GATE EE 2026 Set 1 — Question 18
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Power Systems → Per-Unit System & Load Flow → Gauss-Seidel & Newton-Raphson
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Question
Consider a power system with buses, of which are generator buses and the remaining 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?
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?
Correct answer
(B) (2N - 1 - P) × (2N - 1 - P)
Solution
In the Newton-Raphson load flow method, the size of the Jacobian matrix is determined by the number of unknown variables (voltage phase angles and magnitudes).
For a system with total buses:
Size
Since , we have .
Size .
Thus, the Jacobian matrix is of size . Option (B) is correct.
For a system with total buses:
1.One bus is designated as the slack bus. Both and are known. No equations are needed.
2. are generator buses. Assuming includes the slack bus, there are PV buses. For PV buses, is known, but is unknown. Number of unknowns = .
3. are load buses (PQ buses). Both and are unknown. Number of unknowns = .
Total number of unknowns (and thus the size of the Jacobian) is:Size
Since , we have .
Size .
Thus, the Jacobian matrix is of size . Option (B) is correct.
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