When any fault occurs in power system, current component contains fundamental component & DC decaying component combination of decaying component & harmonics component.
DC offset is an offsetting of a signal from zero. The term originated in electronics, where it refers to a direct current voltage, but the concept has been extended to any representation of a waveform.
DC offset is the mean amplitude of the waveform; if the mean amplitude is zero, there is no DC offset. DC offset is usually undesirable.
In this thesis we have studied the DC decaying component estimation techniques.
It is a filter that reduces the amplitude of signal with frequency higher than cutoff frequency when low frequency signal is passed through the filter.
Highpass Filter Technique:-
It is the opposite of lowpass filter. It reduces the amplitude of signal with frequency lower than cutoff frequency when high frequency signal is passed through the filter.
2. Kalman Filter Technique:-
Kalman Filter design the state space model of plant, process with noise measurement block to provide the optimum solution of continues or discrete estimation problems.
3. FIR/IIR Filter Technique:-
FIR Filter-It is in use when we require the linear phase characteristic within passband of filter in any filtering problem.
IIR Filter-This is quite better than FIR filter. Because it requires lesser complexity problem, requires less parameter & less memory.
To use FIR/IIR filter, it depends upon the nature of problem & specification of desired frequency response.
1.3Need of these Techniques:-
Relay doesn't sense the dc decaying component. Because of that many a time system face many problems. To overcome those problems we are introducing few techniques which are less costly, safer & having lesser complexity structure.
Chapter 2
Lowpass/Highpass filter Technique
2.1Background:-
(A)Lowpass Filter
Definition:-
A low-pass filter is a circuit offering easy passage to low-frequency signals and difficult passage to high-frequency signals.
It is a filter that reduces the amplitude of signal with frequency higher than cutoff frequency when low frequency signal is passed through the filter.
Simple Circuit for Lowpass Filter:
Fig2.1-Simple Lowpass RC Filter
Role of Lowpass Filter:-
In this technique, we have created a Digital FIR model. We used here a Lowpass filter block in order to remove the high frequency noise.
(B)Highpass Filter
Definition:-
A high-pass filter's task is just the opposite of a low-pass filter: to offer easy passage of a high-frequency signal and difficult passage to a low-frequency signal.
It is the opposite of lowpass filter. It reduces the amplitude of signal with frequency lower than cutoff frequency when high frequency signal is passed through the filter.
Simple circuit for Highpass Filter:-
Fig2.2-Simple Highpass RC Filter
Role of Highpass Filter:-
We use the Highpass Filter block to build a model that removes high frequency noise from signal. In this model, use the Highpass Filter is excited using random signal to create high frequency noise.
2.2Simulink Model
Fig2.3- Digital FIR Model contains blocks of Lowpass Filter & Highpass Filter
2.3Parameter Setting
BLOCK
PARAMETER SETTING
ADD
*Icon shape=Rectangular
*List of signs=++
Matrix Concatenate
*No. of inputs=3
*Mode=Multidimensional Array
*Concatenate Dimension=2
Random Source
*Source type=Uniform
*Minimum=0
*Maximum=4
*Sample mode=Discrete
*Sample time=1/1000
*Samples per frame=50
Sine Wave
*Frequency(Hz)=75
*Sample Time=1/1000
*Samples per frame=50
Vector Scope
Scope Properties:
*Input Domain=Time
*Time display span (no. of frames )=1
2.4Result Analysis
Fig2.4-Shows that lowpass filter filters out high frequency noise in the noisy waveform.
NOTE:-
Channel 1: Noisy sine wave
Channel 2: Filtered noisy sine wave
Channel 3: Original sine wave
Chapter 3
Kalman Filter Technique
3.1Background:-
The Kalman filter is named after Rudolph E. Kalman, who in 1960 published his famous paper describing a recursive solution to the discrete-data linear filtering problem (Kalman 1960).
Kalman Filter provides the optimal solution to following continues or discrete estimation problems. Kalman Filter is set of mathematical equation to minimize the estimated error.
3.2 Kalman filter design and simulation:-
Steady-state Kalman filters are considered.
Consider the discrete plant
x[n+1]= Ax[n]+B(u[n]+w[n])
y[n]=Cx[n]
with additive Gaussian noise w[n] on the input u[n] and data
A = [1.1269 -0.4940 0.1129
1.0000 0 0
0 1.0000 0];
B = [-0.3832
0.5919
0.5191];
C = [1 0 0];
Our goal is to design a Kalman filter that estimates the output y[n]given the inputs u[n] and the noisy output measurements
y[n]=Cx[n]+v[n]
where v[n] is some Gaussian white noise.
