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Linear Extended State Observer-Based Distributed Secondary Control of DC Microgrids Under False Data Injection Attacks
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Yiwei FENG, Shuangshuang WANG
CPSS Transactions on Power Electronics and Applications | 2024, 9(3) : 283 - 291
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CPSS Transactions on Power Electronics and Applications | 2024, 9(3): 283-291
• Papers •
Linear Extended State Observer-Based Distributed Secondary Control of DC Microgrids Under False Data Injection Attacks
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Yiwei FENG, Shuangshuang WANG
Affiliations
  • Lanzhou University of Technology Electrical Engineering and Information Engineering Lanzhou 730050 China

    • Yiwei Feng, Professor in Lanzhou University of Technology(LUT), China

    Shuangshuang Wang received the B.E. degree from Panzhihua University, Sichuan, China, in 2022. She is currently pursuing a master's degree in control engineering at Lanzhou University of Technology in Gansu, China. Her main research interests are intelligent microgrids and network attacks.

Published: 2024-09-10 doi: 10.24295/CPSSTPEA.2024.00013
Outline
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To address the issue of unknown False Data Injection (FDI) attacks on controllers in DC microgrids, a distributed secondary controller for DC microgrids based on a linear extended state observer is presented to ensure that the system can achieve the control objectives of voltage regulation and current sharing even when subjected to FDI attacks. Firstly, the secondary control problem of multibus DC microgrids with FDI attacks is transformed into a firstorder MultiAgent System (MAS) faulttolerant consistency problem, and the impact of FDI attacks on the microgrid system is analyzed. Then, a Linear Extended State Observer (LESO) is employed to detect the estimated injected FDI attack signals. Furthermore, a faulttolerant secondary controller is designed to eliminate the adverse effects of FDI attacks on the system. The stability of the proposed method is verified using a Lyapunov function. Finally, the effectiveness and superiority of the proposed control method are experimentally demonstrated.

DC microgrid  /  distributed fault-tolerant secondary control  /  false data injection (FDI) attack  /  linear extended state observer (LESO)
Yiwei FENG, Shuangshuang WANG. Linear Extended State Observer-Based Distributed Secondary Control of DC Microgrids Under False Data Injection Attacks[J]. CPSS Transactions on Power Electronics and Applications, 2024 , 9 (3) : 283 -291 . DOI: 10.24295/CPSSTPEA.2024.00013
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I. INTRODUCTION
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MICROGRIDS have emerged as a promising technology for integrating electronic interface-based renewable and non-renewable Distributed Generators (DGs), Energy Storage Systems (ESSs), and various loads, evolving into cyber-physical systems that incorporate communication, control, sensing, and computing capabilities [1],[2]. In recent years, research on DC and AC microgrids has been in full swing. Compared to AC microgrids, DC microgrids offer the following advantages: straightforward converter control, low power losses, and distributed control can be implemented more directly and efficiently without considering reactive power and harmonics [3]. DC microgrids have attracted widespread attention from researchers in recent years and have witnessed rapid development [4].
In earlier studies, droop control was used to restore DC bus voltages and achieve current sharing, which is the basic goal of DC microgrids [5]-[7]. However, due to the presence of line impedance, droop control alone often fails to achieve accurate current sharing and may also cause voltage drift. Therefore, it is necessary to introduce secondary control to compensate for droop control. Secondary control is mainly applied to centralized or distributed control methods [8]-[10]. Centralized control methods require system information from all DGs to achieve these control objectives over a global communication network, which can lead to high communication burden and operational errors in case of communication breakdowns [11]. Distributed secondary control has been widely studied due to the fact that it does not require global information, and the use of sparse communication networks can reduce the communication burden and make the system more robust [12]. However, this distributed cooperative control approach relies on local sensing and networked control over sparse communication networks, thus making these micro-networks highly vulnerable to malicious attacks and infiltration [13]-[15].
According to the current research on cyber attacks, cyber attacks are mainly categorized into the following types: False Data Injection (FDI) Attacks, Denial of Service Attacks (DoS) and Replay Attacks [16]-[18]. Among these attacks, FDI attacks have the greatest impact on DC microgrid systems, which can lead to problems such as misjudgment and misoperation of the system, thus affecting the stability and safety of the DC microgrid. To address FDI attacks in DC microgrid systems, detecting estimating and isolating the attack signals is one of the main methods to mitigate the impact of FDI attacks on control systems. In [19], a trust-based distributed secondary control method is proposed to detect and isolate the error mesdroopes in the system using a dynamic trust estimation mechanism to minimize the impact of FDI attacks in the communication link on the microgrid system. Literature [20] designed an attack detection method based on uncoordinated elements and proposed an evaluation theory to distinguish network attacks from system failures. However, these methods to suppress FDI attack signals by detecting estimating and isolating the attack signals tend to disrupt the connectivity of the communication network and affect the consistency of the system, thus causing control bias [21]. The fault-tolerant control method, on the other hand, solves this problem. In recent years, many scholars have studied the FDI attack problem with fault-tolerant control methods. In [22], a novel distributed pulse adaptive fault-tolerant controller was constructed to address the challenges posed by spoofing attacks and actuator bias faults. In the literature [23], an adaptive fault-tolerant control scheme is proposed for a class of nonlinear strict feedback cyber-physical systems with spoofing attacks.
