Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 7067025, 7 pages

https://doi.org/10.1155/2017/7067025

## A Combined Markov Chain Model and Generalized Projection Nonnegative Matrix Factorization Approach for Fault Diagnosis

^{1}State Key Laboratory for Alternate Electric Power System with Renewable Energy Source, North China Electric Power University, Beijing 102206, China^{2}School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, China

Correspondence should be addressed to Wang Shilin

Received 9 March 2017; Revised 9 May 2017; Accepted 22 May 2017; Published 13 June 2017

Academic Editor: Honglei Xu

Copyright © 2017 Niu Yuguang 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

The presence of sets of incomplete measurements is a significant issue in the real-world application of multivariate statistical process monitoring models for industrial process fault detection. Since the missing data in the incomplete measurements are usually correlated with some of the available variables, these measurements can be used if an efficient algorithm is presented. To resolve the problem, a novel method combining Markov chain model and generalized projection nonnegative matrix factorization (MCM-GPNMF) is proposed to detect and diagnose the faults in industrial process. The basic idea of the approach is to use MCM-GPNMF to extract the dominant variables from incomplete process data and to combine them with statistical process monitoring techniques. and statistics are defined as online monitoring quantities for fault detection and corresponding contribution plots are also considered for fault isolation. The proposed method is applied to a 1000 MW unit boiler process. The simulation results clearly illustrate the feasibility of the proposed method.

#### 1. Introduction

As industrial process becomes more and more complex, process monitoring and diagnosis techniques are gaining importance for plant safety, maintenance cost, and profit margins. Multivariate statistical process monitoring (MSPM) techniques have been widely used to build statistical models for some unmeasured variables and for establishing online monitoring schemes for industrial process [1, 2]. These models extract a small number of latent variables, which in a manner better summarize the properties contained in the original variables. Monitoring and diagnosis using these latent variables are both simpler and more powerful than using the original variables [3].

It is well known that the traditional MSPM models usually require that the process data on all variables must be complete. In practice, however, one of the challenges in applying these models is to deal with the process data sets that contain some missing observations. Sometimes, more than 50% of industrial process data would contain the missing data [4]. Since the missing data in the incomplete measurements is usually correlated with some of the available variables, the conventional MSPM methods generally eliminate the sample data in the data matrix that contain them, but doing so will leave the corresponding nature of process unknown.

It is of great importance to determine how to use the incomplete process data sets to build the normal operating model. The well-known Markov chain model (MCM), a typical stochastic process model, is one of commonly used prediction approaches. The most basic feature of MCM is “Markov property,” also known as “no aftereffect.” The next process variable state is predicted by the transition probability matrix obtained using MCM to predict the mobility of measurements if the original time series satisfies the conditions of the Markov chain (MC) [5, 6]. The MC prediction method has been widely used in various fields. In Jeon and Lee’s research, MC-based prediction routing methods are proposed to select the optimal behavior nodes [7]. In order to keep balance of electric power system, Yoder et al. use the MCM to predict short-term wind power [8].

Nonnegative matrix factorization (NMF) is a novel multivariate data analysis and dimension reduction technique that has many applications in spectroscopy, data mining, and pattern recognition [9]. NMF and its variant methods are typically applied to high-dimensional data where each element has a non-negative value, and they find a low rank approximation from the historical process data sets. Unlike the traditional MSPM methods, the NMF-based algorithms do not have any assumption about the nature of the process variables except for nonnegativity. The nonnegativity restriction lets only additions in the factorization process. This property makes NMF obtain sparse and part-based subspace representations of the original data sets [10, 11]. Therefore, NMF-based methods have potential superiority to solve the monitoring and diagnosis problem of complex industrial process.

Normally, the NMF-based algorithms require that the measurement data be nonnegative. In practice, however, the process data of industrial process may be not fulfilling this constraint. Due to the difference in unit selection, the collected data is likely to contain negative numbers. Although it is possible by adjusting the unit to make negative data satisfy the nonnegative constraints, in order to make NMF method have a wider range of applications, we want to relax the nonnegative constraints on the original data sets. In this research, we will propose a new variant of NMF to solve the above problem. It can be called generalized projection nonnegative matrix factorization (GPNMF). Then, MCM and GPNMF are combined to extract useful information from incomplete process data and to combine them with statistical process monitoring techniques.

