Optimal Differentiator Filter Banks for PMUs and Their Feasibility Limits

Abstract

In this paper, we present a very general design approach for optimal linear-phase phasor filter bank algorithms for PMUs based on convex semi-infinite optimization. A detailed presentation of the formulation of both the cost functions and constraints is included for the positive-sequence estimation problem. The design method is extremely powerful and flexible as it allows to control precisely the behavior of the system in terms of the total vector error (TVE), frequency error (FE), and rate of change of frequency error (RFE) metrics for several different scenarios. This feature is extremely useful for tailored designs of these filters for different applications. We also determine numerically the uniform feasibility limits of the system as a function of the filter lengths and, in particular, study the required filter lengths for compliance with the IEEE Standards C37.118. 1-2011 and C37.118.1a-2014. It is found that all requirements can be achieved with a significant margin with the exception of the M class FE constraint for interharmonic components, an issue which is also reported in other works and briefly discussed here. Finally, an interesting comparison with the Taylor-Fourier filters is made to illustrate the advantages of our approach.

A Self-Adaptive Contractive Algorithm for Enhanced Dynamic Phasor Estimation

“…This constraint, imposed on the design of the filter A 1 given A 0 , is convex. It is interesting to compare this constraint with the one obtained in [18]. Note that the bound L ω |ω s | is greater than 2πFE STA for all frequencies f s such that |f s | > FE STA /L ω , which in general is in the order of a few mHz.…”

“…The input signal to the SAC algorithm is a complex baseband signal obtained from a real three-phase signal sampled with a sampling period T and by performing an abc-dq transformation. As in [18], the input signal is denoted as Y [n] = X[n] + I[n], where I[n] is a general disturbance signal which includes the harmonics, the negativesequence component, noise, etc. The samples around each reporting time index n r are collected in the input signal vector…”

“…and γ Z k [n r ] are obtained by filtering Z k [n r ] with the current DFB, which is represented by the linear-phase coefficient vectors a 0 ∈ R N/2+1 , a 1 ∈ R N/2 and a 2 ∈ R N/2+1 [18]. The expressions are shown in (13) at the bottom of the next page (the rationale behind these computations can be found in [17]).…”

“…As will be shown, the self-adaptation scheme relaxes most of the performance constraints of the problem. Moreover, in many cases, the constraints imply that, under such cases, the different errors, which are the total vector error (TVE), frequency error (FE), rate of change of frequency error (RFE), damping error (DE) and rate of change of damping error (RDE), converge to zero (see Appendix), which is a much stronger result than a performance bound on them as in the formulation of [18].…”

“…This relaxation of the original harmonic rejection constraints is of critical importance. In fact, if one includes the joint effect of off-nominal frequency conditions and multiple high-level harmonics as constraints in the original CSIP filter design problem, the resulting optimization problems may result unfeasible [18].…”