This paper presents a distributed fault-tolerant secondary control strategy for DC microgrids against FDI attacks. The main contributions of this paper are summarized as follows:
• A standard distributed secondary control method is reviewed, and then, by designing a new state variable, the secondary control problem of the multibus DC microgrid with attack signals is transformed into a first-order multi-intelligent body system (MAS) fault-tolerant consistency problem. The adverse effects of the attack signals on the microgrid system are also analyzed.
• A LESO is designed to detect various types of attack signals, state-space models are established, and the stability of the LESO method in DC microgrid systems is verified by pole configuration and frequency domain analysis.
• A fault-tolerant controller based on LESO is designed. The stability analysis of the global system is carried out, and the results show that the controller can enable the global system to achieve fault-tolerant consensus. Thus, it is theoretically demonstrated that the microgrid system can operate smoothly under the action of the designed controller.
The rest of the paper is structured as follows: Section II reviews distributed secondary control methods for DC microgrids and analyzes the impact of attack signals on the stability of microgrid systems. Section III presents a distributed fault-tolerant secondary control method based on LESO and proves its stability. Section IV verifies the effectiveness and superiority of the proposed control method in DC microgrids through simulation analysis of three cases. The experimental results are given in Section V. Finally, Section VI summarizes the paper.
II. Analysis of DC Microgrid System Based on FDI Attack
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To facilitate the illustration of the proposed control method, this section first reviews a standard distributed secondary control method and then analyzes the impact of FDI attacks on DC microgrids.
A. Graph Theory and Symbols
The communication network of a DC microgrid system can be represented by a graph $G =\left({W, E, A}\right)$ with a node set $W$, an edge set $E \subset W \times W$, and an adjacency matrix $\mathbf{A}= \left\lbrack {a}_{ij}\right\rbrack$. The graph edges represent the information flow from converter $j$ to converter $i$, denoted by $\left({{w}_{j},{w}_{i}}\right)$ and weighted by ${a}_{ij}$. If $\left({{w}_{j},{w}_{i}}\right)$ $\in E$, then ${a}_{ij}= 1$ ; otherwise, ${a}_{ij}= 0$. If $\left({{w}_{j},{w}_{i}}\right)\in E$, then node $j$ is considered to be a neighbor of node $i$. The set of neighbors of node $i$ is denoted by ${N}_{i}= \left\{{j \mid \left\lbrack {\left({{w}_{j},{w}_{i}}\right)\in E}\right\rbrack }\right\}.\mathbf{D}= \operatorname{diag}\left({d}_{i}\right)\in$ ${R}^{N \times N}$, with ${d}_{i}= \mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}$, is called the in-degree matrix. L=D-A denotes the Laplace matrix. G is assumed to be bidirectional, i.e., ${a}_{ij}= {a}_{ji}.\mathbf{L}$ is therefore symmetric. ${f}_{i}$ is the pinning gain of the link from the pilot to the converter $i.{f}_{i}= 1$ if the pilot is connected to the $i$ -converter; otherwise, ${f}_{i}= 0.\mathbf{F}= \operatorname{diag}\left({f}_{i}\right),\forall i =$ $1,\cdots, N$, denotes a diagonal matrix consisting of pinning gains.
B. Secondary Control Method and System Analysis Under FDI Attack
The presented secondary control is based on a multi-bus DC microgrid system. Fig. 1 shows the overall cooperative control scheme of a DG unit in an isolated DC microgrid. In the primary control of isolated DC microgrids, droop control is usually used to realize current sharing [24]. A local reference voltage ${V}_{i}^{* }$ is generated and expressed as:
${V}_{i}^{* }= {V}_{\text{ref }}- {d}_{i}{I}_{\mathrm{o}}^{i}$
where ${V}_{\text{ref }}$ is the output voltage reference, ${d}_{i}$ is the $i$ -th droop factor, and ${I}_{\mathrm{o}}^{i}$ is the output current of the $i$ -th converter.
In order to compensate for the voltage deviation of the droop control, it is necessary to embed the secondary control signal in (1):
${V}_{i}^{* }= {V}_{\text{ref }}- {d}_{i}{I}_{\mathrm{o}}^{i}+ {u}_{i}$
where ${u}_{i}$ is the secondary control signal of the $i$ -th converter, the time derivative of (2) is:
${\dot{u}}_{i}= {\dot{V}}_{i}^{* }+ {d}_{i}{\dot{I}}_{0}^{i}= {e}_{i}$
which ${e}_{i}$ is the control input. The secondary control of DC microgrids is then transformed to a consensus problem for firstorder linear MAS. To achieve voltage regulation and accurate current sharing, the secondary control based on neighboring converter information is:
${e}_{i}= {b}_{i}\left\lbrack {{f}_{i}\left({{V}_{\text{ref }}- {\bar{V}}_{i}}\right)+ \mathop{\sum }\limits_{{i \in {N}_{c}}}{a}_{ij}\left({{d}_{j}{I}_{\mathrm{o}}^{j}- {d}_{i}{I}_{\mathrm{o}}^{i}}\right)}\right\rbrack $
where ${b}_{i}$ is the proportional gain. ${f}_{i}$ is the $i$ -th pinning gain. ${\bar{V}}_{i}$ is the global average voltage estimate for the i-th converter that satisfies the following conditions:
${\dot{\bar{V}}}_{i}= {\dot{V}}_{i}+ {b}_{i}\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{\bar{V}}_{j}- {\bar{V}}_{i}}\right)$
where ${V}_{i}$ is the local output voltage.
As shown in Fig. 1, an unknown attack signal is injected to corrupt the control inputs. The dynamics of the attacked DC microgrid system can be given according to the following equation:
${\dot{u}}_{i}= {e}_{i}^{\prime }= {e}_{i}+ {\chi }_{i}$
One of the control inputs that ${e}_{i}^{\prime }$ is damaged, ${\chi }_{i}$ is the attack signal, which can be modeled as a sum of finite step, ramp, or sinusoidal signals [25].
Assumption 1: For each converter, ${\dot{\chi }}_{i}$ is bounded.
Assumption 2: The attack injections considered in this paper satisfy: ${\chi }_{i}\leq {\bar{\chi }}_{i}$, where ${\bar{\chi }}_{i}$ is a positive constant.
The injected attack signals may destabilize the distributed system and thus damage the microgrid system. The effect of the attack signal on the system is analyzed theoretically below.
The local voltage controller regulates the output voltage of the converter to the reference voltage generated by the droop control with a voltage tracking error of ${\varphi }_{i}= {V}_{i}^{* }- {V}_{i}$, then ${V}_{i}^{* }= {V}_{i}+ {\varphi }_{i}$, the time derivative of ${V}_{i}^{* }$ can be derived as:
${\dot{V}}_{i}^{* }= {\dot{V}}_{i}+ {\dot{\varphi }}_{i}$
Substituting (6) and (7) into (3) yields:
${\dot{V}}_{i}+ {d}_{i}{\dot{I}}_{o}^{i}+ {\dot{\varphi }}_{i}= {e}_{i}+ {\chi }_{i}$
From (3)-(5) and (8), we can get:
$\begin{aligned}\dot{\bar{V}}_{i}+d_{i} \dot{I}_{\mathrm{o}}^{i}= & b_{i} \sum_{j \in N_{i}} a_{i j}\left[\left(\bar{V}_{j}+d_{j} I_{\mathrm{o}}^{j}\right)-\left(\bar{V}_{i}+d_{i} I_{\mathrm{o}}^{i}\right)\right]+ \\& b_{i} f_{i}\left[d_{i} \dot{I}_{\mathrm{o}}^{i}-\left(\bar{V}_{i}+d_{i} I_{\mathrm{o}}^{i}\right)\right]-\dot{\varphi}_{i}+\chi_{i}\end{aligned}$
${d}_{i}{I}_{\mathrm{o}}^{i}$ converges to a constant ${I}_{\mathrm{{ss}}}$ in the steady state. Let ${X}_{i}=$ ${\bar{V}}_{i}+ {d}_{i}{I}_{\mathrm{o}}^{i},{X}_{\text{ref }}= {I}_{\mathrm{{ss}}}$, then (10) can be rewritten as:
${\dot{X}}_{i}= {b}_{i}\left\lbrack {\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{X}_{j}- {X}_{i}}\right)+ {f}_{i}\left({{X}_{\text{ref }}- {X}_{i}}\right)}\right\rbrack -{\dot{\varphi }}_{i}+ {\chi }_{i}$
Remark 1: In this paper, we transform the distributed secondary control problem of a DC microgrid into a first-order MAS consistency problem. This is because in the secondary control, each DG unit needs to exchange its respective voltage and current information, of which ${\bar{V}}_{i}$ is easy to observe and ${d}_{i}{I}_{\mathrm{o}}^{i}$ is hard to observe. In this paper, we design a new state variable ${X}_{i}$ that converts the exchanged information ( ${\bar{V}}_{i}$ and ${d}_{i}{I}_{\mathrm{o}}^{i}$) into ${\bar{V}}_{i}$ and ${X}_{i}$, which allows us to derive the multi-bus DC microgrid model with FDI attack into a first-order MAS model. The MAS model is then used to analyze and design the controller.
