Stability Analysis of Cyber-physical System Under Transmission Delay

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Introduction
Recent advances in power systems have increased attention to the multi-area interconnected power system (MAIPS). To support control schematics in power systems, recent research has focused on applying different virtual inertia emulation methods to link the multi-power generations [1,2,3,4]. The control loops of load frequency control (LFC) send measurements and control data among networks such as the supervisory control and data acquisition (SCADA) system. Load frequency control is a crucial aspect of maintaining the stability of a power system. In power systems, the demand for electricity fluctuates constantly, and the supply must be adjusted accordingly to maintain a constant frequency. LFC is responsible for ensuring that the power supply matches the demand by adjusting the output of the generators in real time. 398 International Journal of Robotics and Control Systems Vol. 3, No. 3, 2023, pp. 396-416 ISSN 2775-2658 delays in state and control input was investigated in [17].
The small gain approach has been widely used by researchers to solve the stabilization problem of distributed systems. In order to maintain the stability of large-scale systems with a limited communication medium, an event-triggered sampling scheme with distributed controllers was investigated in [20]. Moreover, Song et al. proposed a robust pinning synchronization control method to ensure the recovery of the initial state for a complex system under mixed attacks in independent transmission channels [21]. A hierarchical game technique was introduced in [22] to address the control challenge of a wirelessly networked control system subject to a DoS attack.
The subject under consideration entails the design of an output feedback controller and a parameterbased method that ensures stability of a Multi-Area Integrated Power System (MAIPS) subject to load deviation and transmission delay. The present study offers the following contributions: • Firstly, an output feedback controller is proposed to stabilize a multi-area integrated power system in the presence of load deviations and denial of service (DoS) attacks.
• Secondly, this research puts forth a load frequency control (LFC) technique for the power system that incorporates the networking infrastructure features of the sample period and packet delay. A stability condition for a particular control system is established using Lyapunov theory. The stability criterion specifies the sample period and packet delay bands that ensure the power system's stability under the supplied control.
• Thirdly, the stability of a discrete-time MAIPS is discussed and analyzed such that stability is maintained even in the presence of transmission delay and load changes.
• Fourthly, this paper characterizes the limits on the frequency and duration of transmission delay for a MAIPS that implements a typical round-robin communication protocol.
• Lastly, an illustrative example of a typical MAIPS is provided with several scenarios that consider the designed controller with typical and modified sampling intervals, both in the absence and presence of transmission delay. These scenarios are utilized to verify the effectiveness of the proposed approach and controller.
The rest of this paper is organized as follows: Section 2 presents the modeling of a MAIPS. In Section 3, we provide the framework. Then, in Section 4, we discuss the output feedback controller and stability analysis of a MAIPS. In Section 5, we present a simulation of an illustrative example. Finally, in Section 6, we conclude our work.  (1).
Let N denote a set of indexed areas that are connected through tie-lines. In this case, subsystems are connected through a communication network, and the model that describes this is as follows: In this brief, x i (k) = [∆ω, ∆P mech , ∆P vi , ∆P tie,i ] T ∈ ℜ n represents the states of the system's frequency deviation, mechanical power deviation, steam valve position deviation, and deviation of the total tie-line power flow. u i ∈ ℜ m denotes the control signal of each subsystem, while y i ∈ ℜ p denotes the system observation. N refers to the set of connected subsystems, and f i represents the local variation on the demeaned load. In [23] the model is discussed in detail. The system matrices are defined as follows: Table 1 of generation area parameters for multi-area integrated power systems includes various parameters related to generators located in different regions. These parameters are mechanical power, power flow, angular rotating mass frequency, governor time constant, and frequency or load change. They are essential for maintaining the power system's stability and reliability. Mechanical power is the rate of energy production and conversion to electrical power by the generator. Power flow is the amount of electrical power transferred between the areas. The angular rotating mass frequency is the speed of the generator's rotor rotation. The governor time constant is the speed of the governor's reaction to a load variation. The frequency or load change is the system's reaction to changes in power demand. These parameters are applied in various power systems analysis methods, such as transient stability analysis and frequency response analysis, to ensure that the power system operates safely and can cope with disturbances. Considering the MAIPS described by the model (1). The target is to design a control system to guarantee the closed-loop stability of all areas on the power network. N in the model is representing the set of the neighbor's agent j. In this manner, the controller proposes to use the output feedback measurements of the subsystems to compute the input control signals for each time step as a distributed system. The controller's goal is to ensure the stability of the MAIPS. The diagram of the system with three areas was shown in Fig. 1.

