Kalman Filter

The real-time inference of a generator's internal rotor angle and speed states using a Kalman filter and streaming PMU measurements. Unlike static state estimation, which solves a single snapshot of bus voltages, dynamic estimation tracks the electromechanical transients of synchronous machines. The Kalman filter's predict-correct cycle is uniquely suited to this task, fusing a physics-based swing equation model with noisy synchrophasor data to provide operators with visibility into unmeasured internal generator states.

A nonlinear extension of the linear Kalman filter that handles the swing equation dynamics inherent in power systems. The EKF linearizes the nonlinear state transition and measurement functions around the current estimate using a first-order Taylor series expansion (Jacobian matrices). In PMU applications, the EKF processes voltage and current phasor measurements to estimate the rotor angle δ and speed deviation Δω of a synchronous generator, accommodating the sinusoidal nonlinearities in the machine model.

An alternative to the EKF that avoids analytical linearization by propagating a minimal set of carefully chosen sigma points through the true nonlinear system. The UKF captures the posterior mean and covariance accurately to the 3rd order (Taylor series) for any nonlinearity, compared to the EKF's first-order approximation. For generator dynamic state estimation, the UKF often exhibits superior performance during large disturbances where the EKF's linearization error becomes significant, without requiring Jacobian derivation.

A matrix representing the statistical uncertainty in the state transition model of the Kalman filter. In power system dynamic estimation, Q accounts for unmodeled dynamics, parameter errors, and unknown mechanical torque changes. Tuning Q is critical:

• Too small: The filter trusts the model excessively and becomes sluggish in responding to real disturbances
• Too large: The filter overreacts to measurement noise, producing jittery estimates Adaptive techniques can estimate Q online from innovation sequences.

A matrix quantifying the statistical uncertainty of the PMU measurements feeding the Kalman filter. R captures sensor noise, instrument transformer errors, and communication channel degradation. Key considerations:

• PMU Total Vector Error (TVE) directly informs the diagonal elements of R
• Correlated noise across measurement channels requires off-diagonal terms
• Time-varying R matrices can adapt to changing data quality flags in the synchrophasor stream Proper R specification prevents the filter from chasing spurious measurement spikes.