# Fault diagnosis for multiple current sensors in grid-connected inverter based on average modulation voltage

Fan Zhang, Zhiwei Zhang, Guangfeng Jin, Tianzhen Wang, Jingang Han, Hubert Razik, Yide Wang

► To cite this version:

Fan Zhang, Zhiwei Zhang, Guangfeng Jin, Tianzhen Wang, Jingang Han, et al.. Fault diagnosis for multiple current sensors in grid-connected inverter based on average modulation voltage. IET Power Electronics, 2023, 17 (1), pp.54-66. 10.1049/pel2.12614. hal-04337470

HAL Id: hal-04337470

https://hal.science/hal-04337470v1

Submitted on 6 Jun 2025

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Fault diagnosis for multiple current sensors in grid-connected inverter based on average modulation voltage

Fan Zhang<fnref n="1" /> ID

<figure><img src="image_2.png" /></figure>

Zhiwei Zhang<fnref n="2" />

<figure><img src="image_3.png" /></figure>

Guangfeng Jin<fnref n="1" />

<figure><img src="image_4.png" /></figure>

Tianzhen Wang<fnref n="1" /> ID

<figure><img src="image_5.png" /></figure>

Jingang Han<fnref n="1" /> ID

<figure><img src="image_6.png" /></figure>

Hubert Razik<fnref n="3" />

<figure><img src="image_7.png" /></figure>

Yide Wang<fnref n="4" />

**Correspondence**

Tianzhen Wang, Department of Electrical Automation, Shanghai Maritime University, Shanghai, People's Republic of China.

Email: tzwang@shmtu.edu.cn

**Funding information**

National Natural Science Foundation of China,
Grant/Award Number: 61673260

**Abstract**

Grid-connected inverters are the core equipment in the renewable power system. There are multiple current sensors which may affect the driving module of the switch in inverter. In multiple current sensor fault diagnosis, the coupling between fault components makes fault diagnosis difficult. This paper presents an offset fault diagnosis method of multiple current sensors based on the average modulation voltage model. Based on the influence of current sensor offset fault in the system, the average modulation voltage model is established in three-phase stationary coordinates. The difference between the measured value of the current and the actual value is estimated through the model when the offset fault occurs. And then the fault is located. Experimental results show that the fault can be located accurately and fault tolerant control can be performed by this method when there are offset faults in multiple sensors simultaneously.

# 1 INTRODUCTION

With the continuous increase of energy demand, distributed power generation systems of renewable energy have been rapidly developed [1, 2]. The grid-connected inverter is an essential interface between the distributed power generation system and the utility grid [3]. For a power electronic system, reliability is a key performance indicator because faults occurring to the system components may lead to the shutdown of the system [4].

There are two primary sources of faults in a grid-connected inverter, namely, electrical and sensor faults. Electrical faults generally include open-circuit faults and short-circuit faults of power switches [5]. Protection of short-circuit faults is generally based on hardware circuits, and different diagnostic algorithms for the open-circuit faults are proposed [6, 7]. Sensors provide real-time information for the closed-loop control. The precise information returned by sensors has an essential impact on the control performance of the system, and sensor faults can cause an overcurrent in the circuit [8]. If the fault cannot be

located and rectified within a short time, the overcurrent may cause damages to other healthy devices. To improve the system's stability in the case of sensor faults, it is necessary to accurately locate the fault and adopt the corresponding fault-tolerant control method [9, 10].

In recent years, many sensor fault diagnosis methods in power electronic converter systems have been proposed. These methods can be divided into model-based methods, signal-based methods, and data-driven methods. The current sensor cannot accurately feedback the real state of the system due to the fault. The current can be estimated through the system model established. The residual between the estimated current and the measured current can reflect the fault location. For example, [11] proposes a method of monitoring the current sensor by an observer group composed of three independent observers. This method locates the fault location by analyzing the residual signal generated by the observer group. For the fault of current sensor under high-frequency switching, [12] proposed to replace the discrete switching state with the approximately continuous variable of the duty ratio of the switch in the state

This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

© 2023 The Authors. *IET Power Electronics* published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology.

<footnote n="1">¹Department of Electrical Automation, Shanghai Maritime University, Shanghai, People's Republic of China</footnote>

<footnote n="2">²Shanghai Power Industrial and Commercial CO., LTD, Shanghai, People's Republic of China</footnote>

<footnote n="3">³Lab Ampere, University of Lyon, Lyon, France</footnote>

<footnote n="4">⁴IETR Laboratory, University of Nantes, Nantes, France</footnote>
observer, which effectively reduced the output fluctuation of the observer under high-frequency switching state switching, and improved the accuracy of current estimation. For multiple current sensor faults, the state space equation is established in [13]. The output of this equation has no coupling relationship with torque, only related to the current sensor. Ref. [14] analyzed the stuck fault of current sensor under closed-loop structure. Due to the influence of current signal, the mean value of three-phase current absolute value is used to locate the fault current sensor. For complex fault situations such as multiple sensor faults, fault diagnosis methods that only rely on signal processing are difficult to accurately locate the fault location.

In addition to the two broad categories methods, data-driven fault diagnosis methods have attracted more and more attention [15–17]. In data-driven fault diagnosis methods, various signal processing methods and fault feature extraction methods are usually used to amplify fault features. In [14, 16], Fourier transform and wavelet decomposition were used to process the original signal respectively. In [17], Concordia transform is used to transform the three-phase current into αβ-axis, and the slopes of current trajectories are selected as the fault features. In [18], it uses the data-based method to generate the system model, which effectively solves the shortcoming of the difficult parameter determination in the model-based diagnosis method, and also has the ability to quickly locate faults. These methods do not need an accurate mathematical model of the system. They can establish the mapping from fault characteristics to fault category labels only through historical fault data.

The basis of the data-driven and signal-based method is to mine the fault information from various fault signals of the system and then realize the fault diagnosis. When the structure of the system is relatively complex or the fault features are sub-merged due to closed-loop control, a stable and discriminable fault feature method cannot be extracted only through signal processing [19], which also limits the application of the method based only on signal processing in these application field.