Steady-State Design
x[n+1]=Ax[n]+Bu[n]+Bw[n]……..{State Equation}
y[n]=Cx[n]…………………..{Measurement Equation}
Assuming that Q=R=1, you can now design the discrete Kalman filter by
Q = 1; R = 1;
[kalmf,L,P,M] = kalman(Plant,Q,R);
This returns a state-space model kalmf of the filter as well as the innovation gain
M
M =
0.3798
0.0817
-0.2570
The inputs of kalmf are u and yv, and its outputs are the plant output and state estimates ye=y[nIn] and x[nIn].
Kalmf
u ye
yv
x[nIn]
Fig.3.1-Kalman Filter
3.3Block form of kalman filter in MATLAB:-
Fig3.2 Kalman Filter in MATLAB
3.4Circuit to remove the dc decaying component:-
PLANT
KALMAN FILTERye
u
O O y
Process Noise Sensor Noise
Y
Fig3.3-block diagram below shows how to generate both true and filtered outputs
3.5MATLAB Program
The first output of kalmf
kalmf = kalmf(1,:);
kalmf
a =
x1_e x2_e x3_e
x1_e 0.7683 -0.494 0.1129
x2_e 0.6202 0 0
x3_e -0.08173 1 0
b =
u y
x1_e -0.3832 0.3586
x2_e 0.5919 0.3798
x3_e 0.5191 0.08173
c =
x1_e x2_e x3_e
y_e 0.6202 0 0
d =
u y
y_e 0 0.3798
Input groups:
Name Channels
KnownInput 1
Measurement 2
Output groups:
Name Channels
OutputEstimate 1
Sampling time: unspecified
Discrete-time model.
First build a complete plant model with u,w,v as inputs y and yv (measurements) as outputs.
a = A;
b = [B B 0*B];
c = [C;C];
d = [0 0 0;0 0 1];
P = ss(a,b,c,d,-1,'inputname',{'u' 'w' 'v'},...
'outputname',{'y' 'yv'});
Then use parallel to form the following parallel connection.
Pw
v yv
u
KALMAN
FILTER
yv ye
Fig3.4- Parallel connection
ys = parallel(P,kalmf,1,1,[],[])
%Finally, close the sensor loop by connecting the plant output yv to the filter input yv with positive feedback.
% Close loop around input #4 and output #2
SimModel = feedback(sys,1,4,2,1)
% Delete yv from I/O list
SimModel = SimModel([1 3],[1 2 3])
%The resulting simulation model has w, v, u as inputs and y, ye as outputs.
SimModel.inputname
ans =
'w'
'v'
'u'
SimModel.outputname
ans =
'y'
'y_e'
%Generate a sinusoidal input u and process and measurement noise vectors w and v.
t = [0:100]';
u = sin(t/5);
n = length(t)
randn('seed',0)
w = sqrt(Q)*randn(n,1);
v = sqrt(R)*randn(n,1);
%Now simulate with lsim.
[out,x] = lsim(SimModel,[w,v,u]);
y = out(:,1); % true response
ye = out(:,2); % filtered response
yv = y + v; % measured response
% compare the true and filtered responses graphically.
subplot(211), plot(t,y,'--',t,ye,'-'),
xlabel('No. of samples'), ylabel('Output')
title('Kalman filter response')
subplot(212), plot(t,y-yv,'-.',t,y-ye,'-'),
xlabel('No. of samples'), ylabel('Error')
3.6 Result Analysis
Fig-3a.The first plot shows the true response y(dashed line) and the filtered output ye (solid line).
Fig-3b.The second plot compares the measurement error (dash-dot) with the estimation error (solid).
Fig3.5- This plot shows that the noise level has been significantly reduced.
This is confirmed by the following error covariance computations.
MeasErr = y-yv;
MeasErrCov = sum(MeasErr.*MeasErr)/length(MeasErr);
EstErr = y-ye;
EstErrCov = sum(EstErr.*EstErr)/length(EstErr);
The error covariance before filtering (measurement error) is
MeasErrCov ……………..MeasErrCov =1.1138
while the error covariance after filtering (estimation error) is only
EstErrCov…………………EstErrCov =0.2722
Chapter4
Filtering Linear System & Transforms Overview
4.1FIR vs. IIR Filters
Digital filters with finite-duration impulse response (all-zero, or FIR filters) have both advantages and disadvantages compared to infinite-duration impulse response (IIR) filters.
FIR filters have the following primary advantages:
*They can have exactly linear phase.
* They are always stable.
* The design methods are generally linear.
*They can be realized efficiently in hardware.
*The filter startup transients have finite duration.
The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.
