With the increasingly severe global energy crisis and the sharp decline in non-renewable resources, new energy generation will become a major trend of power system (Tan et al., 2023). The reliability of power systems that integrate wind, solar photovoltaic (SPV) and battery energy storage (BES) is also improved (Kundu et al., 2021; Kundu et al., 2023). However, the output uncertainty of new energy generation is large, bringing challenges to the frequency stability of power system (Wang et al., 2022a; Zhang et al., 2023a; Ojha and Maddela, 2023). When the frequency of the power system fluctuates prominently, the equipment of power system is likely to fail to work properly. In severe cases, the power system may collapse, triggering large-scale power outages. In order to avoid the above circumstances, it is particularly vital to maintain the frequency stability of the power system. And LFC is a vital method to confine the frequency deviations of power system within certain pre-specified limits (Ranjitha et al., 2022; Khokhar and Parmar, 2023; Singh and Ramesh, 2024). Therefore, LFC is critical for ensuring the stable operation of power system, which is of great significance and research value for further study.

In order to control the load frequency, numerous researchers and scholars have conducted in-depth studies on LFC, and a number of LFC method strategies have been proposed, such as proportion integration differentiation (PID) control, predictive control, fuzzy control, and so on. In Veerendar et al. (2023), a dual-loop control method was proposed by using teaching-learning optimization (TLO) for the LFC of a multi-area non-heating thermal power system (NRTPS), which prominently improved the response of the system. In Sharma et al. (2022), a proportion integration (PI) controller was designed to determine the parametric uncertainty margin (PUM) to cope with problems caused by the changes of parameters. From Tang et al. (2023), a model predictive control (MPC) technique was presented, and a state feedback MPC controller was designed to deal with wind interference, communication delay, and denial-of-service (DoS) attacks. In Naderipour et al. (2023), based on the fuzzy logic, through a novel meta-heuristic whale algorithm, a self-tuning controller was designed to enable the proposed controller to operate better. In Hasen et al. (2023), a frequency-domain exact method was presented to increase the LFC system stability delay margin. For active/reactive power (AP/RP) regulation and energy storage management of an independent microgrid (MG), a new type-2 fuzzy logic control (T2FLC) was proposed to reduce voltage oscillation created by variation of output power in Mohammadi Moghadam et al. (2022). From Li and Ye (2022), for nonlinear interconnected power systems under a switching topology, a new event-based distributed fuzzy LFC approach was presented to ensure the asymptotic stability of the system. In Zhang et al. (2023b), for large-scale wind farm control, through introducing a multivariate power model, a communication based reinforcement learning method was presented to achieve satisfactory output power and stable performance. Although the above approaches enhance the performance of LFC, they are more demanding in terms of system and more difficult to implement physically. Besides, they are not robust to system uncertainties. Compared with these methods, sliding mode control (SMC) boasts numerous advantages. Firstly, the stability of it is high. When facing external disturbances in power system, appropriate control laws can be chosen to cope with the external disturbances, thus keeping the system frequency within the expected range. Secondly, the response is swift, enabling quick restoration of power system frequency through state transitions on the sliding mode surface. In addition, its physical implementation is simple and does not need a high standard of structure for the system, which can be applied in practice in a short time. So, the SMC is a promising method for LFC and deservesfurther study.

Therefore, an increasing number of researchers and scholars have studied LFC methods based on SMC. For example, an area-based event-triggered (ET) sliding mode control scheme was presented to suppress rapid fluctuations caused by load and wind power generation in Xu et al. (2022). From Guo (2021), a novel sliding mode control approach was presented to reduce the frequency deviation. In Yang et al. (2021), a generalized extended state observer and fractional-order theory based integral sliding mode control strategy were presented to relieve the chattering of frequency deviation and tie-line power deviation. For multiarea interconnected power systems under deception attack, an observer-based event-triggered transmission scheme was proposed to ensure the attacked power system can reach a stable position in Qiao et al. (2021). In Ansari et al. (2023), for two-area thermal interconnected power system, a novel sliding-mode LFC strategy was designed to achieve better tracking performance. From Ge et al. (2021), for a microgrid with hybrid energy storage system (HESS), based on sliding mode method, a frequency coordinated control strategy was designed to avoid unreasonable power output. In Deng and Xu (2022), for multi-area interconnected power systems integrated with wind farms, a derivative and integral terminal sliding-mode-based controller was proposed to eliminate the frequency deviation in each area. However, there are still some drawbacks to these methods. For instance, the variation of some parameters in power system may lead to the degradation of its performance and the phenomenon of oscillation. For the problem of system parameter variation, more scholars have studied the adaptive controller (Mazinan, 2013; Li et al., 2019; Zou, 2020; Wang et al., 2022b).