Therefore, this paper proposes the offset fault diagnosis method of multiple current sensors based on average modulation voltage model. Firstly, the offset fault of the multiple current sensors is modelled. Then the influence of the fault on the three-phase current is analyzed to build the average modulation voltage model. In order to enhance the robustness of the method, the error of the average modulation voltage is introduced into the model. The dynamic analysis of the error and the experimental results show that the model can accurately estimate the offset of the current sensor and achieve accurate fault diagnosis.

## 2 | PROBLEM DESCRIPTION

As shown in Figure 1, according to the grid information $\theta$ provided by the phase-locked loop (PLL), the three-phase grid-connected system transforms the sensed currents $I_x(x = a, b, c)$ from the three-phase axis to the $dq$-axis or $\alpha\beta$-axis for closed-loop control. Through the current controller, the modulation signal $V$ can be obtained. According to the modulation signal

<figure><img src="image_8.png" /><figcaption>FIGURE 1 Block diagram of three-phase grid-connected inverter.</figcaption></figure>

$V$, the Carrier Phase Shifting SPWM (CPS-SPWM) modulation technology is applied to generate the Pulse Width Modulation (PWM) signals to control the power switches. The inverter mathematical model can be established as follows:

$$L \frac{di_x}{dt} + Ri_x + E_x = u_x(x = a, b, c) \quad (1)$$

where $L, R, i_x, E_x, u_x$ represent the filter inductance and its equivalent resistance, the actual current in the circuit, the grid voltage, and the phase voltage of the inverter, respectively.

A current control system is required to make the output power quality reach the grid-connected standard [20]. Control methods are generally implemented in the $dq$-axis or the $\alpha\beta$-axis [21, 22]. At the same time, the closed-loop control reduces or even eliminates the sensor fault characteristics, which increases the difficulty of fault diagnosis [23].

After Clark transformation and Park transformation, the mathematical model of three-phase grid-connected inverter can be transformed from the stationary coordinate system to the two-phase rotating coordinate system. The mathematical model of grid-connected inverter in the $dq$-axis coordinate system is shown in (2).

$$\left\{ \begin{aligned} u_d + \omega L i_q - E_d &= L \frac{di_d}{dt} + Ri_d \\ u_q - \omega L i_d - E_q &= L \frac{di_q}{dt} + Ri_q \end{aligned} \right. \quad (2)$$

where $\omega$ is the angular velocity of the rotating coordinate system.

To eliminate the coupling of d-axis and q-axis components, we adopt the feedforward decoupling control of current. In Figure 2, the right part represents the mathematical model of the inverter in the S-domain, while the left part represents the feedforward decoupling control part of the three-phase inverter. The introduced of PI controller can maintain the stable of closed-loop.

$L$ represents the inductance of the filter in the system, $R$ represents the resistance of the filter. $\omega$ is the angular frequency of the grid. $u_d, u_q, E_d, E_q$ are the components of inverter output
<figure><img src="image_9.png" /><figcaption>FIGURE 2 Feedforward decoupling control of grid connected inverter.</figcaption></figure>

voltage and grid voltage in the two-phase rotating coordinate system, respectively. $u_d$, $u_q$ are the keys to realizing closed-loop control, and they are adjusted by changing the modulation voltage. PI controller is chosen here, which is widely used and most representative. It consists of two parts: proportion and integral.

Under the feedforward decoupling control, the expected output voltage of the inverter is $u_d^*$ and $u_q^*$ respectively, and its calculation formula is shown in (3). Three phase modulation signals can be obtained by inverse coordinate transformation, and PWM signal is generated by carrier layer modulation method to drive the corresponding switch tube.

$$ \left\{ \begin{aligned} u_d^* &= k_{pd}(i_d^* - i_d) + k_{id} \int (i_d^* - i_d)dt - \omega L_i i_d + E_d \\ u_q^* &= k_{pq}(i_q^* - i_q) + k_{iq} \int (i_q^* - i_q)dt + \omega L_i i_q + E_q \end{aligned} \right. \quad (3) $$

In order to further analyze the influence of current sensor fault on grid-connected inverter system, the offset fault of A-phase current sensor is taken as an example to conduct modelling analysis. The mathematical model of offset fault is shown in (4).

$$ I_{a\_offset} = \begin{cases} i_{a\_r}, & 0 \le t < t_0 \\ i_{a\_r} + D_a, & t_0 \le t \end{cases} \quad (4) $$

where $I_{a\_offset}$ is the output signal of the current sensor in the case of offset fault; $i_{a\_r}$ is the actual current signal. $D_a$ reflects the degree to which the measured value of the current sensor deviates from the actual value, which can be regarded as an indicator of the fault degree of the current sensor.

In (4), the A-phase current sensor offset fault occurs at time $t_0$, and the fault will superimpose A DC component $D_a$ in the real current signal. Under the influence of offset fault, even if no current to be measured passes through the original side of the Hall current sensor, the sensor will still have a signal $D_a$.

After the offset fault occurs on the A-phase current sensor, the fault component $D_a$ is fed back to the controller with the coordinate transformation of the current signal. In Figure 3, the current fed back to the controller becomes $I_d$ and $I_q$, as shown

<figure><img src="image_10.png" /><figcaption>FIGURE 3 A-phase sensed current under offset fault condition.</figcaption></figure>

in (5). Fault components of A-phase current sensor under offset fault are shown in (6).

$$ \begin{pmatrix} I_d \\ I_q \end{pmatrix} = \begin{pmatrix} i_d \\ i_q \end{pmatrix} + \begin{pmatrix} f_d \\ f_q \end{pmatrix} \quad (5) $$

$$ \begin{pmatrix} f_d \\ f_q \end{pmatrix} = \frac{2}{3} \begin{pmatrix} D_a \cos \theta \\ -D_a \sin \theta \end{pmatrix} \quad (6) $$

where $i_d$ and $i_q$ are the actual components of the current under the d-q axis, and $f_d$ and $f_q$ are the fault components.

