Data-driven dual-channel dynamic event-triggered load frequency control for multiarea power systems with uniform quantizer

Abstract

Accurate and efficient load frequency control in interconnected multiarea power systems is becoming increasingly challenging due to data quantization effects and communication resource limitations. To overcome these limitations, this paper proposes a dual-channel dynamic event-triggered, data-driven load frequency control strategy based on model-free adaptive control. The controller integrates proportional, differential, and quadratic difference terms to enhance tracking accuracy, while independently designed event-triggering mechanisms for both input and output channels significantly reduce communication and computational burdens. Furthermore, a novel encoding–decoding scheme is designed to mitigate data quantization. Rigorous theoretical analysis confirms that the proposed scheme relies solely on input–output data and ensures asymptotic tracking performance. Extensive simulation results validate the effectiveness and feasibility of the proposed quantized control strategy.

Introduction

The frequency of electrical loads is a critical indicator of power system stability, where deviations can lead to severe consequences such as equipment damage and even large-scale failures. ¹ To mitigate these risks, load frequency control (LFC) has emerged as a central mechanism for maintaining reliable and efficient power system operation amid load fluctuations. ² Numerous model-based LFC approaches have been developed, including fuzzy control,^3–5 model predictive control,^6–9 and sliding mode control.^10–12 These methods are effective when power system dynamics are relatively simple and can be accurately modeled. However, with the growing complexity and nonlinearity of modern power systems due to advancements in the energy field, the effectiveness of model-based techniques becomes limited. ¹³ Consequently, the development of model-free LFC strategies has become increasingly important. ¹⁴

In recent years, data-driven control—which relies solely on system input and output data for controller design—has garnered significant attention.^15–17 Based on the mode of data utilization, data-driven control can be broadly categorized into two main approaches: (1) methods that primarily use online data, such as model-free adaptive control (MFAC)^18–21; and (2) methods that rely on offline data, including virtual reference feedback tuning^22,23 and iterative feedback tuning.^24,25 Among these, MFAC is a prominent online method that employs pseudo-partial derivatives (PPD) to construct a dynamic linearization model between the input and output data of nonlinear systems. Compared to offline methods, MFAC features a simpler structure and requires fewer parameters. ²⁶ However, the MFAC controllers in the aforementioned studies rely exclusively on the current system state, neglecting the differential and quadratic difference information contained in historical data, which reflects short-term dynamics and higher-order output variations. This limitation motivates the development of an enhanced MFAC controller that fully leverages such information to improve control performance.

With the advancement of smart grids, the deployment of remote terminal units has become increasingly widespread. ²⁷ In recent years, the issue of limited communication resources in power networks has garnered considerable scholarly attention. ²⁸ Event-triggered mechanisms have been extensively studied in networked control systems due to their ability to transmit signals only when predefined conditions are met, thereby reducing unnecessary data transmission.^29–31 Building on this concept, Chen et al. ³² proposed a dynamic event-triggered mechanism that adaptively adjusts triggering conditions based on the magnitude of the control error, further minimizing communication frequency. Compared to static and switching event-triggered mechanisms, dynamic event-triggered approaches provide enhanced control performance and improved resource utilization. Nevertheless, most existing and above studies predominantly emphasize the input channel while overlooking the role of the output channel. Consequently, developing a dynamic event-triggered MFAC strategy that simultaneously addresses both input and output channels in power systems remains a critical yet unresolved challenge.

On the other hand, measurement data must be quantized before being transmitted through power networks, which reduces the volume of transmitted data but introduces quantization errors between the actual signals and their quantized representations. ³³ These quantization errors can degrade control performance and, in some cases, compromise the stability of the control system. To date, two primary quantization schemes have been widely studied: static quantization^34,35 and dynamic quantization.^36,37 Feng et al. ³⁸ investigated the influence of quantized communication data rates on control performance and found that the tracking error converges to a finite bound rather than zero. Motivated by this limitation, we propose the design of an encoding–decoding mechanism that achieves zero tracking error under a commonly used uniform quantizer.

This paper investigates the LFC problem in uncertain multiarea power systems subject to uniform quantization. To reduce the communication frequency on both input and output channels, a dual-channel dynamic event-triggered (DDET) data-driven LFC approach is proposed. Additionally, an encoding–decoding mechanism is designed to mitigate the adverse effects of data quantization. The main contributions of this study are summarized as follows:

We develop a novel model-free data-driven LFC scheme for interconnected multiarea power systems that completely eliminates the dependence on precise system models. Compared to existing MFAC-based approaches,^19,20 our controller uniquely integrates proportional, differential, and quadratic difference terms, significantly improving control accuracy and dynamic response.

We propose a DDET control strategy that independently designs triggering conditions for both input and output channels. This asynchronous triggering mechanism differs from traditional synchronous event-triggered methods^31,32 by reducing communication load more effectively without compromising stability or performance.

To tackle the challenges of data quantization in communication channels, we introduce an innovative encoding–decoding mechanism. Unlike prior works,^36,37 which assume known system models or focus on single-channel quantization, our approach is fully data-driven and handles quantization effects in both input and output signals.

The remainder of this paper is organized as follows. The second section formulates the control problem. The third section presents the proposed DDET data-driven LFC strategy, incorporating the encoding–decoding mechanism. The fourth section provides the stability analysis. The penultimate section demonstrates and discusses the simulation results. Finally, the last section concludes the paper.

Preliminary and problem description

Referring to Bu et al. ²⁰ and Shangguan et al. ²⁷ , the dynamic model of $j$ th power system in Figure 1 is described as

$$\{\begin{array}{c}{\dot{x}}_j(t)={\stackrel{⌣}{A}}_j{x}_j(t)+{\stackrel{⌣}{B}}_j{u}_j(t)+{\stackrel{⌣}{F}}_j{\varrho }_j(t)\\ E_j(t)=C_j{x}_j(t)\end{array},$$

(1)

where $j=1,2,\dots ,N$ , $x_j(t)=[\mathrm{\Delta }{f}_j(t)\mathrm{\Delta }{P}_{pj}(t)\mathrm{\Delta }{P}_{\mathrm{tie}-j}(t)\mathrm{\Delta }{P}_{gj}(t){]}^{\mathrm{\top }}$ is the state vector with $|\mathrm{\Delta }{\dot{P}}_{gj}|\le 0.017$ , $u_j(t)=\mathrm{\Delta }{P}_{cj}(t)$ is defined as the system's input variable, and the disturbance vector is ${\varrho }_j^{\mathrm{\top }}(t)=[\mathrm{\Delta }{P}_{dj}(t){\sum }_{v=1,v\ne j}^N{S}_{jv}\mathrm{\Delta }{f}_j(t)]$ . Furthermore,

$$\begin{array}{rl}& {\stackrel{⌣}{A}}_j=\left[\begin{array}{cccc}-{\displaystyle \frac{D_j}{H_j}}& -{\displaystyle \frac{1}{H_j}}& {\displaystyle \frac{1}{H_j}}& 0\\ 0& 0& -{\displaystyle \frac{1}{T_{ti}}}& {\displaystyle \frac{1}{T_{ti}}}\\ 2\pi \sum _{v=1,v\ne j}^N{S}_{jv}& 0& 0& 0\\ -{\displaystyle \frac{1}{R_j{T}_{gj}}}& 0& 0& -{\displaystyle \frac{1}{T_{gj}}}\end{array}\right],\\ & {\stackrel{⌣}{B}}_j={\left[\begin{array}{cccc}0& 0& 0& {\displaystyle \frac{1}{T_{gj}}}\end{array}\right]}^{\mathrm{\top }},C_j=\left[\begin{array}{cccc}-{\mathrm{\Gamma }}_j& 0& -1& 0\end{array}\right],\end{array}$$

and

$${\stackrel{⌣}{F}}_j={\left[\begin{array}{cccc}-{\displaystyle \frac{1}{H_j}}& 0& 0& 0\\ 0& 0& -2\pi & 0\end{array}\right]}^{\mathrm{\top }}.$$

Figure 1.

