Dissipative Discrete PID Load Frequency Control for Restructured Wind Power Systems via Non-Fragile Design Approach

Abstract

This article proposes a discrete proportional-integral-derivative (PID) load frequency control (LFC) scheme to investigate the dissipative analysis issue of restructured wind power systems via a non-fragile design approach. Firstly, by taking the different power-sharing rates of governors into full consideration, a unified model is constructed for interconnected power systems containing multiple governors. Secondly, unlike existing LFC schemes, a non-fragile discrete PID control scheme is designed, which has the performance of tolerating control gain fluctuation and relieving the huge computational burden. Further, by constructing a discrete-type Lyapunov–Krasovskii functional, improved stability criteria with a strict dissipative performance index are established. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed control method.

1. Introduction

Load frequency control (LFC) is an important method for ensuring stability. The two main objectives of its implementation into interconnected power systems are maintaining the frequency and net power interchanges with neighboring areas at the scheduled values, by controlling the area control error (ACE) [1]. In recent years, due to changes in the power system environment and increased complexity and changes in the power system structure, each control area contains different kinds of uncertainties and various disturbances. Abrupt load changes can cause a mismatch between generation and demand, resulting in significant frequency deviations. Some relevant articles have studied the low-frequency oscillation issue of system states [2,3,4]. Therefore, to ensure competitiveness and efficiency in the electricity market, it is necessary to restructure the power system. Generation companies (GENCOs) can choose to participate or not in the task of LFC; there are more forms of cooperation between GENCOs and distribution companies (DISCOs), and any DISCO has the possibility of signing a contract with any GENCO under deregulation. Furthermore, the application of renewable energy is becoming increasingly widespread worldwide. Wind power, as an important renewable energy, makes a great contribution to energy conservation and emission reduction [5]. A restructured power system can help to increase the use of renewable energy sources. When a larger wind power generation replaces the conventional generator, the total inertia of the system decreases, and the equivalent regulation constant increases [6]. Some effective measures have been investigated to solve the previous problems. A novel scheme is developed for wind power modeling regardless of dependency distance in [7] to enhance the system’s ability to handle the variability of wind generation. A supplement controller in a wind power control loop is designed to regulate the rotor speed [8] to remove the fluctuations in output wind power. A novel hidden Markov model is developed to obtain more accurate wind power forecast results [9] to guarantee a safe and reliable system operation. To enhance system damping, an optimized power point tracking controller was proposed [10]. To overcome the problem of high levels of wind power penetration, a model with DFIG-based wind turbines was constructed [11]. To counteract load fluctuations, a robust LFC scheme was designed [12]. Different from the methods mentioned above, the change in the active output in wind power is regarded as a load fluctuation, thus fully mobilizing the regulation capacity of conventional units by controlling the unstable frequency caused by the changes of loads and random fluctuations of wind power. A restructured LFC power system model with wind turbine generator sets is studied in this paper.

Designing suitable methods for LFC is of practical significance to minimize the ACE when load demands fluctuate, and various methods have been proposed in many studies, including a robust delay-dependent PI LFC scheme [13], an event-triggered PI control scheme [14], and a robust decentralized PI control scheme [15]. Considering the practical application, the PID control is more popular in the industry than PI control for its simple structure, strong robustness, and so forth [16]. Various approaches have been taken to design PID control in literature. Singh et al. [17] designed a PID controller with an optimum gain subject to linear matrix inequality (LMI) constraints. J.K. Pradhan and A. Ghosh designed a PID controller with multi-input and multi-output [18]. Pandey et al. [19] proposed a robust iterative PID control scheme. However, these control schemes are unable to deal with the uncertainties in the LFC model. Some researchers have used the non-fragile PID control to solve these problems [20,21]. Non-fragile discrete PID control is designed to provide robust performance by considering uncertainties and disturbances in the system. It can handle parameter variations and disturbances more effectively, leading to improved stability and performance compared to some existing LFC schemes. At the same time, it also limits its effectiveness in capturing complex nonlinear dynamics or dealing with highly uncertain systems for the reason that PID control is based on simple linear models. In addition, there are communication and computational burdens due to some microprocessors with limited computing abilities in the LFC scheme. Discrete control has been surveyed in some articles [22,23,24] because it relieves a huge computational burden and occupies smaller storage space. Therefore, a discrete PID controller is proposed with the time-windows of integral operation over a finite but adjustable length to relieve the computational burden. A non-fragile discrete PID control scheme is designed in this article to provide robust performance and save computing resources.

