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Examoogle • User • info@examoogle.com • EE-2024-8821
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In only active and reactive power injection, 2N number of elements of measurement vector ‘z’ is used to estimate ________ number of elements of state vector ‘x’
N−1
2N−1
N
2N−2
2N−1
Quick Summary: In a power system state estimation problem, the state vector $x$ consists of voltage magnitudes $|V_i|$ and phase angles $\theta_i$ at all $N$ buses. Since one bus is chosen as the reference (slack) bus, where $\theta_{slack} = 0$, there are $N$ voltage magnitudes and $N-1$ unknown angles, totaling $2N-1$ unknown states to be estimated.
In a power system state estimation problem, the state vector x consists of voltage magnitudes ∣Vi∣ and phase angles θi at all N buses. Since one bus is chosen as the reference (slack) bus, where θslack=0, there are N voltage magnitudes and N−1 unknown angles, totaling 2N−1 unknown states to be estimated.
x=[∣V1∣,...,∣VN∣,θ1,...,θN−1]T
n=2N−1
The measurement vector z includes active and reactive power injections at each bus. For N buses, there are N active power measurements and N reactive power measurements. However, because the phase angle is relative, one bus is fixed, removing one variable from the state space. Thus, to estimate 2N−1 states, the system requires a measurement vector of dimension at least 2N−1 for observability.
State estimation requires the system to be observable.
The slack bus phase angle is typically set to 0 radians as the reference.
Measurement vector z relates to state vector x via non-linear equations z=h(x)+e.
A minimum of 2N−1 measurements are required for system observability.
Ensures numerical stability in power flow analysis.
Provides a mathematical basis for Bad Data Detection (BDD).
Requires high computational overhead for large systems.
Depends on the accuracy of the Jacobian matrix calculation.
Energy Management Systems (EMS).
Real-time grid monitoring and contingency analysis.
For a system with N buses, the state vector is of size (2N−1×1).
If 2N measurements are provided, the system is overdetermined, which is beneficial for filtering noise.
B is correct — The state vector comprises N bus voltages and N−1 bus angles, totaling 2N−1 unknown states.
Always remember that in power flow and state estimation, the reference bus angle is always defined as zero, which reduces the degrees of freedom in the angle vector from N to N−1.