Distributed event-triggered robust automatic generation control for networked power system with wind turbines

This paper proposes an event-triggered distributed networked control scheme and its robust stability analysis strategy for the automatic generation control of multi-area interconnected power system with wind turbines. First, for the large-scale distributed networked control systems, considering the networked control systems with parameters uncertainty, effective transmission bandwidth, and time-varying networked-induced delays, a new distributed networked control systems model is proposed. Then, by constructing Lyapunov– Krasovskii functional method, the sufﬁcient conditions for asymptotical stability of the system are obtained, a design method of event-triggered distributed networked control law based on linear matrix inequality is proposed. In order to verify the effectiveness of the proposed scheme, a multi-area power system with wind turbines is considered, and the system performance under different load disturbances is simulated. The simulation results show that the scheme has reliable robust performance for multi-area distributed networked power system and improves the anti-interference ability of the power system.

Distributed AGC problem of a multi-area networked power system with system parameters uncertainty and limited communication resources is studied in this paper. An event-triggered communication mechanism is developed to schedule interconnected subsystem's communication, reduce unnecessary data exchange between subsystems and improve resource utilization. First, for the large-scale DNCSs with considering the system model with parameters uncertainty, effective transmission bandwidth and networked-induced delays, a new DNCSs model is proposed, event-triggered mechanisms (ETMs) are integrated for the DNCSs scheme. By constructing Lyapunov-Krasovskii functional method, based on linear matrix inequality (LMI), the sufficient conditions are proposed for DNCSs asymptotical stability. In order to verify the effectiveness of the proposed scheme, a multi-area power generation system (thermal power station, hydropower station, wind farm) is considered. The performance of the system under different load disturbances is simulated and analyzed. The simulation results show that the method can improve the power system performance in the case of effective transmission bandwidth, parameters uncertainty and time-varying communication delays.
The structure of this paper is as follows. In Section 2, an event-triggered distributed model for DNCSs scheme and the problem to be investigated are formulated. In Section 3, an interconnected controller design method is proposed, sufficient conditions are derived for asymptotical stability of the eventtriggered DNCSs. Section 4 introduces the simulation analysis of typical interconnected system, a multi-area power system is considered and the dynamic performance of the system under different load disturbances is simulated and analyzed. Finally, Section 5 summarizes this paper.