Block diagram of $j$ th area power system.

The physical meaning of all the parameters in the above equation is shown in Table 1. Based on the Euler approximation law with sampling period T, the discrete-time model of $j$ th power system is described as

$$\{\begin{array}{c}{x}_j(k+1)=A_j{x}_j(k)+B_j{u}_j(k)+F_j{\varrho }_j(k)\\ E_j(k)=C_j{x}_j(k)\end{array}$$

(2)

where $A_j=e^{{\stackrel{⌣}{A}}_{j}T}$ , $B_j={\int }_0^T{e}^{{\stackrel{⌣}{A}}_{j}t}{\stackrel{⌣}{B}}_{j}dt$ , and $F_j={\int }_0^T{e}^{{\stackrel{⌣}{A}}_{j}t}{\stackrel{⌣}{F}}_{j}dt.$

Table 1.

Signals of the j area system.

Symbol
$\mathrm{\Delta }{f}_j$	Frequency deviation (Hz)
$\mathrm{\Delta }{P}_{\mathrm{tie}\text{-}j}$	Tie line power deviation (p.u.)
$\mathrm{\Delta }{P}_{\mathrm{p}j}$	Generator output power deviation (p.u.)
$\mathrm{\Delta }{P}_{\mathrm{g}j}$	Governor valve position deviation (p.u.)
$H_j$	Inertia of generator (p.u. s)
$T_{tj}$	Turbine time constants (s)
$T_{\mathrm{g}j}$	Governor time constants (s)
$T_{ij}$	Synchronizing torque coefficient of tie-line between two different areas (p.u./Hz)
$E_j$	Area control error (p.u.)
${\mathrm{\Gamma }}_j$	Frequency bias factor (p.u./Hz)
$D_j$	Generator unit damping coefficient (p.u./Hz)
$R_j$	Speed droop (Hz/p.u.)

The objective of the control problem is to design an event-triggered data-driven controller for power system (2) such that the area control error $E_j(k)$ tracks the desired output trajectory $E_{\mathrm{d}}(k)$ . In this study, the area control error needs to be quantized before being transferred to the controller. The following uniform quantizer is used:

$$Q(\mathcal{X}(k))=\{\begin{array}{c}0,\mathrm{if}-\rho <\mathcal{X}(k)<\rho \\ 2h\rho ,\mathrm{if}(2h-1)\rho <\mathcal{X}(k)<(2h+1)\rho ,\\ -Q(-\mathcal{X}(k)),\mathrm{if}\mathcal{X}(k)\le -\rho \end{array}$$

(3)

where $h\in {\mathbb{N}}^{+}$ , $\rho >0$ denotes quantization density, and $\mathcal{X}(k)$ is the data before quantization. Figure 2 shows the function curve of $Q(\cdot )$ . Obviously, we have $Q(\mathcal{X})-\mathcal{X}\le \rho$ .

Figure 2.

Function curve of $Q(\mathcal{X})$ .

Assumption 1.

In this article, matrices ${\stackrel{⌣}{A}}_j,{\stackrel{⌣}{B}}_j,C_j$ , and ${\stackrel{⌣}{F}}_j$ are supposed to be unknown.

Remark 1.

The choice of the quantization density parameter $\rho$ is closely related to the communication burden of the system. Specifically, heavier communication constraints typically require larger $\rho$ , leading to coarser quantization and lower communication cost, whereas lighter communication constraints allow smaller $\rho$ , resulting in finer resolution and improved control accuracy. In practice, $\rho$ can thus be tuned according to the available bandwidth and system requirements.

Design of DDET-quantized MFAC scheme

In this section, a DDET-quantized MFAC scheme, as shown in Figure 3 and Algorithm 1, is designed.

Figure 3.

System block diagram of the DDET-quantized MFAC scheme. DDET: dual-channel dynamic event-triggered; MFAC: model-free adaptive control.

Data-driven control algorithm

The area control error signal of $j$ th area is given as

$$E_j(k)={\mathrm{\Gamma }}_j\mathrm{\Delta }{f}_j(k)+P_{\mathrm{tie}-j}(k),$$

(4)

where ${\mathrm{\Gamma }}_j=1/R_j+D_j$ denotes the frequency bias factor. From equation (2), one can obtain that

$$E_j(k+1)=C_j{A}_{j}x(k)+C_j{B}_j{u}_j(k)+C_j{F}_j{\varrho }_j(k).$$

(5)

Furthermore, $E_j(k+1)$ can be rewritten as

$$E_j(k+1)=g_j(E_j(k),u_j(k))+{\kappa }_j(k),$$

(6)

where ${\kappa }_j(k)=C_j{F}_j{\varrho }_j(k)$ is a disturbance bounded by ${\overline{\kappa }}_j$ , and $g_j(E_j(k),u_j(k))=C_j{A}_{j}x(k)+C_j{B}_j{u}_j(k)$ is an unknown nonlinear function. Nonlinear system (6) satisfies the following two assumptions.

Assumption 2.

The partial derivative of $g_j(\cdot )$ with respect to $u_j(k)$ is continuous.

Assumption 3

(Qiu et al., ¹⁹ Bu et al. ²⁰ ). Nonlinear system (6) satisfies the generalized Lipschitz condition, that is, for any $k\ge 0$ , and $\mathrm{\Delta }{u}_j(k)\ne 0$ , there is

$$|\mathrm{\Delta }{E}_j(k+1)|\le B_u|\mathrm{\Delta }{u}_j(k)|,$$

(7)

where $\mathrm{\Delta }{E}_j(k+1)=E_j(k+1)-E_j(k)$ , $\mathrm{\Delta }{u}_j(k)=u_j(k)-u_j(k-1)$ , and $B_u>0$ is a constant.

