DOI : 10.17577/IJERTV3IS061236
- Open Access
- Total Downloads : 338
- Authors : Sumit Kumar
- Paper ID : IJERTV3IS061236
- Volume & Issue : Volume 03, Issue 06 (June 2014)
- Published (First Online): 23-06-2014
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
An Improved Method for Efficient Power System Frequency Estimation
Sumit Kumar
Electronics Department, VeermataJijabai Technological Institute
Mumbai, India
AbstractAccurate and efficient estimation of power system frequency is relevant to many power system applications. This paper provides a method for improving the efficiency of power system frequency estimation by combining the two techniques namely Smart Discrete Fourier Transform (SDFT) and Sliding DFT. Smart Discrete Fourier Transform (SDFT) has been proved to be a precise and easy implementable method for power system frequency estimation whereas Sliding DFT establishes a relationship between DFT of successive windowed sequence. Computational efficiency for power system frequency estimation has been analyzed.
KeywordsPower system frequency estimation, Smart Discrete Fourier Transform (SDFT) ,Sliding DFT.
-
INTRODUCTION
The electric power system is base for modern society, a fundamental factor for the development of countries. It consists of a group of apparatuses, wires, and machines that links the power plants to costumers and their needs. The generation scheme of energy (thermal hydroelectric, solar, and wind) at power plants may differ but the power grid signal can be viewed as a single sinusoidal waveform with a fixed frequency (the power system frequency).
Mostly the power generation at the power grid is by
Sliding DFT is a computationally efficient way of finding DFT of a sequence if DFT of previous sequence differing by one sample is known [5][6].
The organization of this paper is as follows: Basic principle of SDFT has been described in section II and Sliding DFT in section III. Section IV describes the proposed method. Section V is comparative analysis for efficiency andfinally, conclusion is given in section VI.
-
SMART DISCRETE FOURIER TRANSFORM (SDFT)
The Smart DFT is a technique to find out the power system frequency by establishing the relation between three consecutive windowed sequences differing by one sample. Consider a sinusoidal input signal of frequency = 2f as follows:
= + (1)
Where X is the amplitude and is the phase angle of the voltage/current signal.
Suppose that x(t) is sampled with a sampling rate (60*N) Hz waveform to produce the sample set x(k).
turbines that work as generators of alternating current. The rotation velocity of these turbines determines the power
=
60
(2)
+ = 0,1,2, , 1
system frequency having nominal value 50/60 Hz. Due to technical reasons this frequency fluctuates in close proximity of its nominal value. It is quite important that for a correct operation of the power system, frequency and phase of all power generation units should remain synchronous within narrow limits. If, for example, a generator drops 2 Hz below
The signal x(t) is conventionally represented by a phasor
(a complex number) x .
= = + (3) Then x(t) can be expressed as
the nominal power system frequency, it will rapidly build up enough heat to destroy itself [1]. Therefore, a tight control is
=
+
2
(4)
needed over operator units and so accurate and efficient estimation of power system frequency is of great importance.
Smart Discrete Fourier Transform (SDFT) has been
where * denotes complex conjugate.
Moreover, the fundamental frequency (60Hz) component of DFT of x(k) is given by
proved as fast and precise in power system frequency
estimation meeting real time specifications [4] incomparision
1
2
2
(5)
to other methods like Prony Method [2], Discrete Fourier Transform (DFT) [3] etc. It estimates the frequency by three consecutive windowed sequences differing by one sample.
=
+
=0
Combing (4) and (5) and taking frequency deviation = 2 (60 +f) into consideration, at last, we obtain:
Where
= + (6)
1
X k(15) = X(15) – x(0) + x(16)
And so
Xk(16) = X k(15) ej2k/N
So, we can relate the DFTs of two successive windowed
=
sin 2
2 +1 +120
60
sequences, each of length N, x (n-1) and x(n) as:
= [ 1 + ]2 /
sin 1
2
sin 2
-
PROPOSED METHOD
2 2 +1 +120 (+1)
=
sin
2
2
60
As we know Sliding DFT relates DFTs of two successive
1=2f/60N
windowed sequences, we will express +1
in terms of
,and +2 in terms of +1 .
If we define
We can write
2=2(120+f)/60N
= 60
( (2+120 )) (7)
+1 = ( + 1 + ( + 1))2/ (11)
+2 = ( +1 + 2 + ( + 2))2/ (12) Now (11) can be written as
=
+ = 1 + 1 (8)
= +12/ + ( + 1 ( + 1))2/ (13)
+1
+1
+1
+2 = +2 + +2 = 2 + 2 (9) Now after some manipulation with (6), (8) and (9) we get
(10)can be written as
= ± + 1 (14)
=
And so
( + +2) ± ( + +2)2 4( +1)2
2( +1 )
(10)
Where
=
( + +2 )
2( +1)
(15)
60
= 60 + = cos1
(11)
From (12),(13) we can write
2
(16)
III. SLIDING DFT
2
= cos +
N
+ D
2 +1
Sliding DFT gives the relation between DFT of two successive time sequence differing by one sample. A discrete-time signal can be considered as a sequence
.., x(-2),x(-1), x(0), x(1), x(2),….x(n), x(n +
1),
Say, x(n) is a time sequence of length N=16.Now x(15) and x(16) will be
x(15) = x(0), x(1), x(2), x(3) .. x(14), x(15)
x(16) = x(1), x(2), x(3), x(4) .. x(15), x(16)
From DFT we can write DFT of x(n)
1
= ()
=0
= 0 + 1 +. . +( 1)(1)
And from DFT properties, if N-DFT of x[n] is X[k] then
Where C= + 1 + 1 and D= + 2 + 2
.
-
COMPARITIVE ANALYSIS
In traditional method of SDFT, first the complex computation is N complex multiplication and N-1 addition per sequence. Then for calculation of p in(15) it will be 3N complex multiplication and 3N-3 complex addition for three DFT sequences and 1 complex addition and one division additional. So total 3N complex multiplication, 3N-2 complex addition and one complex division but when we introduce Sliding DFT concept in SDFT method the complex
calculation reduces to great extent. We need only 2 complex
[]
multiplication and 1 complex addition for central sequence
From above we can observe that first element x(0) in DFT of x(n) is unmodified. Now consider
x (15) = x(16), x(1), x(2), x(3), x(4),.x(15)
If X (15) ,X(15) and X(16) is DFT of x (15), x(15) and x(16) then we can write
and for calculation of p in (16) it is 1 complex multiplication,
1 complex addition and one complex division so total 3 complex multiplication, 2 complex addition and one complex division. Other important point is that the computational cost is fixed and independent of sequence length N.
-
CONCLUSION
-
From comparative analysis of section 5 we can observe that introduction of Sliding DFT concept in Smart Discrete Fourier Transform (SDFT) method enhances the computational efficiency to large extent keeping all the advantages of Smart Discrete Fourier Transform (SDFT) for power system frequency estimation.
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-
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-
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