A Backstepping Control Strategy for Power System Stability Enhancement

Abstract

Secure power system operation relies extensively on the analysis of transient stability and control. The dynamics involved in power system control are often complex and nonlinear. Most of the currently existing works approach these frequent problems with nonlinear control techniques, leading to a requirement for specific controller parameter adjustments. In these veins, this paper proposes a new method for stabilizing electric power systems, using nonlinear backstepping control by optimizing the controller’s parameters. The Jaya algorithm and Genetic algorithm are utilized as a powerful meta-heuristic optimization technique to search parameters of an optimal controller. Improvement in system damping, transient stability, and voltage regulation has been achieved by minimizing the integral time absolute error (ITAE) as the objective function. Numerical simulations on an SMIB power system under different fault conditions showed that the proposed method outperforms classical power system stabilizer (PSS) methods, reducing overshoots and settling times and eliminating steady-state errors. These findings highlight the effectiveness of the proposed approach and its potential contribution to the development of advanced nonlinear control techniques for electric power systems. The suggested optimization methods demonstrate superior performance, compared to classical methods, and achieve a reduction of 27.5% in overshoot and 87% in transient time in addition to complete elimination of static error.

1. Introduction

1.1. Research Background

Excitation control is a voltage output control method of a synchronous power-system generator. This technique is applied by a terminal-voltage regulating excitation system, whereby the generator’s stability and output power are adjusted. Accordingly, the excitation control system proceeds with an identification of the required excitation voltage. It adjusts it in real-time to simultaneously sustainable stability not only of the generator but also of the entire power system. In this respect, the excitation control of the power system’s synchronous generators is investigated regarding the transient and steady states [1]. While the transient state refers to a capacity system withstanding sudden and significant disturbances, the steady state regime deals with eliminating the steady state emanating errors. At this level, the regulation mode is selected in relation to the disturbance nature as well as the status of the system.

1.2. Related Works

Various control theories have been proposed, maintaining the conventional automatic voltage regulation (AVR) process as an elementary component, which regulates the generator terminal voltage. Concerning the dynamic stability, it is often enhanced by means of a power system stabilizer (PSS). An auxiliary signal is introduced to the AVR, which accounts for the slight fault occurrences [2,3]. Actually, the AVR and PSS adopted methods have seemingly contributed to provide noticeable enhancement stability, precisely under normal and minor-disturbance operating conditions. Looking for more optimally effective controller parameters is necessary to further boost the stability status and the damping system under a wide range of different operating conditions. As has been stated, it is worth noting that several novel optimization methods have been explored in the relevant literature, notably the published papers [4,5,6,7], wherein the population-based meta-heuristic algorithm has been utilized for PSS exploitation. In the most recent published papers [8,9], the new teaching–learning-based optimization (TLBO) algorithm has been used for optimizing the PSS parameter setting ends. The supposed efficiency of the PSSs has been successfully tested on single- and multi-machine systems. However, they are proved to be abundant with drawbacks and limitations that are mainly related to frequent fault occurrences as well as noticeable change in the input mechanical power, essentially due to applying approximate linearly mathematical models to the system. They display a highly nonlinear behavior, which requires a cautious handling for more robust control to take place. To overcome such limitations, other researchers have proposed the direct feedback linearization (DFL) technique, as a control technique enabling one to linearize a particular system’s nonlinear dynamics by using feedback [10,11,12,13,14]. However, this approach has frequently been discovered to result in the cancellation of the useful nonlinearities, which undermines the performance of the system. Recently, a number of novel nonlinear stabilizing control mechanisms for power systems have cropped up, namely the Lyapunov-based approach [15,16,17,18], the sliding-mode technique [19,20,21,22], the backstepping method [23,24,25] and the combination of two nonlinear methods [26]. By extending the equilibrium attraction region, nonlinear methods have demonstrated that the power system transient stability is greatly improved. Nonetheless, the effectiveness of these proposed approaches is extensively reliant on the accuracy of the system model, as it is used to design the energy function that determines the stability of the system. If errors occur in the model, they can negatively impact this stability, lose robustness, and degrade the terminal voltage regulation in cases of model uncertainty or system parameter variation. Consequently, the adaptive techniques with nonlinear control have been proposed in many papers [27,28,29,30]. The main drawback of the last proposed solutions is the heavy effort involved in real-time implementation of the control law. Intelligent optimization techniques have been employed so as to optimize the parameters of the controller offline, considering diverse operating conditions and potential disturbances encountered within the power system. The authors reported some powerful meta-heuristics, which have been classified into nine groups based on a combination of biology, physics, swarm, social, music, chemistry, sports, math, and hybrid methods [31]. More importantly, it is worth noting that these groups keep on growing either with simulation or imitation of intelligent phenomena in contemporary life [32]. A number of the recognized evolutionary algorithms are basically genetic, referring to Genetic Algorithm (GA) [33], Non-dominated Sorting Genetic Algorithm NSGA-II [34], and Differential Evolution (DE) [35]. Famous algorithms based on swarm intelligence include Particle Swarm Optimization (PSO) [36], Grey Wolf Optimizer (GWO) [37], and Artificial Bee Colony (ABC) [38]. Indeed, all of these optimization algorithms require the incorporation of common control parameters, such as population, size, generation number, and elite size. In addition to the control parameters, each algorithm stands on the implementation of a proper set of unique parameters. For instance, the GA requires using both mutation and crossover probability [39], while the selection operator, PSO, depends on the exploitation of inertia weight and socio-cognitive parameters. A proper tuning of these algorithm-specific parameters stands as a crucial factor, since they affect the algorithms’ performances. Hence, inaccurate tuning could either result in doubling the computational effort or simply bring about a locally optimal solution. To overcome these indicated deficiencies, devising a number of algorithms void of any algorithm-specific parameter is mainly required. Worth citing among them is the teaching–learning based optimization (TLBO) framework, which does not require any algorithm-specific parameters, except for the common control parameters of population size and number of generations [40]. As an evolved version of the TLBO algorithm, the Jaya Algorithm (JA), initially devised by Rao [41], does not involve any algorithm-specific control parameters. Unlike the TLBO algorithm, the Jaya algorithm involves a single phase only, rather than two phases. It incorporates a straightforward design, thereby remarkably outperforming other optimization algorithms.

1.3. Main Contributions

In general, the nonlinear control methodology requires a particular design, pivoted on several operation scenarios happening in the power system. The achievement of this objective rests on the optimization of the specific controller’s parameters, implying various faults and under steady stressed state conditions. In this context, the purpose of this paper is firstly to explore and later to propose a solution enhancing the transient stability and terminal voltage regulation while eliminating steady-state errors. Specifically, a set of optimized backstepping controller parameters will be implemented to achieve this purpose. The backstepping method is widely recognized as an effective nonlinear control technique that is frequently employed for systems in strict-feedback form [42,43]. However, this method requires an effective determination of the controller’s parameters in order to maintain stability and achieve high-level regulatory performance rapidly and accurately. The control law has been initially designed based on the Lyapunov stability analysis. Later, it is optimized for minimizing any occurring predefined index error. In this way, the envisioned controller targeted challenge of enhancing the stability and accuracy of the power system turns out to be a problem of optimization. With respect to the noticeable limitations predominant in the already proposed traditional optimization techniques, it is worth exploiting the meta-heuristic algorithms. As part of the present work, the Jaya algorithm is adopted for solving the multi-objective optimization problems, which in turn determines the backstepping controller associated parameters. In effect, it could be easily applied to treat both constrained and unconstrained problems without the need for unique parameters, by exclusively relying on standard control ones.