The rest of the article is organized as follows: Section 2 introduces MCM-based prediction method. Section 3 proposes the GPNMF method. In Section 4, the MCM-GPNMF-based process monitoring method is introduced. In Section 5, a case study in a 1000 MW unit boiler process is shown and discussed. Section 6 contains the conclusions.

#### 2. MCM-Based Prediction Method

##### 2.1. Markov Chain Model

Markov chain model is a stochastic process which can be used to estimate the transition probability between the discrete states of the system, and it can be expressed by some parameters. In the first-order Markov chain model, the state at a certain moment only depends on the state of its previous moment [12]. In the further research, the second-order or even higher-order Markov chain process was proposed [12, 13]. In these second-order or higher-order Markov chain processes, the state of a moment depends on two or more states before it. In this paper, we will use the second-order Markov chain model to estimate the missing values in the data set.

For a second-order Markov chain model, let be a stochastic process. The set of all possible states in the stochastic process is called state space. The state space is defined as .

For a given time series , its conditional probability distribution can be expressed as

Markov transition probability is defined as follows:where represents the current state of the moment and and represent the previous and next states of , respectively.

If a stochastic process contains states, the matrix consisting of all transition probabilities is called the state transition matrix of the second-order Markov chain. The state transition matrix can be expressed as follows:

The second-order Markovian transfer matrix has the following property. For any state , the sum of the probability of its transition to all states is 1. Consider

The maximum likelihood estimation of transfer probability for second-order Markov chains is defined as follows:where is the transition frequency that the state is transferred from the previous moment state and the current time state to the next moment state .

The cumulative distribution function (CDF) is also used in the process of generating the time series. The CDF of the second-order Markov chain model is calculated as follows:

The second-order Markov chain model is used to generate the process variable time series, where the initial state is completely random. Generate a random number between 0 and 1 by the random number generator. For the second-order Markov chain model, the current time state and the previous time state are known. If satisfies , then the next time state is .

Only a time series of state is obtained by the above steps. Next, it is necessary to transform the resulting time series into the time series of actual process variable. In addition to the fact that the first and last states do not need to be transformed, all other states need to use formula (7) to transform.where and represent the upper and lower limits of a state, respectively. represents a random number evenly distributed between 0 and 1. represents the actual value of the process variable.

##### 2.2. Markov Property Test

When using the Markov chain to predict the value of a variable, it is not necessary to consider the change of the variable in the past. If the current state of the variable is known, the state of the next moment can be predicted. The Markov property test of the original time series should be carried out before the establishment of the Markov chain model. The specific method of Markov property test is given as follows: we assume that all states of the original time series are . The marginal probability is defined as follows:where stands for the frequency that the variable transitions from state to state .

When the amount of data is large enough, the statistic obeys the distribution and the degree of freedom of distribution is . The statistic can be calculated as follows:where stands for the probability that the variable transitions from state to state .

When the significance level *α* is given, the value of can be obtained by looking up the table. The Markov chain model can be used to process the variable if is satisfied.

##### 2.3. Numerical Example

In this section, the active power of a 1000 MW unit is chosen for numerical experiment. Under the influence of random noise, the power measurement is randomly fluctuating near the set point. Therefore, the active power can be regarded as a stochastic process. The normal operation data of the active power is used as experiment data vector with 500 samples. The maximum and minimum values of power in all samples are 743.86 MW and 749.94 MW, respectively. Therefore, the stochastic process is divided into 16 states. In these states, the power values 743 MW and 750 MW are defined as state 1 and state 16, respectively. The other states are evenly distributed between state 1 and state 16, and the power interval between each two states is 500 kW. The significance level is given by 0.05 in the present work. The conclusion can be easily obtained through the calculation; that is, . Therefore, The Markov chain model can be used to process this time series.

Next, we will select 50 samples from the experimental data randomly and their values will be set to zero. These 50 samples represent the missing data in experiment data vector. Then, the second-order Markov chain model is used to deal with the missing data in experiment data vector. The incomplete data can be replaced by the predicted value, which is shown in Figure 1.