In this paper, wavelet transform and entropy are evaluated using the mathematical analysis concepts of reflexibility, regularity and series obtention, th e s e concept s remark the reason to make a selective reference framework for power quality applications. W ith this idea the paper used the same treatment for the two algorithms (Multiresolution and Multiscale Entropy). The wavelet is denoted to have the most power full consistence to the light off the reflexibility, regularity and series obtention. The paper proposes a power quality technique namely MpqAT.
The natural harmony of Fourier’s analysis in the Hilbert space is demonstrated by Riesz-Fischer’s [
These three concepts are intended to be shown in two ways, firstly, using wavelet transformation, and secondly, through numerical entropy.
In [
In particular, power quality allows identifying the health state of a power system by means of applying processes signal techniques to current and voltage waveforms. Therefore, power quality is used as the concept with which the achieved definition will be tested.
Function φ ∈ L 2 ( R ) is called an orthogonal wavelet, if the family { φ j k = 2 j / 2 φ ( 2 j x − k ) , j , k ∈ Z } is an orthonormal basis of L2(R), this is if
〈 ϕ j k , ϕ e m 〉 = δ j l δ k m = { 0 si j ≠ l or k ≠ m 1 si j = l and k = m (1)
So, if φ it is an orthonormal wavelet, and if f ( x ) ∈ L 2 ( R ) consequently the wavelet series is:
f ( x ) = ∑ j , k = − ∞ ∞ C j k ϕ j k ( x ) = ∑ j = − ∞ ∞ ∑ k = − ∞ ∞ C j k ϕ j k ( x ) (2)
Then
C j k = 〈 f , ϕ j k 〉 〈 ϕ j k , ϕ j k 〉 = 〈 f , ϕ j k 〉 (3)
that is
C j k = ∫ − ∞ ∞ f ( x ) ϕ j k ¯ d x C j k = ∫ − ∞ ∞ f ( x ) 2 j / 2 ϕ ( 2 j x − k ) d x C j k = ( 2 j ) 1 / 2 ∫ − ∞ ∞ f ( x ) ϕ ( x − k / 2 j 2 − j ) ¯ d x C j k = ( 2 − j ) − 1 / 2 ∫ − ∞ ∞ f ( x ) ϕ ( x − k / 2 j 2 − j ) ¯ d x (4)
Then: wavelet transform relative to basic wavelet φ, like the function:
C W T f ( a , b ) = | a | − 1 / 2 ∫ − ∞ ∞ f ( x ) ϕ ( x − b a ) ¯ d x (5)
And thus the Cjk parts of the wavelet series are obtained from the wavelet transform for:
b = k 2 j , a = 1 2 j (6)
The demonstration can be seen in [
From the point of view of physical, entropy is the concept that measures the tendency towards disorder in nature. This concept has had an important development in the applications derived from it, for example: evaluation of the efficiency in electric motors or power systems. Philosophically, the concept has been used, given the implications for understanding of natural elements and their interaction with life.
With regard to information, entropy has made information visible as a message, which must generate a link between sender and receiver by means of propagation or transmission (whether physical or abstract). This characterizes the degree of difficulty in nature as the goal of entropy. That is, these difficulties are noise, interruptions, etc.
They are disturbances in the message during transmission, and can represent loss of information as a result of system conditions, conformed by the source, transmitter and receiver.
In this particular case, the tendency to decrease information can be visualized as loss or disorder, and so, it is visualized as a form of entropy [
Information Entropy is also known as “Shannon’s entropy”. The coding theorem focuses its attention on random behavior of nature, such as disturbing elements or noise [
It is said that an extensive property is one that we can define through the analysis of systems composed by other subsystems; the properties of large systems require varying slopes.
