Further Improvement on Stability Results for Delayed Load Frequency Control System via Novel LKFs and Proportional-Integral Control Strategy

In this work, novel stability results for load frequency control (LFC) system considering time-varying delays, nonlinearly perturbed load, and time-varying disturbance of system parameters are proposed by using proportional-integral control strategy. Considering the nonlinearly exogenous load disturbance and system parameters disturbance, an improved stability criterion in the form of linear matrix inequalities (LMIs) is derived by novel simple Lyapunov–Krasovskii functionals (LKFs). The delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar functions are included in the new simple LKF. Taking the single-area and two-area LFC system installed with proportional-integral (PI) controller as example, our results surpass the previous maximum allowable size of time delay. Meanwhile, the relationship between time delay varying rate, load disturbance degree, gains of PI controller, and delay margin of the LFC system is researched separately. The results can provide guidance to tune the PI controller for achieving maximum delay margin, in which the LFC system can withstand without losing stability. At last, the simulation results verify the effectiveness and superiority of the proposed stability criterion.


Introduction
e usage of load frequency control (LFC) is widely in the power system, which restores the balance of the power system between load demand and generation supply [1][2][3].
e sudden change of load can bring threaten to the safe and economic operation of the power system. us, the analysis and research about LFC scheme of the power system are considered to be essential and important segment to guarantee the safe operation of the power system. e LFC scheme is also e ectively applied in smart grid [4,5].
A wide and dedicated open communication network is needed to transmit control signals, measurements between remote RTUs, and the control center in the conventional LFC system [3,6]. e usage of the communication network leads to a series of inevitable problems, like time delays [7][8][9][10][11], packet losses, and so on. Hence, many researchers focus on the time delay phenomenon of the power control system and the impact of network delay on the communication-based power system [12][13][14][15][16]. From the perspective of stability analysis, it is of signi cant to seek the maximum allowable network delay that the power system with LFC scheme can withstand without losing sable.
Generally speaking, there are mainly two approaches to seek the upper bound of the time delay system, which have di erent restriction and conservatism. One is the direct approach, which is frequency domain method, such as tracing eigenvalue [17,18], or cluster treatment of characteristic roots [19][20][21]. e advantage of these direct methods is that the accurate delay margin can be derived by calculating eigenvalues of the whole system. However, the disadvantage of direct methods is that they can only be applied to constant time delays situation. e other is time domain method, which is based on Lyapunov stability theory and linear matrix inequalities (LMIs) techniques [22][23][24][25][26]. ough the time domain methods are more conservatism than the frequency domain methods, the time domain methods can be applied in both constant and timevarying delays situation. So, the time domain methods based on Lyapunov stability theory are abroad exploited.
e main effort to diminish the conservatism of time domain methods has focused on two aspects, methods of constructing L-K functional and analyzing techniques for bounding the derivatives of L-K functional with regard to time. e former includes delay-division functional, functional with matrices dependent on the time delays [27], functional with two-integral or triple-integral terms [28], and so on. e latter techniques include improved majorization technique, free weighting matrix method [29], integral inequality including Jesen inequality [23], Wirtinger inequality [30], auxiliary function-based integral inequality [31], and reciprocal convex technique [32]. More and more researchers focus on the delay-dependent criteria for stability of the power system because the delay-dependent criteria have less conservatism than delay-independent criteria [22,33].
Until now, there are a few excellent research studies about stability analysis of the delayed LFC systems. Ramakrishnan and Ray [34] focused on the delay-dependent robust stability problem for the LFC system with multiple time-invariant delays in H ∞ framework. e model of the LFC system with both sampling and transmission delay was proposed in [35]. e stability analysis problem was considered in [3], which focused on the LFC system with electric vehicles and time delays. e novel linear operator inequality approach was proposed, and a delay-dependent stability criterion with less conservative was obtained. Unfortunately, the load disturbance and random disturbance of system parameters were not considered either in the stability analysis or simulation in the above literatures. It is worth noting that the bounded nonlinear load perturbation model was built without considering random disturbance of system parameters in literatures [22,36,37]. e augmented L-K functional including single and double integral terms was developed to analyze stability of the LFC system in these literatures. e novel LKF with delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiple by a higher degrees of scalar function proposed by this paper contains more information about state variables than the routine L-K functional adopted in [22,36,37], and the results in this paper are less conservative than those of [22,36,37].
is work proposes a novel stability criterion for the time-varying delay LFC power system considering nonlinearly perturbed and random disturbance of system parameters. Considering unknown exogenous load disturbance and random disturbance of system parameters as constrained time-varying nonlinear function related to current and delayed state vectors, the delayed LFC system with PI controller can be described as a time-varying delay system. en, the novel stability criterion of the LFC power system is expressed in the form of LMIs by a simple LKF and integral inequalities. e main opinion of constructing the new LKF is the application of delay-dependent matrix, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar function. Moreover, the case studies are carried out taking the single-area and two-area LFC system installed with PI controller as example with simulation examples to verify the effectiveness and superiority of the proposed stability criteria.