At present, the dissipativity theory has important applications in many topics such as robust control, regulation, and stabilization. Based on the input–output energy-related theory, strong links between physics and the control theory of systems are given by the dissipativity theory. In order to further analyze and design the control system, a framework is also provided [25]. A number of practical engineering systems have applied the dissipativity theory, for example, mechanical systems [26], robotic manipulators [27], and power systems [28,29]. Additionally, the dissipativity has also been applied in discrete-time networks [30,31] and continuous-time networks [32], respectively. The dissipative concept not only flexibly trades off gain and phase but also provides an appropriate framework for designing less conservative robust controls [33]. The results for a few LFC wind power systems motivated this study in which we consider the dissipative analysis for discrete-time interconnected systems. Dissipativity-based stability conditions consider the energy dissipation rates and their relationship with the system dynamics. By incorporating the dissipative behavior of the system, dissipativity-based stability criteria can capture stability properties more accurately and efficiently, resulting in reduced conservatism [34].

In this work, the main contributions can be summarized as the following three aspects:

1.

A restructured wind power system model is introduced. Compared to some existing models, including single-generator unit models [35], the proposed restructured model introduces new information signals and different controller participation coefficients, resulting in better grid reliability.

2.

A non-fragile discrete PID control scheme for interconnected wind power systems is designed, which can tolerate control gain fluctuation and reduce huge computation costs.

3.

Based on the constructed discrete Lyapunov–Krasovskii functional, strict dissipativity conditions were established for wind power systems. The results can be obtained with lower conservatism and higher computational efficiency.

Notations:$\mathbb{N}$ denotes the set of natural numbers; ${\mathbb{N}}^{+}$ denotes the set of positive numbers. Let ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ represent the n-dimensional Euclidean space and the set of $m\times n$ real matrices, respectively. $A^{-1}$ and $A^{\mathrm{T}}$ denote, respectively, the inverse and the transpose of the matrix A. ${\left[{\mathsf{\Omega }}_{ij}\right]}_{n\times n}$ represents the matrix consisting of $n\times n$ blocks, and the block in ith row and jth column is ${\mathsf{\Omega }}_{ij}$. ${\mathbf{0}}_{m\times n}$ and ${\mathit{I}}_{m\times n}$ denote the zero matrix and the identity matrix with dimensions of $m\times n$. $Q>0$ means that Q is a positive symmetric definite. $\mathbb{E}\{·\}$ stands for the expectation of the stochastic variable $“·”$, and $\mathrm{col}\left[\right]$ represents the column vector. Let $\mathrm{diag}\{·\}$ be a diagonal matrix, i.e., $\mathrm{diag}{\left\{X\right\}}_N=\mathrm{diag}\{\underset{N}{\underbrace{X,\dots ,X}}\}$.

2. Preliminaries

This section develops a model of restructured interconnected wind power systems based on a non-fragile discrete PID LFC scheme. In Section 2.1, a model of interconnected wind power systems is constructed. In Section 2.2, a method is investigated to turn the above-mentioned model into a non-fragile discrete model. The parameters involved in the system are defined in

Table 1. Practical meaning of system parameters.

$T_{{\mathrm{t}}_{ni}}$	Speed governor time constants	${\Delta P_{\mathrm{tie}}}_i$	Total tie-lie power deviation
$T_{{\mathrm{g}}_{ni}}$	Turbine time constants	$\Delta P_{{\mathrm{m}}_i}$	Turbines mechanical output
$R_{ni}$	Governor droop characteristic	$\Delta P_{{\mathrm{d}}_i}$	Load demands
$D_i$	Load damping coefficient	$\Delta f_i$	Deviation of valve position
$T_{ij}$	Coefficient of tie line	$\Delta P_{{\mathrm{loc}}_i}$	Contracted local demand
$T_{{\mathrm{w}}_i}$	Wind generator time constants	$\Delta P_{{\mathrm{L}}_j}$	Other contracted demand
${\beta }_i$	System frequency response coefficient	$\Delta P_{{\mathrm{v}}_i}$	Governors output
$M_i$	Inertia constant	$\Delta P_{{\mathrm{w}}_i}$	Wind generator output
${\alpha }_{ni}$	Participation factors of generator	${\mathrm{ACE}}_i$	Area control error

.

2.1. Restructured LFC Wind Power System Model

Let us define the generation participation matrix (GPM), which shows the participation factor of each Genco in the considered control areas, and each control area is determined by a Disco. For a large-scale power system in Figure 1 with n Gencos, the GPM will have the following structure:

$$\begin{array}{c}\hfill GPM=\left[\begin{array}{ccccc}gp{f}_{11}& gp{f}_{12}& \cdots & gp{f}_{1(N-1)}& gp{f}_{1N}\\ gp{f}_{21}& gp{f}_{22}& \cdots & gp{f}_{2(N-1)}& gp{f}_{2N}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ gp{f}_{(n-1)1}& gp{f}_{(n-1)2}& \cdots & gp{f}_{(n-1)(N-1)}& gp{f}_{(n-1)N}\\ gp{f}_{n1}& gp{f}_{n2}& \cdots & gp{f}_{n(N-1)}& gp{f}_{nN}\end{array}\right],\end{array}$$

where $gp{f}_{ij}$ refers to the generation participation factor and shows the participation factor of Genco i in the load following area j. Meanwhile, ${\sum }_{i=1}^{n}gp{f}_{ij}=1$. From Figure 1, we can see that

$$\begin{array}{c}\hfill w_{1i}=\Delta P_{{\mathrm{loc}}_i}+\Delta P_{\mathrm{d}i}\end{array}$$

(1)

$$\begin{array}{c}\hfill \Delta P_{{\mathrm{tie}}_i}=\Delta P_{{\mathrm{tie}}_i,\mathrm{act}}-w_{2i}.\end{array}$$

(2)

We can obtain the following equations for the generalized scheduled $w_{2i}$ and $w_{3i}$

$$\begin{array}{cc}\hfill & w_{2i}=\sum _{j=1,j\ne i}^N\left(\sum _{k=1}^{n}gp{f}_{kj}\right)\Delta P_{\mathrm{L}j}-\sum _{k=1}^n\left(\sum _{j=1,j\ne i}^{N}gp{f}_{jk}\right)\Delta P_{\mathrm{L}i}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & w_{3i,1}=\sum _{j=1}^{N}gp{f}_{1j}\Delta P_{\mathrm{L}j},\dots ,w_{3i,n}=\sum _{j=1}^{N}gp{f}_{nj}\Delta P_{\mathrm{L}j}.\hfill \end{array}$$

(4)

The total generation can be represented as

$$\begin{array}{c}\hfill \Delta P_{mi}=\sum _{j=1}^{N}gp{f}_{ij}\Delta P_{Lj}.\end{array}$$

(5)

The model of ith area considering the wind power can be expressed as

$$\left\{\begin{array}{c}\Delta {\dot{f}}_i=\frac{1}{M_i}\left(\Delta P_{{\mathrm{m}}_i}-\Delta P_{{\mathrm{d}}_i}-\Delta P_{{\mathrm{tie}}_i}-D_i\Delta f_i+\Delta P_{{\mathrm{w}}_i}\right)\hfill \\ \Delta {\dot{P}}_{{\mathrm{tie}}_i}=2\pi \sum _{j=1,j\ne i}^N{T}_{ij}\left(\Delta f_i-\Delta f_j\right)\hfill \\ \Delta {\dot{P}}_{{\mathrm{v}}_i}=A_{{31}_i}\Delta f_i+B_{3_i}{u}_i\left(t\right)+A_{{33}_i}\Delta P_{{\mathrm{v}}_i}\hfill \\ \Delta {\dot{P}}_{{\mathrm{w}}_i}=\frac{1}{{T_{\mathrm{w}}}_i}\left(\Delta P_{\mathrm{wind}}-\Delta P_{{\mathrm{w}}_i}\right)\hfill \\ \Delta {\dot{P}}_{{\mathrm{m}}_i}=A_{{23}_i}\Delta P_{{\mathrm{v}}_i}+A_{{22}_i}\Delta P_{{\mathrm{m}}_i}.\hfill \end{array}\right.$$

(6)

Due to the presence of numerous components in the power system that introduce time delays, the inclusion of delay terms was considered in the model development to more accurately depict the dynamic behavior of the system. Define the following vectors

$$\begin{array}{cc}\hfill \Delta P_{{\mathrm{v}}_i}& =\mathrm{col}[\Delta P_{{\mathrm{v}}_{1i}},\Delta P_{{\mathrm{v}}_{2i}},\dots ,\Delta P_{{\mathrm{v}}_{ni}}],w_i=\mathrm{col}[\Delta P_{\mathrm{wind}},\Delta P_{{\mathrm{d}}_i},\sum _{j=1,j\ne i}^N{T}_{ij}\Delta f_j]\hfill \\ \hfill \Delta P_{{\mathrm{m}}_i}& =\mathrm{col}[\Delta P_{{\mathrm{m}}_{1i}},\Delta P_{{\mathrm{m}}_{2i}},\dots ,\Delta P_{{\mathrm{m}}_{ni}}],x_i=\mathrm{col}[\Delta f_i,\Delta P_{{\mathrm{m}}_i},\Delta P_{{\mathrm{v}}_i},\Delta P_{{\mathrm{tie}}_i},\Delta P_{{\mathrm{w}}_i}]\hfill \end{array}$$

