Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 3025838, 11 pages
https://doi.org/10.1155/2018/3025838

Research Article

Robust Fault Diagnosis and Adaptive Parameter Identification for Single Phase Transformerless Inverters

Jian Li, Xinxin Guo, and Bo Li

School of Automation Engineering, Northeast Electric Power University, Jilin 132012, China

Correspondence should be addressed to Bo Li; libo@neepu.edu.cn

Received 29 January 2018; Revised 27 June 2018; Accepted 31 July 2018; Published 13 August 2018

Academic Editor: George A. Papakostas

Copyright © 2018 Jian Li 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 paper presents the theoretical analysis and simulation verification of robust fault diagnosis and adaptive parameter identification for single phase transformerless inverters. The fault diagnosis is composed of two parts, fault detection and fault identification. In the fault detection part, a Luenberger observer is designed to realize the detection of faults. Then, we apply a bank of observers to identify the location of faults. Meanwhile, the fault identification observers based estimation along with a gradient descent algorithm are also used in the parameter identification to estimate the actual values of components in a single phase transformerless inverter. Not only we develop the design methodology for the robust fault diagnosis and adaptive parameter identifier but also we present simulation results. The simulation results show the effectiveness of fault diagnosis and the accurate tracking of changes in component parameters for a wide range.

1. Introduction

The reliability of power electronics systems is increasingly important in many applications: industry, transportation, electric power, and so on. Its effective operation directly affects the security and stability of the whole system. Therefore, fault diagnosis for power electronics systems becomes particularly significant, which can timely detect and eliminate the potential faults and guarantee the stable operation of the system. In recent years, fault diagnosis has received the considerable attentions of the scholars; its theory and technique have also achieved rapid development [1]. Many diagnosis methods are proposed, such as model-based method, expert-system-based method, and neural-network-based method. For these methods, there have been great quantity literatures, for instance, the energy equipment [2], the major aspects of power electronics reliability [3–5], and a fast fault diagnosis method for dc-dc converters [6]. We can conclude that it has become very important to diagnose faults at their very inception in order to avoid catastrophic effect in the system and heavy financial losses.

There are a variety of power electronics applying adaptive parameter identification including single-stage boost inverter [7], ac-dc converters [8], and dc-dc converters [9–11]. For instance, real-time system identification [12], fault diagnosis [13, 14], and adaptive predictive control systems [15, 16] have been investigated. In general, the techniques of adaptive parameter identification have been deeply studied [17, 18]. Most of the algorithms for parameter identification [19] estimate the values of system parameters by comparing measurements from the physical system with a model structure in a real-time and online manner. It is confirmed that the accurate parameter estimation can be achieved by classical algorithm for the slow-varied systems when the sample frequency is much lower than the frequency of switching [20, 21]. Nevertheless, the classical methodologies have certain characteristics which limit the effectiveness of parameter identification [22]. The limitations including the failure to track the fast parameter changes [23], only one parameter at a time [24], and high computation hardware requirements and so on.

It is known from the above that the traditional fault diagnosis methods are usually applied to general switched systems; that is to say they are hardly used to power electronics systems. That motivates us to study the fault diagnosis of power electronics. Comparing with the conventional power detection techniques, the fault diagnosis method of the switched system has many superiorities. We apply it to a single phase transformerless inverter.

In this paper, a reasonable robust fault diagnosis scheme and a robust adaptive parameter identification method are presented. Firstly, the fault model of a single phase transformerless inverter, which accurately captures the dynamics characteristics of the inverter, is considered. Subsequently, we design a robust fault detection observer to output residuals between the practical output of the system and the estimated output. The residual evaluation function is calculated to judge the fault. Moreover, for identifying the fault, a series of fault identification observers are designed to ensure the location of failure. Meanwhile, an improved adaptive parameter identification method is proposed. We adapt a generalized gradient descent algorithm to track and estimate the fault parameter.

The remainder of the paper is organized as follows. Section 2 presents the problem formulation. Section 3 presents the fault detection. The fault identification and adaptive parameter estimation are described in Section 4. Sections 5 and 6 present the thresholds computation and the simulation study. The last section concludes this paper.

