Enhanced Nonlinear Robust Control for TCSC in Power System

This paper proposes an enhanced robust control method, which is for thyristor controlled series compensator (TCSC) in presences of time-delay nonlinearity, uncertain parameter, and external disturbances. Unlike conventional adaptive control methods, the uncertain parameter is estimated by using system immersion and manifold invariant (I&I) adaptive control. Thus, the oscillation of states caused by the coupling between parameter estimator and system states can be avoided. In addition, in order to overcome the influences of time-delay nonlinearity and external disturbances, backstepping slidingmode control is adopted to design control law recursively. Furthermore, robustness of TCSC control subsystem is achievable provided that dissipation inequality is satisfied in each step. Effectiveness and efficiencies of the proposed control method are verified by simulations. Compared with adaptive backstepping sliding mode control and adaptive backstepping control, the time of reaching steady state is shortened by at least 11% and the oscillation amplitudes of transient responses are reduced by at most 50%.


Introduction
In modern electrical power industry, security and stability of power system are under constant threat due to increasing power grid capacity, high-voltage transmission lines, and highly complex network configuration.Thyristor controlled series compensator (TCSC) system is often used to improve the stability and transfer capability of power system by scheduling power flow, decreasing voltage asymmetry and net loss, providing voltage support, and damping power oscillation [1][2][3].However, some problems generally exist in TCSC system.TCSC shows the characteristics of timedelay nonlinearity due to the time-delay between triggering time and turn-on time of its thyristor controller [4,5] As uncertain parameter, damping coefficient is difficult to measure accurately in practice; external disturbances can be mainly caused by interference on generator rotor windings and the fluctuation on susceptance.These problems often deteriorate the performance and increase the difficulty in the design of TCSC robust controller significantly.
The proportional-integral-derivative (PID) control is commonly used to control nonlinear systems based on feedback linearization [6,7].PID was applied for TCSC controller design by assuming that accurate system models are available [8].However, feedback linearization not only linearizes the useful nonlinearities but also requires an accurate model which is difficult to obtain.Therefore, when the operating conditions of the system fluctuate widely, the transient stability of the TCSC system cannot be guaranteed if models are not obtained precisely.
Recently, adaptive backstepping is an optional method to mitigate the effects of nonlinearities and external disturbances on the system performance without linearization [9,10].The core concept of the adaptive backstepping is to design a controller recursively by considering some of the state variables as "virtual controls" and designing intermediate control laws for these variables [11][12][13][14].Some TCSC control methods are proposed on the basis of adaptive backstepping [15,16].Nonetheless, this method is flawed with certain shortcomings.The transient stability of a closed-loop system cannot be guaranteed when the parameter estimator is fixed to its limit value and the system running time approaches infinity [17].In other words, if the estimator is fixed, the estimation error will be accumulated in constructing control Lyapunov function (CLF) and the coupling terms between states and estimation error will be also accumulated as running time increase.Thus, the transient stability of the close-loop system cannot be guaranteed.
To avoid this oscillation, a new method named system immersion and manifold invariant (I&I) adaptive control can provide a mean of shaping the dynamic response of the estimation error even if estimators reach the limit of its capacity [18,19].Unlike conventional adaptive control schemes relying on certainty equivalence principle, I&I adaptive control provides an alternative approach which avoids the cancellation of terms in the derivative of the Lyapunov function [20].This method was used for SVC of a SMIB system to enhance the stability and improve the transient response [14].However, I&I adaptive control has not been adopted in estimation of uncertain parameter in nonlinear systems with time-delay nonlinearity and external disturbances.
Sliding mode control can combine with backstepping to design the control law of nonlinear TCSC system, which is insensitive to nonlinearity and external disturbances with matching condition [21][22][23].There are two processes in sliding mode motion: making the orbit approach sliding mode and keeping the orbit in sliding mode surface.As the orbit in sliding mode is achieved, the robustness of control system can be guaranteed on account of the advantage of sliding mode control which is insensitive to parameter perturbation and external disturbances, whereas the robustness of control system cannot be guaranteed before the motion orbit reach to sliding mode surface.By combining the advantages of both robust control and sliding control, robust sliding control method was developed to address this problem and achieve good performance in [24,25].However, the robust sliding mode control has not been adopted for nonlinear TCSC system when time-delay is involved.
In this paper, an enhanced robust control method is proposed to improve stability and robustness of TCSC system by simultaneously addressing the problems involving the existing nonlinear time-delay, uncertain parameter, and external disturbances.This proposed controller consists of the design of adaptive law and control law.As for adaptive law, the I&I adaptive control is first adopted to estimate uncertain parameter in TCSC system and it achieves excellent result in avoiding the oscillation of states caused by the coupling between parameter estimator and system state.With regard to control law, the influences of time-delay nonlinearity and external disturbances are solved by constructing the control law based on backstepping sliding mode control.Simulation results show that better performance in transient and steady state response is achieved compared with adaptive backstepping sliding mode control and adaptive backstepping control.

