Abstract
This study focuses on the protocol-based control for single-area power systems subject to actuator failures and deception attacks. Specifically, actuator failures, network attacks, unreliability, and bandwidth restrictions that emerge in power systems are taken into consideration at the same time. To cut down on the number of broadcast packets, a novel memory-adaptive event-triggered protocol is developed, where the trigger threshold parameter is adaptively changed in accordance with numerous historical sampled signals. Then, in virtue of the proposed algorithm, sufficient stabilization conditions are acquired to ensure the asymptotically stable of power systems with performance. Finally, the efficiency of the proposed control strategy is demonstrated by using a simulation example.
1. Introduction
Due to its potent capacity to change the system frequency to a predetermined value in the presence of load fluctuations, load frequency control (LFC) has been utilized successfully in power systems for several decades [1–3]. Modern LFC in power systems transmits control signals and measured values across open communication networks (OCNs) as opposed to classic LFC, which transfers data over specialized communication channels. As the size of the power system grows, data transmission over a dedicated communication channel in traditional LFC will increase maintenance costs and reduce flexibility; hence, modern LFC with OCNs is becoming more popular [4]. However, as OCNs are prone to data loss, network attacks, and communication delay, data transfer through them might present considerable difficulties. Consequently, there has been a lot of interest in the study and development of LFC schemes for power systems with OCNs (see, for more details, [5–7]).
Actuators are crucial components of networked control systems, and it is a common phenomenon that a variety of malfunctions occur, which may affect the system performance [8]. In this regard, for the purpose of guaranteeing the desired performance, a seemingly natural ideal is to introduce reliable control schemes. As discussed in [9], benefiting from the reliable control schemes, actuator failures can be compensated and avoided. As a consequence, a lot of focus has been placed on the research of actuator failures in an effort to address these shortcomings and boost dependability, and several findings have been published [10, 11]. The finite-time tracking control problem for nonlinear systems with faults has been studied in [10]. Besides, the fault-tolerant control problem for multiagent systems subject to DoS attacks has been exploited in [11]. However, reliable control schemes have not gained sufficient interest in power systems probably due to dynamic complexities including unknown perturbations. To date, despite considerable accomplishments, power systems subject to reliable control schemes are still in their infancies, which remain the first motivation for this study.
It is essential to mention that communication networks are large-scale and decentralized and that the links between each part of a networked power system might make them vulnerable to cyberattacks [12]. According to [13], cyberattacks can have significant negative effects on system performance, such as information leaks, system failures, and financial losses. Additionally, control methods might be used as a compensating strategy since cyber security approaches are insufficient to secure power systems [14]. Among some control methods, denial-of-service (DoS) attacks and deception attacks have generated a lot of research attention [11, 15–18]. Some common concepts or solutions have been carried out for DoS attacks. From the point of adversary, DoS attacks have been described by stochastic models, i.e., the Bernoulli method [19] and Markov method [20]. Additionally, from the standpoint of the attacker, deception attack can alter data to compromise its integrity, and the time-varying attack behavior cannot be detected [21]. As a result, the study of deception attacks becomes more challenging ([14, 18, 22]). Following this trend, a seemingly natural research topic is established for security control of networked power systems, which gives rise to deception attacks.
In light of the limited communication capacity, a range of networked phenomena emerged. In response to communication networks with constrained network resources, two transmission strategies are most adopted: the time-triggered protocol [23–25] and the event-triggered protocol [26, 27]. Note that conventional TTPs may result in wasted network resources; the event-triggered protocol has been developed in recent years to address these issues. Because trigger conditions are carried out after the event as opposed to before it, ETPs typically use fewer system resources than TTP. In general, the existing ETPs can be classified into static event-triggered protocols (SETPs) [15, 28], dynamic event-triggered protocols (DETPs) [29–31], and adaptive event-triggered protocols (AETPs) [32, 33]. For AETPs, threshold functions, based on the evolution of system states, are updated adaptively. Although there are several AETPs with dynamically changed threshold functions, there is still significant potential for development. For instance, in [32, 33], the threshold function is built using a quantitative relationship with the error between the latest transmitted data and the currently sampled data. Inspired by the work of the authors of [34], the aforementioned approach can be improved by containing the historically transmitted packets in the threshold function. Furthermore, memory-based AETPs have not gained proper research interest in power systems, which prompts us to the current study.
In light of the description above, this study examines the based-protocol LFC problem for single-area power systems (SAPSs) that are vulnerable to deception attacks and actuator failures. The following is a summary of our paper’s main points: (1) A memory-adaptive event-triggered protocol (MAETP) is presented for operating SAPSs across communication networks with constrained bandwidth. Meanwhile, the MAETP accomplishes the goal of memory by dynamically altering the adaptive parameters using historical trigger data while preserving the intended control performance. (2) The proposed SAPS results account for a common framework together with the effects of deception attacks, actuator faults, and memory-based event-triggered protocols. (3) For the established power system model, according to the Lyapunov stability method, the asymptotic stability (AS) with preset performance is ensured.
2. Problem Formulations
2.1. System Model
Throughout the study, the dynamic model of a single-area power system is described as follows [35]:where
Table 1 provides a list of parameters’ physical meanings.
Note that, for a single-area power system without power exchange, the ACE signal is written as , where is frequency bias. In actual fact, actuator failures cannot be ignored, so the failure model between the controller and the actuator is represented as follows:where and , in which is unknown, and we assume that and are known. We define and .