System Description and
Problem Formulation e notations of jth area for the LFC system are given in Table 1. e subscript j should be omitted for the single-area LFC system. For example, K Pi can be changed to K P in the single-area LFC system. It is worth noting that ΔP tie � 0 in the single-area LFC system.
Consider the uncertainties in the power parameters of the actual LFC system. e state-space model with disturbance parameters and load disturbance of the multiarea LFC system (N area) is presented as follows: where x(t) represents state vector. A and B represent the system matrices related to the current state and delayed state vectors. ΔA(t) � NF(t)D 1 and ΔB(t) � NF(t)D 2 represent perturbation parameters. F(t) is a time-varying matrix, which satisfies F T (t)F(t) ≤ I. N, D 1 , and D 2 are the constant matrix. h j (t) represents the network time-varying delay in j th area. C represents the known matrix related to the disturbance vector. e initial state can be described by a continuous function defined in t ∈ [− max(h j (t)), 0].

LFC Modeling.
In order to simplify the calculation, it supposes that h j (t) � h(t), which means the multiple delays are supposed to be equal and shown as a unified time- Discrete Dynamics in Nature and Society varying delay h(t). So, the state-space model of multiplearea can be presented as follows: where B � n i�1 B i and ΔB(t) � n i�1 ΔB i (t). h and μ represent the upper bound of time-varying delay and the delay derivative separately. e constraints of h > 0 and 0 < μ < 1 can make time-varying delay systems well-posed because the fast time-varying delay will cause a series of problems, like causality, minimality, inconsistency, and so on.
Furthermore, considering ΔA(t) � NF(t)D 1 and ΔB(t) � NF(t)D 2 , the state-space model of multiple-area can be rewritten as follows: For a typical two-area LFC system, the dynamic model is shown in Figure 1. e − sh(t) is network time-varying delay notation. e state vectors, load disturbance vectors, and system matrices are presented as follows: For the j th area, j � 1, 2, the notations are presented in Table 1.
e dynamic model of the single-area LFC system is shown in Figure 1 without dotted lines. e state vector, disturbance vector, and system matrices of the single-area LFC system are reduced as follows: Remark 1. When there are N(N > 1) subareas connected by tie-lines forming a multiarea power system, the dotted line will be copied by N times. e decentralized control strategy is applied in the multiarea power system, which means every control area is independent and has its own LFC center to maintain the balance of generation and load. As a result, the interactions between i th area and other areas,   (4) is the expanded form of x i (t) in two-area power system. And the system matrices of multiareas are listed as follows:

Disturbance Parameters and Load Disturbance Model of LFC System.
is paper models the unknown disturbance of power system parameters as ), which is nonlinear function related to current and delayed state vector. Owing to the restriction of F(t) T F(t) ≤ I, the following inequalities can be obtained: Similarly, the load disturbance is considered as timevarying nonlinear function with current and delayed state vector as CΔL d (t) � g(t, x(t), x(t − h(t))), which satisfy the following norm-bounded constraints: where ρ ≥ 0 and τ ≥ 0 represent known scalars. G and H represent known constant matrices with appropriate dimensions. ρ, τ, G, and H bound the magnitude of timevarying load disturbance of power system together.

Main Results
Before deriving the main results, Lemma 1 should be introduced. Lemma 1 bounds the integral of quadratic function multiplied by a 1st/2nd degree scalar function [26].
en, we can obtain the delay-dependent robust stability criterion of system (3).
e Lyapunov functionals are adopted as follows: where Computing the derivative of V(t) along the trajectories of system (3): Discrete Dynamics in Nature and Society e sum of all integral terms is expressed by f a (t): Owing , t], the following inequalities can be obtained: Using Lemma 1 to estimate Υ(t): e following inequality holds for any ς ≥ 0 from (9) and (7), respectively: ey can be expressed by the augmented state vector ζ(t), respectively: By substituting (16) into (13) and adding the inequality (18), the following inequality can be obtained: Discrete Dynamics in Nature and Society where G 1 , G 2 , G 3 , G 4 , G 5 , G 6 , H 1 , H 2 , H 3 , Λ, Π have been defined in Appendix, and Finally, considering 0 ≤ h(t) ≤ h and − μ < _ h(t) < μ, inequality (9) is obtained by Schur complement, which ensures the following inequality: Hence, the system is asymptotically stable. e proof is finished.

