Wavelet analysis and consensus algorithm-based fault-tolerant control for smart grids

In this paper, the voltage and frequency regulation problems are investigated for smart grids under the influence of faults. To solve those problem, a wavelet analysis and consensus algorithm-based fault-tolerant control scheme is proposed. Specifically, the wavelet analysis technique is introduced to determine whether there exist faults or not in the smart grids. Then, a distributed fault estimator is designed to estimate the attack signals. Based on this estimator state, a distributed fault-tolerant controller is designed to compensate for the faults. It is theoretically shown that the developed method can achieve the voltage regulation and frequency objectives. Finally, a smart grid with four distributed generations is constructed in MATLAB/Simulink for simulation to validate the effectiveness.

1 Introduction

During the past decades, microgrid (MG) has attracted much attention because of their reliability, efficiency, and so on. AC MG consists of various distributed generation (DG), including energy storage, wind farm, and photovoltaic based on the power inverter. To ensure the stability of the MG, it is imminent for the engineers and scientists to design some reasonable control strategies. Please refer to the classical results on AC MG for details, such as (1–3).

For the AC MG, the hierarchical control method is well-used, i.e., the control is divided into the primary control, the secondary control, and the tertiary control. Specifically, the objective of the primary control is to design the droop control method such that the output voltage and frequency of DG track the corresponding reference signal (4). However, if only the droop control is used, then the deviation of the voltage and frequency will happen. To solve the problem, the secondary controller is usually introduced, which is implemented by introducing a consensus-based algorithm to regulate the voltage and frequency of their reference values. Finally, the tertiary control is to achieve optimal scheduling and economic indicators in a larger time range. This paper mainly focuses on the secondary control layer.

In general, the design methods of the secondary controller can be divided into the centralized control method and the distributed control method. The centralized one is to design the control method in a centralized manner (5–7), i.e., the controller of each DG needs the information of all DGs, which leads to this controller containing some disadvantages such as high computational complexity, high communication cost, and so on. To overcome those drawbacks, distributed secondary control method is proposed (8–11), where the controller of each DG only needs the information from neighbor DGs, which contains the advantages of low communication cost and low computational complexity (12, 13).

Note that as the increase of the number of DGs in AC MG, system faults will inevitably occur, which may lead to the damage of the system performance (14). To ensure the reliability of the AC MG, some fault-tolerant secondary control methods have been developed. The existing fault-tolerant secondary control procedure can be divided into fault detection, fault estimation, and fault-tolerant, which is usually done in the above order. The wavelet analysis method as a fault detection method has been widely used to detect system faults. Based on the detection result, the fault estimation method and the fault-tolerant method will start work. In (16), a robust detector is designed to detect the consistency of measurements through the PMU measurements. In (17), a prototype method is developed to detect faults by using the candidate invariant to compare with the actual invariant. In (18), a vulnerability factor-based distributed fault detection method is proposed to detect the system faults. In addition, some results on fault-tolerant secondary control have been developed. In (19), a distributed fault-tolerant secondary controller is designed to achieve the frequency and voltage restoration by using the adaptive technique. In (20), a fault-tolerant cooperative secondary control scheme is developed for islanded MG with system faults by proposing a hidden layer.

Notice that the results mentioned above focus only on the fault detection or the fault-tolerant of the MGs separately. Besides, different from the faults on the inter-DG communication in MGs, the system faults on the secondary controller are usually more difficult to be detected and eliminated. Although some existing results have been developed to restore the voltage and frequency, the transient performance cannot be ensured and there are large fluctuations which are harmful to MG (22, 23). Based on the above observation, this paper considers the voltage and frequency regulation problems for AC MGs under the influence of faults. To solve this problem, the main method and the main contributions of this paper can be summarized as follows.

• To solve the reliability problem of AC MGs with system faults, a series of methods have been proposed, i.e., a wavelet analysis method is introduced to detect the system faults, the fault estimator is proposed to estimate the unknown faults, and the fault-tolerant controller is designed to eliminate the influence of faults.