PROBLEM DESCRIPTION
NCSs have many coupled subsystems that exchange the data through the communication channel [21]. However, due to limited bandwidth and possible data loss in the communication network, the time-varying networked-induced delays are inevitable in these NCSs. In order to deal with the impacts of communication network uncertainty in designing of networked controller, an efficient DNCSs model is presented. The implementation of effective bandwidth results in network traffic reduction while maintaining acceptable system performance [22]. A node with effective bandwidth compares the value prepared to send to the network, x i , to either the last value sent, x i,sent , or a constant threshold i,t . If the absolute value of the difference between x i and x i,sent is within the effective bandwidth, i x i,sent or i,t , then no update is sent to network For the case when the term i x i,sent is used the threshold changes as a function of the node state and be viewed as a relative threshold. When i,t is used, the threshold remains constant independent of the state.
As the size of the threshold parameters increases, i or i,t , under the assumption that other conditions remain unchanged, the range of judgment conditions for the node to transmit information becomes wider, the node broadcast fewer messages. Implementing a threshold to reduce network traffic produces uncertainty in the state of the system. Since the controller relies on the broadcast state x i,b , to compute the control signal, it is important to determine whether this uncertainty could drive the system to instability. At any given time, the true state of the system x i is: , in case of the controller node, the control signal it sends to the plant is the following: u i = K i x i,b , the event-triggered control scheme consisting of an ETMs and a discrete-time state feedback control law is described as: The influence of model uncertainties on system performance has always been the main direction in automatic control research. In practical application, the assumed dynamic model may not accurately predict the actual system dynamic performance. In addition, due to the aging of equipment and the change of magnetic saturation temperature, parameters such as transient reactance of generator may be time-varying, resulting in large uncertainties of model. These model uncertainties will inevitably affect the analysis and design in the dynamic and steady-state performance. In order to reduce the impact of model uncertainties, some robust control methods are applied to the dynamic state estimation of the power system.
Consider DNCSs S consisting of N subsystems, and subsystem S i , i = (1, 2, … , N ), these subsystems are connected with each other through the network channel. Each subsystem transmits relevant data information through the communication network, including its own data information and the data information of the coupled system. If the subsystem S i is associated with the subsystem S j , the controller j sends the state data of the subsystem S j to the ith controller through the network. S i is related subsystem of S j , the set of all neighbour subsystem of S i is described by N i , N i = {S j |i ≠ j and S j is a neighbor of S i }.
The ith subsystem of large scale interconnected distributed systems based on ETMs is as follow [23]: Therefore, the dynamic model of the ith subsystem S i with modelling uncertainty, time-varying delay and interconnection can be described as the following model: where x i (k),u i (k) are the local state vector and control inputs, respectively. x i (k − i (k)) and x j (k − i j (k))show the local delayed state variables and neighbour delays state variables, respectively. A i ,A di , A i j , and B i are known constant parameter matrices with appropriate dimensions. A i and B i are the system matrices, A di shows the state delay matrix and A i j denotes interconnection matrix, which reflects the relationship between the ith subsystem and the jth subsystem. If the state of the jth subsystem is not coupled with the ith subsystem, matrix A i j will be zero. With consideration of bounded uncertainties, ΔA i , ΔA di , ΔB i and ΔA i j are unknown norm bounded matrix which represent the uncertainty of parameters in the model. For i, j = (1, 2, … , N ), the uncertainty matrix can be described as where, F i and F i j are time-varying uncertainties matrices and satisfying F T i F i ≤ I i and F T i j F i j ≤ I i j , respectively. I k denotes Unit matrix with proper dimension. D i , E ai , E adi , L i j and N i j are known real matrix with proper dimension, describes the structure of uncertain parameters. i (k) and i j (k) for i = 1, …, N, j ∈ N i , are uncertain time-varying state delays and interconnection delays [24] , respectively, meet the conditions:

DESIGN OF DISTRIBUTED NETWORKED ROBUST CONTROLLER BASED ON ETMS
The design of distributed control law for interconnected systems has received considerable attention in recent years. An interconnected system with distributed control architecture is known as distributed control law since the control law for each subsystem does not only depend on its own states but also the states of the other subsystems. Communication networks provide a larger flexibility for the control design of interconnected systems by allowing the information exchange between the local controllers of the subsystems which can be used to improve the overall system performance.
Based on the ETMs proposed in Section 2, a novel state feedback control law that consider the limited bandwidth and networked-induced delays is proposed for DNCSs, for the ith subsystem, the state feedback control law is designed as follows: where K i , K di and K i j are the ith controller feedback gain, delayed state feedback gain and interconnected feedback gain, respectively. i is threshold parameter. The main purpose of this paper is to consider these feedback terms and feedback threshold at the same time, so as to ensure the stability and improve the performance of DNCSs. In general, the stability analysis of NCSs with networked-induced delays can be divided into two categories: delay independent and delay dependent. In delay independent stability analysis, it is not necessary to know the values of delays or even their upper and lower bounds. Therefore, the stability criteria are independent of the time delay. In this paper, delay independent stability analysis is considered. A delay independent theorem based on LMI is proposed, and sufficient conditions for asymptotic stability of closed-loop DNCSs are obtained.
The control law proposed in (3) is applied to the subsystem S i , under the ETMs in Section 2, and the following closed-loop dynamic system is obtained: For convenience, let ΔA i j and ΔB i are known matrix with proper dimension, by the following Theorem 1, a sufficient condition LMI based for the asymptotic stability of the closed-loop DNCSs described in (4) can be derived.
Then the closed-loop DNCSs S is asymptotically stable. Where Proof: Construct the following Lyapunov-Krasovskii function V (k) for the DNCSs as With taking the forward difference along (6) and using (4), we have Closed-loop of four-area interconnected power system In order to reduce the conservatism of stability conditions, slack variable H i is introduced by (7), which aims to reduce con-servatism and increase flexibility [25,26] Thus, we obtain With mathematical manipulation, we have where Θ i is shown in (9).
where index nei stands for the states of neighbouring subsystem S j . From the above analysis, implement Schur complement on (9), we have (5), which means if inequality (5) is satisfied, the closed-loop DNCSs is asymptotically stable. Moreover, inequality (5) is satisfied results in ΔV (k) < 0, the system S is asymptotically stable. This completes the proof. □ In this section, a design method of distributed networked controller suggested in (3) based on ETMs is proposed. Because the inequality (5) is not a LMI, it is difficult to solute this kind of inequality. Though variable substitution, inequality (5) can be transmitted into a line matrix inequality, the designer can use the existing LMI solver and calculate the stable feedback gain K i , K di and K i j . For convenience, the following lemmas are introduced.

