described difficulties. Other elements,
in particular, regulators, flaptraps, REPD-resistors,
are also modeled by piecewise linear conditions of
the marginal signature (Clees et al., 2018a). A con-
flict can arise in any combination of these elements,
and also as a result of direct connection of conflicting
faces with Pset or Qset conditions.
It follows from the above formulas that due to the
ε-regularization, the polyhedron faces corresponding
to the control equation are deformed, and the devia-
tions from the target values on the solution turn out to
Principal Component Analysis in Gas Transport Simulation
183