• A k-step distributed fault estimator is proposed for each DG to estimate the fault signal. The main feature of this estimator is that with the increase of the number of k, the estimation error will be smaller and smaller until it converges to zero.

• Based on the developed k-step estimator, a distributed fault fault-tolerant secondary controller is designed to compensate for the influence of faults in AC MGs. Besides, the stability of the system is proved by the Lyapunov theorem.

The rest sections are organized as follows. In Section 2, the preliminaries are presented. In Section 3, the fault estimation design method is given. Then, a fault-tolerant controller is designed in Section 4. In Section 5, the simulation test is studied. Finally, the conclusion is given in Section 6.

2 Preliminaries

2.1 Graph theory

In this paper, the considered AC MGs is consisted of networks with DGs and leader nodes. The network topology is fixed and is defined by an undirected $\mathcal{G}=(\mathcal{W},\mathcal{E},\mathcal{A})$, where $\mathcal{W}=\{\mathrm{1,2},\dots ,N\}$ represents the set of nodes, $\mathcal{E}\subset \mathcal{W}\times \mathcal{W}$ means the set of edges, and $\mathcal{A}=\left[a_{ij}\right]$ denotes the adjacency matrix. If there exists communication from i to node j, then the edge weight a_ij = 1; otherwise, a_ij = a_ji = 0. The in-degree matrix and the Laplacian matrix are denoted by D = diag(d₁, …, d_N) and $\mathcal{L}=D-\mathcal{A}$, where $d_i={\sum }_{j=1}^N{a}_{ij}$.

Assumption 1. The undirected network topology $\mathcal{G}$ is connected.

2.2 System Model

In this paper, an AC MG is considered. It is assumed that the communication among DGs is ideal and the inductive distribution lines are considered. Besides, the active power and reactive power delivered by DG i at bus i satisfy

$\left\{\begin{array}{c}{P}_i=\frac{v_{\text{bus},i}{v}_{\text{mag},i}\sin \left({\theta }_i-{\beta }_i\right)}{Z_i},\hfill \\ Q_i=-\frac{v_{\text{bus},i}^2}{Z_i}\frac{v_{\text{bus},i}{v}_{\text{mag},i}\cos \left({\theta }_i-{\beta }_i\right)}{Z_i},\hfill \end{array}\right.(1)$

where P_i and Q_i represent the active power and reactive power, respectively. v_mag,i∠θ_i and v_bus,i∠β_i are the output voltage and bus voltage of DG i. Z_i represents the effective collective impedance of the output filter and the connector between distribution network and DG.

2.3 Control objective

The secondary control objective is to design secondary controllers of frequency and voltage regulation such that i) the frequency restores to its reference value ω_ref as frequency reference value; ii) the voltage regulates to $[v_{\mathit{ref}}^1,v_{\mathit{ref}}^2]$, where $v_r^1$ and $v_{\text{ref}}^2$ are the upper and lower bounds of voltage amplitude.

2.4 Cooperative control solution

Applying the droop control to (1), the following control scheme can be achieved

$\left\{\begin{array}{c}{\omega }_i={\omega }_{n_i}-n_{P_i}{P}_i,\hfill \\ v_{oi}=V_{n_i}-n_{Q_i}{Q}_i,\hfill \end{array}\right.(2)$

where ω_i represents the angular frequency. ${\omega }_{n_i}$ and $V_{n_i}$ represent the set points in the secondary controller. $n_{P_i}$ and $n_{Q_i}$ represent droop coefficients chosen according to the power rating of the corresponding DG(17). By using an appropriate reference coordinate transformation, it is reasonable to adjust v_mag,i by v_oi.

Driven by the consensus algorithm, the frequency error and voltage error among DGs are defined as follows.

