Two Novel Approaches of NTSMC and ANTSMC Synchronization for Smart Grid Chaotic Systems
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Abstract
The presence of uncertainties and external disturbances is one of the unavoidable problems with various practical systems which might be unavailable in realtime. Sliding Mode Control (SMC) is one of the effective robust control methods to deal with these uncertainties and external disturbances. In this paper, two novel controllers are designed by using Nonsingular Terminal SMC (NTSMC) and Adaptive Nonsingular Terminal SMC (ANTSMC) methods for synchronization of dual smart grid chaotic systems with various uncertainties and external disturbances. Indeed, both adaptive and nonadaptive controllers based on NTSMC are proposed to provide two alternatives which can adjust by changing operating conditions and dynamics. The concept of SMC method guarantees controller robustness against various uncertainties and external disturbances. Elimination of the undesirable chattering phenomenon is addressed in this study which is one of the common deficiencies with conventional SMC method. Additionally, finite time concept is used to speed up the convergence rate. Finite time stability proof is performed by using Lyapunov stability theory. The numerical simulation is carried out in Simulink/MATLAB to reveal the validity of the proposed controllers for the smart grid chaotic system. A comprehensive comparison is made by performing simulation for the Fractional Order Adaptive Sliding Mode Control (FOASMC) controller and defining three performance criteria, among the proposed controllers in this study and FOASMC controller.
Keywords
ANTSMC NTSMC Synchronization Control Smart gridIntroduction
The traditional power systems are based on centralized generation with their large power plants located far from the power consuming loads [21]. Hence the control and centralized operation of this largescale system is very challenging and complicated task. Smart grids provide smarter operation of conventional power grids by interconnecting the grids in a distributed and interactive manner for the well suitability of distributed multiagent technologies to alleviate these challenges [24, 25]. Indeed, a smart grids is an electricity network which insists on various operational and energy measures including smart appliances, smart meters, renewable energy resources, and energy efficient resources. Smart grids improve efficiency, minimize cost and consumption of energy, and enhance the reliability and transparency of the energy supply by having a proper control, monitoring, communication and analysis within the supply chain [10, 12, 30].
Recently, many efforts have been made to control these networks. In [24], a multiagent based protection framework has been proposed to improve the transient stability of smart grids. In [17], a comprehensive review on the control and communication techniques has been done for the smart grids where the energy efficiency of the smart grids has been focused. In (X. [32]), a novel fault tolerant extended Kalman filter has been proposed for smart grid synchronization. a survey of studying complex network theory has been reported in [7] for modern smart grids. In [6], a CyberPhysical Power System (CPPS) paradigm has been presented for smart transmission grids control, modeling, and monitoring. In [26], a model predictive control (MPC)based approach has been proposed for smart grids with multiple electricvehicle charging stations. In [34], a resonance attacks have been investigated on Load Frequency Control of Smart Grids.
The synchronization refers to the process of precisely matching or coordinating two or more activities, processes, devices, or system in time. The synchronization methods have been used in many applications such as synchronization of robots for collaborative robots [29] and the synchronization of chaotic systems with various goals [33, 35]. In [15], the multiagent cooperative controller has been designed for heterogeneous energy storage devices in smart grids considering their hierarchical control structure with droop controls. The active/reactive power sharing, the frequency/voltage, and the energy of battery energy storage systems (BESSs) have been synchronized by exchanging local information with a few other neighboring BESSs.
Finite time stability is a more comprehensive and recent concept than asymptotic stability. The finite time stability means that the system state variables reach zero at the bounded time. The term “terminal” refers to the concept of finite time stability. Many applications require us to prove the stability in a finite time. Accordingly, various finite time theorems and lemmas have been introduced of which some have been presented in [20, 37]. The finite time stability has been very much considered in the recent literature to speed up the convergence rate and improve the concept of stability [11, 18, 37]. In [2], Finite time concept has been considered to incorporate with optimal robust control for Photovoltaic system using the VSC model in Smart Grid.
SMC method is a wellknown control method because of its main feature which is robustness against a variety of uncertainties and external disturbances [3, 9]. In [27], optimal realtime control based on the SMC method has been investigated for discretetime switched repeated scalar nonlinear systems. This robust control method would guarantee asymptotic stability of the systems. The Terminal SMC (TSMC) method has been presented by incorporating finite time concept and SMC method. The TSMC method accomplishes both robustness against external disturbances and uncertainties and system stability in a finite time. Subsequently, a Nonsingular TSMC (NTSMC) method has been introduced to overcome occurrence singularity as an unwanted problem.
On the other hand, the design of the SMC scheme requires the knowledge of uncertainties and external disturbances bound, which might be unavailable in real time. The adaptive concept provides an effective method to deal with these unknown external disturbances and uncertainties by estimating the upper bound of them [4]. Adaptive NTSMC (ANTSMC) method has been proposed by integration of the concept of adaptive control method and NTSMC method. The ANTSMC offers a robust control method with online parameter estimators to provide the information of the uncertainty upper bound. An ANTSMC method has been proposed in [23] to control an autonomous underwater vehicles (AUVs). In (H. [31]), SteerbyWire Equipped Road Vehicle has been controlled by using adaptive control concept within a finite time. a fractional order adaptive sliding mode controllers (FOASMC) has been proposed in [14] for a fractional order smart grid chaotic system.
The conventional SMC method causes the unwanted chattering phenomenon due to applying some discontinuous terms in the control input. This undesirable problem causes some devastating effects on the system actuators such as reducing the useful life of them. It also reduces the control accuracy and causes deleterious sound in the system. Hence, numerous approaches have been proposed to eliminate or reduce this destructive phenomenon. The undesirable chattering phenomenon has been thoroughly eliminated in [1, 8, 16, 22, 36]. In [13], an SMC method has been proposed for nonlinear fractionalorder systems which has removed the chattering problem completely.