When the current sensor is faulty, the measured current value will change, which contains certain fault information. Therefore, the measured value of current signal can be used as the input of the fault diagnosis method. Figure 3 shows the waveform of the measured current after the offset fault of the A-phase current sensor. After fault occurs at 0.4s, A significant DC component appears in the measured value of A-phase current, which is caused by the offset fault of the A-phase current sensor.

The DC component in Figure 3 is obtained by superimposing the fault component on the DC component of the actual current. Therefore, the value of the DC component in Figure 3 is different from the fault component, and the fault location cannot be diagnosed only from the time-domain characteristics of A-phase current. After A phase current sensor fault, fault component will be through the feedback loop to controller. Then according to the error of measurement information, the controller generates the wrong control signal, leading to spread of fault component of phase A current sensor in the system. As a result, the dc component will appear in
<figure><img src="image_11.png" /><figcaption>FIGURE 4 Simulation results of offset fault of multi-current sensor based on Luenberger observer.</figcaption></figure>

the health current sensor, which cause fault location is difficult. This is one of the difficulties that closed-loop control brings to diagnosis.

In three-phase grid-connected inverter, sensor faults occur in the three-phase coordinate system. Nevertheless, fault diagnosis models are often established in the two-phase coordinate system, which will lead to the coupling of multi-sensor faults in the two-phase coordinate system. The mathematical model of the offset fault can be established in the d-q coordinate system, as follows:

$$ \left( \begin{array}{c} \frac{di_d}{dt} \\ \frac{di_q}{dt} \\ \frac{df_d}{dt} \\ \frac{df_q}{dt} \end{array} \right) = \left( \begin{array}{cc} -\frac{R}{L} & \omega & 0 & 0 \\ -\omega & -\frac{R}{L} & 0 & 0 \\ 0 & 0 & 0 & \omega \\ 0 & 0 & -\omega & 0 \end{array} \right) \left( \begin{array}{c} i_d \\ i_q \\ f_d \\ f_q \end{array} \right) + \left( \begin{array}{cc} \frac{1}{L} & 0 \\ 0 & \frac{1}{L} \\ 0 & 0 \\ 0 & 0 \end{array} \right) \left( \begin{array}{c} \mu_d - E_d \\ \mu_q - E_q \end{array} \right) \quad (7) $$

$$ y = \begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{pmatrix} \begin{pmatrix} i_d & i_q & f_d & f_q \end{pmatrix}^T \quad (8) $$

The rank of the observability matrix of the model is 4, indicating that the system is observable. And the component along the perpendicular axis can be accurately estimated by the sensor fault estimation in the d-q coordinate system. In Figure 4a, A-phase current sensor and B-phase current sensor have offset faults of 1.5 and -1 A at 0.2 s, respectively.

Figure 4a,b shows that the estimated value $\hat{f}_d$, $\hat{f}_q$ of fault component in d-q coordinate system can accurately track the change of actual fault component $f_d, f_q$. However, when multi-

ple sensors fail at the same time, it is not possible to estimate the fault of phase (A/B/C) sensor and locate the fault sensor by coordinate inverse transformation. Figure 4c,d shows that the estimation error of fault component in phase(A/B/C) is large and there is significant static error.

In conclusion, for single-phase sensor faults, the fault current can be accurately estimated by coordinate transformation. However, when multiple sensors fail at the same time, the fault of single-phase sensor cannot be estimated by inverse coordinate transformation. Fault coupling makes current estimation errors occur, resulting in the inability to locate the faulty sensor.

## 3 | MATHEMATICAL MODEL OF AVERAGE MODULATION VOLTAGE

### 3.1 | Model of multiple current sensors offset faults

Multiple current sensors offset faults have an effect on actual current and measured current in the system. Therefore, the offset faults of multi-current sensors under closed-loop structure are modelled and analyzed. After the sensors fault occurs, the current feedback to the controller changes from the actual current component $i_d, i_q$ to $I_d, I_q$ containing the fault component, as shown in left of Figure 2.

Combined with the mathematical model of the main circuit in Figure 2 and (2), (9) can be obtained.

$$ \left\{ \begin{aligned} k_{pd}(i_d^* - i_d - f_d) + k_{id} \int (i_d^* - i_d - f_d)dt - \omega L f_q &= L \frac{di_d}{dt} + R i_d \\ k_{pq}(i_q^* - i_q - f_q) + k_{iq} \int (i_q^* - i_q - f_q)dt + \omega L f_d &= L \frac{di_q}{dt} + R i_q \end{aligned} \right. \quad (9) $$
where $i_d^*$ and $i_q^*$ are the reference currents on the d-q axis, $k_{pd}$ and $k_{id}$ are the ratio and integral coefficient in the d-axis PI controller, $k_{pq}$ and $k_{iq}$ are the proportional and integral coefficient in the q-axis PI controller, respectively.

According to the integral and differential properties of Laplace transform, (9) can be converted into the s domain to solve $i_d(s)$ and $i_q(s)$. The $i_d(s)$ and $i_q(s)$ in the s domain are shown in (10).

$$ egin{cases} i_d^*(s) \left( k_{pd} + \frac{k_{id}}{s} \right) - \left( k_{pd} + \frac{k_{id}}{s} \right) f_d(s) - \omega L f_q(s) \\ i_d(s) = \frac{i_d^*(s) \left( k_{pd} + \frac{k_{id}}{s} \right) + R f_d(s)}{k_{pd} + \frac{k_{id}}{s} + L_s + R} \\ i_q^*(s) \left( k_{pq} + \frac{k_{iq}}{s} \right) - \left( k_{pq} + \frac{k_{iq}}{s} \right) f_q(s) + \omega L f_d(s) \\ i_q(s) = \frac{i_q^*(s) \left( k_{pq} + \frac{k_{iq}}{s} \right) + R f_q(s)}{k_{pq} + \frac{k_{iq}}{s} + L_s + R} \end{cases} \quad (10) $$