$\begin{array}{ll}\hfill e_{\omega ,i}& =c_{\omega }\left(\sum _{j\in {\mathcal{N}}_i}\left({\omega }_j-{\omega }_i\right)+g_i\left({\omega }_r-{\omega }_i\right)\right.\hfill \\ \hfill & \left.+\sum _{j\in {\mathcal{N}}_i}\left(n_{P_j}{P}_j-n_{P_i}{P}_i\right)\right),\hfill \end{array}(3)$

$\begin{array}{ll}\hfill e_{v,i}& =c_v\left(\sum _{j\in {\mathcal{N}}_i}\left(v_{oj}-v_{oi}\right)+g_i\left(v_r-v_{oi}\right)\right.\hfill \\ \hfill & \left.+\sum _{j\in {\mathcal{N}}_i}\left(n_{Q_j}{Q}_j-n_{Q_i}{Q}_i\right)\right),\hfill \end{array}(4)$

where ω_i and v_oi are the frequencies and output voltages of the DGs i-th DG. The positive constants c_ω and c_v are the coupling gains. ω_r and v_r are the frequency and voltage references, respectively.

Together with (2–4), it is shown that

$e_{\omega ,i}=c_{\omega }\left(\sum _{j\in {\mathcal{N}}_i}\left({\omega }_{n_j}-{\omega }_{n_i}\right)+\sum _{\gamma =1}^2{g}_i\left({\omega }_{n,\mathrm{r}\mathrm{e}\mathrm{f}}-{\omega }_{n_i}\right)\right),(5)$

$e_{v,i}=c_v\left(\sum _{j\in {\mathcal{N}}_i}\left(V_{n_j}-V_{n_i}\right)+\sum _{\gamma =1}^2{g}_i\left(V_{n,\mathrm{r}\mathrm{e}\mathrm{f}}^{\gamma }-V_{n_i}\right)\right),(6)$

where ${\omega }_{n,\text{ref}}={\omega }_r+n_{P_i}{P}_i$ and $V_{n,\text{ref}}^{\gamma }=v_{\text{ref\hspace{0.17em}}}^{\gamma }+n_{Q_i}{Q}_i$¹.

To restore the frequency and voltage of all DGs to the reference values, ω_ni and V_ni are provided by the secondary controller, which are driven by the information from neighbors and updated according to the frequency and voltage error terms e_ω,iand e_v,i, respectively. From (2), we have

$\left\{\begin{array}{c}{\dot{\omega }}_{ni}={\dot{\omega }}_i+n_{P_i}{\dot{P}}_i=e_{\omega ,i},\hfill \\ {\dot{V}}_{ni}={\dot{v}}_{\text{oi}}+n_{Q_i}{\dot{Q}}_i=e_{v,i}.\hfill \end{array}\right.(7)$

Inspired by (21), we can design ${\omega }_{n_i}$ to stabilize ω_i and $n_{P_i}{P}_i$ and $V_{n_i}$ to stabilize v_oi and $n_{Q_i}{Q}_i$. Combining (7) with (5, 6), the following distributed secondary controller can be designed

${\dot{\omega }}_{n_i}=c_{\omega }\left(\sum _{j\in {\mathcal{N}}_i}\left({\omega }_{n_j}-{\omega }_{n_i}\right)+\sum _{\gamma =1}^2{g}_i\left({\omega }_{n,\mathrm{r}\mathrm{e}\mathrm{f}}-{\omega }_{n_i}\right)\right),(8)$

${\dot{V}}_{n_i}=c_v\left(\sum _{j\in {\mathcal{N}}_i}\left(V_{n_j}-V_{n_i}\right)+\sum _{\gamma =1}^2{g}_i\left(V_{n,\mathrm{r}\mathrm{e}\mathrm{f}}^{\gamma }-V_{n_i}\right)\right).(9)$

Lemma 1. Under Assumption 1, the designed distributed secondary controllers (8, 9) can regulate the frequency to its reference value and stabilize the voltage amplitude between the upper and lower bounds.

3 Fault estimator design

3.1 Fault Model and analysis

Usually, faults may occur with the increase of system operation time in the secondary controller. In this paper, the fault signal is denoted by f_i, and the following conditions are satisfied.