1.4. Paper Organization

This paper starts with a set of sections. The introductory one is followed by Section 2, which depicts the conceived power system special modeling with more focus on the SMIB power system with an AVR for the generator excitation circuit. Section 3 is mainly an outline of the model introducing the backstepping controller structure. Section 4 details the JA and GA, which are applied to the controller’s performance enhancement purposes. The last section reveals the achieved simulation results and the relevant discussions highlighting the developed controller design contributions.

2. System Dynamic Modeling

Further to the seventh-order detailed dynamic model of a synchronous generator (SG), the widely applied third-order model stands as a mechanism of paramount importance for analyzing stability and control once the generator is connected to the power system, as stated in [1,2]. Hence, the state equation describing the dynamics of an SG connected to an infinite bus by means of a transformer and two transmission lines is expressed (per unit) as follows:

$$\left\{\begin{array}{c}\frac{d\delta \left(t\right)}{dt}=\omega \left(t\right)\hfill \\ \frac{d\omega \left(t\right)}{dt}=-\frac{K_D}{2\mathrm{H}}\omega \left(t\right)+\frac{{\omega }_s}{2\mathrm{H}}\left(P_m\left(t\right)-P_e\left(t\right)\right)\hfill \\ \frac{d{E}_q^{\prime }\left(t\right)}{dt}=\frac{1}{T_{d0}^{\prime }}\left(E_{fd}\left(t\right)-E_q\left(t\right)\right) \hfill \end{array}\right.$$

(1)

where δ denotes the generator rotor angle, ω designates the speed deviation between the generator and the synchronism, $E_q^{\prime }$ is the transient q-axis EMF, and $P_e$ is the generator delivered electrical output power. $P_m$ and $E_{fd}$ are the two inputs of the system corresponding, respectively, to the mechanical power and the excitation voltage. The constants $K_D$, $\mathrm{H},$ and $T_{d0}^{\prime }$ signify the system parameters corresponding, respectively, to the dumping torque coefficient, inertia constant, and excitation circuit time constant.

Other algebraic electrical equations are determined through phase or diagram analysis [2] as follows:

$$P_e\left(t\right)=\frac{V_s E_q\left(t\right)\sin \left(\delta \left(t\right)\right)}{X_{ds}}$$

(2)

$$E_q\left(t\right)=\frac{X_{ds}}{X_{ds}^{\prime }}{E}_q^{\prime }\left(t\right)-\frac{\Delta X_d}{X_{ds}^{\prime }}{V}_s\cos \left(\delta \left(t\right)\right)$$

(3)

$$V_t\left(t\right)=\frac{1}{X_{\mathrm{ds}}}{\left(X_s^2{E}_q^2\left(\mathrm{t}\right)+V_s^2{X}_d^2+2X_{\mathrm{s}}{X}_{\mathrm{d}}{V}_{\mathrm{s}}{E}_q\left(\mathrm{t}\right)\cos \left(\delta \left(t\right)\right)\right)}^{0.5}$$

(4)

with $X_s=X_T+\frac{X_L}{2}$, $X_{ds}^{\prime }=X_d^{\prime }+X_s$, $X_{ds}=X_d+X_s$, $X_{qs}=X_q+X_s$, $\Delta X_d=X_d-X_d^{\prime }$, $V_s$ is the infinite bus voltage, and $V_t\left(t\right)$ denotes the generator terminal voltage, $X_d$ and $X_q$ are the d-axis and q-axis synchronous reactance, $X_d^{\prime }$ is the d-axis transient reactance, $X_T$ and $X_L$ mean the transformer and the single line reactance. Actually, the generator is coupled to the infinite bus using two parallel lines. In this study, the assumption that the generator is connected to an infinite bus is obviously an idealization, which is a detailed examination of the behavior of the single generator, the analysis of the transient process and the design of an adequate controller.

The generator internal transient voltage, $E_q^{\prime }\left(t\right)$, is physically unmeasurable, whereas the active electrical power, $P_e\left(t\right)$, is actually measurable in practice. Thus, the differential of electrical power (see Equation (5)) could substitute the differential of $E_q^{\prime }\left(t\right)$ in the dynamic system associated with Equation (1):

$$\frac{d{P}_e\left(t\right)}{dt}=\frac{1}{T_{d0}^{\prime }}\frac{V_s}{X_{ds}^{\prime }}\sin \left(\delta \right)E_{fd}\left(t\right)+\frac{\Delta X_d}{X_{ds}{X}_{ds}^{\prime }}{V}_s^2\omega si{n}^2\left(\delta \right)+\left(\omega \cot \left(\delta \right)-\frac{X_{ds}}{X_{ds}^{\prime }{T}_{d0}^{\prime }}\right)P_e\left(t\right)$$

(5)

In the present work, the SG is equipped with an excitation system of type 1, as assumed in [3]. The following equation describes the model-related excitation system:

$$\frac{d{E}_{fd}\left(t\right)}{dt}=\frac{1}{T_A}\left(K_A\left(V_{ref}-V_t\left(\mathrm{t}\right)+U_S\left(t\right)\right)-E_{fd}\left(t\right)\right)$$

(6)

where $K_A$ and $T_A$ denote, respectively, the exciter gain constant and the exciter time constant, and $U_S\left(t\right)$ is the auxiliary control voltage. The dynamic system equation for an SG coupled to an infinite bus equipped with excitation system type 1 is as follows:

$$\left\{\begin{array}{l}\frac{d\delta (t)}{dt}=\omega (t)\\ \frac{d\omega (t)}{dt}=\mathrm{a}\omega (t)+\mathrm{b}\left(P_m(t)-P_e(t)\right)\\ \frac{d{P}_e(t)}{dt}=\mathrm{c}\sin (\delta (t))E_{fd}(t)+\mathrm{d}\omega (t){\mathit{sin}}^2(\delta (t))+(\omega (t)\cot (\delta (t))+e)P_e(t)\\ \frac{d{E}_{fd}(t)}{dt}=\mathrm{f}\left(V_{\mathit{ref} }-V_t(t)\right)+\mathrm{g}{E}_{fd}(t)+\mathrm{f}{U}_S(t)\end{array}\right.$$

(7)

with $\mathrm{a}=-\frac{K_D}{2\mathrm{H}}$, $b=\frac{{\omega }_s}{2\mathrm{H}}$, $\mathrm{c}=\frac{1}{T_{d0}^{\prime }}\frac{V_s}{X_{ds}^{\prime }}$, $\mathrm{d}=\frac{\Delta X_d}{X_{ds}{X}_{ds}^{\prime }}{V}_s^2$, $\mathrm{e}=\frac{-X_{ds}}{X_{ds}^{\prime }{T}_{d0}^{\prime }}$, and $\mathrm{f}=\frac{K_A}{T_A}$; $\mathrm{g}=\frac{-1}{T_A}$.

