Multiple Fault-Tolerant Control of DC Microgrids Based on Sliding Mode Observer

Abstract

Different locations and types of faults affect the safe and reliable operation of DC microgrids. Therefore, this paper proposes a secondary multiple fault-tolerant control scheme for a DC microgrid based on a sliding mode observer to ensure the voltage is restored to the rated value and realize the proportional current sharing of all sources. Firstly, the secondary control model of the DC microgrid is established, considering the multiple faults of actuators and sensors simultaneously. Secondly, the system model is transformed into two subsystems by bilinear coordinate transformation, and multiple faults decoupling between the sensor and actuator is realized. Then, two sliding mode observers are designed for the two transformed subsystems. The sliding mode variable structure equivalent principle is used to reconstruct the faults at different positions without knowing the fault models in advance, which is convenient for subsequent processing. Then, the fault-tolerant controller based on the sliding mode observer is designed, which uses the reconstructed value to offset the influence of sensor and actuator faults on the DC microgrid and realizes the fault-tolerant control of the DC microgrid. Finally, the effectiveness of the proposed control strategy is verified by experiments.

1. Introduction

Microgrid technology is an important component of future intelligent power distribution systems, which is of great significance for promoting energy conservation, emission reduction, and achieving sustainable energy development [1,2]. Compared with traditional AC microgrids, DC microgrids have advantages such as cost saving, loss reduction, flexibility, and efficiency due to the elimination of some AC/DC conversion devices. Meanwhile, due to the absence of power quality issues such as frequency and reactive power in DC microgrids, their controllability and reliability are greatly improved [3,4]. At present, DC microgrids are widely used in small-capacity, close-range power systems such as electric propulsion aircraft, hybrid vehicles, electric ships, and intelligent buildings [5,6,7].

In a DC microgrid, a hierarchical control architecture is usually used to control the distributed generation (DG) of each distributed power source [8]. The first-layer control is mainly used to achieve voltage and current control, as well as preliminary current distribution between DGs. In the first-layer control, droop control has been widely applied and studied due to its advantages of no communication and plug-and-play [9]. However, droop control leads to voltage drop and uneven current distribution among sources, so it is necessary to introduce two-layer control to eliminate voltage deviation and improve the accuracy of current distribution. According to the different communication forms in the second-layer control, it can be divided into centralized, distributed, and decentralized control [10,11,12]. In recent years, consensus control has become a research hotspot [13]. Consensus control overcomes the shortcomings of centralized and decentralized control, requiring only local measurement information and neighboring node measurement information to make the control variables of the entire system tend to be consistent.

With the large-scale application of power electronic equipment, the operating environment of DC microgrids has become more complex, and the probability of internal faults in the system has greatly increased [14]. The actuator failure and sensor failure, respectively, interfere with the output instructions and input information of the controller, resulting in the inability to achieve control objectives. When the system adopts secondary control, the faults that occur in a single DG may propagate to the entire system and evolve into faults for the entire system [15]. Therefore, actuator and sensor faults seriously disrupt the stable operation and safety reliability of the microgrid. Thus, it is of great significance to study the handling of actuator and sensor faults in DC microgrids.

Some literature has proposed relevant solutions for actuator and sensor faults in microgrids. The first method is to detect, locate, and identify faults in real time based on data and signal analysis and isolate faults. Ref. [16] used the method of generating synthetic data from measured line currents during normal operation, where fault detection is achieved by training an integrated model. Ref. [17] used wavelet-based power spectral density to detect faults, identify fault phases, and locate faults in microgrids. Ref. [18] first decomposed the measured signals (bus voltage and active power) and then extracted certain features to represent the distribution characteristics of the decomposed eigenvalues. Fault identification in microgrid by extreme learning machine. Although these strategies can ensure the operational performance of microgrids, they have problems of low efficiency and difficult data acquisition.

The second method is to design a fault-tolerant controller to ensure that the microgrid can still operate within the allowable performance range under fault conditions. Ref. [19] proposed the adaptive fault-tolerant secondary voltage and frequency control of isolated AC microgrids when dealing with sensor faults, but the sensor fault boundary needs to be known. Ref. [20] proposed an adaptive micro-grid actuator bias fault tolerance algorithm independent of the global information of the network graph and fault factor boundary. Ref. [21] proposes a distributed energy storage micro-grid fault tolerance algorithm with actuator bias and proportional faults, but the above method has a limited fault processing range. The strategy of using adaptive compensation mentioned above is passive fault-tolerant control, which offsets the impact of faults through adaptive terms. In order to ensure stable operation of the system in the worst-case scenario, the designed controller is often more conservative.

The active fault-tolerant strategy has the additional function of control reconstruction, making it suitable for handling different types of faults in the system, which has attracted great interest from researchers in recent years. Ref. [22] used a disturbance observer to deal with the constant disturbance of heterogeneous nodes of the microgrid. Ref. [23] designed extended state observers to deal with constant disturbance and time-varying disturbance. These observers can only handle certain types of disturbances and faults. Sliding mode variable structure control is a special type of nonlinear control. Due to the independence of system characteristics and parameters from external disturbances, sliding mode variable structure control has good robustness and is widely used in the research of power system state observation and fault tolerance. Ref. [24] proposed two secondary fault-tolerant control schemes for AC microgrid based on sliding mode observers. The first scheme includes a central fault-tolerant controller, and the second scheme adopts a decentralized scheme, with each DG having a separate fault-tolerant controller. Ref. [25] proposed a fault-tolerant control based on second-order sliding mode for different DGs in a DC microgrid, which can effectively realize secondary fault tolerance of actuator faults in the shortest possible time. Ref. [26] detected multiple sensor faults that occur simultaneously or separately by designing a sliding mode observer. The above research mainly focuses on the fault tolerance control of actuator fault or sensor fault, and the processing range is limited.

