Robust H ∞ Control of Networked System In Microgrid Under Random Packet Loss

The desire for sustainable development has led to the advancement of renewable energy generation through microgrids, and the control and stability of microgrids pose serious challenges to the popularization of microgrids. This article focuses on the stability and performance of the microgrid, and due to its limited channel bandwidth and other factors, may result in the loss of random packets from the sensor to the controller and from the controller to the actuator. To solve the problem, this paper designed a feedback controller based on the observer and robust H ∞ control method and has established Bernoulli packet loss model. The proposed controller enables the networked control system robust exponential stability in the sense of mean square. Fi-nally, the correctness and eﬀectiveness of the proposed controller is veriﬁed by MATLAB simulation.


Introduction
Microgrid is a complex system composed of various systems, and it is a trend that cannot be ignored in the distribution network. Nowadays, some hospitals, factories, remote work sites and individual users are equipped with microgrid systems, so the reliable and stable operation of microgrids becomes more and more important. In other words, as the penetration rate of microgrids continues to increase, control issues have become the primary issue of microgrids. Centralized and distributed controllers were kicked out [1] [2]. A fully centralized controller relies on data collected in a dedicated central controller, which performs the required calculations and determines the control actions of all units at a single point, requiring extensive communication between the central controller and the controlled units. On the other hand, in a completely decentralized control, each unit is controlled by its local controller, which only receives local information and does not fully understand other system variables or the actions of other controllers. The [3] proposed a robust decentralized control strategy for the islanding operation of microgrid. The island microgrid is composed of two distributed generation (DG) units. The main DG controls the voltage and frequency of the load, and the slave DG uses the traditional dq current control strategy to adjust its power components. A new type of distributed controller for secondary frequency and voltage control of island microgrid. Inspired by cooperative control technology, the controller uses local information and nearest neighbor communication to jointly perform secondary control actions and was proposed in [4]. The [5] proposes a control architecture that is a compromise between a completely centralized and a completely decentralized control scheme, which can be realized by a hierarchical control scheme composed of three control levels: primary, secondary and tertiary. The main control is related to the voltage, current, virtual impedance and the local layer of the drop loop in DG. The core idea of realizing secondary control is to compensate for the voltage and frequency changes of the microgrid caused by the primary control loop. Contrary to the main control, the secondary control uses a feedback communication system to pass the microgrid parameter value to each DG unit. However, the above documents do not consider the impact of data loss on the system during signal transmission. The transmission of signal data is crucial to the control and stable operation of the microgrid and should be paid attention to. The networked control system is a communication system that connects devices and controllers over a long distance. Because of its low cost and high reliability, it is introduced as a successful control strategy into the microgrid [6] [7]. However, many studies have focused on the impact of delay on the system or only assume that the packet loss problem only occurs in the channel from the sensor to the controller [8]. And the modeling of the packet loss phenomenon of the networked control system is basically divided into three methods. For example, one approach is to treat packet loss as a binary switching sequence specified by the conditional probability distribution in the signal transmission channel. The binary exchange sequence obeys the white sequence of Bernoulli distribution, and its values are 0 and 1. The second method is to use a discrete linear system with Markov jump parameters to represent the random packet loss model of the network [9]. The third method is to replace the packet loss with zeros, and then construct an incomplete matrix in the measurement. The [10] considered the dual-channel packet loss in the controller of the networked control system, but did not conduct in-depth research on the uncertainty in the model. However, there are still few documents considering the packet loss problem in the networked control system in the microgrid. This paper applies the designed networked control system controller to the microgrid system. It is assumed that there is interference and network packet loss from the sensor to the controller, and from the controller to the actuator. Using the idea of robust control, the packet loss process is modeled as a Bernoulli packet loss model, and an observer-based feedback controller is designed to make the networked system robust and exponentially stable in the sense of mean square.The Lyapunov function is used to analyze stability, and finally the effectiveness of the proposed LMI method is verified by a simulation example.
This article is organized as follows. The second section gives the basic idea and mathematical model for the establishment of microgrid. The third section expounds the expression of the microgrid networked control problem in the form of state space. The main result of the stability analysis is taken as the LMI condition and an optimization problem is proposed. The simulation parameters and results will be given in Section five. Finally, the work is summarized in Section six.

