Estimation of power system frequency using a recurrent scheme

Abstract

This paper presents a new numeric technique to estimate the operating power system frequency. The technique employs a recurrent scheme which consists of a tunable input FIR filter, frequency calculator and an averaging output filter. The recurrent structure ensures that power system frequency is efficiently tracked while minimizing signal distortions arising from harmonic and noise effects. The performance of the developed technique has been thoroughly investigated using computer simulations, and the results are provided. The effectiveness of the proposed technique is demonstrated by comparing it with three different methods reported in the literature. An experimental setup is successfully developed to evaluate the practical operation of the estimator corroborating the performance during simulations. It was confirmed that the proposed estimator performs well during static and dynamic conditions which makes it useful for the estimation of online power system frequency.

Appendices

Choice of the functions g(n) and h(n)

Let us define u(n) as:

$$\begin{aligned} u(n)&=y^2(n-k)+y^2(n-M-k) \end{aligned}$$

(21)

using (9) we get:

$$\begin{aligned} u(n)=&A^2\,\hbox {cos}^2((n-k)\,w_p\,T+\alpha )\nonumber \\&+A^2\,\hbox {cos}^2((n-k)\,w_p\,T+ \alpha -M\,w_p\,T) \end{aligned}$$

(22)

Defining $(n-k)\,w_p\,T+\alpha$ as another variable K and using double angle identity we get:

$$\begin{aligned} u(n)&= A^2(1+\frac{1}{2}\,(\hbox {cos}\,(2K)+\hbox {cos}{\,2(K-M\,w_p\,T)}) \end{aligned}$$

(23)

$$\begin{aligned} u(n)&= A^2(1+\hbox {cos}{\,(2\,K-M\,w_p\,T)}\,{\hbox {cos}(M\,w_p\,T)}) \end{aligned}$$

(24)

u(n) takes values in the range $A^2(1-\hbox {cos}{(M\,w_p,\,T)})$ to $A^2(1+\hbox {cos}{(M\,w_p,\,T)})$. In the presence of noise the sinusoidal expression becomes:

$$\begin{aligned} \hbox {sin}\left( \frac{\pi \triangle f}{2 f_o}\right) =1-\frac{g^2(n)+h^2(n)+\epsilon _1}{2\, (u(n)+\epsilon _2)} \end{aligned}$$

(25)

where $\epsilon _1$ and $\epsilon _2$ are random errors in the numerator and denominator terms arising due to errors in the samples. The error in frequency estimate is minimized, if the sinusoid value is well approximated in the presence of noise and that occurs when we minimize the error in z(n) defined as:

$$\begin{aligned} z(n)=\frac{g^2(n)+h^2(n)}{u(n)} \end{aligned}$$

(26)

In the presence of noise the expression becomes:

$$\begin{aligned} z(n)+\epsilon _z=\frac{g^2(n)+h^2(n)+\epsilon _1}{u(n)+\epsilon _2} \end{aligned}$$

(27)

To minimize $\epsilon _z$, the fractional errors of the numerator and denominator terms need to be minimized.

Adding squares of both functions g(n) and h(n) as done in (15), for a given frequency we get:

$$\begin{aligned} g^2(n)+h^2(n)= \lambda \,u(n) \end{aligned}$$

(28)

where $\lambda$ is a constant. So to minimize the error $\epsilon _z, u(n)$ needs to be maximized. The minimum value u(n) takes is $A^2(1-\hbox {cos}{(M\,w_p,\,T)})$ which can be maximized by taking $M=N$/4. Since $\hbox {cos}(\frac{N\,w_p\,T}{4})=-\hbox {sin}(\frac{\pi \,\triangle \,f}{2\,f_o})$ from (17), (24) becomes:

$$\begin{aligned} u(n)=A^2\left( 1-\hbox {cos}{\,\left( 2\,K-\frac{N\,w_p\,T}{4}\right) }\,\hbox {sin}\left( \frac{\pi \,\triangle f}{2\,f_o}\right) \right) \nonumber \\ \end{aligned}$$

(29)

u(n) takes values in the range $A^2(1-\hbox {sin}({\frac{\pi \,\triangle f}{2\,f_o}}))$ to $A^2(1+\hbox {sin}({\frac{\pi \,\triangle \,f}{2\,f_o}}))$. Since $\triangle f$ is supposed to be small in an actual power system, the values of u(n) remain close to $A^2$ and the resulting error is minimal. The values of u(n) can approach to zero only when $\triangle f = f_o$ which is not possible in a power system.

Convergence analysis

For analysis (18) can be rewritten as:

$$\begin{aligned} \triangle f_{n+1} =\frac{2}{\pi }\,f_o\,\hbox {sin}\left( \frac{\pi \, \triangle f}{2\,f_o}\right) +\frac{\pi ^2\,{\triangle f_n}^3 }{24\,f_o^2} \end{aligned}$$

(30)

where $\triangle f$ is the actual frequency deviation and $\triangle f_n$ is the nth iterate of the recursion. For larger frequency deviation the higher-order terms become more dominant so convergence analysis for (30) is performed considering $\triangle f=\pm \,2.5$. The recursive equation in (30) is of the form $x_{n+1}=g(x_n)$. For a bounded closed interval [0, 3], g(x) becomes a contraction on the interval [0, 3] with g(x) ranging in the interval [2.497, 2.502]. So from contraction mapping theorem (30) has a unique fixed point $\xi$ in the interval [0, 3]. Given $g'(x)=\frac{\pi ^2 x^2}{8 f_o}$ and $g''(x)=\frac{\pi ^2 x}{4 f_o}$, $g'(x)$ is monotonically increasing on [0, 3]. Given g(x) is a contraction and considering the mean value theorem yields:

$$\begin{aligned} |g(x)-g(y)|=g'(\eta )|x-y|\le L |(x-y)| \end{aligned}$$

(31)

with $0<g'(n)<L=\frac{\pi ^2 3^2}{8\,50^2}$. This leads to upper limit of number of iterations n, needed to ensure certain accuracy $\varepsilon$, given in [25] as:

$$\begin{aligned} n \le \left[ \frac{ln|x_1-x_o|-ln(\varepsilon (1-L))}{ln(1/L)}\right] +1 \end{aligned}$$

(32)

where [x] represents largest integer less than equal to x.

For $\triangle f=\pm \,2.5$ and the accuracy $\varepsilon =10^{-4}$ results in $n\le 2$. Carrying out iteration with $n=1$ is equivalent to ignoring all the higher-order terms in the Maclaurin series expansion leading to accuracy of 2 decimal places. With $n=2$ accuracy of 5 decimal places is achieved. Further accuracy can be achieved by performing further iterations resulting in more computational cost.