In order to solve the above problems better, LFC based on adaptive global sliding mode control (AGSMC) is studied in this article. Through this approach, the advantages of adaptive control, global control, and SMC are combined, significantly improving the control capability of LFC. When the external load disturbances occur or the parameters of rhe system change, through this method, the parameters of controller can be adjusted to adapt to the changes of external disturbance. And the specified characteristics can be maintained under certain conditions. This method demonstrates excellent anti-disturbance ability, strong robustness, rapid response, and high fault tolerance of the model. Therefore, this method can play a crucial role in achieving better control performance, which is worthy of in-depth study.

In this section, an improved Lyapunov function and the Bessel-Legendre (B-L) inequality are applied to study the stability criteria for the multi-area LFC.

Lemma 1 (Lu et al., 2021): For an integer N > 0, and there exists [α, β] → R ⁿ , where α, β are given real scalars with α < β. For any symmetric matrices R > 0, the following Eq. 19 holds: ∫ α β υ ̇ T s R υ ̇ s d s ≥ 1 β − α υ ̃ N T ς N T Ψ N T R ̃ N Ψ N ς N υ ̃ N (19) where: ς N = I − I 0 0 ⋯ 0 0 − I I 0 ⋯ 0 0 − I 0 2 I ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ 0 − I 0 0 ⋯ N I Ψ N = I 0 0 ⋯ 0 − 1 1 I − 1 1 q 1 1 I 0 ⋯ 0 − 1 2 I − 1 2 q 1 2 I − 1 2 q 2 2 I ⋯ 0 ⋮ ⋮ ⋮ ⋱ ⋮ − 1 N I − 1 N q 1 N I − 1 N q 2 N I ⋯ − 1 N q N N I R ̃ N = d i a g R , 3 R , ⋯ , 2 N + 1 R υ ̃ N = c o l υ β , υ α , ε 1 , ⋯ , ε N , ε i = 1 β − α i ∫ α β β − s i − 1 υ s d s , i = 1,2 , ⋯ , N q j i = − 1 j i j i + j j , i j = i ! i − j ! j ! Lemma 2: For a given positive matrix R > 0, and differentiable function {φ(u)|u ∈ [a, b]}, the following Eq. 20 and Eq. 21 can be hold as: ∫ a b ∫ β b φ ̇ T α R φ ̇ α d α d β ≥ 2 Ω 1 T R Ω 1 + 4 Ω 2 T R Ω 2 (20) ∫ a b ∫ a β φ ̇ T α R φ ̇ α d α d β ≥ 2 Ω 3 T R Ω 3 + 4 Ω 4 T R Ω 4 (21) where: Ω 1 = φ b − 1 b − a ∫ a b φ α d α Ω 2 = φ b + 2 b − a ∫ a b φ α d α − 6 b − a 2 ∫ a b ∫ β b φ α d α d β Ω 3 = φ a − 1 b − a ∫ a b φ α d α Ω 4 = φ a − 4 b − a ∫ a b φ α d α + 6 b − a 2 ∫ a b ∫ β b φ α d α d β Lemma 3: For a real scalar α ∈ (0, 1), symmetric matrices γ ₁ > 0, γ ₂ > 0 and arbitrary matrices ϑ ₁ > 0, ϑ ₂ > 0, the following matrix inequality holds: 1 α γ 1 0 0 1 1 − α γ 2 ≥ γ 1 + 1 − α Γ 1 1 − α ϑ 1 + α ϑ 2 ∗ γ 2 + α Γ 2 (22) where: Γ 1 = γ 1 − ϑ 2 γ 2 − 1 ϑ 2 , Γ 2 = γ 2 − ϑ 1 γ 1 − 1 ϑ 1