Lemma 1

(Qiu et al., ¹⁹ Bu et al. ²⁰ ). Under Assumptions 1 and 2, when $|\mathrm{\Delta }{u}_j(k)|\ne 0$ , the system can be transformed into the following compact-form dynamic linearization model:

$$\mathrm{\Delta }{E}_j(k+1)={\psi }_j(k)\mathrm{\Delta }{u}_j(k)+\mathrm{\Delta }{\kappa }_j(k),$$

(8)

where ${\psi }_j(k)$ is a time-varying parameter called pseudo-partial derivative (PPD), and $\mathrm{\Delta }{\kappa }_j(k)={\kappa }_j(k)-{\kappa }_j(k-1)$ . ${\psi }_j(k)$ is an unknown time-varying parameter, which reflects the dynamic evolution relationship between $u_j(k)$ and $E_j(k)$ . Furthermore, since system (6) satisfies the generalized Lipschitz condition, we have $|{\psi }_j(k)|\le B_u$ for any instant k.

To design the data-driven LFC algorithm, the linearization data model in Lemma 1 is rewritten as follows:

$$E_j(k+1)=E_j(k)+{\psi }_j(k)\mathrm{\Delta }{u}_j(k)+\mathrm{\Delta }{\kappa }_j(k).$$

(9)

Consider the following criterion function of the control input $u_j(k)$ :

$$J(u_j(k))=E_j^2(k)+{\alpha }_j(u_j(k)-u_j(k-1){)}^2,$$

(10)

where ${\alpha }_j>0$ is a weighting factor used to limit the changes of $u_j(k)$ .

Substitute equation (9) into the criterion function and set the derivative of the criterion function with respect to $u_j(k)$ be zero. Accordingly, the following control algorithm with uniform quantizer can be obtained as

$$u_j(k)=u_j(k-1)+{\displaystyle \frac{Q({\psi }_j(k))}{{\alpha }_j+{|Q({\psi }_j(k))|}^2}Q(E_j(k)).}$$

(11)

It is worthwhile to note that the above control algorithm only utilizes the error information at instant k, which inspires us to further consider the impact of multistep historical error information. Consequently, the improved control algorithm is designed as

$$u_j(k)=u_j(k-1)+{\displaystyle \frac{{\nu }_{j}Q({\psi }_j(k))}{{\alpha }_j+{|Q({\psi }_j(k))|}^2}Q({\eta }_j(k)),}$$

(12)

where ${\nu }_j\in (0,1]$ is a step-size factor added to make the algorithm more general. ${\eta }_j(k)={\lambda }_{aj}{E}_j(k)+{\lambda }_{bj}(E_j(k)-E_j(k-1))+{\lambda }_{cj}(E_j(k)-2E_j(k-1)+E_j(k-2))$ denotes the feedback term that comprises three constituents: the proportional control term, the differential control term, and the quadratic differential control term. ${\lambda }_{aj}$ , ${\lambda }_{bj}$ , and ${\lambda }_{cj}$ are positive control coefficients.

To implement control algorithm (12), the value of PPD ${\psi }_j(k)$ is needed to be known in advance. However, because the parameter matrices of the system are unknown, and PPD is a time-varying parameter, its exact value is difficult to obtain. As a result, an estimation algorithm of PPD is designed, and the criterion function is set as

$$\begin{array}{rl}& J({\psi }_j(k))=|E_j(k)-E_j(k-1)|\\ & -{\psi }_j(k)\mathrm{\Delta }{u}_j(k-1)+{\beta }_j({\psi }_j(k)-{\hat{\psi }}_j(k-1){)}^2,\end{array}$$

(13)

where ${\beta }_j>0$ is a weighing factor. ${\hat{\psi }}_j(k)$ is the estimation value of ${\psi }_j(k)$ . The method similar to one above is adopted to minimize the criterion function equation (10). Then, an estimation algorithm of PPD ${\psi }_j(k)$ can be obtained as

$$\begin{array}{rl}& {\hat{\psi }}_j(k)={\hat{\psi }}_j(k-1)+(\mathrm{\Delta }{E}_j(k)-{\hat{\psi }}_j(k-1)\\ & \times Q(\mathrm{\Delta }{u}_j(k-1)))\times {\displaystyle \frac{{\varsigma }_{j}Q(\mathrm{\Delta }{u}_j(k-1))}{{\beta }_j+Q{(\mathrm{\Delta }{u}_j(k-1))}^2},}\end{array}$$

(14)

where ${\varsigma }_j\in (0,1]$ denotes a step-size factor. Equation (12) can be rewritten as

$$u_j(k)=u_j(k-1)+\mathrm{\Delta }{u}_j(k),$$

(15)

where

$$\mathrm{\Delta }{u}_j(k)={\displaystyle \frac{{\nu }_{j}Q({\hat{\psi }}_j(k))}{{\alpha }_j+{|Q({\hat{\psi }}_j(k))|}^2}Q({\eta }_j(k)).}$$

(16)

Remark 2.

Clearly, equations (14) and (15) do not incorporate any model information, where they rely solely on system output data and control input data. Furthermore, internal state information, such as $\mathrm{\Delta }{P}_{\mathrm{tie}-j}$ and $\mathrm{\Delta }{P}_{pj}$ , is not required. Hence, this control scheme is data-driven.

Dual-channel dynamic event-triggered mechanism

The design of dynamic thresholds plays a critical role in the event-triggered mechanism. Generally, the triggering threshold should meet the following criteria: when the control input or triggering error exhibits a large magnitude, the threshold should increase accordingly to extend triggering intervals, thereby alleviating communication congestion caused by excessive events. Conversely, when the control input or triggering error is small, the threshold should be reduced to increase the triggering frequency, ensuring the elimination of steady-state errors. Based on the above analysis, the input triggering conditions are designed as follows:

$${\stackrel{⌣}{k}}_{j,p}=inf\{k\in \mathbb{N}k>{\stackrel{⌣}{k}}_{j,p-1},|\mathrm{\Delta }{\hat{u}}_j(k)|\ge {\stackrel{⌣}{\delta }}_j|u_j({\stackrel{⌣}{k}}_{j,p-1})|+{\stackrel{⌣}{\phi }}_j\}$$

(17)

where ${\stackrel{⌣}{\phi }}_j>0$ and $0<{\stackrel{⌣}{\delta }}_j<1$ denote positive triggering parameters, and ${\stackrel{⌣}{k}}_{j,p}$ ( $p=1,2,\dots$ ) denotes the input triggering instant. Then, the control algorithm with triggering strategy is given as

$$u_j(k)=\{\begin{array}{cc}{u}_j({\stackrel{⌣}{k}}_{j,p-1})+\mathrm{\Delta }{u}_j(k),& k={\stackrel{⌣}{k}}_{j,p}\\ u_j({\stackrel{⌣}{k}}_{j,p-1}),& k\in ({\stackrel{⌣}{k}}_{j,p-1},{\stackrel{⌣}{k}}_{j,p})\end{array}$$

(18)

Further, the output triggering conditions are designed as follows:

$${\overline{k}}_{j,q}=inf\{k\in \mathbb{N}k>{\overline{k}}_{j,q-1},|{\hat{E}}_j^q(k)|\ge {\overline{\delta }}_j|{\varpi }_j(k)|+{\overline{\phi }}_j\}$$

