Ill-Conditioning

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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}1&1\\1&1.0001\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\\end{bmatrix}}={\begin{bmatrix}2\\2\\\end{bmatrix}}} ... (1)

and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 & 1 \\ 1 & 1.0001 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \end{bmatrix} = \begin{bmatrix} 2 \\ 2.0001 \\ \end{bmatrix} } ... (2)

The solution to system (1) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1} = 2 \,} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{2} = 0 \,} , while the solution to system (2) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{1} = x_{2} = 1 \,} .

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [J] \Delta \boldsymbol{x} = -\Delta \boldsymbol{S} \, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [J] \, } is the power flow Jacobian matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \boldsymbol{x} \, } is the bus voltage (magnitude and angle) correction vector

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \boldsymbol{S} \, } 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)