1 Introduction

In the past decades, static var compensator (SVC) has been used in power system to regulate the system voltage and improve power system stability [1–3]. SVC has many virtues over traditional reactive power system compensators. Many meaningful results have been reported for SVC in the past decades. Based on exact linearization scheme, the SVC controllers have been developed in [4, 5]. Although these controllers can guarantee system have good control performance, they may cause unsatisfactory performances when operating states are far from given operating points. Furthermore, on the basis of nonlinear system model, some control approaches have been proposed. In [6], the adaptive fuzzy controller is presented for SVCs based on backstepping control methods. The adaptive fault-tolerant controller is proposed in [7]. When system subjects to external disturbances, the methods in [6, 7] may obtain unsatisfactory performance. In [8], an adaptive backstepping sliding mode \(H_\infty \) controller is given, where \(H_\infty \) control scheme is used to attenuate external disturbances. In [9], a nonlinear robust controller is investigated for the SVC system with external disturbances and parameter uncertainties using modified adaptive backstepping and minmax scheme.

Although these methods have good control performance of the SVC system, the disturbance rejection and robustness performance of these controllers are achieved at a price of sacrificing the nominal control performance. When subjecting to strong disturbances, these approaches may lead to poor performance, for example, the dynamic process of the closed-loop system may become sluggish and even unstable. It is because most of above control schemes reject disturbances merely via the action of feedback regulation in a relatively slow way and do not consider active and direct disturbance rejection in the controller design. Disturbance observer based control (DOBC) is an effective method to reject external disturbances and improve robustness against uncertainties [10–25]. So far, DOBC method can be used to cope with both matched disturbances [10–18] and mismatched disturbances [22–25].

In order to guarantee system have a faster convergence rate and a stronger disturbance rejection performance of systems with mismatched disturbances, some finite time composite anti-disturbance control schemes are proposed. Based on finite time disturbance observer and non-singular terminal sliding mode control techniques, a finite time composite controller is developed for rejecting mismatched disturbances in [26]. In [27], a continuous finite time composite anti-disturbance controller is proposed for a class of linear system with mismatched disturbance via finite time disturbance observer and added power integrator methods. Inspired by the above reference, we devote to investigating the finite time anti-disturbance control problem for SVC system with mismatched disturbances. Because the SVC system is a complex nonlinear system with mismatched disturbances, the previous results are difficult to directly apply. This motives us to develop this study.

In this paper, the problem of finite time composite anti-disturbance control for SVC system with mismatched disturbances is addressed. By using finite time disturbance observer, the mismatched disturbances are estimated. Based on disturbance estimation, some novel virtual control laws are constructed to compensate the mismatched disturbances. Then finite time stability is established via Lyapunov function theory. With the proposed composite control method, the system output can converge to zero in finite time in spite of mismatched disturbances and the disturbance rejection ability of system is improved without sacrificing the nominal performance of the original control strategy. Finally, a simulation result is employed to demonstrated the effectiveness of the proposed control scheme.

2 Model and problem formulation

The dynamics of single-machine infinite-bus (SMIB) electrical power system with SVC can be depicted by the following nonlinear equation [4]

$$\begin{aligned}&\dot{\delta }=\omega -\omega _0,\nonumber \\&\dot{\omega }=-\frac{D}{H}(\omega -\omega _0) +\frac{\omega _0}{H}\left( P_m-E_q'V_sy_{\mathrm{SVC}}\sin \delta \right) +w_2,\nonumber \\&\dot{y}_{\mathrm{SVC}}=\frac{1}{T_{\mathrm{SVC}}}(-y_{\mathrm{SVC}}+y_{\mathrm{SVC}0}+u)+w_3, \end{aligned}$$
(1)

where \(\delta \) and \(\omega \) represent the angle and speed of the generator rotor, respectively; H, \(P_m\), D, \(E_q'\), \(V_s\) and \(T_{\mathrm{SVC}}\) are the inertia constant, the mechanical power on the generator shaft, the damping coefficient, the inner generator voltage and infinite bus voltage, the time constant of SVC regulator, respectively. \(y_{\mathrm{SVC}}=\frac{1}{X_1}+X_2+X_1X_2(B_L+B_C)\) denotes the susceptance of the overall system, and \(y_{\mathrm{SVC}0}\) is the initial stable value of \(y_{\mathrm{SVC}}\); \(X_1=X_d'+X_T+X_L,X_2=X_L,\) \(X_d'\) \(X_T,\) and \(X_L\) mean the direct axis transient reactance of the generator, the reactance of the transformer, and the line reactance, respectively; \(B_L\), \(B_C\), and \(B_L+B_C\) show the susceptance of the inductor in SVC, the susceptance of the capacitor in SVC, and the equivalence reactance of SVC, respectively; u denotes the equivalence input of SVC regulator; \(w_2\) and \(w_3\) mean the external disturbances.

Let \((\delta _0,\omega _0,y_{\mathrm{SVC}0})\) denote an operating point of the power system. Define the system state variables as \(x_1=\delta -\delta _0,\) \(x_2=\omega -\omega _0,\) \(x_3=y_{\mathrm{SVC}}-y_{\mathrm{SVC}0}\). Furthermore, letting \(\frac{\omega _0}{H}P_m=a_0,\) \(-\frac{\omega _0}{H}E_q'V_s=k,\) system (1) is rewritten as

$$\begin{aligned}&\dot{x}_1=x_2\nonumber \\&\dot{x}_2=\theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+w_2,\nonumber \\&\dot{x}_3=-\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+w_3,\nonumber \\&y=\left[ \begin{array}{cc} q_1x_1\\ q_2x_2 \end{array} \right] , \end{aligned}$$
(2)

where y is the regulated output, \(q_1\) and \(q_2\) are nonnegative weighted coefficients, \(\theta =-\frac{D}{H}\).