(19)

where ${\hat{E}}_j^q(k)=E_j(k)-E_j({\overline{k}}_{j,q-1})$ , ${\overline{\phi }}_j>0$ , and $0<{\overline{\delta }}_j<1$ denote positive triggering parameters, and ${\varpi }_j(k)$ is a dynamic event-triggered function designed as

$${\varpi }_j(k+1)={\delta }_j{\varpi }_j(k)+{∍}_j-|{\hat{E}}_j^q(k)|$$

(20)

where ${\delta }_j$ and ${∍}_j$ denote positive parameters. Then, the estimation law of PPD ${\psi }_j(k)$ with an event-triggered strategy is designed as

$$\begin{array}{rl}& {\hat{\psi }}_j(k)=\{\begin{array}{c}\hat{\psi }({\overline{k}}_{j,q-1})+(\mathrm{\Delta }{E}_j(k)-{\hat{\psi }}_j(k-1)\\ \mathrm{\Delta }{u}_j(k-1))\times {\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }{u}_j(k-1)}{{\beta }_j+\mathrm{\Delta }{u}_j{(k-1)}^2},k={\overline{k}}_{j,q}}\\ {\hat{\psi }}_j({\overline{k}}_{j,q-1}),k\in ({\overline{k}}_{j,q-1},{\overline{k}}_{j,q})\end{array}\\ & {\hat{\psi }}_j(k)={\hat{\psi }}_j(1),\mathrm{if}|{\hat{\psi }}_j|\le \epsilon ,\mathrm{or}|\mathrm{\Delta }{u}_j(k)|\le \epsilon \end{array}$$

(21)

where $\epsilon$ denotes a small positive constant and ${\hat{\psi }}_j(1)$ denotes the initial PPD value.

Remark 3.

The results in Feng et al. ³⁸ have shown that data quantization in equation (16) can lead to a decrease in the system's tracking performance, which motivates us to design an encoding–decoding mechanism to compensate for the influence of data quantization.

Encoding–decoding mechanism

To mitigate the effects of data quantization, data needs to be encoded before transmission over the network. The output of the designed encoder is

$$\mathcal{E}(\mathcal{X}(k))={\displaystyle \frac{\mathcal{X}(k)-\xi (k-1)}{\zeta (k)}}$$

(22)

where $\mathcal{X}(k)$ is the input of the encoder in instant k, $\xi (k)$ is the internal state of the encoder with $\xi (1)=0$ , $\zeta (k)$ is a positive scaling sequence that adjusts the magnitude of the encoder output, and $\xi (k)$ iterates as follows:

$$\xi (k)=\zeta (k)Q(\mathcal{E}(\mathcal{X}(k)))+\xi (k-1).$$

(23)

Then, the corresponding decoder can be designed as

$$\mathcal{D}(\mathcal{X}(k))=\mathcal{D}(\mathcal{X}(k-1))+\zeta (k)Q(\mathcal{E}(\mathcal{X}(k)))$$

(24)

where $\mathcal{D}(0)=0$ . To simplify the expression, the output of the decoder $\mathcal{D}(\mathcal{X}(k))$ is rewritten as $\mathcal{X}*(k)$ as shown in Figure 3. Thus, by substituting equation (22) into equation (24), we have

$$\begin{array}{c}\mathcal{X}*(k)=\mathcal{X}*(k-1)+\zeta (k)Q\left({\displaystyle \frac{\mathcal{X}(k)-\xi (k-1)}{\zeta (k)}}\right)\\ =\mathcal{X}*(k-1)+\zeta (k)\left({\displaystyle \frac{\mathcal{X}(k)-\xi (k-1)}{\zeta (k)}+\omega (k)}\right)\\ =\mathcal{X}*(k-1)+\mathcal{X}(k)-\xi (k-1)+\zeta (k)\omega (k)\end{array}$$

(25)

where $\omega (k)=Q(\mathcal{E}(\mathcal{X}(k)))-\mathcal{E}(\mathcal{X}(k))\le \rho$ . Moreover, according to equations (23) and (24), we have

$$\begin{array}{c}\mathcal{X}*(k)-\xi (k)=\zeta (k)Q(\mathcal{E}(\mathcal{X}(k)))+\mathcal{X}*(k-1)\\ -\zeta (k)Q(\mathcal{E}(\mathcal{X}(k)))-\xi (k-1)\\ =\mathcal{X}*(k-1)-\xi (k-1)=\dots \\ =\mathcal{X}*(0)-\xi (0)=0\end{array}$$

(26)

Thus, equation (25) can be rewritten as

$$\begin{array}{c}\mathcal{X}*(k)=\mathcal{X}*(k-1)+\mathcal{X}(k)-\xi (k-1)+\zeta (k)\omega (k)\\ =\mathcal{X}(k)+\zeta (k)\omega (k)\end{array}$$

(27)

Finally, the quantized DDET data-driven LFC algorithm can be designed as

$$\mathrm{\Delta }{u}_j*(k)={\displaystyle \frac{{\nu }_j{\hat{\psi }}_j*(k)}{{\alpha }_j+{|{\hat{\psi }}_j*(k)|}^2}{\eta }_j*(k),}$$

(28)

$$u_j*(k)=\{\begin{array}{cc}{u}_j*({\stackrel{⌣}{k}}_{j,p-1})+\mathrm{\Delta }{u}_j*(k),& k={\stackrel{⌣}{k}}_{j,p}\\ u_j*({\stackrel{⌣}{k}}_{j,p-1}),& k\in ({\stackrel{⌣}{k}}_{j,p-1},{\stackrel{⌣}{k}}_{j,p})\end{array}$$

(29)

$$\begin{array}{rl}& {\hat{\psi }}_j*(k)=\{\begin{array}{c}\hat{\psi }({\overline{k}}_{j,q-1})+(\mathrm{\Delta }{E}_j(k)-{\hat{\psi }}_j(k-1)\\ \mathrm{\Delta }{u}_j*(k-1))\times {\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }{u}_j*(k-1)}{{\beta }_j+\mathrm{\Delta }{u}_j*{(k-1)}^2},k={\overline{k}}_{j,q}}\\ {\hat{\psi }}_j({\overline{k}}_{j,q-1}),k\in ({\overline{k}}_{j,q-1},{\overline{k}}_{j,q})\end{array}\\ & {\hat{\psi }}_j*(k)={\hat{\psi }}_j(1),\mathrm{if}|{\hat{\psi }}_j*|\le \epsilon ,\mathrm{or}|\mathrm{\Delta }{u}_j*(k)|\le \epsilon .\end{array}$$

(30)

The nomenclature of all parameters involved in the proposed Algorithm 1 is provided in Table 2.

Table 2.

Parameters of the $j$ th controller.

Symbol
${\nu }_j$	Step-size factor in equation (12)
${\alpha }_j$	Weighting factor in equation (10)
${\varsigma }_j$	Step-size factor in equation (14)
${\beta }_j$	Weighting factor in equation (13)
${\stackrel{⌣}{\delta }}_j$	Input triggering factor in equation (17)
${\stackrel{⌣}{\phi }}_j$	Input triggering threshold in equation (17)
${\overline{\delta }}_j$	Output triggering factor in equation (19)
${\overline{\phi }}_j$	Output triggering threshold in equation (19)
${\delta }_j$	Dynamic event-triggered function factor in equation (20)
${∍}_j$	Dynamic event-triggered function bias in equation (20)
$\zeta$	Scaling sequence in equation (22)

Algorithm 1.

