Stability analysis of wide area power system under the influence of interval time-varying delay

ABSTRACT In view of the problem that time delay always existing in wide area power system can cause severe effects on the operation performance of the whole system, this paper studies the stability of the wide area power system with interval time-varying delays. Firstly, the model of wide area power system with interval time-varying delay is established, based on that, a new augmented vector and new Lyapunov-Krasoskii functional (LKF) are constructed. Then, the delay-partitioning approach, Wirtinger integral inequality, free-matrix-based inequality and convex combination approach are used to estimate the derivative of the functional, and as a result, a less conservative stability criterion for the delayed power system is obtained. Finally, numerical simulations of the typical second-order system, the single machine system and two-area four-generator power system are given to illustrate that the proposed method in this paper expands the stability margin of the system effectively.


Introduction
With the expansion of the scale of modern power system and the interconnection of power grids, the dynamic process of the power system becomes more and more complex. The traditional local control method was unable to meet the requirements of security and stability in the current wide area power system. In recent years, the wide area measurement system (WAMS) based on phasor measurement unit (PMU) has rapidly developed and widely used in power system, which promoted the development of the wide area control in power system (Hadidi & Jeyasurya, 2013;Manousakis, Korres, & Georgilakis, 2012;Yan, Govindarasu, Liu, Ming, & Vaidya, 2015). In the wide area environment, time delay exists in the process of signal transmission and processing, especially in long distance transmission. It has been shown that even small time delay can cause serious negative effects to the stable operation of the power system (Zhang, Zhan, Wei, Shi, & Xie, 2016). So, it is of great practical significance to study the stability of power system under the influence of time delay (Hailati & Wang, 2014;Yang & Sun, 2014).
There are two main methods for analyzing the stability of power systems with time delay: frequency domain method and time domain method. The frequency domain method is mainly based on the transformation of the characteristic equation and the distribution of eigenvalues to determine the stability of the system (Hua, Jian, CONTACT Wei Qian qwei@hpu.edu.cn & Liu, 2013;Li, 2015), the necessary and sufficient conditions for the stability of the system can be obtained by this method, but the calculation process is so complicated that it is difficult to be applied when the operation state of power system jumps or contains time-varying parameters. Compared with the frequency domain method, the time domain method has obvious advantages (Ma, Li, Li, Zhu, & Wang, 2015;Liu, Ding, Wang, & Zhou, 2011), and it is the main method for the stability analysis of time delay power systems. The LKF method based on the Lyapunov stability theory is used most widely in time domain method. This method gives the sufficient conditions for the stability of the system, which leads to a certain conservatism. Therefore, how to reduce the conservatism to expand the stable operation area of the system becomes a hot issue in recent years, and different research methods have been proposed by many researchers. In aspect of LKF construction, by constructing one-integral LKF and double integral LKF (Chen & Cai, 2009;Sun et al., 2015), augmented LKF (Li, Sun, & Wei, 2017), the stability and controller design of power systems with constant time delay and time-varying delay are studied. In aspect of estmating functional derivatives, many new methods are proposed such as free matrices method (Jia, An, & Yu, 2010), the Jesen integral inequality (Dong, Jia, & Jiang, 2015), the free-weighting matrix approach (Huang, Guo, & Sun, 2014), the generalized eigenvalue method (Ma, Li, Gao, & Wang, 2014), the convex combination approach (Qian & Gao, 2015), the Wirtinger integral inequality method (Qian, Jiang, & Che, 2016), to study the stability analysis and control of the wide area time delay power system. Although the above literatures reduce the conservatism of stability criterion for the time delay power system, they still have some shortcomings, such as the simple LKF, the limitations of the analytic method in reducing conservatism and so on, all of which cause the conservatism of the stability criteria. Motivated by the discussion mentioned above, the main purpose of this paper is to study the stability of the wide area power system with interval time-varying delays. By establishing the model of wide area power system with interval time-varing delay, construting new augmented vector and a new LKF with triple integral terms, dividing the delay interval into two parts, using wirtinger integral inequality, free-matrix-based inequality and convex combination approach to estimate the derivative of the functional, the less conservative stability conditions are proposed. The numerical examples are also given to show that the proposed method expands the stable operating area of the system effectively.

Model of power system with time delay
In this section, based on the traditional power system model, by introducing time-varying delay to describe, the model of of power system with time-varying delay is established.
To the power system, the dynamic model of generator is described as: where The meanings of parameters in the differential equations are given in (Li, 2015). In order to ensure the reliability of the power system, AVR excitation control method is used. Considering the time delay existing in the system, the dynamic equations of the excitation system can be expressed as follows: According to (1) and (2), the model of time delay power system can be expressed as follows: Linearizing the equation (3) at the equilibrium point can obtain: where x(t) ∈ R n is the state vector of power system, the initial condition φ(t) is a continuously differen- Considering the disturbance in the system, the system (4) should be expressed as: where A, A 1 are disturbance terms satisfying [ A, , and H, E a , E b are known constant matrices, F 0 is free matrix satisfying F T 0 F 0 ≤ I. In order to obtained the main result, the following lemmas are needed.