The controlled turbine serves to provide mechanical power. The SG input power has a much slower variation than its instantaneous active electrical power and is, therefore, supposed to be constant $\left(P_m=P_{m0}\right)$. In a steady state, $E_{fd}\left(t\right)=E_{qref}$; $P_e\left(t\right)=P_m; \omega \left(t\right)=0, \mathrm{and} \mathsf{\delta }\left(\mathrm{t}\right)={\delta }_{ref}$. The desired amount ${\delta }_{ref}$, corresponding to the stable equilibrium conditions defined in Equation (8), is attained using the rotor angle expression provided in Equation (9) [4], wherein $V_t\left(t\right)$ is substituted with its desired value $(V_{tref}=V_s)$, and $P_e\left(t\right)$ with the corresponding amount of $P_m$. The excitation voltage corresponding to the steady state operating condition is equal to the q-axis EMF $E_{qref}$, as defined by Equation (2), wherein $P_e\left(t\right)=P_m$ and $\mathsf{\delta }\left(\mathrm{t}\right)={\delta }_{ref}$. What remains is to determine each of the power system bus-corresponding steady-state conditions; the Load Flow Program (LFP) is investigated. Hence:

$$E=\left\{\mathsf{{\rm X}}\in {ℝ}^4; \mathsf{\delta }\in \left[0 \frac{\pi }{2}\right[, \omega \left(t\right)=0, P_e\left(t\right)=P_{m0},E_{fd}\left(t\right)=E_{qref}\right\}$$

(8)

$$\mathsf{\delta }\left(\mathrm{t}\right)=\mathrm{ar} \mathrm{cotg}(\frac{V_s}{X_s{P}_e\left(t\right)}(\sqrt{V_t^2\left(t\right)-\frac{X_s^2}{V_s^2}{P}_e^2\left(t\right)}-\frac{X_d{V}_s}{X_{ds}}))$$

(9)

The plant is modeled by a set of nonlinear differential Equation (7). This modeling is the strict-feedback form (triangular form), necessary to design a nonlinear backstepping controller [42,43]. In effect, applying a nonlinear control helps to solve the dynamics of the power system and the transient stability problem. As illustrated by Figure 1, the objective of this study consists in generating the supplementary signal $U_S\left(t\right)$ to exciter through application of the backstepping controller optimized parameters.

3. Design of a Backstepping Controller Fit for an SMIB System

The backstepping approach generates a recursive control method for stabilizing the system in strict-feedback form [42,43]. The resultant closed-loop system is stable at any equilibrium point. This stability is achieved by implementing a candidate Lyapunov function for the entire system. The main concept is embodied by applying some of the state variables as virtual controls, while drawing intermediate control laws based on each state’s respective dynamics. This method is intended to maintain the SG steady synchronism state, which operates in a comfortable stability margin under large power systems with disturbed conditions. To this end, four recursive steps are developed to simultaneously regulate the electric power $P_e$, the speed deviation ω(t), and the rotor angle δ(t) within their respective reference values. The virtual reference of $\omega \left(t\right), P_e\left(t\right),$ and $E_{fd}\left(t\right)$ is determined by the controller as an auxiliary control ${\alpha }_1$, ${\alpha }_{2,}$ and ${\alpha }_3$. The desired $\delta \left(t\right)\in \left[0 \frac{\pi }{2}\right[$ corresponding to the stable equilibrium steady-state condition is ${\delta }_{ref}$, as already determined. Hence, for the constant input $P_m=P_{m0}$ and excitation voltage to be equal to $E_q$, the states are made to be at an equilibrium point, as defined above.

The following section is aimed at developing a clear and accurate description of the relevant experimental results, interpretations, and conclusions.

3.1. Controller Design

At this level, the backstepping scheme serves to compute the excitation signal as the final control law through three auxiliary controls (${\alpha }_1$, ${\alpha }_2,$ and ${\alpha }_3$) on a step-by- step basis. The error variables are presumed to be:

$${\mathsf{\xi }}_1\left(\mathrm{t}\right)=\delta \left(t\right)-{\delta }_{ref}\left(t\right)$$

(10)

$${\mathsf{\xi }}_2\left(\mathrm{t}\right)=\omega \left(t\right)-{\alpha }_1$$

(11)

$${\mathsf{\xi }}_3\left(\mathrm{t}\right)=P_e\left(t\right)-{\alpha }_2$$

(12)

$${\mathsf{\xi }}_4\left(\mathrm{t}\right)=E_{fd}\left(t\right)-{\alpha }_3$$

(13)

Step 1: 

The derivative of ${\mathsf{\xi }}_1$ can be drawn from the system Equation (7) on a differentiation basis:

$${\dot{\mathsf{\xi }}}_1=\frac{d\delta \left(t\right)}{dt}-\frac{d{\delta }_{ref}}{dt}=\omega \left(t\right)-{\dot{\delta }}_{ref}$$

(14)

By adding and subtracting the auxiliary control ${\alpha }_1$ in (14), while maintaining the rotor angle reference constant (${\dot{\delta }}_{des}=0$), the final relation is quantified as presented below:

$${\dot{\mathsf{\xi }}}_1={\mathsf{\xi }}_2+{\alpha }_1$$

(15)

Considering the first Lyapunov function: $V_1\left(\mathrm{x}\right)=\frac{1}{2}{ \mathsf{\xi }}_1^2$, the derivative of $V_1$ along the ξ₁ trajectory is computed as:

$${\dot{V}}_1={\mathsf{\xi }}_1{\mathsf{\xi }}_2+{\mathsf{\xi }}_1{\alpha }_1$$

(16)

By Lyapunov’s direct stability method, the system is stable when ${\dot{V}}_1$ conditionally contains a negative derivative. This can only be achieved by setting ${\mathsf{\xi }}_2$ to 0 and maintaining ${\alpha }_1$ as $-k_1{\mathsf{\xi }}_1$, where $k_1$ is a positive constant [44]. Hence, the control objective consists in maintaining ${\mathsf{\xi }}_2$ equal to zero. Once the last expression of ${\alpha }_1$ in Equation (15) is substituted, the first error variable corresponding derivative can be represented as:

$${\dot{\mathsf{\xi }}}_1={\mathsf{\xi }}_2-k_1{\mathsf{\xi }}_1$$

(17)

Step 2: 

This step involves the backstepping methodology, which consists in calculating the second error variable ${\mathsf{\xi }}_2$ corresponding derivative using Equations (7), (11), and (17), such as:

$$\dot{{\mathsf{\xi }}_2}=\left(k_1+a\right)\omega \left(t\right)+\mathrm{b}\left(P_m\left(t\right)-P_e\left(t\right)\right)$$

(18)

The simultaneous addition and subtraction of the quantity $\mathrm{b}{\alpha }_2$ leads to:

$$\dot{{\mathsf{\xi }}_2}=\left(k_1+a\right)\omega \left(t\right)+\mathrm{b}{P}_{m0}-\mathrm{b}\left({\mathsf{\xi }}_3+{\alpha }_2\right)$$

(19)

At this level, defining the Lyapunov function for Step 1 is considered as: $V_2\left(\mathrm{x}\right)=V_1\left(\mathrm{x}\right)+\frac{1}{2}{ \mathsf{\xi }}_2^2$. The derivative of $V_2$ along the system trajectory, computed by Equation (19), is designed as:

$${\dot{V}}_2=-k_1{\mathsf{\xi }}_1^2+{\mathsf{\xi }}_2\left[{\mathsf{\xi }}_1+\left(k_1+a\right)\omega \left(t\right)+\mathrm{b}{P}_{m0}-\mathrm{b}\left({\mathsf{\xi }}_3+{\alpha }_2\right)\right]$$

(20)