Due to the offset fault, there is a deviation between the measured current and the actual current fed back to the controller. Combining (5) and (10), the expression of measured current in s domain can be obtained as follows:

$$ egin{cases} I_d(s) = \frac{i_d^*(s) \left( k_{pd} + \frac{k_{id}}{s} \right) + (L_s + R) f_d(s) - \omega L f_q(s)}{k_{pd} + \frac{k_{id}}{s} + L_s + R} \\ I_q(s) = \frac{i_q^*(s) \left( k_{pq} + \frac{k_{iq}}{s} \right) + (L_s + R) f_q(s) + \omega L f_d(s)}{k_{pq} + \frac{k_{iq}}{s} + L_s + R} \end{cases} \quad (11) $$

In the offset fault of multi-current sensors, $f_d, f_q$ is obtained by $f_x(x = a, b, c)$ after coordinate transformation, so the relationship between them is as follows:

$$ \left( \begin{matrix} f_d \\ f_q \end{matrix} \right) = \frac{2}{3} \left( \begin{matrix} \cos \omega t & \cos \left( \omega t - \frac{2\pi}{3} \right) & \cos \left( \omega t + \frac{2\pi}{3} \right) \\ -\sin \omega t & -\sin \left( \omega t - \frac{2\pi}{3} \right) & -\sin \left( \omega t + \frac{2\pi}{3} \right) \end{matrix} \right) \left( \begin{matrix} f_a \\ f_b \\ f_c \end{matrix} \right) \quad (12) $$

(12) can be expressed as a differential equation:

$$ egin{cases} \frac{d f_d}{d t} = \omega f_q \\ \frac{d f_q}{d t} = -\omega f_d \end{cases} \quad (13) $$

According to the differential property of Laplace transform, (13) can be transformed into the s domain to obtain (14). By substituting (14) into (11), the expressions of $I_d(s)$ and $I_q(s)$ can be further simplified into (15). The equivalent resistance between the power grid and the inverter is small, and the ampli-

tude of the fault component caused by $\frac{R f_d(s)}{k_{pd} + \frac{k_{id}}{s} + L_s + R}$ is relatively small, which can be ignored in order to simplify the analysis.

$$ \left\{ \begin{aligned} s f_d(s) - \omega f_q(s) &= 0 \\ s f_q(s) + \omega f_d(s) &= 0 \end{aligned} \right. \quad (14) $$

$$ egin{cases} I_d(s) = \frac{i_d^*(s) \left( k_{pd} + \frac{k_{id}}{s} \right) + R f_d(s)}{k_{pd} + \frac{k_{id}}{s} + L_s + R} \\ I_q(s) = \frac{i_q^*(s) \left( k_{pq} + \frac{k_{iq}}{s} \right) + R f_q(s)}{k_{pq} + \frac{k_{iq}}{s} + L_s + R} \end{cases} \approx \frac{i_d^*(s) \left( k_{pd} + \frac{k_{id}}{s} \right)}{k_{pd} + \frac{k_{id}}{s} + L_s + R} \approx \frac{i_q^*(s) \left( k_{pq} + \frac{k_{iq}}{s} \right)}{k_{pq} + \frac{k_{iq}}{s} + L_s + R} \quad (15) $$

In (15), it is found that in the case of offset fault, the current feedback to the controller is consistent with the normal current. This means that under the action of closed-loop control, $I_d(t)$ and $I_q(t)$ are constantly approaching the reference values $M$ and $0$ of the current. Then the actual current in the circuit is approximated by (16). The actual current can be obtained by (17), which is obtained by inversely transforming (16) to the three-phase stationary coordinate system. (17) shows that under the offset fault of multiple current sensors, the actual current is mainly composed of power frequency component and DC component, which lays a foundation for the proposed average modulation voltage model.

$$ \left\{ \begin{aligned} i_d(t) &\approx M - f_d(t) \\ i_q(t) &\approx -f_q(t) \end{aligned} \right. \quad (16) $$

$$ \left\{ \begin{aligned} i_a(t) &\approx M \cos(\omega t) - \frac{2}{3} D_a + \frac{1}{3} D_b + \frac{1}{3} D_c \\ i_b(t) &\approx M \cos(\omega t - 2\pi/3) - \frac{2}{3} D_b + \frac{1}{3} D_a + \frac{1}{3} D_c \\ i_c(t) &\approx M \cos(\omega t + 2\pi/3) - \frac{2}{3} D_c + \frac{1}{3} D_a + \frac{1}{3} D_b \end{aligned} \right. \quad (17) $$

The calculation results of (17) and simulation results of Simulink model are shown in Figure 5. The specific parameters in simulation are shown in Table 1.

$\tilde{i}_a$ in Figure 5 is calculated by the transfer function, and $i_a$ is the current waveform obtained in the simulation. The comparison verifies (17) can accurately reflect the current composition under the multi-current sensor offset faults. In order to make the simulation closer to the actual application field, measurement noise is added to the current sensors in the simulation model, which is also the reason for the difference between $\tilde{i}_a$ and $i_a$. The root mean square error is selected to evaluate the error between $\tilde{i}_a$ and $i_a$. The root mean square error of d-axis current and q-axis current are 0.065 and 0.09, respectively. It shows that the simplified model has a high consistency with the simulation
<figure><img src="image_12.png" /><figcaption>**FIGURE 5** Verification of fault current composition under multi-current sensor faults.</figcaption></figure>