Entropy is an extensive property. The information contained in two information channels should equal the sum of the information carried by the two channels individually [
Entropy is defined as a measure of uncertainty for a random variable [
Shannon’s entropy H(X) is defined as:
H b ( X ) = − ∑ x ∈ Θ p ( x i ) log b p ( x i ) (7)
where X represents the random variable with Θ set of values, and probability density function p ( x i ) = P { X = x i } , x i ∈ Θ . The equation is generally calculated in binary logarithm. In this case, entropy is expressed in (for example, the entropy of throwing a die is 0.1870 bits). Note that – p log ( p ) ≥ 0 because 0 ≤ p ≤ 1 , therefore, entropy is strictly positive, as observed in reference [
For a time series representing the output of a stochastic process, which is an ordered sequence of n random variables { X i } = { X 1 , ⋯ , X n } , with a set of values Θ 1 , ⋯ , Θ n respectively, n- dimensional entropy is defined as:
H n = − ∑ x 1 ∈ Θ 1 ⋯ ∑ x n ∈ Θ n p ( x 1 , ⋯ , x n ) log p ( x 1 , ⋯ , x n ) (8)
where p ( x 1 , ⋯ , x n ) is the joint probability for n variables X 1 , ⋯ , X n .
The state of a system at a certain moment Xn, is partly determined by its history, Xn, X 2 , ⋯ , X n − 1 . However, each new state of the system brings a certain amount of new information with it.
Theorem 1: Reflexibility:
Let E be a Hilbert space, with E' as its dual. Denoted by the duality between E’, E and ‖ . ‖ ∗ the dual norm of ‖ . ‖ E , then Riesz’s theorem, better known as the concept of reflexibility, says:
If f Î E' a unique element, uf Î E, exists, such that:
Part A:
{ 〈 f , v 〉 = ( v , u f ) E ∀ v ∈ E ‖ f ‖ * = ‖ u f ‖ E (9)
Similarly, each element of u Î E defines an element of fn Î E' such that:
Part B:
{ 〈 f u , v 〉 = ( v , u ) E ∀ v ∈ E ‖ f u ‖ * = ‖ u ‖ E (10)
This is constituted by the following definition:
Part A:
C W T f ( a , b ) = | a | − 1 / 2 ∫ − ∞ ∞ f ( x ) ϕ ( x − b a ) ¯ d x (11)
Part B:
Involves demonstrating the existence of the inverse transform [
This is constituted by definition:
H b ( X ) = − ∑ x ∈ Θ p ( x i ) log b p ( x i ) (12)
But the sample is not recoverable; it cannot be obtained its inverse transformation, and there is no recoverable application.
The Parseval theorem and the central limit: by definition, regularity indicates the variation of a number, with respect to its mean.
In wave transformation, it is used to characterize the regularity of f in L2, as measured by the Sobolev norm, which indicates the Parseval identity.
In other words:
Applying the Parseval identity:
∫ − ∞ ∞ ∫ − ∞ ∞ | C W T f ( a , b ) | 2 ⋅ d a d b a 2 = ∫ − ∞ ∞ ( 1 2π ∫ − ∞ ∞ | P ( ω ) | 2 d ω ) d a | a | ∫ − ∞ ∞ ( 1 2 π ∫ − ∞ ∞ | Ψ ∗ ( a ω ) | 2 | F ( ω ) | 2 d ω ) d a | a | = 1 2 π ∫ − ∞ ∞ | F ( ω ) | 2 ∫ − ∞ ∞ | Ψ ( a ω ) | 2 | a | d a d ω (13)
NOTE: The change in integration is performed in accordance with Fubini, and the second integral on the right side is Cψ.
Applying the Parseval equality again:
1 C ψ ∫ − ∞ ∞ | C W T f ( a ω ) | 2 d a d b | a | = 1 C ψ C ψ 2π ∫ − ∞ ∞ | F ( ω ) | 2 d ω = ∫ − ∞ ∞ | f ( t ) | 2 d t (14)
With the amount of energy from the signal as a function of the energy is decomposed in each element or component in frequency bands (wavelets).
The definition of entropy comes from the central limit theorem, that is:
Suppose that m is a measure of probability in the data from the real signal,
We have:
∫ x m ( d x ) = 0 and ∫ x 2 m d x = σ 2 (15)
Then, for any interval of A:
lim n ( m ∗ m ∗ m ∗ ⋯ ∗ m ) ( A n ) = 1 2 π σ 2 ∫ e − x 2 σ 2 d x (16)
Convergence in (16) shows the definition of entropy.