At present, most of the research on fault-tolerant control of microgrid mainly focuses on the processing of faults in limited locations or limited types, and some even need to establish accurate fault models. Therefore, this article focuses on the secondary fault tolerance problem of DC microgrid with sensor and actuator faults. This article is arranged as follows: the first section is the introduction, the second section is the problem description, the third section is the design of fault sliding mode observers, the fourth section is the fault-tolerant control scheme, the fifth section is experimental analysis, and the sixth section is the conclusion. The contributions of this article are as follows:

(1)

A multiple faults reconstruction framework is proposed to accurately reconstruct multiple faults at different locations such as sensors and actuators to better analyze the composition of faults.

(2)

In the case of unknown fault-related information, the sliding mode controller uses the reconstructed value to offset the influence of different positions and types of faults on the DC microgrid to ensure that the microgrid can still operate within the permissible performance range under the fault condition. The method is simple and easy to implement.

2. Graph Theory

The communication topology between nodes in a microgrid can be represented by a directed graph $G\left(v,E\right)$, where $v=\left\{1,2,\cdots N\right\}$ is a finite set of non-empty nodes and $E\subseteq v\times v$ is a set of ordered pairs of nodes, i.e., an edge set. The edge set represents the communication link for data exchange. If node j can obtain information from node i, then $\left(i,j\right)\subseteq E$. The communication weight between nodes in the directed graph can be represented by the adjacency matrix $A=\left[a_{ij}\right]\in R^{N\times N}$. If $\left(i,j\right)\in E$, then $a_{ij}\begin{array}{l}\end{array}>\begin{array}{l}\end{array}0$; otherwise, $a_{ij}=0$. When the communication weight is not considered, $a_{ij}=1$ can be taken to be $\left(i,j\right)\in E$. $D=diag\left\{d_1,d_2,\cdots ,d_N\right\}$ is the degree matrix of a directed graph, whose elements $d_i$ represent the total communication weight of node i, that is, $d_i={\displaystyle \sum _{j=1}^N{a}_{ij}}$. When communication weight is not considered, $d_i$ is the number of all nodes communicating with node i. The topology of a communication network can be represented by a Laplace matrix $L=\left[l_{ij}\right]\in R^{N\times N}$, which is defined as:

$$L = D-A\Rightarrow \left\{\begin{array}{cc}{l}_{ij} = -a_{ij}& i \ne j\\ l_{\mathrm{ii}} = {\displaystyle \sum _{j=1}^N{a}_{ij}}& i = j\end{array}\right.$$

(1)

3. Problem Description

The hierarchical control of a DC microgrid includes primary control and secondary control, as shown in Figure 1. Each DC source is connected to the microgrid through the boost converter. The goal of primary control is to quickly distribute the power of each source, usually using droop control, but droop control is differential regulation. Secondary control has two control objectives, namely, adjusting the system voltage to the rated value and realizing load sharing in proportion. According to the hierarchical control strategy of microgrid, droop control is applied to the outer loop control of the distributed power supply, which maintains the stability of the system in the primary control. At the same time, the appropriate voltage reference value is provided for the droop controller in the secondary control.

The droop method (applied to DG i) is described by the following voltage droop characteristics:

$$v_i^d{ = v}_{n,i}-{ki}_i$$

(2)

where $v_i^d$ is the given voltage, $i_i$ is the current measured at the DG i terminal, $v_{n,i}$ is the DG i voltage reference, and k is the droop coefficient.

The secondary protocol structure of secondary control is derived based on the method outlined in [24], as follows:

$${\dot{v}}_i^d = {\dot{v}}_{n,i}-k{\dot{i}}_i$$

(3)

Define ${\dot{v}}_i^d = {\dot{v}}_{n,i}-k{\dot{i}}_i$ as a consensus control structure; the secondary control in a microgrid can be designed as ${\dot{v}}_i^d=u_i$. The control law satisfies the following equation to ensure that the DG voltage can track a given reference value.

$$\underset{t\to \infty }{lim}{ v}_i^d-v_r^d = 0$$

(4)

Next, consider a microgrid consisting of N DGs and a reference point, where the consensus protocol is used to adjust the dynamics of each DG. The nominal dynamics of the ith DG in the microgrid are as follows:

$$u_i^n = {\displaystyle \sum _{j=1, j\ne i}^N{a}_{ij}\left(v_i^d-v_j^d\right)+b_i\left(v_i^d-v_r^d\right)}$$

(5)

where i = 1,2, … N, $a_{ij}$, and $b_i$ are the connection weights.

The secondary control protocol in the above equation does not consider the possible faults that may occur in each DG; however, there is a possibility of faults occurring in microgrid actuators and sensors. When the execution structure fails, it is reflected in the deviation of the collaborative control law. When the deviation of the actuators exceeds a certain value, it will affect the stability of the system. When a sensor malfunctions, it is reflected in the error in receiving information from neighboring intelligent agents. Equation (5) can be rewritten as:

$$\begin{array}{cc}\hfill u_{v,i}& ={\displaystyle \sum _{j=1, j\ne i}^N{a}_{ij}\left(v_i^d-v_j^d-f_{v,j}^s\right){+b}_i\left(v_i^d-v_r^d\right)}+m_{v,i}{f}_{v,i}^a\hfill \\ & ={\displaystyle \sum _{j=1, j\ne i}^N{a}_{ij}\left(v_i^d-v_j^d\right)-{\displaystyle \sum _{j=1, j\ne i}^N{a}_{ij}{f}_{v,j}^s+}{b}_i\left(v_i^d-v_r^d\right)}+m_{v,i}{f}_{v,i}^a\hfill \\ & =u_i^N-A{q}_{v,j}{f}_{v,j}^s+p_{v,i}{f}_{v,i}^a\hfill \end{array}$$

(6)

where $f_{v,j}^s$ and $q_{v,j}$ represent sensor faults and distribution weights and $f_{v,i}^a$ and $p_{v,i}$ represent actuator fault and distribution weights. When a sensor fault occurs, $q_{v,i}=1$; when an actuator fault occurs, $p_{v,i}=1$; otherwise, $q_{v,i}=0$, $p_{v,i}=0$. A is the connection matrix associated with the diagram describing the microgrid structure.