Assumption 1

The disturbances satisfy the following condition \(|\dddot{w}_i(t)|\le L_i,\) where \(L_i\) are known constants, \(i=2,3.\)

According to [28], the following assumption is required for controller design.

Assumption 2

The angle \(\delta \) satisfies \(0^{\circ }<\delta <180^{\circ }\).

Remark 1

If \(\sin (x_1+\delta _0)=0,\) then \(\delta =k\pi \), \(k=0,1,2,3,\ldots \), which implies that the power systems do not maintain synchronism. Therefore, the normal region of the power system is \(0^{\circ }<\delta <180^{\circ }\).

In order to obtain a finite-time composite controller, some lemmas that will play a key role in the subsequent control development and analysis are revisited as follows.

Lemma 1

[29] If \(0<\ell =\frac{\ell _1}{\ell _2}<1,\) then \(|x^\ell -y^\ell |\le 2^{1-\ell }|x-y|^{\ell },\) where \(\ell _1\) and \(\ell _2\) are positive odd integers.

Lemma 2

[30] The inequality \((|x_1|+\cdots +|x_n|)^p\le |x_1|^p+\cdots +|x_n|^p\) holds for \(x_i\in R,\) \(i=1,\ldots ,n,\) where p is a real number satisfying \(0<p\le 1\).

Control object In this paper, the problem of finite time output regulation is investigated for system (2) under mismatched disturbances using the finite time disturbance observer and finite time control technique. We aim to design a composite controller such that system outputs converge to zero in finite time with the mismatched disturbances.

3 Composite controller design and stability analysis

3.1 Composite controller design

The problem of finite time output regulation for system (2) with mismatched disturbances is investigated by using a composite finite time controller. The detailed design method is given step by step as follows.

Part I Finite time disturbance observer design.

Borrowed from [31, 32], a finite time disturbance observer (FTDO) is presented as

$$\begin{aligned} \dot{z}_{0j}= & {} \nu _{0j}+{\hat{x}}_{j+1}, \dot{z}_{ij}=\nu _{ij},\ldots ,\dot{z}_{3j}\nonumber \\= & {} -\lambda _{3}L_j\text {sign}(z_{3j}-\nu _{2j}), \nonumber \\ \nu _{0j}= & {} -\lambda _0L_j^\frac{1}{4}|z_{0j}-x_j|^{\frac{3}{4}}\text {sign}(z_{0j}-x_j)+z_{1j},\nonumber \\ \nu _{ij}= & {} -\lambda _iL_j^{1/(4-i))}|z_{ij}-\nu _{(i-1)j}|^{\frac{3-i}{4-i}}\text {sign}(z_{ij}\nonumber \\&-\,\nu _{(i-1)j})+z_{(i+1)j}, \end{aligned}$$
(3)

where \(i=1,2,\) \(j=1,2,3\), \({\hat{x}}_{2}=x_2,\) \({\hat{x}}_3=\theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0),\) \({\hat{x}}_4=-\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u\), \(\lambda _0,\lambda _1, \lambda _2,\lambda _3\) are the observer coefficients to be designed, \(z_{0j}, z_{1j}, z_{2j}, z_{3j}\) are the estimates of \(x_j, w_j,\dot{w}_j, \ddot{w}_j,\) respectively.

Define the observer errors \(e_{0j}=z_{0j}-x_j,\) \({e}_{1j}=z_{1j}-w_j,\) \({e}_{2j}=z_{2j}-\dot{w}_j,\) and \({e}_{3j}=z_{3j}-\ddot{w}_j,\) where \(w_1=\dot{w}_1=\ddot{w}_1=0.\) The observer error dynamics are presented as

$$\begin{aligned} \dot{e}_{0j}= & {} -\lambda _0L_j^\frac{1}{4}|e_{0j}|^{\frac{3}{4}}\hbox {sign}(e_{0j})+{e}_{1j},\nonumber \\ \dot{e}_{ij}= & {} -\lambda _iL_j^\frac{1}{4-i}|{e}_{ij}-\dot{e}_{(i-1)j}|^{\frac{3-i}{4-i}}\hbox {sign}({e}_{ij}-\dot{e}_{(i-1)j})\nonumber \\&+\,{e}_{(i+1)j}, \nonumber \\ \dot{{e}}_{3j}= & {} -\lambda _{3}L_j\hbox {sign}({e}_{3j}-\dot{{e}}_{2j})-\dddot{w}_j. \end{aligned}$$
(4)

It can be obtained from [31, 32] that the observer error system is finite time stable, that is, there exists a finite time instant \(t_1\) such that \(e_{0j}\equiv 0\), \(e_{1j}\equiv 0, e_{2j}\equiv 0, e_{3j}\equiv 0\), \(j=1,2,3,\) for \(t> t_1\).

Remark 1

According to [31], the parameters \(\lambda _0,\) \(\lambda _1,\lambda _2, \lambda _3\) are selected. The convergence rate of FTDO is determined by the values of \(L_j\), i.e., the faster convergence rate of the FTDO, the larger the parameters \(L_j\) required. However, the parameters \(L_j\) can not be selected too large to avoid resulting in an excessive transient peaking.

When \(t> t_1,\) the system (3) is changed to

$$\begin{aligned} \dot{z}_{0j}=z_{1j}+{\hat{x}}_{j+1},\quad \dot{z}_{1j}=z_{2j}, \quad \dot{z}_{2j}=z_{3j}, \end{aligned}$$
(5)

where \(j=1,2,3\).

Part II Composite controller design.

Combining disturbance estimation values, system (2) is rewritten as

$$\begin{aligned}&\dot{x}_1=x_2\nonumber \\&\dot{x}_2=\theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12}-e_{12},\nonumber \\&\dot{x}_3=-\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+z_{13}-e_{13},\nonumber \\&y=\left[ \begin{array}{cc} q_1x_1\\ q_2x_2 \end{array} \right] . \end{aligned}$$
(6)

Since the disturbance estimation errors satisfy \(e_{12}=0, e_{13}=0\) for \(t>t_1\), system (6) boils down to

$$\begin{aligned} \dot{x}_1= & {} x_2,\nonumber \\ \dot{x}_2= & {} \theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12},\nonumber \\ \dot{x}_3= & {} -\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+z_{13},\nonumber \\ y= & {} \left[ \begin{array}{cc} q_1x_1\\ q_2x_2 \end{array} \right] . \end{aligned}$$
(7)

In the next, the composite finite time controller will be developed for system (7). The stability analysis will be given in the next subsection.

Step 1 Consider the first equation in (7), i.e.,

$$\begin{aligned} \dot{x}_1=x_2. \end{aligned}$$
(8)

Choose a Lyapunov function as

$$\begin{aligned} V_1=\int _0^{x_1}(s^{\frac{1}{r_1}})^{2-r_2}\hbox {d}s, \end{aligned}$$
(9)

where \(r_1=1, r_2=r_1+\tau ,\) \(-\frac{1}{4}<\tau =-\frac{q}{p}<0,\) q is a positive even integer, p is a positive odd integer. Computing the first derivative of (9) along system (8), yields

$$\begin{aligned} \dot{V}_1=x_1^{\frac{2-r_2}{r_1}}x_2=x_1^{\frac{2-r_2}{r_1}}x_2^{*} +x_1^{\frac{2-r_2}{r_1}}(x_2-x_2^{*}), \end{aligned}$$
(10)

where \(x_2^{*}\) is a virtual control law. The virtual control law \(x_2^{*}\) is designed as

$$\begin{aligned} x_2^{*}=-\beta _1x_1^{r_2}, \end{aligned}$$
(11)

where \(\beta _1>\beta _1^{*}=3.\) Combining (10) and (11), we have

$$\begin{aligned} \dot{V}_1\le -3x_1^2+x_1^{\frac{2-r_2}{r_1}}(x_2-x_2^{*}). \end{aligned}$$
(12)

Step 2 Consider

$$\begin{aligned} \dot{x}_2= & {} \theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12}.\nonumber \\ \end{aligned}$$
(13)

The Lyapunov function is selected as

$$\begin{aligned} V_{2}=V_1+\int _{{x}_2^*}^{{x}_2} \left( s^{\frac{1}{r_2}}-({x}_2^*)^{\frac{1}{r_2}}\right) ^{2-r_2-\tau }\hbox {d}s. \end{aligned}$$
(14)

Computing the first derivative of (14), we have

$$\begin{aligned} \dot{V}_2\le & {} -3x_1^2+x_1^{\frac{2-r_2}{r_1}} (x_2-x_2^{*})+\xi _2^{2-r_2-\tau }(\theta x_2\nonumber \\&+\,a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12})\nonumber \\&-\,(2-r_2-\tau )\int _{{x}_2^*}^{{x}_2} \left( s^{\frac{1}{r_2}}-({x}_2^*)^{\frac{1}{r_2}}\right) ^{1-r_2-\tau }\nonumber \\&\times \,\hbox {d}s\frac{\partial {{x}_2^*}^{\frac{1}{r_2}}}{\partial x_1}x_2, \end{aligned}$$
(15)

where \(\xi _2=x_2^{\frac{1}{r_2}}-({x}_2^*)^{\frac{1}{r_2}}.\)

Note that \(0<r_2\le 1,\) by Lemma 1 and Young’s inequality, we have

$$\begin{aligned}&x_1^{\frac{2-r_2}{r_1}}(x_2-x_2^{*})\le 2^{1-r_2}|x_1|^{\frac{2-r_2}{r_1}}|\xi _2|^{r_2}\le \frac{1}{2}x_{1}^2+{\hat{c}}_{21}\xi _2^2,\nonumber \\&\quad -\,(2-r_2-\tau )\int _{{x}_2^*}^{{x}_2}\left( s^{\frac{1}{r_2}}-({x}_2^*)^ {\frac{1}{r_2}}\right) ^{1-r_2-\tau }\hbox {d}s\frac{\partial {{x}_2^*}^{\frac{1}{r_2}}}{\partial x_1}x_2\nonumber \\&\quad \le (2-r_2-\tau )\left| \frac{\partial {{x}_2^*}^{\frac{1}{r_2}}}{\partial x_1}\right| |\xi _2|^{1-r_2-\tau }(|x_2-x_2^{*}|)(|x_2\nonumber \\&\quad \quad -\,x_2^{*}|+ |x_2^*|)\nonumber \\&\quad \le (2-r_2-\tau )2^{1-r_2}\left| \frac{\partial {{x}_2^*}^{\frac{1}{r_2}}}{\partial x_1}\right| |\xi _2|^{1-\tau }(2^{1-r_2}|\xi _2|^{r_2}\nonumber \\&\quad \quad +\,\beta _1|x_1|^{r_2})\nonumber \\&\quad \le \frac{1}{2}|x_1|^2+{\hat{c}}_{22}|\xi _2|^2, \end{aligned}$$
(16)

where \({\hat{c}}_{21}>0\) and \({\hat{c}}_{22}>0\). Combining (15) and (16), yields

$$\begin{aligned} \dot{V}_2\le & {} -2x_1^2+\xi _2^{2-r_2-\tau } (\theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\nonumber \\&\sin (x_1 +\,\delta _0)+z_{12})+({\hat{c}}_{21}+{\hat{c}}_{22})\xi _2^2\nonumber \\= & {} -2x_1^2+\xi _2^{2-r_2-\tau }(\theta x_2+a_0+k(x_3-x_3^{*}+x_3^{*}\nonumber \\&+\,y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12})+({\hat{c}}_{21}+{\hat{c}}_{22})\xi _2^2,\nonumber \\ \end{aligned}$$
(17)

where \(x_3^{*}\) is a virtual control law. The virtual control law \(x_3^{*}\) is designed as

$$\begin{aligned} x_3^{*}= & {} \frac{1}{k\sin (x_1+\delta _0)} \left( -\theta {x_2}-a_0-ky_{\mathrm{SVC}}\sin (x_1+\delta _0)\right. \nonumber \\&\left. -\,z_{12}-\beta _2\xi _2^{r_3}\right) , \end{aligned}$$
(18)

where \(\beta _2>\beta _2^{*}=2+{\hat{c}}_{21}+{\hat{c}}_{22},\) \(r_3=r_2+\tau .\) Substituting (18) into (17), leads to

$$\begin{aligned} \dot{V}_2\le & {} -2x_1^2-2\xi _2^2+\xi _2^{2-r_2-\tau }k(x_3-\,x_3^{*})\sin (x_1+\delta _0).\nonumber \\ \end{aligned}$$
(19)

