Prediction method for monitoring performance of power plant instruments
View Patent ↗Disclosed is a prediction method for monitoring performance of power plant instruments. The prediction method extracts a principal component of an instrument signal, obtains an optimized constant of a SVR model through a response surface methodology using data for optimization, and trains a model using training data. Therefore, compared to an existing Kernel regression method, accuracy for calculating a prediction value can be improved.
1. A prediction method for monitoring performance of power plant instruments, comprising:
displaying measurement data in a matrix;
normalizing the measurement data into a normalized measurement data set;
trisecting the normalized measurement data set into three data sets, wherein the three trisected data sets comprising a normalized training data set, a normalized optimization data set, and a normalized test data set;
extracting principal components of each of the three trisected data sets;
calculating an optimal constant of a Support Vector Regression (SVR) model, based on the normalized measurement data set and the extracted principal components thereof, to optimize prediction value errors of data for optimization with a response surface method;
generating the SVR training model with the optimal constant;
obtaining a Kernel function matrix with the normalized test data set as an input and predicting an output value of the SVR training model; and
de-normalizing the output value into an original range to obtain a predicted value of a variable,
wherein the displaying of the measurement data in the matrix is represented by the following equation:
X
=
[
X
1
,
1
X
1
,
2
…
X
1
,
m
X
2
,
1
X
2
,
2
…
X
2
,
m
⋮
⋮
⋱
M
X
3
n
,
1
X
3
n
,
2
…
X
3
n
,
m
]
=
[
X
1
X
2
…
X
m
]
X
ts
=
[
X
3
i
+
1
,
1
X
3
i
+
1
,
2
…
,
X
3
i
+
1
,
m
]
=
[
X
ts
1
X
ts
2
…
X
tsm
]
X
tr
=
[
X
3
i
+
2
,
1
X
3
i
+
2
,
2
…
,
X
3
i
+
2
,
m
]
=
[
X
tr
1
X
tr
2
…
X
trm
]
X
op
=
[
X
3
i
+
1
,
1
X
3
i
+
1
,
2
…
,
X
3
i
+
3
,
m
]
=
[
X
op
1
X
op
2
…
X
opm
]
where:
X is a matrix representing the measurement data,
X tr , X op , and X ts are matrices respectively representing a data set for training, a data set for optimization, and a data set for test,
n is a positive integer,
m is the number of the power plant instruments, and
i=0, 1, 2, . . . , n−1.
2. The prediction method of claim 1 , wherein the normalizing the measurement data is performed through the following Equation:
Z
i
=
X
i
-
min
(
X
i
)
max
(
X
i
)
-
min
(
X
i
)
where i=1,2, . . . , 3n
where X i is a measurement variable and Z i is a normalized variable of X i .
3. The prediction method of claim 1 , wherein the trisecting the normalized measurement data set is performed through the following equation, each of the three trisected normalized data sets having an n×m dimension:
Z ts =[Z 3i+1,1 Z 3i+1,2 . . . Z 3i+1,m ]
Z tr =[Z 3i+2,1 Z 3i+2,2 . . . Z 3i+2,m ]
Z op =[Z 3i+3,1 Z 3i+3,2 . . . Z 3i+3,m ]
where i=0, 1, 2, n−1, the normalized measurement data set being referred to as Z, the normalized training data set being referred to as Z tr , the data set for optimization being referred to as Z op , and the data set for test being referred to as Z ts .
4. The prediction method of claim 1 , wherein
the normalized training data set, the normalized optimization data set, and the normalized test data set are referred to as Z tr , Z op , and Z ts , respectively,
the principal components of each of the normalized training data set Z tr , the normalized optimization data set Z op , and the normalized test data set Z ts are extracted,
dispersion of the principal components, which is an eigenvalue of a covariance matrix, is arranged according to a size of the principal component, and
sets of principal components P re , P op , and P ts , corresponding to Z tr , Z op , and Z ts are selected until a cumulative sum reaches greater than 99.5%, starting with a principal component having the largest percentage dispersion value, to extract the principal components of each of the trisected normalized data sets Z tr , Z op , and Z ts .
5. A prediction method for monitoring performance of power plant instruments, comprising:
displaying measurement data in a matrix;
normalizing the measurement data into a normalized measurement data set;
trisecting the normalized measurement data set into three data sets, wherein the three trisected data sets comprising a normalized training data set, a normalized optimization data set, and a normalized test data set;
extracting principal components of each of the three trisected data sets;
calculating an optimal constant of a Support Vector Regression (SVR) model, based on the normalized measurement data set and the extracted principal components thereof, to optimize prediction value errors of data for optimization with a response surface method;
generating the SVR training model with the optimal constant;
obtaining a Kernel function matrix with the normalized test data set as an input and predicting an output value of the SVR training model; and
de-normalizing the output value into an original range to obtain a predicted value of a variable, wherein:
the normalized training data set, the normalized optimization data set, and the normalized test data set are referred to as Z tr , Z op , and Z ts , respectively,
the principal components of each of the normalized training data set Z tr , the normalized optimization data set Z op , and the normalized test data set Z ts , are extracted,
dispersion of the principal components, which is an eigenvalue of a covariance matrix, is arranged according to a size of the principal component, and
sets of principal components P re , P op , and P ts corresponding to Z tr , Z op , and Z ts are selected until a cumulative sum reaches greater than 99.5%, starting with a principal component having the largest percentage dispersion value, to extract the principal components of each of the trisected normalized data sets Z tr , Z op , and Z ts ,
a matrix obtained by subtracting an average value of each variable from each of the data sets Z tr Z op , and Z ts , is called a matrix A and is represented by the following equation:
A=Z tr − Z tr .
