Frequency estimation and tracking by two-layered iterative DFT with re-sampling in non-steady states of power system

DFT can provide the amplitude and phase angle of phasor and the frequency by way of differentiating the phase angles. Suppose the nominal voltage or current signal in a nominal power system iswhere ω and A are the angular frequency and amplitude respectively. ω is supposed to be 2πf₀, f₀ is the nominal frequency, and φ₀ is the initial phase angle. In order to precede the DFT calculation, signal is always truncated and finite samples should be taken by kinds of windows. The process of making a phasor estimation will require sampling the waveform over some interval of time which can lead to some confusion if the number of samples is not an integer in a window. The magnitude is compensated by dividing the magnitude with a sine at the actual signals frequency. The 2-cycle triangular window produces a faster roll off than a standard 1s-cycle rectangular window, but the frequency deviation is spread with an additional factor of 1.625 to increase compensation [25]. A simple rectangular window is adopted in this study. By the way, the length of a rectangular window could be selected easily and arbitrarily according to the requirements of accuracy and computational burden. DFT converts the equally spaced and uniform samples into a finite combination of complex sinusoids, ordered by their frequency. According to DFT, a phasor is calculated bywhere N is number of samples, x(n) = A cos(2πn/N + φ₀), (n = 0,1, 2,…,N − 1) are the samples taken uniformly within the length of a rectangular window; k is the order of harmonic; especially if k = 1, it stands for the phasor of the nominal component. The amplitudes and phase angles of kth harmonics arewhere atan stands for the arctan function. X_{k_real} and X_{k_imag} are real part and imaginary part of X_k. According to Eq. 2 and Eq. 3, if k = 1, the phasor of the nominal signal is calculated bywhere X_{1_real} is the real part of the phasor of nominal component, and X_{1_imag} is the imaginary part. In front of the summation sign, a multiplier of 2/N is used to generate the normalized amplitude of DFT, i.e., 1 p.u.. The phasor of nominal signal is expressed as

Definition of synchronous phasor is shown in Fig. 1, in which the peak of the sine/cosine signal coincides with the time-tag, so the angular is 0°; if the signal crosses zero at the time-tag, it produces − 90° according to the IEEE Std. C37.118 [24‐26].

The synchronous phasor (synchrophasor) is defined as a complex number, representing the fundamental frequency component of a voltage or current, with an accompanying time-tag defining the time instant for which the phasor measurement is performed [27]. Serial synchrophasors are calculated by shifting windows at each sampling time t_r = rT_s, r = 0,1,2,…,+∞, where T_s is the sampling interval, T_s = 1/f_s. A synchrophasor at time t_r is represented as

Amplitudes of serial synchrophasor at steady state are A, if the signal is a nominal one. The phase angles of serial synchrophasor are {φ₀, φ₀ + 2π/N, φ₀ + 4π/N,…, φ₀ + 2πr/N,…}. In the following, we use X^ras the nominal phasor instead of \( {X}_1^r \) for simplicity. Frequency is defined as the speed of rotation of a phasor and can be calculated by two consecutive measured phaseswhere φ^r is the phase angle calculated by DFT. If φ^r is not the nominal one, f ^r would be inaccurate.

It was reported that dynamic movement of rotors of generators and motors following power system disturbances causes the electromechanical transients or electromechanical non-steady states [24]. Generally, the frequency of an off-nominal signal is always represented as f₀ + Δf. If Δf is a fixed frequency deviation. The representation of input signal iswhere x(n) are the samples taken in one window with length of NT_s, x(n) = A cos[2πf₀nT_s + 2πΔfnT_s + φ₀], (n = 0,1, 2,…,N − 1). The nominal phasor is recorded as\( {X}_{\mathrm{nomi}}^r={Ae}^{j{\varphi}_{\mathrm{nomi}}^r} \), where \( {\varphi}_{\mathrm{nomi}}^r={\varphi}_0+2\pi r/N \)at time t_r. The measured one is\( {X}_{\mathrm{meas}}^r={A}_{\mathrm{meas}}^r{e}^{j{\varphi}_{\mathrm{meas}}^r} \), and the measured phasor \( {X}_{\mathrm{meas}}^r \)for the off-nominal signal x(t) according to DFT iswhere Η(Δf, N, T_s) is constant if the N, Δf, and T_s are all constant (detail formula derivations are in Appendix); and symbol “*” denotes the conjugation of a complex number.

If Δf→0, \( \underset{\Delta f\to 0}{\lim}\mid \mathrm{H}\left(\Delta f,N,{T}_s\right)\mid \to 1 \) as shown in Fig. 18 in Appendix. The phase angle \( {\varphi}_1^r \) at time t_r in Eq. 9 is

If Δf→0, \( \underset{\Delta f\to 0}{\lim}\mid \mathrm{H}\left(2{f}_0+\Delta f,N,{T}_s\right)\mid \to 0 \) is also shown in Fig. 18 in Appendix. \( {\varphi}_2^r \)is a very small phase angle as shown in Fig. 2 and can be calculated as

The measured phase angle \( {\varphi}_{\mathrm{meas}}^r \) is

According to Eq. 7, for the first two phase angles calculated by DFT, we have

Without loss of generality, since N> > 4 and Δf < <f₀, and suppose φ₀ = 0, such in-equation of 0 < cos[(2N + 1)πΔfT_s] < 1 can be fulfilled. We have

From analysis above, the measured frequency f_meas would approach f₀ + Δf more and more closely if we change the sampling frequency f_s = Nf_meas consecutively. Especially when the sampling frequency becomes closely enough to the value of N(f₀ + Δf), DFT calculated iteratively would give a measured frequency f_meas as the input and off-nominal frequency f₀ + Δf exactly. It means that there are integral samples in 1 cycle of the input frequency again. The process of iterative DFT algorithm within 1 cycle (i.e., inner-layered iteration) is shown in Fig. 3.

In the first cycle of DFT calculation, the sampling frequency f_s is set to be Nf₀ and two phasor are calculated to get the frequency f_meas. Then in the following cycle, new \( {f}_{\mathrm{meas}}^{\hbox{'}} \) is gotten according to the sampling frequency of \( {f}_s^{\hbox{'}}={Nf}_{\mathrm{meas}} \), until difference of two successive frequencies \( {f}_{\mathrm{meas}}^{\hbox{'}} \) and f_meas is less than a threshold δ. In Fig. 4, samples are taken by a rectangular overlapping shifting window at time t_r = rT_s (r = 0 and 1 respectively) as shown in Fig. 4. The first rectangle window contains N samples s₀,s₁,…,s_N − 1 but not the sample of S_N; however, the length of this window is N*T_s, where T_s is the sampling period. In the second rectangle window, N samples s₁, s₂,…,s_N are taken with the same length N*T_s and so on.

In general, voltage and current waveforms are not always nominal sinusoids or cosine wave, particular in a distributed power system. Researchers have done some works on correcting this asynchronous effect [13, 28, 29]. Transients are non-unsteady states that occur in the power system. They are electrical transients and electromechanical transients generally.

The former are caused by faults and other switching operations, while the latter ones are generated by dynamic movement of rotors of generators and following power system disturbances [26, 30]. Phasors calculated in electrical transients often display a step change in phase angles and amplitude, but not the frequency. However, the motor speed in modern power systems may deviate from synchronous speed by 0.1~5 Hz, the phase angle behavior during the phasor estimation window is approximately linear [26, 30]. Recommend by IEEE Std. C37.118-2005 [30], three step-changing models are adopted.