Power system having N number of buses, the state variable ‘x’ may be defined as _______number of bus

Question

Power system having N number of buses, the state variable ‘x’ may be defined as _______number of buses voltage angle and __________number of bus voltage magnitude

Examoogle · Accepted Answer

$N-1, N$

Explanation: In a power system with N buses, the state vector is defined by the set of independent variables that describe the network state. The voltage angle at the slack bus is taken as the reference (set to 0°), resulting in N-1 independent angle variables, while the voltage magnitude is typically defined for all N buses (including the slack bus).

Working Principle: State estimation involves calculating the system state based on available measurements. Since the slack bus voltage angle is fixed as the reference point, it is not an unknown. Therefore, there are N-1 unknown angles and N unknown magnitudes, totaling 2N-1 state variables.

Key Formulas: $N_{angles} = N - 1$; $N_{magnitudes} = N$; $N_{total} = 2N - 1$

Key Points: The slack bus acts as the phase reference, hence its angle is known.; Voltage magnitude at the slack bus is typically specified as a known value in load flow, but treated as a variable in generalized state estimation depending on the model.; The standard state vector in Newton-Raphson load flow is of size 2N-1.

Advantages: Reduces the number of unknowns in system analysis.; Provides a consistent mathematical framework for Newton-Raphson methods.

Disadvantages: Requires identifying the slack bus accurately for convergence.; State estimation complexity increases linearly with the number of buses.

Applications: Power system load flow analysis.; Optimal power flow (OPF) studies.; Real-time grid monitoring and supervisory control.

Additional Info: The slack bus (or swing bus) is essential for balancing system losses which are unknown until the end of the calculation.. If the slack bus voltage magnitude were also treated as a variable, N could become N-1 for magnitudes, but standard convention treats magnitudes at all N buses as part of the state space.

Answer: B is correct — For a system of N buses, the slack bus angle is used as a reference, leaving N-1 unknown angles, and all N bus voltage magnitudes are defined as system state variables.

Exam Tip: Always remember that in power flow studies, the number of independent equations equals the number of unknowns ($2N-1$). Check if the question asks for independent variables (N-1, N) or total parameters.

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