6. The prediction method of claim 5 , wherein eigenvalues λ of A T A and a singular value matrix S of A are obtained through the following equations:
A
t
A
-
λ
I
=
0
s
1
=
λ
1
,
s
2
=
λ
2
,
…
,
s
m
=
λ
m
,
(
λ
1
≥
λ
2
≥
Λ
≥
λ
m
)
S
=
[
s
1
0
…
0
0
s
2
…
0
⋮
⋮
⋱
0
0
0
…
s
m
]
where:
the eigenvalues λ except for 0 are arranged in a descending order and the arranged eigenvalues λ are called λ 1 , λ 2 , . . . , λ m , respectively, and
each of the three trisected normalized data sets has an n×m dimension.
7. The prediction method of claim 6 , wherein an eigenvector of AA T that is an n×n matrix is obtained, and then a unitary matrix U is obtained, and a particular eigenvalue λ is obtained through the following Equation A and then is substituted into the following Equation B to obtain an eigenvector e 1 , e 2 , e m of n×1 with respect to the particular eigenvalue λ:
| AA T −λI|= 0 Equation A
( AA T −λI ) X= 0 Equation B.
8. The prediction method of claim 7 , wherein the dispersion (σ p ) of each of the principal components of each of the data sets Z tr , Z op , and Z ts is obtained through the following Equation:
σ
P
-
(
[
s
1
s
2
…
s
m
]
n
-
1
)
2
.
9. The prediction method of claim 8 , wherein the percentage is obtained by dividing dispersion of each principal component by the sum of dispersions (σ p — tot ) of the principal components of each of the data sets Z tr , Z op , and Z ts through the following equations:
σ
p_tot
=
sum
(
[
s
1
s
2
…
s
m
]
n
-
1
)
2
%
σ
p
=
(
σ
p
σ
p_tot
)
×
100.
10. The prediction method of claim 9 , wherein a number is selected such that the selected number of the principal components cumulatively satisfy preferred percentage dispersion by performing a cumulative calculation starting with the largest percentage dispersion %σ p .
11. The prediction method of claim 10 , wherein
p denotes the selected number, and
the principal components P tr of the data set Z tr are extracted through the following equation
P tr =└s 1 e 1 s 2 e 2 . . . s p e p ┘
where e 1 , e 2 , . . . , and e p are obtained with the Equation B of claim 8 .
12. The prediction method of claim 11 , wherein compressing input variables x i , x 2 , x n , of an m-dimension into the principal components θ 1 , θ 2 , . . . , θ m of a p-dimension is performed through the following equations:
θ
1
=
q
11
x
1
+
q
12
x
2
+
⋯
+
q
1
m
x
m
θ
2
=
q
21
x
1
+
q
22
x
2
+
⋯
+
q
mm
x
m
⋯
θ
p
=
q
p
1
x
1
+
q
p
2
x
2
+
⋯
+
q
p
m
x
m
where q i1 , q i2 , . . ., g im , satisfy the following two conditions:
q i1 2 +q i2 2 +. . . +q im 2 =1 , i =1, 2 , . . . , m
q i1 q j1 +q i2 q j2 +. . . +q im q jm =0 , i ≠j.
13. The prediction method of claim 12 , wherein an Optimum Regression Line (ORL) f k (θ) obtained as the Support Vector Regression (SVR) with respect to a k th output is represented by the following equation:
f k (θ)= w K T +b k
where:
k is 1, 2, . . . , m,
θ=[θ 1 θ 2 . . . θ p ] T ,
w k is a k th p-dimensional weight vector, and
b k is a k th bias term.