Framework
Let's consider the discrete MAIPS expressed in Section 2. In (1), we find the mathematical model of the i − th subsystem on MAIPS. A wireless communication network is used to transfer data and measurements. A feedback controller uses these measurements to determine control signals and forward them to the systems' actuators to maintain a certain frequency of the grid at the desired frequency. In an ideal situation, signals arrive at the control unit with no time delay in a sample-andhold manner. For example, y i (k r i ) where k r i refers to received communication attempts.

Remark 1
We assume that output feedback gains K i and L ij exist that press all eigenvalues of matrix A ii to have a norm strictly less than one. In other words, each feedback area is Schur stable.

Transmission Delay
In this section, we introduce a time delay in the multi-area interconnected power system. To represent this time delay on the MAIPS, we consider the sampling time T s with a constant sampling period T s = t k+1 − t k and a stochastic time delay d k . We rewrite the sampling period at the controller as in Assumption 1.
Assumption 1 Let h be the fixed ideal period of sampling and the delay d k be random and limited to d k < d max .
where d k and h k are restricted to this condition 0 < d k < h k and 0 < h min < h k < h max . However, d k is independent and has a known distribution with an occurring frequency f k .
Where k ≥ τ for all k and τ .
We remark that the communication network generated a time delay d k that is less than h k but the packets were correctly received. By considering discrete-time dynamic (1) with measurements delay d k . The subsystem dynamic is written as: Remark 2 Each control input u i affecting subsystem i consists of two parts. The first part depends on the area output with a gain of K i , the second one depends on the area neighbors' output with a gain of L ij , such control signal is given by: Where K i and L ij denote the controller gains.

Output Feedback Controller and Stability Analysis
The aim is to administrate the stability of the MAIPS in the normal situation or with the presence of the communication time delay. Mainly we address the stabilization problem of MAIPS connected by tie-line in the nominal situation and under the influence of transmission delay.
At each transmission instant, let e i (k) refer to the error between the received states x d i (k) and the actual states x i (k) in each i-area. So, it could be written as: The objective of this work is to design an output feedback controller as in (6) that guarantee the stability of the close loop model described in (1). On the coming section, the Lyapunov theory is used to ensure the system (1) is exponentially stable.
The i − th subsystem dynamic could be rewritten by substituting (7) and (6) in (1), such as , and e j (k) the interconnected neighbors and errors, respectively.

Remark 3
As illustrated in (8) stability can be achieved in a weak couplings situation and within a small error e i (k). Besides, a design parameter σ has been introduced to clarify the "smallness" of e i (k). Assumption 3 (Inter-sampling interval). With nonexistence transmission delay there is an intersampling interval (∆) satisfying: σ i here is holds as a design factor. According to [24], the selection and design of ∆ and σ i is essential to guarantee the stability of the distributed systems.
Remark 4 Even if ∆ is not shown explicitly in (9), it can be noticed from the definition of the error (7) that the inter-sampling interval affects the stability of the system. Referring to Remark 3, Assumption 3 guarantees the "smallness" of the error by selecting a proper inter-sampling interval.
Remark 5 It is worth mentioning that designing the inter-sampling interval, ∆ in (9) is a considerable problem. An inter-sampling interval satisfying limits as (9) could be precisely resolved for centralized settings [24]. While some literature computes and applies a lower bound of time elapsed between two events to avoid Zeno behavior in asymptotically stable distributed/decentralized systems [25], [26].

Static Output Feedback Control Design
In this section, the main objective is to design a static output feedback controller in the form of (6) to achieve the asymptotic stability for nominal distributed systems (1). For static output feedback control design, the following two theorems are established.
Theorem 1 If the controller gains K i and L ij in (6) is given. The system in (1) is asymptotically stable if positive matrices P i exist and satisfy the following: By manipulating ∆V i (k)) using (11) we obtain: The aforementioned expression (13) is rewritten in a compact form using (14) as : The main result is summarized by the following theorem: and positive scalars ϵ i such that the following bilinear matrix inequality (BMI) is satisfied Then, the system (8) is asymptotic stable with the designed observer controller in (6) where K i , and L ij are given as: Proof 2 Ξ i in LMI (10) can be rewritten as: So, (18) is formulated using Schur complements as: and Y ij := R ij C j , the BMI can be obtained as in (16).
Remark 6 Theorem 2 was established in view of the work presented by Mahmoud and Nounou in [27].

