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In some power systems, the Jacobian matrix can be ill-conditioned, leading to difficulties in reaching a valid a power flow solution.

Ill-Conditioned Matrices

A matrix is considered to be ill-conditioned if it is very sensitive to small changes. The classic illustration of ill-conditioning is the following two linear systems of the form Ax = b:

... (1)


... (2)

The solution to system (1) is , , while the solution to system (2) is .

What this shows is that a tiny change in the 4th decimal place of the vector b in the system Ax = b can lead to relatively large changes in the solution vector x. As a result, inaccuracies in the data (such as rounding errors) can have large consequences when systems are ill-conditioned. The condition number of a matrix is often used to describe the degree of ill-conditioning.

Ill-Conditioned Power Flow Problems

In power systems, the power flow problem is said to be ill-conditioned if the Jacobian matrix is ill-conditioned. This is because in the Newton-Raphson algorithm, each iteration has the following linear form:

where is the power flow Jacobian matrix

is the bus voltage (magnitude and angle) correction vector
is the active and reactive power mismatch vector

Therefore, if the Jacobian matrix is ill-conditioned, the solution to the power flow iteration can become wildly unstable or divergent.

The most common characteristics that lead to ill-conditioned power flow problems are as follows:

  • Heavily loaded power system (i.e. voltage stability problem where system has reached nose point of PV curve)
  • Lines with high R/X ratios
  • Large system with many radial lines
  • Poor selection of the slack bus (e.g. in a weakly supported part of the network)