Step 3 Consider

$$\begin{aligned} \dot{x}_3=-\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+z_{13}. \end{aligned}$$
(20)

We choose the following Lyapunov function

$$\begin{aligned} V_3=V_2+\int _{{\bar{x}}_3^*}^{{\bar{x}}_3} \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{2-r_3-\tau }\hbox {d}s, \end{aligned}$$
(21)

where

$$\begin{aligned} {\bar{x}}_3= & {} k\sin (x_1+\delta _0)x_3+\theta x_2+a_0\nonumber \\&+\,ky_{\mathrm{SVC}}\sin (x_1+\delta _0)+z_{12},\nonumber \\ {\bar{x}}_3^{*}= & {} k\sin (x_1+\delta _0)x_3^{*}+\theta x_2+a_0\nonumber \\&+\,ky_{\mathrm{SVC}}\sin (x_1+\delta _0)+z_{12}=-\beta _2\xi _2^{r_3}. \end{aligned}$$
(22)

Computing the first derivative of (21) gives rise to

$$\begin{aligned} \dot{V}_3\le & {} -2x_1^2-2\xi _2^2+\xi _2^{2-r_2-\tau } k(x_3-x_3^{*})\sin (x_1+\delta _0)\nonumber \\&+\,\xi _3^{2-r_3-\tau }\dot{{\bar{x}}}_3-(2-r_3-\tau ) \int _{{\bar{x}}_3^*}^{{\bar{x}}_3}\nonumber \\&\times \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{1-r_3-\tau }\hbox {d}s\left( \frac{\partial {\bar{x}}_3 ^{*\frac{1}{r_3}}}{\partial x_1}\dot{x}_1+\frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_2}\dot{x}_2\right) \nonumber \\= & {} -2x_1^2-2\xi _2^2+\xi _2^{2-r_2-\tau }k(x_3-x_3^{*})\sin (x_1+\delta _0)\nonumber \\&+\,\xi _3^{2-r_3-\tau } \left( \frac{\partial {\bar{x}}_3}{\partial x_1}x_2+ \frac{\partial {\bar{x}}_3}{\partial x_2}(k\sin (x_1+\delta _0)x_3\right. \nonumber \\&+\,\theta x_2+a_0+ky_{\mathrm{SVC}}\sin (x_1+\delta _0)+z_{12})\nonumber \\&\left. +\,\frac{\partial {\bar{x}}_3}{\partial x_3}\left( -\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+z_{13}\right) +\frac{\partial {\bar{x}}_3}{\partial z_{12}}z_{22}\right) \nonumber \\&-\,(2-r_3-\tau )\int _{{\bar{x}}_3^*}^{{\bar{x}}_3} \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{1-r_3-\tau }\nonumber \\&\times \,\hbox {d}s \left( \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_1}\dot{x}_1+\frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_2}\dot{x}_2\right) , \end{aligned}$$
(23)

where \(\xi _3={\bar{x}}_3^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}},\) \(\frac{\partial {\bar{x}}_3}{\partial x_1}=k\cos (x_1+\delta _0)(x_3+y_{\mathrm{SVC}}),\) \(\frac{\partial {\bar{x}}_3}{\partial x_2}=\theta \), \(\frac{\partial {\bar{x}}_3}{\partial x_3}=k\sin (x_1+\delta _0),\) \(\frac{\partial {\bar{x}}_3}{\partial z_{12}}=1.\)

Combining (22) and Lemma 1 plus Young’s inequality, we obtain

$$\begin{aligned}&\xi _2^{2-r_2-\tau }k(x_3-x_3^{*})\sin (x_1+\delta _0)\nonumber \\&\quad =\xi _2^{2-r_2-\tau }({\bar{x}}_3-{\bar{x}}_3^{*})\nonumber \\&\quad \le 2^{1-r_2}|\xi _2|^{2-r_2-\tau }|\xi _3|^{r_3} \le \frac{1}{2}|\xi _2|^2+{\hat{c}}_{31}|\xi _3|^2, \end{aligned}$$
(24)

where \({\hat{c}}_{31}>0.\) Now, we estimate the last term on the right-hand side of (23).

First, it follows Lemma 1 and Young’s inequality that

$$\begin{aligned}&-(2-r_3-\tau )\int _{{\bar{x}}_3^*}^{{\bar{x}}_3} \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{1-r_3-\tau }\hbox {d}s \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_1}\dot{x}_1\nonumber \\&\quad \le (2-r_3-\tau )\left| \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_1}\right| |\xi _3|^{1-r_3-\tau }|{\bar{x}}_3-{\bar{x}}_3^{*}||x_2|\nonumber \\&\quad \le (2-r_3-\tau )\left| \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_1}\right| |\xi _3|^{1-r_3-\tau }|{\bar{x}}_3 -\,{\bar{x}}_3^{*}| \nonumber \\&\quad \quad \times \left( |x_2-x_2^{*}|+|x_2^*|\right) \nonumber \\&\quad \le (2-r_3-\tau )\left| \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_1}\right| |\xi _3|^{1-\tau }\nonumber \\&\qquad \times \,\left( 2^{1-r_2}|\xi _2|^{r_2}+\beta _1|x_1|^{r_2}\right) \nonumber \\&\quad \le \frac{1}{3}x_1^2+\frac{1}{4}\xi _2^2+{\hat{c}}_{31}\xi _3^2+{\hat{c}}_{32}\xi _3^2, \end{aligned}$$
(25)