Assumption 2. The fault is changed slowly, i.e., the fault signal and its derivative are norm bounded as two positive scalars a and b (‖f‖ ≤ a and $\Vert \dot{f}\Vert \le b$).Under the influence of faults, the secondary controller of voltage regulation comes

$\begin{array}{ll}\hfill {\dot{V}}_i& =c_v\left(\sum _{j\in {\mathcal{N}}_i}\left(V_j-V_i\right)+\sum _{\gamma =1}^2{g}_i\left(V_r-V_i\right)\right)+f_i\hfill \\ \hfill & =c_v\left(\sum _{\gamma =1}^2{g}_i{V}_r-\left(\left(d_i+\sum _{\gamma =1}^2{g}_i\right)V_i-\sum _{j\in {\mathcal{N}}_i}{V}_j\right)\right)+f_i.\hfill \end{array}(10)$

Denote $V={\left[V_1,\dots ,V_N\right]}^T$, and $f={\left[f_1,\dots ,f_N\right]}^T$. Then, (10) can be rewritten as follows,

$\begin{array}{ll}\hfill \dot{V}& =c_v\left(G_1\left({\mathbf{1}}_N\otimes v_r\right)+G_2\left({\mathbf{1}}_N\otimes V_{\text{ref}}^2\right)\right.\hfill \\ \hfill & \left.-\left(L+G_1+G_2\right)V\right)+f.\hfill \end{array}(11)$

Define ${\mathrm{\Theta }}_{\gamma }=\frac{1}{2}L+G_{\gamma },\gamma =\mathrm{1,2}.$ Then, it has $L\left({\mathbf{1}}_N\otimes V_{\text{ref\hspace{0.17em}}}^{\gamma }\right)=(D-A)\left({\mathbf{1}}_N\otimes V_{\text{ref\hspace{0.17em}}}^{\gamma }\right)=0,\forall \gamma =\mathrm{1,2}$. Thus, (10) is rewritten as

$\begin{array}{ll}\hfill \dot{V}& =c_v\left(\left(\frac{1}{2}L+G_1\right)\left({\mathbf{1}}_N\otimes v_r\right)+\left(\frac{1}{2}L+G_2\right)\right.\hfill \\ \hfill & \left.\times \left({\mathbf{1}}_N\otimes V_{\text{ref}}^2\right)-\left(\frac{1}{2}L+G_1+\frac{1}{2}L+G_2\right)V\right)+f\hfill \\ \hfill & =c_v\left(\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\left({\mathbf{1}}_N\otimes V_r\right)-\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }V\right)+f.\hfill \end{array}(12)$

It can be seen that the normal secondary controller cannot achieve the voltage regulation objective under the influence of faults.

3.2 Fault estimators design

To estimate the voltage and fault of DG i, the following fault estimators can be designed.

$\left\{\begin{array}{cc}\hfill {\dot{\widehat{V}}}_{i,0}=& c_v\left(\sum _{j\in {\mathcal{N}}_i}\left({\widehat{V}}_{j,0}-{\widehat{V}}_{i,0}\right)+g_i\left(V_r-{\widehat{V}}_{i,0}\right)\right)+b_v{\tilde{V}}_{i,0}+{\widehat{f}}_{i,0},\hfill \\ \hfill {\dot{\widehat{f}}}_{i,0}=& -d_v\left(\sum _{j\in {\mathcal{N}}_i}\left({\tilde{V}}_{j,0}-{\tilde{V}}_{i,0}\right)-g_i{\tilde{V}}_{i,0}\right),\hfill \end{array}\right.(13)$

where the positive constants b_v and d_v are coupling gains. ${\widehat{V}}_{i,0}$ and ${\widehat{f}}_{i,0}$ are the initial estimator states of DG i and ${\tilde{V}}_{i0}=V_i-{\widehat{V}}_{i0}$ is the estimation error of V_i.

During the above estimator, the effect of fault-dependent term $\dot{f}$ is ignored, which may result in estimation bias. To eliminate this influence, the following k-step estimators are introduced.