Theorem 2: For given positive integers N ∈ {1, 2, 3, ⋯ }, scalars d ₂ > d ₁ > 0, disturbance attenuation level γ > 0, the closed-loop system is asymptotically stable under the event-triggering scheme. If there exist positive definite matrices P, Q, R, S and real matrices S _N , U _N , such the following inequalities holds: Θ d t = d 1 | α = 0 = ∑ k = 1 5 Φ k + Π 3 T Π 3 Ψ 1 N T U N ∗ − R 1 N + R 4 N < 0 Θ d t = d m | α = 1 = ∑ k = 1 5 Φ k + Π 3 T Π 3 Ψ 2 N T S N T ∗ − R 1 N + R 3 N < 0 (23) where: Π 1 = e 1 υ 1 e 1 T − e 2 Q 2 e 2 T − e 3 υ 3 e 3 T Π 2 = ∑ k = 1 4 Φ k + 1 − α Ψ 1 N T U N R 1 N + R 4 N − 1 U N T Ψ 1 N + α Ψ 2 N T S N T R 1 N + R 3 N − 1 S N Ψ 2 N Π 3 = C e 1 υ 1 = Q 2 + Q 3 + d m R 2 υ 2 = d m 2 R 1 + d m 2 2 S 1 + d m 2 2 S 2 υ 3 = 1 − d ̇ t Q 3 − λ t k h Ω Ψ 1 N = Ψ N ς N c o l e 1 , e 3 , H 2 e 9 , … , H 2 e N + 8 Ψ 2 N = Ψ N ς N c o l e 3 , e 2 , H 3 e 9 , … , H 3 e N + 8 R 1 N = d i a g R 1 , 3 R 1 , … , 2 N + 1 R 1 R 3 N = d i a g S 1 , 3 S 1 , … , 2 N + 1 S 1 R 4 N = d i a g S 2 , 3 S 2 , … , 2 N + 1 S 2 h 3 = e 1 − e 4 , e 1 + 2 e 4 − 3 e 6 , e 3 − e 5 , e 3 + 2 e 5 − 3 e 7 φ 3 = d i a g − 2 S 1 , − 4 S 1 , − 2 S 1 , − 4 S 1 h 4 = e 3 − e 4 , e 3 − 4 e 4 + 3 e 6 , e 2 − e 5 , e 2 − 4 e 5 + 3 e 7 φ 4 = d i a g − 2 S 2 , − 4 S 2 , − 2 S 2 , − 4 S 2 Φ 1 = − 2 − α Ψ 1 N T R 1 N + R 3 N Ψ 1 N − 1 + α Ψ 2 N T × R 1 N + R 4 N Ψ 2 N − S y m Ψ 1 N T 1 − α S N + α U N Ψ 2 N + Ψ 1 N T R 3 N Ψ 1 N + Ψ 2 N T R 4 N Ψ 2 N Φ 2 = − h 3 φ 3 h 3 T , Φ 3 = − h 4 φ 4 h 4 T Φ 4 = Π 1 + 2 P χ 1 + χ 1 υ 2 χ 1 T Φ 5 = − γ 2 e 9 T e 9 χ 1 = A e 1 + B K C e 3 Proof: Defining the Lyapunov function as: V t = V 1 t + V 2 t + V 3 t + V 4 t (24) where: V 1 t = x T t P x t V 2 t = ∫ t − τ M t x T s Q 2 x s d s + ∫ t − d t t x T s Q 3 x s d s V 3 t = d M ∫ − d M 0 ∫ t + α t x ̇ T s R 1 x ̇ s d s d α + ∫ − d M 0 ∫ t + α t x T s R 2 x s d s d α V 4 t = ∫ − d M 0 ∫ β 0 ∫ t + α t x ̇ T s S 1 x ̇ s d s d α d β + ∫ − d M 0 ∫ − d M β ∫ t + α t x ̇ T s S 2 x ̇ s d s d α d β P , Q , R , S a r e p o s i t i v e d e f i n i t e m a t r i c e s .

Calculating the derivation of V(t) obtains the following Eq. 25: Δ V t = Δ V 1 t + Δ V 2 t + Δ V 3 t + Δ V 4 t (25) where: Δ V 1 t = 2 x ̇ T t P x t Δ V 2 t = x T t Q 2 x t − x T t − d M Q 2 x t − d M + x T t × Q 3 x t − 1 − d ̇ t x T t − d t Q 3 x t − d t Δ V 3 t = d M 2 x ̇ T t R 1 x ̇ t − d M ∫ t − d M t x ̇ T α R 1 x ̇ α d α + d M x T t R 2 x t − ∫ t − d t t x T α R 2 x α d α Δ V 4 t = d M 2 2 x ̇ T t S 1 x ̇ t − ∫ − d t 0 ∫ t + α t x ̇ T s S 1 x ̇ s d s d α − ∫ − d M − d t ∫ t + α t − d t x ̇ T s S 1 x ̇ s d s d α − d M − d t ∫ t − d t t x ̇ T α S 1 x ̇ α d α + d M 2 2 x ̇ T t × S 2 x ̇ t − ∫ − d t 0 ∫ t − d t t + α x ̇ T s S 2 x ̇ s d s d α − ∫ − d M − d t ∫ t − d M t + α x ̇ T s S 2 x ̇ s d s d α − d t ∫ t − d M t − d t x ̇ T s S 2 x ̇ s d s