Following the Lyapunov criteria, Equation (20) is expressed as: ${\dot{V}}_2=-k_1{\mathsf{\xi }}_1^2-k_2{\mathsf{\xi }}_2^2$, where $k_2$ is a positive constant. Hence, based on the control objective ${\mathsf{\xi }}_3=0$, the second auxiliary control ${\alpha }_2$ can be computed through the following equation:

$${\alpha }_2=\frac{1}{\mathrm{b}}\left(k_2{\mathsf{\xi }}_2+{\mathsf{\xi }}_1+\left(k_1+a\right)\omega \left(t\right)\right)+P_{m0}$$

(21)

By introducing this expression of ${\alpha }_2$ in Equation (19), the second error variable dynamics are possibly simplified as:

$$\dot{{\mathsf{\xi }}_2}=-{ \mathsf{\xi }}_1-k_2{\mathsf{\xi }}_2-{\mathrm{b}\mathsf{\xi }}_3$$

(22)

Step 3: 

This step involves calculating the derivative of the next error variable ${\mathsf{\xi }}_3$ regarding the state and error variables. Using Equations (7), (12) and (21), the derivative of ${\mathsf{\xi }}_3$ is expressed by:

$$\dot{{\xi }_3}=A \omega \left(t\right)+B{P}_e\left(t\right)+C P_{m0}+D E_{fd}\left(t\right)$$

(23)

with $A=-\frac{1}{b}\left(\left(a-1\right)k_1+a{k}_2+a^2+1\right)+d si{n}^2\delta \left(t\right), B=\omega \left(t\right)\cot \left(\delta \left(t\right)\right)+k_1+k_2+a+e$, $C=\left(k_1+k_2+a\right)$, and $D=c\mathit{sin}\left(\delta \left(t\right)\right)$.

The Lyapunov function candidate, defined in step 2, is: $V_3\left(\mathrm{x}\right)=V_2\left(\mathrm{x}\right)+\frac{1}{2}{ \mathsf{\xi }}_3^2$. The derivative of the third auxiliary function, $V_3$, is calculated in the light of the third and fourth error variables, as follows:

$${\dot{V}}_3=-k_1{\mathsf{\xi }}_1^2-k_2{\mathsf{\xi }}_2^2+{\mathsf{\xi }}_3\left(A \omega \left(t\right)+B{P}_e\left(t\right)+C P_{m0}+D \left({\alpha }_3+{\mathsf{\xi }}_4\right)\right)$$

(24)

With reference to the Lyapunov stability theory, ${\dot{V}}_3$ is required to be a negative defined function. Thus, it is written as: ${\dot{V}}_3=-k_1{\mathsf{\xi }}_1^2-k_2{\mathsf{\xi }}_2^2-k_3{\mathsf{\xi }}_3^2+D{\mathsf{\xi }}_3{\mathsf{\xi }}_4$, with $k_3>0$. Hence, based on the control objective ${\mathsf{\xi }}_4=0$, the third auxiliary control is calculated as:

$${\alpha }_3=\frac{-1}{D}\left(A \omega \left(t\right)+B{P}_e\left(t\right)+C P_{m0}+k_3{\mathsf{\xi }}_3\right)$$

(25)

Like step 2, by substituting the last expression of ${\alpha }_3$ into Equation (23), the third error variable dynamics appear to be determined as:

$$\dot{{\xi }_3}=D {\xi }_4-k_3{\xi }_3$$

(26)

Step 4: 

The ultimate step of the Lyapunov methodology involves calculating the derivative of the last error variable, ${\mathsf{\xi }}_4$, in relation to the state variables, prior to representing the system dynamics model by means of the new error variables:

$$\begin{array}{c}\dot{{\xi }_4}=\mathrm{f} \left(V_{ref}-V_t\left(t\right)\right)+\mathrm{g} E_{fd}\left(t\right)+\mathrm{f}{U}_S\left(t\right)+\frac{1}{D}(E {\omega }^2\left(t\right)+F)-\frac{k_3^2}{D}{\mathsf{\xi }}_3+k_3{\mathsf{\xi }}_4\\ +\frac{\omega \left(t\right)}{D}\left(A \omega \left(t\right)+B P_e\left(t\right)+C P_{m0}-k_3{\mathsf{\xi }}_3\right)cot\left(\delta \left(t\right)\right)\end{array}$$

(27)

where $E=d\cos \left(\delta \left(t\right)\right)-\frac{1}{si{n}^2\left(\delta \left(t\right)\right)}, \mathrm{and} F=\left(\mathrm{a} \omega \left(t\right)+\mathrm{b}\left(P_m\left(t\right)-P_e\left(t\right)\right)\right)\left(A+cot\left(\delta \left(t\right)\right)\right)$.

Using the new coordinates $\left({\mathsf{\xi }}_1,{ \mathsf{\xi }}_2,{ \mathsf{\xi }}_3,{ \mathrm{and} \mathsf{\xi }}_4\right)$, the dynamic system given in (7) is demonstrated as follows:

$$\left\{\begin{array}{c}{\dot{\mathsf{\xi }}}_1={\mathsf{\xi }}_2-k_1{\mathsf{\xi }}_1\\ \dot{{\mathsf{\xi }}_2}=-({\mathsf{\xi }}_1+k_2{\mathsf{\xi }}_2+{\mathrm{b}\mathsf{\xi }}_3)\\ \dot{{\xi }_3}=D {\xi }_4-k_3{\xi }_3\\ \dot{{\xi }_4}=\mathrm{f} \left(V_{ref}-V_t\left(t\right)\right)+\mathrm{g} E_{fd}\left(t\right)+\mathrm{f}{U}_S\left(t\right)+\frac{1}{D}(E {\omega }^2\left(t\right)+F)-\frac{k_3^2}{D}{\mathsf{\xi }}_3+k_3{\mathsf{\xi }}_4\\ +\frac{\omega \left(t\right)}{D}\left(A \omega \left(t\right)+B{P}_e\left(t\right)+C P_{m0}-k_3{\mathsf{\xi }}_3\right)cot\left(\delta \left(t\right)\right)\end{array}\right.$$

(28)

3.2. Development of Stability Analysis and Control Law Design for System Optimization

The stability of the closed-loop system is analyzed through the implementation of a Lyapunov function candidate, as defined in step 4, for the purpose of assessing its behavior over time: $V_4\left(\mathrm{x}\right)=V_3\left(\mathrm{x}\right)+\frac{1}{2}{ \mathsf{\xi }}_4^2$. The derivative of $V_4$ is computed by using the new dynamic system model (28):

$$\begin{array}{c}{\dot{V}}_4=-k_1{\xi }_1^2-k_2{\xi }_2^2-k_3{\xi }_3^2+{\xi }_4(f (V_{ref}-V_t\left(t\right))+g E_{fd}\left(t\right)+\frac{1}{D}(E {\omega }^2\left(t\right)+F)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+ k_3{\xi }_4+\frac{\omega \left(t\right)}{D}\left(A \omega \left(t\right)+B P_e\left(t\right)+C P_{m0}\right)cot\left(\delta \left(t\right)\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+ f U_S\left(t\right))-(\frac{k_3^2}{D}+k_3\frac{\omega \left(t\right)}{D}cot\left(\delta \left(t\right)\right)){\xi }_3{\xi }_4\hfill \end{array}$$

(29)

Based on Lyapunov stability theory, ${\dot{V}}_4$ is presumably a negative defined function. In this process, it is applied as: ${\dot{V}}_4=-k_1{\mathsf{\xi }}_1^2-k_2{\mathsf{\xi }}_2^2-k_3{\mathsf{\xi }}_3^2-k_4{\mathsf{\xi }}_4^2$, with $k_4>0$, while the control law $U_s\left(t\right)$ is calculated as:

$$U_s\left(t\right)=\left(V_{ref}-V_t\left(t\right)\right)-\frac{1}{f}\left(\mathrm{g} E_{fd}\left(t\right)+\frac{1}{D}\left(E {\omega }^2\left(t\right)+F\right)+\frac{G}{D}\omega \left(t\right)cot\left(\delta \left(t\right)\right)+H {\mathsf{\xi }}_4\right)$$

(30)

where $G$ and H are provided by: $G=\left(A \omega \left(t\right)+B{P}_e\left(t\right)+C P_{m0}\right)$ and $H=k_3+k_4-\frac{k_3}{D}(k_3+{\mathsf{\xi }}_3\omega \left(t\right)cot\left(\delta \left(t\right)\right)$.