TCSC System Model and Control Objective
Figure 1 shows the single-machine infinite-bus system with TCSC.
The dynamic model of the TCSC system is expressed by the following nonlinear differential equations [5,26,27]: Figure 1: Single-machine infinite-bus power system with TCSC.
where , , and  TCSC are state variables denoting generator rotor angle, generator rotor angular speed, and equivalent susceptance, respectively.The physical significance of parameters in (1) is shown in Physical Significance of Parameters.
In Physical Significance of Parameters, time-delay  is caused by the differences between triggering time and turnon time of its thyristor controller.Damping coefficient  is viewed as an uncertain parameter, and the relational expression  = −/ is also the uncertain parameter.  is the equivalence input of SVC regulator.In the equation  TCSC = 1/ 1 +  2 −  1  2 (  −   ),   is the susceptance of the capacitor in the TCSC.  is the susceptance of the inductor of the TCR. 1 =    +   +  1 is the total impedance from the generator to the injection of the SVC device with    being the transient reactance of the generator on the direct axis and   being the reactance of the transformer. 2 is the total impedance from the injection of the SVC device to the infinite bus.The two external disturbances are as follows:  1 is defined as the interference on generator rotor windings and  2 is the fluctuation on susceptance, respectively [14,27].As a result, the controlled TCSC is an uncertain and nonlinear system involving time-delay nonlinearity, uncertain parameter, and external disturbances.
To simplify (1), three state variables are redefined as  1 =  −  0 ,  2 =  −  0 , and  3 =  TCSC −  TCSC0 , respectively, and we have The output of TCSC system is expressed as y = [q 1 x 1 q 2 x 2 ] T , where  1 and  2 are nonnegative weighted coefficients which represent the weighted proportions of the state variables  1 and  2 in output.
The proposed method involves an adaptive law ( θ ) and a control law (  ).The objective of designing TCSC robust controller is to guarantee that all the state variables can attain stability and eventually converge to equilibrium points in the nonlinear TCSC system.In addition, by using the proposed robust controller, the robustness of the TCSC robust controller is guaranteed, the performance and speed of transient responses is improved, and the time of reaching steady state is reduced.

Design of Robust Controller for Nonlinear TCSC
It divided three sections to introduce our proposed method in designing robust controller.In Section 3.1, I&I adaptive control is adopted for adaptive law θ design, which can ensure that the estimation error of θ −  gradually converged to zero.In Section 3.2, backstepping sliding mode control is used for control law   design recursively.The dissipation inequality is constructed in each step for guaranteeing the robustness of the subsystem.In Section 3.3, the stability and robustness of the TCSC control system are verified.