Notation. For a matrix , denotes its transpose. For a symmetric matrix, () and () denote positive-definiteness (positive semidefinite matrix) and negative-definiteness (negative semidefinite matrix), respectively. The Hermitian part of a square matrix is denoted by . represents a matrix in which all elements are one with appropriate dimensions. The symbol within a matrix represents the symmetric entries. 0 and denote the zero matrix and unit matrix with appropriate dimensions.

2. Problem Formulation

In this section, we present the modeling framework of a single phase transformerless inverter and describe kinds of fault cases in the inverter. Firstly, linear-switched model of the single phase transformerless inverter is described. Secondly, its faults are formulated. Finally, our design objective and the scheme of the proposed method are presented.

2.1. System Configuration

The single phase transformerless inverter is shown in Figure 1. Using a similar modeling approach to [25], it can be modeled by a linear-switched system as follows:

Figure 1: Single phase transformerless inverter.

In the upper equation, , , and are the state vector, output vector, and input vector, respectively; is the continuous-time switching signal. , , and are the collection of state space models, which vary as a function of . Table 1 shows the possible for the switching signal , if switch is open, and if is close, .

Table 1: Possible open/close switch positions for .

1
2

1
0

0
1

0
1

1
0

Example 1. Consider a power electronic system shown in Figure 1; the linear-switched model of a single phase transformerless inverter is given by

2.2. Fault Model

Next, we model the component fault dynamics in the power electronic inverter. For the system of Figure 1, we consider the inductance fault and the resistance fault. Once the components experience faults, the matrix parameters and are not the original values. Meanwhile, the state and the output are also influenced. We describe the system with these kinds of faults as follows:

where with appropriate dimensions; , is the number of fault types; is the fault invertor when the fault happens.

Example 2. Consider a fault in the inductance that the value of the inductance changes to , where describes the magnitude of the fault. Thus, the system dynamics can be described as

where and . Then, the upper equation can be rewritten as

where / , .

A similar development follows for this case; we can describe other kinds of faults in Table 2, where represents the parameter of component when the fault type occurs.

Table 2: The fault signal.

Parameter
Fault events
Fault signal

p=1

p=2

p=3

2.3. Design Objective

Consider the single phase transformerless inverter system modeled as switched system (3); the objective is to design a fault detection observer and a bank of fault identification observers with adaptive parameter identification. The scheme of the proposed method is involved in Figure 2.

Figure 2: Scheme of the proposed method.

2.3.1. Fault Detection

The fault detection observer is designed for an arbitrary switching signal and generates a residual signal to satisfy the robust performance. When a fault occurs, the residual signal is sensitive and changes rapidly. Then, the residual evaluation function is larger than a predefined threshold ; the fault could be detected as quickly as possible.

2.3.2. Fault Identification and Adaptive Parameter Identification

Once the fault has been detected, the fault identification determines the location of the component where the fault originated; then several fault identification observers work to estimate the specified fault components. If the estimated fault component matches the occurred fault, the output of the fault identification observer will converge to system (3), and the residual signal between them will converge to zero. If not, there exists the error between the output of the fault identification observer and system (3). Hence, we compare the evaluation function to identify the location of the fault. Moreover, an adaptive law is designed to estimate accurately the values of the components.

3. Fault Detection

This section proposes the design methodology for the fault detection. We design a Luenberger observer to detect the system fault [26].

The linear-switched fault detection observer is given by the following equations:

where is the state estimation vector; is the observer gain matrix; and is the observer residual vector which is the difference between the actual output and the estimated output .

For fault detection, it is not necessary to estimate the fault in (3). It is interested in the fault signal of a certain frequency interval. Thus, we can formulate the estimated fault as the weighted fault , where is a given stable weighted matrix. A minimal realization of is supposed to be

where is the state of the weighted fault, is the original fault, and is the weighted fault. , , , and are known constant matrices.

Denote , , and . The augmented switched system is described as

where

Now, the frameworks of fault detection design can be formulated as follows: considering system (8), dynamic observer gains are obtained for an arbitrary switching signal. Then system (8) satisfies the following requirements:

(i)System (8) is asymptotically stable.(ii)For detection objective, the effects from the faults to the residual error signal are minimized; i.e., the FD observer satisfies the following index with

For obtaining the gain of the fault detection observer and satisfying the performance (10), the following theorem is given to design the fault detection observer.

Theorem 3. Given the constant , if there exist matrix variables , , , , , , , satisfying the inequalities

where

then system (8) under arbitrary switching is asymptotically stable and guarantees the robust performance (10). Moreover, if (11) is feasible, then the observer gains in form of (6) can be given by .