The auxiliary control signals $({\alpha }_1, {\alpha }_2, \mathrm{and} {\alpha }_3$) and the control law $U_s$ are designed to set the error variables toward zero, which proves that the system is operating within the desired set point or target. If and only if ${\dot{V}}_4\le 0$ and $V_4\left(t\right)\le V_4\left(0\right)$, then ${\mathsf{\xi }}_1,{ \mathsf{\xi }}_2,{ \mathsf{\xi }}_3 \mathrm{and} {\mathsf{\xi }}_4$ are implicitly bounded. In fact, if we consider $\mathrm{\Phi }=-{\dot{V}}_4\left({\mathsf{\xi }}_{\mathrm{i}}\right)$ with $i=1\dots 4$ taken into consideration, it follows that:

$${{\displaystyle \int }}_0^t\varphi \left(\tau \right)d\tau =V_4\left({\mathsf{\xi }}_1\left(0\right),{\mathsf{\xi }}_2\left(0\right),{\mathsf{\xi }}_3\left(0\right), {\mathsf{\xi }}_4\left(0\right)\right)-V_3\left({\mathsf{\xi }}_1\left(t\right),{\mathsf{\xi }}_2\left(t\right),{\mathsf{\xi }}_3\left(t\right) ,{\mathsf{\xi }}_4\left(t\right)\right)$$

(31)

Since $V_4\left(0\right)$ is solely bounded and $V_4\left(t\right)$ is non-increasing and bounded at the same time $\left(V_4\left(t\right)\le V_4\left(0\right)\right)$, $\underset{t\to \infty }{\lim }{\int }_0^t\varphi \left(\tau \right)d\tau <\infty$, adding to them the bounded $\dot{\mathrm{\Phi }}$, with reference to Barbalat’s Lemma, it can be shown that $\underset{t\to \infty }{\lim }\varphi \left(t\right)=0$ [44]. It signifies that ${\mathsf{\xi }}_1,{ \mathsf{\xi }}_2,{ \mathsf{\xi }}_3 \mathrm{and} {\mathsf{\xi }}_4$ converge to zero while $t\to \infty$. Accordingly, the backstepping controller plays the role of maintaining the dynamic system, as described in Equation (28). It is able to reach a stable state as time progresses toward infinity. Considering the definition of error variables ${\mathsf{\xi }}_1, {\mathsf{\xi }}_2, {\mathsf{\xi }}_3 \mathrm{and} {\mathsf{\xi }}_4$, which converge to zero, it is clear that ${\alpha }_1\to 0, {\alpha }_2\to P_{m0} \mathrm{and} {\alpha }_3\to E_{qref}$, hence:

$$\underset{t\to \infty }{\lim }\left(\begin{array}{c}\delta \left(t\right)\\ \omega \left(t\right)\\ P_e\left(t\right)\\ E_{fd}\left(t\right)\end{array}\right)=\left(\begin{array}{c}{\delta }_{ref}\\ 0\\ P_{m0}\\ E_{qref}\end{array}\right)$$

(32)

4. Design Controller Optimization

4.1. Problem Formulation

In the present work, the controller parameters ($k_j; j=1\dots 4$) are maintained to ensure a high transient stability and attain an optimal regulation of the power system. With a randomly selected parameter setting, however, the backstepping controller would fail to achieve the designed stability and regulation. The backstepping regulator would, therefore, become inadequate to cope with the shifting equilibrium point. In this case, the controller parameters’ workability is determined by taking a wide range of power system functionality conditions into account. The most optimal values of the parameters are demonstrated by considering three objectives. The first two objectives consist in reducing the stabilization time and minimizing the maximum overshoot after a large disturbance. The third objective is achieved by adjusting the voltage regulation to a desired value. The timing solution of the power system’s transient stability problem with a minimum overshoot is possibly achieved by optimizing the integral of the time-multiplied absolute value of the speed error (ITAE) as a first objective function $J_1$, and equally by optimizing the integral of the time-multiplied absolute terminal voltage error as a second objective function $J_2$ [45,46]. For this purpose, the Jaya optimization algorithm is used as a means of minimizing the objective function $J$, incorporating the sum of the two preceding functions, which are effectively weighed by the coefficients $\alpha$ and $\beta$. These coefficients are determined through a trial/error process, while the robustness requirement is met by introducing multiple operating scenarios in the design phase. Hence

$$J\left(k_j; j=1,\dots ,4\right)=\alpha J_1\left(k_j\right)+\beta J_2\left(k_j\right)$$

(33)

with

$$J_1\left(k_j; j=1,\dots ,4\right)={{\displaystyle \int }}_0^{tsim}t \left|\Delta V_t\right|.dt$$

(34)

$$J_2\left(k_j; j=1,\dots ,4\right)={{\displaystyle \int }}_0^{tsim}t \left|\Delta \omega \right|.dt$$

(35)

where $\Delta \omega$ denotes the speed deviation of the SMIB system, $\Delta V_t$ is the terminal voltage error, and $tsim$ is the time range of the simulation. The constants $\alpha$ and $\beta$ are identified by the range of [0, 1] and selected by the designer to weigh the importance of each output error $\Delta V_t$ or $\Delta \omega$ component in the control process. Here, the main target is operated by reducing the value of the objective function J, while satisfying the set constraints indicating the upper and lower limits of each parameter.

$$k_{\mathrm{j}}^{min}<k_{\mathrm{j}}<k_{\mathrm{j}}^{max}; j=1,\dots ,4$$

(36)

4.2. Optimization Techniques

4.2.1. The Jaya Algorithm

Initially introduced by Rao in 2016 [41], the JA represents solely an evolution of the TLBO algorithm [40]. The latter rests on the classroom modeling procedure, wherein the teacher influences the learners, who in turn interact to improve their results. The basic algorithm is constituted by two phases: the “teacher phase” and the “learner phase”. The teacher phase implies the learning motivated by the teacher; while the learner phase relates to how learning is carried out through the interaction between the learners per se. The improved performance in the class environment stems from the collaborative efforts made by the teacher and the learners. Unlike the two-phase TLBO algorithm, the JA proceeds in a single-phase manner. Its operation method differs from the TLBO’s since it optimizes the objective function by gravitating toward the optimal solution, while avoiding subpar ones.