Problem describes
First, we consider a distributed energy resource (DER) unit, which is a DG unit that is electronically coupled to the grid through coupling [11]. The DG device is coupled to the grid by means of electronic coupling. The DG unit consists of a DC voltage source, a voltage source converter, a series filter and a transformer, and the RLC network is used as a local load. Using the state space method to model the system, the state space expression can be obtained as ẋ = Ax + Bu are respectively the state, the control input, the output.v d is the only output signal which has to be controlled. In order to complete the control of v d , the input signals v td and v tq are needed. However, since all state variables and load parameters are unknown and uncertain, it is difficult to calculate v tq .Therefore, we assume that v tq is a disturbance variable that can be set to zero. In other words, in order to control v d , only one of v td and v tq is required to meet the requirements. Discretize model (1) and transform it into the following discrete-time linear steady model where x(k) ∈ R n and u(k) ∈ R m represent the plant states and control input respectively. It is assumed that random data packet loss occurs at the same time during the transmission of data packet data from the sensor to the controller or the controller to the actuator. The sensors, controllers and actuators here are all driven by clocks and data packets will not be out of order. The structure is shown in Figure 2. Suppose the system is also disturbed, denoted by w k . Since we assume that the data is transmitted from the sensor to the controller or the controller to the actuator will cause packet loss, we add an intermediate link z k to realize the representation of the two parts. In addition, we also consider the parameter uncertainty in the system, for example, ∆A 0 and ∆C 0 . The disturbed system (2) is asfollows where x k ∈ R n , u k ∈ R m , z k ∈ R r and y k ∈ R p are the state, the control input, the control output and measured output, respectively.
where, H 1 , H 2 and E are known real constant matrices with appropriate dimensions, and F k is an unknown real-time variable matrix satisfying F k F T k ≤ I. Next, this paper establishes a communication random packet loss model α k ∈ R is a random variable representing the random Bernoulli distribution of packet loss from the sensor to the controller. Suppose it is a white noise sequence of linear random variables with the following properties The dynamic observer-based control method of system (5) is described aŝ wherex k ∈ R n is the state estimate of the system (7),û k ∈ R m is the control input with transmission missing, L ∈ R n×p is the observer gain and K ∈ R m×n is the controller gain. β k ∈ R is a random variable representing the random Bernoulli distribution of packet loss from the controller to the actuator. Hence α k and β k are independent of each other. Suppose it is a white sequence of linear random variables with the following properties Definition e k is the deviation between the system measured value and the estimated value, the expression is The closed-loop networked states are given as We will rewrite (10) in compact form where The goal of this paper is to design a controller (7) for the system (5) when the closed-loop system (11) has random packet loss, so that the closedloop system (11) is exponentially mean square stable and H ∞ performance constraint is satisfied. Under the condition of ensuring that the system is stable in the mean square, it is also necessary to ensure that under the zero initial condition, the control output z of the system satisfies For all non-zero w k , where γ > 0 is a prescibed scalar.

