Load Frequency Control of Renewable Energy Power Systems Based on Adaptive Global Fast Terminal Sliding Mode Control

Featured Application

This study investigates improved frequency control strategies for multi-area power systems, aiming to enhance stability and performance under varying load conditions.

Abstract

In this paper, the load frequency control (LFC) of multi-area power systems incorporating photovoltaic (PV) and energy storage systems (ESSs) is studied. First, the model of the LFC system encompassing PV and ESS is established. Then, a novel LFC scheme based on adaptive global fast terminal sliding mode control (AGFTSMC) is proposed. To make the system robust globally, an adaptive sliding mode control law and a new type of global fast terminal sliding mode surface containing a nonlinear time-varying function are designed. Moreover, by utilizing the improved Lyapunov function, the stability of the system is analyzed. Finally, two simulation experiments incorporating the two-area LFC system and IEEE 39-bus test power system are presented to validate the effectiveness of the proposed method. The simulation results show that adopting the AGFTSMC can significantly reduce steady-state error and stabilization time. This makes it a promising solution for maintaining frequency stability.

1. Introduction

To achieve these dual carbon goals, the large-scale development of renewable energy power systems has become a primary task [1]. However, due to variations in natural conditions, the variable power output from renewable energy power plants may adversely impact grid frequency stability [2,3]. When combined with the maximum power point tracking (MPPT) control in renewable energy systems, high renewable penetration scenarios exacerbate challenges of strong uncertainty and system fragility, leading to reduced inherent inertia of existing power systems [4]. Load disturbances may induce significant frequency deviations, potentially leading to system instability. The primary role of load frequency control (LFC) is to maintain real-time power balance between generation and load demand by regulating tie-line power and system frequency within permissible thresholds [5]. As such, LFC is critical for ensuring stable power system operation and preventing equipment malfunctions caused by frequency fluctuations [6].

To address stability challenges arising from equipment failures and load demand fluctuations, energy storage systems are proven effective in enhancing grid resilience and dynamic regulation capabilities. Compared to traditional frequency regulation methods, energy storage systems (ESSs) exhibit superior advantages. Notably, with the escalating environmental challenges posed by conventional energy sources [7], the penetration of renewable energy generation in power systems is continuously increasing. Consequently, this paper develops a coordinated LFC control framework incorporating photovoltaic (PV) and ESS, aiming to address the frequency regulation capacity gap in high-penetration renewable energy grids through PV-ESS dynamic complementarity mechanisms. As the core safeguard for stable system operation, this architecture effectively suppresses frequency exceedance while significantly reducing frequency regulation costs and carbon emission intensity, thereby providing a low-carbon stability solution for modern power systems [8].

However, relying solely on PV-ESS for frequency regulation often necessitates large-capacity storage devices, substantially increasing construction costs. Thus, PV-ESS should operate synergistically with other control strategies. Over past decades, numerous control methods have been proposed for frequency regulation. PID controllers were applied to a five-area interconnected power system to minimize area control error to zero [9]. Simulation results indicate that the PID controller exhibits sluggish response speeds, with excessive response time under load transients or frequency deviations. A hybrid optimization technique combining bacterial foraging optimization (BFO) [10,11] and particle swarm optimization (PSO) [12,13,14] was proposed to suppress frequency and tie-line power deviations [15]. And a rat swarm optimization method [16] and reinforcement learning [17] were proposed to improve the dynamic response of the system.

Nevertheless, as renewable penetration rates increase, the design of PID controller gains to achieve desired LFC performance becomes increasingly challenging. Consequently, advanced control strategies have emerged. Davidson, R.A. and Ushakumari, S. proposed a decentralized LFC for two-area regulated power systems using H-infinity loop shaping, but its applicability to non-regulated power systems remains unverified [18]. A fuzzy logic controller for electric vehicles (EV) and PV integrated frequency regulation was developed [19], yet its performance under uncertainty and nonlinear dynamics requires further validation. Distributed model predictive control (MPC) [20] was implemented to enhance coordinated control with reduced computational burdens inherent to centralized schemes, but linearized models were employed. Alhejji, A. introduced an L1 adaptive controller to robustly balance generation-load demand with rapid convergence to zero steady-state error [21], yet only validated on single-input-single-output (SISO) isolated systems with fixed non-reheat turbine parameters, omitting multi-input-multi-output (MIMO) scenarios and parameter adaptability analysis. A novel PID controller based on the honey badger algorithm (HBA) was designed to reduce frequency deviations significantly [22]. To address the limitations of LFC under diverse operating conditions, superior advanced control strategies must be developed to enhance dynamic performance and robustness.

