Adaptive Dynamic Surface Integral Sliding Mode Fault-Tolerant Control for Multimachine Excitation Systems with SVC

In this paper, an adaptive dynamic surface integral sliding mode fault-tolerant controller is designed for the multimachine power system with static var compensator (SVC) to overcome the problem of actuator failure. The main features of the proposed method are as follows: (1) By combining the dynamic surface control (DSC) method with integral sliding mode (ISM), the tracking errors of the system converge to the neighborhood of zero within a finite time, and the convergence speed, tracking accuracy, and anti-interference ability of the system are also significantly improved. (2) By introducing the failure factors, an adaptive fault-tolerant controller is designed to ensure the stability of the entire system after partial failure of the actuator. (3) By estimating the norm of the ideal weight vector of the radial basis function neural networks (RBFNNs), the computational burden of the controller is reduced. Finally, the simulation results show the effectiveness of the proposed control scheme.


Introduction
e modern power system has the characteristics of large power grid, large unit, ultrahigh voltage, long distance power transmission, and high automatic control, which brings about great challenges to the stability operation of the power system [1][2][3][4]. e important role of generator excitation control in improving the stable operation of the power system has attracted widespread attention from researchers worldwide [5][6][7][8]. e traditional power system stabilizer (PSS) [9,10] and linear optimal excitation control [11] are based on the approximate linearization model. Although the small disturbance stability problem can be improved, the effective suppression of large disturbances cannot be achieved. Moreover, due to parameter uncertainty, external interference, multidimensional and strong coupling, and strong nonlinear characteristics, designing a control system is a very challenging task. erefore, advanced control strategies such as backstepping [12][13][14][15], sliding mode variable structure [16][17][18], and feedback linearization [19] have been proposed to solve the problem of power system excitation control. e adaptive backstepping technique can effectively handle the case where the parameters are unknown [20][21][22]. In [20], the infinite bus voltage and transmission line parameters of the single-machine infinite-bus (SMIB) power system are regarded as unknown, and an adaptive backstepping controller is proposed. A robust nonlinear adaptive backstepping excitation controller is designed in [21] by considering the influence of external disturbance based on the parameter uncertainty. However, the problem of "explosion of complexity" exists in the backstepping method. In order to solve this problem, the dynamic surface control method is used [23][24][25][26][27]. In [24], for a class of uncertain nonlinear systems with asymmetric dead zone nonlinearity, an antidisturbance dynamic surface control scheme is proposed, which effectively solves the problems of nonmatching external disturbance and unknown asymmetric dead zone nonlinearity. A first-order filter is introduced on the traditional backstepping method for the power system, and the controller based on the dynamic surface method is designed in [26]. e device effectively improves the stability of the power system. For large-scale multimachine systems with static var compensator (SVC), the decentralized dynamic surface quantization control scheme is proposed in [4] and the fuzzy dynamic surface sliding mode controller is designed in [7], which improves the robustness of the system. e partial feedback linearization (PFBL) technique is widely used in the design of excitation control systems by overcoming the limitations of direct feedback linearization (DFBL) and exact feedback linearization (EFBL) on rotor angle measurement problem. In [28], the nonlinear system model is transformed into a reduced-order linear part and a nonlinear dynamic autonomous part. e optimal control theory is used to design a linear state feedback stability controller for the reduced-order linear part, which enhances the transient stability limit of the power system. However, the feedback linearized excitation controller has a parameter sensitivity problem. Sliding mode control (SMC) overcomes the parameter sensitivity problem due to being less sensitive to changes in parameters and external disturbances [29][30][31][32][33][34]. In [35], for the multimachine power system with external disturbance, the adaptive method and the sliding mode variable structure method are combined based on the feedback linearization to design a decentralized coordinated adaptive sliding mode stabilizer. e adaptive gain of the super-twist algorithm is constructed by using the equivalent control theory, and the finite time stability of the closed-loop power system is realized in [33]. In [36], by combining adaptive fuzzy control with sliding mode control, the chattering problem caused by system uncertainty and dynamics is overcome, and the tracking error is converged to zero better and faster. In addition, intelligent controllers [37][38][39][40][41][42][43] are also widely used. e fuzzy logic rules [40] and the neural networks [44,45] are utilized to design adaptive controller for nonlinear system with parameters uncertainty and external disturbances; the robustness and anti-interference ability of control system are improved. e static var compensator (SVC) control is also one of the effective and economical means of improving the stability of power systems. Generally, the generator excitation controller and the SVC controller are designed independently of each other, but the uncoordinated control of the SVC and the excitation will have a negative effect and even lead to system instability. e research on the excitation control problem of SMIB power system with SVC has achieved remarkable results. In [46], the sliding mode dynamic surface method and disturbance attenuation technique are adopted. Considering the external disturbance and parameter uncertainty, a nonlinear adaptive robust coordination controller for SVC and generator excitation is designed. e energy-based coordinated stability controller is constructed by using the Hamiltonian function method in [47,48], which effectively improves the transient stability and voltage regulation performance of the power systems. Immersion and invariant (I&I) control method is widely used in the coordinated control of SVC and excitation. An I&I adaptive control method is adopted to ensure that all single-machine infinitebus power system states are globally bounded and converge to a new stable equilibrium [3,49]. However, in the actual operation of the power grid and power generation system, the system is more complicated and cannot be simply equivalent to a SMIB. erefore, the excitation control of multimachine infinite-bus power systems with SVC has become the research emphasis of this paper. Inspired by the above research works, the objective of this paper is to design a new adaptive controller for generators with SVC to ensure the stability of the system and the performance of the transient power system when the system has parameter uncertainties and actuator failures. e controller design process includes two partial steps. First, the dynamic surface is combined with the integral sliding mode to design the fault-tolerant controller of the generator excitation controller. Second, the SVC controller is designed. e main contributions of the proposed method are summarized as follows: (1) By introducing integral sliding mode surface in the dynamic surface control process, the problem of "differential explosion" in the traditional backstepping method is solved, and the structure of the controller is simplified.
e introduction of the integral sliding mode surface makes the tracking error of the system converge to zero within a finite time, thereby ensuring the convergence speed and tracking accuracy of the closed-loop system, and the anti-interference ability of the system is also improved.
(2) By introducing the actuator failure factor, an adaptive fault-tolerant controller is designed to solve the problem of actuator failure and ensure that the system can still be stable after partial actuator failure. (3) By adopting the minimum parameter learning method, the norm of ideal weight vector of radial basis function neural networks (RBFNNs) is estimated online, the "dimension disaster" problem caused by the traditional neural network due to the estimated weight vector is avoided, and the calculation burden of the controller is reduced. e rest of this paper is organized as follows. In Section 2, the mathematical model of the multimachine infinity-bus power systems with SVC is described. Coordination controller for generator excitation and SVC is designed in Section 3. e stability analysis is presented in Section 4. e effectiveness of the proposed scheme is illustrated by simulation in Section 5. e conclusion is drawn in Section 6.