By choosing the Lyapunov function
Then each subsystem i should satisfy: thus λ min (P i ) is the smallest eigenvalue of of P i and λ max (P i ) is the largest. To maintain the stability of the system, σ i must be selected based on the Lemma 1.
Lemma 1 For MAIPS described by (1) that controlled by (6). Let µ ∈ R N + to be any column vector satisfy µ T A < 0. Then, the MAIPS is asymptotically stable if there is σ i ∀i ∈ N such that Where for l i refers to the i-th vector of L := µ T (−A+Ψ)and j i refers to the j-th vector J := µ T Γ. The matrices A, Γ and Ψ are defined below.
Proof 3 Evaluating the Lyapunov function difference (13) based on (12), taking the norm and utilizing Young's inequalities lead us to By rewriting the inequalities (22) in matrix form: Where α i , α ij , γ ii , γ ij , and ψ i are written as Selecting δ > 0 to satisfy α i > 0 and the minimum eigenvalue of Q i refers as λ min (Q i ), i is the the i-th subsystem.
Let combine the vectors as follows: The inequality (23) could be compactly rewritten as: Assuming the spectral radius satisfies r(A −1 ii A ij ) < 1 then,there is a µ > 0 ∈ R n + such that µ T A < 0 [28]. The Lyapunov function selected as V (x(k)) := µ T V vec (x i (k)). Then ∆V yields: The demand load f i directly affects the frequency of the power generation unit [29], which is represented by the output in our model y = C i x i (k) as described in Section 2. So, for small f i we assume that where ϵ is a positive or negative depending on the load's type if capacitive or inductive loads.
Since,l i and j i are an entry of L and J vectors. So, we rewrite (34) as follows: leading to asymptotic stability with σ i < l i j i

Stability Analysis of a MAIPS subject to transmission delay
To achieve the asymptotic stability of the MAIPS in a normal situation with a round-robin protocol, we investigate selecting σ i to deal with the error limits. However, MAIPS stability is not guaranteed if the system is experiencing transmission delays. Our aim in this part is to address the stability of MAIPS subject to transmission delay.
Theorem 3 A multi-area interconnected power system (MAIPS) consists of N area as described in (1). The static feedback controller as in (6) is applied to control the frequency of MAIPS. Also, a sampling interval ∆ satisfying Assumption 3. The MAIPS under the influence of the transmission delay with frequency and duration satisfying Assumption 2. this system is asymptotically stable if where and ∆ * = N ∆, l i , j i , µ i , and σ i are as in Lemma 1. Lyapunov function derivative is given as: where ω 1 ; = min{ delay period), the Lyapunov function rewritten as: 2. During delay. Introducing z i m as the last attempt successfully transmitted on the channel prior to the Trans. By considering e i (k) as mentioned previously, this leads to and Calling (35), we can conclude that Thus, for all k ∈ H n (transmission delay interval) If i∈N ∥x i (h n )∥ 2 ≤ i∈N ∥x i (k)∥ 2 , the difference of the Lyapunov function rewritten as: with ω 2 := 4max{j i } min{µ i λ min (P i )} . Also, ∀k ∈ H n with i∈N ∥x i (h n )∥ 2 > i∈N ∥x i (k)∥ 2 , one has So, (44) and (45) implying that the Lyapunov function in the transmission delay period H n satisfy the following equation During the transaction between stable and unstable modes.we will reflect the the protocol waiting time, the Lyapunov function in this instant In conclusion, the overall behavior of the closed-loop system could be treated as a switching system with two modes. So, when simple iterations are applied to the Lyapunov function in and out of the presence of the transmission delay status, we will get To assure the stability of the last equation, (36) is obtained easily.

Remark 7
Theorem 3 offers a criterion to characterize the stability of the distributed system in the form of (1) with an output feedback controller in the form of (6) and in the presence of transmission delay. The transmission delay is assumed to have constrained frequency and duration as described in Assumption 2. The signals are exchanged over a communication channel with sampling interval ∆ satisfying Assumption 3.