Adaptive Law Design.
Figure 2 shows the mapping between the trajectories of the controlled and target system based on the notions of system immersion and manifold invariance.
The purpose of I&I methodology is to achieve stabilization by immersing the plant dynamics into a stable (lowerorder) target system [20,28].The I&I adaptive control proposes an alternative approach avoiding the cancellation of terms in the derivative of the Lyapunov function and provides a mean of shaping the dynamic response of the estimation error [29].
Define a manifold where  is an uncertain parameter, θ is the estimation value of , and   is an estimation error function.( 1 ,  2 ) is a continuous function.The dynamics of (4) along with system (2) are where  1 is a bounded function denoting the interference on generator rotor windings.
By cancelling out all the parameter-independent terms, an adaptive law θ is designed as Substituting ( 6) into ( 5), we have Design a CLF as where ℎ 1 is the least upper bound of external disturbance  1 , that is, ℎ 1 ≥ | 1 |, due to the physical significance of external disturbance on generator rotor windings [14,27,30]; ℎ 2 is the least upper bound of generator rotor angular  2 due to the limitation of generator rotor, that is, with  > 0, and the dynamics of ( 8) is Since and then have According to Lyapunov stability theorem, the proposed adaptive law θ and the selected continuous function ( 1 ,  2 ) = (1/2) 2 2 can ensure that   converges to zero in finite time.
Remark 1. Unlike the conventional adaptive control based on certainty equivalency principle, the proposed adaptive law can introduce a continuous function ( 1 ,  2 ) to compensate the residual estimation error θ−.Furthermore, by requiring the estimation error to converge to zero, the stability and convergence of system (7) are guaranteed based on the proposed adaptive law and the selected continuous function.Therefore, the error accumulation of the coupling terms is avoided even if the parameter estimates are fixed.
Remark 3. The selected ( 1 ,  2 ) and the designed adaptive law are not only guarantee that the estimation error   converges to zero but also ensure that the parametric manifold is invariant and attractive [28,29].

Control Law Design.
In this section, the control law is designed by using backstepping sliding mode control method in TCSC system with time-delay nonlinearity and external disturbances.Three steps are constructed to design the control law recursively.In each step, the dissipation inequality is satisfied to guarantee the robustness of TCSC control system.
Based on dissipation theory, an inequation external disturbances is constructed as () is an energy storage function.(Δ) = ‖Δ‖ 2 − ‖‖ 2 is an energy supply function, where Δ = ( 1 ,  2 )  is external disturbance,  is a nonnegative constant, and is the output of the TCSC system.From (11), the dissipation inequality is achievable provided that the  2 gain from the disturbance to the output of the system is smaller than or equal to , where  is disturbance attenuation constant.
According to the definitions above, the control law is designed as follows.
Define error state variables   ( = 1, 2, 3) where  2 * and  3 * are virtual control law.The derivative of  1 along with (2) is Step 1.The first CLF is and the derivative of  1 along with ( 13) is To satisfy Lyapunov stability theorem, the virtual control law  2 * is where  1 is a positive constant.Substituting ( 16) into (15), it can be seen clearly that V1 ≤ 0 when  2 = 0.However, when  2 ̸ = 0, we can construct the second CLF in Step 2.

Proof of System Stability.
The dynamics of closed-loop error system are expressed as Substituting virtual control input  2 * and  3 * into (32), we can obtain The asymptotic stability of the closed-loop error system (33) is discussed in two different conditions.
Firstly, when the external disturbances  1 ̸ = 0 and  2 ̸ = 0, a relationship according to (26) and ( 31) is constructed as Let () = 2 3 (); an inequation with regard to V3 is expressed as By integrating both sides of (35), the dissipation inequality is The dissipation inequality (34) is satisfied and the robustness of TCSC controller is guaranteed by using the proposed method.This implies that all the increased energy of the system from  = 0 to  is always equal to or less than the ones from outside.Therefore, the energy of power systems has been decreasing.Obviously, the proposed method consisting of the designed law   and the adaptive law θ can guarantee that the closed-loop error system is globally asymptotically stable.
Secondly, when the external disturbances are not involved, we can construct Lemma 4 and prove it as below.
Lemma 4. Based on the above derivations, one can conclude that the dynamics of closed-loop error system are asymptotically stable when  1 = 0 and  2 = 0.