$\left\{\begin{array}{cc}\hfill {\dot{\widehat{V}}}_{i,k}& =c_v\left(\sum _{j\in {\mathcal{N}}_i}\left({\widehat{V}}_{jk}-{\widehat{V}}_{i,k}\right)+g_i\left(V_r-{\widehat{V}}_{i,k}\right)\right)+b_v{\tilde{V}}_{i,k}+{\widehat{f}}_{i,k},\hfill \\ \hfill {\dot{\widehat{f}}}_{i,k}& =-d_v\left(\sum _{j\in {\mathcal{N}}_i}\left({\tilde{V}}_{jk}-{\tilde{V}}_{i,k}\right)-g_i{\tilde{V}}_{i,k}\right)+{\dot{\widehat{f}}}_{ik-1},\hfill \end{array}\right.(14)$

where ${\widehat{V}}_{i,k}$ and ${\widehat{f}}_{i,k}$ are the k-step estimator states of DG i. ${\tilde{V}}_{i,k}=V_i-{\widehat{V}}_{i,k}$ represents the estimation error.

Denote ${\widehat{V}}_0=\left[{\widehat{V}}_{\mathrm{1,0}},\dots ,{\widehat{V}}_{N,0}\right]$ and ${\widehat{f}}_0=\left[{\widehat{f}}_{\mathrm{1,0}},\dots ,{\widehat{f}}_{N,0}\right]$. The dynamics of (13) can be rewritten as,

$\left\{\begin{array}{cc}\hfill {\dot{\widehat{V}}}_0& =c_v\left(\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\left({\mathbf{1}}_N\otimes V_r\right)-\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\widehat{V}}_0\right)+b_v{\tilde{V}}_0+{\widehat{f}}_0,\hfill \\ \hfill {\dot{\widehat{f}}}_0& =d_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_0.\hfill \end{array}\right.(15)$

Define ${\widehat{V}}_k=\left[{\widehat{V}}_{1,k},\dots ,{\widehat{V}}_{N,k}\right]$ and ${\widehat{f}}_k=\left[{\widehat{f}}_{1,k},\dots ,{\widehat{f}}_{N,k}\right]$. The dynamic of (14) can be rewritten as

$\left\{\begin{array}{cc}\hfill {\dot{\widehat{V}}}_k=& c_v\left(\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\left({\mathbf{1}}_N\otimes V_r\right)-\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\widehat{V}}_k\right)+b_v{\tilde{V}}_k+{\widehat{f}}_k,\hfill \\ \hfill {\dot{\widehat{f}}}_k=& d_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_k+{\dot{\widehat{f}}}_{k-1}.\hfill \end{array}\right.(16)$

Define ${\tilde{f}}_0=f-{\widehat{f}}_0$. From (15), it yields

$\left\{\begin{array}{cc}\hfill {\dot{\stackrel{̃}{V}}}_0& =-\left(c_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }+b_v\right){\tilde{V}}_0+{\tilde{f}}_0,\hfill \\ \hfill {\dot{\stackrel{̃}{f}}}_0& =-d_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_0+\dot{f}.\hfill \end{array}\right.(17)$

Define ${\tilde{f}}_k=f-{\widehat{f}}_k$, for k = 1, 2, …. It follows from (16) that

$\left\{\begin{array}{cc}\hfill {\dot{\stackrel{̃}{V}}}_k& =-\left(c_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }+b_v\right){\tilde{V}}_k+{\tilde{f}}_k,\hfill \\ \hfill {\dot{\stackrel{̃}{f}}}_k& =-d_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_k+{\dot{\stackrel{̃}{f}}}_{k-1}.\hfill \end{array}\right.(18)$

From the above estimators, the following theorem can be achieved.

Theorem 1. Under Assumptions 1 and 2, the proposed iterative estimators (13, 14) ensure that the estimation errors converge to zero as the iteration step k tends to ∞, i.e.,

$\underset{k\to \infty }{\text{lim}}\left(f-\frac{1}{k+1}\sum _{s=0}^k{\widehat{f}}_s\right)=0.(19)$

Proof. For k = 0, 1, 2, …, it follows from the estimation errors dynamics (17, 18) that

$\frac{{\tilde{f}}_k\left(t\right)}{k+1}=-d_v{\int }_0^t\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\frac{{\sum }_{s=0}^k{\tilde{V}}_s\left(\tau \right)}{k+1}d\tau +\frac{{\int }_0^t\dot{f}\left(\tau \right)d\tau }{k+1}.(20)$