Mathematical
Models. e dynamic model of multimachine infinite-bus power systems with SVC is given in this section, in which the dynamics of the i th generator are described by a third-order differential equation, and the SVC is described by a first-order inertia equation. en, the mathematical models of power system can be described as follows [3,5]: with a SVC model 2 Complexity and the electrical equations are as follows: where the explanation of symbols used in the dynamic model (1)-(8) is given in Table 1.

Remark 1.
Generally, in order to accurately describe a largescale multimachine power system, a high-order complex nonlinear dynamic equation is required [28], which makes it very difficult to analyze the stability of the power system and design the excitation controller [3,5]. In order to overcome such difficulties, a third-order power system model simplified by the integral manifold method is proposed in [5,50]. is model makes the control system design and stability analysis feasible; and it has been widely used in the design of excitation controllers for multimachine power system [7,26,[51][52][53]. e third-order power system mode and the first-order static reactive power compensator model adopted in this paper are derived from [3,5].
In order to reduce the complexity of the controller design process in Section 3, E qi ′ (t) in the generator electrical dynamics is eliminated by differentiating P ei (t) in (5): Let ΔP ei � P ei − P mi ; we have e multimachine power systems model can be transformed into the following mathematical model [1,5]: where d i1 and d i2 are bounded uncertainties, including modeling errors, measurement errors, and external disturbances. u i is the control signal of the generator with By considering the failure fault of the actuator at time t f , the generator excitation voltage E fi (t) can be defined as where β i is the failure factor. So (14) So the multimachine power systems model by considering the failure fault can be described as follows: Here, the bound of the interconnection term c i (δ, ω) satisfies where with p 1ij and p 2ij being the constants with values of either 1 or 0. P ei (t) and Q ei (t) are readily measurable variables. From (5)-(8), we can obtain that Define the following state variables for coordinate transformation: where P mi � P mi0 is a constant and where V mi is the accessing point voltage and V refi is the corresponding reference voltage of the SVC: where X 2i and X Ti are the transmission line reactance and the transformer reactance, respectively. According to (2) and Assumption 1. e signs of g ij , i � 1, 2, j � 2, 3, 4, are unknown bounded positive parameters, and there are known positive constants g max and g min such that g max ≥ g ij ≥ g min ; e reference signal x i1 d is bounded and smooth function, its first and second derivatives both exist, and there is a positive real number B i0 that satisfies Remark 2. Since g ij , i � 1, 2, j � 2, 3, 4, are unknown bounded positive parameters and the reference signal x i1 d is a bounded smoothing function, Assumptions 1 and 2 are reasonable and common assumptions in [24,26].