Remark 3. Our work has investigated the stability condition of LFC system (3) under the limitation 0
at is to say, the stability condition should be satisfied under the four situations, < 0 must be guaranteed under the four situations. us, inequality (9) under β 1 − β 4 in eorem 1 must be satisfied simultaneously for system asymptotic stability.

Remark 4.
e main contribution of this section is to propose a novel eorem 1 in the form of LMIs which can derive a less conservative result of maximum allowable network delay for LFC system (3). e LKF constructed by literature [36] is x T (s)R _ x(s)dsdθ. Compared with that, the LKF proposed by this paper has four improvement points. [36], which can reduce the conservatism by increasing the link between variables. (ii) e time-delay dependent matrix P 1 + h(t)P 2 is introduced in quadratic term, rather than constant matrix P in [36], which can reduce the conservatism by introducing more messages about time-varying delay. (iii) e cross terms of variable s) also can reduce the conservatism by increasing the link between variables. (iv) e 4th, 5th, and 6th terms of the novel LKF multiplied by 1st, 2nd, and 3rd degrees of a scalar function (h − t + s). e increasing of (h − t + s) by one signifies the increasing the number of integral by one. Compared with the LKF of literatures [22,36,37], we increase the quadratic terms multiplied by 2 nd and 3rd degrees of the scalar function (h − t + s), which may reduce the conservatism compared with literatures [22,36,37].

Remark 5.
If there is a small disturbance or no disturbed load in the LFC system, the delay-dependent robust stability criterion can be developed from eorem 1 by setting ρ � 0, τ � 0. If there is no or very small disturbance in LFC system parameters, set N � 0.

Case Studies
In this section, the typical single-area and two-area LFC systems are carried out to analyze the delay margins based on eorem 1. e parameters of the real two-area LFC system are listed as Table 2. For verifying the less conservatism of our proposed method, the parameters of the twoarea LFC system given in [37] are presented in Table 2

Single-Area LFC System.
e parameters of Area 1 in Table 2 are adopted as single-area LFC system parameters.
Setting K P as very small, we investigate the effect of Integral (I) controller on delay margin under three situations of load disturbance (ρ � 0, τ � 0; ρ � 0, τ � 0.025; and ρ � 0.025, τ � 0.025) for both constant and timevarying delays. Suppose G and H in (9) and D 1 and D 2 in (8) as 0.1I n , where n represents the dimension of the system state vector x(t). It can be seen that n � 4 in the single-area LFC system from (5). eorem 1 is adopted to calculate the delay margins.
e results are shown in Table 3 and Figures 2-4. e following conclusions can be obtained from Table 3 and Figures 2-4: (i) e increasing of ρ and τ represents that the load fluctuation becomes more pronounced. With the increase in ρ and τ, the delay margin that PI controller can withstand without decrease in losing stability under the same value of μ and K I . e effect of load disturbance on delay-dependent stability of the LFC system can be obtained. (ii) It can be seen from Figures 2-4 that the increasing of μ causes the decreasing of delay margin h with the same value of K I under the three situations of load disturbance (ρ � 0, τ � 0; ρ � 0, τ � 0.025; ρ � 0.025, τ � 0.025). e effect of time delay changing rate on delay-dependent stability of the LFC system can be obtained. (iii) Simultaneously, it can be seen from Figures 2-4 that maximum allowable network delay decreases with the increasing of K I with the same value of μ under the three situations of load fluctuation.
As for the nominal system condition (ρ � 0, τ � 0) with constant (μ � 0) and time-varying delay (μ � 0.9), the maximum allowable network delays are listed in Table 4 compared with the other results from literatures [22,36,37]. It can be seen that the results calculated by eorem 1 are less conservative than those of [22,36,37], owing to the progressiveness of Lyapunov functional and Lemma 1 adopted by this paper. erefore, when compared with [22,36,37], the results proposed by this paper are more realistic.
Results of the maximum allowable network delays with regard to gains of PI controller (K P , K I ) are listed in Table 5 and Figure 5 with constant delays (μ � 0) and no load disturbance (ρ � 0, τ � 0) and in Table 6 and Figure 6 with time-varying delays (μ � 0.9) and load disturbance (ρ � 0.025, τ � 0.025), separately. Results demonstrate that smaller h is obtained under the bigger K P , K I , and a relatively larger h is acquired for the smaller K P , K I . h is reduced with the increasing of K P when K I ≤ 0.05. For K I ≥ 0.1, h increases at first and then decreases with the increasing of K P . is results implies that for a given K I , there is a optimal value of K P which can acquire the maximum h, which can give guidance to tune PI controller to achieve maximum delay margins. Moreover, it can be found that h of the timevarying delays is smaller and decreases faster than that of constant delays.