Notably, among existing advanced control methods, sliding mode control (SMC) stands out for its inherent robustness against parametric uncertainties and external disturbances [23,24,25]. Furthermore, SMC achieves rapid transient response, accommodates nonlinear dynamics, and allows customization through diverse sliding surface designs for multi-objective optimization. Consequently, SMC has been widely adopted in nonlinear control domains such as aircraft and robotic systems, demonstrating superior performance. In power systems, SMC can be synergistically integrated with other control architectures to enhance grid stability. Mondai, R. and Rahman, M.M. synthesized a novel sliding mode fuzzy logic controller that significantly improves dynamic performance and stability in interconnected systems [26].

While conventional SMC is widely utilized due to its robustness and simplicity, its asymptotic convergence to equilibrium points limits transient performance. In contrast, terminal sliding mode controllers (TSMC) enable finite-time convergence, substantially accelerating settling speeds. This breakthrough has spurred extensive research. To mitigate chattering effects, An, B.; Wang, Y.; Liu, L. and Hou, Z. designed an intelligent TSMC based on PSO while maintaining disturbance rejection capabilities [27]. Fractional-order TSMC (FO-TSMC) achieved uniform voltage regulation under wide-ranging load variations and input voltage fluctuation [28]. Fast terminal sliding mode control (FTSMC) further enhances convergence rates near equilibrium by incorporating linear terms into sliding surfaces, outperforming conventional TSMC in dynamic response. And a novel global sliding mode control method was proposed to make the system robust globally in [29,30] and the Lyapunov stability theory was employed to analyze stability [31]. In [32], a learning-based load frequency control approach was proposed to balance generation costs and frequency stability. In [33], a novel algorithm was designed based on an event-triggered mechanism, which can cause the frequency deviation of the power system to converge to a bounded value. In [34], a novel approach based on multi-event collaborative triggering mechanism was proposed, improving system reliability and computational efficiency. And a model-free adaptive dynamic programming (ADP) algorithm was designed, utilizing measured data to learn optimal control gains in [35]. Error serves as a critical metric for evaluating algorithm performance [36] and a novel algorithm deep neural networks trained by the twin-delayed deep deterministic gradient reinforcement learning policy was designed, significantly reducing steady-state error in [37]. In [38], an optimized approach combining model order reduction techniques with teaching learning-based optimization was proposed, achieving a 56.8% reduction in integral square error compared to conventional methods. In [39], a novel algorithm was designed based on particle swarm optimisation and deep artificial neural network, significantly reducing mean square error and improving the response speed.

Based on the above analysis, extensive studies have been conducted on LFC for interconnected power systems with energy storage devices. However, these studies have not incorporated high renewable penetration scenarios. Furthermore, LFC methodologies based on PID controllers, advanced control strategies, and SMC with its extensions have been thoroughly investigated. In contrast, adaptive global fast terminal sliding mode control (AGFTSMC) exhibits exceptional robustness against system parametric uncertainties and external disturbances, while guaranteeing fast and precise finite-time convergence of system states to the equilibrium point.

Motivated by the above discussions, the contributions of this work are specified as follows:

• To investigate the impact of PV generation uncertainty on load frequency deviation, this paper establishes a two-area LFC model incorporating PV and ESS.
• A continuous control law without switching terms is proposed to suppress chattering. Furthermore, a novel global fast terminal sliding mode surface is designed by introducing nonlinear terms and a nonlinear time-varying function. This enhances the convergence rate toward equilibrium states and ensures global robustness of the system.
• Considering load demand variations, a novel AGFTSMC method is developed to enhance the robustness of power system LFC. This method guarantees rapid and precise finite-time convergence of system states to equilibrium; additionally, an adaptive sliding mode control law is introduced to dynamically suppress frequency variations induced by continuous random load disturbances.

2. Modeling of Load Frequency Control in New Energy Power Systems

The LFC system model of the i-th area is presented in Figure 1. The parameters of the i-th control area are listed in

Table 1. Explanation of symbols for the multi-area LFC system.