where \({\hat{c}}_{32}>0.\) And

$$\begin{aligned}&-(2-r_3-\tau )\int _{{\bar{x}}_3^*}^{{\bar{x}}_3} \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{1-r_3-\tau }\hbox {d}s \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_2}\dot{x}_2\nonumber \\&\quad \le (2-r_3-\tau )\left| \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_2}\right| |\xi _3|^{1-r_3-\tau }|{\bar{x}}_3-{\bar{x}}_3^{*}|\beta _2^{r_3}\nonumber \\&\qquad \times |x_2|^{\frac{1}{r_2}-1}|\theta x_2+a_0+k(x_3+y_{\mathrm{SVC}0})\sin (x_1+\delta _0)+z_{12}|\nonumber \\&\quad =\beta _2^{\frac{1}{r_3}}\beta _2^{r_3}(2-r_3-\tau )|\xi _3|^{1-r_3-\tau }|{\bar{x}}_3-{\bar{x}}_3^{*}||x_2|^{\frac{1}{r_2}-1}|{\bar{x}}_3|\nonumber \\&\quad =\beta _2^{\frac{1}{r_3}}\beta _2^{r_3}(2-r_3-\tau )|\xi _3|^{1-r_3-\tau }|{\bar{x}}_3-{\bar{x}}_3^{*}||x_2|^{\frac{1}{r_2}-1}\nonumber \\&\qquad \left( |{\bar{x}}_3-{\bar{x}}_3^{*}|+|{\bar{x}}_3^{*}|\right) \nonumber \\&\quad \le \beta _2^{\frac{1}{r_3}}\beta _2^{r_3}(2-r_3-\tau )2^{1-r_3}|\xi _3|^{1-\tau }|x_2|^{\frac{1}{r_2}-1}\nonumber \\&\qquad \left( 2^{1-r_3}|\xi _3|^{r_3}+\beta _2|\xi _2|^{r_3}\right) . \end{aligned}$$
(26)

Using Lemmas 1 and 2, we have

$$\begin{aligned} |x_2|^{\frac{1}{r_2}-1}= & {} |x_2-x_2^{*}+x_2^{*}|^{\frac{1}{r_2}-1}\nonumber \\\le & {} |x_2-x_2^{*}|^{\frac{1}{r_2}-1}+|x_2^{*}|^{\frac{1}{r_2}-1}\nonumber \\\le & {} 2^{1-r_2}|\xi _2|^{1-r_2}+\beta _1^{\frac{1}{r_2}-1}|x_1|^{1-r_2}. \end{aligned}$$
(27)

Substituting (27) into (26) and using Lemma 1 leads to

$$\begin{aligned}&\beta _2^{\frac{1}{r_3}}(2-r_3-\tau )\int _{{\bar{x}}_3^*}^{{\bar{x}}_3} \left( s^{\frac{1}{r_3}}-({\bar{x}}_3^*)^{\frac{1}{r_3}}\right) ^{1-r_3-\tau }\hbox {d}s \frac{\partial {\bar{x}}_3^{*\frac{1}{r_3}}}{\partial x_2}\dot{x}_2\nonumber \\&\quad \le \beta _2^{\frac{1}{r_3}}\beta _2^{r_3}(2-r_3-\tau )2^{1-r_3}|\xi _3|^{1-\tau } \left( 2^{1-r_2}|\xi _2|^{1-r_2}\right. \nonumber \\&\quad \quad \left. +\beta _1^{\frac{1}{r_2}-1}|x_1|^{1-r_2}\right) \left( 2^{1-r_3}|\xi _3|^{r_3}+\beta _2|\xi _2|^{r_3}\right) \nonumber \\&\quad \le \beta _2^{\frac{1}{r_3}}\beta _2^{r_3}(2-r_3-\tau )2^{1-r_3} \left( 2^{2-r_2-r_3}|\xi _2|^{1-r_2}|\xi _3|^{1-\tau }\right. \nonumber \\&\quad \quad +\,2^{1-r_2}|\xi _2|^{1-r_2+r_3}|\xi _3|^{1-\tau }\nonumber \\&\quad \quad +\,2^{1-r_3}\beta _1^{\frac{1}{r_2}-1}|x_1|^{1-r_2}|\xi _3|^{1-\tau +r_3}+\beta _1^{\frac{1}{r_2}-1}\beta _2\nonumber \\&\quad \quad \times \,\left. |x_1|^{1-r_2}|\xi _2|^{r_3}|\xi _3|^{1-\tau }\right) \nonumber \\&\quad \le \frac{1}{4}\xi _2^2+{\hat{c}}_{33}\xi _3^2+\frac{1}{4}\xi _2^2+{\hat{c}}_{34}\xi _3^2+\frac{1}{3}x_1^2+{\hat{c}}_{35}\xi _3^2\nonumber \\&\quad \quad +\,\frac{1}{3}x_1^2+\frac{1}{4}\xi _2^2+{\hat{c}}_{36}\xi _3^2, \end{aligned}$$
(28)

where \({\hat{c}}_{33},{\hat{c}}_{34},{\hat{c}}_{35},\) and \({\hat{c}}_{36}\) are positive constants. Combining (23), (25) and (28), yields

$$\begin{aligned} \dot{V}_3\le & {} -x_1^2-\xi _2^2-lx_1^{2-\tau }-l\xi _2^{2-\tau } +\xi _3^{2-r_3-\tau }\left( \frac{\partial {\bar{x}}_3}{\partial x_1}x_2\right. \nonumber \\&+\,\frac{\partial {\bar{x}}_3}{\partial x_2}{\bar{x}}_3+\frac{\partial {\bar{x}}_3}{\partial x_3}\left( -\frac{1}{T_{\mathrm{SVC}}}x_3+\frac{1}{T_{\mathrm{SVC}}}u+z_{13}\right) \nonumber \\&\left. +\,\frac{\partial {\bar{x}}_3}{\partial z_{12}}z_{22}\right) \nonumber \\&+\, ({\hat{c}}_{31}+{\hat{c}}_{32}+{\hat{c}}_{33}+{\hat{c}}_{34}+{\hat{c}}_{35}+{\hat{c}}_{36})\xi _3^2. \end{aligned}$$
(29)