According to the fact that ${\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }$ is a positive definite matrix and c_v is a positive constant, it yields that the matrix $H=\left[\begin{array}{cc}\hfill -\left(C_v{\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }+b_v\right)\hfill & \hfill I_N\hfill \\ \hfill -d_v{\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\hfill & \hfill 0_N\hfill \end{array}\right]$ is Hurwitz. Since $\dot{f}$ is bounded, then $\tilde{f}$ and $\tilde{V}$ are bounded, which implies that

$\underset{k\to \infty }{\text{lim}}\frac{{\tilde{f}}_k\left(t\right)}{k+1}=0.(21)$

Under Assumption 2, it is shown that f is bounded, which yields

$\underset{k\to \infty }{\text{lim}}\frac{{\int }_0^t\dot{f}\left(\tau \right)d\tau }{k+1}=\underset{k\to \infty }{\text{lim}}\frac{f\left(t\right)-f\left(0\right)}{k+1}=0.(22)$

Together with (20–22), it yields

$\underset{k\to \infty }{\text{lim}}{\int }_0^t\frac{1}{k+1}\sum _{s=0}^k\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_s\left(\tau \right)d\tau =0.(23)$

Then, it is shown that

$\underset{k\to \infty }{\text{lim}}\frac{1}{k+1}\sum _{s=0}^k\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_s=0,(24)$

which implies that

$\underset{k\to \infty }{\text{lim}}\frac{1}{k+1}\sum _{s=0}^k\widetilde{V_s}=0,(25)$

during which we have used the fact that ${\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }$ is positive definite. From ${\tilde{V}}_s=V-{\widehat{V}}_s$, it follows that

$\underset{k\to \infty }{\text{lim}}\sum _{s=0}^k\frac{1}{k+1}\left(V-{\widehat{V}}_s\right)=0.(26)$

Differentiate the equation as

$\underset{k\to \infty }{\text{lim}}\sum _{i=1}^k\frac{1}{k+1}\left(\dot{V}-{\dot{\widehat{V}}}_s\right)=0.(27)$

According to (11, 15, 16), one has

$\underset{k\to \infty }{\text{lim}}\frac{1}{k+1}\sum _{s=0}^k\left(-d_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }{\tilde{V}}_s-b_v{\tilde{V}}_s-{\widehat{f}}_s\right)+f=0.(28)$

It further follows that

$\underset{k\to \infty }{\text{lim}}\frac{1}{k+1}\sum _{s=0}^k{\widehat{f}}_s=f.(29)$

$\underset{k\to \infty }{\text{lim}}\frac{1}{k+1}\sum _{s=0}^k{\tilde{f}}_s=0,(30)$

which implies that the mean value of ${\tilde{f}}_i$, i = 0, 1, 2, …, k, can converge to zero when k → ∞. □

4 Fault-tolerant controller design

In this section, a fault-tolerant controller is designed to recover the voltage to its reference value by compensating for the influence of unknown fault f_i. According to the conclusion in Theorem 1, we can define the estimation of the signal of fault as f_i,k, i.e.,

$f_{i,k}=\frac{1}{k+1}\sum _{s=0}^k{\widehat{f}}_{i,s}.(31)$

Based on f_i,k, the following fault-tolerant controller $V_i^d$ is designed,

${\dot{V}}_i^d={\dot{V}}_i-m_v{f}_{i,k}.(32)$

Define $V^d={\left[V_1^d,\dots ,V_N^d\right]}^T$. Based on (31), the following dynamic can be achieved

${\dot{V}}^d=\dot{V}-M_v{F}_k,(33)$

where $F_k={\left[f_{1k},\dots ,f_{Nk}\right]}^T$ and satisfies $F_k=\frac{1}{k+1}{\sum }_{s=0}^k{\widehat{f}}_s$ according to (30).