Regarding the programming algorithm, it is assumed that $J\left(X\right)$ represents the objective function, which requires a minimization. At any iteration $i$, it is premised that there are $m$ number of design variables, and $n$ number of candidate solutions (population size). The optimal candidate solution yields the lowest value of the objective function $J\left(X\right)$. The inferior candidate solution brings about the highest value of $J\left(X\right)$ among all the supposedly candidate solutions. The algorithm is initiated by generating first solutions and evaluating the objective function of each one. If $X_{j,k,i}$ is the value of the $j^{th}$ variable for the $k^{th}$ candidate throughout the $i^{th}$ iteration, it is updated following Equation (37):

$$X_{j,k,i}^{\prime }=X_{j,k,i}+r_{1,i,j}\left(X_{j,best,i}-\left|X_{j,k,i}\right|\right)-r_{2,i,j}\left(X_{j,worst,i}-\left|X_{j,k,i}\right|\right)$$

(37)

where $X_{j,k,i}^{\prime }$ is the updated value of $X_{j,k,i}$.$X_{j,best,i}$, and $X_{j,worst,i}$ is the value of the variable $j$, respectively, for the best candidate and the worst candidate during the $i^{th}$ iteration. $r_{1,i,j}$ and $r_{2,i,j}$ are randomly generated numbers between 0 and 1 for the $j^{th}$ variable during the $i^{th}$ iteration. The term $r_{1,i,j}\left(X_{j,best,i}-\left|X_{j,k,i}\right|\right)$ indicates the solution tendency to move closer to the best solution, while the term $r_{2,i,j}\left(X_{j,worst,i}-\left|X_{j,k,i}\right|\right)$ shows the solution tendency to avoid the worst solution. In case the updated value $X_{j,k,i}^{\prime }$ violates its range (Equation (34)), it will be set to its limit. The value of $X_{j,k,i}^{\prime }$ is accepted only if it yields a better objective function value. With the application of the Jaya algorithm, the solution obtained for an optimization problem is iteratively improved by approaching the best candidate and avoiding the worst one. The random numbers $r_{1,i,j}$ and $r_{2,i,j}$ ensure an effective exploration of the search space.

The following section deals with applying the JA algorithm to a backstepping controller. The process of implementing the JA to determine the optimal parameters fits into the backstepping controller, which involves the following steps:

Step 1: 

Select the number m of the design variables (c$\mathrm{ontroller} \mathrm{parameters}$), the population size ($n$), the minimum and maximum limits of the design variables and the maximum number of iterations ($ite{r}^{max}$).

Step 2: 

Generate the initial solution as random values within a range of design parameters’ min and max values. The equation of this initial solution is designed by:

$$X_{j,k,0}=\min \left(X_j\right)+r_{j,k,0}\left(\mathrm{Max}\left(X_j\right)-\min \left(X_j\right)\right)$$

(38)

where $\min \left(X_j\right)$ and $\mathrm{Max}\left(X_j\right)$ are, respectively, the minimum and maximum for $j^{th}$ variable; $r_{j,k,0}$ are the random figures in the range of [0, 1] for the $j^{th}$ variable for the $k^{th}$ population candidate at initial iteration.

Step 3: 

Simulate the SMIB system with the proposed excitation controller and calculate the objective functions $J\left(X_{j,k,0}; j=1\dots m\right)$ for each member of the population $\left(k\right)$ under different types of possible faults considered at separate time intervals throughout the simulation time ($t_{sim}$).

Step 4: 

Identify the best and worst solutions in the population corresponding, respectively, to the minimum and the maximum of the objective functions for each member of the population $\left(k\right)$, and update the population using Equation (35).

Step 5: 

For each candidate $\left(k\right)$ of the new population, re-simulate the SMIB system with the proposed excitation controller and compute the new objective functions $J(X_{j,k,i}^{\prime }; j=1\dots m)$.

Step 6: 

Compare the objective function vectors and retain the candidate that displays the best optimal objective function to form a new population.

Step 7: 

Reiterate steps 3 to 6 successively until the stopping criteria are verified. In this study, the optimization algorithm ends when the maximum number of iterations is reached or the objective function cannot be improved, as described by inequality (39).

$$\left|J\left(X\_\left(j,best,i+1\right)\right)-J\left(X\_\left(j,best,i\right)\right)\right|<Va{l}_{min}$$

(39)

Step 8: 

Report this solution as the best global one for the design variables: $k_{\mathrm{j}}=X_{j,best}; j=1\dots m$.

The envisioned algorithm architecture, designed to optimize the backstepping controller, is illustrated in Figure 2.

4.2.2. Genetic Algorithm (GA)

The GA is mainly exploited by the strategy of the “survival of the fittest” to evolve a population of solutions toward the most effectively optimal ones. The algorithm is based on favoring the fittest individuals to reproduce and pass on their traits to the next generation, leading to an enhanced improvement over time. Still, the inferior ranked individuals may also survive and reproduce, creating diversity in the population. The implementation of GA requires determining six fundamental issues: chromosome representation, selection function, genetic operators, initialization, termination, and evaluation function [33,39].

To assess the JA attained results, the GA is usually utilized as a benchmark for evolutionary algorithms. In the first step, the JA attained optimization results are compared to the GA, determining the optimal backstepping control parameters. In the second step, the newly proposed auxiliary control design is compared to the traditional PSS auxiliary control. The optimization of the backstepping controller or the PSS parameters with GA can be summed up in terms of the following successive steps:

Step 1: 

Specify the population size $\left(N_{pop}\right)$, mutation probability $\left(p_m\right)$, crossover probability $\left(p_c\right)$, limits of the controller parameters, selection operator (tournament selection), and the maximum number of generations ($ite{r}^{max}$).

Step 2: 

As the JA, the initial population is generated, while the power system time–domain simulation is fixed ($t_{max}$), and various types of fault scenarios are considered at different simulation times.

Step 3: 

Simulate the SMIB with the selected excitation controller (backstepping or PSS) for each individual of the current population with the last condition and compute the corresponding fitness function.

Step 4: 

Check the stopping condition of the algorithm. The same termination criteria as those for JA are adopted.

Step 5: 

If the stopping criterion is not verified, apply GA operators: selection, crossover and mutation to obtain a new population. Then, return to step three with the updated parameters of the new population.

Step 6: 

If the stopping criterion is satisfied, retain the best solution of the controller parameters $\left(k_{\mathrm{j}}; j=1\dots m\right)$ corresponding to the minimum of $J_{min}$ in the last population.

The flowchart of GA-based backstepping and PSS controllers is presented in Figure 3.

5. Simulation Studies and Results

In this section, the effectiveness and robustness of the new scheme controller, outlined in Figure 1, are evaluated using a single generator connected to an infinite bus through a transformer and two parallel transmission lines having the same impedance $X_L$. The test system parameters are detailed in

Table 1. Power system parameters.

	Machine						Transformer	Transmission Line	AVR
Parameter	$X_d$	$X_d^{\prime }$	$X_q$	$T_{d0}^{\prime }$	$H$	$D$	$X_T$	$X_L$	$K_A$	$T_{A }$
Value	1.863	0.257	1.49	6.9	4	6	0.127	0.242	200	0.05
Unit	(p.u)	(p.u)	(p.u)	(s)	MJ/MVA	(p.u)	(p.u)	(p.u)		(s)

.

The physical limits of the excitation voltage are taken as follows.