The controller u is designed as

$$\begin{aligned} u= & {} x_3-T_{\mathrm{SVC}}z_{13}-\frac{T_{\mathrm{SVC}}}{\frac{\partial {\bar{x}}_3}{\partial x_3}}\left( \frac{\partial {\bar{x}}_3}{\partial x_1}x_2+\frac{\partial {\bar{x}}_3}{\partial x_2}{\bar{x}}_3\right. \nonumber \\&\left. +\,\frac{\partial {\bar{x}}_3}{\partial z_{12}}z_{22}+\beta _3\xi _3^{r_4}\right) , \end{aligned}$$
(30)

where \(\beta _3\ge \beta _3^*=1+{\hat{c}}_{31}+{\hat{c}}_{32}+{\hat{c}}_{33}+{\hat{c}}_{34}+{\hat{c}}_{35}+{\hat{c}}_{36}, r_4=r_3+\tau .\)

Remark 2

In order to obtain the finite time controller, the auxiliary state \({\bar{x}}_3\) and the auxiliary virtual control law \({\bar{x}}_3^{*}\) in (22) are defined.

3.2 Stability analysis

Theorem 1

Consider system (2). If Assumptions 1 and 2 hold, the composite controller (30) can guarantee the closed-loop system (2), (4) and composite controller (30) is globally finite time stable.

Proof

The stability analysis of the closed-loop system is divided into two parts. First, the finite time stability of system (7) and (30) is established when \(t>t_1\). Next, we will prove that system states of (3) and (6) will not escape to the infinity in any time interval \([0, t_1]\).

At first step: substituting (30) into (29) gives rise to

$$\begin{aligned} \dot{V}_3\le -x_1^2-\xi _2^2-\xi _3^2. \end{aligned}$$
(31)

It can be verify that

$$\begin{aligned} V_3\le \frac{1}{c}\left( x_1^{2-\tau }+\xi _2^{2-\tau }+\xi _n^{2-\tau }\right) , \end{aligned}$$
(32)

where \(c>0.\) Let \(\lambda _1=\frac{1}{2}c^{\frac{2}{2-\tau }}\). Using Lemma 2, we can derive from (31) and (32) that

$$\begin{aligned} \dot{V}_3+\lambda _1 V_3^{\frac{2}{2-\tau }}\le & {} -(x_1^2+\xi _2^2+\xi _3^2)+\frac{\lambda _1}{c^{\frac{2}{2-\tau }}}(x_1^2+\xi _2^2+\xi _3^2)\nonumber \\= & {} -\frac{1}{2}(x_1^2+\xi _2^2+\xi _3^2). \end{aligned}$$
(33)

According to the finite time stability definition [33], we obtain that system (7) and (30) is finite time stable.

Next, we will prove that system states of (3) and (6) will not escape to the infinity in any time interval \([0, t_1]\). By coordinate transform, we obtain the following system

$$\begin{aligned} \dot{x}_1= & {} x_2=x_2-x_2^{*}+x_2^{*},\nonumber \\ \dot{\xi }_2= & {} x_2^{\frac{1}{r_2}-1}({\bar{x}}_3-e_{12})+\beta _1^{\frac{1}{r_2}}x_2\nonumber \\= & {} x_2^{\frac{1}{r_2}-1}\left( {\bar{x}}_3-{\bar{x}}_3^{*}-\beta _2\xi _2^{r_3}-e_{12}\right) +\beta _1^{\frac{1}{r_2}}x_2,\nonumber \\ \dot{\xi }_3= & {} {\bar{x}}_3^{\frac{1}{r_3}-1}\left( -\rho _3\xi _3^{r_4}-e_{13}\right) +\beta _2^{\frac{1}{r_3}-1} \left( x_2^{\frac{1}{r_2}-1}\right. \nonumber \\&\left. \quad \left( {\bar{x}}_3-\,{\bar{x}}_3^{*}-\beta _2\xi _2^{r_3}-e_{12}\right) +\beta _1^{\frac{1}{r_2}}x_2\right) . \end{aligned}$$
(34)

A finite time bounded function is selected as

$$\begin{aligned} B(x_1,\xi _2,\xi _3)=\frac{1}{2}(x_1^2+\xi _2^2+\xi _3^2). \end{aligned}$$
(35)

Taking the first derivative of (35), we have

$$\begin{aligned}&\dot{B}(x_1,\xi _2,\xi _3)\nonumber \\&\quad =x_1\dot{x}_1+\xi _2\dot{\xi }_2+\xi _3\dot{\xi }_3\nonumber \\&\quad =x_1(x_2-x_2^{*}+x_2^{*})+\xi _2 \left( x_2^{\frac{1}{r_2}-1}({\bar{x}}_3-e_{12})+\beta _1^{\frac{1}{r_2}}x_2\right) \nonumber \\&\quad \quad +\,\xi _3 \left( {\bar{x}}_3^{\frac{1}{r_3}-1}\left( -\beta _3\xi _3^{r_4}-e_{13}\right) + \beta _2^{\frac{1}{r_3}-1}\left( x_2^{\frac{1}{r_2}-1}\left( {\bar{x}}_3\right. \right. \right. \nonumber \\&\quad \quad \left. \left. \left. -{\bar{x}}_3^{*}-\beta _2\xi _2^{r_3}-e_{12}\right) +\beta _1^{\frac{1}{r_2}}x_2\right) \right) . \end{aligned}$$
(36)
Fig. 1
figure 1