Symbol	Description
$\Delta P_{di}$	Load deviation
$\Delta P_{mi}$	Generator mechanical output deviation
$\Delta P_{vi}$	Valve position deviation
$\Delta P_{bi}$	Battery output power deviation
$\Delta P_{tie-i}$	Tie-line active power deviation
$\Delta P_{pvi}$	PV output deviation
$\Delta f_i$	Frequency deviation
$M_i$	Moment of inertia of the generator
$D_i$	Generator damping coefficient
$T_{gi}$	Time constant of the governor
$T_{chi}$	Time constant of the turbine
$R_i$	Speed drop
${\beta }_i$	Frequency bias factor
$T_{ij}$	Tie-line synchronizing coefficient
${\lambda }_{pi}$	PV gain factor
${\lambda }_{bi}$	ESS gain factor
$k_1$	ESS proportional factor
$k_2$	Turbine proportional factor
$k_3$	PV proportional factor

. The LFC system model of the multi-area power system can be described as shown in Equation (1) [30]:

$$\left\{\begin{array}{c}\dot{x}(t)=Ax(t)+Bu(t)+F\omega (t)\\ y(t)=Cx(t)\end{array}\right.$$

(1)

where the following is true:

$$\begin{gathered} x_i(t)={\left[\begin{array}{cccccccc}\Delta f_i& \Delta P_{mi}& \Delta P_{vi}& \Delta P_{tie-i}& \int AC{E}_i& \Delta P_{bi}& \Delta P_{pvi}& \Delta P_{pi}\end{array}\right]}^T, \\ x(t)={\left[\begin{array}{ccccc}{x}_1(t)& x_2(t)& x_3(t)& \dots & x_n(t)\end{array}\right]}^T,{\omega }_i(t)=\Delta P_{di}, \\ \omega (t)={\left[\begin{array}{ccccc}{{\omega }_1}^T(t)& {{\omega }_2}^T(t)& {{\omega }_3}^T(t)& \dots & {{\omega }_n}^T(t)\end{array}\right]}^T,y_i(t)={\left[\begin{array}{cc}\Delta f_i& {\displaystyle \int AC{E}_i}\end{array}\right]}^T \\ y(t)={\left[\begin{array}{ccccc}{y}_1(t)& y_2(t)& y_3(t)& \dots & y_n(t)\end{array}\right]}^T,u(t)={\left[\begin{array}{ccccc}{u_1}^T(t)& {u_2}^T(t)& {u_3}^T(t)& \dots & {u_n}^T(t)\end{array}\right]}^T, \\ A=\left[\begin{array}{ccc}{A}_{11}& \dots & A_{1n}\\ ⋮& \ddots & ⋮\\ A_{n1}& \dots & A_{nn}\end{array}\right],B={[\begin{array}{cccccccc}0& 0& {\displaystyle \frac{k_2}{T_{gi}}}& 0& 0& {\displaystyle \frac{k_1{\lambda }_{bi}}{T_{bi}}}& {\lambda }_{pi}{k}_3& {\lambda }_{pi}{k}_3\end{array}]}^T, \\ {A}_{ii}=\left[\begin{array}{cccccccc}-{\displaystyle \frac{D_i}{M_i}}& {\displaystyle \frac{1}{M_i}}& 0& -{\displaystyle \frac{1}{M_i}}& 0& {\displaystyle \frac{1}{M_i}}& {\displaystyle \frac{1}{M_i}}& 0\\ 0& -{\displaystyle \frac{1}{T_{chi}}}& {\displaystyle \frac{1}{T_{chi}}}& 0& 0& 0& 0& 0\\ -{\displaystyle \frac{1}{R_i{T}_{gi}}}& 0& -{\displaystyle \frac{1}{T_{gi}}}& 0& 0& 0& 0& 0\\ 2\pi {\displaystyle \sum _{j=1,j\ne i}^n{T}_{ij}}& 0& 0& 0& 0& 0& 0& 0\\ {\beta }_i& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -{\displaystyle \frac{1}{T_{bi}}}& 0& 0\\ 0& 0& 0& 0& 0& 0& -c& b-a\\ 0& 0& 0& 0& 0& 0& 0& -a\end{array}\right], \end{gathered}$$

$$\begin{gathered} A_{ij}=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 2\pi {\displaystyle \sum _{j=1,j\ne i}^n{T}_{ij}}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right], \\ C=\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\end{array}\right],F={\left[-{\displaystyle \frac{1}{M_i}}\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]}^T \end{gathered}$$