From this definition comes:
Derived from Shannon’s work, Pincus [
ApEn algorithm description:
Given an N sample, time series X N = { x 1 , ⋯ , x i , ⋯ , x N } , two input parameters m and r, must be incorporated. These belong to parameters from the correlation dimension postulated by Grassberger and Procaccia [
According to length value m, vectors u m ( 1 ) , ⋯ , u m ( N − m + 1 ) are created, where each vector is expressed as u m ( i ) = [ u ( i ) , u ( i + 1 ) , ⋯ , u ( i + m − 1 ) ] . These vectors represent m consecutive values of time series x, starting with the first event-tracking element, as shown in
The distance between vectors um(i) and um(j) is defined as the maximum of the absolute value of the difference between vector components:
d [ u m ( i ) , u m ( j ) ] ≤ max k = 1 , ⋯ , m ( | u ( i + k ) − u ( j + k ) | ) (17)
If C i m ( r ) it is the probability that vector um(j) is close to vector um(i), i.e. the number of j( 1 ≤ j ≤ N − m + 1 ) such that d [ u m ( i ) , u m ( j ) ] ≤ r divided by the
number of vectors extracted from the time series. For ( 1 ≤ i ≤ N − m + 1 ), the probability of being within the range is given by:
C i m ( r ) = ( Numberof j suchthat d [ u m ( i ) , u m ( j ) ] ≤ r ) N − m + 1 (18)
Each element of C i m ( r ) then measures the regularity, or frequency, of similar values, within length m with r tolerance [
C i m ( r ) is constructed by (19):
C m ( r ) = 1 N − m + 1 ∑ i = 1 N − m + 1 C i m ( r ) (19)
Φ m ( r ) is defined as log that of each C i m ( r ) element average of i, and is expressed as follows:
Φ m ( r ) = 1 N − m + 1 ∑ i = 1 N − m + 1 log C i m ( r ) (20)
Therefore, ApEn is estimated as follows:
A p E n ( m , r , N ) = Φ m ( r ) − Φ m + 1 ( r ) (21)
Sample Entropy (SamPEn)
An improvement to the A p E n ( m , r , N ) algorithm was presented by Richman and Moorman [
U m ( r ) = 1 N − m ∑ i = 1 N − m C i m ( r ) (22)
U m + 1 ( r ) = 1 N − m ∑ i = 1 N − m C i m + 1 ( r ) (23)
Equations ((22) and (23)) define vector SampEn elements using the number of pairs um(i), um(j) that comply with parameter r, so long as d [ u m ( i ) , u m ( j ) ] ≤ r . That said, i ¹ j, and so the pairing of a vector with itself is not taken into account.
Richman and Moorman defined the sample entropy as:
S a m p E n ( m , r ) = lim N → ∞ − ln U m + 1 ( r ) U m ( r ) (24)
Which is estimated in statistics [
S a m p E n ( m , r , N ) = − ln U m + 1 ( r ) U m ( r ) (25)
With the definition of regularity, it is necessary to introduce an approach to series from a signal to show the components that can be disaggregated and have the same degree of regularity of these components.
An intuitive idea for the division of the spectrum by series of discrete waves, using filters is represented in
A multiresolution structure is a sequence of subspaces {Vj} in L2(R), such that:
1 ) V j + 1 ⊆ V j ∀ j > 0 2 ) ∪ j V j Itisdensein L 2 ( R ) 3 ) ∩ j V j = { 0 } 4 ) V j = V j + 1 ⊕ W j + 1 ∀ j > 0 5 ) f ( t ) ∈ V j + 1 ⇔ f ( 2 t ) ∈ V j (26)
The symbol Å should be interpreted as the orthogonal sum of two subspaces.
From
Theorem 2: any succession of spaces satisfying the five equations in definition (26), shows that there is an orthonormal basis for L2(R) such that:
ψ m , n ( t ) = 2 − m / 2 ψ ( 2 − m t − n ) , m , n ∈ Z (27)
Beginning with: {ψm,n}, nÎZ an orthonormal basis for Wm, where Wm is the orthogonal complement of Vm in Vm − 1. A demonstration can be seen in [
By virtue of the previous theorem, the simple choice of a0 = 2 and b0 = 1 generates an orthonormal basis of functions.