Due to the existence of these deviations, the secondary control target cannot be achieved, the voltage deviates from the rated value, and the power cannot be distributed according to the preset proportion of each source. Moreover, when these deviations exceed a certain value, the stability of the system will be affected. Therefore, a centralized multiple fault-tolerant secondary control scheme for DC microgrid based on sliding mode observer (SMO) is proposed in this paper, fault tolerance is placed in the second layer control to reduce the impact of faults on the system, achieve voltage recovery, and achieve the goal of power distribution according to the preset proportion of sources.

4. Sensor and Actuator Faults Reconfiguration

For the sake of simplicity of symbols, the following contents are defined as in (7):

$$\begin{array}{l}{F}_v^a=\left[\begin{array}{c}{f}_{v,1}^a\\ ⋮\\ f_{v,N}^a\end{array}\right],F_v^s=\left[\begin{array}{c}{f}_{v,1}^s\\ ⋮\\ f_{v,N}^s\end{array}\right],x=\begin{array}{l}\end{array}\left[\begin{array}{c}{v}_1^d-v_r^d\\ ⋮\\ v_N^d-v_r^d\end{array}\right],\begin{array}{l}\end{array}{u}_{cv}=\left[\begin{array}{c}{u}_{cv,1}\\ ⋮\\ u_{cv,N}\end{array}\right],\\ P_v=\left[\begin{array}{ccc}{p}_{v,1}& & \\ & \ddots & \\ & & p_{v,N}\end{array}\right],Q_v=\left[\begin{array}{ccc}{q}_{v,1}& & \\ & \ddots & \\ & & q_{v,N}\end{array}\right],\\ B=\left[\begin{array}{ccc}{b}_1& & \\ & \ddots & \\ & & b_N\end{array}\right], A{Q}_v=-H_v\end{array}$$

(7)

The dynamic equation for secondary control of microgrids can be rewritten as:

$$\left\{\begin{array}{c}\dot{x}=Ax+H_v{F}_v^s+P_v{F}_v^a\hfill \\ y=Cx+Q_v{F}_v^s\hfill \end{array}\right.$$

(8)

where $A=L+B$, L is a Laplace matrix associated with a graph describing the structure of the microgrid.

Considering a general situation where not all outputs fault, which means that the dimension of sensor faults is less than the dimension of the system output, the following linear transformation was made to the output equation of the system. The output signal $y\in R^p$ of (8) of the system has a linear transformation $\varsigma \in R^{p\times p}$, such that:

$$\varsigma \left[C\begin{array}{c}\begin{array}{}\end{array}{Q}_v\end{array}\right]\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cc}{C}_1& 0\\ C_2& Q\end{array}\right]$$

(9)

$$\varsigma y=\left\{\begin{array}{c}{y}_1\begin{array}{l}\end{array}=C_{1}x\hfill \\ y_2\begin{array}{l}\end{array}=C_{2}x+Q{F}_v^s\hfill \end{array}\right.$$

(10)

where $y_1\in R^{p-m}$, $y_2\in R^m$, $C_1\in R^{\left(p-m\right)\times n}$, $C_2\in R^{m\times n}$, and $Q\in R^{m\times m}$ are invertible constant matrices. It can be seen that the output signal $y_1$ is an output signal without sensor faults, and $y_2$ is an output signal containing sensor faults.

Rewrite the system equation as:

$$\left\{\begin{array}{c}\dot{x}=Ax{+H}_v{F}_v^s+P_v{F}_v^a\hfill \\ y_1=C_{1}x\hfill \end{array}\right.$$

(11)

4.1. Fault Decoupling at Different Locations

Assumption 1. 

$H_v$ is a column full rank matrix, and $rank(C_1{H}_v)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}rank(H_v)$.

Explanation 1. 

If the sensor fault distribution matrix $A{H}_v$ does not satisfy column full rank, i.e., $rank(H_v)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{q}_1<q$, then the matrix $H_v$ can be decomposed into $H_v=H_{v1}^{\prime }{H}_{v2}^{\prime }$, where $H_{v1}^{\prime }$ is the column full rank matrix and $d_1^{\prime }(t)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{H}_{v2}^{\prime }{d}_2^{\prime }(t)$ can be considered as a new unknown input interference, $d_1^{\prime }(t)\in R^{q_1}$. Assumption 1 indicates that multiple actuator faults in the microgrid are independent of each other.

Assumption 2. 

$(A,C_1)$ is observable.