The output power of the governor is described as follows [40]:

$$\Delta {\dot{P}}_{vi}={\displaystyle \frac{k_2}{T_{gi}}}u(t)-{\displaystyle \frac{1}{T_{gi}{R}_i}}\Delta f_i-{\displaystyle \frac{1}{T_{gi}}}\Delta P_{vi}$$

(2)

The output power of the turbine is expressed as follows [40]:

$$\Delta {\dot{P}}_{mi}={\displaystyle \frac{1}{T_{chi}}}\Delta P_{vi}-{\displaystyle \frac{1}{T_{chi}}}\Delta P_{mi}$$

(3)

The transfer function of the PV system is described as follows [40]:

$$G_{pvi}(s)={\displaystyle \frac{{\lambda }_{pi}}{s+a}}\cdot {\displaystyle \frac{s+b}{s+c}}$$

(4)

The output power of the PV system is as follows [40]:

$$\Delta {\dot{P}}_{pi}={\lambda }_{pi}{k}_{3}u(t)-a_{1i}\Delta P_{pi}$$

(5)

$$\Delta {\dot{P}}_{pvi}=-{\lambda }_{pi}{k}_{3}u(t)-c_{1i}\Delta P_{pvi}+(b_{2i}-a_{1i})\Delta P_{pi}$$

(6)

The output power of the ESS system is as follows [40]:

$$\Delta {\dot{P}}_{bi}={\displaystyle \frac{k_1{\lambda }_{bi}}{T_{bi}}}u(t)-{\displaystyle \frac{1}{T_{bi}}}\Delta P_{bi}$$

(7)

For each control area, the area control error (ACE) is a significant parameter in power systems, encompassing frequency deviation and tie-line active power deviation. It is defined as follows [40]:

$$AC{E}_i={\beta }_i\Delta f_i+\Delta P_{tie-i}$$

(8)

3. Design of Global Fast Terminal Sliding Mode Control

The conventional fast terminal sliding surface is designed as follows:

$$s(t)={\dot{x}}_1+n{x}_1^{\nu }+l{x}_1$$

(9)

To enhance the global robustness of the system and accelerate its convergence to the steady state, a nonlinear time-varying function $g(t)$ is incorporated into Equation (9). Consequently, the improved global fast terminal sliding mode surface can be designed as follows:

$$s(t)={\dot{x}}_1+m{\displaystyle {\int }_0^t{x}_1(\tau )d}\tau +n{x}_1^{\nu }+l{x}_1+g(t)$$

(10)

where $m$, $n$, and $l$ are the controller parameters to be set; $m>0$, $n>0$, $l>0$, $\nu ={\displaystyle \frac{p}{q}}$, $p$, and $q$ are positive odd numbers. The function $g(t)$ meets the following conditions: (1) it possesses a first-order derivative; (2) when $t=0$, $s(t)=0$; (3) when $t\to \infty$, $g(t)\to 0$.

Given the above conditions, $g(t)$ is formulated as a monotonically decreasing exponential function, as shown in Equation (11):

$$g(t)=k{e}^{-\lambda t}$$

(11)

where $k$ and $\lambda$ are the controller parameters to be designed according to the sliding mode surface, $\lambda >0$.

The global fast terminal sliding mode control (GFTSMC) controller is designed as follows:

$$\begin{array}{cc}u(t)& =-{\alpha }_{1}s(t)-{\alpha }_2\mathrm{sgn}(s(t))-{\alpha }_3|s(t){|}^{\nu }\mathrm{sgn}(s(t))+Lx(t)\\ & +\left[\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)Q_1+\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)Q_2\right]\mathrm{sgn}(s(t))\\ & +k\lambda {\mathrm{e}}^{-kt}\end{array}$$

(12)

where ${\alpha }_1$, ${\alpha }_2$, and ${\alpha }_3$ are positive numbers; $L$ is a constant matrix; $\alpha >0$, $Q_1$, and $Q_2$ are constants; and $‖\omega (t)‖\le Q_1$, $‖\dot{\omega }(t)‖\le Q_2$.