Defining the state variable error as ${\varepsilon }_{i}= {X}_{i}- {X}_{\text{ref }}$, the time derivative of ${\varepsilon }_{i}$ is:
${\dot{\varepsilon }}_{i}= -{b}_{i}\left\lbrack {\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{\varepsilon }_{i}- {\varepsilon }_{j}}\right)- {f}_{i}{\varepsilon }_{i}}\right\rbrack -{\dot{\varphi }}_{i}+ {\chi }_{i}$
Let $\varepsilon ={\left\lbrack {\varepsilon }_{1},{\varepsilon }_{2},\ldots,{\varepsilon }_{n}\right\rbrack }^{\mathrm{T}}$, then the global form of (11) can be expressed as:
$\dot{\varepsilon }= -\mathbf{B}\left({\mathbf{L}+ \mathbf{F}}\right)\varepsilon -\mathbf{\psi }+ \mathbf{X}$
where $\mathbf{B}= \operatorname{diag}\left({b}_{i}\right),\mathbf{\psi }= {\left\lbrack {\dot{\varphi }}_{1},{\dot{\varphi }}_{2},\cdots,{\dot{\varphi }}_{n}\right\rbrack }^{\mathrm{T}},\mathbf{X}= {\left\lbrack {\chi }_{1},{\chi }_{2},\cdots,{\chi }_{n}\right\rbrack }^{\mathrm{T}}$, $\mathbf{F}= \operatorname{diag}\left({f}_{i}\right)$. The dynamic expression can be derived:
$\varepsilon \left( t\right)= \exp \left\lbrack {-\mathbf{B}\left({\mathbf{L}+ \mathbf{F}}\right) t}\right\rbrack \cdot \varepsilon \left({t}_{0}\right)+ \\{\int }_{{t}_{0}}^{t}\exp \left\lbrack {-\mathbf{B}\left({\mathbf{L}+ \mathbf{F}}\right)\left({t -\tau }\right)}\right\rbrack \cdot \left\lbrack {\mathbf{X}\left(\tau \right)- \mathbf{\psi }\left(\tau \right)}\right\rbrack \mathrm{d}\tau $
Without loss of generality, the attack signal is assumed to be positive, $\chi \left(\tau \right)> {\chi }_{0}> 0$. It is assumed that the system can converge to the reference voltage, i.e., $\mathop{\lim }\limits_{{t \rightarrow \infty }}\varphi \left( t\right)= 0$, by means of a local voltage controller. Moreover, since $\mathbf{B}(\mathbf{L}+$ $\mathbf{F}$) is a positive definite invertible matrix, $\exp \lbrack -\mathbf{B}\left({\mathbf{L}+ \mathbf{F}}\right)$ $t\rbrack \cdot \varepsilon \left({t}_{0}\right)$ can converge to zero. Therefore, it is possible to obtain:
$\begin{aligned}\lim _{t \rightarrow \infty} \varepsilon(t)= & \lim _{t \rightarrow \infty} \int_{t_{0}}^{t} \exp [-\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F})(t-\tau)] \cdot \boldsymbol{X}(\tau) \mathrm{d} \tau> \\& \exp [-\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F}) t] \cdot\left\{\exp [\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F}) t]-\exp \left[\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F}) t_{0}\right]\right\} \cdot \\& {[\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F})]^{-1} \chi_{0}=[\boldsymbol{B}(\boldsymbol{L}+\boldsymbol{F})]^{-1} \chi_{0} \geqslant 0 }\end{aligned}$
From (14), the state error of the system fails to converge to zero under the influence of the FDI attack signal, i.e., it fails to achieve the desired goal of voltage regulation and current sharing.
III. A Distributed Fault-Tolerant Control Method BASED ON LESO IS GIVEN
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This section details the given distributed fault-tolerant secondary control method based on LESO. Based on the system analysis, this paper transforms the control problem of DC microgrids into a first-order MAS fault-tolerant consistency problem. Firstly, LESO is designed to detect the estimated FDI attack signals. Then, a distributed fault-tolerant controller based on LESO is given to compensate and suppress the FDI attack. Finally, the stability of the designed LESO and fault-tolerant controller is verified.
A. Linear Extended State Observer
In order to detect and estimate FDI attack signals, the model of LESO is given in this part, as shown in Fig. 2.
From (6), considering ${u}_{i},\chi$ as system state variables, the corresponding state space equation can be constructed as [26]:
$\left\{\begin{array}{l}{\dot{x}}_{1}= {x}_{2}+ {b}_{\mathrm{o}}u \\{\dot{x}}_{2}= {x}_{3}\\{\dot{x}}_{3}= h \\ y ={x}_{4}\end{array}\right.$
where ${x}_{1}= {u}_{i},{x}_{2}= \chi,{b}_{\mathrm{o}}= 1$, and $e$ is the control input $u$. Extending the derivative term of the attack as a new state variable, that is $\dot{\chi }= {x}_{3}$, to increase bandwidth and reduce the number of communications. Then (15) is written in matrix form as:
$\left\{\begin{array}{l}\dot{x}= \mathbf{A}x +\mathbf{B}u +\mathbf{E}h \\ y =\mathbf{C}x \end{array}\right.$
where $A=\left[\begin{array}{lll}0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{array}\right], B=\left[\begin{array}{l}1 \\0 \\0\end{array}\right], E=\left[\begin{array}{l}0 \\0 \\1\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 0\end{array}\right]$.
By the introduction of the extended state, the original system is changed from the original second-order system to the thirdorder system, then the design of the extended state observer can be expressed as:
$\left\{\begin{array}{l}\dot{z}= \mathbf{A}z +\mathbf{B}u -\mathbf{L}\left({{z}_{1}- y}\right)\\ y =\mathbf{C}z \end{array}\right.$
where the observation matrix $\mathbf{L}= {\left\lbrack {\beta }_{1}{\beta }_{2}{\beta }_{3}\right\rbrack }^{\mathrm{T}}$, then the error matrix relationship can be expressed as:
$\dot{e}= {A}_{e}e -{Eh}$
where ${\mathbf{A}}_{e}= \mathbf{A}- \mathbf{{LC}}= \left\lbrack \begin{array}{lll}- {\beta }_{1}& 1 & 0 \\- {\beta }_{2}& 0 & 1 \\- {\beta }_{3}& 0 & 0 \end{array}\right\rbrack$.
The system is stable if the chosen roots ${A}_{\mathrm{e}}$ are in the left half plane. To make the design process easy to implement, assuming that the observer poles are all located at $-{\omega }_{\mathrm{o}}$, the characteristic polynomial can be designed as:
$ u\left( s\right)= \left|{{sI}- {A}_{e}}\right|= {s}^{3}+ {\beta }_{1}{s}^{2}+ {\beta }_{2}s +{\beta }_{3}= {\left( s +{\omega }_{o}\right)}^{3}$
where ${\omega }_{\mathrm{o}}$ is the bandwidth of the LESO. ${\beta }_{1}= 3{\omega }_{\mathrm{o}},{\beta }_{2}= 3{\omega }_{\mathrm{o}}^{2},{\beta }_{3}$ $={\omega }_{0}^{3}$.
B. Designed A Distributed Fault-Tolerant Controller
This section details the designed distributed fault-tolerant secondary controller and verifies the stability of the control system. Based on the derived MAS and the observed FDI attack signals, the distributed fault-tolerant secondary controller is designed to eliminate the adverse effects of the attack signals on the system.
For the sake of convenience, we denote:
${\delta }_{i}= {b}_{i}\left({\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{X}_{j}- {X}_{i}}\right)+ {f}_{i}\left({{X}_{\text{ref }}- {X}_{i}}\right)}\right)$
To ensure voltage regulation and proportional load sharing under FDI attacks, we design the following distributed faulttolerant secondary controller [27]:
${e}_{i}= {b}_{i}\left\lbrack {{f}_{i}\left({{V}_{\text{ref }}- {\bar{V}}_{i}}\right)+ \mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{d}_{j}{I}_{\mathrm{o}}^{j}- {d}_{i}{I}_{\mathrm{o}}^{i}}\right)}\right\rbrack +{\Gamma }_{i}$
${\Gamma }_{i}= \frac{{\delta }_{i}{\phi }_{i}}{\left|{\delta }_{i}\right|+ {\mathrm{e}}^{-{\alpha }_{i}t}}$
${\dot{\phi }}_{i}= {u}_{i}\left|{\delta }_{i}\right|$
where ${\Gamma }_{i}$ is the fault-tolerance compensation term, ${\phi }_{i}$ is the fault-tolerance parameter, ${\alpha }_{i}$ and ${\mu }_{i}$ are positive constants. For convenience, set ${\mu }_{i}\geq 1$. The consistent continuous function ${\mathrm{e}}^{-{\alpha }_{i}t}$ provides a smooth control method for practical implementation. Next, the stability of the designed controller is analyzed.