Response curves of system state \(\delta \) with external disturbances

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Fig. 2
figure 2

Response curves of system state \(\omega \) with external disturbances

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Fig. 3
figure 3

Curves of system input with external disturbances

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Fig. 4
figure 4

Curves of disturbances and disturbance estimation

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Fig. 5
figure 5

Response curves of system state \(\delta \) with model uncertainties

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Fig. 6
figure 6

Response curves of system state \(\omega \) with model uncertainties

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Fig. 7
figure 7

Curves of system input with model uncertainties

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Using Lemma 1 plus Young’s inequality, the following inequalities are true

$$\begin{aligned}&x_1(x_2-x_2^{*})\le 2^{1-r_2}|x_1||\xi _2|^{r_2}\le |x_1|^{2+\tau }+h_{11}|\xi _2|^{2+\tau },\nonumber \\&x_1x_2^{*}\le |x_1||\beta x_1^{r_2}|=\beta _1|x_1|^{2+\tau },\nonumber \\&\xi _2x_2^{\frac{1}{r_2}-1} \nonumber \\&\quad \le 2^{1-r_2}|\xi _2|(|\xi _2|^{1-r_2}+\beta _1^{\frac{1}{r_2}-1}|x_1|^{1-r_2})\nonumber \\&\quad \le 2^{1-r_2}(|\xi _2|^{2-r_2}+\beta _1^{\frac{1}{r_2}-1}|x_1|^{1-r_2}|\xi _2|)\nonumber \\&\quad \le 2^{1-r_2}(2|\xi _2|^{2-r_2}+h_{12}|x_1|^{2-r_2}),\nonumber \\&\xi _2x_2^{\frac{1}{r_2}-1}({\bar{x}}_3-e_{12})\nonumber \\&\quad \le 2^{1-r_2}(2|\xi _2|^{2-r_2}+h_{12}|x_1|^{2-r_2})(2^{1-r_3}|\xi _3|^{r_3}\nonumber \\&\qquad +\,\beta _3|\xi _2|^{r_3}+e_{12})\nonumber \\&\quad \le x_1^{2+\tau }+h_{21}\xi _2^{2+\tau }+h_{22}\xi _{3}^{2+\tau }+2^{1-r_2}(2|\xi _2|^{2-r_2}\nonumber \\&\qquad +\,h_{12}|x_1|^{2-r_2})|e_{12}|,\nonumber \\&\xi _2\beta _1^{\frac{1}{r_2}}x_2\nonumber \\&\quad \le \beta _1^{\frac{1}{r_2}}|\xi _2|(2^{1-r_2}|\xi _2|^{r_2}+\beta _1|x_1|^{r_2})\nonumber \\&\quad \le x_1^{2+\tau }+h_{23}|\xi _2|^{2+\tau },\nonumber \\&\xi _3{\bar{x}}_3^{\frac{1}{r_3}-1}\nonumber \\&\quad \le 2^{1-r_3}|\xi _3|(|\xi _3|^{1-r_3}+\beta _2^{\frac{1}{r_3}-1}|\xi _2|^{1-r_3})\nonumber \\&\quad \le 2^{1-r_3}(|\xi _3|^{2-r_3}+\beta _2^{\frac{1}{r_3}-1}|\xi _2|^{1-r_3}|\xi _3|)\nonumber \\&\quad \le 2^{1-r_3}(2|\xi _3|^{2-r_3}+h_{31}|\xi _2|^{2-r_3}),\nonumber \\&\xi _3{\bar{x}}_3^{\frac{1}{r_3}-1}(-\beta _3\xi _3^{r_4}-e_{13})\nonumber \\&\quad \le |\xi _2|^{2+\tau }+h_{32}|\xi _3|^{2+\tau }+2^{1-r_3}(2|\xi _3|^{2-r_3}\nonumber \\&\qquad +\,h_{31}|\xi _2|^{2-r_3})|e_{13}|,\nonumber \\&\xi _3 \left( \beta _2^{\frac{1}{r_3}-1}(x_2^{\frac{1}{r_2}-1}({\bar{x}}_3-{\bar{x}}_3^{*}-\beta _2\xi _2^{r_3}-e_{12})+\beta _1^{\frac{1}{r_2}}x_2)\right) \nonumber \\&\quad \le x_1^{2+\tau }+h_{41}\xi _2^{2+\tau }+h_{42}\xi _3^{2+\tau }+(x_1^{2-r_2}\nonumber \\&\qquad +\,h_{43}\xi _2^{2-r_2}+h_{44}\xi _3^{2-r_2})|e_{12}|, \end{aligned}$$
(37)

where \(h_{11},h_{12},h_{21},\) \(h_{22},\) \(h_{31},\) \(h_{32},\) \(h_{41},\) \(h_{42},\) \(h_{43},\) and \(h_{44}\) are positive numbers. Since \(w_2\) and \(w_3\) are estimated in finite time, i.e., the estimation errors \(e_{12}\) and \(e_{13}\) converge to zero in finite time, then \(e_{1i}\) is bounded. We denote \(e_{1i}\le {\bar{e}}_i\le {\bar{e}}\), where \({\bar{e}}> 0\) is a constant. On the one hand if

$$\begin{aligned} \eta =\sqrt{x_1^2+\xi _2^2+\xi _3^2}\ge {\hat{\eta }}>1, \end{aligned}$$
(38)

then we obtain \(|x_1|^{2+\tau }\le \eta ^{2+\tau }\le \eta ^2,\) \(|\xi _2|^{2+\tau }\le \eta ^{2+\tau }\le \eta ^2,\) \(|\xi _3|^{2+\tau }\le \eta ^{2+\tau }\le \eta ^2\), \(|x_1|^{2-r_i}\le \eta ^2,\) \(|\xi _2|^{2-r_i}\le \eta ^2,\) \(|\xi _3|^{2-r_i}\le \eta ^2.\) With this in mind, we obtain