Simulations
The MATLAB software is performed in TCSC simulations.According to operating characteristics of TCSC system, the time-delay between the triggering time and turn-on time is equal to or less than 0.04 s.The damping coefficient is viewed as the uncertain parameter.The external disturbances in the system are  1 =  −2 sin(5) and the unknown fluctuation on susceptance is  2 =  −2 cos(5), respectively [14,27].The parameters used in the simulation are given in Table 1.
The generator rotor angle , generator rotor angular speed , and equivalent susceptance  TCSC are the state variables whose transient response trajectories are tracked.The initial operating values are  0 = 57.1 ∘ ,  0 = 314.159rad/s, and  TCST0 = 0.4 pu.
Four case studies are performed to simulate the nonlinear TCSC systems with different delay times  or attenuation coefficient .Cases 1-3 aim to compare the stability and robustness of TCSC control systems designed by the proposed adaptive immersion and invariance backstepping sliding mode (AIIBSM) method, adaptive backstepping sliding mode (ABSM) method, and adaptive backstepping method (AB). and  are set to 0.01 s and 2 in case 1, 0.02 s and 2 in case 2, and 0.04 s and 2 in case 3.In case 4, simulations of our designed TCSC system with  4 = 0.02 s are performed at  1 = 2 and  2 = 5, respectively.
Figure 3 shows the comparisons between the proposed method and the two existing methods in case 1, where  1 = 0.01 and  1 = 2.The transient responses of three state variables, that is, rotor angle, rotor angular speed, and equivalent susceptance, are investigated, respectively.It can  Figure 4 shows the transient responses of the three state variables in case 2, where  2 = 0.02 s and  1 = 2.The transient trajectories of the AIIBSM controller fluctuate less strongly and quickly converge to steady state than compared to those of the ABSM and AB controllers in finite time, suggesting that the proposed AIIBSM method results in better system performance.
Figure 5 shows the transient responses of the three state variables in case 3, where  3 = 0.04 and  1 = 2.The result of case 3 reveals that the proposed method can guarantee that the state variables of the nonlinear TCSC system are globally bounded and transient responses will eventually converge to a stable value regardless of what delay time is considered.A comparison of the transient responses of the AIIBSM controller in the three cases shows that the transient responses in case 3 converge more slowly and the system stability is attained in a longer finite time.Similar comparative results are obtained for the ABSM and AB controllers.
Figure 6 shows transient responses of the three state variables in case 4. The simulation results are compared to investigate the effect of the attenuation coefficient  on the stability and robustness of TCSC control systems.It can be seen that the AIIBSM controller attains more rapid speed responses and stronger robustness at  1 = 2 than at  2 = 5.Therefore, the delay time  is a crucial nonlinear factor impacting the transient and steady performance of the TCSC system.A smaller  can lead to better transient stability and stronger robustness.Furthermore, the attenuation coefficient  is also a key factor impacting the performance of the control system.A smaller  tends to result in stronger robustness and smaller oscillation.This result is consistent with the theoretical analysis.

Conclusions
This paper investigates the problems involving time-delay nonlinearity, uncertain parameter, and external disturbances in the controller design of TCSC system.An uncertain parameter is estimated in the adaptive law design based on I&I adaptive control.The oscillation of states caused by the coupling between parameter estimator and system states is avoided, due to the fact that the errors of parameter estimation can gradually converge to zero by using the designed adaptive law.In addition, backstepping sliding mode control is adopted to solve the problems caused by time-delay nonlinearity and external disturbances in the control law design.Moreover, by satisfying the dissipation inequality, the robustness of the proposed TCSC control system is achieved.Simulation results have confirmed that, by using the proposed method, all state variables are globally Mathematical Problems in Engineering

Figure 3 :
Figure 3: Transient responses in case 1: (a) transient responses of rotor angle; (b) transient responses of rotor angular speed; (c) transient responses of equivalent susceptance.

Figure 4 :
Figure 4: Transient responses in case 2: (a) transient responses of rotor angle; (b) transient responses of rotor angular speed; (c) transient responses of equivalent susceptance.

Figure 5 :
Figure 5: Transient responses in case 3: (a) transient responses of rotor angle; (b) transient responses of rotor angular speed; (c) transient responses of equivalent susceptance.

Figure 6 :
Figure 6: Transient responses in case 3: (a) transient responses of rotor angle (b); transient responses of rotor angular speed; (c) transient responses of equivalent susceptance.

Table 1 :
Parameters for nonlinear TCSC system simulations.beseen that the transient responses of the state variables of the AIIBSM controller are faster and the AIIBSM system tends to attain stability more rapidly, whereas the transient responses of the ABSM and AB controllers fluctuate faster and the ABSM and AB systems tend to reach stability in a longer finite time.Especially, three state variables cannot achieve the steady state in finite time.