The time derivative of (20) can be obtained using (5),(6), and (21).
${\dot{\delta }}_{i}= {b}_{i}\left\lbrack {\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{\dot{X}}_{j}- {\dot{X}}_{i}}\right)+ {f}_{i}\left({{\dot{X}}_{\text{ref }}- {\dot{X}}_{i}}\right)}\right\rbrack \\= -{b}_{i}\left({{d}_{i}+ {f}_{i}}\right){\dot{X}}_{i}+ {b}_{i}\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}{\dot{X}}_{j}\\= -{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left({{\delta }_{i}+ {\chi }_{i}+ {\Gamma }_{i}}\right)+ {b}_{i}\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{\delta }_{i}+ {\chi }_{i}+ {\Gamma }_{i}}\right)$
Denote ${\Upsilon }_{i}= {\chi }_{i}- \frac{1}{\left({d}_{i}+ {f}_{i}\right)}\mathop{\sum }\limits_{{j \in {N}_{i}}}{a}_{ij}\left({{\delta }_{j}+ {\chi }_{j}+ {\Gamma }_{j}}\right)$, then
(24) can be rewritten as:
${\dot{\delta }}_{i}= -{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left({{\delta }_{i}+ {\chi }_{i}+ {\Gamma }_{i}}\right)$
Consider the following Lyapunov candidate function:
${V}_{i}= \frac{1}{2}{\left(\left|{\delta }_{i}\right|+ \frac{\mathrm{d}\left|{\Upsilon }_{i}\right|}{\mathrm{d}t}\right)}^{2}$
Its time derivative is given as:
${\dot{V}}_{i}= \left({\left|{\delta }_{i}\right|- \frac{\mathrm{d}\left|{\Upsilon }_{i}\right|}{\mathrm{d}t}}\right)\left({\frac{\mathrm{d}\left|{\delta }_{i}\right|}{\mathrm{d}t}- \frac{{\mathrm{d}}^{2}\left|{\Upsilon }_{i}\right|}{\mathrm{d}{t}^{2}}}\right)$
where $\frac{\mathrm{d}\left|{Y}_{i}\right|}{\mathrm{d}t}= \frac{{Y}_{i}{\dot{Y}}_{i}}{\left|{Y}_{i}\right|} \leq \left|{\dot{Y}}_{i}\right|$, note that
$\frac{\mathrm{d}\left|{\delta }_{i}\right|}{\mathrm{d}t}= \frac{{\delta }_{i}{\dot{\delta }}_{i}}{\left|{\delta }_{i}\right|} =\frac{-{b}_{i}\left({{d}_{i}+ {f}_{i}}\right){\delta }_{i}\left({{\delta }_{i}+ {\chi }_{i}+ {\Gamma }_{i}}\right)}{\left|{\delta }_{i}\right|} $
$= -{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left({\left|{\delta }_{i}\right|+ \frac{{\delta }_{i}{Y}_{i}}{\left|{\delta }_{i}\right|} +\frac{{\delta }_{i}{\Gamma }_{i}}{\left|{\delta }_{i}\right|}}\right)$
Substituting (28) into (27) yields:
${\dot{V}}_{i}= \left({\left|{\delta }_{i}\right|- \frac{\mathrm{d}\left|{\Upsilon }_{i}\right|}{\mathrm{d}t}}\right)\times \\\left\lbrack {-{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left|{\delta }_{i}\right|- \frac{{\mathrm{d}}^{2}\left|{Y}_{i}\right|}{\mathrm{d}{t}^{2}}- {b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left({\frac{{\delta }_{i}{Y}_{i}}{\left|{\delta }_{i}\right|} +\frac{{\delta }_{i}{\Gamma }_{i}}{\left|{\delta }_{i}\right|}}\right)}\right\rbrack $
Use (22) to get:
$\begin{array}{l}-b_{i}\left(d_{i}+f_{i}\right)\left(\frac{\delta_{i} \Upsilon_{i}}{\left|\delta_{i}\right|}+\frac{\delta_{i} \Gamma_{i}}{\left|\delta_{i}\right|}\right)= \\-b_{i}\left(d_{i}+f_{i}\right) \frac{\delta_{i} \Upsilon_{i}}{\left|\delta_{i}\right|}-b_{i}\left(d_{i}+f_{i}\right) \frac{\left|\delta_{i}\right| \phi_{i}}{\left|\delta_{i}\right|+\mathrm{e}^{-\alpha, t}} \leqslant \\b_{i}\left(d_{i}+f_{i}\right)\left|Y_{i}\right|-b_{i}\left(d_{i}+f_{i}\right) \frac{\left|\delta_{i}\right| \phi_{i}}{\left|\delta_{i}\right|+\mathrm{e}^{-\alpha, t}} \leqslant \\b_{i}\left(d_{i}+f_{i}\right) \frac{\left|Y_{i}\right|\left|\delta_{i}\right|+\mathrm{e}^{-\alpha, t}\left|Y_{i}\right|-\left|\delta_{i}\right| \phi_{i}}{\left|\delta_{i}\right|+\mathrm{e}^{-\alpha, t}}\end{array}$
Thus, $\exists {\tau }_{1}> 0$, that makes for all $t >{\tau }_{1}$, we have:
$\left|{Y}_{i}\right|\left|{\delta }_{i}\right|+ {\mathrm{e}}^{-{\alpha }_{i}}\left|{Y}_{i}\right|- \left|{\delta }_{i}\right|{\phi }_{i}\leq 0 $
Then,(30) and (31) are utilized to get:
$- {b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left({\frac{{\delta }_{i}{\Upsilon }_{i}}{\left|{\delta }_{i}\right|} +\frac{{\delta }_{i}{\Gamma }_{i}}{\left|{\delta }_{i}\right|}}\right)< 0, t >{\tau }_{1}$
Combining (29) and (32), the selection
$\left|{\delta }_{i}\right|\geq -\frac{1}{{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)}\frac{{\mathrm{d}}^{2}\left|{Y}_{i}\right|}{\mathrm{d}{t}^{2}}\text{ yields:} \\{\dot{V}}_{i}\leq \left({\left|{\delta }_{i}\right|- \frac{\mathrm{d}\left|{\Upsilon }_{i}\right|}{\mathrm{d}t}}\right)\left\lbrack {-{b}_{i}\left({{d}_{i}+ {f}_{i}}\right)\left|{\delta }_{i}\right|- \frac{{\mathrm{d}}^{2}\left|{\Upsilon }_{i}\right|}{\mathrm{d}{t}^{2}}}\right\rbrack \\\leq 0,\forall t >{\tau }_{1}$
${\dot{V}}_{i}= 0$ if and only if $\left|{\delta }_{i}\right|= 0$. Therefore, the designed fault-tolerant controller can ensure the fault-tolerant consistency of the MAS, i.e., the DC microgrid system under FDI attack can maintain stable operation.
IV. Simulation Results
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In order to verify the effectiveness of the given fault-tolerant control method, an islanded multi-bus DC microgrid system with three DG units is constructed using MATLAB/Simulink as shown in Fig. 3. The parameters of the DGs and controllers are shown in Table I. The attack signals of each DG in different cases are shown in Table II [25], where tattack is the time interval of FDI attack.
A. Case 1: Performance Comparison Under FDI Attack
In this section, the effectiveness of the given controller under network attack is simulated. The simulation process takes $1\mathrm{\;s}$. A step attack signal ${\chi }_{1}= {10}$ is injected into DG1 at $t ={0.2}\mathrm{\;s}$ and a sinusoidal attack signal ${\chi }_{2}= {10}\sin \left({100\pi t}\right)$ is injected into DG3 at $t ={0.4}\mathrm{\;s}$. In this experiment, the given control strategies are compared with and without fault-tolerant controllers. The experimental results are shown in Figs. 4-6.
As shown in Fig. 4, injecting the attack signal into the control input channel at ${0.2}\mathrm{\;s}$ and ${0.4}\mathrm{\;s}$, the DC microgrid system undergoes significant fluctuations. However, after adopting the fault-tolerant control method, as shown in Fig. 5, the system quickly converges to a steady-state value. Moreover, the output voltage of each converter is maintained at ${48}\mathrm{\;V}$ in the presence of the attack signal and the output current is accurately shared. The observed signals of the attack as well as the observation errors are shown in Fig. 6. It can be seen that the given LESO can track the attack signal quickly.
Therefore, the given fault tolerant control method is effective for general attack signals and ensures voltage regulation and current sharing.
B. Case 2: Communication Interruption
This section simulates and analyzes the performance of the designed fault-tolerant controller under communication interruption. In order to better analyze the performance of the fault-tolerant controller in the case of communication interruption, a sinusoidal attack signal ${\chi }_{1}= {20}\sin \left({200\pi t}\right)+ {20}$ is injected into DG1 at $t ={0.2}\mathrm{\;s}$ and a ramp attack signal ${\gamma }_{2}= -{20t}$ is injected into DG2 at $t ={0.4}\mathrm{\;s}$. At $t ={0.7}\mathrm{\;s}$, the communication between DG2 and DG3 is interrupted. The experimental results are shown in Figs. 7-9.