Theorem 2. If the condition in Theorem 1 is satisfied and the step k is large enough, then the controller designed based on iterative mean estimation information in (31) can ensure that the voltage restores to the normal value under the influence of faults.Proof. Denote $\bar{V}=V^d-{({\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma })}^{-1}{\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }({\mathbf{1}}_N\otimes V_r)$. Under Assumption 1, we know that the matrix ${\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }$ is non-singular and positive-definite. It follows from (32) that the derivative of $\bar{V}$ is

$\begin{array}{ll}\hfill \dot{\stackrel{̄}{V}}& =c_v\left(\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\left({\mathbf{1}}_N\otimes V_r\right)-\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }V\right)+f-F_k\hfill \\ \hfill & =-c_v\sum _{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }\bar{V}+f-F_k.\hfill \end{array}(34)$

Define $\bar{\mathrm{\Psi }}={\mathrm{\Psi }}^T+\mathrm{\Psi }$ and $\mathrm{\Psi }=-c_v{\sum }_{\gamma =1}^2{\mathrm{\Theta }}_{\gamma }$. Then, formula (33) can be rewritten as

$\dot{\stackrel{̄}{V}}=\mathrm{\Psi }\bar{V}+f-F_k.(35)$

Construct the Lyapunov function as $W={\bar{V}}^T\bar{V}$. The time derivative of W along (34) is

$\dot{W}={\bar{V}}^T\bar{\mathrm{\Psi }}\bar{V}+2{\bar{V}}^T\left(f-F_k\right).(36)$

Using Young’s inequality leads to

$\begin{array}{ll}\hfill 2{\bar{V}}^T\left(f-F_k\right)& \le {\bar{V}}^T\left(-\frac{1}{2}\bar{\mathrm{\Psi }}\right)\bar{V}+{\left(f-F_k\right)}^T\left(-2\bar{\mathrm{\Psi }}\right)\left(f-F_k\right)\hfill \\ \hfill & \le -\frac{1}{2}{\bar{V}}^T\bar{\mathrm{\Psi }}\bar{V}+2\Vert \bar{\mathrm{\Psi }}\Vert \cdot \Vert f-F_k{\Vert }^2.\hfill \end{array}(37)$

Substituting (36) into (35) yields

$\dot{W}\le \frac{1}{2}{\bar{V}}^T\bar{\mathrm{\Psi }}\bar{V}+2\Vert \bar{\mathrm{\Psi }}\Vert \cdot \Vert f-F_k{\Vert }^2.(38)$

From $\bar{\mathrm{\Psi }}<0$ and f − F_k → 0 as k → ∞, it shows that $\bar{V}\to 0$ as k → ∞. Thus, it is shown that the voltage belongs to the set $[v_{\mathit{ref}}^1,v_{\mathit{ref}}^2]$. □

Remark 1. Similarly, the developed can be also used to solve the frequency restore problem of MG under the influence of faults. Specifically, the frequency controller can be designed as follows,

$\left\{\begin{array}{cc}\hfill {\dot{\widehat{\omega }}}_{i,l}& =c_{\omega }\left(\sum _{j\in {\mathcal{N}}_i}\left({\widehat{\omega }}_{j,l}-{\widehat{\omega }}_{i,l}\right)+g_i\left({\omega }_r^1-{\widehat{\omega }}_{i,il}\right)+g_i^2\left({\omega }_r^2-{\widehat{\omega }}_{i,l}\right)\right)+b_i{\tilde{\omega }}_{i,l}+m_w{\widehat{f}}_{i,l},\hfill \\ \hfill {\dot{\widehat{f}}}_{i,l}& =-d_w\left(\sum _{j\in {\mathcal{N}}_i}\left({\tilde{\omega }}_{j,l}-{\tilde{\omega }}_{i,l}\right)-\sum _{\gamma =1}^2{g}_i{\tilde{\omega }}_{i,l}\right)+{\dot{\widehat{f}}}_{i,l-1},\hfill \end{array}\right.(39)$

where ${\dot{\widehat{f}}}_{i-1}=0$. The corresponding conclusions and stability analysis process are similar to that of voltage regulation and they are omitted here.