$E_{fd}^{min}=-5 \mathrm{p}.\mathrm{u}$., and $E_{fd}^{max}=5 \mathrm{p}.\mathrm{u}$. The steady state condition of the system is as follows.

${\delta }_0=1.423 \mathrm{rad}$, $E_{q0}=2.252 \mathrm{p}.\mathrm{u}.$, $E_{fd0}=2.252 \mathrm{p}.\mathrm{u}.$, $E_{q0}^{\prime }=2.157 \mathrm{p}.\mathrm{u}.$, $P_{e0}=0.9 \mathrm{p}.\mathrm{u}.$, $P_{m0}=0.9 \mathrm{p}.\mathrm{u}.$, $V_{t0}=1 \mathrm{p}.\mathrm{u}.$ and $V_s=1 \mathrm{p}.\mathrm{u}$.

The results were obtained by using MATLAB 2017 environment running on a laptop computer with a 3.1 GHz Intel Core i5 processor with 8 GB RAM. The JA and GA were programmed using m-file, whereas the SMIB model was designed and simulated using Simulink with the following solver options: type (fixed step), solver (ode4 Runge-kutta) and the fundamental sample time (${10}^{-3} \mathrm{s}$).

The transient stability improvement and performance of the proposed controllers were thoroughly estimated based on three faulty cases, which are described as follows.

• Case 1: The system is subjected to a three-phase short-circuit fault appearing at t = 1 s with a duration of ∆t = 50 ms. Once the fault is resolved, the system resumes its original configuration.
• Case 2: A 20% step change in the input power of the machine (mechanical power) is applied.
• Case 3: A three-phase fault with a duration of ∆t = 150 ms occurs at t = 1 s. The fault location is in one of the parallel lines and close to the terminal bus. The fault is cleared by opening the faulty line.

To evaluate the advantages and disadvantages associated with the examined optimization techniques, and to highlight the effectiveness of the optimized backstepping controller, the latter was compared to the standard PSS controller optimized using GA technique.

5.1. Identification of JA and GA Parameters

To preserve consistency for all investigated optimization methods, population size (n) and maximum number of iterations $ite{r}^{max}$ are set as follows [47]:

$$n=10m$$

(40)

$$ite{r}^{max}=20m-50$$

(41)

with m is the number of search variables. It is worth noting that the number of function evaluations (NFE) is $n\times ite{r}^{max}$.

With regard to the probabilities of the genetic operators, there is no consensus in the state-of-the art of the values of the crossover and mutation probabilities. Referring to [48], 840 combinations of these two parameters applied to 10 complex problems were studied. By investigating the obtained results, it was concluded that the most suitable values of crossover and mutation probabilities were 95% and 1%, respectively. This proves that optimal solutions are usually obtained when using high crossover and low mutation rates. Accordingly, a statistical analysis for different combinations of these rates around the aforementioned values was investigated in this study to select the appropriate combination for the studied problem. The statistical findings comprising best and standard deviation (Std) results were obtained for a crossover probability ranging from 0.9 to 0.95 and a mutation probability ranging from 0.01 to 0.05. In this study, the GA real-code was adapted to avoid the shortcomings of the binary coding, such as poor local search capability and its high calculation time due to the encoding/decoding phases.

Table 2. Statistical results.

${\mathit{p}}_{\mathit{c}}$	${\mathit{p}}_{\mathit{m}}$	$\mathbf{Best} (\times {10}^{-3})$	Std		${\mathit{p}}_{\mathit{c}}$	${\mathit{p}}_{\mathit{m}}$	$\mathbf{Best} (\times {10}^{-3})$	$\mathbf{Std} (\times {10}^{-3})$
0.90	0.01	0.2172	3.50 × 10−6		0.95	0.01	0.2594	1.12 × 10−6
	0.02	0.2751	3.56 × 10−5			0.02	0.2455	1.62 × 10−4
	0.03	0.6444	5.90 × 10−6			0.03	0.2662	1.16 × 10−3
	0.04	0.2332	6.11 × 10−5			0.04	0.2715	9.50 × 10−4
	0.05	0.2219	6.80 × 10−6			0.05	0.3903	9.51 × 10−4
0.91	0.01	0.3007	2.41 × 10−4		0.96	0.01	0.2407	9.22 × 10−5
	0.02	0.2326	2.48 × 10−4			0.02	0.2479	3.28 × 10−5
	0.03	0.2298	2.75 × 10−4			0.03	0.2745	1.19 × 10−4
	0.04	0.3020	2.34 × 10−4			0.04	0.2343	1.51 × 10−4
	0.05	0.2330	4.4 × 10−5			0.05	0.4364	4.96 × 10−5
0.92	0.01	0.2340	2.93 × 10−4		0.97	0.01	0.2931	7.14 × 10−5
	0.02	0.3452	3.12 × 10−4			0.02	0.3123	8.42 × 10−5
	0.03	0.3452	2.30 × 10−4			0.03	0.2301	9.00 × 10−6
	0.04	0.3224	2.44 × 10−4			0.04	0.2440	2.70 × 10−5
	0.05	0.3271	2.18 × 10−4			0.05	0.2288	1.40 × 10−5
0.93	0.01	0.3002	2.19 × 10−4		0.98	0.01	0.2188	1.40 × 10−4
	0.02	0.2367	2.50 × 10−4			0.02	0.2501	7.12 × 10−6
	0.03	0.2313	2.47 × 10−4			0.03	0.2467	1.50 × 10−6
	0.04	0.2484	2.22 × 10−4			0.04	0.2223	1.25 × 10−5
	0.05	0.2477	2.31 × 10−4			0.05	0.2306	8.70 × 10−6
0.94	0.01	0.3452	2.50 × 10−4		0.99	0.01	0.2501	6.32 × 10−6
	0.02	0.2297	2.50 × 10−4			0.02	0.2501	3.89 × 10−5
	0.03	0.2308	2.50 × 10−4			0.03	0.2501	9.11 × 10−5
	0.04	0.2572	2.36 × 10−4			0.04	0.2356	6.50 × 10−6
	0.05	0.23554	2.50 × 10−4			0.05	0.2501	7.58 × 10−6

shows the statistical results obtained after 30 runs of the GA where the objective function is described in Equation (31). More importantly, referring to its content, it can be clearly noticed that the lowest best minimum value of the objective function was obtained when the crossover probability ($p_c$) was equal to 0.9. Moreover, the minimum standard deviation for this value of $p_c$ corresponds to $p_m=0.01$. Thus, these values of $p_c$ and $p_m$ could be selected for the GA. Based on these results, values of JA and GA control parameters could be tabulated as shown in

Table 3. Values of optimization methods’ parameters.

Parameters/Method	JA	GA
$m$	$4$	4
$n$	40	40
$ite{r}^{max}$	30	30
$p_c$	-	$0.90$
$p_m$	-	$0.01$
NFE	1200	1200

.

Convergence of the objective function described by Equation (31) for BS-JA, BSGA and PSS-GA controllers is depicted in Figure 4. To be more specific, these results were obtained by using the parameters’ values given in

Table 4. Performance indices for case 1: first overshoots $D_1\%$, settling time $t_s\left(s\right)$, static error ${\epsilon }_s$.