4.2Windowing Method
Consider the ideal, or "brick wall," digital lowpass filter with a cutoff frequency of ω0 rad/s. This filter has magnitude 1 at all frequencies with magnitude less than ω0, and magnitude 0 at frequencies with magnitude between ω0 and π. Its impulse response sequence h(n) is
h(n)=1/2Ï€= 1/2Ï€dw= wo/Ï€ sinc(wo /Ï€)
This filter is not implementable since its impulse response is infinite and noncausal. To create a finite-duration impulse response, truncate it by applying a window. By retaining the central section of impulse response in this truncation, you obtain a linear phase FIR filter. For example, a length 51 filter with a lowpass cutoff frequency ω0 of rad/s is
b = 0.4*sinc(0.4*(-25:25));
The window applied here is a simple rectangular window. By Parseval's theorem, this is the length 51 filter that best approximates the ideal lowpass filter, in the integrated least squares sense.
The following command displays the filter's frequency response in FVTool:
fvtool(b,1)
4.3 Hamming Window
Apply a length 51 Hamming window to the filter and display the result using FVTool:
b = 0.4*sinc(0.4*(-25:25));
b = b.*hamming(51)';
fvtool(b,1)
Fig4.1-Magnitude Response in Hamming window
Fig4.2-Date Sheet for Hamming window
4.4 Multiband FIR Filter Design with Transition Bands
Basic Configurations:-
The default mode of operation of firls and firpm is to design type I or type II linear phase filters, depending on whether the order you desire is even or odd, respectively.
Case Study- A lowpass example with approximate amplitude 1 from 0 to 0.4 Hz, and approximate amplitude 0 from 0.5 to 1.0 Hz is
n = 20; % Filter order
f = [0 0.4 0.5 1]; % Frequency band edges
a = [1 1 0 0]; % Desired amplitudes
b = firpm(n,f,a);
From 0.4 to 0.5 Hz, firpm performs no error minimization; this is a transition band or "don't care" region. A transition band minimizes the error more in the bands that you do care about, at the expense of a slower transition rate. In this way, these types of filters have an inherent trade-off similar to FIR design by windowing.
To compare least squares to equiripple filter design, use firls to create a similar filter. Type
bb = firls(n,f,a);
and compare their frequency responses using FVTool:
fvtool(b,1,bb,1)
Fig4.3- Magnitude Response in multiband FIR filter
Fig4.4. Data Sheet for multiband FIR filter
4.5Anti-Symmetric Filters / Hilbert Transformers:-
When called with a trailing 'h' or 'Hilbert' option, firpm and firls design FIR filters with odd symmetry, that is, type III (for even order) or type IV (for odd order) linear phase filters. An ideal Hilbert transformer has this anti-symmetry property and an amplitude of 1 across the entire frequency range.
Case Study-Using Hilbert transformers and plot them using FVTool:
b = firpm(21,[0.05 1],[1 1],'h'); % Highpass Hilbert
bb = firpm(20,[0.05 0.95],[1 1],'h'); % Bandpass Hilbert
fvtool(b,1,bb,1)
Fig. 4.5Magnitude Response for Hilbert Transformers
Fig4.6. Data Sheet For Hilbert Transformers
4.6Constrained least square multiband FIR filter design:-
Design a CLS filter with impulse response order 129 that meets these specifications:
n = 129;
f = [0 0.3 0.5 0.7 0.9 1];
a = [0 0.5 0 1 0];
up = [0.005 0.51 0.03 1.02 0.05];
lo = [-0.005 0.49 -0.03 0.98 -0.05];
h = fircls(n,f,a,up,lo);
fvtool(h,1)
Fig4.7. Magnitude Response for Least Square FIR filter Design
Fig.4.8 Data Sheet for Least Square FIR filter Design
4.7 Constrained least square filter design for lowpass and highpass linear phase filters:-
To approach this design problem using fircls1,
commands:
n = 61;
wo = 0.3;
dp = 0.02;
ds = 0.008;
h = fircls1(n,wo,dp,ds);
fvtool(h,1)
Fig4.9-Magnitude Response for Constrained least square filter design for lowpass and highpass linear phase filters
Fig10-Data Sheet for Constrained least square filter design for lowpass and high pass linear phase filters
Chapter 5
FINAL CONCLUSION
Conclusion
This thesis has demonstrated the techniques through which we can estimate the dc decaying component from the fault current/fault signal.
*In Low/Highpass Filter Technique we found that it is the easiest method to estimate the dc decaying component. The result analysis shows that we have estimated the dc decaying component correctly.
*In kalman Filter Technique we found that it's full of mathematical operation. We have designed the Kalman filter using state equation & measurement equation. We have considered here the steady state condition.
*In last chapter Filtering Linear system. We have discussed here the different conditions/different transforms & analyze that for every other method/condition we get the different responses.
