ACE 방법과 가상 계측기 정보를 이용한 공간변수의 비선형 추정방법

ACE 방법과 가상계측기 정보를 이용한 공간변수의 비선형 추정방법
박문규, 이은기, 김용희
한국전력공사 전력연구원
Nonlinear Fitting of Spatially Distributed Variables using Virtual Detectors
based on ACE Algorithm
Moon Ghu Park , Eun Ki Lee, Yong Hee Kim
Korea Electric Power Research Institute, KEPCO
INTRODUCTION
The estimation or reconstruction of the unmeasured spatial variables has been
widely applied in various scientific and engineering fields, especially for the m
onitoring systems of safety critical and expensive detectors are required. In this
paper, a concept of Virtual Detector Method (VDM) is introduced to enhance th
e estimation accuracy of the spatially distributed variables without additional cos
t of detectors. Normally, one can reconstruct the continuous spatial variable by
synthesizing the discrete measurements from spatially distributed detectors by Fo
urier, spline, and/or nonlinear fitting method like neural networks. However, it is
not easy to get satisfactory accuracy with limited number of detectors in spatia
l domain. The objective of this paper is to give a constructive method of applyi
ng virtual detectors. The key factor of VDM is to justify the signal from virtual
detectors. In this paper, the signal transferred to virtual detectors are extracted
from the neighboring real detectors via the correlation specifying the accurate an
d robust relationships between them. Then the desired continuous form of the m
easurement can be synthesized via the fitting methods with enhanced accuracy
with increased number of detectors.
ACE ALGORITHM
The optimal correlation between measured signal and virtual detector is extra
cted by using Alternating Conditional Expectation (ACE) algorithm. 1,2,3 The ACE
method is a generalized nonlinear regression algorithm that yields an optimal r
elationship between a dependent variable and multiple independent variables. By
using the ACE algorithm, we can get transformations that produce the best-fitti
ng additive model. Generally, this objective can be achieved by treating each va
lue of the transformed dependent variable as the expectation of several realizatio
ns of the sum of transformed independent variables. Initially, it is not required t
o guess explicit functional forms of the transformations. This property is one of
the remarkable advantages of the ACE algorithm. Once the optimal transformati
ons are found by iteration, one can determine the coefficients of functional form
for the transformed dependent and independent variables through the simple reg
ression analysis.
By introducing the ACE algorithm, we can get considerable advantages over
traditional nonlinear regression techniques;
1. The ACE algorithm guarantees the convergence of the transformations,
2. An initial selection of the functional forms is not required. Thus, indepen
dent of the order of the fitting function or the sizes of data sets, the ACE algo
rithm can be applied to get complex correlation between variables.
3. Iterative modifications of the functional relationships are not needed.
Virtual Det
ectors
Real Detec
tors
Fig. 1.
Spatial variation to
be reconstructed
One-dimensional spatial detector system with 5-real/4virtual detector
s.
Let's consider a problem to reconstruct the spatial variation as depicted in
Fig. 1. The ACE algorithm is applied to find out the optimal correlation betwee
n virtual detector signal (dependent variable), Pv, and real detector signal (indep
endent variables), {Di, i=1,_,5}. For the systems which have difficulties in extra
cting the optimal relationship between output y and input variables xi by conven
tional statistical methods, the transformation techniques can be generally applicab
le. For multivariate regression problem with a set of data {Pvi, D1i, D2i,_, D5i, i
=1,_,N}, the optimal transformations of multivariate problems are readily derived
by ACE algorithm as follows,
5


 n ( Dn )  S ( Pv )    d ( Dd ) Dni  (n  1,...,5)
d n


(1)
and
5

S   d ( Dd ) pvi 
 d 1

 5

S   d ( Dd ) pvi 
 d 1
 ,
( Pv ) 
 
5
5




S   2d ( Dd ) pvi 
S   2d ( Dd ) pvi 
 d 1

 d 1

(2)
where S[T|qi] means a conditional expectation at qi and is determined by evaluat
ing an weighted expectation about T around the neighboring values in the interv
al [i-M, i+M] for a given user defined value, M. All transformations should be
mean zero functions. The transformations of each variables are coupled each oth
er and solved by alternative-iteration procedure to minimize the square error of
regression,
2
5


1
e   ( Pv )    d ( Dd ) .
N j 
d 1

2
(3)
There are several weighting or smooth techniques such as Histogram, Nearest ne
ighber, kernel, regression, and supersmoother. In this paper, kernel method is use
d with weighting function defined on the real line with maximum at z = 0.
Basically the ACE algorithm produces the discrete optimal transformtions fo
r given discrete data sets. Therefore, after the optimal transformations are calcul
ated, we construct a simple regression model based on the least square method
to obtain the continuous polynomial functions describing each transformation. To
consider the extrapolation of data, each transformation was devided into 3 regi
ons.
SYNTHESIS OF THE DISCRETE MEASUREMENTS
The ACE algorithm is introduced to construct the signal at virtual detectors
by setting up optimal correlation between real and virtual detectors at each spati
al location. Analytic function corresponding to each optimal transformation was
calculated by simple regression model. Finally, both signals from real and virtua
l detectors are synthesized to get more accurate measurements. We demonstrated
the synthesis method using Fourier fitting method (FFM) compared with refere
nce data set.
The key feature of VDM is how to get the optimal correlation between rea
l 5 detectors and each of 9 virtual detector information. After the 9 detector in
formation is determined by optimal correlation, output variable is computed by u
sing 5 points FFM. In FFM, the following equation is used to generate axial po
wers;
ND
P( z )   a n cos nBc ( z  0.5)  bn sin nBc ( z  0.5) 
(4)
n 1
where ND=no. of detectors, n = no. of mode (n=1,_,ND), an(bn) = Fourier coeff
icients to be determined with 5 detectors , and Bc = fitting parameters minimizi
ng the average RMS error. The Fourier coefficients are determined by following
equations;
zhm
PDm   m P( z)dz ,
zl
(m= 1,..., ND),
(5)
where PDm, Zl and Zh mean the detector signal, relative lower and upper height
at the m-th detector level, respectively.
NUMERICAL RESULTS
The application to a collection of massive data set shows the apparent impr
ovements in mean, root mean square, and maximum estimation errors. The conc
ept of virtual detector can be a viable approach to enhance the estimation accur
acy of the spatially distributed variables without additional detector cost. To veri
fy the usefulness of the method, we have reconstructed a total of 21,000 cases
of one-dimensional axial profile fitted by using 5 real detectors and 4 virtual de
tectors located between each real detectors.
Fig. 2 shows an example of relationships between 8 th virtual detector infor
mation and each normalized 5 detectors. Fig. 2 shows that it is very difficult to
find initial trial functions due to the diffuse and complex nature of the relation
ship between real and virtual. Fig. 3 shows the avg. absolute RMS error of VD
M. The avg. error line of 5-FFM is ~0.03 while ~0.01 of VDM with 9-FFM.
Fig. 4 shows the result for a sample profile indicating the problem of curr
ent 5-FFM for which profile gives some fluctuating signal due to sinusoidal bas
is function with 5-FFM. This phenomenon is certainly improved with 9-FFM ba
sed VDM.
REFERENCES
4
8TH VIRTUAL DETECTOR POWER
8TH VIRTUAL DETECTOR POWER
4
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4TH REAL DETECTOR POWER
3RD REAL DETECTOR POWER
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Fig. 4
Sample profile of reconstructed variable 9-FFM and 5-FFM