Event-triggered load frequency control of smart grids under deception attacks

This paper shows a result for the load frequency control with event-triggering mechanism under deception attacks. Speciﬁcally, the event-triggering mechanism and deception attacks are considered in the dynamic model of power systems. The event-triggering is used to reduce the frequency of controller update and communication between nodes. The H ∞ controller is designed and the stability of the system is guaranteed by utilising the Lyapunov–Krasovskii functional method and truncated Bessel–Legendre inequality. Finally, a three-area interconnected networked power system is given and the simulated results are presented to show the effectiveness of the developed theoretical results.


INTRODUCTION
Driven by green energy conservation awareness, smart grid has become a key area of competing development in the world, which has the advantages of high controllability, high energy efficiency, and self-healing [1][2][3]. However, at the same time, privacy and security problems in the smart grid are gradually exposed and become an important factor restricting its further development [4][5][6]. With the development of the smart power grid, a mass of smart terminals are introduced into the smart power grid. In the aspect of network attacks, Denial-of-Service (DoS) attacks [7][8][9] and deception attacks [10,11] are the most concerned at present. Deception attacks are a kind of attack commonly adopted by attackers. The attacker will inject wrong data information into the target system, making the data transmitted in the sensor produce errors, and then make biased decisions [12]. Compared with DoS attacks, the attacker can keep deception attacks hidden and not easy to be detected, and deception attacks have randomness. Therefore, to ensure the safe and stable operation of the smart grid, it is of great significance to study During the past few years, event-triggered control, also called event-based control or event-driven control has been proposed to reduce the date transmission, minimize the network pressure and save the network bandwidth in the network system. Wen et al. [22] considered the load frequency control for power systems with communication delays via an eventtriggering mechanism to reduce the communication burdens and lower the control updating frequency. In this trend, an LFC architecture based on auxiliary adaptive dynamic planning is proposed in [23], and event-triggered management will play an important role in reducing communication and computing costs. Considering the influence of network induced delay, signal quantization, and data loss in the communication link between the device and the controller, a design framework of an event-triggered network control system based on passivity is introduced in [24]. For a multi-area closed-loop power system, a scheme of flexible event-triggered communication is designed in [7]; this scheme allows DoS attacks to cause a certain degree of packet loss advantage of improving transaction efficiency. However, LFC of smart grid subject to random deception attacks has not been paid enough attention especially when the event-triggered control protocol is simultaneously concerned.
In response to the above discussion, we endeavour to investigate the event-triggered LFC of smart grids under deception attacks in this paper, since deception attacks will randomly reduce the integrity of transmission packets and affect the performance of smart grid. By employing the stochastic analysis techniques and matrix inequalities, sufficient conditions are established to guarantee the H ∞ performance and the desired controller gain is given in terms of some linear matrix inequalities. The main contributions of this paper are highlighted as follows.

A closed-loop time-delay LFC dynamic model is established
within which load frequency control problem can be conveniently handled in the presence of event-triggering mechanism and random deception attacks. 2. In the process of proving the stability criterion, we utilize the truncated Bessel-Legendre integral inequality, which is less conservative than Jensen inequality and Wirtinger inequality. Simultaneously, we construct appropriate multiple Lyapunov functionals, which is less conservative than the single Lyapunov functional method. 3. Based on the stochastic analysis techniques, Lyapunov-Krasovskii (L-K) theory and matrix inequalities, sufficient conditions are derived in terms of LMIs to achieve asymptotical stability and H ∞ performance for the smart grid under deception attacks.
The rest of this paper is organized as follows. In Section 2, we propose our time-delay LFC dynamic model with an eventtriggering mechanism under deception attacks. In Section 3, we analyse the stability of the system and design the controller. A numerical example and simulated studies are conducted to verify the proposed results in Section 4. Finally, Section 5 concludes this paper.
Notation. The following notations are used throughout this paper. {⋅} represent mathematical expectation. We use ℝ n to denote the n-dimensional Euclidean space and ℝ n×m the set of all n × m matrices. diag{⋅} representative diagonal matrix. Let ‖x‖ and ‖A‖ be the Euclidean norm of a vector x and a matrix A, respectively. For a symmetric block matrix, we use ⋆ to denote the terms introduced by symmetry.