<table><caption>**TABLE 1** Table of simulation parameters of three-phase cascaded H-bridge grid-connected inverter.</caption><thead><tr><th>Symbol</th><th>Physical quantity</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>U<sub>dc</sub></td><td>The DC input voltage of the H-bridge unit</td><td>100</td><td>V</td></tr><tr><td>E<sub>x_rms</sub></td><td>Effective value of grid voltage</td><td>220</td><td>V</td></tr><tr><td>L</td><td>Filtering inductance</td><td>6</td><td>mH</td></tr><tr><td>R</td><td>Equivalent resistance</td><td>0.3</td><td>Ω</td></tr><tr><td>k<sub>pd</sub>, k<sub>pq</sub></td><td>Proportional factor of the controller</td><td>5.5</td><td>-</td></tr><tr><td>k<sub>id</sub>, k<sub>iq</sub></td><td>Integral factor of the controller</td><td>3000</td><td>-</td></tr><tr><td>f<sub>a</sub></td><td>Current offset of phase A</td><td>0.5</td><td>A</td></tr><tr><td>f<sub>b</sub></td><td>Current offset of phase B</td><td>1</td><td>A</td></tr><tr><td>f<sub>c</sub></td><td>Current offset of phase C</td><td>0.75</td><td>A</td></tr><tr><td>i<sup>*</sup><sub>d</sub></td><td>Reference current of d axis</td><td>5</td><td>A</td></tr><tr><td>i<sup>*</sup><sub>q</sub></td><td>Reference current of q axis</td><td>0</td><td>A</td></tr></tbody></table>

model, and the grid-connected inverter can be analyzed on the basis of the simplified model.

## 3.2 | Model of average modulation voltage and error analysis

Part A mainly analyzes the composition of the actual current under the multi-current sensor offset faults, and this part establishes the average modulation voltage model on the basis of Part A. (18) can be obtained after discretization of (1), where $T_s$ is the sampling period, and $i_x(g)$, $E_x(g)$, $u_x(g)$ are the actual currents the grid voltage and the inverter output phase voltage at g, respectively.

$$u_k(g) = L \frac{i_k(g + 1) - i_k(g)}{T_s} + R i_k(g) + E_k(g) (k = a, b, c) \quad (18)$$

Equation (17) has shown that the actual current in the circuit mainly includes the power frequency component and the DC component caused by the offset faults, when the offset faults of multi-current sensors occur. The DC component of current is eliminated in the process of current derivative. The power frequency component is a standard cosine function, and its average

in a grid cycle is zero. In the case of the ideal power grid, average voltage of a grid cycle is zero. Therefore, (18) can be obtained by calculating the mean value of (19) within a power grid cycle.

$$\tilde{u}_k = R \tilde{i}_k \quad (19)$$

Equation (19) shows that the mean value $\tilde{i}_k$ of the actual current can be estimated by $\tilde{u}_k$. The output phase voltage of the inverter needs to be obtained during the calculating $\tilde{u}_k$, nevertheless, the output phase voltage of the inverter is not measured in the common three-phase grid-connected inverter. Therefore, in order to avoid additional voltage sensors as much as possible, the transfer function $K_{pwm}$ from the modulated signal input to the inverter bridge output $u_k$ is introduced at low frequencies [24]. The modulating signal and DC voltage form the modulating phase voltage $U_k$. The equivalent gain $K_{pwm}$ is shown in (20).

$$K_{pwm} = \frac{u_k(s)}{U_k(s)} = \frac{U_{dc}}{V_{tri}} \quad (20)$$

where $U_{dc}$ is the DC voltage and $V_{tri}$ is the amplitude of the carrier.

In (20), the relationship between the DC voltage $U_{dc}$ and the amplitude of the carrier $V_{tri}$ is linear. Therefore, the mean value of the actual current in a power grid period can be estimated through (21).

$$\tilde{i}_k = K_{pwm} \frac{\tilde{U}_k}{R} \quad (21)$$

Hence, the relationship between the actual current $\tilde{i}_k$ and phase voltage of modulation $\tilde{U}_k$ is linear. However, modulated phase voltages $U_k$ and current sensor output currents $I_k$ can actually be measured.

The zero offsets caused by the offset faults of the current sensors can be regarded as a constant in a power grid cycle. Therefore, the mean value in a power grid cycle can be obtained as follows:

$$\tilde{I}_k = \tilde{i}_k + D_k \quad (22)$$

The estimated value of offsets $\hat{D}_x$ can also be obtained:

$$\hat{D}_k = \tilde{i}_k - \hat{\tilde{i}}_k = \tilde{i}_k - \frac{K_{pwm}}{R} \tilde{U}_k \quad (23)$$

The estimated value of offsets $\hat{D}_x$ makes modulated phase voltages $U_k$ and current sensor output currents $I_k$ nonlinear. (24) can be got after discretizing (21):

$$\tilde{i}_k(g) = K_{pwm} \frac{\tilde{U}_k(g)}{R} \quad (24)$$

In order to realize real-time monitoring of faults and shorten the time for fault diagnosis, the sliding window as shown in Figure 6 is used to calculate $\tilde{U}_k$. The sliding step is $T_s$, which is the control period of the system, and the length of the sliding window is one power grid period.
<figure><img src="image_13.png" /><figcaption>FIGURE 6 Schematic diagram of sliding sampling principle.</figcaption></figure>

The model shown in (25) is very simple and easy to implement. Whereas the initial state of the system is unknown, which will cause the accuracy of the estimation to be greatly affected by the disturbance. Because the initial state $x_0$ of the system is unknown, and in general $x_0 \neq \hat{x}_0$, it's hard to make sure that $e$ keeps approaching to 0 in this case. In order to improve the accuracy of estimation, the error term of modulation voltage is introduced to improve the robustness of the system to various disturbances.

$$ \left\{ \begin{array}{l} \hat{i}_k(g+1) = K_{\text{PWM}} \frac{\bar{U}_k(g+1)}{R} + W(\bar{U}_k(g) - \hat{U}_k(g)) \\ \\ \hat{U}_k(g) = \frac{R}{K_{\text{PWM}}} \hat{i}_k(g) \end{array} \right. \qquad (25) $$

where $\bar{i}_k(g+1)$ is the mean value in the first sliding window; $\hat{U}_k(g)$ represents the estimated value of the mean value of modulation voltage in the first sliding window; $\bar{U}_k(g) - \hat{U}_k(g)$ is the error between the actual value and the estimated value of the modulated voltage mean; $W$ is a gain term.