Radial Basis Function Neural Networks (RBFNNs).
e unknown nonlinear continuous functions are approximated by the RBFNNs in this study. e general form of RBFNNs can be expressed as where ξ ∈ R n is the input of RBFNNs, y(ξ) ∈ R is the output of RBFNNs, W ∈ R N is the adjustable weight vector, and ε(ξ) ∈ R N is the nonlinear vector function. In general, a continuous nonlinear function F: Ω ξ ⟶ R with Ω ξ ∈ R n being a compact set and an approximation error where σ * is the approximation error and satisfies |σ * | ≤ σ m . e optimal weight vector W * for analytical purposes can be defined as

Process of the Controller Design
e design process of the controller is divided into two parts: the design of control law for multimachine excitation systems and the design of control law for SVC. e detailed design procedure is described in the following steps.
Step 1 and 2 apply the dynamic surface method, and Step 3 uses integral sliding mode control. According to (24), we know that the model of SVC is of the first order. erefore, the dynamic surface control method is used to design the controller of SVC in Step 4. e design scheme of the adaptive dynamic surface integral sliding mode fault-tolerant controller is shown in Figure 1, and the design steps are displayed in Table 2.

Complexity 5
In Table 2, z ij (j � 1, 2, 3, 4) are the jth surface errors with x i1 d being the desired power angle of the ith generator. S i3 represents the designed integral sliding mode surface. x i2 and x i3 are the virtual control laws in Step 1 and Step 2. (T2.3) and (T2.7) are first-order low-pass filters, which are used to overcome the differential explosion problem in backstepping method. τ i2 and τ i3 are the time constants of (T2.3) and (T2. 7). v i2 , v i3 , v i4 , and θ gi3 are adaptive laws. u i and u Bi ′ are the actual control laws of generator excitation and SVC, respectively. c ij (j � 1, 2, 3, 4), r il , λ il (l � 2,3,4)ρ gi3 , δ i3 , and k i3 are positive design parameters Remark 3. A neural network based adaptive robust controller is developed by using the traditional backstepping design method in [2,54]. Although the semiglobal ultimate uniform boundedness of all signals in the control system is achieved, there is a problem of "explosion of complexity." In this paper, the introduction of a first-order low-pass filter overcomes the problem.

Remark 4.
e values of time constants τ i2 and τ i3 should be chosen to be as small as possible. According to the actual condition, their values are usually between 0.001 and 0.1.

Remark 5.
In order to achieve better transient performance while maintaining the tracking accuracy of the control, the integral sliding mode surface S i3 is adopted in Step 3, which greatly improves the robustness of the system.

Remark 6.
e saturation function sat(·) is used instead of the symbol function sgn(·) to attenuate the chattering phenomenon of sliding mode control. e expressions of the sat(·) function can be written as sat

Remark 7.
e adaptive fault-tolerant controller proposed in this paper is based on the dynamic surface control (DSC) technique [25]. Subsystems (23) and (24) are third-order system and first-order system, respectively. erefore, the controller design process consists of 4 steps, and the actual control input signals (T2.10) and (T2.14) are obtained in the 3rd and 4th steps. In addition, the detailed information of the DSC method is described in some literatures, such as [13,15,24]. In order to make the article more concise, the