Multi-area LFC model in smart grid
The power system is a complex dynamical system with nonlinear and time-varying terms. Inspired by [10], the framework of network-based LFC of the ith area power system is given in Figure 1. From Figure 1, the following relation can be obtained (1) where s is the Laplace variable. Δ f i is the system deviation value of the ith area, ΔP mi represents the mechanical power deviation value, ΔP vi represents the position quantity of regulating valve, and ΔP di represents the load of the ith area. R i is the speed drop coefficient, M i is the moment of inertia of the generator, D i is the damping coefficient of the generator, T chi and T gi are the steam capacity time constant and the governor time constant respectively. i represents the conversion coefficient of system power and frequency. ACE i (t ) is the area control error signal of the ith area. △P tie−i is the power deviation value of the sub-area tie-line in the ith area, T i j is the synchronization power coefficient of the tie-line between the ith and j th control areas [10]. According to equation (1), the LFC dynamic model of the multi-area power system can be described as follows Using ACE as the input of the controller, the PI load frequency controller is designed as follows where K = diag{K 1 , … , K N }, K i = [K Pi K Ii ], K Pi and K Ii are the proportional and integral gains, respectively. We assume the controller works under an open network communication. It means that the transmission is vulnerable to network attacks. Therefore, an attacker sends false system information to the controller or sensor, including inaccurate measurements, faulty controller output, or wrong timestamps. In this scenario, corrupted information gives rise to deception, which disrupts data transmission. In order to account attacks in our formulation, we write the control rule as where (t ) is an energy-bounded signal belonging to 1} is a stochastic variable that defines the occurrence of attacks and obeys Bernoulli distribution. More specifically, if (t ) = 1, the controller u(t ) = (t ), which means that the deception attacks occurred in transmission. If (t ) = 0, the controller becomes u(t ) = −BKCx(t ), indicating that the sampling measurement has been transmitted successfully. We assume in addition that , then the state equation can be rewritten as

Event-triggered control
Motivated by [7], we introduce the event-triggering mechanism as follows: where and t k+1 h are the sampling times of two adjacent signals transmitted to the controller before and after satisfying the triggering conditions, respectively. The event-triggering parameter is a preset constant. The trigger matrix is a positive definite matrix to be solved, and h is the sampling period of LFC.
Remark 1. The sampling period h can significantly affect the frequency stability of the power systems. In [28], the transmission delay margin of the control schemes under different sampling period is studied. The results show that the proposed control schemes can provide good robustness under small sampling period (h = 0.1), and the designed controllers can tolerate large parameter uncertainties. If the sampling period of the power systems is too large, the control signal packet will be lost when the communication or physical fault occurs. In such a large sampling period (e.g. h = 11.6 s given in [28]), the stable operation of the systems cannot be guaranteed if multiple packets are con-tinuously lost within tens of seconds. Therefore, we had better choose a smaller sampling period to reduce the burden of the communication network and ensure the stable operation of the systems in the case of communication failure.
By substituting the output y(t ) of (5) into the threshold condition (6), the triggering condition of the event-triggering mechanism can be obtained as follows: Based on the communication scheme (6), it is known that the sampled-data are not transmitted over the communication networks unless the threshold condition is satisfied. For introducing the proposed event-triggered communication scheme (6) at each sampling instant to determine whether the currently sampled data should be transmitted through the communication networks, the holding interval Ω = [t k h + t k , t k+1 h + t k+1 ) of the ZOH is also divided into the following subsets Ω l , Remark 2. It should be mentioned that t * represents the time delay existing when all released signals x(i l h) arrive at the actuator. t k represents the time delay generated from time 0 to time t k+1 − t k − 2, and t k+1 represents the time delay generated from time t k+1 − t k − 1.
Define (t ) = t − i l h, t ∈ Ω l , output feedback u(t ) can be rewritten as Substituting (8) into (2) we have that the initial state model (2) of the multi-area power system with LFC can be rewritten as the following ACE -dependent time-delay model For sake of simplicity, we split (9) into two parts, i.e. where Definition 1. If the following conditions are satisfied, the closed-loop system (9) is asymptotically stable in the security sense, and the disturbance rejection level of H ∞ is .
1. When (t ) = 0 and (t ) = 0, the system (9) is asymptotically stable, that is, in the neighbourhood of equilibrium state, there is a continuous first-order partial derivative of V (x(t )) and V (x(t )) is positive definite and {V (x(t ))} is negative definite, then the system is asymptotically stable in equilibrium; 2. Under the zero initial condition, i.e. x(t ) = 0, t ∈ [−̄, 0], for any non-zero (t ) ∈ L 2 [0, ∞] and (t ) ∈ L 2 [0, ∞], for a given , the following inequality hold: Remark 3. Asymptotically stable in the security sense means that the closed-loop system (9) can be achieved asymptotical stability performance even if the measurement and control signals transmission fails under deception attacks.