Data-driven dual-channel dynamic event-triggered load frequency control algorithm.

Stability analysis

This section studies the stabilization and convergence of the multiarea interconnected power systems.

Theorem 1.

For system (6), the proposed DDET data-driven LFC scheme with input design-based compensation can guarantee the boundedness of system signals for $j=1,2,\dots ,N$ , as follows:

The estimation error ${\tilde{\psi }}_j(k)={\hat{\psi }}_j*(k)-{\psi }_j(k)$ is bounded.

The dynamic event-triggered function ${\varpi }_j(k)$ is bounded.

The area control error $E_j(k)$ is bounded.

Proof: The boundedness of ${\psi }_j*(k)$ is first proved in Part 1, then we proved the boundedness of ${\varpi }_j(k)$ in Part 2, and the boundedness of $E_j(k)$ is proved in Part 3.

Part 1: When $k={\overline{k}}_{j,q}$ , the PPD ${\psi }_j(k)$ updates according to equation (14), one has

$$\begin{array}{c}{\tilde{\psi }}_j(k)={\tilde{\psi }}_j(k-1)-\mathrm{\Delta }{\psi }_j(k)+{\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }{u}_j*(k-1)}{{\beta }_j+\mathrm{\Delta }u{{}_j*}^2(k-1)}}\\ \times (\mathrm{\Delta }{E}_j(k)-{\hat{\psi }}_j*(k-1)\mathrm{\Delta }{u}_j*(k-1))\\ ={\tilde{\psi }}_j(k-1)-\mathrm{\Delta }{\psi }_j(k)+{\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }u{{}_j*}^2(k-1)}{{\beta }_j+\mathrm{\Delta }u{{}_j*}^2(k-1)}}\\ \times \left({\psi }_j(k-1)-{\hat{\psi }}_j*(k-1)+{\displaystyle \frac{\mathrm{\Delta }{\kappa }_j(k-1)}{\mathrm{\Delta }{u}_j*(k-1)}}\right)\\ ={\tilde{\psi }}_j(k-1)-\mathrm{\Delta }{\psi }_j(k)+{\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }u{{}_j*}^2(k-1)}{{\beta }_j+\mathrm{\Delta }u{{}_j*}^2(k-1)}{\tilde{\psi }}_j(k-1)+D_j(k),}\end{array}$$

(31)

where $D_j=({\varsigma }_j\mathrm{ucgreek}\mathrm{\Delta }{\mathrm{u}}_{\mathrm{j}}^{*}\left(\mathrm{k}-1\right)\mathrm{ucgreek}\mathrm{\Delta }{\kappa }_{\mathrm{j}}\left(\mathrm{k}-1\right))/({\beta }_{\mathrm{j}}+\mathrm{ucgreek}\mathrm{\Delta }{\mathrm{u}}_{\mathrm{j}}^{*2}\left(\mathrm{k}-1\right))$ is a bounded term and ${\kappa }_j$ is a bounded disturbance. Taking the absolute value of equation (31), one has

$${\tilde{\psi }}_j(k)\le |\mathrm{\Delta }{\psi }_j(k)|+\left|1-{\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }u{{}_j*}^2(k-1)}{{\beta }_j+\mathrm{\Delta }u{{}_j*}^2(k-1)}}\right|\times |{\tilde{\psi }}_j(k-1)|+|D_j|.$$

(32)

where the term $({\varsigma }_j\mathrm{\Delta }{u}_j^{*2}(k-1))/({\beta }_j+\mathrm{\Delta }{u}_j^{*2}(k-1))$ is obviously monotonically increasing when we set ${\varsigma }_j\in (0,1]$ and ${\beta }_j>0$ for the variable $\mathrm{\Delta }{u}_j^{*2}$ , and we have $\mathrm{\Delta }{u}_j^{*2}\ge {\epsilon }^2$ according to equation (30). Thus, the following inequality holds:

$$\left|1-{\displaystyle \frac{{\varsigma }_j\mathrm{\Delta }u{{}_j*}^2(k-1)}{{\beta }_j+\mathrm{\Delta }u{{}_j*}^2(k-1)}}\right|\le 1-{\displaystyle \frac{{\varsigma }_j{\epsilon }^2}{{\beta }_j+{\epsilon }^2}\triangleq {ϵ}_j<1}$$

(33)

From Lemma 1, we have ${\psi }_j(k)\le B_u$ . Thus, $|\mathrm{\Delta }{\psi }_j*(k)|\le 2(B_u+\zeta (k)\omega (k))$ . Then, according to equation (31), we have

$$\begin{array}{c}{\tilde{\psi }}_j(k)\le {ϵ}_j|{\psi }_j(k-1)|+2(B_u+\zeta (k)\omega (k))+D_j\\ \le {ϵ}_j^2|{\psi }_j(k-2)|+2{ϵ}_j{B}_u+2(B_u+\zeta (k)\omega (k))+{ϵ}_j{D}_j+D_j\\ \le \cdots \le {ϵ}_j^{k-1}|{\psi }_j(1)|+{\displaystyle \frac{1-{ϵ}_j^{k-1}}{1-{ϵ}_j}\times (B_u+\zeta (k)\omega (k)+D_j).}\end{array}$$

(34)

If the choice of $\zeta (k)$ satisfies $0<\zeta (k)\le 1$ and $\underset{k\to \mathrm{\infty }}{lim}\zeta (k)=0$ , one has $\underset{k\to \mathrm{\infty }}{lim}{ϵ}_j^{k-1}=0$ and $\underset{k\to \mathrm{\infty }}{lim}\zeta (k)\omega (k)=0$ . Thus, equation (34) can be rewritten as

$$\underset{k\to \mathrm{\infty }}{limsup}{\tilde{\psi }}_j(k)\le {\displaystyle \frac{2B_u+D_j}{1-{ϵ}_j}}$$

(35)

Considering the boundedness of ${\psi }_j(k)$ and ${\tilde{\psi }}_j(k)$ , it is obvious that ${\hat{\psi }}_j*(k)$ is also bounded.

For $k\in ({\overline{k}}_{j,q-1},{\overline{k}}_{j,q})$ , the PPD ${\psi }_j(k)$ will not change and is obviously bounded.