Proof. Firstly, consider the stability for system (8), we rewrite the system as when and choose the common Lyapunov functions: . Then it has

We consider the following performance index: . For any nonzero and the under zero-initial condition, we have

It notes that

where

Thus, if , it follows form (13) that , which implies that converges to zero as . The switched system (8) under arbitrary switching is asymptotically stable. Moreover, it also implies . Then, it has the robust performance (10).
It notes that can be rewritten as

By Projection theorem, (17) is equivalent to

where are the matrix variables of appropriate dimensions, and let and . By denoting and applying Schur complement formula, and after some matrix manipulation, (18) becomes (11).

4. Fault Identification and Adaptive Parameter Identification

In this section, the fault identification and an improved adaptive parameter identification method [27] are presented for the single transformerless inverter. After a fault is detected, a bank of fault identification observers is activated to recognize the location of fault. Each observer in the bank corresponds to an inverter component to real time estimate the numerical parameter values. When the certain component has a fault, its residual signal generated by the corresponding observer is minimum of all the evaluation functions for the residual vectors generated by the other observers.

Now we assume that the fault is detected; the fault identification observers are activated. As Table 2, the faults exist in inductance , resistance or inductance and resistance . We denote the parameters , , , ; then we design two observers. We assume that the parameter has a fault; the switched system (1) can be rewritten as

where and are the system matrices which use in and .

The parameter values of the system contained in matrices and should be precisely known for accurate state estimation of system. However, and are not precisely known when the values of components change and cannot be measured accurately because of faults. Consider this case; the joint state and parameter estimation is encapsulated by the following equations:

where and are the observer state and output, respectively. is the estimate signals of . is a chosen symmetric positive-definite matrix such that the dynamics of evolve much slower than the state dynamics . is the dimension of . is the additive vector variable. is the residual signal of fault identification observer, and is the observer gain.

Denote and . Then, the error system is described as

where .

For obtaining the gain of the observer and realizing the adaptive parameter identification, the following theorem is given.

Theorem 4. If there exist matrix variables , satisfying the inequalities

then system (21) under arbitrary switching is asymptotically stable, and is determined according to the adaptive law:

Moreover, if (22) is feasible, then the fault identification observer gain in form of (20) can be given by .

Proof. Similar to [28], the classical Lyapunov’s direct method is adapted to analyze the stability of the system (21). According to this, we denote and choose the common Lyapunov function: . Then from the derivative of along system (21), it has

Choose the appropriate adaptive law that is presented in (23); it has

Hence, it has . If (22) holds, . Thus, when the chosen adaptive law is adapted, system (21) is asymptotically stable.

Example 5. Consider the failure of parameter , according to Table 2: , the observer can be designed as

where

We can obtain the error system as

where

According to Theorem 4, the observer gain can be calculated by conditions (22). The parameter can be precisely estimated according to the designed adaptive parameter identification.

5. Thresholds Computation

In this section, we compute the residual evaluation functions and determine the thresholds after obtaining the observer gains. The residual evaluation function for the fault detection and the residual evaluation function for the fault identification can be chosen as

where is the length of the evaluation window and and are residual signals for the fault detection observer and for the fault identification observers. The residual evaluation functions and denote the average value from to and from to , respectively.

Fault Detection. We can set the threshold according to the selected evaluation function. The threshold is selected in the no-fault condition. Thus, the other reasons will not let exceed . The threshold is designed as

We can detect the faults by comparing and as follows:

Fault Identification. For the fault identification, we compare the evaluation function for all fault identification observers. As shown in Table 2, the inductance fault corresponds to parameter , the resistance fault corresponds to parameter , and the inductance and resistance all fault correspond to parameter . The parameter is chosen to show the result of fault identification as

6. Simulation Study

In this section, we first provide the design of the observers. Subsequently, a simulation study of fault diagnose and adaptive parameter identification for the single transformerless inverter is presented. The complete specifications of the invert parameters are presented in Table 3.

Table 3: Single phase transformerless inverter parameters.

