H ∞ performance for load frequency control systems with random delays

This paper investigates the problem of performance analysis for PI-type load frequency control (LFC) of power systems with random delays. By taking the probability distribution characteristic of communication delays into account in the LFC design, the power systems with a PI controller are modelled as stochastic time-delay systems. Furthermore, a delay-product-type augmented Lyapunov-Krasovskii functional (LKF) is constructed, and a new extended reciprocally convex matrix inequality combining Wirtinger-based integral inequality with convex combination approach is utilized to reduce the conservatism of main results. As a result, less conservative performance criteria are derived, which guarantee the asymptotically stable in the mean-square of the considered systems. Numerical examples are also provided to illustrate the superiority of our proposed methods.


Introduction
Load frequency control (LFC) has been widely utilized in large interconnected power systems with multiple control areas, which is one of the major measures to maintain the balance between the load and generation in a specified control area (Sharma et al., 2019;Wen et al., 2016;Yan & Xu, 2019). It should be noted that, dedicated communication channels have been made use of to transmit control signals between remote terminal units (RTUs) and a control centre in traditional centralized LFC schemes, the problems due to communication delays have been ignored by most previous research work (Fu et al., 2020;Jiang et al., 2012;Xiong et al., 2018;C. K. Zhang et al., 2013). However, with the emergence of numerous private networks and the employment of open communication networks, some challenging problems have appeared on account of limited network bandwidth, such as communication delays and data losses, which exerts potential threats on the stable operation of power systems (Peng et al., 2018;Sargolzaei et al., 2016;Singh et al., 2016). As a matter of fact, for a given communication channel based on transmission control protocol/internet protocol, communication delay is often random, and varies in an interval (Peng & Zhang, 2016). Generally speaking, random delays are characterized by means of Bernoulli-distributed stochastic variable, and this type of random interval delays could occur with a high probability in one subinterval and the opposite probability of CONTACT Xiaozhuo Xu xxzhpu@163.com occurring in another subinterval (Jia et al., 2019). Consequently, the delay intervals and corresponding occurrence probability should be fully considered, which is significant to obtain less conservative results. However, the probability distribution characteristic of communication delays was rarely considered in most existing studies. Therefore, it is of great significance to study the influence of random delays on LFC systems. To LFC systems with random delays and load disturbance, the H ∞ performance level and the upper bounds of time delay are two major factors to judge the conservatism of the derived criteria. In order to further reduce the conservatism, sustained efforts have been made mainly on two aspects, one is to construct an appropriate LKF, the other is to estimate the derivatives of the LKF more accurately, such as delay-partitioning approach (Ko et al., 2018), augmented LKF (W. I. Lee et al., 2018;Zeng et al., 2019), LKF with triple-integral and quadrupleintegral terms , LKF with delayproduct terms (Li et al., 2019;Qian, Xing, et al., 2020;C. F. Shen et al., 2020;, Jensen's inequality (Qian, Li, Zhao, et al., 2020), Wirtingerbased integral inequality (Qian et al., 2019), free-matrixbased integral inequality (Zeng et al., 2015), auxiliaryfunction-based inequality (P. G. Park et al., 2015), Bessel-Legendre inequality (W. I. Lee et al., 2018;Seuret & Gouaisbaut, 2018) and reciprocally convex combination techniques in different forms (P. G. Park et al., 2011;R. M. Zhang et al., 2019). Furthermore, various H ∞ performance criteria for LFC systems have been put forward and researches on this problem are still going on. For instance, in Wen et al. (2016), by integrating the communication delays and event triggered control in the formulated model, and utilizing freeweighting matrix approach, the H ∞ performance criteria of LFC systems were derived. In Peng et al. (2018), an adaptive time-delay LFC model was developed, and reciprocally convex combination technique was applied in the derivation of main results, which can obtain improved H ∞ performance criteria and reduce the number of decision variables. By introducing the single and double integral items in LKF construction, and employing Jensen's inequality along with reciprocally convex combination approach, the delay-distribution-dependent H ∞ performance and stability criteria were presented in Peng and Zhang (2016). In Cheng et al. (2020), by considering transmission delays and denial-of-service attacks in the LFC design, and employing piecewise LKFs together with novel analysis methods, sufficient conditions were developed with H ∞ performance. In H. Zhang et al. (2020), a new model based on the area control error and timevarying delays was established, then an suitable LKF and extended Wirtinger's inequality was used, which had better H ∞ performance by the number of packets sent and average sampling period. By building an accurate model with a degree of packet losses and introducing an appropriate LKF, then exploiting Wirtinger-based inequality to estimate the integral terms, the desired H ∞ performance index of multi-area LFC systems was attained in Peng et al. (2017). It should be noted that, there is still plenty of room in how to coordinate LKF construction with estimating techniques efficiently, which helps to get H ∞ performance criteria with less conservative.
Inspired by above discussion, we further explore the H ∞ performance for PI-type LFC of power systems with random delays in this paper. The aim is to apply novel LKFs and explore new optimal analysis methods, by which the less conservative delay-dependent conditions and the desired H ∞ performance level can be obtained. The main advantages of this paper can be listed as follows: • Delay-product-type functional approach is utilized in LKF construction, which make full use of the information about time-varying delay and its derivatives. By introducing state nonintegral terms with timevarying delay-dependent matrices and multiple integral terms, a novel augmented LKF is constructed. Meanwhile, the constraint that every Lyapunov matrix should be positive is relaxed, all of which contribute to reduce the conservatism of the main results.
• In order to estimate the infinitesimal operators of constructed LKF more accurately, the integral terms in single and double forms are separated precisely by using delay-partitioning method. Then the single integral terms are estimated by an extended reciprocally convex matrix inequality together with Wirtinger-based integral inequality, and the double integral items are estimated by Jensen's inequality, by which the constructed LKF and the estimating methods fit together effectively to reduce the conservatism of the main results.
Notation: Throughout this paper, R n and R n×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. P > 0(< 0) means that P is a positive (negative) definite matrix. E{•} is the mathematical expectation of •. A T and A −1 represent the transpose and the inverse of A. I n×n and 0 n×n stand for n × n identity matrix and n × n zero matrix, respectively. * in the matrix denotes the symmetric term. diag{· · ·} denotes a block diagonal matrix, col{x 1 ,

