Question

Consider a power system with \(N\) buses, of which \(P\) are generator buses and the remaining \(Q\) are load buses (where there is no generation).

Assume that there are no reactive power-limit violations at the generator buses.

What is the size of the Jacobian matrix in the Newton-Raphson load flow method?

• A. \(2N\times2N\)
• B. \((2N-1-P)\times(2N-1-P)\)
• C. \((2N-Q)\times(2N-Q)\)
• D. \((P+Q)\times(P+Q)\)

Hint:

Count angle unknowns (all buses except slack, giving N-1) and magnitude unknowns (only load buses, giving Q=N-P), then add and simplify using N=P+Q.

Answer 1

The Correct Option is B

Step 1: Set up the bus classification.
One of the \(P\) generator buses is the slack (reference) bus; the remaining \(P-1\) generator buses are PV buses. The \(Q\) load buses are PQ buses. All buses together number \(N=P+Q\).
Step 2: Count the voltage angle unknowns.
Every bus except the slack bus has an unknown voltage angle in Newton-Raphson load flow, since only its scheduled real power is fixed. This gives
\[ N-1 \]
angle unknowns.
Step 3: Count the voltage magnitude unknowns.
Only the PQ (load) buses have an unknown voltage magnitude, since every generator bus (the slack and every PV bus) has its voltage magnitude specified and held fixed (no reactive power-limit violation means no PV bus is ever converted to a PQ bus during the iteration). This gives exactly
\[ Q \]
magnitude unknowns.
Step 4: Add the two counts.
Total number of unknowns, which equals the size of the (square) Jacobian matrix:
\[ (N-1)+Q \]
Step 5: Substitute \(Q=N-P\).
Since \(N=P+Q\), we have \(Q=N-P\). Substituting,
\[ (N-1)+(N-P)=2N-1-P \]
Step 6: Rule out the other options.
Option (A), \(2N\times2N\), overcounts by treating every bus as having both an unknown angle and an unknown magnitude, ignoring that generator buses have a fixed voltage magnitude. Option (C), \((2N-Q)\times(2N-Q)\), does not simplify to the correct count; substituting \(Q=N-P\) gives \(2N-Q=N+P\), which is too large. Option (D), \((P+Q)\times(P+Q)=N\times N\), also does not match, since it does not separate angle unknowns from magnitude unknowns correctly.
Step 7: Final Answer.
\[ \boxed{(2N-1-P)\times(2N-1-P)} \]