Part 2: Now we prove the boundedness of ${\varpi }_j(k)$ . According to equation (20), one has

$$\begin{array}{rl}& |{\varpi }_j(k+1)|\le {\delta }_j|{\varpi }_j(k)|+{∍}_j+|{\hat{E}}_j^q(k)|\\ & \le {\delta }_j|{\varpi }_j(k)|+{∍}_j+{\overline{\delta }}_j|{\varpi }_j(k)|+{∍}_j\\ & \le ({\delta }_j+{\overline{\delta }}_j)|{\varpi }_j(k)|+2{∍}_j\\ & \le ({\delta }_j+{\overline{\delta }}_j{)}^2|{\varpi }_j(k-1)|+2({\delta }_j+{\overline{\delta }}_j){\varpi }_j+2{∍}_j\\ & \le \cdots \le ({\delta }_j+{\overline{\delta }}_j{)}^k|{\varpi }_j(1)|+{\displaystyle \frac{2{∍}_j[1-{({\delta }_j+{\overline{\delta }}_j)}^k]}{1-({\delta }_j+{\overline{\delta }}_j)}.}\end{array}$$

(36)

When ${\delta }_j$ and ${\overline{\delta }}_j$ are set to satisfy $0<{\delta }_j+{\overline{\delta }}_j<1$ , we have $\underset{k\to \mathrm{\infty }}{lim}({\delta }_j+{\overline{\delta }}_j{)}^k\to 0$ . Thus, ${\varpi }_j(k)$ can converge to the following set:

$$\left[{\displaystyle \frac{-2{∍}_j}{1-{\delta }_j-{\overline{\delta }}_j},{\displaystyle \frac{2{∍}_j}{1-{\delta }_j-{\overline{\delta }}_j}}}\right].$$

Part 3: Now we prove the boundedness of $E_j(k)$ . When $k={\stackrel{⌣}{k}}_{j,p}$ , the control input updates according to equation (15). The area control error $E_j(k+1)$ can be rewritten as

$$\begin{array}{c}{E}_j(k+1)=E_j(k)+{\psi }_j(k)\mathrm{\Delta }{u}_j*(k)+\mathrm{\Delta }{\kappa }_j(k)\\ =E_j*(k)-(({\lambda }_{aj}+{\lambda }_{bj}+{\lambda }_{cj})E_j*(k)\\ +({\lambda }_{bj}+2{\lambda }_{cj})E_j*(k-1)\\ -{\lambda }_{cj}{E}_j*(k-2)){\psi }_j(k){\varphi }_j(k)+\mathrm{\Delta }{\kappa }_j(k),\end{array}$$

(37)

where ${\varphi }_j(k)=({\nu }_j{\hat{\psi }}_j*(k))/({\alpha }_j+{\hat{\psi }}_j^{*2}(k))$ . Defining $L_1(k)={\psi }_j(k){\varphi }_j(k)({\lambda }_{aj}+{\lambda }_{bj}+{\lambda }_{cj})$ , $L_2(k)={\psi }_j(k){\varphi }_j(k)({\lambda }_{bj}+2{\lambda }_{cj})$ , and $L_3(k)={\psi }_j(k){\varphi }_j(k){\lambda }_{cj}$ , one has

$$\begin{array}{c}|E_j(k+1)|\le |1-L_1(k)||E_j(k)|+|L_2(k)||E_j(k-1)|+|L_3(k)||E_j(k-2)|\\ \le L_{\mathrm{sum}}(k)\max \{E_j(k-h)\}+\mathrm{\Delta }{\overline{\kappa }}_j\end{array}$$

(38)

for $h=0,1,2$ , where $L_{\mathrm{sum}}(k)=|1-L_1(k)|+L_2(k)+L_3(k)\ge 0$ .

Then, one has

$$\begin{array}{c}0<L_1(k)+L_2(k)+L_3(k)\\ =({\lambda }_{aj}+2{\lambda }_{bj}+4{\lambda }_{cj}){\psi }_j(k){\varphi }_j(k)\\ \le ({\lambda }_{aj}+2{\lambda }_{bj}+4{\lambda }_{cj})(B_u+\zeta (k)\xi (k)){\displaystyle \frac{{\hat{\psi }}_j*(k)}{2\sqrt{{\alpha }_j}{\hat{\psi }}_j*(k)}.}\end{array}$$

(39)

If we satisfy $0<\zeta (k)\le 1$ , $\underset{k\to \mathrm{\infty }}{lim}\zeta (k)=0$ , ${\lambda }_{aj}+2{\lambda }_{bj}+4{\lambda }_{cj}<4\sqrt{{\alpha }_j}/(B_u+\zeta (k)\xi (k))$ , and ${\lambda }_{aj}>2{\lambda }_{cj}$ , we have $L_1(k)>L_2(k)+L_3(k)\ge 0$ and $0<L_1(k)+L_2(k)+L_3(k)<2L_1(k)<2$ . Based on the above conclusion, we get the boundedness of $L_{\mathrm{sum}}(k)$ :

$$\begin{array}{rl}& 0\le L_{\mathrm{sum}}(k)=|1-L_1(k)|+L_2(k)+L_3(k)\\ & <1-L_1(k)+L_1(k)=1.\end{array}$$

(40)

Thus, there exists $\mathcal{V}$ satisfying $0<L_{\mathrm{sum}}(k)<\mathcal{V}<1$ , and equation (38) is rewritten as

$$\begin{array}{c}|E_j(k+1)|\le \mathcal{V}\max \{E_j(k-h)\}+\mathrm{\Delta }{\overline{\kappa }}_j,h=0,1,2\\ \le {\mathcal{V}}^2\max \{\max \{E_j(k-h-1)\}\}+\mathrm{\Delta }{\overline{\kappa }}_j,h=0,1,2,3,4\\ \le \dots \\ \le {\mathcal{V}}^{n+1}\max \{\dots \max \{E_j(k-h-n)\}\dots \}+\mathrm{\Delta }{\overline{\kappa }}_j,h=0,1,2,\dots ,2n+2,\end{array}$$

(41)

where $n\in {\mathbb{Z}}_{+}$ . It is obvious that if $k\to \mathrm{\infty }$ , $n\to \mathrm{\infty }$ . As $0<\mathcal{V}<1$ , we have $lim\underset{k\to \mathrm{\infty }}{sup}|E_j(k+1)|\le \mathrm{\Delta }{\overline{\kappa }}_j$ , the boundedness of $E_j(k)$ is proven.

For $k\in ({\stackrel{⌣}{k}}_{j,p-1},{\stackrel{⌣}{k}}_{j,p})$ , the control input will not change, and $E_j(k)$ is only affected by bounded disturbance ${\kappa }_j(k)$ , so $E_j(k)$ is also bounded.

Remark 4.

According to the stability analysis, the controller parameters must satisfy specific conditions to ensure the convergence of the tracking error. These conditions also provide practical guidelines for parameter tuning:

Choose ${\nu }_j,{\varsigma }_j\in (0,1]$ and ${\alpha }_j,{\beta }_j>0$ ;

Ensure ${\lambda }_{aj}+2{\lambda }_{bj}+4{\lambda }_{cj}<4\sqrt{{\alpha }_j}(B_u+\zeta (k)\xi (k){)}^{-1}$ and ${\lambda }_{aj}>2{\lambda }_{cj}$ . It is recommended to initially select ${\lambda }_{aj}=1$ and ${\lambda }_{bj}={\lambda }_{cj}=0$ (corresponding to the traditional MFAC controller), and then gradually increase ${\lambda }_{bj}$ and ${\lambda }_{cj}$ until no further performance improvement is observed;

Choose $0<{\delta }_j+{\overline{\delta }}_j<1$ and ${\stackrel{⌣}{\phi }}_j,\overline{{\phi }_j}>0$ ;

Select a monotonically decreasing sequence $\zeta (k)$ that satisfies $\zeta (k)\in (0,1]$ and $\underset{k\to \mathrm{\infty }}{lim}\zeta (k)=0$ .