6.1. Design Result of the Observer

As Figure 1, we give the parameters of the single phase transformerless inverter system as Table 3. For the fault detection to estimate the fault , we choose a kind of high-pass filter as parameters , , , and and give , , and . By solving the LMI (11), the fault detection observer gain is shown as

Consider the parameters in Table 2; two observers are designed for the inductance and the resistance , respectively. We adapt the identical observer gain in two observers. By solving (22), the fault identification observer gain is given as

6.2. Fault Diagnosis Results

In this subsection, we simulate the residual signals for three types of faults by Piecewise Linear Electrical Circuit Simulation toolbox in MATLAB/Simulink environment.

6.2.1. Fault in the Inductance

In this case, the fault occurs in the inductance which value reduces at . The residual evaluation function is shown in Figure 3(a), which is quick over the threshold . Thus, the inductance fault has been injected as presented in Figure 3(b). The fault identification observers are activated after fault detection. As the output of these observers in Figures 3(c) and 3(d), we can identify that the observer for the inductance matches the fault system and decide that the parameter of fault identification flag is one in Figure 3(e).

(a) for FD observer

(b) FD flag

(c) for FI observer

(d) for FI observer

(e) FI flag

Figure 3: (a) and (b) are FD results as L experiences fault. (c), (d), and (e) are simulation results for FI as L experiences fault.

6.2.2. Fault in the Resistance

The fault occurs in the resistance which value changes from to . Similarly, Figures 4(a) and 4(b) present the fault detection observer residual and fault detection flag, respectively, for fault. From these results, the fault for resistance can be detected immediately at because the residual evaluation function exceeds the threshold . Figure 4(e) shows that the fault identification flag ; in other words, the fault is identified by comparing the outputs of fault identification observers shown in Figures 4(c) and 4(d).

(a) for FD observer

(b) FD flag

(c) for FI observer

(d) for FI observer

(e) FI flag

Figure 4: (a) and (b) are FD results as R experiences fault. (c), (d), and (e) are FI results as R experiences fault.

6.2.3. Fault in the Inductance and Resistance

We consider the case that the inductance and resistance have all occurred faults, changes to , and increases to at . Figure 5(a) presents the residual evaluation function of fault detection; then the fault is detected as shown in Figure 5(b). The outputs of fault identification observers presented in Figures 5(c) and 5(d) all converge to zero. From the two figures, we can identify that the third kind fault occurs according to Table 2 as shown in Figure 5(e).

(a) for FD observer

(b) FD flag

(c) for FI observer

(d) for FI observer

(e) FI flag

Figure 5: (a) and (b) are FD results as and all experience faults. (c), (d), and (e) are simulation results for FI as and all experience faults.

6.3. Adaptive Parameter Identification

Once a component fault happens, the value of this component changes. Hence, the section of fault identification and adaptive parameter identification design an adaptive law to reestimate the value of the component that has experienced fault. In this subsection, the simulation results of the adaptive parameter identification are presented. The simulation figures exhibit the tracking capability and convergence speed of the proposed parameter identification.

For inductance , we considers the different degree fault from to when resistance does not vary. Figure 6 shows the results of this test. As shown, the fault value of inductance can be estimated accurately. As shown in Figure 7, we inject a different step perturbation in the resistance : , , , and . The figure testifies the effectiveness of adaptive parameter identification. Finally, Figure 8 presents the tracking of multiple parameters inductance and resistance . We can learn that the convergence for each parameter is independent of other parameter conditions from these simulation results.

Figure 6: Parameter identification as changes from to .

Figure 7: Parameter identification as changes from to .

Figure 8: Parameter identification as changes from to and changes from to .

7. Conclusion

This paper deals with the fault diagnosis and adaptive parameter identification of a single phase transformerless inverter. A fault detection observer and a bank of fault identification Luenberger observers, which are based on linear-switched, are designed by using principles of robust. Through the outputs of these observers, the fault can be detected and identified accurately. Moreover, the proposed adaptive parameter identification for a single phase transformerless inverter is developed digitally and validated. In the last section, the simulation results show accurate fault diagnose and fast convergence speed for parameter identification.

Data Availability

The results simulations are based on MATLAB; readers can access all the original data, including parameters and programs, by sending an email to Dr. Bo Li at libo@neepu.edu.cn.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China [Grant nos. 61503071 and 61703091; the Natural Science Foundation of Jilin Province [Grant no. 20180520211JH]; and the Science Research of Education Department of Jilin Province [Grant nos. 201693 and JJKH20170106KJ].

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