Alternative variables
Step 1

Stability Analysis
e stability analysis of the multimachine power systems with SVC will be given in this section. First of all, define the filter errors y i2e and y i3e : From (T2.3) and (29), we have Similarly, from (T2.7) and (30), we have ere are nonnegative continuous functions B i2 and B i3 , and where Consider the following Lyapunov function condition: In addition, the integral sliding mode surface shown in (T2.9) can eliminate the steady-state error of the system tracking and ensure the robustness of the system.
Proof. e specific proof can be seen in the Appendix.
In order to verify the effectiveness of the designed controller, the tracking performances of the following four controllers are compared in     Figure 3 shows the tracking error of power angle δ i under four controllers; it can be seen that, compared to the controllers designed by the other three methods, the dynamic surface integral sliding mode method adopted in this paper has better dynamic performance in terms of overshoot and stable operation time.
e time required to achieve stabilization is shortened by 1∼2 s, and the fluctuation range is also significantly improved. According to Figure 3, the steady state (9∼10) can be selected to obtain the maximum value of tracking error (MVTE) and the root mean square value of tracking error (RMSVTE) for the four control schemes as shown in Table 3. Figures 6-11 show the power angles δ 1 , δ 2 , the rotated speeds ω 1 , ω 2 , the electrical powers P e1 , P e2 , the control inputs u 1 , u 2 of the excitation system, the accessing voltages V m1 , V m2 , and control inputs u B1 , u B2 of SVC. It is proven by Figures 6-11 that the controller designed by combining dynamic surface and integral sliding mode has good control effect.

Case 2.
e operating points are selected as follows:

Complexity
In the second case, it is assumed that the short circuit fault occurred on the transmission line at t � 5 s and disappeared after 0.2 seconds. Figures 4 and 12-17 show the simulation results in this case. Figure 4 shows the power angle tracking error z i1 , i � 1, 2, of the four control schemes in the case of the short circuit; it can be seen that, after the short circuit fault disappears, the method adopted in this paper can make the tracking error quickly converge to the working point, so as to achieve the expected tracking performance of power angles. Similarly, according to Figure 4, the MVTE and RMSVTE in Case 2 can be given in Table 4. Figures 12-17 illustrate the curves of the power angles δ 1 , δ 2 , the rotated speeds ω 1 , ω 2 , the electrical powers P e1 , P e2 , the control inputs u 1 , u 2 of the excitation system, the accessing voltages V m1 , V m2 , and the control inputs u B1 , u B2 of SVC under the case that the unexpected short circuit fault occurred on the transmission line at t � 5 s and disappeared at t � 5.2 s.
In this case, consider that an actuator failure fault occurs at time t f � 4.8 s, so u fi can be assumed such that u fi � (1 − β i )u fi , t ≥ t f , u fi is the input of the SCR amplifier when failure occurs, and the multimachine excitation systems models with SVC in the case of failure fault are equations (23) and (24). e failure factors β 1 � 0.2, β 2 � 0.4. Figures 5 and 18-23 show the simulation results in Case 3. Figure 5 illustrates the power angle tracking error z i1 , i � 1, 2, of the four control schemes in the case of the failure fault; it can be seen that when the failure fault occurs, the method designed in this paper has good tracking performance. Similarly, Table 5 shows the MVTE and RMVSTE in Case 3. Figures 18-21 show the curves of the

Conclusion
In this paper, the method of combining adaptive dynamic surface with integral sliding mode fault-tolerant control is proposed to solve the excitation control problem of multimachine infinite-bus power systems with SVC by considering parameter uncertainties, external disturbances, and actuator failure. Applying the RBFNNs to approximate the unknown nonlinear function not only overcomes the uncertainty of the model but also reduces the computational   burden. e designed controller solves the "explosion of complexity" problem of the traditional backstepping method and makes the tracking error converge to zero in a limited time, thereby improving the steady-state tracking accuracy of the system, while ensuring that the system still has good stability after the failure. Simulation results show the effectiveness of the proposed control algorithm. Future work will focus on the design of a disturbance observer based adaptive output feedback control scheme for multimachine power systems by introducing a fixed-time control method to improve the robustness of the system.

Complexity
Substituting (T2.14) and (T2.15) into (A.25), one has and consider the sets (A.27) e continuous functions B i2 and B i3 have maximums in the compact set Υ 1 × Υ 2 . According to Young's inequalities, we have the following inequalities: