Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 6, No. 1, 2023 

 

103 

Fault Detection Based on Complete Information 
Principal Component Analysis for Electric Submersible 
Pump 
Jie Yuan*, Yinchang Du, Jiankui Xu, Jianshen Liu 
Designing Institute of Engineering, Offshore Oil Engineering Co.,Ltd, Tianjin, China 
* Corresponding author: yuanjie7@cnooc.com.cn 

 
Abstract: Electric submersible pump (ESP) is the key production equipment for the offshore oil industry. As a result of poor 
working condition, fault and failures happen frequently and affect the production effectiveness. It is significant to detect the fault 
and failure in time. The fault detection for ESP mainly depends on the expert experience for decades. The data-driven fault 
detection methods are widely utilized in recent years with the rapid development of sensor technology, where the PCA is the 
most widely used methods. However, the data collected from the front line faces the imprecision problem, which reduces the 
accuracy of data-driven fault detection methods. The interval arithmetic is an effective way to handle imprecision data. However, 
traditional PCA and interval PCA algorithms cannot handle nonlinear problem. Aiming at this problem, the complete information 
principal component analysis methods are adopted for the fault detection of electric submersible pump in this paper. Compared 
with PCA and CPCA methods, the CIPCA method has better detection robustness when more inaccurate point value data. 

Keywords: Interval algorithm, Principal component analysis, Electric submersible pump, Fault detection. 
 

1. Introduction 
As the key equipment of offshore oil industry, electric 

submersible pump (ESP) transport quantities of oil from 
underground to the surface. However, ESP works under high 
temperature and high pressure conditions which leading to 
failures frequently. Severe economic losses will happen once 
the electric submersible pump failure occurs. Efficient fault 
detection of electric submersible pump can help production 
workers to judge faults quickly and accurately, reduce the 
equipment maintenance time, and help producers to find early 
failures of electric submersible pump in time. Therefore, it is 
of great significance to study the fault detection of electric 
submersible pump(1). 

Traditional fault detection of electric submersible pumps in 
offshore oil and gas wells mainly relies on manual calculation 
and expert experience(2). With the rapid development of 
computing technologies and sensors, data-driven methods 
have become effective approaches for addressing the fault 
detection problem and have become increasingly popular in 
recent years. Zhao et al(3). proposed a full-condition 
monitoring method with cointegration and slow feature 
analysis for nonstationary dynamic chemical processes. It is 
not necessary to extract the process knowledge for these 
methods because all the information is stored in the process 
data. The reliability of the fault detection model is strongly 
dependent upon the quality of the available data. Fault 
detection methods such as principal component analysis 
(PCA) (4)and kernel PCA (KPCA)(5) assume that the process 
data are precisely known and point-valued. However, in 
practice, the data are typically contaminated with 
uncertainties due to various factors, which include 
measurement noise, sensor degradation from age, and severe 
working environments (6). Imprecise measurements with low 
reliability are prominent and may deteriorate the 
identification performance of the data-driven methods that are 
discussed above. The development of fault detection methods 

for the process industry with imprecise data is an urgent 
problem. 

A substantial amount of work has been done by researchers 
on the processing of imprecise data. However, classic data 
analysis cannot satisfactorily deal with data with variability 
and uncertainty. Symbolic data analysis (SDA) introduced 
several representations for data types with data variability, 
among which the interval representation is typical. Intervals 
naturally arise from the description of ranges of values, such 
as daily temperature and daily stock price variations. 
Numerical examples were provided by Lev V. to demonstrate 
that imprecise interval information may improve the 
robustness performance of the SVM algorithm(7). A new 
clustering algorithm that is based on the learning of a self-
organizing map for interval data is proposed in the literature, 
in which the number of clusters is determined automatically 
and no a priori hypothesis is required. Petridis et al. used 
interval numbers to predict annual sugar production based on 
populations of measurements and to improve the prediction 
accuracy(8). Several artificial neural networks were extended 
to interval fields and have exhibited satisfactory performance 
in industry applications with imprecise measurement data. In 
addition, The interval PCA approaches were introduced by 
Cazes et al. and Chouakria et al. known as the vertices PCA 
(VPCA) and the centers PCA (CPCA), respectively. The 
interval PCA methods are employed for process monitoring 
and yielded more robust results(9). 

In this paper, aiming at improving effectiveness of PCA 
method when facing the imprecision measurement problem, 
several interval PCA methods are applied for the fault 
detection of electric submersible pump. Sections 2 is a brief 
introduction of interval PCA. Section 3 discusses the 
simulation of interval PCA to electric submersible pump FDI. 
Section 4 is the conclusion that the IPCA performs better than 
the point valued PCA when the collected data is imprecise. 



 

104 

2. Preliminary 
2.1. PCA 

In recent years, principal components analysis(10) is 
widely used to extract the main relationship of variables in a 
multidimensional data set. It reduces the complexity of the 
multivariate data via a projection into a subspace which 
preserves maximum variance of the original space in less 
number of dimensions. Let us consider n mX R  the data 
matrix   containing n samples of m process variables. Before 
applling PCA, matrix X must be normalized to zero mean and 
unit variance. Then, the PCA linear projection can be 
expressed as 

  ,    TT XP X TP             (1) 
 

where n mT R  is the principal component matrix and   
m mP R  contains the principal vectors which are the 

eigenvectors associated with the eigenvalues of the 
covariance matrix  of X . 

 
1

1
0

     
0

T

T
lT l

l m l T
m l m l

X X
N

P
P P P P

P

      (2) 

 
where T T

mPP P P I , and is a diagonal matrix that 
contains in its diagonal the eigenvalues of  sorted in 
decreasing order. Then, the most important step in performing 
PCA modeling is the determination of the number of 
components to be retained. Several ways can be applied to 
determine the number of main components, in this paper, the 
percentage of accumulated contribution of variances can be 
used as the parameter.  

The transformation matrix m lP R   is generated by 
choosing l   eigenvectors of P corresponding to largest l
eigenvalues. The reduced dimension space X   can be 
denoted as 

 
T

l lX XPP                    (3) 
 
 The residual space, also denoted E  , is the residual 

between X and X . 
 

 E X X                    (4) 
 
At each sampling time k, the measurement vector is 

1 2 . . . mx k x k x k x k  , and its estimation is 

1 2 . . . mx k x k x k x k ,and estimation error can be 

expressed as: 
 

e k x k x k                (5) 
 
When it comes to the fault detection, the established PCA 

model describes the relationship between normal process data 
in principal and residual spaces. Fault events will be detected 

by referencing the observed behavior against this model 
especially in residuals which are expected to be null in the 
fault-free case, and different from zero in the presence of 
faults. 

2.2. IPCA 
In this section, several interval PCA methods are 

introduced. As the interval PCA approach is an extended 
version of the classical PCA approach applied to interval data. 
the interval data are introduced first. 

2.2.1 interval data 
Interval arithmetic is the foundation of many interval 

algorithms(11). Interval numbers are the basis of arithmetic 
operations and are defined as , ,X x x I R x x  , 
where x   and x   are real numbers and X   is an interval 
number. If the width of X   is denoted as w X  , then 

w X x x  . The median of X   can be denoted as

2m X x x .  

If n mX R is the conventional data matrix containing n  
samples of m process variables in normal condition, where 
jx k   is the kth observation of the jth variable. All the 

observation data is considered imprecise and expressed in 
interval format, then the new interval matrix [X] is formalized 
as a set of interval valued observation as below 

 

1 1

1 1

1 ,  1 1 ,  1

,  ,  

m m

m m

x x x x

x n x n x n x n

X
m

x nmx n

      (6) 

 
When defining an interval model of the measured data, the 

radius of the interval observation is given by the measurement 
error r

jx k , while the center of the  interval is given by the 

measurement c
jx k  . Thus, yielding the following interval 

construction formulas, representing respectively, standard 
form and midpoint-radius form.  

 

       ,c r c r
j j j j jx k x k x k x k x k    (7) 

 

where 
2

j jc
j

x k x k
x k and

2
j jr

j

x k x k
x k . 

2.2.2. Centers PCA 
Most interval PCA methods are also based on the analysis 

of transformed matrices from initial interval data as vertices 
of hyper-cubes or as centers and radii. CPCA(12) decomposes 
the correlation matrix of the centers coded matrix CX  and 
it projects the vertices as supplementary points in the factorial 
subspace.

 



 

105 

1

1

1 1c c
m

C

c c
m

x x
X

x n x n

1c
m 1mx

c
mx nm
cx n

           (8) 

 
where c

jx k  is the midpoint or center of the interval at 
hand. The CPCA algorithm for interval data can be resumed 
in the following steps: 

• Step 1: Calculate the centers coded matrix CX  as in (16). 
• Step 2: Apply a classical PCA on the centers matrix, based 

on the symbolic covariance given in (9) and (10), with 
1,  . . .,  mT T  being the principal components of this PCA, and 

1,  . . .,  mp p  their corresponding eigenvectors. 
• Step 3: Determine the new interval principal components 

as: 
 

1, 0 1, 0

1, 0 1, 0

ij ij

ij ij

m m

ij ij i ij
i p i p

m m

j ii ij ij
i p i p

t k x k p x k p

t k x k p x k p
     (9) 

 
where ijp is the ith element of the jth column of eigenvector 

matrix P . 
• Step 4: Compute interval estimates based on CPCA model 

for the first l  components as: 
 

1, 0 1, 0

1, 0 1, 0

ij ij

ij ij

m m

ij ij i ij
i c i c

m m

j ii ij ij
i c i c

x k x k c x k c

x k x k c x k c
       (10) 

 
where ijc is the ith element of the jth column of matrix C  . 
In a word, CPCA performed on interval input data is based 

on a numerical centers codification of the data, a treatment 
with classical PCA analysis technique, and finally a 
transformation of classical results into interval description. 

3. Methodology
3.1. Complete information PCA 

In order to seize more information within interval 
observations, Huiwen et al. proposed a new interval PCA 
method called complete information based PCA (CIPCA). By 
defining the inner product of hyper-cubes divided into 
informative grid data, and based on a rather analytic approach, 
CIPCA accomplishes the derivation of interval-valued 
principal components and transforms PCA modeling into the 
computation of some inner products. Thus, leading to more 
accurate results and providing an efficient and effective way 
for conducting PCA on large-scaled interval data. 

According to CIPCA, given any two interval-valued 

variables jX  and 
j

X , the inner product is defined as 

 

'''

1,
4

j jj j jj
X X x k x k x k x k   (11) 

 

Based on the above definitions of interval norm and inner 
product, and with all data units pre-processed, the covariance 
matrix of n mX  is given by 

 

1 1 1 2 1

2 1 2 2 2

1 2

, , ,

, , ,1

, , ,

m

m

m m m m

X X X X X X

X X X X X X
n

X X X X X X

X111111

2X 22222

XmXX

(12) 

 
The CIPCA method for interval-valued data can be 

summarized in the following steps: 
• Step 1: Compute the covariance matrix of the interval 

data matrix X  using (12). 
• Step 2: Perform an eigen-decomposition of the covariance 

matrix  , where 1 2, , , m, m, and 1 2, , , mp p pmp,   are 
the resulting eigenvalues and eigenvectors respectively. 

• Step 3: Calculate the interval principal components using 
the following formulas  

 

1

1

1

1

m

ij ij i
i
m

j iij i
i

t k p x k x k

t k p x k x k
    (13) 

 
with 

0,  0

1, 0
ij

ij

p

p
 

Step 4: Compute the interval estimates from CIPCA model 
as: 

 

    
1

1

1

1

m

qj lqj q
q

m

j qlqj q
q

x k C x k x k

x k C x k x k
    (14) 

 
with the same condition on , and given that T

l l lC PP . 
Based on the interval arithmetic, the estimated error 

between the estimated values and the process data, 
 

ˆ

ˆ
ij ij ij

ij ij ij

e x x
e x x

(15)

 

3.2. Off-line model building based on CIPCA 
The CIPCA method is to perform interval PCA on the 

principle space and the residual space. The establishment 
steps of modeling are as follow, 

Step 1: Convert the collected imprecision data n mx n m

into interval form, 
 

[ ] [ , ]n mn m n mx x x           (16) 
 
Step 2: Standardize the interval value data, 



 

106 

 
[ ] ([ ]) ([ ]) ([ ])

,
([ ]) ([ ]) ([ ])

n m j n m j n m j

j j j

x E x E x E

D D D

x x x
x x x

  (17) 

 
Step 3: perform CIPCA introduced in section A on the 

standardize matrix [ ]n mx . 
Step 4: Two monitoring statistics can be obtained, which 

are denoted as statistics and statistics respectively. The 
specific calculation formula is as follows 

 
2

,:2
1 2

,

[ ]i
i

CI l

IT
t

                (18) 

 
2

2
,:

1
[ ] e

m

i i ij
j

ISPE e               (19) 

 
where 1 2( , , )ldiag )l  is the diagonal matrix with 

the principle components eigenvalues as diagonal elements.  
Step 5: After all the off-line monitoring statistics of 

modeling samples are calculated, kernel density estimation 
method can be used to obtain the control limit. 

3.3. On-line fault detection based on CIPCA 
The online detection steps of CIPCA method can be 

expressed as, 
Step 1: Convert the online testin imprecision data 

1 m
newx 1 m into interval form, ,, ,[ ] [ , ]j newj new j newx x x .  
Step 2: Standardize the interval value data with Equ(17). 
Step 3: perform CIPCA introduced in section A on the 

standardize sample ,[ ]j newx . 
Step 4: 2

iIT  and iISPE   can be obtained based on 
Equations (18) and (19), respectively.  

Step 5: Compare the statistics with control limit and get the 
fault detection conclusion. 

The offline modeling and online fault detection can be 
expressed in Fig.1, 

 

Transform into
interval format

start

end

N
O

Yes

Off-line
modeling

steps

Fault
detection

steps

start

end

Normal
Fault

detection

Transform into
interval format

Standarlization of

Calculate the
covariance matrix

Obtaining the
online data

Calculate and

2 2
new

new 1?
UCL

UCL

IT IT

ISPE ISPE

Normal
Process data

,1:CI lP

CI

x

[x]

[x]

newx

new[x ]

Establish the CIPCA
model

Calculate the Statistics
and2IT ISPE

Calculate the limit of
statistics

Standarlization of new[x ]

Perform CIPCA on ˆ ]new[x

2
newIT newISPE

 
Figure 1. Flowchart of modeling and online fault detection 

 
4. Simulation 

In this section, a few simulations are performed to show the 
effectiveness of interval PCA when it faces the problem of 
data imprecision. 

 Firstly, 1000 pieces of production records are selected 
from history production database. The records are collected 

from 700 electric submersible pumps in offshore oil industry.  
Among the 1000 records, 800 are normal operation condition 
while other 200 are with different faults. All the records are 
of unequal length, high dimensional and low imprecision. 
Before the records are utilized for fault detection, necessary 
procedures are carried out to make each record equal length 
with 300 sample points. Twelve variables are selected as 



 

107 

character variables to identify the fault of ESP. The variables 
are listed in TABLE.1 

 
Table 1. Selected Features variables of ESD 

Variable Unit Variable Unit 

Pressure of Oil Mpa, Rotational Speed 
of ESD r/min 

Outlet 
Pressure of 

ESD 
Mpa, Acceleration of 

ESD g 

Intlet Pressure 
of ESD Mpa, Gas Oil Ratio % 

Fluid Pressure Mpa, Oil Gas Ratio % 
Current of 

ESD A Water Gas Ratio % 

Voltage of 
ESD V Oil Density g/cm3 

 
There are various of fault for the ESD. Despite the manual 

operation fault, the leaking of oil pipe, low load of the ESD, 
three-phase unbalanced of the ESD, over load of the ESD, 
insufficient of the water supply are the most common fault 
caused by unclear inner mechanic reason. The common faults 

are listed in TABLE.2 
 

Table 2. Common faults of ESD 
Index Faults 

1 Oil Pipe Leaking 
2 Low Load of ESD 
3 Three-phase Unbalanceof ESD 
4 Over Load of ESD 
5 Insufficient Water Supply 

 
Firstly, three data-models are established. 100 samples are 

selected as the modeling data for PCA model, CPCA model, 
CIPCA model are checked by expert and are believed precise, 
relatively. The interval data for CPCA and CIPCA is 
transformed from the same model-building data as that in 
PCA model based on GMM algorithm with one time variance. 
While for the online cases, the test data contains more 
imprecision problem with 3 times variances. 

Two fault cases are chosen from the fault records to verify 
the online fault detection ability.  The Oil pipe leaking fault 
occurs in the first case from 201 sample point to the 300 
sample point. The fault detection results of PCA, CPCA and 
CIPCA models are shown in Fig.2, Fig.3 and Fig.4 

 

 
Figure 2. Fault detection results of PCA model for case1 

 

 
Figure 3. Fault detection results of CPCA model for case1 



 

108 

 
Figure 4. Fault detection results of CIPCA model for case1 

 
As the test data contains much noise, the traditional PCA 

model regards the samples as fault sample. Therefore, nearly 
all the samples are tested fault sample as shown in Fig.2. For 
the interval model can handle the imprecision problem well 
in Fig.3. Almost all the normal samples are tested. However, 
the fault samples have low identification. The CIPCA utilized 
more data information and identify the fault in time and 

accurately as shown in Fig4. 
In case 2, the three-phase unbalanced fault is introduced 

from 200 sample point to the end. The traditional PCA cannot 
separated the fault from the normal point. The CPCA can 
identify most fault sample point with a few false alarm. The 
CIPCA identify most the fault samples. 

 

 
Figure 5. Fault detection results of PCA model for case2 

 

 
Figure 6. Fault detection results of CPCA model for case2 



 

109 

 
Figure 7. Fault detection results of CIPCA model for case2 

 
5. Conclusion 

Most of the data-driven fault detection methods of electric 
submersible pump are affected by the data imprecision. The 
interval data algorithms provide a way in handling this 
problem. The CPCA method can identify more fault and 
perform more sensitive than the point valued traditional PCA. 
However, the traditional and modified PCA algorithms 
mainly handle the linear relationship between variables. Some 
nonlinear relationship cannot be identified by the PCA 
algorithms. In this paper, the complete information PCA is 
applied to identify the faults and all the information between 
variables are considered and reserved. Better detection results 
are received. 

Acknowledgment 
This work is supported by the National Natural Science 

Foundation of China (No.61903272), the Natural Science 
Foundation of Tianjin (No.20JCQNJC01670) and Science 
and Technology on Space Intelligent Control Laboratory (No. 
HTKJ2019KL502008). 

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