The Lyapunov function can be expressed as follows:

$$V(t)={\displaystyle \frac{1}{2}}{s}^2(t)$$

(13)

Taking the derivative of $V(t)$, the result is given below:

$$\dot{V}(t)=s(t)\dot{s}(t)$$

(14)

The derivative of $s(t)$ can be written as follows:

$$\begin{array}{cc}\dot{s}(t)& ={\ddot{x}}_1+l{\dot{x}}_1+m{x}_1+n\nu x_1^{\nu -1}{\dot{x}}_1+k\lambda {\mathrm{e}}^{-kt}\\ & =-{\displaystyle \frac{D_i}{M_i}}{\dot{x}}_1+{\displaystyle \frac{1}{M_i}}{\dot{x}}_2-{\displaystyle \frac{1}{M_i}}{\dot{x}}_4\\ & +{\displaystyle \frac{1}{M_i}}{\dot{x}}_6+{\displaystyle \frac{1}{M_i}}{\dot{x}}_7-{\displaystyle \frac{1}{M_i}}\dot{\omega }(t)\\ & +l[-{\displaystyle \frac{D_i}{M_i}}{x}_1+{\displaystyle \frac{1}{M_i}}{x}_2-{\displaystyle \frac{1}{M_i}}{x}_4\\ & +{\displaystyle \frac{1}{M_i}}{x}_6+{\displaystyle \frac{1}{M_i}}{x}_7-{\displaystyle \frac{1}{M_i}}\omega (t)]\\ & +m{x}_1+n\nu x_1^{\nu -1}{\dot{x}}_1+k\lambda {\mathrm{e}}^{-kt}\end{array}$$

(15)

By integrating Equations (1), (12), (14) and (15), the following is obtained:

$$\begin{array}{cc}\dot{V}(t)& =s(t)\dot{s}(t)\\ & =s(t)[-{\alpha }_{1}s(t)-{\alpha }_2\mathrm{sgn}(s(t))-{\alpha }_3|s(t){|}^{\nu }\mathrm{sgn}(s(t))\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)\omega (t)+k\lambda {\mathrm{e}}^{-kt}-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)\dot{\omega }(t)\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)Q_1\mathrm{sgn}(s(t))-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)Q_2\mathrm{sgn}(s(t))\\ & -k\lambda {\mathrm{e}}^{-kt}]\\ & \le -{\alpha }_1{s}^2(t)-{\alpha }_2{s}^2(t)-{\alpha }_3|s(t){|}^{1+\nu }\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)Q_1\left|s(t)\right|-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)Q_2\left|s(t)\right|\\ & -\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)Q_1\left|s(t)\right|+\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)Q_2\left|s(t)\right|\\ & \le -{\alpha }_1{s}^2(t)-{\alpha }_2{s}^2(t)-{\alpha }_3|s(t){|}^{1+\nu }\\ & \le -({\alpha }_1+{\alpha }_2)s^2(t)-{\alpha }_3|s(t){|}^{1+\nu }<0\end{array}$$

(16)

Based on the above analysis, it can be concluded that the system is stable.

4. Design of Adaptive Global Fast Terminal Sliding Mode Control

To prove that the system based on AGFTSMC is stable, the following assumption is made.

Assumption 1.

The unknown load disturbances are bounded, setting $‖\omega (t)‖\le Q_1$.

To guarantee the stable operation of each area in an interconnected power system and mitigate the impact of unknown load disturbances, adaptive laws are designed to estimate the unknown upper bound:

$${\dot{\widehat{Q}}}_1=\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)s(t)$$

(17)

$${\dot{\widehat{Q}}}_2=\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)s(t)$$

(18)

The AGFTSMC controller can be designed as follows:

$$\begin{array}{cc}u(t)& =-{\alpha }_{1}s(t)-{\alpha }_2\mathrm{sgn}(s(t))-{\alpha }_3|s(t){|}^{\nu }\mathrm{sgn}(s(t))+Lx(t)\\ & +\left[\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_1+\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_2\right]\mathrm{sgn}(s(t))\\ & +k\lambda {\mathrm{e}}^{-kt}\end{array}$$

(19)

where ${\widehat{Q}}_1$ and ${\widehat{Q}}_2$ are the estimate of $Q_1$ and $Q_2$, respectively, and ${\tilde{Q}}_1=Q_1-{\widehat{Q}}_1$ and ${\tilde{Q}}_2=Q_2-{\widehat{Q}}_2$ are estimation errors.