Simulation results

This section verifies the effectiveness of the proposed strategy in three interconnected power systems, as shown in Figure 4. Table 3 presents the selected system parameter values. Moreover, the sampling period is 0.005 s, $S_{12}=S_{21}=0.25$  p.u./Hz, $S_{13}=S_{31}=0.35$  p.u./Hz, and $S_{23}=S_{32}=0.22$  p.u./Hz. It is important to note that all the parameters are unknown to the controller. The simulation is implemented in MATLAB/Simulink on an AMD R7 7700 computing platform. Case I executed in an average of 0.50 s, and case II in 0.51 s, demonstrating the feasibility of real-time implementation.

Figure 4.

Three-area power system.

Table 3.

Signals of the jth area system. ²⁰

Parameters	Area #1	Area #2	Area #3
D (p.u./HZ)	0.015	0.016	0.015
H (p.u. ·s)	0.1567	0.1846	0.1241
R (Hz/p.u.)	2.38	2.49	2.41
$T_{\mathrm{g}}(\mathrm{s})$	0.05	0.07	0.06
$T_t(\mathrm{s})$	0.41	0.42	0.38
$\mathrm{\Gamma }$ (p.u./Hz)	0.3479	0.3826	0.3689

Case I

In this section, we demonstrate the control effectiveness of the proposed DDET data-driven LFC method and its ability to save communication resources under step load disturbances. The controller parameters are shown in Table 4. According to stability analysis, it is noted that the setting value of the controller parameters must meet the following conditions, as stated in Remark 4. Other parameters are adjusted according to the actual control performance. The power system's initial states are set to zero, and 0.03 p.u. load step disturbances are added 10 s later.

Table 4.

Controller parameters of the jth area system.

Parameters	Area #1	Area #2	Area #3
$\nu$	0.8	0.9	0.8
$\alpha$	1.5	1.4	1.3
$\varsigma$	2.38	2.49	2.41
$\beta$	1.3	1.1	1.2
${\lambda }_a$	1.2	1.2	1.2
${\lambda }_b$	1.0	1.1	1.0
${\lambda }_c$	0.5	0.5	0.5
$\stackrel{⌣}{\delta }$	0.1	0.08	0.11
$\stackrel{⌣}{\phi }$	2 $\times {10}^{-5}$	2 $\times {10}^{-5}$	2 $\times {10}^{-5}$
$\overline{\delta }$	0.21	0.2	0.22
$\overline{\phi }$	7 $\times {10}^{-5}$	7 $\times {10}^{-5}$	7 $\times {10}^{-5}$
$\delta$	0.05	0.04	0.05
$∍$	0.7	0.6	0.7
$\epsilon$	${10}^{-5}$	${10}^{-5}$	${10}^{-5}$
$\hat{\psi }(1)$	0.1	0.1	0.1
$\rho$	0.03	0.03	0.03
$\zeta$	1/k	1/k	1/k

Figures 5 to 7 depict the output responses of frequency deviation $\mathrm{\Delta }f$ , generator output power deviation $\mathrm{\Delta }{P}_{\mathrm{p}}$ , and tie-line power deviation $\mathrm{\Delta }{P}_{\mathrm{tie}}$ across the three interconnected power areas. The results demonstrate that the proposed control method effectively suppresses frequency deviations under load disturbances. For comparison, Figure 8 shows the response of $\mathrm{\Delta }f$ without the incorporation of the encoding–decoding mechanism. In this case, the frequency deviation exhibits persistent oscillations within a bounded range, primarily due to the effects of data quantization. These observations are consistent with the findings reported in Feng et al., ³⁸ further validating the efficacy of the proposed encoding–decoding mechanism in mitigating quantization-induced degradation in control performance.

Figure 5.

Response of $\mathrm{\Delta }f$ in three areas (case I).

Figure 6.

Response of $\mathrm{\Delta }{P}_{\mathrm{tie}}$ in three areas (case I).

Figure 7.

Response of $\mathrm{\Delta }{P}_{\mathrm{p}}$ in three areas (case I).

Figure 8.

Response of $\mathrm{\Delta }f$ without encoding–decoding mechanism (case I).

As illustrated in Figures 9 and 10, the horizontal axis denotes the triggering instants, while the vertical axis represents the corresponding intertriggering intervals. The proposed DDET mechanism initiates 2478, 2123, and 2445 triggering events in the input channels, and 1607, 1505, and 1535 events in the output channels, respectively—resulting in an overall communication resource savings of 83.76%. These results clearly demonstrate that the proposed DDET strategy can significantly reduce communication load while maintaining effective control performance.

Figure 9.

Triggering instants and intervals in input channels (case I).

Figure 10.

Triggering instants and intervals in output channels (case I).

Case II

In this section, a dynamic load disturbance, defined as $\mathrm{\Delta }{P}_{dj}(k)=0.03+0.004\sin (5k\pi /n)$ , is introduced 10 s after the simulation begins, where n represents the total number of sampling points. The controller parameters are consistent with case I.

In Figures 11 to 13, the output curves of $\mathrm{\Delta }f$ , $\mathrm{\Delta }{P}_{\mathrm{p}}$ , and $\mathrm{\Delta }{P}_{\mathrm{tie}}$ in three power systems are plotted respectively, which illustrate that the proposed method can effectively reduce frequency deviation under dynamic load disturbance.

Figure 11.

Response of $\mathrm{\Delta }f$ in three areas (case II).

Figure 12.

Response of $\mathrm{\Delta }{P}_{\mathrm{tie}}$ in three areas (case II).

Figure 13.

Response of $\mathrm{\Delta }{P}_{\mathrm{p}}$ in three areas (case II).

As shown in Figures 14 and 15, the proposed DDET mechanism activates 7072, 5550, and 7132 times in the input channels, and 5189, 3654, and 5222 times in the output channels, making 53.03% reduction in communication resource usage. Compared with case I, the total trigger number has significantly increased, primarily due to the higher control frequency caused by the dynamic load disturbances.

Figure 14.

Triggering instants and intervals in input channels (case II).

Figure 15.

Triggering instants and intervals in output channels (case II).

Sensitivity analysis

In this section, we present the effect of different parameters on the control performance.

In Figure 16, the frequency deviation is plotted for different controller parameters in equation (12). Table 5 lists five standard performance criteria: overshoot (OS), regulation time (RT), steady-state error (SSE), integral of absolute error (IAE), and integral of time-weighted absolute error (ITAE) to assess control performance. The results indicate that both ${\nu }_j$ and ${\alpha }_j$ influence the regulation time and steady-state error, with ${\alpha }_j$ having a more significant impact on overshoot.

Figure 16.

Response of $\mathrm{\Delta }{f}_1$ with different ${\nu }_1$ and ${\alpha }_1$ (case I).