14. The prediction method of claim 13 , wherein, when an ε-insensitive Loss Function L k (y k ) with respect to a k th output variable y (k) is defined by the first line of the following equations, and an optimization equation for obtaining an ORL f k (θ) with respect to (k) i y (k) is represented by the second through fifth lines of the following equations:
L
k
(
y
k
)
=
{
0
,
f
k
(
θ
)
-
y
(
k
)
<
ɛ
k
f
k
(
θ
)
-
y
(
k
)
-
ɛ
k
,
elsewhere
Minimize
Φ
(
w
k
,
ξ
k
)
=
1
2
w
k
T
w
k
+
C
k
∑
i
=
1
n
(
ξ
k
,
i
+
ξ
k
,
i
*
)
s
.
t
.
y
i
(
k
)
-
w
k
T
θ
i
-
b
≤
ɛ
k
+
ξ
k
,
i
w
k
T
θ
i
+
b
-
y
i
(
k
)
≤
ɛ
k
+
ξ
k
,
i
*
ɛ
k
,
ξ
k
,
i
,
ξ
k
,
i
*
≥
0
for
i
=
1
,
2
,
…
,
n
where:
k is 1, 2, . . . , m and ζ ki and ζ ki * are slack variables,
ε k is a k th error term,
C k is a k th coefficient term, and
b is a bias term.
15. The prediction method of claim 14 , wherein the optimization problem is represented as a dual problem by the following equation:
max
λ
k
,
λ
k
*
{
-
1
2
∑
i
=
1
n
∑
j
=
1
n
(
λ
k
,
i
-
λ
k
,
i
*
)
θ
i
T
θ
j
+
∑
i
=
1
n
[
λ
k
,
i
(
y
i
(
k
)
-
ɛ
k
)
-
λ
k
,
j
*
(
y
i
(
k
)
-
ɛ
k
)
]
}
s
.
t
.
0
≤
λ
k
,
i
,
λ
k
,
j
*
≤
C
k
for
i
=
1
,
2
,
…
,
n
∑
i
=
1
n
(
λ
k
,
i
-
λ
k
,
j
*
)
=
0
where:
k is 1, 2, . . . , m,
j is 1, 2, . . . , n, and
λ k, i and λk k,j * are Lagrange multipliers.
16. The prediction method of claim 15 , wherein the Lagrange multipliers λ k, i , and λ k,j * are substituted into the following equation to determine an ORL f k (θ) with respect to a k th output variable of Auto Associative Support Vector Regression (AASVR):
f
k
(
θ
)
=
w
k
*
T
θ
+
b
k
*
=
∑
i
=
1
n
(
λ
k
,
i
-
λ
k
,
j
*
)
θ
i
T
θ
+
b
k
*
where:
w k * is an optimized k th p-dimensional weight vector, and
b k * is an optimized k th bias term.
17. The prediction method of claim 16 , wherein, when the result of nonlinearly mapping from primal data into a space of high-dimension is called a vector Φ (•), that is the result of the nonlinear mapping:
f
k
(
0
)
=
∑
i
=
1
n
(
λ
k
,
i
-
λ
k
,
j
*
)
K
(
θ
i
,
θ
)
+
b
k
*
.
18. The prediction method of claim 17 , wherein the optimized k th bias term is calculated with θ r and θ s , which are arbitrary support vectors, through the following equation:
b
k
*
=
-
1
2
∑
i
=
1
n
(
λ
k
,
i
-
λ
k
,
j
*
)
[
K
(
θ
i
,
θ
r
)
+
K
(
θ
i
,
θ
s
)
]
.
19. A prediction method for monitoring performance of power plan instruments, comprising:
displaying measurement data in a matrix;
normalizing the measurement data into a measurement data set;
trisecting the normalized measurement data set into three data sets, wherein the three trisected data sets comprising a normalized training data set Z tr , a normalized optimization data set Z op , and a normalized test data set Z ts ;
extracting principal components of the normalized measurement data set;
calculating an optimal constant of a Support Vector Regression (SVR) model , based on the normalized measurement data set and the extracted principal components thereof, to optimize prediction value errors of data for optimization with a response surface method;
generating the Support Vector Regression (SVR) model with the optimal constant;
obtaining a Kernel function matrix with the normalized measurement data set as an input and predicting an output value of the SVR model; and
de-normalizing the output value into an original range to obtain a predicted value of a variable, wherein:
the calculating of the optimal constant of the SVR model includes:
determining a plurality of optimal constant values,
obtaining a bias constant of an SVR model for each optimal constant value using the normalized measurement data set, and
calculating a mean squared error (MSE) of the SVR model for each optimal constant value by inputting the principal components to the SVR model for each optimal constant value,
a matrix obtained by subtracting an average value of each variable from each of the data sets Z tr , Z op , and Z ts , is called a matrix A and is represented by the following equation:
A=Z tr − Z ts
wherein eigenvalues λ of A T A and a singular value matrix S of A are obtained through the following equations:
A
t
A
-
λ
I
=
0
s
1
=
λ
1
,
s
2
=
λ
2
,
…
,
s
m
=
λ
m
,
(
λ
1
≥
λ
2
≥
…
≥
λ
m
)
S
=
[
s
1
0
…
0
0
s
2
…
0
⋮
⋮
⋱
0
0
0
…
s
m
]
where:
the eigenvalues λ except for 0 are arranged in a descending order and the arranged eigenvalues λ are called λ 1 , λ 2 , . . . , λ m , respectively, and
each of the three trisected normalized data sets has an n×m dimension.