Assumption 2 indicates that the state of the microgrid is observable. In order to decouple actuator faults and sensor faults in system (11), based on Assumption 1, two transformation matrices, T and S, can be constructed, such that:

$$x(t)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{T}^{-1}z(t)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{T}^{-1}\left[\begin{array}{c}{z}_1(t)\\ z_2(t)\end{array}\right]$$

(12)

$$y_1(t)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{S}^{-1}\left[\begin{array}{c}{v}_1(t)\\ v_2(t)\end{array}\right]$$

(13)

Then, the system (11) can be rewritten as:

$$\left\{\begin{array}{c}\dot{z}(t)=\left[\begin{array}{c}{\dot{z}}_1(t)\\ {\dot{z}}_2(t)\end{array}\right]=TA{T}^{-1}z(t)+T{H}_v{F}_v^s(t)+T{P}_v{F}_v^a(t)\hfill \\ v(t)=\left[\begin{array}{c}{v}_1(t)\\ v_2(t)\end{array}\right]=S{C}_1{T}^{-1}z(t)\hfill \end{array}\right.$$

(14)

where $S{C}_1{T}^{-1}\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cc}{C}_{11}& 0\\ 0& C_{22}\end{array}\right]$ and $C_{22}$ are invertible matrices.

The transformation matrix is constructed as follows:

$$T\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cc}{I}_{n-q}& -H_{v1}{H}_{v2}^{-1}\\ 0& I_q\end{array}\right]$$

(15)

So, the coefficient matrix in (14) is:

$$TA{T}^{-1}\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cc}{\overline{A}}_{11}& {\overline{A}}_{12}\\ {\overline{A}}_{21}& {\overline{A}}_{22}\end{array}\right],\begin{array}{l}\hfill \end{array}T{H}_v\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{c}0\\ {\overline{H}}_{2v}\end{array}\right],\begin{array}{l}\hfill \end{array}T{P}_v\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{c}{\overline{P}}_{1v}\\ {\overline{P}}_{2v}\end{array}\right]$$

(16)

Then, the system (11) can be transformed into two subsystems, as follows:

$$\left\{\begin{array}{c}{\dot{z}}_1(t)={\overline{A}}_{11}{z}_1(t)+{\overline{A}}_{12}{z}_2(t)+{\overline{P}}_{1v}{F}_v^a(t)\hfill \\ v_1(t)=C_{11}{z}_1(t)\hfill \end{array}\right.$$

(17)

$$\left\{\begin{array}{c}{\dot{z}}_2(t)={\overline{A}}_{21}{z}_1(t)+{\overline{A}}_{22}{z}_2(t)+{\overline{H}}_{2v}{F}_v^s(t)+{\overline{P}}_{2v}{F}_v^a(t)\hfill \\ v_2(t)=C_{22}{z}_2(t)\hfill \end{array}\right.$$

(18)

The goal of transforming matrices $T$ and $S$ is to achieve complete decoupling between sensor and actuator faults through coordinate transformation, effectively separating actuator faults and sensor faults. Among them, one subsystem only has actuator faults without sensor faults, while the other subsystem has both actuator and sensor faults. Adopting a sliding mode variable structure observer and applying the principle of sliding mode variable structure equivalence can not only achieve accurate reconstruction of actuator faults, but can also achieve accurate reconstruction of sensor faults.

4.2. Design of Sliding Mode Observer

The task of the sliding mode observer is described as follows: The sliding mode observer is designed for two subsystems decoupled from faults at different locations, and the multiple faults reconstruction are realized. The detailed design of the two sliding mode observers corresponding to the above two subsystems is as follows.

Assumption 3. 

Both $({\overline{A}}_{11},C_{11})$ and $({\overline{A}}_{22},C_{22})$ can be observed.

Assumption 3 indicates that the state of the two subsystems after transformation is observable. Based on the transformation system structures (17) and (18), the following sliding mode variable structure observers are constructed:

$$\left\{\begin{array}{c}{\dot{\widehat{z}}}_1(t)={\overline{A}}_{11}{\widehat{z}}_1(t)+{\overline{A}}_{12}{\widehat{z}}_2(t)+{\overline{P}}_{1v}{r}_1(t)+{\zeta }_1(v_1(t)-{\widehat{v}}_1(t))\hfill \\ {\widehat{v}}_1(t)=C_{11}{\widehat{z}}_1(t)\hfill \end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{\dot{\widehat{z}}}_2(t)=\begin{array}{l}\end{array}{\overline{A}}_{21}{\widehat{z}}_1(t)+{\overline{A}}_{22}{\widehat{z}}_2(t)+{\overline{P}}_{2v}{r}_2(t)+{\zeta }_2(v_2(t)-{\widehat{v}}_2(t))\hfill \\ {\widehat{v}}_2(t)\begin{array}{l}\end{array}=C_{22}{\widehat{z}}_2(t)\hfill \end{array}\right.$$

(20)

where the observed values of the corresponding variables in the equation are represented by a superscript “^”, and the two parameter matrices to be designed are ${\zeta }_1$ and ${\zeta }_2$, respectively.

The input signals of the two sliding mode variable structures are $r_1(t)$ and $r_2(t)$, as shown below:

$$\left\{\begin{array}{c}{r}_1(t)={\rho }_{1}sign\left({\chi }_1(v_1(t)-{\widehat{v}}_1(t)\right)\hfill \\ r_2(t)={\rho }_{2}sign\left({\chi }_2(v_2(t)-{\widehat{v}}_2(t)\right)\hfill \end{array}\right.$$

(21)

where ${\chi }_1$ and ${\chi }_2$ are the gain matrices of the two observers to be designed, and ${\rho }_1$ and ${\rho }_2$ are two normal numbers to be designed.

Assumption 4. 

