Jules at 07:31, 22 November 2020

2020-11-22T07:31:56Z

Jules: Created page with "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..."

2020-11-16T13:47:19Z

Created page with "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..."

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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 [http://en.wikipedia.org/wiki/Condition_number 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_Power_Flow|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 [[AC_Power_Transmission#Steady-State_Voltage_Stability_Limits|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)

[[Category:Modelling/Analysis]]

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Ill-Conditioning - Revision history

Jules at 07:31, 22 November 2020

Jules: Created page with "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..."