5 Simulation study

In this simulation, AC MG systems are considered, which consist of 4 DGs and 2 leaders as shown in Figure 1. The parameters of all DGs are shown in Table 1. To show the effectiveness of our developed fault-tolerant controller, the faults considered are displayed in Table 2, where f_v,i and f_w,i represent the attack signal on the voltage and frequency controller of DG i, respectively. Then, the following two cases are introduced to show the effectiveness of the developed method.

FIGURE 1

FIGURE 1. Simulation test system.

TABLE 1

TABLE 1. The AC MG parameters and the fault-tolerant controller parameters.

TABLE 2

TABLE 2. The considered faults.

Case I. (Voltage and frequency regulation with distributed secondary controllers (Zhang et al., 2021) and (Anand et al., 2012)) In this case, the secondary controllers are added at t = 2 s. For the secondary controllers (Zhang et al., 2021) and (Anand et al., 2012), the trajectories of voltage and frequency under faults are shown in Figure 2. The simulation result can be summarized from the following stages.

1) Stage 1 (0–2 s): Only primary control is actuated. The frequency ω_i is deviated from its reference value 50 Hz and the voltage V_i is also deviated from its reference range, i.e., [310,311].

2) Stage 2 (2–5 s): The secondary controllers (Zhang et al., 2021) and (Anand et al., 2012) are added at t = 2 s, which can restore the frequency ω_i to its reference value and can also control the voltage V_i to [310,311].

3) Stage 3 (5–9 s): The trajectories of frequency ω₁ deviate from its reference value and the trajectories of voltage V₁ cannot stay within its scope under faults signal f_v,1 and f_w,1 as shown in Table 2.

4) Stage 4 (9–12 s): The trajectories of frequency ω₁ and ω₂ deviate from its reference value and the trajectories of voltages V₁ and V₂ cannot stay within its scope with the addition of fault signals f_v,2 and f_w,2.

5) Stage 5 (12- s): All trajectories of frequency ω_i deviate from its reference value and the trajectories of voltage V_i cannot stay within its scope with the addition of fault signals f_v,2 and f_w,2.

FIGURE 2

FIGURE 2. Trajectories of voltage and frequency with the distributed secondary controller (8)–(9) under faults.

In summary, under the distributed secondary controllers (Zhang et al., 2021) and (Anand et al., 2012)), the frequencies and voltages cannot be adjusted to the proper values under the influence of faults.

Case II. (Voltage and frequency regulation using distributed fault-tolerant controllers (31) and (38))

First, the parameters in the fault-tolerant controllers (31) and (38) are displayed in Table 3. Based on the distributed fault-tolerant controllers (31) and (38), the trajectories of voltage and frequency under the influence of faults are shown in Figure 3. Similar to case I, the following stages are shown.

1) Stage 1 (0–5 s): The conclusions are same to those in Stage 1 of Case I.

2) Stage 2 (5–16 s): The distributed fault-tolerant controllers (31) and (38) ensure that the frequency voltage are restored to their range.

TABLE 3

TABLE 3. Parameters of the designed fault-tolerant controller.

FIGURE 3

FIGURE 3. Trajectories of voltage and frequency with the secondary fault-tolerant controller (31) and (38) under faults.

Under the influence of the considered faults, the developed fault-tolerant control scheme can achieve the frequency recovery and voltage regulation effectively.

6 Conclusion

The distributed secondary fault-tolerant control problems of voltage and frequency regulation have been studied for smart grids under faults. A wavelet analysis and consensus algorithm-based fault-tolerant control method has been developed to solve the considered problems. In particular, the wavelet analysis technique is firstly introduced to detect the considered faults. Then, distributed secondary fault-tolerant controllers are designed to compensate for the influence of faults by using a newly introduced distributed fault estimate method. It is shown that the designed scheme ensures that both frequency regulation and voltage recovery objectives can be achieved. Finally, AC MGs with four distributed generations are used to validate the effectiveness of the developed fault-tolerant schemes.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

All the paper has been written by YH.

Funding

The work was supported by the basic scientific research project of Department of Education in 2022 of Liaoning Provincial of China under Grant LJKMZ20221993.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

¹The details of (5) and (6) are given in the Appendix.

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