Lemma 3. [29]: For given real matrices H and E with appro-
priate dimensions, and the matrix F = diag{F 1 , F 2 , … , F l }, which satisfies the condition F T F ≤ I , the following matrix inequality holds: where F i ∈ R h i ×e i , Γ h = diag{ 1 I h1 , 2 I h2 , … , l I hl }, Γ e = diag{ 1 I e1 , 2 I e2 , … , l I el }, 1 , … , l is a set of positive scalars.
Theorem 2. Given DNCSs S i described in (2) under ETMs in Section 2, there is the state feedback controller (3) in such a way that the closed- loop system (4) is asymptotically stable, if there exist matrices R i j > 0, Q i > 0, X i > 0,H i > 0,K i ,K di and K i j for j ∈ N i of appropriate dimensions, such that satisfy where and K i j . j ∈ N i are obtained directly.
Proof: By applying Lemma 1 to inequality (5), it can be seen that ΔV(k) < 0 if the following inequality is satisfied: If inequality (11) holds, we can get (12) by Lemma 2.
by Lemma 3, a sufficient condition for matrix inequality (11) to hold for all allowed uncertainties is that there exists a set of positive scalars i , i1 , j = (1, 2, … , N ), such that Then inequality (11) holds where By defining X i = P −1 i , and multiplying diag{I P −1 i P −1 i I I I I} to both sides of (11), the LMI (10) is obtained. This completes the proof. □ Therefore, by solving LMI (10) using one of the existing convex programming tools such as LMI toolbox MATLAB, the asymptotically stable feedback controller will be straightforwardly designed for each subsystem.

AGC SYSTEM DESIGN
In this section, as is shown in Figure 1, a four-area interconnected power system with wind turbines is adopted to demonstrate the design effectiveness of the proposed event-triggered AGC scheme for load frequency control system. The system parameters of the four-area interconnected power system used in simulations are as follows [20] (see also Table 1).
The system is composed of a thermal power plant, variable speed wind turbines (VSWTs) and hydropower plant, which is used to design a distributed networked AGC system. As shown in Figures 1 and 2 wind turbines with three blades, a horizontal axis and variable speed. In addition, area 4 is a thermal power plant, area 2 and area 3 are hydropower plants. The details of the regional composition and mathematical representation of wind turbines, thermal power plants and hydroelectric power plants can be seen in [20]. In addition, area 1 includes an aggregated wind model consisting of 30 VSWTs units of 2 MW.
The state vectors for area i are defined as follows: To obtain the control law (3), according to Wang et al. [30], the sampling time T s = 20 ms of four-area interconnected system is chosen in this paper. The changes in parameter uncertainties are within 20%. The corresponding controller gain K i , K di and K i j could be obtained based on MATLAB LMI solver. In order to verify the effectiveness of the proposed method, the following two cases are considered: Case 1: ETMs, sensors node and actuators node with constant threshold si , ai , load disturbance change 0.01 p.u. under the system parameter uncertainties.
Case 2: ETMs, sensors node and actuators node with constant threshold si and ai , load disturbance change 0.1 p.u. under the system parameter uncertainties.
A simultaneous step load change of 0.01 and 0.1 p.u. in four areas. Figures 3 and 4 show the frequency deviation and tie-line