Load Frequency Control for Power Systems with Actuator Faults within a Finite-Time Interval

+is paper is concerned with the issue of finite-time H∞ load frequency control for power systems with actuator faults. Concerning various disturbances, the actuator fault is modeled by a homogeneous Markov chain. +e aperiodic sampling data controller is designed to alleviate the conservatism of attained results. Based on a new piecewise Lyapunov functional, some novel sufficient criteria are established, and the resulting power system is stochastic finite-time bounded. Finally, a single-area power system is adjusted to verify the effectiveness of the attained results.


Introduction
Load frequency control (LFC), as an integral part of automatic generation control in power systems, has been adopted to regulate the frequency deviation and tie-line power exchanges [1][2][3]. Added by the LFC strategy, the high-quality electric energy can be maintained over a certain range [4]. In general, constant frequency deviation may lead to unreliable frequency devices, transmission lines overload, etc. Meanwhile, owing to the large size of the power grid, it raises the difficulty in frequency control. erefore, it is a tough task to design suitable frequency control law. In practical applications, the loads are unexpected and unmeasurable, which indirectly regulate the system frequency. Accordingly, through the LFC strategy, the system performance can be guaranteed without affecting the generation capacity or frequency deviation. Up to now, the research on the LFC for power system gradually becomes a hot topic [5][6][7].
In networked control systems, various faults can be encountered due to the long-term utilization of components [8][9][10]. Note that the actuator faults are the source of instability and performance deterioration. To overcome the above shortage and improve the dependability, a great deal of attention has been shifted to actuator faults, and plenty of results have emerged [11,12]. However, the actuator faults are assumed to be time-unchanged, which limits the potential applications. As stated in [13], the so-called failure probability is common in the reliability industry, where failure rates can be governed by the Markov switching chain [14][15][16]. Despite the significant achievement has attained, no suitable attention has been devoted to the power systems.
On the other hand, Lyapunov asymptotic stability is most common in the literature, where asymptotic behavior can be expected over the infinite-time domain. Nevertheless, in reality, the desirable transient performance is very important in many physical systems, which causes the inapplicability of the Lyapunov stability. Following this trend, finite-time stability (FTS) has been studied [17][18][19], which concerns the dynamic behavior within bound over a fixed time interval instead of an asymptotical case. As is well known that FTS is different from the Lyapunov case, it gives more solutions of transient performance control. Owing to the merits of the FTS, many valuable achievements have been made over the past years [20]. However, to our knowledge, most of the previous results are assumed that the data communication keeps continuous between sensors and controllers. In the fields of sampled-data control law, this assumption is not accurate. In general, with respect to the demand of actual systems, the sampler may encounter component aging, data losses, etc. [21][22][23]. ese shortages may lead to unreliable periodic sampling. Fortunately, the aperiodic sampled-data control strategy is presented [24,25], which can efficiently deal with the aforementioned issues. However, the finite-time aperiodic sampled-data control for power systems remains unsettled, not mentioned to the LFC, which motivates us for this study.
Inspired by the above observations, we focus on the finite-time H ∞ load frequency control for power systems with actuator faults over the finite-time interval in this study. e main contributions can be summarized as follows: (1) different from the previous studies, to fully describe the randomly occurring actuator fault, the actuator fault is characterized by a homogeneous Markov chain. (2) To better characterize the actual demands of practical dynamics, a generalized framework of the actuator constraint is considered. (3) Apart from the traditional Lyapunov asymptotic stability, this study exploits the FTS for power systems and focuses on the finite-time control issue. By resorting to the piecewise Lyapunov theory, some novel results over the finite-time interval are reached. Finally, a numerical example is manifested to reveal the validity of the gained results. e remainder of this study is listed as follows. Section 2 provides a description of the problem. Section 3 presents the main results, and the simulation validation is exhibited in Section 4. Section 5 concludes the study.

Notations.
e notations of this paper are standard. ‖ · ‖ means the Euclidean norm. R indicates a set of n-dimensional matrix. E refers to the mathematical expectation. (λ max (A)/λ min (A)) means the largest/smallest eigenvalue of matrix A. Pr · { } means the occurrence probability. diag · { } represents a block-diagonal matrix.