$$\begin{aligned} \dot{B}(x_1,\xi _2,\xi _3)\le K\eta ^2=\textit{KB}(x_1,\xi _2,\xi _3), \end{aligned}$$
(39)

where \(K=\beta _1+4+h_{11}+h_{21}+h_{23}+h_{41}+h_{22}+h_{32}+h_{42}+2*2^{1-r_2}{\bar{e}}+2^{1-r_2}{\bar{e}}h_{12}+2*2^{1-r_3}{\bar{e}}+2^{1-r_3}{\bar{e}}h_{31}+(1+h_{43}+h_{44}){\bar{e}}.\) On the other hand, if \(\eta <1\), there exists a constant \({\bar{L}}\) such that

$$\begin{aligned} \dot{B}(x_1,\xi _2,\xi _3)\le {\bar{L}}. \end{aligned}$$
(40)

Thus we obtain

$$\begin{aligned} \dot{B}(x_1,\xi _2,\xi _3)\le \textit{KB}(x_1,\xi _2,\xi _3)+{\bar{L}}. \end{aligned}$$
(41)

Solving the inequality (41), we have \(B(x_1,\xi _2,\xi _3)\le (B(x_1(0),\xi _2(0),\xi _3(0))+\frac{{\bar{L}}}{K})e^{Kt}-\frac{{\bar{L}}}{K}.\) When \(t\le t_1,\) the system states \(x_1,\) \(\xi _2\) \(\xi _3\) of (34) are bounded.

According to the above analysis and [34], we obtain that the system consisting of system (6), the estimation error (4) and the control law (30) is finite time stable.

Remark 3

In the absence of disturbances, it is derived from (3) and (4) that

$$\begin{aligned} \dot{e}_{0j}= & {} -\lambda _0L_j^{1/4}|e_{0j}|^{\frac{n}{n+1}}\hbox {sign}(e_{0j})+z_{1j},\nonumber \\ \dot{z}_{1j}= & {} -\lambda _1L_j^{1/3}|z_{1j}-\dot{e}_{0j}|^{\frac{2}{3}}\hbox {sign}(z_{1j}-\dot{e}_{0j})+z_{2j},\nonumber \\ \dot{z}_{2j}= & {} -\lambda _2L_j^{1/2}|z_{2j}-\dot{z}_{1j}|^{\frac{1}{2}}\hbox {sign}(z_{2j}-\dot{z}_{1j})+z_{3j},\nonumber \\ \dot{z}_{3j}= & {} -\lambda _{3}L_j\hbox {sign}(z_{3j}-\dot{z}_{2j}), j=1,2,3, \end{aligned}$$

which implies that \(z_1, z_2, z_3\) equal to zero all the time. Then the composite controller (30) degenerates to the traditional finite time controller, which means that the proposed method does not sacrifice the nominal performance. This good property will be verified via simulation results in the next section.

4 Simulation result

The SVC system has the following parameters [8, 35]:

$$\begin{aligned}&H=5.9\,\hbox {s},\quad D=1.0, \quad V_s=1.0\,\hbox {pu},\\&T_{\mathrm{SVC}}=0.02, \quad X_1=0.84\,\hbox {pu},\\&X_2=0.52\,\hbox {pu},\quad B_L+B_C=0.3\,\hbox {pu},\\&q_1=0.4,\quad q_2=0.6. \end{aligned}$$

The controller parameters are chosen as

$$\begin{aligned}&r_1=1,\quad \tau =-\frac{2}{15},\quad \beta _1=5, \quad \beta _2=10, \\&\quad \beta _3=25, \quad L_1=L_2=L_3=10. \end{aligned}$$

The following operating point is considered

$$\begin{aligned}&\delta _0=57.1\,\hbox {deg}, \quad \omega _0=314.159\,\hbox {rad/s}, \nonumber \\&\quad y_{\mathrm{SVC}0}=0.4\,\hbox {pu}. \end{aligned}$$
(42)

Case I External disturbance rejection ability

The external disturbances in the SVC system are taken as

$$\begin{aligned} w_1(t)= & {} \left\{ \begin{array}{ll} 0, &{}\quad t<6\quad \hbox {and}\quad 10\le t<15,\\ 1.5, &{}\quad 6\le t<10,\\ 3\sin (t), &{}\quad t>15,\\ \end{array}\right. , \\ w_2(t)= & {} \left\{ \begin{array}{ll} 0, &{}\quad t<6\quad \hbox {and}\quad 10\le t<15,\\ 1, &{}\quad 6\le t<10,\\ 2.4\cos (t), &{}\quad t>15.\\ \end{array}\right. \end{aligned}$$

The simulation results are presented in Figs. 1, 2, 3 and 4. Response curves of system states \(\delta \) and \(\omega \) are shown in Figs. 1 and 2. It can be observed that the system outputs can achieve their control object in the presence of mismatched disturbances. In order to illustrate the effectiveness of disturbance observer, the curves of disturbances and disturbance estimation are presented in Fig. 4. The control input is depicted in Fig. 3.

Case II: Robustness against model uncertainties

The robustness against model uncertainties of the proposed scheme is verified in this part. To investigate the performance of robustness, we choose the model uncertainties as follows. \(\theta \) has variation of \({+}20\,\%\).

Curve of system outputs under the proposed control method is given in Figs. 5 and 6. It can be seen that the closed-loop system has a good robustness performance and the outputs have a satisfactory performance. The control input is presented in Fig. 7.

5 Conclusion

In this paper, the problem of finite time composite anti-disturbance control for SVC system has been investigated. Based on finite time disturbance observer and finite time control techniques, a finite time composite controller has been proposed. Using Lyapunov function theory, the finite time stability of the closed-loop system has been analyzed. Finally, the simulation result has been presented to show the effectiveness of the developed method.