Probability-Density-Dependent Load Frequency Control of Power Systems With Random Delays and Cyber-Attacks via Circuital Implementation

This paper studies the decentralized <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> secure load frequency control issue and circuital realization of multi-area networked power systems subject to random transmission delays and deception attacks. To make full use of the stochastic feature of the network-induced transmission delay, its distribution described by the probability density function is utilized. Due to this feature, the normal control signal and the injected deceptive attack signal transmitted over the network can be formed as two distributed delay terms. Then, the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>th load frequency control area is established as a new distributed delay system, in which the delay probability density is treated as the distributed kernel. By utilizing an integral inequality dependent on the kernel, new sufficient controller design conditions are derived to guarantee the system stability with given <inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> performance. Moreover, a physical execution approach is addressed to transfer the load frequency control systems into electrical analogy circuits. The effectiveness of the proposed approach is illustrated via the professional simcape toolbox in Simulink/MATLAB for circuit simulations.


I. INTRODUCTION
A S WE known, there are basically three layers of the control framework of power systems, which are primary control (conventional droop control), secondary control (voltage/frequency regulation), and tertiary control (economic dispatch and power flow optimization) [1]. As a representative secondary control, load frequency control (LFC) is a critical strategy to maintain the frequency stable for power systems and much effort has been devoted to studying it [2]- [8]. In a multi-area power system, the centralized strategy of LFC utilizing global system information and the distributed control method using the neighbor subsystem information are effective strategies to regulate the system frequency. Nevertheless, the utilization of power interchange through tie-lines among subsystems increases the complexity and difficultly of frequency regulation. Compared with them, the decentralized LFC scheme investigated in [9], [10] is more practical and easier for implementation because only the local area information is utilized to attenuate the frequency deviation. In recent years, as the trend of the integration of physical power systems and communication network, networked power systems (NPSs) have received much attention from a lot of researchers.
With the introduction of communication network, the network-induced delay is a common phenomenon, which could degrade the system performance, even make system unstable [11]. However, some published results of LFC systems using proportional-integral (PI) controller have not considered the network communication delay [12]. To tackle this problem, a decentralized LFC method for power systems with multi constant communication delays is developed in [13], where the constant delays are treated as multiplicative attack [21], [22], deception attack [23], [24]. Concretely speaking, a resilient event-triggered LFC approach for NPSs subject to DoS attack with finite energy is addressed in [20]. Different from DoS attack by blocking the connections of nets, deception attack aims to change the real value of control signal by injecting a false data, which makes it more deceptive. Considering multi-area NPSs with time-varying communication delay and deception attack, a resilient LFC approach is addressed in [25], in which the deception attack is described as a time-varying delay-dependent function. Reference [26] studies the H ∞ LFC issue of multi-area NPSs with DoS attack and deception attack, which is also modeled as a time-varying delay-dependent function. In [25], [26], the timevarying delay-dependent deception attack model is established only by using the bounds of delays. Compared to the conventional interval time-varying delay method, using not only the upper and lower bounds, but also the delay probability distribution is potential for obtaining less conservative results.
It is known that testing the effectiveness of control strategies in some practical systems is costly and difficult. Therefore, it is worthy to develop suitable hardware simulators for some real control systems. Due the fact that many control devices of practical systems are executed by electrical equipments, it is natural and reasonable to transform the physical systems into analogy circuits to verify the validity of control method and save the testing cost. For instance, a circuital implementation method is studied to convert the robust adaptive fault tolerant control schemes into electrical circuits in [27]. For uncertain nonlinear multi-agent systems with actuator faults, the adaptive control issue is developed and its circuital realization is studied in [28].
Based on the above motivations, this article addresses the H ∞ load frequency regulation of NPSs with random delays and cyber-attacks in a decentralized method. A circuital realization approach is proposed to simulate the dynamics of power system components and the controller via equivalent analogy circuits. The following new challenging problems are introduced in developing the model and obtaining the stability and synthesis conditions. 1) Since the existing methods [15], [25], [26] based on an interval time-varying delay model do not take into account the probability distribution of random delay, how to model the random delay by making use of the distribution information is the first challenging problem.
2) Considering the first challenging problem, a new distributed delay model with the kernel representing the delay distribution will be proposed in this paper. However, the existing method [29] based on Legendre polynomials to handle this new model will cause approximation error and further lead to conservative results. Thus, how to remove such approximation error and reduce conservativeness is the second challenging problem.
3) There exist stochastic transmission delays when the attackers inject attack signals into network, therefore, how to model and deal with the deception attack will also bring the challenging problem similar with handling the random delay.
To solve these challenging issues, the main contributions of this work are given below: 1) To make efficient use of the stochastic feature of the network-induced transmission delay, a novel distributed delay model with kernel is established, where the kernel is used to represent the delay probability density. To model the random deception attacks, not only the stochastic feature induced by attacker, but also the stochastic feature caused by random delays are considered in this paper. Then, a new representation of random delay and deception attack is proposed for the first time. This model is more general than the traditional one based on interval time-varying delay method [15], [25], [26].
2) In the existing result [29], the distributed delay with kernel is approximated by Legendre polynomials and handled by Bessel-Legendre inequality, which could lead to approximation error and design conservativeness. To solve this problem, the distributed kernel is utilized directly to construct a new Lyapunov-Krasovskii functional (LKF) in this paper. An integral inequality based on the kernel is used to deal with the distributed delay term directly without any approximation error. Then, new sufficient and less conservative conditions formed by linear matrix inequalities (LMIs) are derived to design the decentralized H ∞ secure controller.
The organization of this paper is provided as follows. The preliminaries of modeling the closed-loop NPS is given in Section II. Then the stability and controller design conditions are obtained in Section III. Some simulation results are implemented in Section IV. Section V summaries the conclusions and presents some future investigations.
Notation: In this paper, we define He

II. PRELIMINARIES
In the LFC problem of power systems, there exist some complicate nonlinear dynamics. Due to the fact that the load variations of power system are usually small, it is permissible and useful to linearize them around the operating point [30], [31]. In many existing results [6]- [8], [31], the dynamics of governor, turbine and power generator are linearized as the following first-order transfer functions: where T gi , T ti , D i , M i and R gi are the governor and turbine time constants, damping coefficient, inertia moment of generator and droop coefficient of the ith control area, respectively.
In this paper, following the above linearization method, the structure of the linear dynamic model of the ith control area in a multi-area NPS is illustrated in Fig. 1 , P ti (t) and P di (t) represent the deviations of system frequency, tie-line power exchange, valve position, mechanical output of turbine and load disturbance of the ith control area, respectively. Then, the differential equations to describe the system dynamics of the ith control area [6], [25] are given as: . (1) The ith area control error signal used to maintain zero steady-state error for frequency deviation is defined as where β i denotes the frequency bias factor. Selecting the same state, output and disturbance in [25] as , one can represent the ith LFC system in the form of state-space by: in which z i (t) is the performance output and In order to keep the frequency f i (t) at the prescribed value, the following decentralized LFC scheme based on the conventional PI control is utilized: where K i = [K Pi K Ii ] and K Pi , K Ii are the PI controller gains to be designed. When the control signal is transmitted over the communication network, it is highly possible that the transmitted signal is delayed and damaged by sudden congestion and malicious attackers. Considering these factors, the output of controller with delay and deception attack is represented as where τ (t) ∈ [0, h] means the random transmission delay; h is the upper bound of the delay; λ(t) ∈ {0, 1} is a Bernoulli variable to describe the occurrence of deception attack by λ(t) = 1 and no attack λ(t) = 0; the mathematical expectations of λ(t) = 1 and λ(t) = 0 are E{λ(t) = 1} =λ and E{λ(t) = 0} = 1 −λ, respectively. In some existing results [8], [26], the deception attack is usually bounded and modeled by a Lipschitz nonlinear function related with system state or output. Following the same way in [8], [26], the deception signal ϕ i (t) is assumed to satisfy where H i is a given constant matrix used to describe the upper bound of deception attack. It is noted that the random transmission delay τ (t) usually meets some probability distributions. For convenience, we abbreviate τ (t) as τ . For example, based on the described scenarios in [18], [32] to simulate the network channel, the communication delay with upper bound h = 0.2s and its normalized distribution histogram are drawn in Fig. 2.
To utilize the random information of delay τ , its probability density described by the kernel g(·) with h 0 g(s)ds = 1 is considered, which can be accessible by statistical method. In Fig. 2, a probability density function g(τ ) is used to approximate the probability distribution, which is chosen as Then, we define g 0 (s) = g(s) and construct a new vector g(s) = [g 0 (s), g 1 (s), . . . , g i (s), . . . , g (s)] T with ∈ N, where all the elements g i (s) are independent. According to [33], the basic principle of choosing g i (s) is that the chosen g i (s) should make the vector g(s) satisfy the property: which means the elements g i (s) are the solutions of linear homogeneous differential equations with constant coefficients in the matrix G ∈ R × . By utilizing the delay distribution information, the stochas- with the probability density of stochastic delays is expressed as: Based on (8), the ith closed-loop LFC cyber-physical power system becomes ⎧ ⎪ ⎨ ⎪ ⎩ẋ Remark 1: In this work, a new model of LFC system with random communication delays is established as a distributed delay system with kernel. In contrast to the conventional model based on an interval time-varying delay system only using the upper and lower bounds of delay, the delay probability density is taken into consideration in system modeling. Then, the presented distributed delay modeling approach has the potential to obtain less conservative results.
Remark 2: To describe the feature induced by random behaviors of adversaries, a Bernoulli variable λ(t) is adopted to represent the control signal is attacked or not. Besides, the random feature caused by the delayed deceptive signal is also utilized in (8), which is not considered in [25], [26]. Thus, the presented model of deception attacks is more general than these existing results.
This article aims to design the controller (8) such that 1) The system (9) is mean-square asymptotically stable for Before further proceeding, a useful lemma is given as below.
Lemma 1 [33]: For a symmetric matrix Y > 0 ∈ R n×n and the vector g(s) given in (7), it gives

Remark 3:
The Bessel-Legendre inequality in [29] uses Legendre polynomials to approximate the probability density, which leads to approximation error and further yields more conservative results. Different from this method, our method based on (10) can handle the probability density directly without any approximation error. Thus, less conservative results are obtained by the integral inequality (10) in Lemma 1 than the Bessel-Legendre inequality in [29].

III. MAIN RESULTS
To derive the controller design conditions, the Lyapunov method is employed and the following Lyapunov-Krasovskii functional (LKF) is constructed as where Based on G(s), G xi (t) and G ϕi (t) defined in (11), one has with I 1 = [ I 5 0 5,5 ] and I 2 = [ I 2 0 2,2 ]. By substituting (12) and (13) into (9), the ith closed-loop power system is further expressed as To simplify the derivation, ζ i (t) and I a are defined as where Then, some sufficient H ∞ stability analysis criterions for the system (14) are provided in the following theorem.
According to the stability results in Theorem 1, the controller design conditions of system (14) are derived in Theorem 2.
Theorem 2: For given scalars μ 1 , μ 2 , h, ρ, the system (14) is asymptotically stable in mean-square with required for i = 1, 2, . . . , n, wherẽ Moreover, the controller gains are computed by K i = L i N −1 i . Proof: Adopting Schur complement to (18) Left-and right-multiplying Utilizing the inequality −N T is not a strict inequality. Then the problem of tackling the  nonlinear item X T i B i K i C i in Theorem 1 can be transformed as a W-problem [35].
In terms of By using Schur complement to (40), one can get which equals the condition (37) with a sufficient small constant > 0. Then, the proof is fulfilled.

IV. CIRCUITAL IMPLEMENTATION
In this section, a specific circuital execution method is used to imitate the considered LFC power system, where the system components and controller are realized by some equivalent analogy circuits based on resistors, capacitors and operational amplifiers. For simplification of presentation, the first control area (i = 1) of a multi-area NPS with three control areas is discussed below.
It is noted that the dynamics of system components are described via some first-order inertial elements in Fig. 1. By defining the parameters as: r 1 = r 2 , T gi = r 1 c 1 , r 3 = r 4 , T ti = r 3 c 2 , D i = r 6 r 5 , M i = r 5 c 3 , the equivalent analogy circuits of them are shown in Fig. 3 (a), (b) and (c), respectively. The droop constant can be viewed as a proportion element, and its equivalent analogy circuit is shown in Fig. 3 (d) by choosing 1 R gi = r 8 r 7 . The circuital realization of the PI controller is drawn in Fig. 4, where a proportion circuit, an integration circuit and an adding circuit are used to represent u i (t) = K Pi ACE i (t) + K Ii ACE i (t)dt with K Pi = r 10 r 9 and K Ii = 1 r 11 c 4 .  The signal ACE i (t) of (2) shown in Fig. 5 is the sum of the signal β i f i (t) and the signal P tie−i (t), which can be realized by adding circuit with OA and some resistors with the same resistance values. The signal β i f i (t), the product of the bias β i and f i (t), can be achieved by proportion circuit with β i = r 19 r 18 . The signal P tie−i (t) equals to the sum of T 12 ( f 1 (t) − f 2 (t)) and T 13 ( f 1 (t) − f 3 (t)), which can be derived by integration circuits with r 14 c 5 = 1 and r 17 c 6 = 1. Moreover, the signal T 12 ( f 1 (t) − f 2 (t)) is obtained by subtraction circuit and proportion circuit with T 12 = r 13 r 12 . The same process of deriving the signal T 13 ( f 1 (t) − f 3 (t)) is executed with T 13 = r 16 r 15 . Remark 5: The equivalent analogy circuits for power systems can be viewed as low-cost simulators, which are used to test the effectiveness of the proposed control method. This is helpful for avoiding the serious hazard induced by potential drawbacks of designed controller to expensive devices of practical systems. Meanwhile, the cost of testing can be saved dramatically.

V. EXAMPLE
The parameters of the considered NPS are borrowed from [34], which are presented in Table I.
In this example, the random transmission delays shown in Fig. 2 are considered. In order to correspond to the form of   2 based on |tanh(y i1 (t))| ≤ 1 and |tanh(y i2 (t))| ≤ 1. Under the given H ∞ index γ i = 20, by choosing = 0.01, μ 1 = 0.1, μ 2 = 0.5, ρ = 0.3 and ν = 10, the controller gains of three power systems solved by Theorem 2 for different probabilities of the attack occurrencē λ = 0.5 are derived as: The control system is simulated via circuital realization method in Simulink/MATLAB. The system components and the PI controller can be realized by equivalent electrical circuits. The professional Simcape toolbox of Simulink is used to realize the governor, turbine and power generator of Area 1, which is given in Fig. 6. Due to page limitation, the other simulation diagrams are omitted here. The communication network is executed by a variable delay module and a random number module generating the random delay given by Simulink. According to Section IV, the values of resistors and capacitors of analogy circuits for the Area 1 are given in Table II. Due to page limitation, the other two areas are omitted here.
In the simulation, the sampling period is 0.01s and the initial condition is zero. The following two different cases of load variations are considered.
For the considered deception attacks and the load variations in Case 1, the frequency trajectories by using the designed controller gains are drawn in Fig. 7. Meanwhile, the corresponding curves for Case 2 are drawn in Fig. 8. These figures denote that for the above two different situations of load variations, the designed controllers are effective for maintaining the system frequencies to be stable even when control signals are attacked by deceptive data.
To show the advantages of the proposed method over some existing ones, three groups of comparison results are provided as below.

A. The First Comparison Group
The following comparison further shows the better control performance of our designed controller against deception attacks than the existing controller. Two cases are considered: The controller is designed without taking into account the deception attacks, which is called a 'standard' controller. In this case, the controller design conditions can be obtained by removing the terms related to ϕ i (t). With the above parameters andλ = 0.3, the controller gains are derived as: Case 4: The controller is designed by our method considering the deception attacks, which is called a 'resilient' controller. Namely, the designed resilient controller is able to ensure the system stability even if the system is attacked. In this case, withλ = 0.3 and the same parameters used in Case 3, the controller gains solved by Theorem 2 are: In this simulation, the load disturbances are taken as P d1 (t) = 0.1cos(π t)e −0.5t , P d2 (t) = 0.1sin(π t)e −0.5t and P d3 (t) = 0.1e −0.5t for 0 < t < 8s (ω(t) = 0 for t ≥ 8s) for three areas. The responses of system frequencies in Case 3 and Case 4 are drawn in Fig. 9 and Fig. 10, respectively. By comparing the curves in Fig. 9 and Fig. 10, it is observed that our proposed controller considering the deception attacks (Case 4) can produce better control performance than the controller designed without considering the deception attacks (Case 4). From Fig. 9, it is seen that the standard controller is not able to stabilize the system frequencies when the transmitted signals are attacked by injecting some deceptive data. However, Fig. 10 shows our designed resilient controller can still guarantee the stability of system frequencies even deception attacks happen. This means that our proposed control strategy is resilient to the deception attacks and can outperform the standard control method.

B. The Second Comparison Group
To illustrate the advantage of the delay modeling approach based on probability density, the comparison results of the H ∞ index for three power systems obtained by our approach and the conventional interval time-varying delay approach in [15] are given in Table III, where three different random delay cases are considered.
To be specific, the first case is g(s) = −10000se 100s , s ∈ [−h, 0] and h = 0.1; the second case is g(s) = −2500se 50s , s ∈ [−h, 0] and h = 0.2; the third case is g(s) = −1110se 33s , s ∈ [−h, 0] and h = 0.3. And the second case is the same with the random delay given in Fig. 2. According to this table, all the optimized γ i of three control areas obtained by our method are smaller than the results derived by traditional interval time-varying delay approach, which indicates better stability performance will be achieved. Specifically, for h = 0.1 and i = 1, 2, 3, the values of γ i are reduced by 5.6%, 6.5% and 5.5%; for h = 0.2 and i = 1, 2, 3, the reductions of the values of γ i are 22.3%, 19.4% and 21.9%; for h = 0.3 and i = 1, 2, 3, the corresponding reductions are 32.6%, 27.7% and 27.5%. In addition, it is also seen that the corresponding values of γ i increase as the growth of the values of delay bound h.

C. The Third Comparison Group
To show the merit of our approach based on Lemma 1 to handle the distributed delay with kernel, the comparison results of the H ∞ index derived by our method and the Bessel-Legendre inequality in [29] are obtained in Table IV.
The second case of random delay with h = 0.2 in the above second comparison group is considered here. Following the method in [29], we utilize Legendre polynomials to approximate the distributed kernel g(τ ). For given , g(τ ) can be approximated byĝ(τ ): The figures of the distributed kernel g(τ ) and its approximationĝ(τ ) with different are shown in Fig. 11. Furthermore, the optimized H ∞ index γ and the number of decision variables (℘ = 5 2 (2 + 3)(1 + 5(2 + 3))) in P i with our method and the method in [29] are shown in Table IV. In terms of this figure and Table IV, it is indicated that ℘ increases and the approximation error e(τ ) decreases as the growth of . Meanwhile, one can also observe that our approach needs fewer decision variables to deal with the random delay and deception attack without any approximation error and leads to better H ∞ performance than the existing approach in [29].
Based on these comparison results, it is shown that the proposed LFC method for NPSs subject to random delay and deception attacks in this manuscript is more effective than some existing methods.

VI. CONCLUSION
This work studied the decentralized H ∞ secure LFC of NPSs with random communication delays and deception attacks. A novel model of random transmission delays has been presented by using the delay probability density. Based on this model, the normal control signals and deception attacks were represented as two distributed delay terms, in which the probability density is treated as the distributed delay kernel. Then, new sufficient probability-density-dependent conditions were proposed by Lyapunov method to design the decentralized H ∞ secure controller to maintain the system frequencies. The dynamics of power system were realized by electrical circuit systems, which were executed by the professional Simcape toolbox of Simulink/MATLAB. The simulation examples demonstrated the effectiveness of the developed approach. In addition, how to extend the developed approach to handle the LFC issue of NPSs with nonlinear dynamics or integrated with wind power and solar power deserves further investigations in the future.