Observation: From the multiresolution analysis, we have, therefore, two spaces and for each of them, a set of generating functions.
As V1 Í V0 and W1 Í V0, the functions of these subspaces are boundaries (in L2) FOR linear combinations of the base function V0. There is a sequence {v(k)}, such that:
ϕ ( t ) = ∑ k v ( k ) ϕ ( 2 t − k ) ( ratiooftwoscales ) (28)
ϕ ( t ) = 2 1 / 2 ∑ k v ( k ) 1 / 2 sin c ( 2 π ( t − k / 2 ) / 2 ) (29)
Reorganizing internal parentheses:
ϕ ( t ) = 2 1 / 2 ∑ k v ( k ) 1 / 2 sin c [ 2 π ( 1 / 2 ) ( t − k / 2 ) ] (30)
In accordance with the sampling theorem results:
v ( k ) = 2 1 / 2 ϕ ( k / 2 ) = sin ( π k / 4 ) / π k (31)
And since f(t) satisfies the equation between two scales, it is called scale function.
In the same way, {w(k)} is a sequence such that:
ψ ( t ) = ∑ k w ( k ) φ ( 2 t − k ) 2 1 / 2 (32)
Resulting in:
w ( k ) = 2 1 / 2 ψ ( k / 2 ) = 2 1 / 2 ( 2 φ ( k ) − φ ( k / 2 ) ) (33)
The following relationships result from orthogonality:
〈 φ ( t ) , φ ( t − m ) 〉 = δ ( m ) (34)
〈 ψ ( t ) , ψ ( t − m ) 〉 = δ ( m ) (35)
〈 φ ( t ) , ψ ( t − m ) 〉 = 0 (36)
where δ(m) is a generalized function or a Dirac delta [
The V spaces are generated by scale functions f(t), and similarly, W spaces are generated by wavelet functions ψ(k).
In other words, wavelet functions and scale functions are used as blocks on which to construct or decompose the signal at different levels of resolution. Wavelet functions will generate different versions of details of the composite signal and the scale function will generate the approximate version of the signal, object of the decomposition. This can be mathematically represented by the following equation:
f ( t ) = ∑ k c ( k ) φ ( t − k ) + ∑ k ∑ j = 0 J − 1 d j ( k ) 2 j / 2 ψ ( 2 j t − k ) (37)
where, c is the coefficient of the scale, dj is the coefficient of the wavelet in scale j, f(t) and ψ(t) are the functions scale and wavelet, respectively, and k is the coefficient of translation.
Partial conclusion: This main result proposed by French mathematician Yves F. Meyer was the core for posterior (section VII) assessment in power quality. Equation (37) has all three central elements proposed in this article: reflexivity, regularity, and it is a series.
With algorithms ApEn [
Description:
From succession { x 1 , ⋯ , x i , ⋯ , x N } , a new series y(τ) emerges, whose terms are the average of the consecutive elements of the original series, without overlapping. τ corresponds to a scale factor. Each element generated in the time series is calculated by Equation (38):
y j τ = 1 τ ∑ i = ( j − 1 ) τ + 1 j τ x i , 1 ≤ j ≤ N τ (38)
For scale one (τ = 1), time series y(1) is simply the original time series. The length of each new time series generated is equal to the length of the original series, divided by factor τ.
Finally, each new time series represents a new τ (factor scale function), which is processed by the SampEn, thus obtaining the entropy of the signal at multiple scales, or MSE.
The three characteristics cited, reflexivity, regularity, and series, are indispensa-
ble properties for the design of a quality indicator.
Quality itself can be classified as the valuation that is given to a physical object that comes from the production process of another such object. This representation shows its own degree of excellence. For this, it is necessary to have an instrument which allows measurement of a signal from the physical object to be evaluated. This signal must be able to be placed in a comparative framework, where it is demonstrated that the degree of deviation from a reference is measurable. This deviation speaks to its degree of quality.
It is possible to normalize the workspace with a scalar type value, this will be a representation of the degree of excellence of a measured point versus its reference value, it is a normalized value, and since the signal analysis is equivalent to the signal noise ratio.