The Lyapunov function is developed as follows:

$$V(t)={\displaystyle \frac{1}{2}}{s}^2(t)+{\displaystyle \frac{1}{2}}{\tilde{Q}}_1^2+{\displaystyle \frac{1}{2}}{\tilde{Q}}_2^2$$

(20)

By deriving Equation (20), the following is obtained:

$$\begin{array}{cc}\dot{V}(t)& =s(t)[-{\alpha }_{1}s(t)-{\alpha }_2\mathrm{sgn}(s(t))-{\alpha }_3|s(t){|}^{\nu }\mathrm{sgn}(s(t))\\ & +k\lambda {\mathrm{e}}^{-kt}+\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)\omega (t)-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)\dot{\omega }(t)\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_1\mathrm{sgn}(s(t))-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_2\mathrm{sgn}(s(t))\\ & -k\lambda {\mathrm{e}}^{-kt}]-{\tilde{Q}}_1{\dot{\tilde{Q}}}_1-{\tilde{Q}}_2{\dot{\tilde{Q}}}_2\\ & \le -{\alpha }_1{s}^2(t)-{\alpha }_2{s}^2(t)-{\alpha }_3|s(t){|}^{1+\nu }\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)\left({\tilde{Q}}_1+{\widehat{Q}}_1\right)s(t)-\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_1\left|\left.s(t)\right|\right.\\ & +\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)\left({\tilde{Q}}_2+{\widehat{Q}}_2\right)s(t)-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_2\left|\left.s(t)\right|\right.\\ & -{\tilde{Q}}_1{\dot{\tilde{Q}}}_1-{\tilde{Q}}_2{\dot{\tilde{Q}}}_2\\ & \le -({\alpha }_1+{\alpha }_2)s^2(t)-{\alpha }_3|s(t){|}^{1+\nu }\\ & +\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\tilde{Q}}_{1}s(t)+\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_1\left|\left.s(t)\right|\right.\\ & -\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_1\left|\left.s(t)\right|\right.+\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\tilde{Q}}_{2}s(t)\\ & +\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_2\left|\left.s(t)\right|\right.-\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right){\widehat{Q}}_2\left|\left.s(t)\right|\right.\\ & -{\tilde{Q}}_1{\dot{\tilde{Q}}}_1-{\tilde{Q}}_2{\dot{\tilde{Q}}}_2\\ & \le -({\alpha }_1+{\alpha }_2)s^2(t)-{\alpha }_3|s(t){|}^{1+\nu }\\ & +\left[-{\dot{\widehat{M}}}_1+\left({\displaystyle \frac{D_i}{M_i}}+{\displaystyle \frac{1}{M_i}}\right)s(t)\right]{\tilde{Q}}_1\\ & +\left[-{\dot{\widehat{M}}}_2+\left({\displaystyle \frac{\alpha }{M_i^2}}+{\displaystyle \frac{1}{M_i}}\right)s(t)\right]{\tilde{Q}}_2\end{array}$$

(21)

By integrating Equations (17) and (18), Equation (22) can be obtained:

$$\dot{V}(t)\le -({\alpha }_1+{\alpha }_2)s^2(t)-{\alpha }_3|s(t){|}^{1+\nu }<0$$

(22)

Therefore, the preceding analysis demonstrates that the system can remain stable under the action of the proposed controller.

5. Simulation and Analysis

To validate the performance of the LFC system with AGFTSMC, the model of a two-area LFC system with AGFTSMC, which incorporates an ESS and PV generation, is established using the MATLAB/Simulink R2022a toolbox. For the above LFC system model under load disturbances, the frequency deviations are analyzed based on their response curves. To further demonstrate the effectiveness of the proposed method, the results are compared with other controllers, such as the PID controller, SMC controller, and GFTSMC controller.

5.1. Case Study 1

The model of the two-area LFC system with an ESS and PV generation is shown in Figure 2, and the parameters of the two-area power system are listed in

Table 2. Parameters of the two-area power system model.

Parameters	Area1	Area2	Parameters	Area1	Area2
$T_{gi}$	0.08	0.09	$k_1$	0.3	0.4
$T_{chi}$	0.35	0.4	$k_2$	0.6	0.5
$D_i$	0.02	0.01	$k_3$	0.1	0.1
$M_i$	0.3	0.3	${\lambda }_{pi}$	120	120
${\beta }_i$	0.4	0.4	$a$	99.5	98
$R_i$	5	5	$b$	−50	−50
$T_{ij}$	0.2	0.22	$c$	0.5	0.5
$T_{bi}$	1	1	${\lambda }_{bi}$	10	15

.

In this subsection, the model of the two-area LFC system with an ESS and PV generation is taken as an example to examine the impact of large frequency perturbations and continuous random disturbances on the performance of the LFC system. First, the large frequency perturbation of area 1 and area 2 is set to 0.016 per unit (p.u) at $t=5\mathrm{s}$. Furthermore, amplitude-limited continuous random disturbances of area 1 and area 2, with the upper limit set to 0.001 p.u, are added, occurring at $t=5\mathrm{s}~15\mathrm{s}$. The simulation runs for $80\mathrm{s}$, providing sufficient time to analyze the frequency response.