There are two symmetric positive-definite matrices, $G_1$ and $G_2$, and the matrices to be designed are ${\chi }_1$ and ${\chi }_2$, which satisfy the following equation:

$$\left\{\begin{array}{c}{G}_1{\overline{P}}_{1v}=C_{11}^T{\chi }_1^T\hfill \\ G_2{\overline{P}}_{2v}=C_{22}^T{\chi }_2^T\hfill \end{array}\right.$$

(22)

Based on Assumption 3, there are also matrices ${\zeta }_1$ and ${\zeta }_2$, which can make matrices $A_{10}$ and $A_{20}$ stable matrices:

$$\left\{\begin{array}{c}{\overline{A}}_{11}-{\zeta }_1{\overline{C}}_{11}=A_{10}\hfill \\ {\overline{A}}_{22}-{\zeta }_2{\overline{C}}_{22}=A_{20}\hfill \end{array}\right.$$

(23)

The construction observer errors $e_1$ and $e_2$ are:

$$\left\{\begin{array}{c}{e}_1(t)=z_1(t)-{\widehat{z}}_1(t)\hfill \\ e_2(t)=z_2(t)-{\widehat{z}}_2(t)\hfill \end{array}\right.$$

(24)

The construction output errors $e_{v1}$ and $e_{v2}$ are:

$$\left\{\begin{array}{c}{e}_{v1}(t)=v_1(t)-{\widehat{v}}_1(t)=C_{11}{e}_1(t)\hfill \\ e_{v2}(t)=v_2(t)-{\widehat{v}}_2(t)=C_{22}{e}_2(t)\hfill \end{array}\right.$$

(25)

$$e={(\begin{array}{cc}{e}_1\hfill & e_2\hfill \end{array})}^T$$

(26)

From (17) to (20), it can be obtained that:

$$\left\{\begin{array}{c}{\dot{e}}_1(t)=({\overline{A}}_{11}-{\zeta }_1{C}_{11})e_1(t)+{\overline{A}}_{12}{e}_2(t)+{\overline{P}}_{1v}\left(F_v^a(t)-r_1(t)\right)\hfill \\ {\dot{e}}_2(t)=({\overline{A}}_{22}-{\zeta }_2{C}_{22})e_2(t)+{\overline{A}}_{21}{e}_1(t)+{\overline{P}}_{2v}(F_v^a(t)-r_2(t))+{\overline{H}}_{2v}{F}_v^s(t)\hfill \end{array}\right.$$

(27)

Theorem 1. 

The proof process is shown in Appendix A. Considering the observation dynamic error (25) and Assumption 4, if the following inequality can be satisfied:

$${\overline{A}}^{T}G+G\overline{A}<0$$

(28)

and the normal numbers ${\rho }_1$ and ${\rho }_2$ to be designed satisfy the following requirements:

$$\left\{\begin{array}{c}{\rho }_1>{\kappa }_2\hfill \\ {\rho }_2>{\kappa }_2+{\displaystyle \frac{‖{\overline{H}}_{2v}‖}{‖{\overline{P}}_{2v}‖}}{\kappa }_1\hfill \end{array}\right.$$

(29)

then, the observer errors $e_1$ and $e_2$ gradually stabilize and converge to zero:

$$\left\{\begin{array}{c}\underset{t\to \infty }{\lim }{e}_1(t)=0\hfill \\ \underset{t\to \infty }{\lim }{e}_2(t)=0\hfill \end{array}\right.$$

(30)

Explanation 2. 

Theorem 1 states that both $e_1$ and $e_2$ are bounded, that is, there exists a time point  $t_0$, when $t>t_0$:

$$\left\{\begin{array}{c}‖e_1‖\le {\delta }_1\hfill \\ ‖e_2‖\le {\delta }_2\hfill \end{array}\right.\Rightarrow ‖e‖\begin{array}{l}\hfill \end{array}\le \begin{array}{l}\hfill \end{array}\delta$$

(31)

where ${\delta }_1$, ${\delta }_2$, and $\delta$ are normal numbers. When time approaches infinity, ${\delta }_1$, ${\delta }_2$, and $\delta$ all tend to zero.

Theorem 2. 

The proof process is shown in  Appendix B. Select sliding surfaces $s_1\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\chi }_1{e}_{v1}$ and $s_2\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\chi }_2{e}_{v2}$, respectively, based on Assumption 4 and (31), if the two normal numbers to be designed satisfy (32):

$$\left\{\begin{array}{c}{\rho }_1\begin{array}{l}\end{array}>{\displaystyle \frac{‖{\chi }_1{C}_{11}‖\left({\delta }_1‖{\overline{A}}_{11}-{\zeta }_1{C}_{11}‖+{\delta }_2‖{\overline{A}}_{12}‖+{\gamma }_2‖{\overline{P}}_{1v}‖\right)+{\theta }_1}{{\lambda }_{\min }\left({\overline{P}}_{1v}^T{G}_1{\overline{P}}_{1v}\right)}}\hfill \\ {\rho }_2>{\displaystyle \frac{‖{\chi }_2{C}_{22}‖\left({\delta }_2‖{\overline{A}}_{22}-{\zeta }_2{C}_{22}‖+{\delta }_1‖{\overline{A}}_{21}‖+{\gamma }_2‖{\overline{P}}_{2v}‖+{\gamma }_1‖{\overline{H}}_{2v}‖\right)+{\theta }_2}{{\lambda }_{\min }\left({\overline{M}}_{2v}^T{P}_2{\overline{M}}_{2v}\right)}}\hfill \end{array}\right.$$

(32)

within a finite time, the system satisfies the conditions of sliding mode existence and reachability.