Table 5.

Performance comparison between different parameters ( ${\nu }_1$ , ${\alpha }_1$ ).

${\nu }_1$	${\alpha }_1$	OS (Hz)	RT (s)	SSE (Hz)	IAE	ITAE
1.2	1.5	0.0951	21.52	2.99 $\times {10}^{-6}$	0.16	2.0
0.8	1.5	0.0962	18.55	5.29 $\times {10}^{-6}$	0.16	1.9
0.6	1.5	0.0971	21.54	2.41 $\times {10}^{-5}$	0.18	2.1
0.4	1.5	0.0978	22.77	3.07 $\times {10}^{-4}$	0.22	2.8
0.2	1.5	0.0985	36.53	3.13 $\times {10}^{-3}$	0.37	6.2
0.8	0.5	0.0916	27.12	2.23 $\times {10}^{-5}$	0.19	2.6
0.8	3	0.0978	22.76	3.07 $\times {10}^{-4}$	0.22	2.8
0.8	9	0.0988	38.92	6.51 $\times {10}^{-3}$	0.63	10.3

OS: overshoot; RT: regulation time; SSE: steady-state error; IAE: integral of absolute error; ITAE: integral of time-weighted absolute error. The optimal values are highlighted in bold.

In Figure 17, the frequency deviation is plotted for different triggering parameters in equation (19). Table 6 shows that ${\overline{\delta }}_j$ primarily influences control performance, while ${\overline{\phi }}_j$ has a more significant impact on triggering times. It is noted that the frequency deviation fails to converge to zero due to insufficient triggering times when ${\overline{\phi }}_1=7\times {10}^{-4}$ . In comparison with the dynamic event-triggered mechanism in Chen et al., ³² the proposed DDET mechanism further reduces the number of triggers without compromising control performance.

Figure 17.

Response of $\mathrm{\Delta }{f}_1$ with different ${\overline{\delta }}_1$ and ${\overline{\phi }}_1$ (case I).

Table 6.

Performance comparison between different parameters ( ${\overline{\delta }}_1$ , ${\overline{\phi }}_1$ ).

${\overline{\delta }}_1$	${\overline{\phi }}_1$	Triggering times	OS (Hz)	RT (s)	SSE (Hz)	ITAE
Method in Chen et al. 32		6142	0.0962	22.82	2.31 $\times {10}^{-5}$	2.2
0.2	7 $\times {10}^{-4}$	302	0.0994	–	3.42 $\times {10}^{-2}$	63.7
0.2	7 $\times {10}^{-5}$	2478	0.0962	18.55	5.29 $\times {10}^{-6}$	1.9
0.2	7 $\times {10}^{-6}$	8156	0.0931	27.66	4.33 $\times {10}^{-6}$	2.5
0.1	7 $\times {10}^{-5}$	2792	0.0962	20.32	1.19 $\times {10}^{-5}$	2.0
0.7	7 $\times {10}^{-5}$	2307	0.0986	22.74	7.71 $\times {10}^{-5}$	3.7

OS: overshoot; RT: regulation time; SSE: steady-state error; ITAE: integral of time-weighted absolute error. The optimal values are highlighted in bold.

In Figure 18, the frequency deviations corresponding to various controller parameters ${\lambda }_{a1}$ , ${\lambda }_{b1}$ , and ${\lambda }_{c1}$ are depicted. As summarized in Table 7, ${\lambda }_{a1}$ primarily affects the regulation time, ${\lambda }_{b1}$ significantly influences the steady-state error, and ${\lambda }_{c1}$ has a notable impact on overshoot. Compared to the MFAC method presented in Bu et al., ²⁰ in which the controller is designed as

$$u_j(k)=u_j(k-1)+{\displaystyle \frac{{\psi }_j(k)}{{\alpha }_j+{|{\psi }_j(k)|}^2}{E}_j(k).}$$

Figure 18.

Response of $\mathrm{\Delta }{f}_1$ with different ${\lambda }_{a1}$ , ${\lambda }_{b1}$ , and ${\lambda }_{c1}$ (case I).

Table 7.

Performance comparison between different parameters ( ${\lambda }_{a1}$ , ${\lambda }_{b1}$ , ${\lambda }_{c1}$ ).

${\lambda }_{a1}$	${\lambda }_{b1}$	${\lambda }_{c1}$	OS (Hz)	RT (s)	SSE (Hz)	IAE	ITAE
Method in Bu et al. 20			0.0984	23.81	9.34 $\times {10}^{-4}$	0.30	4.1
12	1.2	0.5	0.0962	18.55	5.29 $\times {10}^{-6}$	0.16	1.9
0.8	1.2	0.5	0.0970	21.50	4.41 $\times {10}^{-5}$	0.20	2.5
2.0	1.2	0.5	0.0947	22.79	1.62 $\times {10}^{-5}$	0.17	2.1
1.2	0.5	0.5	0.0962	24.75	2.47 $\times {10}^{-7}$	0.18	2.0
1.2	2	0.5	0.0966	19.79	2.69 $\times {10}^{-6}$	0.17	2.0
1.2	1.2	0.1	0.0979	22.74	3.43 $\times {10}^{-4}$	0.26	3.3
1.2	1.2	1.5	0.0927	36.35	1.53 $\times {10}^{-5}$	0.36	4.5

OS: overshoot; RT: regulation time; SSE: steady-state error; IAE: integral of absolute error; ITAE: integral of time-weighted absolute error. The optimal values are highlighted in bold.

The results show that the proposed method achieves superior control performance, attributed to the integration of proportional, differential, and quadratic difference terms in the control law defined by equation (12).

Figure 19 illustrates the frequency deviations under different quantization densities $\rho$ . As shown in Table 8, increasing $\rho$ reduces the control capability. However, the system is still able to achieve effective frequency regulation.

Figure 19.

Response of $\mathrm{\Delta }{f}_1$ with different $\rho$ (case I).

Table 8.

Performance comparison between different parameter $\rho$ .

$\rho$	OS (Hz)	RT (s)	SSE (Hz)	IAE	ITAE
0.03	0.0962	18.55	5.29 $\times {10}^{-6}$	0.16	1.9
0.06	0.0967	22.74	1.61 $\times {10}^{-4}$	0.25	2.8
0.12	0.0971	28.03	3.43 $\times {10}^{-3}$	0.28	3.2

OS: overshoot; RT: regulation time; SSE: steady-state error; IAE: integral of absolute error; ITAE: integral of time-weighted absolute error. The optimal values are highlighted in bold.

Conclusion

This paper presented a data-driven LFC approach for multiarea interconnected power systems. A DDET mechanism, along with an encoding–decoding scheme, is developed to optimize resource efficiency and mitigate the effects of data quantization. The effectiveness of the proposed algorithm is demonstrated through extensive simulations, and the impact of various parameters on control performance is thoroughly analyzed. Future research will focus on exploring prescribed-time LFC strategies for uncertain multiarea interconnected wind power systems with the virtual inertia control system subject to multiple cyber-attacks, which remains a significant challenge.