In order to analyze the error in fault estimation, the error term of the mean current is defined as:

$$ e_k(g+1) = \bar{i}_k(g+1) - \hat{i}_k(g+1) \qquad (26) $$

Substituting (25) into (26):

$$ e_k(g+1) = W \frac{R}{K_{\text{PWM}}} (\bar{i}_k(g) - \hat{i}_k(g)) \qquad (27) $$

On the basis of (27), the dynamic change of error can be expressed as (28). When $T_s$ is small enough relative to the grid period, (28) can be regarded as a state differential equation.

$$ \frac{e_k(g+1) - e_k(g)}{T_s} = \frac{\left( W \frac{R}{K_{\text{PWM}}} - 1 \right) e_k(g)}{T_s} \qquad (28) $$

The result that $e_k$ continuously approaches to zero is equivalent to the stability of the system shown in (28). $W \frac{R}{K_{\text{PWM}}} - 1$ is a constant, so according to the stability judgment of the linear time-invariant system, if all the features of the system matrix are in the left semi-open complex plane, it will continue to approach zero. The above conditions are equivalent to that all features of

$W \frac{R}{K_{\text{PWM}}} - 1$ have negative real parts, and the goal of $e_k$ continuously approaching zero can be achieved by choosing the gain $W$, as shown in (25).

$$ \text{real} \left( \lambda \left( W \frac{R}{K_{\text{PWM}}} - 1 \right) \right) < 0 \qquad (29) $$

### 3.3 | Model parameter identification based on least squares

In practical application field, the parameters $R$ in (25) are equivalent resistance which are unknown. Therefore, the relationship between current and voltage in (25) can be estimated by least square method. The least squares method finds the best functional match for the current and voltage by minimizing the squared error. That is, finding the sum of vectors $a$ and $b$ such that the sum $Q$ is minimized.

$$ Q = \sum_{i=1}^{n} (y_i - ax_i - b)^2 \qquad (30) $$

where $x_i$ and $y_i$ are the observations of current and voltage, $Q$ is the squared differences between the observations $y_i$ and $ax_i + b$, $a$ and $b$ are the linear fitting parameters.

When solving the best vector, the binary function is used to find the extreme value, and the partial derivative of $Q$ with respect to $a$, $b$ is set to 0:

$$ \left\{ \begin{array}{l} \frac{\partial Q}{\partial a} = -2 \sum_{i=1}^{n} (y_i - ax_i - b)x_i = 2a \sum_{i=1}^{n} x_i^2 + 2b \sum_{i=1}^{n} x_i + 2 \sum_{i=1}^{n} x_i y_i = 0 \\ \\ \frac{\partial Q}{\partial b} = -2 \sum_{i=1}^{n} (y_i - ax_i - b) = 2a \sum_{i=1}^{n} x_i - 2 \sum_{i=1}^{n} y_i + 2nb = 0 \end{array} \right. \qquad (31) $$

Write (26) in matrix form:

$$ \begin{pmatrix} \sum_{i=1}^{n} x_i^2 & \sum_{i=1}^{n} x_i \\ \sum_{i=1}^{n} x_i & n \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} \sum_{i=1}^{n} x_i y_i \\ \sum_{i=1}^{n} y_i \end{pmatrix} \qquad (32) $$

When the requirements of (28) are met, the only solution of (27) is (29), and both $a$ and $b$ in $y_i = ax_i + b$ have been determined in (30).

$$ \left| \begin{array}{l} \sum_{i=1}^{n} x_i^2 \quad \sum_{i=1}^{n} x_i \\ \sum_{i=1}^{n} x_i \quad n \end{array} \right| \neq 0 \qquad (33) $$
$$ egin{cases} a = \frac{n \sum_{i=1}^{n} x_i y_i - \sum_{i=1}^{n} x_i \sum_{i=1}^{n} y_i}{n \sum_{i=1}^{n} x_i^2 - \left( \sum_{i=1}^{n} x_i \right)^2} \\ b = \frac{\sum_{i=1}^{n} x_i^2 \sum_{i=1}^{n} y_i - \sum_{i=1}^{n} x_i \sum_{i=1}^{n} x_i y_i}{n \sum_{i=1}^{n} x_i^2 - \left( \sum_{i=1}^{n} x_i \right)^2} \end{cases} \quad (34) $$

As a consequence, the linear regression equation of actual current $i_k$ and modulation phase voltage $U_k$ in (25) can be estimated by least square method.

In (23), The estimated value of offsets $\hat{D}_x$ is the difference between output currents $I_k$ of the current sensor and actual current $i_k$. $I_k$ can be obtained by current sensor. And the actual current $i_k$ can be obtained by modulation phase voltage $U_k$ in (25).

## 4 | FAULT DIAGNOSIS METHOD BASED ON AVERAGE MODULATION VOLTAGE MODEL

The flowchart of fault diagnosis based on average modulation voltage model is shown in Figure 7.

In the three-phase three-wire system, according to Kirchhoff's current law, the vector sum of the three-phase currents $I_{sum}$ is zero. Whereas the fault of the current sensor will make the sum of the three-phase currents no longer zero. In this part, the sum $I_{sum}$ of the three-phase currents is selected for fault detection.

After the fault is detected, modulation phase voltage and current sensor output currents are collected. Then the offset $\hat{D}_x$ is estimated by the average modulation voltage model, and the fault location is located based on the estimator $\hat{D}_x$ in fault diagnosis.

The fault sensor is located through (31). When the absolute value of the estimated value exceeds the threshold, the current sensor fails.

$$ F_k = \begin{cases} 1 & \text{if } (|\hat{D}_k| \ge O_k) \\ 0 & \text{if } (|\hat{D}_k| < O_k) \end{cases} \quad (35) $$

where $O_k$ is the pre-set threshold; $F_k$ is a fault marker which used to indicate the location of the fault sensor.