After the system state reaches the sliding mode surface $s_i\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0\begin{array}{l}\hfill \end{array}(i\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}1,2)$, based on the principle of sliding mode equivalence, $s_i\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\dot{s}}_i\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0$:

$$\left\{\begin{array}{c}{\chi }_1{e}_{v1}={\chi }_1{\dot{e}}_{v1}={\chi }_1{C}_{11}{\dot{e}}_1=0\hfill \\ {\chi }_2{e}_{v2}={\chi }_2{\dot{e}}_{v2}={\chi }_2{C}_{22}{\dot{e}}_2=0\hfill \end{array}\right.$$

(33)

4.3. Reconstruction of Actuator Faults and Sensor Faults

According to Theorem 1, after a period of time, the observer errors $e_1\left(t\right)$ and $e_2\left(t\right)$ asymptotically converge to zero. According to Theorem 2, the deviation equation of the system starts from any point and reaches the sliding mode surfaces $s_1\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\chi }_1{e}_{v1}$ and $s_2\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\chi }_2{e}_{v2}$. Based on the above conclusions and the basic principles of sliding mode control, precise reconstruction of actuator and sensor faults can be achieved.

From (33), it can be obtained that:

$${\chi }_1{C}_{11}\left(({\overline{A}}_{11}-{\zeta }_1{C}_{11})e_1(t)+{\overline{A}}_{12}{e}_2(t)+{\overline{P}}_{1v}\left(F_v^a(t)-r_1(t)\right)\right)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0$$

(34)

According to Theorem 2, when t tends towards infinity, $e_1\left(t\right)$ and $e_2\left(t\right)$ converge to zero. It can be obtained that,

$${\chi }_1{C}_{11}{\overline{P}}_{1v}\left(F_v^a(t)-r_1(t)\right)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0$$

(35)

Since ${\overline{P}}_{1v}$ is a non-zero matrix, ${\chi }_1{C}_{11}{\overline{P}}_{1v}\begin{array}{l}\hfill \end{array}\ne \begin{array}{l}\hfill \end{array}0$, it can be obtained that:

$${\widehat{F}}_v^a(t)\begin{array}{l}\hfill \end{array}\approx \begin{array}{l}\hfill \end{array}{r}_1(t)\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\rho }_1\mathrm{sgn}({\chi }_1{e}_{v1})$$

(36)

$${\chi }_2{C}_{22}(({\overline{A}}_{22}-{\zeta }_2{\overline{C}}_{22})e_2(t)+{\overline{A}}_{21}{e}_1(t)+{\overline{P}}_{2v}{F}_v^a(t)-{\overline{P}}_{2v}{r}_2(t)+{\overline{H}}_{2v}{F}_v^s(t))\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0$$

(37)

It can be obtained that:

$${\chi }_2{C}_{22}({\overline{P}}_{2v}{F}_v^a(t)-{\overline{P}}_{2v}{r}_2(t)+{\overline{H}}_{2v}{F}_v^s(t))\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}0$$

(38)

$${\widehat{F}}_v^s(t)\begin{array}{l}\hfill \end{array}\approx \begin{array}{l}\hfill \end{array}{\overline{H}}_{2v}^{-1}({\overline{P}}_{2v}{r}_2(t)-{\overline{P}}_{2v}{F}_v^a(t))\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\overline{H}}_{2v}^{-1}{\overline{P}}_{2v}({\rho }_2\mathrm{sgn}({\chi }_2{e}_{v2})-{\rho }_1\mathrm{sgn}({\chi }_1{e}_{v1}))$$

(39)

5. Fault-Tolerant Control

The fault-tolerant control of a DC microgrid with actuator and sensor faults is described as follows: When sensor and actuator faults occur in the system, additional compensation $u_{cv}\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}{\left[u_{cv,1},\cdots u_{cv,n}\right]}^T$ are designed and each source is coordinated to control so as to reduce the influence of multiple faults on the microgrid, ensure that the voltage is restored to the rated value, and realize the proportional sharing of the current of each source.

$$\left\{\begin{array}{c}\dot{x}=Ax{+H}_v{F}_v^s+P_v{F}_v^a{+i}_N{u}_{cv}\hfill \\ y_1=C_{1}x\hfill \end{array}\right.$$

(40)

It is worth noting that under normal operation of the microgrid, the fault-tolerant strategy proposed in this article will not be activated. The controller will only be reconfigured when, and only when, a microgrid fault is detected. Design $u_{cv}$:

$$u_{cv}\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}-\left(Ax{+H}_v{F}_v^s+P_v{F}_v^a\right)-ks-\epsilon \mathrm{sgn}\left(s\right)$$

(41)

In this paper, sliding mode control adopts exponential reaching law, which not only shortens the reaching time, but also makes the speed of moving point reach the switching surface very small. By substituting the observed values of the system state, actuator fault reconstruction values ${\widehat{F}}_v^a(t)$ (36), and sensor fault reconstruction values ${\widehat{F}}_v^s(t)$ (39) into the designed control rate $u_{cv}$ (41), the adjusted online fault-tolerant control rate $u_{cv}$ is:

$$u_{cv}\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}-\left(A\widehat{x}{+H}_v{\widehat{F}}_v^s+P_v{\widehat{F}}_v^a\right)-k{s}_3-\epsilon \mathrm{sgn}\left(s_3\right)$$

(42)

Theorem 3. 