Controllers/Performances	${\mathit{D}}_1\mathbf{\%}$				${\mathit{t}}_{\mathit{s}}\left(\mathit{s}\right)$	${\mathit{\epsilon }}_{\mathit{s}}$
	$\delta$	$\omega$	$E_q^{\prime }$	$V_t$	$\delta , \omega , E_q^{\prime }, V_t$	$\delta \left(°\right)$	$\omega$	$E_q^{\prime }$	$V_t\left(p.u\right)$
BS-JA	38.3	1.8	54	18	1	0	0	0	0
BS-GA	39.2	2	74	35	2	5.7	0	0	0.01
PSS-GA	40.4	2.1	70	42	4.5	10.3	0	0	0.09

.

Figure 4 shows that BS-JA outperforms BS-GA and PSS-GA in point of view minimum value of the ITAE and convergence rate. The worst results were provided by PSS-GA.

5.2. Faults Simulation Results

To test the effectiveness of the suggested Jaya algorithm, which is based on the backstepping (BS-JA) controller, nonlinear time domain simulation was performed under the aforementioned fault scenarios. The obtained results using BS-JA were compared with those reached by using GA, based on the optimized PSS controller (PSS-GA), and JA based optimized PSS controller (PSS-JA).

5.2.1. Case 1

In this sub-section, the performances of the proposed BS-JA, PSS-GA and JA-PSS controllers are evaluated under the fault of case 1. Figure 5 shows the time responses of the state variables as well as the terminal voltage when the studied controllers were applied. From this figure, it can be deduced that the BS-JA outperformed the two other controllers in supplying the best damping characteristics. Additionally, the terminal voltage response remained stable and maintained its reference value even in close proximity to the fault.

To further show the robustness and superiority of the BS-JA compared to PSS-GA and JA-PSS, the amplitude of the first overshoots (D₁), the settling time, and the steady state error were calculated for the system state variables and the terminal voltage. The obtained results are shown in Table 4. This table indicates that the proposed BS-JA controller had the lowest amplitude of the first overshoots and the settling time, compared to the other controllers. Moreover, the BS-JA managed to eliminate the steady state error (static error), which represented the difference between the desired quantity and the instantaneous quantity, in the steady-state operation, for all of the state variables (${\epsilon }_s=\underset{t\to \infty }{\lim }\left(X_i\left(t\right)-X_i^d\right)=0$), where the desired quantities are defined as: ${\delta }^d={\delta }_{ref}$, ${\omega }^d=0$, $E_q^{\prime }{}^d=\frac{X_{ds}^{\prime }}{X_{ds}}{E}_{qref}+\frac{X_{ds}^{\prime }}{\Delta X_d}{V}_{ref}\cos \left({\delta }_{ref}\right)$ and $V_t{}^d=V_{ref}$.

Nevertheless, the static error value, which corresponded to the backstepping optimized by the genetic algorithm BS-GA, was much smaller than that of the PSS-GA controller. This criterion implies the high performance of the nonlinear backstepping control in the post fault regulation and voltage recovery.

The excitation voltage $E_{fd}$, is illustrated in Figure 6. It is worth noting, at this level, that the saturation bloc ($\pm 5 p.u$) was applied to limit the voltage excitation level for coil protection purposes.

5.2.2. Case 2

This case illustrates the system evolution when a 20% step increase in the input mechanical power of the generator is applied. Figure 7 displays the rotor angle, the speed deviation, the transient voltage, and the terminal voltage of the generator before, during, and after the faulty state. It is noted that the proposed controller was able to adapt itself to the new operating conditions, which enhanced its transient stability performance. The BS-JA controller was proved to perform effectively even under changing operating conditions. In this respect, both the BS-JA and PSS-GA control schemes were proved to exhibit closely similar terminal voltage performances. The PSS-GA overall performance tended to dwindle noticeably in accordance with the other variables, displaying recurrent oscillations for a long transient phase. However, the BS-GA controller managed to eliminate all kinds of oscillations, and its transient stability enhancement capacity was justified by the first overshoot indices, as shown in

Table 5. Performance indices for case 1: first overshoots $D_1\%$, settling time ${\mathit{t}}_{\mathit{s}}\left(\mathit{s}\right)$, static error ${\mathit{\epsilon }}_{\mathit{s}}$.

Controllers/Performances	${\mathit{D}}_1\mathbf{\%}$				${\mathit{t}}_{\mathit{s}}\left(\mathit{s}\right)$	${\mathit{\epsilon }}_{\mathit{s}}$
	$\delta$	$\omega$	$E_q^{\prime }$	$V_t$	$\delta , \omega , E_q^{\prime }, V_t$	$\delta \left(°\right)$	$\omega$	$E_q^{\prime }$	$V_t\left(p.u\right)$
BS-JA	0	0.2	19	2	1	0	0	0	0.10
BS-GA	0	0.13	29.5	5	10	5.7	0	0	0.30
PSS-GA	14	0.5	31	9	5	10.3	0	0	0.03

, even though the stabilization time was proved to increase to 10 s and the static errors to grow under the new operating condition.

5.2.3. Case 3

The robustness of the BS-JA controller was also evaluated under the severe fault conditions of case 3 and under the extreme steady-state condition ($\delta \to \frac{\pi }{2}$).

Referring to the results presented in Figure 8, the design of the nonlinear controllers ensured the power system instability problem faced sudden and large disturbances and improved the stability margin of the generator operating conditions. This point can be explained by the extension of the equilibrium attraction region. The successfully effective oscillation damping was achieved for both JA and GA optimization methods. However, it is noteworthy that the PSS-GA controller was proved to be ineffective in damping out the oscillations.

Relying on the results presented above, the robustness of the suggested BS-JA controller design was verified and proved for all of the investigated fault cases. Moreover, the proposed optimized backstepping controller acting as supplementary signal to the exciter illustrated its effectiveness in comparison to the existing PSS-GA for all of the fault scenarios. Consequently, retaining the useful nonlinearities in designing the control law and the optimization of the specific controller parameters increased the performance of the system. It seems that this design was only tested by using a single generator system, which cannot ensure the stability of a multi-machine system. In order to overcome this limitation, a multi-machine power system can be transformed into different SMIB sub-systems as presented in [49]. Each generator is modeled as a single machine coupled to the rest of the power system, which necessitates the optimization of the controller parameters for each SMIB subsystem.

6. Conclusions

In this paper, an optimal design of a backstepping controller was developed for power system stability improvement. The proposed controller was employed instead of the conventional PSS to add a supplementary damping signal to the AVR excitation system. In order to provide sufficient damping of system oscillations subjected to a severe disturbance, the controller parameters were optimally tuned with the JA method. For this purpose, the controller design problem was modeled by a minimization problem where an ITAE-based objective function was considered.

To test the effectiveness of the proposed JA method, the optimal design of the backstepping controller was also achieved with the GA technique. Then, a SMIB system under various disturbances, such as three-phase short-circuits and a step change in the mechanical power, was used to show the robustness of the suggested BS-JA controller in the improvement of system stability and voltage regulation. The performance and effectiveness of the BS-JA controller were compared with those of the BS-GA and PSS-GA controllers through a nonlinear dynamic response analysis. The obtained results in terms of system oscillation damping, peak overshoot ratio and static error showed that BS-JA greatly enhanced the damping characteristics of the system oscillations in comparison to BS-GA and PSS-GA. Moreover, the optimized GA-based PSS regulator provided the worst results, which proved the superiority of the backstepping controller in comparison with conventional regulators such as PSSs.

Thanks to its promising results, the proposed BS-JA can be extended to an interconnected power network including renewable energy resources.