Ill-Conditioning

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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:

[math]\begin{bmatrix} 1 & 1 \\ 1 & 1.0001 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \\ \end{bmatrix} [/math] ... (1)

and

[math]\begin{bmatrix} 1 & 1 \\ 1 & 1.0001 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} = \begin{bmatrix} 2 \\ 2.0001 \\ \end{bmatrix} [/math] ... (2)

The solution to system (1) is [math]x_{1} = 2 \,[/math], [math]x_{2} = 0 \,[/math], while the solution to system (2) is [math]x_{1} = x_{2} = 1 \,[/math].

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:

[math] [J] \Delta \boldsymbol{x} = -\Delta \boldsymbol{S} \, [/math]

where [math] [J] \, [/math] is the power flow Jacobian matrix

[math] \Delta \boldsymbol{x} \, [/math] is the bus voltage (magnitude and angle) correction vector
[math] \Delta \boldsymbol{S} \, [/math] 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)