Remark 8
The stability of the MAIPS is affected by the sampling interval (∆) of the Round-robin protocol since it delimits when the overall system is able to repair communication. In the case where the bounded duration and frequency of the transmission delay subjected to the MAIPS with applying appropriate Round-robin inter-sampling time diminish the left-hand side of (36) that ensures the stability of the MAIPS. However, this is at the expense of high communication facilities.

Simulation
The proposed static output control scheme is evaluated through a comprehensive simulation analysis, using a standard power system model Fig. 1 as a test case. A simulation result of three scenarios is shown in this section. The model incorporates various factors that affect the dynamics of the system, such as parameter changes, modeling errors due to disturbances in load and generation, time lag, and conventional generation sources. A MATLAB software version 2020a is used to construct a large-scale power system model with different tie-line dynamics to emulate a realistic system and test the performance of the proposed controller. The interconnected power system considered in (1) comprised of three subsystems are tested with parameters listed in Table 2. The controller gain calculated using YALMIP is given by

Designed Controller in Nominal Situation
As can be seen from Fig. 3, the first states (frequencies) of the system in the three areas are restored to the acceptable ranges within 20 seconds under a nominal condition. Fig. 3 also demonstrates that the MAIPS is equilibrium under standard conditions with the controller gain given by (48). The controller applied can eliminate the frequency deviation whenever there is a load variation.

Under Transmission Delay
In this scenario, a transmission delay was implemented on the system with the Assumption 2. In Fig. 4 Fig. 4. As the time delay increases, the system frequency exhibits more oscillations and becomes unstable, failing to converge to its steady state value within the acceptable range. The maximum deviation of frequency is observed to be large.

Stabilization under Transmission Delay
Figs. 5 and 6 demonstrate that the system frequency of three interconnected areas can be stabilized by the proposed controller algorithm in the presence of time delay using a modified sampling interval of T s = 0.01sec. The frequency deviation, overshoot, and settling time of the three interconnected power systems under nominal transmission conditions and two different sampling periods are presented in Figs. 3 and 5. The responses of the three areas in Fig. 5, have less overshoot and settle faster than those with the nominal sampling interval. Fig. 5 shows that the proposed design can achieve system performance to balance the load and generation with the adjusted sampling period.
Using Lemma 1 and YALMIP, we found the following:   Then we obtain:

Conclusion
This study examines novel output feedback for a cyber-physical system that includes power networks. It analyzes the stability of a round-robin communication system when faced with mixed cyberattacks and load variations. A static feedback controller with an adjusted sampling time is designed to maintain the stability of a multi-area interconnected power system (MAIPS) while minimizing the required performance function. The stability of the MAIPS is then determined when it is subjected to transmission delays, taking into account pre-established parameters for the delay's duration and frequency. Our findings indicate that time delays can influence system stability and that choosing an appropriate sampling interval is necessary to ensure the desired stability of the system. The proposed approach's feasibility is demonstrated through an example involving three interconnected power network areas under various scenarios.
Based on the conclusion of this study, here are some additional future suggestions and directions for research: • Investigating the impact of multiple cyberattacks and unknown loads: The study focused on mixed cyberattacks and load variations, but did not consider the effects of multiple cyberattacks or unknown loads on system stability. Future research could explore the stability of the system under multiple simultaneous cyberattacks and how unknown loads affect system performance.
• Dynamic event-triggered output feedback: The study used a static feedback controller with an adjusted sampling time, but dynamic event-triggered output feedback may be a more effective approach for maintaining stability. Future research could investigate the use of dynamic eventtriggered output feedback and compare its performance to that of the static feedback controller used in this study.
• Fault isolation in networked cyber-physical systems: The study did not examine the problem of fault isolation in networked cyber-physical systems. Future research could investigate fault isolation techniques that can detect and isolate faults in the system, ensuring that any failures do not propagate and lead to system instability.
• Developing real-time monitoring and control techniques: As power networks become more complex and interconnected, real-time monitoring and control become increasingly important for maintaining stability. Future research could explore the development of advanced monitoring and control techniques that can detect and respond to changes in the system in real time.
• Incorporating renewable energy sources: The study did not specifically consider the impact of renewable energy sources on system stability. Future research could examine how renewable energy sources, such as solar or wind power, affect system stability and explore ways to optimize the integration of renewable energy sources into the power network.