In Eq. 26, we define the following augmenting state variable ξ(t): ξ t = c o l x t , e t , x t − d m , x t − d t , 1 d t ∫ t − d t t x α d α , 1 d m − d t ∫ t − d m t − d t x α d α , 2 d t 2 ∫ − d t 0 ∫ t + β t x α d α d β , 2 d m − d t 2 ∫ − d m − d t × ∫ t + β t − d t x α d α d β , w t , ψ 1 t , … , ψ N t ψ i t = c o l ∫ t − d 1 t t − s i − 1 d 1 i x s d s , ∫ t − d t t − d 1 t − d 1 − s i − 1 d t − d 1 i × x s d s , ∫ t − d 2 t − d t t − d t − s i − 1 d 2 − d t i x s d s , i = 1,2 , … , N θ i t = ∫ t − d 2 t − d 1 t − d 1 − s i − 1 d 12 i x s d s (26)

Thus, the following inequality can be yielded: Δ V t ≤ ξ T t Π 1 ξ t + 2 x ̇ t P x T t + x ̇ t υ 2 x ̇ T t + Δ V ̃ 1 t + Δ V ̃ 2 t + Δ V ̃ 3 t (27) where: Δ V ̃ 1 t = − d m ∫ t − d m t x ̇ s R 1 x ̇ T s d s − d m − d t × ∫ t − d t t x ̇ s S 1 x ̇ T s d s − d ̇ t ∫ t − d m t − d t x ̇ s S 2 x ̇ T s d s Δ V ̃ 2 t = − ∫ − d t 0 ∫ t + α t x ̇ s S 1 x ̇ T s d s d α − ∫ − d m − d t ∫ t + α t − d t x ̇ s S 1 x ̇ T s d s d α Δ V ̃ 3 t = − ∫ − d t 0 ∫ t − d t t + α x ̇ s S 2 x ̇ T s d s d α − ∫ − d m − d t ∫ t − d m t + α x ̇ T s S 2 x ̇ s d s d α where: Δ V ̃ 1 ( t ) can be rewritten. The item − d m ∫ t − d m t x ̇ ( s ) R 1 x ̇ T ( s ) d s in the Equation will be decomposed into the following form: − d m ∫ t − d m t x ̇ s R 1 x ̇ T s d s = − d m ∫ t − d t t x ̇ s R 1 x ̇ T s d s − d m ∫ t − d m t − d t x ̇ s R 1 x ̇ T s d s (28)

Applying Lemma 1 and assuming α = d ( t ) d m , β = d m − d ( t ) d m = 1 − α , then Eq. 28 is less than the following Equation: − 1 α ξ T t Ψ 1 N T R 1 N Ψ 1 N ξ t − 1 1 − α ξ T t Ψ 2 N T R 1 N Ψ 2 N ξ t (29)

Rewrite the relevant items in Eq. 29 into the following form: − Ψ 1 N Ψ 2 N T M 1 Ψ 1 N Ψ 2 N ≤ − Ψ 1 N Ψ 2 N T M 2 Ψ 1 N Ψ 2 N + 1 − α Ψ 1 N T × U N R 1 N − 1 U N T Ψ 1 N + α Ψ 2 N T S N T R 1 N − 1 × S N Ψ 2 N (30) where: M 1 = 1 α R 1 N 0 0 1 1 − α R 1 N M 2 = 2 − α R 1 N 1 − α S N + α U N * 1 + α R 1 N Applying Lemma 1 to the other two terms of Δ V ̃ 1 ( t ) yields the following inequality: − d m − d t ∫ t − d t t x ̇ s S 1 x ̇ T s d s ≤ 1 − 1 α ξ T t Ψ 1 N T R 3 N Ψ 1 N ξ t , − d t ∫ t − d m t − d t x ̇ s S 2 x ̇ T s d s ≤ 1 − 1 β ξ T t Ψ 2 N T R 4 N Ψ 2 N ξ t (31)