After the fault location and degree are determined concurrently, the output of the fault sensor is corrected by means of compensation current to realize fault tolerant control.

The calculation method of compensation current $i_{com_k}$ is shown in (32). When no current sensor fault occurs $i_{com_a} = 0$, the corrected current $\hat{i}_k$ in (33) feeds back to the controller instead of the output $I_k$ of the fault sensor.

<figure><img src="image_14.png" /><figcaption>FIGURE 7 Flow chart of fault diagnosis based on average modulation voltage model.</figcaption></figure>

$$ i_{com_x} = \begin{cases} -\hat{D}_x & \text{if } F_x = 1 \\ 0 & \text{if } F_x = 0 \end{cases} \quad (36) $$

$$ \hat{i}_x = I_x + i_{com_x} \quad (37) $$
<figure><img src="image_15.png" /><figcaption>FIGURE 8 Experimental platform setup.</figcaption></figure>

<table><caption>TABLE 2 System parameters.</caption><thead><tr><td>System parameters</td><td>Values</td></tr></thead><tbody><tr><td>DC voltage</td><td>100 V</td></tr><tr><td>Filter inductance(L)</td><td>4 mH</td></tr><tr><td>Switching frequency</td><td>5 kHz</td></tr><tr><td>Grid voltage RMS</td><td>110 V</td></tr><tr><td>Grid frequency</td><td>50 Hz</td></tr><tr><td>Transformer ratio</td><td>1:2</td></tr><tr><td>Rated current RMS</td><td>5.656 A</td></tr></tbody></table>

## 5 EXPERIMENTAL RESULT

A three-phase cascaded H-bridge five-level grid-connected inverter system is developed in the laboratory, and the overall experimental setup is shown in Figure 8. DC side is composed of six 1.5 kVA DC power supplies. The main circuit of the cascaded five-level inverter consists of six H-bridge inverter units. The current control is implemented by the dSPACE HIL controller (DS1202). The inverter's main circuit is connected to the power grid through a filter circuit and a 6 kVA isolation transformer. The specific system parameters are set in Table 2.

Table 3 shows the collection Settings of fault samples. A total of three phase current $I_x$ and three phase modulated voltage $U_x$ are collected under the condition of no fault and 30 kinds of sensor faults. 200 fault samples are collected under each fault condition, and the collection length of each sample is one power grid cycle. In total, 6200 samples are collected. After obtaining data samples for various fault conditions, the least squares method is used to estimate the parameters.

Taking offset fault of current sensor in phase A as an example, sample distribution under various fault conditions is shown in Figure 9. Each red cluster represents a fault condition, which contains 200 fault samples. The coefficients in (21) were solved by least squares, and the linear regression equation was

<table><caption>TABLE 3 Data collection in different fault states.</caption><tr><td>Fault setting of phase A current sensor</td><td>-3 A: -1 A/0.5 A<br>1 A: 3 A/0.5 A</td></tr><tr><td>Fault setting of phase B current sensor</td><td>-3 A: -1 A/0.5 A<br>1 A: 3 A/0.5 A</td></tr><tr><td>Fault setting of phase C current sensor</td><td>-3 A: -1 A/0.5 A<br>1 A: 3 A/0.5 A</td></tr><tr><td>Sampling frequency</td><td>20 kHz</td></tr><tr><td>Number of samples for each failure condition</td><td>200</td></tr><tr><td>The total number of samples</td><td>6200</td></tr><tr><td>Acquisition variable</td><td>I<sub>x</sub>, U<sub>x</sub></td></tr></table>

<figure><img src="image_16.png" /><figcaption>FIGURE 9 Fault data distribution of phase A current sensor and its linear regression curve.</figcaption></figure>

$\bar{I}_a = 65.17 \bar{U}_a + 0.03574$. The determination coefficient of linear regression is 0.9985, which proves that there is a strong linear relationship between the mean value of three-phase current and the mean value of modulation voltage. This lays a foundation for the subsequent estimation of offset.
<figure><img src="image_17.png" /><figcaption>**FIGURE 10** Fault diagnosis and fault tolerance after phase A current sensor faults.</figcaption></figure>

## 5.1 | Offset fault diagnosis and fault-tolerant control of single current sensor

An error term is led in expressing the impact of the initial state of the system on the estimation. The gain term *W* is selected as -30 in experiments on the basis of (25). The performance of the proposed method is verified in the case of single current sensor failure. An offset variable *D*<sub>a</sub> of 1A is introduced into the phase A current sensor, and the B-phase and C-phase current sensors still operate normally. Figure 10 shows the waveform of variables in the process of fault detection, diagnosis and fault-tolerant control after the offset fault of A-phase current sensor occurs.

Figure 10a shows current waveforms of three phase. Figure 10b shows fault currents, where *f*<sub>a</sub> is actual current of phase A; *D*<sub>a</sub>, *D*<sub>b</sub>, *D*<sub>c</sub> are fault current estimation values of the proposed method. According to the sum of the three-phase currents, fault can be detected in a short time. Then the fault estimation of all current sensors is triggered. The estimated values of all current sensor faults rise rapidly. However, the estimated values of healthy sensors in phase B and C show small fluctuations around the zero point without exceeding the threshold. After 1/3 cycle of the fault occurrence, the estimated offset of the A-phase sensor exceeds the threshold of 0.4 A. This indicates that the current A-phase current sensor has an offset fault.

After locating the fault location, the compensation current *i*<sub>com_a</sub> is generated to correct the phase A current sensor based on the fault estimate. However, because the estimated value is not accurate and the feedback current to the controller is suddenly changed, the three-phase current at this time still has obvious distortion. The estimated value of the fault fluctuates around the true value until the fault occurs after one grid cycle. At this point, the three-phase current returns to normal. Figure 11 shows the distribution of estimation errors of A-phase current sensors.

