Abstract
This paper is concerned with the issue of finite-time load frequency control for power systems with actuator faults. Concerning various disturbances, the actuator fault is modeled by a homogeneous Markov chain. The 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.
1. 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 [1–3]. 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. Therefore, 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 [5–7].
In networked control systems, various faults can be encountered due to the long-term utilization of components [8–10]. 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 [14–16]. 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 [17–19], 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. [21–23]. These 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 load frequency control for power systems with actuator faults over the finite-time interval in this study. The 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.
The 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.
1.1. Notations
The notations of this paper are standard. means the Euclidean norm. indicates a set of -dimensional matrix. refers to the mathematical expectation. means the largest/smallest eigenvalue of matrix . means the occurrence probability. represents a block-diagonal matrix.
2. 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:whereand the system parameters are expressed in Table 1.
In single-area, the area control error (ACE) is interpreted as 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 aswhere , and each element . More specifically, is identified as a right-continuous Markov chain taking values over a set with generator , and its transition probabilities are inferred aswhere and , for and for each .
Taking the ACE as the desired controller input of LFC, the output of the proportional-integral (PI) controller is asserted aswhere and signify the proportional and integral gains of the area, respectively.
Let , , the power model (1) is reformulated aswhere
The purpose of this study is to solve the output feedback control for power system (6) with data sampling. Therefore, the sampling sequence attained at a set of time instants. Added by the data sampling technique, only the measured signal 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 is time-varying with the upper sampling period. Thus, one defines . Based on the input delay technique, we have that with for . Summarizing the above discussion, we have .
Letting , 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.
Assumption 1 (see [18]). The external disturbance belongs to , and it is assumed that there exists a parameter such that .
Assumption 2. From the viewpoint of the physical limitation of actuators in power systems, the control torque is assumed to meet
Definition 1 (see [26]). Given parameters , , time interval , and matrix , the closed loop power system (10) is called stochastic finite-time stability (SFTS) with respect to and , if inequality holds for .
Definition 2 (see [26]). Given parameters , , , time interval , and matrix , the closed loop power system (10) is called stochastic finite-time boundedness (SFTB) with respect to , if inequality holds for .
The object of this study is to design sampled-data-based controller (9) such that(1)The closed-loop power system (10) is SFTS with .(2)When disturbances , the power system (10) is called SFTB with performance index such that
Lemma 1 (see [27]). For any vectors and , scalar , a matrix , and symmetric matrices and , the following inequality holds:
3. Main Results
Theorem 1. For given parameters , , , , , , and matrix , the closed-loop power system (10) is called SFTB with respect to , if there exists matrix , , , , and , such that where
Proof. Establishing a Lyapunov functional as , wherewhereThe weak infinitesimal operator can be inferred asApplying the operator along the power system (10), which yieldsBased on Lemma 1, the following inequality can be devised:whereIt is well known that for any matrices , one getsThe aforementioned condition can be rewritten asOn the other hand, for any matrices and , it is clear thatSubstituting (23)–(32) into (20), it can be deduced thatwhereNote that (33) is a convex combination of and , in accordance with Schur complement; one can deduce that if and only if (15) and (16) hold. Therefore, one can see thatBy integrating the both sides of (35) from to and simple derivation, it yieldsRecalling the Lyapunov functional (20), we can getSubstituting (37) and (38) into (36), we can obtainIn light of (17), it can be concluded from (39) that . Thus, 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 hasRecalling Assumption 2, it yieldsAccording to Schur complement, (18) can be guaranteed by (41), which completes the proof of Theorem 1.
Theorem 2. For given parameters , , , , , , and matrix , the closed-loop power system (10) is called SFTB with respect to and meet an performance index , if there exists matrix , , , , , and matrices , with suitable dimensions, such that whereThe controller gain is deduced as
Proof. By resorting to the same Lyapunov functional as displayed in (20), and adopting the similar derivations in the proof of Theorem 1, for , the following inequality can be realized:Integrating the both sides of (49) from to , it yieldsClearly, (50) can be rewritten asUnder the zero-initial conditions, one gets . It follows from (51) thatSince and , (52) signifieswith . Therefore, the performance index is ensured.
Meanwhile, in light of Lemma 1 of [28], conditions (42) and (43) can be easily attained. Similarly, (46) can be ensured. The proof is completed.
4. Computational Experiments
In this example, a single-area power system is studied for simulating the derived results. Similar to [6], the parameters of system (1) are expressed in Table 2.
Taking the actuator faults into consideration, which is characterized by a Markov process, the possible fault coefficients are , , and . Furthermore, the transition probability matrix among faults is given by
On the other hand, other parameters are selected as , , , , , , , and . The control input is supposed to be constrained by . By solving the linear matrix inequalities of Theorem 2, the desired PI-type controller is derived as
For graphically verifying the achieved results, we select the initial state disturbance as
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 is expressed in Figure 7. One can be observed from Figure 7 that the state of closed-loop system stays in the prefixed region, which implies the resulting system is SFTB. Meanwhile, the input constraint is also satisfied.
5. 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
The authors declare that they have no conflicts of interest.