Synchronization of dual smart grid chaotic systems by using two novel adaptive and nonadaptive control methods.

Elimination of undesirable chattering phenomenon.

Robust control with the goal of synchronization of two identical chaotic smart grid systems in presence of various uncertainties and external disturbances.

Estimation of the upper bound of the uncertainties and external disturbances and using their estimations in the control input in the adaptive controllers.

Guarantee the global finite time stability for all proposed controllers in this study.

Provide a numerical comparison among the proposed controllers in this study and the FOASMC controller by using three wellknown performance criteria, IAE, ITAE, and ISV.
The remaining of this note is organized in the following manner. Section “Mathematical Preliminaries” presents the mathematical preliminaries about the finite time stability theorems and the related lemmas. Section “Model Description of Chaotic Smart Grid and Problem Statement” is dedicated to the model description of the smart grid chaotic system and problem statement. In section “Controller Design”, the adaptive and nonadaptive controllers basedon NTSMC are proposed for the smart grid chaotic system along with an explanation of detailed methodology. In section “Results and Discussion”, numerical simulation results of proposed controllers are firstly given. Then, comparison and discussion are provided by using three performance criteria. Section “Conclusion” is devoted to the conclusion.
Mathematical Preliminaries
Lemma 1: consider the nonlinear system as \( \dot{x}=f(x),f(0)=0, x\epsilon {\mathfrak{R}}^n \) with initial conditions x(0) = x_{0}. Suppose there exists candidate Lyapunov function V(x) which is globally positive definite, radially unbounded and only at x = 0 is zero, such that; the time derivative of the candidate Lyapunov function will be as \( \dot{V}(x)\le {\rho}_1{V}^{\rho_2}(x) \), where ρ_{1} is a positive number and ρ_{2} is a constant between zero and one. Then, the variable x of the system from any initial conditions, it reaches zero in a finite time, and since then it remains exactly equal to zero, i.e. \( \underset{t\to T}{\lim }x\to 0 \) and the upper bound of the settling time, T, will be as \( T\le \frac{V^{1{\rho}_2}\left({x}_0\right)}{\rho_1\left(1{\rho}_2\right)} \) [23].
Lemma 1: Consider the nonlinear system \( \dot{x}=f(x)+g(x)u+d \), where d is the model of the uncertainties and external disturbances of the system which is estimated at any moment of time as \( h\le \widehat{h} \). At any moment of time, there exist positive constants h^{∗}, such that \( \widehat{h}\le {h}^{\ast } \) [23].
Model Description of Chaotic Smart Grid and Problem Statement
Controller Design
Two Novel Approaches of Terminal Sliding Mode Control
Theorem1 (NTSMC1)
Proof
According to Lemma 1, the system reaches to the sliding surfaces in a finite time as \( {T}_s\le 4\frac{{\left(V\left(x(0)\right)\right)}^{\rho_2}}{\rho_1\left(1{\rho}_2\right)} \).
As a result, the system finite time stability proof is completed and the system stability time is as T = T_{r} + T_{s}. ∎.
Note that using the sgn function in the designed control input in this section is likely to cause the undesirable chattering phenomenon. To overcome this deficiency, we will design control input in the next section in such a way that we use of integral of sgn function in the control input, which can reduce or eliminate this unwanted problem.
Theorem 2 (NTSMC2)
Proof
According to lemma 1, the system reaches to the sliding surfaces in a finite time as \( {T}_s\le 4\frac{{\left(V\left(x(0)\right)\right)}^{\rho_2}}{\rho_1\left(1{\rho}_2\right)} \).
