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

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.


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, expertsystem-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][4][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][10][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,  > 0 ( ≥ 0) and  < 0 ( ≤ 0) denote positive-definiteness (positive semidefinite matrix) and negative-definiteness (negative semidefinite matrix), respectively.The Hermitian part of a square matrix  is denoted by () fl +  .  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.

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.

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: 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 (),   = 0 if switch   is open, and  Example 1.Consider a power electronic system shown in Figure 1; the linear-switched model of a single phase transformerless inverter is given by

𝜎(𝑡)
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;  = 1, . . ., ,  is the number of fault types;   () is the fault invertor when the  ℎ fault happens.
Example 2. Consider a fault in the inductance  that the value of the inductance  changes to  − Δ, where Δ describes  the magnitude of the fault.Thus, the system dynamics can be described as where 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  ℎ component when the  ℎ fault type occurs.

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.

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  ℎ ; the fault could be detected as quickly as possible.

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  ℎ estimated fault component matches the occurred fault, the output of the  ℎ 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  ℎ 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.

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 x() is the state estimation vector; L is the observer gain matrix; and () is the observer residual vector which is the difference between the actual output () and the estimated output ŷ().
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 f() =   ()  (), where   () is a Mathematical Problems in Engineering given stable weighted matrix.A minimal realization of f() =   ()  () is supposed to be where   () ∈ R  is the state of the weighted fault,   () ∈ R  is the original fault, and f () ∈ R  is the weighted fault.  ,   ,   , and   are known constant matrices. Denote and   () = () − f ().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. 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 L =  −1 1 L.
It notes that Λ can be rewritten as By Projection theorem, ( 17) is equivalent to where W 1 W 2 are the matrix variables of appropriate dimensions, and let . By denoting L =   1 L and applying Schur complement formula, and after some matrix manipulation, ( 18) becomes (11).

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  11 = 1/,  21 = ,  31 = 1/,  32 = ; then we design two observers.We assume that the parameter   has a fault; the switched system (1) can be rewritten as ẋ () =  () (  )  () +  () (  )  () ,  () =  () (19) 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: ẋ  () =  () ( θ )  () +  () ( θ )  () + (L  + Γ ()   Γ  ()   )   () , ŷ () = x () ,   () =  () − ŷ () (20) where x () and ŷ () are the  ℎ observer state and output, respectively.θ is the estimate signals of   . ∈  × is a chosen symmetric positive-definite matrix such that the dynamics of θ  () evolve much slower than the state dynamics x(). is the dimension of θ  ().Γ() is the additive vector variable.  () is the residual signal of fault identification observer, and L  is the observer gain.
Denote x () = () − x () and θ =   − θ .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 L  ,  > 0 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 L  =  −1 L  .
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 () = x () − Γ() θ 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 22) holds, V() < 0. Thus, when the chosen adaptive law is adapted, system ( 21) is asymptotically stable.
Example 5. Consider the failure of parameter , according to Table 2:  11 = 1/, the observer can be designed as where We can obtain the error system as where According to Theorem 4, the observer gain L  can be calculated by conditions (22).The parameter  can be precisely estimated according to the designed adaptive parameter identification.

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 () = () − ŷ() and   () = () − ŷ () 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  ℎ is selected in the no-fault condition.Thus, the other reasons will not let (()) exceed  ℎ .The threshold is designed as We can detect the faults by comparing (()) and  ℎ as follows: ‖ ( ())‖ ≤  ℎ ⇒ the system has fault, ‖ ( ())‖ >  ℎ ⇒ the system has no fault.
(32) 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  = 1, the resistance  fault corresponds to parameter  = 2, and the inductance  and resistance  all fault correspond to parameter  = 3.The parameter  is chosen to show the result of fault identification as min   (  ()) ⇒ Parameter  = ℎ, ℎ ∈ {1, 2, 3} ⇒ identify that the  ℎ fault has occurred. (33)

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.

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   = Mathematical Problems in Engineering 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 L is given as  3(a), which is quick over the threshold  ℎ = 15.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).From these results, the fault for resistance  can be detected immediately at  = 0.2 because the residual evaluation function (()) exceeds the threshold  ℎ .Figure 4(e) shows that the fault identification flag  = 2; in other words, the fault is identified by comparing the outputs of fault identification observers shown in Figures 4(c) and 4(d).

Fault in the Resistance
6.2.3.Fault in the Inductance  and Resistance .We consider the case that the inductance and resistance have all occurred faults,  changes to 50%, and  increases to 0.4Ω at  = 0.2.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).

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 67% to 10% 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 : 10%, 33%, 50%, and 67%.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.

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.

Figure 2 :
Figure 2: Scheme of the proposed method.

Figure 3 :
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.
.The fault occurs in the resistance  which value changes from 0.3Ω to 0.03Ω.Similarly, Figures4(a) and 4(b) present the fault detection observer residual and fault detection flag, respectively, for  fault.

Figure 5 (
a) presents the residual evaluation function of fault detection; then the fault is detected as shown in Figure 5(b).

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

Figure 5 :
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.

Table 1 :
Possible open/close switch positions for   .

Table 2 :
The fault signal.
Fault in the Inductance .In this case, the fault occurs in the inductance  which value reduces 33% at  = 0.2.The residual evaluation function (()) is shown in Figure