A total of 400 samples have been counted in Figure 11. The blue bar represents the number of samples within the current error range, and the red line indicates the cumulative proportion of errors. From the curve of cumulative proportion, it can be seen that the fault estimation is mainly concentrated in the range

<figure><img src="image_18.png" /><figcaption>**FIGURE 11** Distribution of estimation error in phase A current sensor fault.</figcaption></figure>

between [-0.08, 0.06], which further shows the accuracy of the estimation error of the proposed method.

## 5.2 | Offset fault diagnosis and fault-tolerant control of multiple current sensors

In the multi-current sensor fault, coordinate transformation makes fault component coupled to each other. Nevertheless, the controller cannot separate each fault component. This results that three-phase modulation voltage output by the controller is affected by the fault component. Therefore, it is difficult to extract stable and discriminative fault features by single analysis of modulated voltage or three-phase current.

In consequence, fault diagnosis method based on current averaging is not suitable for multi-sensor simultaneous faults. The proposed fault diagnosis method based on the average modulation voltage model reveals the relationship between the modulation voltage and the three-phase current. In order to verify the performance of the proposed method in the case of multi-phase sensor faults, the offsets *D*<sub>a</sub>, *D*<sub>b</sub> of the phase A current sensor and the phase B current sensor are set to 1.5 and 1 A, respectively, as shown in Figure 12.

As shown in Figure 12, after the fault occurs, the estimated value $\hat{D}_a$, $\hat{D}_b$, $\hat{D}_c$ of the three-phase sensor rises rapidly. And the estimated value $\hat{D}_c$ of the healthy phase C sensor continues to fluctuate around the zero point without exceeding the threshold of 0.4A. After one cycle, the estimated value of each phase current sensor fluctuates at a low frequency near the actual value. And the three-phase current can be restored to normal operation by compensating the current. Figure 13 shows the distribution of estimation errors of phase A and B current sensors.

The estimation errors of phase A current sensors are mainly concentrated between [0, 0.06], while those of phase B current sensors are mainly concentrated between [-0.04, 0.06], indicating the accuracy of fault estimation results in the experiment.
<figure><img src="image_19.png" /><figcaption>**FIGURE 12** Fault diagnosis and fault tolerance results after faults of phase A and B current sensors.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**FIGURE 13** Estimation error distribution under faults of phase A and B current sensors. (a) Fault estimation distribution of phase B current sensor. (b) Fault estimation distribution of phase B current sensor.</figcaption></figure>

## 5.3 | Fault diagnosis and fault-tolerant control in changing currents condition

When the three-phase inverter is connected to the grid, the reference current may change suddenly or slowly. Figure 14 shows

<figure><img src="image_21.png" /><figcaption>**FIGURE 14** Fault diagnosis and fault tolerance after sudden changes in the reference currents. (a) The waveform of currents. (b) Fault component of currents.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**FIGURE 15** Fault diagnosis and fault tolerance after slow changes in the reference currents. (a) The waveform of currents. (b) Fault component of currents.</figcaption></figure>

the waveform of variables in the process of fault diagnosis and fault-tolerant control after sudden changes in the reference currents.

As shown in Figure 14a, the reference current changes from 5 to 10 A suddenly at 0.2 s. And as shown in Figure 14b, the offset fault of A-phase current sensor occurs at 0.3 s. The
sudden change in the reference currents cause a small change in the fault component of currents $\hat{D}_a$. This change does not reach the threshold of 0.4 A for fault diagnosis. The sudden changes in the reference currents will not be considered as a fault. The estimated offset of the A-phase sensor at 0.3 s exceeds the threshold of 0.4 A. This indicates that the current A-phase current sensor has an offset fault.

The waveform of variables in the process of fault diagnosis and fault-tolerant control after slow changes in the reference currents is shown in Figure 15.

As shown in Figure 15a, the reference current changes from 5 A at 0.2 s to 6 A at 0.3 A slowly. And as shown in Figure 15b, the offset fault of A-phase current sensor occurs at 0.3s. The slow change in the reference currents cause no special change in the fault component of currents $\hat{D}_a$. The slow change in the reference currents will not be considered as a fault. Similarly, offset faults can be diagnosed.

As a consequence, when the current is changing slowly or suddenly, it is not identified as a fault in the proposed method. The fault diagnosis in this paper is verified the ruggedness.

## 6 | CONCLUSIONS

Here, a fault diagnosis method based on average modulation voltage model is proposed for offset faults of multi-current sensors. The offset fault of multi-current sensor is first modelled and the actual and measured current values are calculated. On the basis of fault modelling, the average modulation voltage model is established to estimate the offset in the sensor. In order to enhance the robustness and accuracy of the estimation, an error term of the mean modulation voltage is introduced. The error analysis further shows the effectiveness of the proposed method, and the experimental results show the accuracy of the fault estimation, and under the action of the compensation current, the three-phase grid-connected inverter returns to normal operation. In addition, the fault diagnosis here is not affected by sudden or slow changes in current.

## AUTHOR CONTRIBUTIONS

**Fan Zhang**: Software; validation; writing—review and editing. **Zhiwei Zhang**: Validation. **Guangfeng Jin**: Software; writing—original draft. **Tianzhen Wang**: Resources; writing—review and editing. **Jingang Han**: Writing—review and editing. **Hubert Razik**: Supervision. **Yide Wang**: Supervision.

## ACKNOWLEDGEMENTS

This research was funded by the National Natural Science Foundation of China (No. 61673260).

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

## DATA AVAILABILITY STATEMENT

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

## ORCID

Fan Zhang https://orcid.org/0000-0002-2311-7557

Tianzhen Wang https://orcid.org/0000-0002-7525-8466

Jingang Han https://orcid.org/0000-0002-6347-1972

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**How to cite this article:** Zhang, F., Zhang, Z., Jin, G., Wang, T., Han, J., Razik, H., Wang, Y.: Fault diagnosis for multiple current sensors in grid-connected inverter based on average modulation voltage. IET Power Electron. 17, 54-66 (2024).

https://doi.org/10.1049/pel2.12614