Mathematical Problems in Engineering

Volume 2015, Article ID 971257, 11 pages

http://dx.doi.org/10.1155/2015/971257

## A Fault Diagnosis Method of Power Systems Based on Gray System Theory

College of Information Science and Engineering, Chongqing Jiaotong University, Chongqing 400074, China

Received 22 August 2014; Revised 9 November 2014; Accepted 13 November 2014

Academic Editor: XiaoSheng Si

Copyright © 2015 Huang Darong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To provide some decision-making suggestions for fault diagnosis in power systems, a new model for identifying fault component is constructed by using Gray theory. Firstly, the basic concepts of Gray theory are introduced and explained in detail. And then the recognition algorithm of the power supply interrupted districts and the assignment principle of fault state vectors are depicted according to the working principle of protective relays (PRs) and circuit breakers (CBs). Secondly, based on the concept of the Gray correlation degree, the fault information explanation degree model is constructed and the judging method of malfunction and rejection for PRs and CBs is established. Meanwhile, to achieve the goal of the fault diagnosis, the fault diagnosis procedure that determined which components malfunction is designed for power systems. Finally, some simple experiments have already verified that the proposed method and model are effective and reasonable and the trend of further research is analyzed and summarized.

#### 1. Introduction

Fault diagnosis of power systems is a method, which uses the information collected from protective relays and circuit breaker to recognize fault component and malfunctioned or tripped PRs and CBs. It is generally known that the fault component recognition is the crucial problem in engineering application [1]. In recent year, many fault diagnosis methods of power systems are proposed, including expert system [2], artificial neural network [3], optimization technology [4], rough set theory [5, 6], Petri net [7], and Bayesian network [6]. In order to meet certain preconditions in the existing diagnosis methods, some assumptions are made for them. But in some cases these assumptions may be contrary to reality and may even cause error diagnosis results. For example, when the information of protective device is incomplete, the fault diagnosis’ conclusion may be incorrect. So the crucial difficulty of fault diagnosis is how to guarantee the corrective and effective diagnosis results for power systems with incomplete information [8]. In other words, if we want to get the corrective and effective diagnosis results, we have to know how to obtain the unknown information by utilizing the known information under the condition of incomplete information [9–11].

Fortunately, the Gray system theory can simulate and determine the unknown information according to the known information of systems. In addition, using the Gray systems theory to construct the fault diagnosis model involves a smaller sample and little information. The process is easier to be operated. However, the anomalies contrary to qualitative analysis will not be produced in the process of Gray correlation analysis. Hence, under the condition of information incomplete and not explicit, the Gray system theory has its unique superiority; that is, it is an effective method in case of incomplete and uncertain information.

Based on the above, application of the Gray theory to the fault diagnosis of power systems is proposed in this paper. The layout of the rest of the paper is arranged as follows. in Section 2 the Gray system theory and Gray correlation analysis were introduced briefly. Section 3 described the fault diagnosis method and model of power systems based on Gray system theory and designed the diagnostic procedure for fault components. In Section 4 we have discussed the simulative results by experiments. Finally, Section 5 concluded this paper with inferences and directions for future work.

#### 2. Basis Introduction of Gray System Theory and Gray Correlation Analysis

##### 2.1. Basal Principle of Gray System Theory

Gray system theory, which is established by Chinese scholar Professor Deng Ju-Long in 1982, has been used to research uncertain problem with lack of data and information. In general speaking, some information is known and other pieces of information are unknown in uncertain data systems. The main idea of Gray system theory is to describe correctly the systems’ evolution law and to monitor effectively their running behavior by extracting valuable information from the known information. Over forty years later, the structure system of Gray system theory has basically taken shape. The theoretical basis includes Gray Matrix, Gray Algebra, and Gray Equation. And the Gray model (GM) may also implement the analysis, evaluation, prediction, and control decision of uncertain data systems by using spatial association rule and sequence generation method. For future understanding of the advantages of Gray system theory, six basic principles are introduced as follows.

###### 2.1.1. Differential Information Principle

There is the difference among differential information. In other words, the difference is the information. For example, if there are two differential objects or systems, everyone has unique information that is not similar to another piece of information. One of the most basic pieces of information which the human society perceives is that the world comes from the difference between matter and matter.

###### 2.1.2. Nonuniqueness of Solutions

The solution is nonunique under the condition of information being incomplete and vague. The uncertainty of systems leads to the existence of uncertain information and then causes nonuniqueness of solutions.

###### 2.1.3. Smallest Information Principle

The basic idea of Gray system theory is to utilize the smallest information achieving from known knowledge data to accomplish a given task. The smallest information getting by researching the uncertain problem with the small samples and poor information is the fundamental basis of making a distinction between Gray area and no-Gray.

###### 2.1.4. Cognitive Foundation Principle

*The Foundation of Cognition Is Information*. The accurate and complete cognition is determined according to definite and precise knowledge; the uncertain and incomplete knowledge may also lead to vague cognition. Correspondingly, if there is no information of systems, the cognition of systems is not also completed. Thus the cognition should be studied based on information.

###### 2.1.5. Innovation Priority Principle

The function of new information is more important than old information for cognitive behaviors, because the new information directly reflects the current states of the system and mainly influences the future trend of the development.

###### 2.1.6. Gray Indestructibility Principle

Notably, the incomplete and uncertain information is very much strongly entrenched in real systems. As new information is continuously generated in real engineering, the cognition conclusion is improved gradually and the level of cognition will tend to rationality and correctness. Accordingly, the Gray systems do not disappear.

Based on the above principles, the theoretical model implementing a given task may be constructed according to small sample and poor information. However, the running states and trend of systems are determined by many factors in real application. So we need to discern the primary factors and lesser factors. And fortunately, the Gray correlation analysis, which is the important part of Gray system theory, can judge the connection in accordance with the approximation degree of the two-dimensional curve in time domain and frequencies domain. The higher the similarity is, the greater the correlation is. For further analysis, the Gray relational axioms are introduced firstly.

##### 2.2. Gray Correlation Axiom

Suppose that the behavior sequence of the system is ; thus the corresponding factor series is as follows:For a given real , if satisfies(1)normalization: , and ;(2)integrity: let ; if , then (3)even symmetry: if , ;(4)accessibility: the value of the Euclid distance is inversely proportional to the value of the given real .Thus is called as Gray relational degree, where represents the correlation coefficient of each pair of variables and . Four conditions (1–4) are also regarded as Gray relational four axioms.

Notice that four conditions just do positively mean these things. Normalization illustrates that there is a correlation between two arbitrary behavior sequences. Integrity describes that the Gray relational degree is influenced by the external environment. If the outside environment is changed, the relational degree is also varied. So the symmetry principle is not necessarily true. Meanwhile, even symmetry represents that the symmetry principle is true while the set of factors contains just two factors. Accessibility may constrain the relational quantization.

##### 2.3. Gray Correlation Analysis

To get the computing formula of Gray related degree, the distance measure between vectors and is defined as follows:

And suppose that

According to the formulas (2)-(3), the correlation coefficient between vectors and vector is defined as follows:that is,where is called the resolution coefficients and the values of are usually restricted to a certain range .

Notice that the discriminatory power varies depending on the different correlation coefficients: the smaller the is, the higher the differences between two correlation coefficients are and the stronger the discriminatory power is.

Let , and then we defineand is considered as the Gray correlation degree between reference sequence and compare sequence . Obviously, satisfy Gray relational four axioms (1–4).

##### 2.4. Computing Algorithm of Gray Correlation Degree

By the definition of Gray correlation degree, the computational steps of Gray correlation are made as follows.

(1) Collect the evaluation data on the evaluation index system; then the sequences of data may be stated in matrix form as follows:where represents the number of indexes. And , .

(2) Apply dimensionless method to the original data sequences; let the dimensionless model be Thus the new data sequences processed by dimensionless model may be rewritten as

(3) Define the reference sequence . The reference sequence consists of the most optimal value or the worst value of every index. That is, . Accordingly, the rest of the data is as compare sequence.

(4) Compute the distance measure between the corresponding elements of the reference sequence and compare sequence ; that is, .

(5) Calculate using the formula (3); that is,

(6) Compute the correlation coefficient between vector and vector by formulas (4) and (5); that is, Notice that is the resolution coefficients and the values of are usually restricted to a certain range . To keep things simple, put in this paper.

(7) Compute the correlation degree between reference sequence and compare sequence by formula (6), and then obtain the evaluation conclusion by comparing the size of correlation degree.

#### 3. Model and Algorithm of Fault Diagnosis of Power Systems Based on Gray System Theory

##### 3.1. Overall Description of Fault Diagnosis of Power Systems

Firstly, when some relevant components of power systems malfunction, the corresponding protective relays and circuit breakers will work actively and the fault components will be disconnected with the power supplier. So the power supply interrupted districts will be formed. And then analyzing and eliminating breakdown will be also made in the power supply interrupted districts. Secondly, in the blackout area, the working principle of protective relays and circuit breakers will be analyzed in detail, and the information of the protective relays and circuit breakers, which are in fault state, may be deduced. And based on what is mentioned above, we may establish the state vector of the fault modes and quarantined modes of the components and give the assignment principle of the value for each member of the vector. Finally, the correlation degree between the reference component and compare component will be calculated and sorted. The discriminant criterion is that the component with maximum correlation value is as the fault component. Moreover, the numbers of fault components increase constantly. When the information explanation degree of fault components reaches a specified threshold and the numbers of protective relays and circuit breakers which fail to operate or refused operation do not increase, the number of fault components is determined. According to the order of correlation degree, those components which have same account of maloperation and refused operation are regarded as the fault component. So the ultimate goal of the fault diagnosis is also accomplished.

Obviously, some basic concepts should be explained such as the power supply interrupted districts, fault information explanation degree, and judging rule of maloperation and refused operation. Next, these related concepts and model will be explained step by step.

##### 3.2. Quick Recognition for the Power Supply Interrupted Districts

The formation mechanism of the power supply interrupted districts is established on the difference of the topological structures of power systems before and after fault occurrence. When the fault happens, the protection action will be implemented to trip the relevant circuit breakers and the fault components can be isolated from the systems for avoiding the expansion of accident. On the basis of the real time information of the circuit breaker, the topologies of systems before and after fault occurrence are recognized by the real time topology analysis on power systems. And the fault components can automatically form some passive networks according to the difference of the information of the topological structures. So these passive networks are regarded as the power supply interrupted districts. Thus, the recognition of fault components may be limited to the blackout area.

The specific steps [12] are shown as follows.(1)Based on topology analysis for the normal power systems, the corresponding equivalent power or the generator are remarked as the active nodes.(2)The topology analysis is made for the faulty system again, and several subsystems are achieved at the same time according to the different information of the topological structures.(3)The nodes of each subsystem are searched for one by one, and every component connected with each node is active or is not judged. If the component is active, the subsystem is in the normal state, and the search ends; otherwise, if all components in a subsystem are searched for and no active components are found, the subsystem is regarded as the power supply interrupted district.

##### 3.3. Construction for Fault State Vectors and Assignment Principle for Vector Elements

The state vectors are established by means of the protective relays and circuit breakers. If there are circuit breakers and protective relays, the form of state vectors can be expressed aswhere is the state value of protective relays and is the state value of circuit breakers. The assignment principle of vector elements is as follows.(1)If the protective relay run in active mode, then , ; otherwise .(2)Analogously, if the circuit breaker is kept in off state, then , ; otherwise .(3)When the state of the element possesses two modes simultaneously, the information of the protective relays conflicts with the information of the circuit breakers. That is to say, the value of state is both 1 and 0. In that case, the value of and is given as 0.5; that is, , ; , .

##### 3.4. Modeling the Fault Information Explanation Degree

Fault information explanation degree (FIED) is the matching degree of quarantined state vector and fault state vector of multicomponent system. In the other words, it is used for explaining the behavior of protective relays and circuit breakers when the fault of multicomponent systems happens.

Let , whose sizes of dimension are all , be the fault state vector, respectively, to single-fault components . is quarantined state vector with dimension.

Therefore, the fault information explanation degree (FIED) of single component can be expressed by the following formula:If there are more than one faulty component in power system, FIED can be presented as follows:where the symbol “” denotes that the greater value is selected for the values of corresponding elements.

To locate out the fault components in systems, we need to compare the size of FIED. Once FIED satisfies , the fault components may be determined by formulas (13) and (14).

##### 3.5. Judging Method of Malfunction and Rejection for Protective Relays and Circuit Breakers

According to the assumption and analysis above, the judging rule is designed as follows.

(1) If the value of element of the vector () is negative, then the corresponding protection of protective relays or action of circuit breakers is regarded as malfunction for single-component fault. Otherwise, the protection of protective relays or trip of circuit breakers is regarded as rejecting action.

(2) If there is more than one fault component in the system, the judging rule is as follows. The protection of protective relays or action of circuit breakers is regarded as malfunction while the value of the vector is negative; otherwise the action is regarded as rejecting action.

Notice that when failures happen, the main protection of PRs is first implemented to trip the circuit breakers. If the main protective relays do not work, then the backup protective relays begin to work to trip the circuit breakers. Therefore, the situation of the backup PRs deduced by (1) and (2) was the rejecting action; it may be that the action of PRs has already tripped the action of CBs. So we concern only the main protective relays and the circuit breakers and then give priority to main PRs when the malfunction and rejecting action are considered in this paper.

##### 3.6. Fault Diagnosis Model of Single Component

The problem of recognition for single-fault component can be solved by seeking for the fault assumptions, which can best explain the alarming information and can be expressed by the following maximization problem:where stands for the total number of the single-fault, and its value is determined by adding the number of components and the number of lines; denotes the number of vector elements, and its value is the number of protective relays and circuit breakers; is on behalf of the resolution coefficient; indicates the quarantined faulty state vector; means the fault state vector of single-line or single component and also is called the single-fault state vector; represents the similarity degree between the quarantined fault state vector and single-fault state vector and also is called correlation degree.

Notice that if the state vector of single-fault component is , respectively, then the dimension of these state vectors is and the dimension of the quarantined fault state vector is also . Meanwhile, we may calculate the correlation degree and sort the calculated results. Based on above discussion, the recognition rule of fault component is given as follows.

*Rule 1. *If the correlation degree between the reference state vector and the compare state vector is the largest, the corresponding part is fault component or fault line.

*Rule 2. *Let the threshold of fault diagnosis be the mean of correlation degree. If the correlation degree between the reference state vector and new compare state vector is more than the threshold, the part is regarded as the fault component or fault line.

Therefore, the possibility that fault happens may be determined for every single component according to Rules 1 and 2.

##### 3.7. Diagnostic Procedure for Fault Components

In the actual project, there is more than one fault component when faults occur. But the diagnostic procedure may be designed on the basis of the fault diagnosis model of single component. So the specific steps of fault diagnosis are shown as follows.

###### 3.7.1. Data Preprocessing

The data, which contains the information of protective relays and circuit breakers and the topology of the power system, is written in the system.

###### 3.7.2. Recognition for the Power Supply Interrupted Districts

The blackout area of power systems is recognized, and the tripping principle of protective relays and circuit breaker in blackout area is listed.

###### 3.7.3. Recognition of State Information

Under single-fault state, the state information of protective relays and circuit breaker is obtained in accordance with the working principle of protective relays and circuit breakers.

###### 3.7.4. Construction for Fault State Vectors

The state vector of fault mode for each component is given by the assignment principle of vector elements, and then the quarantined fault state vector is loaded.

###### 3.7.5. Calculating and Sorting the Correlation Degree

The correlation coefficients are calculated, respectively, and then these results are sorted. If the correlation degree between quarantined state vector and reference state vector is the highest, the probability that component malfunctions is also the largest.

###### 3.7.6. Computing Fault Information Explanation Degree

FIED is calculated by formulas (13) and (14) and is sorted. If FIED of some fault components satisfies , go to next step.

###### 3.7.7. Fault Diagnosis Analysis

According to Rules 1 and 2, the fault component can be recognized and diagnosed. And then judge the malfunction and rejection for protective relays and circuit breakers.

The specific flowchart for fault diagnosis is shown as in Figure 1.