Problem formulation and preliminaries
The schematic diagram of one-area delayed LFC systems with proportional-integral (PI) controller is presented in Figure 1,and its state-space equation is indicated as: and f , P m , P v , P d are the deviations of frequency, the turbine/generator mechanical output, generator valve position and the disturbance of load, respectively. M, D,R, T ch , T g denote the moment of inertia of the generator, the generator damping constant, speed droop, time constant of the turbine and time constant of the governor, respectively. As we all know, there is no net tie-line power exchange in single-area power systems. It can be seen from Figure 1 that, as the output of the system (1), the area control error (ACE) is denoted as:ỹ where β > 0 is frequency bias factor. Moreover, ACE is also acted as the input of the designed controller, so the following PI-based controller can be designed: where K P and K I are proportional and integral gains, and ACE is the integration of ACE. As depicted in Figure 1, the communication delay from ACE to the PI-based controller (3) is defined by an exponential block e −sh(t) . Denote the PI-based controller (3) can be further written as: where h(t) is a time-varying delay satisfying: where h and μ < 1 are constants.
In this paper, the information about the probability distribution of time-varying delay h(t) is employed.
Assumption 2.1: To describe the probability distribution of the time-varying delay h(t), define two sets and functions by Obviously, 1 ∪ 2 =R + and 1 ∩ 2 =∅. It is easy to know that t ∈ 1 means the event h(t) ∈ [0, h 0 ) occurs and t ∈ 2 means the event h(t) ∈ [h 0 , h] occurs. Therefore, a stochastic variable α(t) can be defined as Remark 2.1: As is well known, the real power system is a system with high nonlinearity and time-varying characteristics. In practice, modern power systems usually require a wide area open communication network to transmit information concerned. The usage of these networks causes inevitable unreliable factors, such as time delays, packet losses, latent faults, and etc. Similarly, these nonlinear disturbances may occur randomly as a result of some environment reasons. Therefore, the stochastic variable α(t) is introduced in this paper to describe such randomly occurring phenomenon, which has universality and application prospect.
According to the above analysis, the following delaydistribution-dependent PI controller can be taken to replace the general form shown in (5): Defining x(t) = [ f P m P v ACE] T and substituting (6) into (1), it can be obtained: where The main objective of this paper is to derive the less conservatism H ∞ performance criteria, and guarantee system (7) is asymptotically mean-square stable. In order to obtain main results, the following definition and lemmas are required.

Definition 2.1 (B. Shen et al., 2011):
Given a scalar γ > 0, under the zero initial condition, the system (7) is said to be asymptotically mean-square stable with H ∞ performance level γ , if the following inequality holds

Main results
In this section, by constructing a novel delay-product augmented LKF and employing appropriate analytical methods, some improved H ∞ performance and stability criteria for the considered systems are given. Some notations are shown to simplify the representation of the following parts: Theorem 3.1: For some given positive scalars α 0 , h 0 , h, μ 1 , μ 2 , γ and a matrix K, system (7) is asymptotically mean- and Q j ∈ R 2n×2n (j = 1, 2, . . . , 8), and any matrices T k , S k (k = 1, 2, 3, 4), N 1 and N 2 with appropriate dimensions, such that the following LMIs hold: Proof: Define the Lyapunov-Krasovskii functional candidate as follows: where with Remark 3.1: As we all know, in order to improve the H ∞ performance level, choosing an appropriate LKF is crucial. In this paper, the single integral terms x(s) ds dθ are augmented in V 1 (x t ), which establishes more relations among some new cross items. Moreover, the augmented delay-product nonintegral items are introduced in V 2 (x t ) and delay-product-type functional method is extended to single integral terms in V 3 (x t ). The matrices P, L i (t) (i = 1, 2, 3, 4) in the constructed LKF are just symmetrical, not positive definite, and Q j (t) (j = 1, 3, 5, 7) are delay-dependent. Different from the existing constant variable matrices L i and Q j , delay-dependent matrices can fully capture more information of time delay. Furthermore, x(t) andẋ(t) are augmented in the single, double and triple integral terms of V m (x t ) (m = 3, 4, 5), so that the relationships between LKF and state information is deepened, all of which play a vital role in obtaining new H ∞ performance conditions with less conservatism.
First, in order to ensure the positive definiteness of V(x t ), V 1 (x t ) + V 2 (x t ) can be written together and expressed as below: A 4 > 0 can be obtained. Therefore, by utilizing Lemma 2.2, for any matrices T i (i = 1, 2, 3, 4), can be further calculated as below: Hence, we have the following inequalities: Based on the above analysis, by using convex combination approach, V 1 (x t ) + V 2 (x t ) > ε x(t) 2 can be ensured for a sufficiently small ε > 0 if [h 1 (t),h 2 (t)] > 0 holds. In consequence, the positive definiteness of V(x t ) can be guaranteed by W m , M m (m = 1, 2) > 0 and conditions (12), (13).

Remark 3.2:
It can be clearly discovered that, the delayproduct nonintegral terms can be selected differently depend on actual situations. In LKF construction, delayproduct nonintegral terms such as η T 2 (t)L 1 (t)η 2 (t) and η T 3 (t)L 2 (t)η 3 (t) are introduced in V 2 (x t ), which fully utilizes the information of time delay in the coefficients before symmetric matrices L i (i = 1, 2, 3, 4). Moreover, by considering V 1 (x t ) and V 2 (x t ) together, we can obtain that From the above inequality, we can see that conditions P > 0 and L i (i = 1, 2, 3, 4) > 0 are relaxed as [h 1 (t),h 2 (t)] > 0. In other words, the delay-product-type functional method can make the constructed LKF have a more general form since the restrictions of some conditions are defined loosely. It is worth to mention that the delayproduct-type functional approach has not been applied to deal with the problem of random delays for LFC systems before.
Defining the infinitesimal operator L of V(x t ) as follows it can be obtained η T 6 (s)W 1 η 6 (s) ds, η T 6 (s)W 2 η 6 (s) ds, t θ η T 6 (s)M 1 η 6 (s) ds dθ , η T 6 (s)M 1 η 6 (s) ds, η T 6 (s)M 2 η 6 (s) ds dθ , The nonintegral terms in LV 4 (x t ) and LV 5 (x t ) are defined in 4[h 1 (t),h 2 (t)] . By taking single integral terms in (27), (28) and (29) into consideration together, we have By choosing the integral inequality introduced in Lemma 2.1 to bound J 1 , J 2 , J 3 and J 4 , the following inequalities can be attained thus from (31) to (34), it follows that Besides, according to Lemma 2.2, there exists constant matrix S j (j = 1, 2, 3, 4) with appropriate dimensions such that Then, applying Jensen's inequality to estimate the double integral items Z 5 , Z 6 , Z 8 and Z 9 in (29) yields the following For any matrices N 1 and N 2 with appropriate dimensions, from the system (7), the following zero equality holds Combining the equalities and inequalities from (25) to (39) and taking the expectation, we can derive that To analyze the H ∞ performance, we introduce the following performance index By considering the zero initial condition, it can be obtained Then by considering the inequality (42) and employing Schur complement lemma, when inequalities (8)-(11) hold, J(t) < 0 can be obtained. Letting t → ∞, the condition in Definition 2.1 is guaranteed. Therefore, the closedloop system (7) is asymptotically mean-square stable with H ∞ performance γ . This completes the proof.
When α(t) = 1, that is, there is only one delay interval with h(t) = h 1 (t),ḣ(t) ≤ μ, system (7) decreases to ⎧ ⎨ By making use of the similar methods in the derivation of Theorem 3.1, we have Corollary 3.1.