As a consequence, a PI controller is inferred as
Furthermore, we define state vectors as and measured output as ; the dynamic model of the single-area power system is redescribed aswhere
2.2. Memory-Adaptive Event-Triggered Protocol
By using some historical data, a sampling-based MAETP is proposed in [34]:and . , , and describe the sampling interval, instant, and latest broadcast instant, respectively. The matrix . Meanwhile, . In the sequel, the adaptive parameter yieldswhere and indicate two bounds of the adaptive threshold parameters, , and is the number of recent released packets.
Remark 1. Note that the adaptive threshold function of MAETP (7) taken into consideration in this study, which substitutes the latest trigger sample with the arithmetic mean of historically trigger data, can lessen the sensitivity of to the most latest transmitted data . As in [18], the proposed controller gain number is , which increases the computing cost if the amount of historical data is too large. To avoid this situation, the MAETM proposed in this research adds the memory feature to the adaptive rule.
Remark 2. In addition to providing flexibility in modifying the trigger threshold, the MAETP (7) also improves control performance. Nevertheless, the MAETP (5) provided in this research is more inclusive and covers the majority of the current protocols. When , MAETPs (7) reduce to AETPs [32]. When and is a constant, the MAETP (7) degenerates to SETPs [15]. Following this fact, the designed DMETP is more appropriate than the current SETP/AETP to describe the actual scenario.
Considering the delay in transmitting data, we define the transmission interval as , and one gets and . We set , which yieldswhere is the upper bound of .
Summarizing the aforementioned analysis, we let . Under the MAETP, the PI controller is rewritten asAs a follow-up, the measurement output is assumed to be attacked by random deception attacks. In this regard, the load frequency controller is remodelled aswhere the nonlinear function indicates the deception signal. is a Bernoulli random variable, from which one has
Assumption 1 (see [15]). It is assumed that the nonlinear function can be characterized by the following condition:where is a known matrix.
Substituting (11) into (5), the closed-loop power system can be established asIn order to solve static output-feedback control questions, we let , in which has [36]. Then, we introduce a new variable ; system (15) is reformulated aswhere , , , and .
The goal of this paper is to construct the controller (11) in such a way that it satisfies asymptotically stable subjects to preset performance for the closed-loop power system (16) under MAETPs (7). In particular, the following requirements are met:1)The closed-loop power system (16) with is AS.2)Under zero initial conditions, it holds that for all and prescribed .
Lemma 1 (see [37]). For any matrices and , we satisfy , , with , and the vector function ; the following condition holdswhere and .
3. Main Results
Theorem 1. For given scalars ,,,,, and , the closed-loop power system (16) is AS in the sense of the performance index if there exist matrices ,,, and and matrices and such thatwhere
Proof. Considering the following Lyapunov function,whereTaking the derivative of , one hasBased on Lemma 1, it followswhereReviewing the established MAETP in (7), it holds thatRecalling (15), for any proper matrix , it holds thatThen, based on Assumption 1, one hasTaking (20)–(27) into account, we have thatwhere , .
Applying the Schur complement to (19), one hasIntegrating (29) from to yieldsUnder zero initial conditions, the following inequality holds:Furthermore, we get . When , we can deduce the closed loop power system (16) AS by applying condition (19). This ends the proof.
Next, Theorem 2 presents essential conditions for designing the controller gain based on the conclusion of Theorem 1.
Theorem 2. For given scalars ,,,,, and , the closed-loop power system (16) is AS in the sense of the performance index if there exist matrices ,,, and and matrices and with compatible dimensions such that (18) holds andwhere
Furthermore, the gains can be computed as
Proof. Let , and it is clear thatwhere is a diagonal matrix. On account of the Schur complement, (32) is ensured in virtue of (16).
4. Numerical Examples
This section exhibits a simulation example in order to evaluate the effectiveness of the given methodology, Figure 1 describes dynamic modes of the single-area power system. Note that the parameters are explained in Table 2 [35], and the matrix is presupposed to be .
We set the deception attack as , which . The actuator failure is supposed to be with and . Other parameters are chosen as , , , , , , , and the sampling period .
In light of Theorem 2, the controller gain and the event-triggered matrix can be calculated:
Specifically, we select the load disturbance asand the initial value is set as .
The numerical simulations of the system (16) are presented in Figures 2–6 using the aforementioned parameters and assumptions. Figures 2 and 3 depict the control input and the trend of the adaptive parameter . Figure 4 illustrates the deception attacks signals with the expectation . Additionally, to demonstrate the superiority of the proposed MAETP, we compared it with the METP of literature [18], as shown in Figures 5–7, where Figures 5 and 8 depict the power system state trajectory curves under different ETPs, respectively, and Figures 6 and 7 correspond to their triggering instants. In Figures 5 and 8, it can be seen that the system curve takes less time to reach stability with the proposed MAETP. Furthermore, 108 packets have been transmitted under METS, while 68 packets have been transmitted under MAETS. Therefore, by comparison, it can be found that the proposed MAETP not only achieves good control performance but also saves transmission resources to a certain extent.
5. Conclusions
In this paper, a protocol-based LFC issue for SAPSs has been discussed. In light of the network’s actual state, the networked power system is expanded to include actuator failures and deception attacks. Furthermore, by adaptively modifying the trigger thresholds based on historical sample data, MAETPs are developed to conserve network resources. By using Lyapunov function theory, sufficient criteria have been forwarded to guarantee the AS of SAPSs. In the end, an example has been applied to demonstrate the viability of the presented control scheme.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.