#### Abstract

In this study, we study the load frequency control (LFC) problem for interconnected multiarea power systems (IMAPSs) with quantization and actuator failure. To effectively reduce the amount of data in the channel, input signals will be quantized before being transmitted from a controller to a system through the digital communication channel. To reveal the asynchronous phenomenon between the original plant and LFC with actuator failure, a hidden semi-Markov model is formulated. In addition, the stability of the jump system under network attack is discussed. On the basis of the Lyapunov theory, sufficient conditions are derived to ensure the stochastic stability of IMAPSs. Finally, the validity of the theoretical results is tested via a simulation example.

#### 1. Introduction

The power system is a complex nonlinear system, which has developed into a multiregional interconnected power system (PS) since the Industrial Revolution. To deal with the low-frequency little oscillations of interconnected PSs, the LFC was proposed in [1], which has been effectively applied to PSs [2–4]. According to the LFC technique, the frequency can be adjusted at a desired level, which guarantees the stability of entire PSs. Over the past few decades, researchers have proposed a number of techniques concerning with LFC, such as use of the integral control law [5], PI case [6], and PID case [7]. These approaches have been verified to improve the control performance of interconnected PSs [8, 9].

In practical applications, the dynamic systems may undergo sudden changes in their parameters or structures due to component failures, sudden environmental changes, etc. In this case, Markov chains are widely adopted to model the variations in PS states. In [10], the Markov chain was employed to describe the random mutations of the discrete-time PS. In [11], the uncertain Markov chain was applied for the decentralized control of the PS. However, in most existing literature concerning Markov PSs, the residence time of Markov processes obeys a memory-free random distribution, in which the probability of the transition rate is time-independent. As signified in [12–14], compared with the conventional Markov chain, the semi-Markov chain is more general in approximating practical dynamics owing to its time-varying transition rate. Consequently, it is meaningful to study the LFC problem for PSs with semi-Markov jumping parameters, the so-called semi-Markov PSs. To the best of our knowledge, quite a few theoretical results have been applied to semi-Markov PSs due to their inherent difficulty, and this motivates this article.

In reality, the signal is communicated through a limited bandwidth network [13, 15, 16]. Note that massive signals are transmitted via the limited network, which may lead to channel congestion, thus reducing the system performance. To overcome this shortcoming, we quantify the control inputs, in which quantization stands for the process of mapping the continuous values of a signal to a limited number of discrete values [17, 18]. By means of quantization, the amount of data in the channel and load of the channel can be effectively reduced. The quantizers can be roughly categorized into linear quantizers [19], and logarithm quantizers [20, 21]. The problem of stabilizing a continuous-time switched system affected by the time-varying delay and data quantization has been addressed in [22]. In the quantitative input multiarea, however, results of the power system are very few; in order to fill the gap, this article takes into account the interconnected more regional power system with quantitative input, and the quantification method is used as the basis of the design, making the obtained quantitative instruments with the crudest density and further reducing the burden of transmission. We would like to mention that, in practice, capturing system information is a tricky task. Therefore, the asynchrony between the system mode and the controller mode cannot be omitted. Nevertheless, the asynchronous control of the semi-Markov IMAPS has not been researched thoroughly, which partially motivates the current work.

Motivated by the above considerations, this work considers the LFC problem of the IMAPS subject to quantization input. Different from the existing homogeneous Markov IMAPS, the semi-Markov chain is employed to describe the dynamic behavior of the IMAPS. Aiming to describe the asynchronous phenomenon between the original plant and LFC, the hidden semi-Markov model is formulated. By resorting to the Lyapunov theory, sufficient conditions are derived to ensure the stochastic stability of the resulting dynamic. In the end, one numerical example is inferred to show the correctness of the proposed method. The general structure is rendered as follows: the second section describes the asynchronous LFC of the semi-Markov PS with quantization form. In section 3, sufficient conditions of random stability are given. A numerical example is given in section 4.

Notations: means a block-diagonal matrix; ; means is positive definite; implies occurrence probability; indicates the mathematical expectation; signifies the Euclidean vector norm. signifies the identity matrix; and stand for the transpose and inverse matrix, respectively.