The proof process is shown in Appendix C. Selecting the sliding mode surface  $s_3=e_x$, $e_x={\left[v_1^d-v_r^d,\cdots ,v_N^d-v_r^d\right]}^T$, based on Equation (42), the existence and reachability conditions of the sliding mode for the fault-tolerant system can be satisfied when the constants satisfy the following inequalities:

$$\left\{\begin{array}{c}k>0\hfill \\ \epsilon >\begin{array}{l}\end{array}‖GA{T}^{-1}\delta ‖+‖G{H}_v{\overline{H}}_{2v}^{-1}{\overline{P}}_{2v}\left({\rho }_1+{\rho }_2\right)‖+‖G{P}_v{\rho }_1‖\hfill \end{array}\right.$$

(43)

The active fault-tolerant control strategy flow of the DC microgrid proposed in this article is shown in Figure 2, which includes multidimensional fault decoupling, multidimensional fault sliding mode observer, and sliding mode controller. It is worth noting that the fault-tolerant control strategy proposed in this paper will not be activated when the microgrid is operating normally. The controller is reconfigured only when a power supply system fault is detected. The system configuration process is shown in Figure 3, which can improve the processing efficiency of the system. The fault-tolerant control process of DC microgrid described in this paper is shown in

Table 1. Algorithm steps.

Faults Reconstruction and Fault-Tolerant of DC Microgrid
(1)	Establish a secondary control Equation (8) for a DC microgrid with multiple actuator and sensor faults.
(2)	By using coordinate transformation (10), the output is transformed into a signal without sensor faults (11).
(3)	Completely decouple sensor and actuator faults (17) and (18) through coordinate transformation (12) and (13).
(4)	For the decoupled system, design two sliding mode observers (19) and (20) to reconstruct actuator faults (36) and sensor faults (39).
(5)	Design a sliding mode controller (42) using reconstructed values to offset the impact of faults on secondary control.

.

6. Experimental Results and Analysis

In this section, the effectiveness of the proposed multiple fault reconstruction and fault-tolerant control strategy for DC microgrids is verified through experiments on a semi-physical platform. The fault detection and control strategy are calculated in DSP, while the microgrid model is simulated in OPAL-RT OP4510. The voltage and current signals of the model are obtained through the OPAL-RT system terminals and converted into analog signals for transmission to the DSP controller. The DSP controller outputs the PWM signal to drive the switching tube in the power converter model. The relevant information is in the paper. This article considers an interconnected four DG microgrid system as shown in Figure 4, which also describes a communication network for data exchange. The rated voltage of the microgrid system is 100 V, and each DG is connected to the local load through output impedance, where only DG1 can obtain a voltage reference signal. To demonstrate the performance of this scheme, the voltage of each DG can be measured locally and transmitted back to the control center. At the same time, the control signal can be input to each DG. The main parameters of each DG, as well as the voltage loop, current loop, and filter parameters in the primary control, are shown in

Table 2. System parameters.

Parameter	Explanation	Value
$f_{sw}$	switching frequency	20 kHz
R1, R2, R3, R4	line resistance	0.1, 0.1, 0.15, 0.15 Ω
$C_f$	filter capacitor	2200 μF
$L_f$	filter inductance	20 mH
R	load	6 Ω/3 Ω
#1, #2, #3, #4	Boost converter	60 v/100 v

. The parameters of each DG controller are shown in

Table 3. Controller parameters.

Parameter	Explanation	Value
$v_{ref}$	reference voltage	100 V
k1, k2, k3, k4	droop coefficient	1, 1.5, 3, 4
$k_{pi},k_{ii}$	voltage loop	0.52, 33.8
$k_{pv},k_{iv}$	current loop	0.504, 4.16
G1, G2	observer parameter	[−1 1;0 0;−1 1], [1 −1]
χ1, χ2	observer parameter	[1 1]T, [1 1]T
$k,\epsilon$	controller parameter	4, 2

.

A consensus control between DGs can be obtained:

$$\left[\begin{array}{c}{\dot{v}}_1^d-{\dot{v}}_r^d\\ {\dot{v}}_2^d-{\dot{v}}_r^d\\ {\dot{v}}_3^d-{\dot{v}}_r^d\\ {\dot{v}}_4^d-{\dot{v}}_r^d\end{array}\right]\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cccc}2& 0& 0& -1\\ -1& 1& 0& 0\\ 0& -1& 1& 0\\ 0& 0& -1& 1\end{array}\right]\left[\begin{array}{c}{v}_1^d-v_r^d\\ v_2^d-v_r^d\\ v_3^d-v_r^d\\ v_4^d-v_r^d\end{array}\right]+\left[\begin{array}{cc}-1& 0\\ 0& 0\\ -1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{l}{f}_1^s\\ f_2^s\end{array}\right]+\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{f}_1^a\\ f_2^a\end{array}\right]+I_4\left[\begin{array}{c}{u}_{cv1}\\ u_{cv2}\\ u_{cv3}\\ u_{cv4}\end{array}\right]$$

(44)

$$\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\\ y_4\end{array}\right]\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]x+\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{l}{f}_1^s\\ f_2^s\end{array}\right]$$

(45)

Among them, actuator fault 1 simulates a bias signal, actuator fault 2 simulates a slowly changing signal, sensor fault 1 simulates a sudden change signal, and sensor fault 2 simulates a noise signal:

$$\begin{array}{l}{f}_1^a=8square\left(0.25\pi ,0.5\right),f_2^a=8sin\left(80t\right)+8sin\left(10t\right)\\ f_1^s=20square\left(0.1\pi ,0.02\right),f_2^s=wgn(1,1,0.2)\end{array}$$

(46)

Design matrix:

$$T\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 1\end{array}\right],\begin{array}{l}\hfill \end{array}\varsigma \begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cccc}1& 0& 1& 0\\ 0& -1& 0& 1\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\begin{array}{l}\hfill \end{array}S\begin{array}{l}\hfill \end{array}=\begin{array}{l}\hfill \end{array}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

(47)

This paper sets up four working scenarios: Scenario 1: Secondary Control, Scenario 2: Load Change, Scenario 3: DG Switching, Scenario 4: Fault Testing. To verify the effectiveness of the proposed method, five examples are presented. Example 1 verifies the effectiveness of the proposed secondary control strategy in different scenarios and without faults. Example 2 illustrates the performance of conventional secondary control under the influence of a fault. Example 3 verifies the effectiveness of the proposed fault-tolerant control strategy in different scenarios and fault conditions. In Example 4, the proposed method is compared with the fault-tolerant strategy based on direct observer compensation to illustrate the advanced nature of the proposed method. Example 5 verifies the effectiveness of the proposed observer for different locations and different types of fault reconstruction.