To demonstrate the effectiveness of the proposed method under disturbances of different magnitudes, the large frequency disturbance of area 1 and area 2 is set to 0.02 p.u at $t=5\mathrm{s}$. Moreover, continuous random disturbances of area 1 and area 2, with the upper limit set to 0.002 p.u, are introduced, occurring at $t=5\mathrm{s}~15\mathrm{s}$. The simulation results are depicted in Figure 3a–d. Key performance metrics derived from simulations are compared in

Table 3. Comparison of simulation results when the large frequency disturbance is set to 0.016 p.u.

Method	Area 1		Area 2
	Steady-State Error (Hz)	Response Time (s)	Steady-State Error (Hz)	Response Time (s)
PID	$7.56\times {10}^{-4}$	47.75	$7.34\times {10}^{-4}$	45.62
SMC	$5.18\times {10}^{-4}$	42.48	$4.96\times {10}^{-4}$	39.57
GFTSMC	$2.47\times {10}^{-4}$	22.81	$2.16\times {10}^{-4}$	21.14
AGFTSMC	$9.52\times {10}^{-5}$	16.48	$7.88\times {10}^{-5}$	13.36

and

Table 4. Comparison of simulation results when the large frequency disturbance is set to 0.02 p.u.

Method	Area 1		Area 2
	Steady-State Error (Hz)	Response Time (s)	Steady-State Error (Hz)	Response Time (s)
PID	$9.12\times {10}^{-4}$	48.36	$8.79\times {10}^{-4}$	47.18
SMC	$5.92\times {10}^{-4}$	42.71	$5.72\times {10}^{-4}$	40.07
GFTSMC	$3.58\times {10}^{-4}$	23.93	$3.21\times {10}^{-4}$	23.41
AGFTSMC	$1.82\times {10}^{-4}$	17.50	$1.63\times {10}^{-4}$	14.03

.

As can be seen from Figure 3, the maximum frequency deviations of the system based on PID and SMC are about −0.323 Hz and −0.276 Hz, respectively, and the response times are about 47.75 s and 42.48 s, respectively. The response time of the system based on GFTSMC is roughly 22.81 s and is relatively short compared with the preceding two control schemes. The response time of the system based on AGFTSMC is approximately 16.48 s and is the shortest among these four methods; it is about 66.42% shorter than that of the system based on PID. From Figure 3b, it can be observed that the maximum frequency deviation of the system based on AGFTSMC is 37.78% smaller than that of the system based on PID, which demonstrates notable superiority in terms of overshoot.

It can be seen from Figure 3a,b and Table 3, the response speed of the system adopting the PID method is the slowest under disturbances. The maximum frequency deviation and steady-state error of the system adopting the PID method are the largest. Due to the fixed gains, PID controller lacks real-time adaptability to handle uncertain load frequency fluctuations, resulting in inferior performance. Adopting the SMC method can suppress the frequency fluctuation of the system, but the response time of the system is relatively long, while adopting the GFTSMC method can significantly shorten the system response time. Compared with the preceding three control schemes, adopting the AGFTSMC approach significantly reduces the overshoot and steady-state error, while the response time decreases dramatically, demonstrating the superiority of this method.

As shown from Figure 3d, by adopting the AGFTSMC approach, the stabilization time dramatically decreases to roughly 14.03 s and it is about 68.87% shorter than that of the system based on PID, which demonstrates a significant advantage in terms of stabilization time. Therefore, the system adopting the AGFTSMC method demonstrates the strongest ability to restore stability when dealing with disturbances, reflecting the fastest response speed.

Compared with PID, SMC, and GFTSMC, adopting the AGFTSMC approach significantly reduces the overshoot and stabilization time. Under the control of AGFTSMC, two-area power system shows excellent control performance under large frequency perturbations and continuous random disturbances. Therefore, from the above analysis, it can be concluded that the AGFTSMC method can be applied to the load frequency control of complex two-area power systems.