Lemma 2.
[25] (Convex property) For any symmetric positive integer n, m and scalar ∈ (0, 1), given the n × n-dimensional matrix R > 0, two matrices W 1 ∈ ℝ n×m and W 2 ∈ ℝ n×m , all vectors ∈ ℝ m are defined, and the function Θ( , R) is defined as If there is a matrix X ∈ ℝ n×n that satisfies [ R X * R ] > 0, then the following inequality holds min ∈(0,1)

MAIN RESULTS
In this section, we first analyse the asymptotical stability and H ∞ performance of the LFC dynamic model of the power system (9), which takes into account the event-triggered mechanism and deception attacks. Then a PI controller design method based on event-triggered is presented and the controller gain is derived.
Theorem 1. For given disturbance rejection level > 0, 0 ≥ 0, (9) is asymptotically stable, if there exist real symmetric matrices P > 0, Q 1 > 0, Q 2 > 0, R > 0, W > 0, Φ > 0 with appropriate dimensions and an arbitrary matrix N satisfying the following LMI where Proof. Consider the following Lyapunov-functional candidate Remark 4. It should be mentioned that we explicitly construct multiple Lyapunov functionals (13) to deal with smart grids under deception attacks, which is less conservative than the single Lyapunov functional method used in many existing papers.
The derivative of V (t ) can be obtained as follows ℒV (x(t )) = ℒV 1 (x(t )) + ℒV 2 (x(t )) + ℒV 3 (x(t )), Take the expectation for both sides of derivative ℒV (x(t )), and then we have that By Lemma 1 we can obtain that Remark 5. It should be noticed that in this paper truncated Bessel-Legendre integral inequality is used as described above, which is less conservative than Jensen inequality and Wirtingerbased inequality. As a result, we can obtain a new stability criterion in terms of matrix inequalities, which is less conservative than those results in [7, 11 29].
, i.e. . By using Lemma 2 (Convex theorem), we have By (15) and (16), we can obtain that Event-triggered communication strategy can ensure Substituting (18) into (17), we have From the Schur theorem, the following holds Integrating both sides of (20), we have At zero initial condition, then there are when (t ) = 0, (t ) = 0, we have Then, there exists a positive scalar > 0 that makes the following inequality hold By above, when (t ) = 0 and (t ) = 0, we have {ℒV (x(t ))} ≤ − {‖x(t )‖ 2 }, the closed-loop system (9) is asymptotically stable in the security sense. When (t ) ≠ 0 and (t ) ≠ 0, we proved that under the zero initial condition, the closed-loop system (9) has H ∞ disturbance suppression performance and the disturbance rejection level is . This completes the proof. □ Remark 6. By employing the truncated Bessel-Legendre inequality, which is more compact than Wirtinger inequality, to estimate the expectation of L-K function, we propose a new delay-dependent stability criterion for LFC smart grids under deception attacks. Besides, the convex lemma is used in the closed-loop stability analysis. Hence, the method of combining the truncated Bessel-Legendre inequality and the convex lemma can be regarded as unique feature of our study.
Since the controller gain cannot be calculated directly from Theorem 1 due to the coupling terms, we give the following theorem to obtain the controller gains.
Since the matrix C is not invertible, in order to solve the controller gain, we define it as follows where U , V are matrices of appropriate dimensions. And then we can get Remark 7. In order to solve the stability condition described in Theorem 1 and obtain a feasible LMI solution, inspired by [30] we can transform (29) into a minimization problem. In such a way, by utilizing Schur complement we can get inequality (26) and can be solved as a minimum positive scalar. However, this paper does not give a specific value range for but restrict → 0, since we can easily find a sufficiently small when (26) is satisfied.
According to Equation (28), the controller gain matrix K may be described as K = VU −1 . This completes the proof. □

NUMERICAL EXAMPLE
This section analyses a three-area interconnected power system to verify the effectiveness of the proposed load frequency control method. Table 1 shows the parameters of the three-area interconnected power system [22]. Figure 2 shows a schematic

FIGURE 2
Schematic diagram of the relationship between the three-area interconnected power systems diagram of the relationship between the three-area interconnected power systems. Based on the results of Theorem 2, we set the sampling period h = 0.01, PI-based controller gains K Pi = 0.1, K Ii = 0.1, and T 12 = 0.2(pu/rad), T 13 = 0.12(pu/rad), T 23 = 0.25 (pu/rad). From Table 1 and the modelling part mentioned earlier in this article, we can get the parameter matrices of each region as shown in [27].
The evolutions of random variable (t ) and random attack signals (t ) (t ) are shown in Figures 3 and 4, respectively. It can be seen the control input in Figure 5 and state trajectories  Figure 7 that the frequency deviation of the controlled interconnected power system can be stabilized at about 15 s, which shows the effectiveness of the proposed method. The trigger frequency under the event-triggered mechanism is shown in Figure 6. Not only a lot of unnecessary continuous information interaction is avoided, but also certain attacks can be resisted, which effectively saves the limited network communication bandwidth.

CONCLUSION
This paper mainly solved the problem of LFC under the combination of event-triggered communication and random attack signals. By using the convex property, we have derived the asymptotical stability and H ∞ performance criteria. The It has been proved that in the environment with random attack signals, the event-triggering mechanism is still effective and the control of the system can still achieve the expected effect, that is, the entire control process does not produce unnecessary frequency fluctuation. In future, our work will focus on anti-GDB (anti-GRC) schemes for load frequency control of power systems and renewable energy systems.