As a result, the system finite time stability proof is completed and the system stability time is as T = T_{r} + T_{s}. ∎
Two Novel Approaches of Adaptive Nonsingular Terminal Sliding Mode Control
Theorem 3 (ANTSMC1)
Proof 3
According to Lemma 1, the system reaches to the sliding surfaces in a finite time as \( {T}_s\le 4\frac{{\left(V\left(x(0)\right)\right)}^{\rho_2}}{\rho_1\left(\frac{1}{2}\right)} \).
The second part of the system stability proof is investigating the finite time stability of the sliding surface in Eq. (10) (which is repeated in Eq. (29)). In Eqs. (16) and (17) the finite time stability proof of the sliding surfaces of Eq. (29) or Eq. (10) has been performed. Also, its stability time has been presented in Eq. (18).
As a result, the system finite time stability proof is completed and the system stability time is as T = T_{r} + T_{s}. ∎
Theorem 4 (ANTSMC2)
Proof 4
According to Lemma 1, the system reaches to the sliding surfaces in a finite time as \( {T}_s\le 4\frac{{\left(V\left(x(0)\right)\right)}^{\rho_2}}{\rho_1\left(\frac{1}{2}\right)} \).
The second part of the system stability proof is investigating the finite time stability of the sliding surface in Eq. (19) (which is repeated in Eq. (37)). In Eqs. (25) and (26) the finite time stability proof of the sliding surfaces of Eq. (37) or Eq. (19) has been performed. Also, its stability time has been presented in Eq. (27).
As a result, the system finite time stability proof is completed and the system stability time is as T = T_{r} + T_{s}. ∎
Results and Discussion
Numerical Simulation
In this section, the numerical simulations of the four proposed control approaches have been done for the smart grid chaotic system. The numerical simulation results are carried out in Simulink/MATLAB by using the ode45 solver and with a simulation step size of 0.001. The initial conditions of the master system have been considered as in Section “Model Description of Chaotic Smart Grid and Problem Statement”. All initial conditions of the slave systems have been considered zero, i.e. X_{s}(0) = [0, 0, 0, 0]^{T}. The selected control parameters for this simulation are presented in Table 1.
Comparison and Discussion
 (a)
Integral of the absolute value of the error (IAE)
 (b)
Integral of the time multiplied by the absolute value of the error (ITAE)
 (c)
Integral of the square value (ISV) of the control input
The IAE and ITAE are used as the goal numerical measures of tracking performance for an entire error curve, where t_{f} shows the total running time. The IAE criterion will provide an intermediate result. In the ITAE criterion, time appears as a factor, which will deeply emphasize the errors that occur late in time. The ISV criterion represents the consumption of energy [19].
To reveal the effectiveness of the designed controllers in this study, we have performed the numerical simulation by using the control input of FOASMC which has been presented in [14] for the considered smart grid chaotic system in Eq. (3). In fact, the numerical simulation of FOASMC has been done to calculate the numerical value of the performance criteria and to make a comparison with our proposed control methods. The considered control parameters for the numerical simulations of the FOASMC has been presented in [14]. Note that the exact same initial conditions have been considered for all numerical simulations in this study which have been presented in [14] and Sections “3 and 5” of this article.
Selected control parameters for the proposed controllers
NTSMC1  NTSMC2  ANTSMC1  ANTSMC2  

α _{11}  5.99  60  0.03  60 
α _{12}  0.04  60  0.03  60 
α _{13}  0.04  60  0.03  60 
α _{14}  0.04  60  0.03  60 
α _{21}  103/101  7/5  7/5  7/5 
α _{22}  103/101  7/5  7/5  7/5 
α _{23}  103/101  7/5  7/5  7/5 
α _{24}  103/101  7/5  7/5  7/5 
Performance criteria for different methods
NTSMC1  NTSMC2  ANTSM1  ANTSM2  FOASMC  

IAE1  0.1142  0.2717  0.0056  0.1332  0.0242 
IAE2  0.0119  0.2427  0.0032  0.1042  0.0152 
IAE3  0.0057  0.2072  0.0012  0.0685  0.0067 
IAE4  0.0679  0.4098  0.0288  0.2705  0.0877 
ITAE1  0.9792  0.5115  0.0021  0.0142  0.2761 
ITAE2  0.0930  0.5102  0.0017  0.0128  0.1716 
ITAE3  0.0858  0.5084  0.0014  0.0107  0.0736 
ITAE4  0.2904  0.5160  0.0079  0.0189  1.0290 
ISV1  46.5377  13.6650  22.2144  14.5287  16.7790 
ISV2  90.6944  18.1175  24.7743  16.9378  22.8813 
ISV3  5843.8  3949.6  4631.2  3872.5  4622.5 
ISV4  84.8396  154.8209  36.1542  159.3897  68.5617 

As shown in Fig. 15, both proposed adaptive control methods (ANTSMC1 and ANTSMC2) and the proposed NTSMC2 method have a smaller value for the ISV than two other methods. Note that, The ISV criterion is related to the amplitude of the control input and lower values of ISV can be deduced as the lower cost of construction and lower consumption of energy. Accordingly, ANTSMC1, ANTSMC2, and NTSMC2 methods are more costeffective than two other methods in terms of constructing control inputs.

In terms of the value of ITAE criterion, both proposed adaptive control methods (ANTSMC1 and ANTSMC2) and the proposed NTSMC1 method are more appropriate (due to a smaller value of ITAE) than two other methods.

In terms of the value of the IAE criterion, the proposed ANTSMC1 method provides the best controller for the smart grid chaotic system (due to a smaller value).

The NTSMC1 method in its control input creates the chattering problem as it is predicted (see Figs. 9 to 12). However, this destructive phenomenon is eliminated thoroughly in the proposed NTSMC2, ANTSMC1, and ANTSMC2 methods.

In terms of constructing control inputs, the NTSMC1 and ANTSMC1 methods (where at their sliding surfaces the integral elements are used) in comparison to the other two methods, NTSMC2 and ANTSMC2 (where at their sliding surfaces the derivative elements are used) are superior. Note that the derivative elements not only boosts the noise, but it is also difficult to construct an ideal derivative element.
Conclusion
In this paper, two robust finite time controllers are designed to synchronize dual smart grid chaotic systems by using nonadaptive and adaptive control methods basedon NTSMC. The adaptive and nonadaptive controllers based on NTSMC are proposed to provide two alternatives which can adjust by changing operating conditions and dynamics as well as comparing them. The numerical simulation results are carried out to show the validity of the four proposed controllers for synchronization objective. Then, by performing the simulation for the FOASMC controller and using three performance criteria, a comprehensive comparison is made among the proposed controllers in this study and FOASMC controller. In terms of the numerical values of the performance criteria, our proposed control methods (especially the proposed adaptive control methods) outperform FOASMC method. The key features of the proposed control methods in this study are finite time stability and robustness against a variety of uncertainties and external disturbances. The elimination of undesirable chattering problem is also addressed in this study. Additionally, the upper bounds of uncertainties and external disturbances are estimated in a finite time in the proposed adaptive control methods.
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