Two-Area LFC System.
Maximum allowable network delay (h) is calculated by eorem 1 for the two-area LFC system listed in Table 2 by simplifying the multiple delay as a single delay like (3) and gains of PI controller for two areas as same (K Pi � K P , K Ii � K I ). In the two-area LFC system, both the values of G, H, D 1 , and D 2 are taken as 0.1I n , where n � 9.
In two-area LFC system case, for more clearly comparisons with literatures [22,36,37], the results are summarized in Table 7 with the condition of (ρ � 0, τ � 0) under constant (μ � 0) and time-varying delay (μ � 0.5). Similarity, the listed results are less conservative than those of [22,36,37]. Also, the results of Table 7 show the effect of PI controller gains on the delay margins.
Also, the results of delay margin with respect to gains of PI controller are listed in Table 8 and Figure 7 with constant delay (μ � 0) and no load fluctuation (ρ � 0, τ � 0) and in Table 9 and Figure 8 with time-varying delay (μ � 0.9) and load fluctuation (ρ � 0.025, τ � 0.025), respectively. e delay margin changing trend with regard to gains of PI controller of the two-area LFC system is consistent with that of the single-area LFC system. Hence, the details can be omitted here.

Simulation Verification.
e models of the single-area and two-area LFC system are built in MATLAB/Simulink separately. e simulation results verify the above calculation results. e state vectors (Δf) are selected as the observation since the load fluctuation directly affects the frequency of the LFC system.  Table 4 that the maximum allowable network delay calculated by eorem 1 is h � 6.64 under the condition of ρ � τ � 0, μ � 0.9, K P � 0.1, K I � 0.2. e time-varying delay h(t) is supposed to be a sine function h(t) � 3.32 + 3.32sin(0.27t). Figure 9 shows that the state vector Δf converges asymptotically to stable point for h � 6.64; however, the delay margin calculated by [22]  Moreover, if h exceeds beyond 6.64, the system becomes unstable as shown in Figure 10 for h � 7s.

Two-Area LFC System.
At first, when h(t) � 0, the state-space model of two-area LFC system becomes _ x(t) � (A + ΔA + B + ΔB)x(t) + CΔL d (t). Figure 11 shows that the delay-free system is asymptotically stable. It can be seen from Table 9 that the maximum allowable delay Discrete Dynamics in Nature and Society  Figure 6: h ∝ (K P , K I ) (single-area with μ � 0.9, ρ � 0.025, τ � 0.025).  calculated by eorem 1 is h � 6.78 under the condition of ρ � τ � 0.025, μ � 0.9, K P � K I � 0.2. en, it can be seen from Figure 11 that the two-area LFC system is asymptotically stable under h � 6.78. Moreover, if h exceeds beyond 6.78, the system becomes unstable as shown in Figure 12 for h � 7s.

Conclusion
e stability criterion for the time-varying delay LFC system with nonlinearly load disturbance and random disturbance of system parameters has been investigated by adopting a novel simple LKF and integral inequality (Lemma 1). Considering the exogenous load disturbance and random disturbance of system parameters as constrained nonlinearly function, a time-varying delay system is modeled to describe the delayed LFC system. e novel simple LKF includes delay-dependent matrix in quadratic term, cross terms of variables, and quadratic terms multiplied by 1st, 2nd, and 3rd degrees of scalar function. Lemma 1 is adopted to bound the quadratic terms multiplied 1 st and 2nd degrees of scalar function emerging in the derivative of LKF. e results of the novel simple LKF contributing to reduce conservatism and the relationships between the time delay varying rate, load disturbance degree, the gains of PI controller, and delay     margins are derived by the case studied. Meanwhile, simulation verifies the effectiveness and superiority of our results. Nevertheless, the security threats in actual communication network such as denial of service (DOS) attacks and deception attacks are not considered in our work. In the future, how to analyze the stability of the LFC system under the simultaneous existence of various network attack, such as DOS and deception attacks, will be the focus of our following work.