Stability analysis of time varying delay power system
In this section, based on the established model of time varying delay power system, by constructing new augmented terms and new Lyapunov-Krasoskii functional, applying less conservative methods to dealing with the derivatives of the functional, the developed stability criteria of the wide area power system with time varying delays is obtained.
Firstly, let: The following LKF is constructed for system (4): Remark 1: Different from the existing references, in this paper, a new augmented term is constructed as x T (u)duds and the proposed LKF contains triple integral term V 5 (t), so, more information of the time delay is employed, which play a key role in the further reduction of conservation.
Taking the time derivatives of V(t) along the trajectory of system (4) yield:V Deviding the integral interval −h t t−hẋ T (s)Zẋ(s)ds inV 4 , and using lemma 1 and lemma 3, the following can be obtained: , t], compared with the existing works (Ma et al., 2014;Qian et al., 2016;Sun et al., 2015), more information of the time delay is employed, which efficiently reduces the conservatism of the proposed approach. Then, Wirtinger integral inequality is applied to tackling with the integral terms on two subintervals, compared with the methods in (Park, Kwon, Park, & Lee, 2011;Qian & Gao, 2015;Seuret & Gouaisbaut, 2013) which use Jensen integral inequalities, this method is beneficial to expand the stable operating area of the system.
(16) Deviding the integral into two parts inV 5 , we can get: Using the lemma 2 to deal with Using the lemma 4 to deal with t t−h(t)ẋ T (s)Fẋ(s)ds can be obtained: Remark 3: Similar to the method dealing withV 4 , the delay interval of the functionalV 5 is also decomposed into two subintervals. To the double integral terms, double Wirtinger integral inequality and Free-matrix-based integral equality are used to reduce the estimating error, so that the obtained result is much less conservative than those in (Chaibi & Tissir, 2012;Seuret & Gouaisbaut, 2013;Zeng et al., 2015).

Numerical simulations
In this section, four typical numerical examples are given to show the less conservatism and the effectiveness of the proposed method in this paper.
Example 1: Consider system (4) with time-varying delay and the parameters as follows: The purpose of this example is to compare the con-  servatism of the stability conditions by applying different existing methods. When h 1 = 0, μ = 0.5, the upper bound of time delay given in (Chen & Cai, 2009) is 2.42, using the method of this paper, the upper bound of time delay is 2.63. When μ = 0, 0.1, 0.5, 0.8, by Theorem 1 of this paper, the obtained upper delay bounds are 6.0594, 4.7261, 2.6317, 2.2539 respectively. Table 1 lists the results of the maximum allowable delay bounds when μ(μ = −μ 1 = μ 2 )takes different values. From Table 1, it can be clearly seen that the results obtained by Theorem 1 are less conservative than the methods presented in (Ariba & Gouaisbaut, 2009;Park, Kwon, et al., 2011;Qian & Gao, 2015). Figure 1 gives stability margin of typical second-order system with time delay by different methods, it can be seen that the larger stability margin of typical of the system is obtained in this paper compared with the results in (Ariba & Gouaisbaut, 2009;Park, Kwon, et al., 2011;Qian & Gao, 2015).
Example 2: Consider system (4) with time-varying delay and the parameters as follows: For different μ, Table 2 lists the results of the maximum allowable delay bounds obtained by Theorem 1 in this paper and results in (Ariba & Gouaisbaut, 2009;Park, Kwon, et al., 2011;Seuret & Gouaisbaut, 2013). Figure 2  gives stability margin of typical second-order system with time-varying delay. It is clear that the proposed method in this paper has less conservatism and larger stability margin than those in (Ariba & Gouaisbaut, 2009;Park, Kwon, et al., 2011;Seuret & Gouaisbaut, 2013).

Example 3:
The single-machine infinite-bus system is choosed to verify the effectiveness of Theorem 1, and the parameters of the system resource from (Qian et al., 2016) ( Figure 3).  Table 3 lists the maximal allowable time delays of the single-machine infinite-bus system by using different methods when there is no disturbance in the system. The upper time-delay bound obtained by Theorem 1 is 71.9 ms, which is lager than the results in (Chen & Cai, 2009;Jia et al., 2010;Qian et al., 2016;Sun et al., 2015). It is clear that the proposed method in this paper has less conservatism and larger stability margin than those in (Chen & Cai, 2009;Jia et al., 2010;Qian et al., 2016;Sun et al., 2015). Further, if there is random disturbance in the system, the excitation amplification factor should be: where, K A : Excitation amplification factor setting value; K A : Excitation amplification factor with disturbance; r: A scalar that reflects the disturbance to K A .
Let the matrix H, E a , E b as follows: Table 4 lists the maximal allowable time delays obtained by different methods when r = 0.5, 1, 1.5, 2 . . . , 10 (takeing 0.5 as an interval), and Figure  4 gives stability margin of the single-machine infinite-bus system in this paper and (Jia et al., 2010;Qian et al., 2016;Sun et al., 2015). It can be seen, with the increase of disturbance term r, the stability margin of the system with time-varying delay becomes smaller. It is clear that our results has less conservatism and larger stability margin than those in (Jia et al., 2010;Qian et al., 2016;Sun et al., 2015).
Example 4: In this example, the two-area-four-machine power system shown in Figure 5 is used to verify the effectiveness of the main result in this paper. The detailed parameters of the two-area-four-machine power system are given in (Chen & Cai, 2009 Table 5 shows the upper bounds of time-delay for the two-area-four-machine power system by different methods, it can be seen that Theorem 1 in this paper has less conservatism than those in (Chaibi & Tissir, 2012;Ma et al., 2014;Yang & Sun, 2014).

Conclusion
This paper studies the stability of the wide area power system with interval time-varying delays. By establishing the model of the wide area power system with interval time-varying delay, a novel LKF with augmented vector is constructed. Then, the delay-partitioning approach, wirtinger-based integral inequality, free-matrix-based inequality and convex combination approach are used to estimate the derivative of the functional, and as a results, the new stability criterion with less conservatism is obtained. Finally, the numerical examples are given to show the effectiveness of the proposed method in this paper.