5.2. Case Study 2

To demonstrate the frequency modulation effect of the ESS, the simulation is performed with AGFTSMC in two cases: with and without ESS in the system. First, the large frequency disturbance of area 1 and area 2 is set to 0.02 p.u at $t=5\mathrm{s}$. Moreover, continuous random disturbances of area 1 and area 2, with the upper limit set to 0.002 p.u, are introduced, occurring at $t=5\mathrm{s}~15\mathrm{s}$. The simulation results are presented in Figure 4a,b. A quantitative comparison of the simulation results is depicted in

Table 5. Comparison of simulation results.

Method	Area 1		Area 2
	Maximum Value of Frequency Deviation (Hz)	Response Time (s)	Maximum Value of Frequency Deviation (Hz)	Response Time (s)
Without ESS	0.391	25.64	0.371	22.48
With ESS	0.297	17.50	0.283	14.03

.

From Figure 4a,b and Table 5, compared with the case without ESS in the system, the maximum frequency deviation of the system with ESS is −0.297 Hz, which is 37.78% smaller than that of the system without ESS. The stabilization time of the system with ESS is roughly 17.50 s, about 31.75% shorter than that of the system without ESS. As shown from Figure 4a,b, the maximum frequency deviation and the response time of the system can be reduced significantly when ESS is applied. Thus, the dynamic compensation capability of ESS is verified.

5.3. Case Study 3

To verify the effectiveness of the proposed AGFTSMC method in a more realistic power system model, IEEE 39-bus system model is established. This model consists of 10 generators, 19 loads, 34 transmission lines, and 12 transformers. Gen 4 is the selected generator in Area 1, Gen 8 is the chosen generator in Area 2, and Gen 10 is the selected generator in Area 3. The schematic diagram of the IEEE 39-bus test system and the division of the control regions are shown in Figure 5.

At t = 5 s, a 0.032 p.u, disturbance is added to bus 8 in area 1; at t = 35 s, a 0.072 p.u, disturbance is added to bus 16 in area 2; and at t = 70 s, a 0.054 p.u disturbance is added to bus 3 in area 3. The simulation runs for $100\mathrm{s}$. The frequency deviations and tie-line power deviations of the system in the three areas are shown in Figure 6.

From Figure 6a,b, it can be observed that when a 0.032 p.u disturbance is added to bus 8 in area 1, the maximum frequency deviation of the system in area 1 is approximately $2.527\times {10}^{-5}\mathrm{H}\mathrm{z}$, making the response time about 10.48 s. Additionally, the maximum tie-line power deviation of the system between area 1 and area 2 is about $58.16\mathrm{W}$. At t = 35 s, when a 0.072 p.u perturbation is added to bus 16 in area 2, the maximum frequency deviation of the system in area 2 is roughly $6.606\times {10}^{-5}\mathrm{H}\mathrm{z}$, making the response time about 12.82 s. Additionally, the maximum tie-line power deviation of the system between area 1 and area 3 is roughly $62.18\mathrm{W}$. When a 0.054 p.u disturbance is added to bus 3 in area 3, the maximum frequency deviation of the system in area 3 is about $3.914\times {10}^{-5}\mathrm{H}\mathrm{z}$, making the response time about 11.32 s. In addition, the maximum tie-line power deviation of the system between area 1 and area 3 is approximately $40.18\mathrm{W}$. It can be seen from Figure 6a,b that adopting the proposed AGFTSMC approach can cope with load disturbances effectively, which makes frequency deviation and tie-line power deviation of the system converge to zero in a short time. The proposed control method has the characteristics of fast response and low overshoot, demonstrating superior control performance.

6. Conclusions

In this paper, an LFC system model that incorporates ESS and PV is established and a novel method based on AGFTSMC is proposed. The proposed approach includes a nonlinear time-varying function and adaptive sliding mode control laws, which are designed to cope dynamically with frequency variations induced by continuous random load disturbances. By utilizing an enhanced Lyapunov function, the stability of the system is analyzed. Furthermore, simulation of a two-area LFC system is conducted to verify the proposed approach. Compared with PID, SMC, and GFTSMC, adopting the proposed method can significantly reduce overshoot and shorten the response time, demonstrating superior performance in terms of both overshoot and response speed. Finally, the effectiveness of the proposed AGFTSMC scheme in a realistic power system is validated using the IEEE 39-bus test system.

Despite the demonstrated effectiveness of the AGFTSMC method, this approach still faces limitations, including substantial computational requirements. Nevertheless, the method still holds numerous promising avenues for future development. Integrating event-triggered schemes and artificial intelligence, such as deep learning, holds great potential to significantly enhance dynamic frequency regulation capabilities.