#### 2. Problem Formulations

##### 2.1. System Model

In this study, the dynamic model of the multiarea LFC is described as follows:where , , , and represent the state, the output, the control input, and the disturbance, respectively, and

with , , and . Each signal is described as a linear combination of the tie-line power exchange and frequency deviation, i.e., , where refers to the net exchange of the tie-line power. The nomenclature of other parameters is shown in Table 1.

In view of the uncertain fault time of the power system, the semi-Markov process is adopted. For all ,

Then, is said to be a renewal process if where with

Meanwhile, the probability distribution function can be described as follow:

According to the aforementioned observation, we have

Hence, by simple calculation, the transition rate matrix can be defined by , whereand .

##### 2.2. Asynchronous Control Input with Quantized

As exhibited in Figure 1, the control input is required to be quantized before sending it to the power system. Inspired by this fact, the logarithmic quantizer can be described as follows:where the th subquantizer satisfying , . The set of the logarithmic quantization level can be described aswhere and on behalf of the quantizer density and the initial quantization, respectively .And . Then, we define the subquantizer as

Then, we have thatwhere , .

In this work, the -th control area of an asynchronous PI controller is given bywhere , and are the -th proportional and integral gains, respectively, and . The variable that refers to the Markov chain belongs to the space , whose conditional probability matrix is inferred withwith . According to equations (11) and (12), the control signal can be devised as follows:where , and .

##### 2.3. Model Transformation

Let and , substituting equations (14) into (1), the closed-loop IMAPS is formulated as follows:

For analysis convenience, based on the compatible matrix , we can obtain a full rank matrix . We define , we have thatwhere , , , and . It is worth noting that linear transformation is invertible. Thus, the overall IMAPS can be inferred aswhere , , , , and

*Definition 1. *([23]) The interconnected power system with is called stochastic stability if the following equation holds:Under the zero initial condition, the system with and , the performance index is satisfied:

Lemma 1. *([21]) For the given matrix and matrices and with appropriate dimensions, if inequality holds for all , for any scalar , such that .*

#### 3. Main Results

Theorem 1. *For given scalars and and the matrix , the IMAPS is stochastic stability with preset performance, such thatwhere*

*Proof. *Establish the semi-Markov-based Lyapunov function as follows:It follows thatOn the basis of equation (17), for any proper matrix , such thatIt follows from equations (24) and (25) thatwhere , andIn case of , we have , by equation (21), we have , when , , . Subsequently, we havewhere , we can further haveFurthermore, for , according to equation (24), it yieldsNote that , from which one can obtainwhich indicatesthis completes the proof.

Theorem 2. *For given scalars , , and , system equation (13) is stochastic stability with the preset performance index , if there exist matrices and , such thatwhere , **Furthermore, controller gain matrices are given by .*

*Proof. *Let , with the purpose of structure of the matrix . In detail,According to Lemma 1 and by using the Schur complement in equation (21)(10) the proof is completed.

#### 4. Numerical Example

In this section, to show the effectiveness of the attached methodology, a numerical example of the 3-area interconnected semi-Markov switching power system is presented. Suppose that the coefficients are selected , , and , . Furthermore, for any , set

Let , , , and . Other nominal parameters are listed in Table 2 [3, 24].

Specifically, the transition rate function can be written , and the Weibull distribution density function is given by , where and indicate the parameters of the shape and scale, respectively. Noting that when , we choose and . Otherwise, for , we set and . Accordingly, the transition rate matrix can be expressed as

Thus, we have

In order to better describe the asynchronous phenomenon, the condition probability matrix is set as follows . According to Theorem 2, the asynchronous controller gains can be easily devised. We select the initial state , , , and the load disturbance , . Added by the above-derived gains, the mode evolution of and is shown in Figure 1. The trajectories of the state are plotted in Figure 2. Figure 3 thatdepict the state trajectories and the measured output. These simulation results verify the effectiveness of the proposed method.

#### 5. Conclusions

In this study, the problem of the asynchronous load frequency control problem for semi-Markov interconnected multi-area power systems with the quantization effect has been addressed. In this case, the system under consideration is modeled as the semi-Markov jump system. The merit of this work is to tackle the asynchronous phenomenon between the control and semi-Markov interconnected multi-area power systems. By designing an asynchronous controller with quantized form, the quantized closed-loop system has stochastic stability under the specified performance. At last, the effectiveness of the developed method has been tested by the simulation result.

#### Data Availability

The data used to support the findings of this study are included in the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.