As can be seen in Figs. 7 and 8, the system stabilizes under the fault-tolerant controller at ${0.2}\mathrm{\;s}$ and ${0.4}\mathrm{\;s}$ injection attacks. Under the effect of ${0.7}\mathrm{\;s}$ communication interruption, both Fig. 7 and Fig. 8 have short fluctuations, indicating that the DC microgrid system without fault-tolerant controllers is inherently robust. In addition, it can be seen from Fig. 9 that the attack signal is successfully tracked.
C. Case 3: Load Variation
In this section, the system response of the given fault-tolerant secondary controller under load variation and network attack is simulated and analyzed. A step attack signal ${\chi }_{1}= 6$ is injected into DG2 at $t ={0.2}\mathrm{\;s}$ and a sinusoidal attack signal ${\chi }_{2}= {30}\sin \left({{300\pi t}+ \pi }\right)+ {30}$ is injected into DG3 at $t ={0.4}\mathrm{\;s}$. The load resistance of DG3 is reduced to ${2.5\Omega }$ at $t ={0.7}\mathrm{\;s}$. The experimental results are shown in Figs. 10-12.
As can be observed from Fig. 10(a) and Fig. 11(a), it can be concluded that the system output voltage stabilizes under the fault-tolerant controller at ${0.2}\mathrm{\;s}$ and ${0.4}\mathrm{\;s}$ injection attacks; from Fig. 10(b) and Fig. 11(b), there is a sudden change in the output current of DG3 due to the load change. Under the effect of ${0.7}\mathrm{\;s}$ communication interruption, there is a short fluctuation in both Fig. 7 and Fig. 8, which also illustrates that the DC microgrid system without fault-tolerant controller is inherently robust. In addition, it can be seen from Fig. 12 that the tracking signal of the attack is good.
V. Experimental Results
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The performance of the proposed fault-tolerant control scheme has been experimentally verified on a microgrid semiphysical platform consisting of 3 DGs and the YXSPACE2000 control platform, as shown in Fig. 13. The nominal voltage for the DC microgrid is ${48}\mathrm{\;V}$. Besides, the line impedance of DGs are ${r}_{ij}= {5\Omega }$. The control loop uses the scheme shown in Fig. 1 to regulate the voltage and current of each DG.
For convenience, all the results described above are represented in a single experiment via a microgrid semi-physical platform. In Figs. 14 and 15, the communication between DG2 and DG3 is interrupted at 0.5s, at this time, the voltage and current fluctuate slightly. The load resistance of DG3 is reduced to ${2.5\Omega }$ at $t = 1\mathrm{\;s}$, there is a fluctuation in voltage and the current in DG3 doubles. At $t ={1.5}\mathrm{\;s}$, a sinusoidal attack signal is injected into DG1, the system continues to fluctuate. At $t = 2\mathrm{\;s}$, the system incorporates the fault tolerant controller given in this paper, the system is restored to stability. It can be seen that system normal regulation signal do not continuously affect the system stability. When the attack signal is added, the system is dysregulated and external intervention is required to stabilize the system. Therefore, the fault-tolerance of the given scheme is verified.
VI. Conclusion
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This paper presents a distributed LESO-based fault-tolerant secondary control method for DC microgrids, which enables voltage regulation and accurate current sharing in the presence of FDI attacks. LESO is utilized to estimate the attack signals in the DG, and a distributed fault-tolerant secondary controller is employed to compensate for the adverse effects of the attack signals on the system, with the capability to detect and suppress FDI attacks. A rigorous proof based on the Lyapunov function is provided. Moreover, the controller is robust against communication interruptions and load variations, further demonstrating the effectiveness and superiority of the method in DC microgrids. The controller designed in this paper primarily focuses on the attack signals in the actuators, and the attack signals in the sensors will be analyzed and investigated in the following step.
Funding
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  • Major Science and Technology Special Project of Gansu Province in China(23ZDGA007)
  • Industrial support plan project of Gansu Province in China(2024CYZC-18)
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Appendix
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Year 2024 volume 9 Issue 3
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Cite this Article
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Article Info
doi: 10.24295/CPSSTPEA.2024.00013
  • Receive Date：2024-02-29
  • Online Date：2025-07-05
  • Published：2024-09-10
Article Data
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History
  • Received：2024-02-29
  • Revised：2024-06-08
  • Accepted：2024-07-07
Funding
Major Science and Technology Special Project of Gansu Province in China(23ZDGA007)
Industrial support plan project of Gansu Province in China(2024CYZC-18)
Affiliations
    Lanzhou University of Technology Electrical Engineering and Information Engineering Lanzhou 730050 China

Corresponding:

Yiwei Feng.
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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