Main results
In order to prove the convenience and generality of the main results in this section, we will make the following definitions and quote lemmas. Definition 1: The closed-loop system (11) is said to be exponentially meansquare stable if, with w k = 0, the exist constants φ > 0 and τ ∈ (0, 1), so that Lemma 1 : Let V (δ k ) be a Lyapunov funtional. If there exist real scalars λ ≥ 0, µ > 0, v > 0 and 0 < ψ < 1, so that and Then the sequence δ k satisfies Next, we give a sufficient condition in the form of the theorem to prove that the closed-loop network system (11) is exponentially mean-square stable and realize the H ∞ performance constraint (12).
Theorem 1: Given communication channel parameters 0 <ᾱ < 1, 0 < β < 1. If there are positive definite matrices P > 0 and Q > 0, the closedloop network system (11) has exponential mean square stability, and the H ∞ performance constraint (12) can be achieved for all non-zero w k , real matrices K and L satisfy the following matrix inequalities, The form of K and L can refer to Theorem 3 in the [10]. if LMI is feasible, the controller design problem under consideration is solvable. Proof: The proof in this part will be divided into two parts. The first part is the proof of stability in the case of disturbance is zero in the network control system. The second part is that the network control system can still satisfy the constraints of H ∞ performance in the presence of disturbances. Define a Lyapunov functional where P and Q are the positive definite solutions to (17). Then Using the properties of the expecting value of the Bernoulli distributions α k and β k , the following results are obtained. E (α k −ᾱ) Assuming w k = 0, we can obtain From (4.17) we know that Ω < 0, hence where 0 < α < min {−λ min (−Ω), σ} , σ = max {λ max (P ), λ max (Q)} . It will lead to the following (23) Therefore, from Definition 1 and Lemma 1, it can be proved that the closedloop system (11) is exponentially mean-square stable when the disturbance is zero. Next, we will prove the existence of disturbance.
For non-zero w k and output is z k , the following equation holds By using Shure's complement, Γ can be rewritten as It can be concluded from (24) that Summing up (26) from 0 to ∞. With respect to k yield Since ξ 0 = 0 and the system (11) is exponentially mean square stable, it is easy to conclude that That is, the specified H ∞ norm constraint is realized, and the proof is complete.
The matrix inequality given in (25) is bilinear on P , Q, K and L. let M = P −1 , N = Q −1 and pre and post multiplied by the congruential transformation matrix diag I I I P −1 −Q −1 I P −1 Q −1 . We can get The uniform transformation used in this method separates the controller gains K and L from the system parameters.
Observer-based controller (7) for the network control system (5) under the H ∞ performance constraint (12) with the minimum γ min . The following optimization problems need to be solved.
Under the H ∞ performance constraint (12) with the minimumγ min , the observer-based feedback controller (7) for the network control system (11).
The following optimization problems need to be solved.

Simulation Example
The penetration rate of microgrid in daily life is getting higher and higher, and ensuring the stable operation of microgrid has become the primary issue. This paper uses system (5) for simulation, and the microgrid parameters are shown in Table (1). Our goal is to control the inverter in the presence of load interference with strong robustness to keep the output AC voltage at the desired value without distortion. The control signal is transmitted through a network cable, but due to limited network bandwidth or other factors, signal transmission may experience signal loss (packet loss), which affects the performance of the network system. Therefore, in the presence of random packet loss, we aim to design an observer-based feedback controller to make the network system robust and exponentially stable in the mean square sense and achieve the specified H ∞ performance. First, the observer result is shown in figure 3, and the initial condition is set to x 0 = 0.5 0.2 −0.1 0 T ,x = 0 0 0 0 T controller to actuator packet loss rate to be the same, that is,ᾱ =β, which is also the same setting in the simulation. It can be seen that the observer is effective and able to observe the state of the system. The observer shown in the figure is under the condi-  (2). We can see that under different packet loss rates, the control effect is differ-   that when the packet loss rate is smaller, such asᾱ =β = 0.95 , the output of the controller is closer to the output of the system controller without packet loss. When the packet loss rate gradually increases, the deviation between the output of the controller and the output of the controller without packet loss becomes larger, and the performance of the robust controller becomes worse, such as γ min increases.

Conclusions
This paper studies the robust H ∞ problem of networked control systems with random communication packet loss in microgrids, and proposes an observerbased dynamic feedback controller for microgrid networked control based on a random packet loss phenomenon, and proves the controller has robust exponential stability in the mean square sense. The effectiveness of the controller design is verified by simulation and comparison of controller performance under different packet loss situations. In the future work, we will focus on the stability of the microgrid networked control system due to the simultaneous existence of delay and packet loss, and the severe test of the microgrid communication system brought about by the interconnection of multiple microgrids.