Problem Formulations
Block diagram of single-area LFC power model is exhibited in Figure 1 [6]. Accordingly, the dynamic equation of power model can be listed as follows: and the system parameters are expressed in Table 1.
In single-area, the area control error (ACE) is interpreted as y(t) � βΔf due to the unaccessiblity of the tie-line power exchange. In reality, the actuator faults cannot be neglected for long-term utilization of components, which can be expressed as where . . , f). More specifically, r t , t ≥ 0 is identified as a right-continuous Markov chain taking values over a set S � 1, 2, . . . , S { } with generator Π � [π pq ] S×S , and its transition probabilities are inferred as where Δt > 0 and (lim Δt⟶0 o(Δt)/Δt) � 0, for q ≠ p and π pp � − q≠p π pq for each p ∈ S. Taking the ACE as the desired controller input of LFC, the output of the proportional-integral (PI) controller is asserted as where K P and K I signify the proportional and integral gains of the area, respectively. Let e purpose of this study is to solve the output feedback control for power system (6) with data sampling. erefore, the sampling sequence attained at a set of time instants. Added by the data sampling technique, only the measured signal y(t k ) can be released to the controller. Specifically, the sampling instants are represented as In light of periodic sampling instants, in this study, we consider the aperiodic sampling case. Following this trend, the sampling interval [t k , t k+1 ) is time-varying with the upper sampling period. us, one defines Letting K � K p K I , the PI-based sampled data LFC can be designed as Substituting (3) and (9) into (6), the closed loop power system can be governed by Before further derivation, some important contents are stated as follows.

r t . (26)
Based on Lemma 1, the following inequality can be devised: where It is well known that for any matrices H, one gets e aforementioned condition can be rewritten as On the other hand, for any matrices T 1 and T 2 , it is clear that

Mathematical Problems in Engineering
Substituting (23)-(32) into (20), it can be deduced that where Note that (33) is a convex combination of t − t k and τ − (t − t k ), in accordance with Schur complement; one can deduce that Θ(t) + (t − t k )H T Q − 1 1 H < 0 if and only if (15) and (16) hold. erefore, one can see that E LV t, r t < ρV t, r t + c 2 ω T (t)ω(t), t ∈ t k , t k+1 .

(35)
By integrating the both sides of (35) from t k to t and simple derivation, it yields Recalling the Lyapunov functional (20), we can get 6 Mathematical Problems in Engineering Substituting (37) and (38) into (36), we can obtain In light of (17), it can be concluded from (39) that E δ T (t)Rδ(t) < c 2 .
us, from Definition 2, we have to derive that power system (10) is SFTB over the time interval In the following, the actuator constraints (18) will be discussed. In light of (9), one has Recalling Assumption 2, it yields According to Schur complement, (18) can be guaranteed by (41), which completes the proof of eorem 1. □ Theorem 2. For given parameters ρ > 0, c 1 > 0, c 2 > 0, T > 0, ω > 0, u max , and matrix R, the closed-loop power system (10) is called SFTB with respect to (c 1 , c 2 , ω, T, R) and meet an , if there exists matrix P p > 0, On the other hand, other parameters are selected as τ � 0.2, c � 0.6, ρ � 0.2, c 1 � 0.1, c 2 � 1, ω � 0.8, R � I 4×4 , and T � 8. e control input u(t) is supposed to be constrained by |u(t)| ≤ u max � 2. By solving the linear matrix inequalities of eorem 2, the desired PI-type controller is derived as For graphically verifying the achieved results, we select the initial state disturbance ω(t) as

Mathematical Problems in Engineering
Added by the aforementioned controller, the simulation results are plotted in Figures 2-7. Figure 2 plots the simulated frequency, and Figure 3 displays the evolution of ACE. Meanwhile, the mode switching of actuator faults is shown in Figure 4, and control output is presented in Figure 5. Furthermore, with the disturbance given in Figure 6, the evolution of δ T (t)Rδ(t) is expressed in Figure 7. One can be observed from Figure 7 that the state of closedloop system stays in the prefixed region, which implies the resulting system is SFTB. Meanwhile, the input constraint is also satisfied.

Conclusions
In this study, the finite-time LFC problem for power systems with actuator fault has been considered. To better reflect the actual demands of practical dynamics, a generalized framework of the actuator constraint has been studied. Given the randomly occurring actuator fault, a homogeneous Markov chain-based actuator fault has been studied. Together with the piecewise Lyapunov theory, sufficient conditions have been attained. In the end, a numerical example has been applied to verify the effectiveness of the developed results.

Data Availability
No data were used to support the current work.

Conflicts of Interest
e authors declare that they have no conflicts of interest.