Q I = ∑ x i 2 x r 2 (39)
In Equation (39), xi indicates the components that are deviated from reference xr.
In this way, the definition of the quality index is reached. You can use continuous parameter or discrete parameter space {Ln, ln}. As it is a work that can be carried to the transformed frame, the measure and integral within will be defined according to Lebesgue [
Q I = ∑ k ∑ j = 0 J − 1 ( d j ( k ) 2 j / 2 ψ ( 2 j t − k ) ) 2 ∑ k c ( k ) φ ( t − k ) 2 (40)
With this definition, the level of deviation of the detailed energy values, with respect to energy values from the thick part of the signal, or low frequency, is measured.
Q I = ∑ k ∑ j = 1 J ( M S C j ) 2 ( M S C ) k 2 (41)
where MSC is multiscale entropy, k is the scale where the energy of major importance is concentrated against the rest. This will be the level of quality deviation.
Much work has been done on the classification and characterization of disturbances [
Here a modification to the power quality Analysis Technique-pqAT [
The signal analysis method starts from the definition of the instantaneous power tensor, and the transformation is then performed on the frame of the transformed wavelet [
the classification of events in power systems.
In
1) Block 1 measures voltage, and current is measured by an instrument or by a SCADA system (as is done in some systems at present) [
2) In Block 3, the quality index is calculated.
3) Block 4 represents the database, with which planning and operation of the monitored system can be achieved. This is used to feed the inference engine, which is the block indicated by the number four. It defines the premises upon which the load identification and classification of transient events are made. This block, called the system database, also feeds the calibration block.
4) Block 5: This block examines events and has to do with a decrease in voltage value. The main events characterized here are:
・ Line energization.
・ Motor ignition.
・ Capacitor bank start-up.
・ In the block, there may be a classification of voltages, due to errors, which shows all aspects of error characterization (possibly followed by the performance of the protection system).
・ These premises, accompanied by the inference engine, produce results presented by the classification block.
5) Block 6: In the calibration block, two parameters are set, on which the entire multiresolution analysis depends, making the method entirely dependent on them. These parameters are: the sampling frequency and the number of decomposition levels (in the case of wavelets or signal multiscale in the case of entropy). In the case of any number of levels, because it is a dyadic decomposition (division of the frequency axis into octaves), the signal size will also be limited to a multiple number of samples, which agrees with most instruments that use the FFT.
The most current literature shows significant advances in the treatment of information from the point of view of classification techniques, using waveform transform. From pioneering studies [
Two traditionally used techniques have been compared in signal analysis, entropy and wavelets. The comparison has focused on three criteria: reflexivity, regularity, and series construction.
The article showed that for the case of wavelet theory, these three criteria are perfectly fulfilled. In the case of entropy, the concept of a series in an “artificial” signal shape is introduced, but the case of reflexibility is not fulfilled. Consequently, entropy is a valuable tool for regularity measurement only.
Additionally:
・ Quality has been defined from two main points of view, as a series of attributes of a physical object, and a degree of excellence that must be qualified according to that set of attributes.
- The first part of the definition involves decomposing the attribute into a measurable series using the property of regularity. The second proposes the idea of quantifying and the degree of excellence through definition of quality indexes. This is based on those conservative-type parameters that are determined through energy definitions in the transformed frame―Parseval’s theorem.
- Finally, any technique that exhibits decomposition in reflexivity, regularity, or series is a candidate for use as quality evaluation framework.
This article will close with a proposal to evaluate power quality using the structure of an expert system, dedicated to the measurement and classification of perturbations, a system called MpqAT. The novelty of this technique is that, through use of the same structure, analysis of both transient and stationary perturbation in any type of frame of reference is unified. System topology has been considered in the most general way possible, and is based on the results obtained by the series criteria, regularity, and reflexivity.
The authors would like give the thanks to the power and distribution network group (GREDyP) in the Universidad Nacional de Colombia, Manizales Branch.
Cano-Plata, E.A., Ustariz-Farfán, A.J. and Estrada-Estrada, J.H. (2017) Wavelets and Entropy for Power Quality Assessment. American Journal of Computational Mathematics, 7, 276-290. https://doi.org/10.4236/ajcm.2017.73022