Example 1. 

Performance of the proposed secondary strategy without fault conditions. This example is simulated and tested in scenarios 1, 2, and 3 to verify the effectiveness of the secondary control strategy proposed in this paper in the absence of faults. The dynamic response of the system is shown in Figure 5. Initiate the proposed secondary control strategy at 0 s and set the droop coefficients of each DG to 1, 2, 3, and 4, respectively. When t ∈ [0.5, 1) s, the load decreases from 6 $\Omega$ to 3 $\Omega$, and DG4 is cut off at 3 s. When t ∈ [0, 0.5) and [1,1.5) s, each source current is 2.3 A, 3.2 A, 4.6 A, and 9.3 A, respectively. When t ∈ [0.5, 1)s, each source current is 4.6 A, 6.2 A, 9.3 A, and 18.6 A, respectively. When t = 1.5–2 s, each source current is 0 A, 3.3 A, 5.2 A, and 10.6 A, respectively. In all three scenarios, the actual values of each DG current can be accurately distributed according to the droop coefficient, and the actual values of the bus voltage can accurately track the rated value. According to the test results of this example, the control strategy proposed in this article can achieve the secondary control goal of the microgrid without faults. It not only has good robustness to load changes, but also has the characteristics of DG plug-and-play.

Example 2. 

Performance of the proposed secondary strategy under fault conditions. This example is simulated and tested in scenarios 1 and 4 to verify the impact of faults on the secondary control strategy proposed in this paper. The dynamic response of the system is shown in Figure 6. When t∈ [0.5, 1.5) s, actuator faults occurred in DG1 and DG2, and sensor faults occurred in DG3 and DG4. Due to the use of consensus-based secondary control and the fact that each DG did not initiate fault-tolerant control at this time, the current of each DG was not only affected by local faults, but also by faults generated by other DGs. At the same time, the maximum fluctuation of bus voltage is 16 v, reaching 16%, which also produces a large deviation. Therefore, the proposed secondary control strategy did not perform well under fault conditions.

Example 3. 

Performance of fault-tolerant strategies proposed under fault conditions. This example is simulated and tested in scenarios 1–4 to verify the effectiveness of the fault-tolerant control strategy proposed in this paper under fault conditions. The dynamic response of the system is shown in Figure 7 and Figure 8. At 0.5 s, DG1 and DG2 have actuator faults, and DG3 and DG4 have sensor faults. When t ∈ [1, 2) s, each DG initiates fault-tolerant control, and the maximum voltage fluctuation is 1.4 v, which is reduced by 14.6% compared with the non-starting time. The fluctuation of voltage and current ratio is very small, and all are within the allowable range. As shown in Figure 7, the proposed fault-tolerant control strategy still exhibits good robustness against load changes and DG switching events. Compared with the test results of Example 1, the control strategy proposed in this article can ensure secondary control performance and robustness to large disturbance events such as load changes and DG switching in the event of multiple faults occurring simultaneously in the actuator and sensor.

Example 4. 

Compare existing control strategies. In this example, the fault-tolerant strategy proposed in [24] that directly compensates with the observer is simulated and tested in four scenarios, and the dynamic response of the system is obtained, as shown in Figure 9. The same as Example 2, at 0.5 s, DG1 and DG2 had actuator faults, while DG3 and DG4 had sensor faults. When t ∈ [1, 2) s, each DG initiates fault-tolerant control. It can be seen that the fault-tolerant results have also achieved good results. However, since this strategy directly uses observer results to compensate, the observer error will have a significant impact on the control results. The maximum voltage fluctuation is 6.5 v, which is 5.1% higher than the proposed strategy.

Example 5. 

Observer fault signal waveform. This example verifies the observation ability of the control strategy proposed in this article on the waveform of sensor actuator fault signals. Obtain the multiple actuator and sensor faults signal waveforms observed by the proposed strategy in the experimental test results of examples 2 and 3 and compare the actual fault waveforms. The experimental results are shown in Figure 10, Figure 11, Figure 12 and Figure 13. The absolute error integrals of the four fault observations are 1.57, 1.03, 4.43, and 0.83, respectively. The results indicate that the control strategy proposed in this article can effectively estimate the fault signals of multiple sensors and actuators and is not affected by the fault type, which is conducive to the subsequent judgment of fault severity and the development of processing plans.

7. Conclusions

To address the issue of DC microgrid performance being compromised by faults at various positions, this paper proposes a secondary fault-tolerant control scheme based on sliding mode observer. Firstly, through the design of linear transformation and sliding mode observer, multiple faults decoupling and reconstruction for sensors and actuators are achieved. Subsequently, employing a sliding mode controller, the impact of faults at different positions on the DC microgrid is mitigated, restoring the voltage to its rated value and ensuring proportional current sharing among sources. Finally, the results show that the proposed scheme can effectively offset the influence of unknown sensor and actuator faults on the DC microgrid, where the bus voltage fluctuation is reduced by 14.6%. The absolute integral of error for different types of fault interference such as bias, gradient, mutation, and noise reconstruction is 1.57, 1.03, 4.43, and 0.83, respectively, indicating that the method has good adaptability to different faults and disturbances. In addition, communication fault is also a problem that needs further research.