Numerical simulations and analysis
In this segment, a second-order example and one-area LFC system are provided to illustrate the effectiveness of main results. Moreover, the time delay in one-area LFC systems is considered as a random delay with the probability distribution characteristic, which shows a significant improvement in the stable operating regions and the disturbance attenuation ability of power systems.
Example 4.1: Consider the following parameters in the system (43) with ω(t) = 0: The purpose of this example is to compare the admissible upper bounds h by various approaches, which can check the conservatism of the stability conditions. Table 1 lists the admissible upper bounds h obtained by different methods for various μ. When μ = 0.8, by applying the methods in W. I. Lee et al. (2018) and Chen and Chen (2019), the admissible upper bounds are h = 2.735 and h = 2.899, and the result achieved by Corollary 3.2 is h = 3.361. Hence, it can be seen obviously that the admissible upper bounds of Corollary 3.2 are larger than those in above works, which verifies the progressiveness of our applied methods.

Example 4.2:
For one-area closed-loop LFC system, the following parameters are considered:

A. Result comparison and analysis
For various controller gains K P and K I , Table 2 shows the maximum delay upper bounds h of system (43) with ω(t) = 0 based on Corollary 3.2. It can be discovered that PI controller gains have a significant impact on affecting delay margins. When K P is fixed, the maximum delay upper bound h decreases with the increase of K I . However, the relationship between delay upper bound h and K P is more complicated. When K I is fixed, in most situations h decreases first and then increases with the increase of K P . Therefore, all of these regulations can be regarded as auxiliary conditions for designing PI controllers, which have a positive effect on obtaining larger stable operating regions for power systems. Table 3 gives more comparative results of the maximum delay upper bounds h with Jiang et al. (2012) and Peng and Zhang (2016) based on Corollary 3.2. We can see clearly that the results obtained by our methods are obviously larger than that acquired by other methods, which means that the methods applied in this work have distinct advantages in calculating the delay margins of real networks. For the given conditions of μ = 0.5 and γ = 1, Table 4 provides the maximum delay upper bounds h under various K P and K I based on Corollary 3.1. It should be pointed out that delay upper bound h becomes smaller with the increase of K P and K I , which reveals the stable operating regions of power systems is closely related to PI-based controller gains. Table 5 presents the maximum delay upper bounds h with γ = 1 under controller gains K P = 0.4, K I = 0.4. The maximum delay upper bound achieved by Corollary 3.1 is h = 0.594. Comparing with the obtained results by Theorem 3.1, it is easily found that the larger maximum delay upper bounds h can be obtained by taking the probability distribution characteristic of time delay into consideration, which confirms the accuracy of our results.
For the prescribed conditions of μ = 0.5 and h = 2, Table 6 lists the allowable minimum γ min by different K P and K I based on Corollary 3.1. It is worth to mention that allowable minimum γ min becomes larger with the increase of K P and K I , which reflects the disturbance attenuation ability of power systems is also closely contact with PI-based controller gains.     Table 7. Allowable minimum γ min with K P = 0.2, K I = 0.6.
For K P = 0.15, K I = 0.1, μ 1 = μ 2 = 0.5 and the same delay upper bound h = 2, the allowable minimum H ∞ performance index γ min based on different methods are listed in Table 8. Through the comparative results with Jiang et al. (2012) and Peng and Zhang (2016), it can be seen apparently that our results are much smaller than those obtained by other methods, which show the less conservative of our methods.

B. Simulation verification
For purpose of validating the accuracy of our theoretical results, we utilize MATLAB/Simulink for simulations based on delayed LFC systems with/without considering      (7) with considering probability distribution characteristic. It can be seen clearly from the simulation results in Figure 2-5 that all the state variables converge to their equilibrium points, which confirm the veracity of our theoretical results.

Conclusion
In this paper, H ∞ performance for PI-type LFC of power systems with random delays have been investigated. By introducing new vectors and delay-dependent matrices, a delay-product-type augmented LKF has been constructed, and a novel extended reciprocally convex matrix inequality combining with Wirtinger-based integral inequality have been employed to tackle with the integral terms effectively, which can utilize more information of time delay and improve the estimation accuracy. According to applied optimal analysis methods, less conservative delay-dependent H ∞ performance and stability criteria have been developed. Finally, two numerical examples have been carried out to illustrate the effectiveness of our theoretical results and the improvement of the proposed methods. In the future, we will consider the influence of other stochastic factors in power systems, construct more reasonable LKF and propose new integral inequalities to further cut down the conservatism of main results.