and $y_i={\mathrm{ACE}}_i$, we arrive at

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+B_i{u}_i\left(t\right)+A_{{\tau }_i}{x}_i(t-{\tau }_i\left(t\right))+{\tilde{F}}_i{w}_i\left(t\right)\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right),\hfill \end{array}\right.$$

(7)

where

$$\begin{array}{cc}\hfill & A_i=\left[\begin{array}{cccc}-\frac{D_i}{M_i}& A_{{12}_i}& {\mathbf{0}}_{1\times n}& A_{{14}_i}\\ {\mathbf{0}}_{n\times 1}& A_{{22}_i}& A_{{23}_i}& {\mathbf{0}}_{n\times 2}\\ A_{{31}_i}& {\mathbf{0}}_{n\times n}& A_{{33}_i}& {\mathbf{0}}_{n\times 2}\\ A_{{41}_i}& {\mathbf{0}}_{2\times n}& {\mathbf{0}}_{2\times n}& A_{{44}_i}\end{array}\right],{\tilde{F}}_i=\left[\begin{array}{ccc}0& -\frac{1}{M_i}& 0\\ {\mathbf{0}}_{2n\times 1}& {\mathbf{0}}_{2n\times 1}& {\mathbf{0}}_{2n\times 1}\\ 0& 0& -2\pi \\ \frac{1}{{T_{\mathrm{w}}}_i}& 0& 0\end{array}\right],A_{{44}_i}=\left[\begin{array}{cc}0& 0\\ 0& -1/{T_{\mathrm{w}}}_i\end{array}\right]\hfill \\ \hfill & A_{{41}_i}=\left[\begin{array}{c}2\pi \sum _{j=1,j\ne i}^N{T}_{ij}\\ 0\end{array}\right],A_{{12}_i}=\left[\begin{array}{c}1/M_i,\dots ,1/M_i\end{array}\right],A_{{33}_i}=\mathrm{diag}\{-\frac{1}{T_{{\mathrm{g}}_{1i}}},\dots ,-\frac{1}{T_{{\mathrm{g}}_{ni}}}\}\hfill \\ \hfill & A_{{23}_i}=-A_{{22}_i}=\mathrm{diag}\{\frac{1}{T_{{\mathrm{t}}_{1i}}},\dots ,\frac{1}{T_{{\mathrm{t}}_{ni}}}\},C_i=\left[\begin{array}{cccc}{\beta }_i& {\mathbf{0}}_{1\times 2n}& 1& 0\end{array}\right],A_{{14}_i}=\left[\begin{array}{c}-1/M_i,1/M_i\end{array}\right]\hfill \\ \hfill & A_{{31}_i}=\mathrm{col}\left[\begin{array}{c}-\frac{1}{T_{{\mathrm{g}}_{1i}}{R}_{1i}},\dots ,-\frac{1}{T_{{\mathrm{g}}_{ni}}{R}_{ni}}\end{array}\right],B_i=\mathrm{col}\left[\begin{array}{c}{\mathbf{0}}_{(n+1)\times 1},B_{3_i},0,0\end{array}\right],B_{3_i}=\mathrm{col}\left[\begin{array}{c}\frac{{\alpha }_{1i}}{T_{{\mathrm{g}}_{1i}}},\dots ,\frac{{\alpha }_{ni}}{T_{{\mathrm{g}}_{ni}}}\end{array}\right].\hfill \end{array}$$

When taking into account the dotted line connections representing new load demands based on deregulated contracts, one can obtain that

$$\left\{\begin{array}{c}\Delta {\dot{f}}_i=\frac{1}{M_i}\left(\Delta P_{{\mathrm{m}}_i}+\Delta P_{{\mathrm{w}}_i}-\Delta P_{{\mathrm{tie}}_i}-D_i\Delta f_i-\Delta P_{{\mathrm{tie}}_i}-w_{1i}-w_{2i}\right)\hfill \\ \Delta {\dot{P}}_{{\mathrm{tie}}_i}=2\pi \sum _{j=1,j\ne i}^N{T}_{ij}\left(\Delta f_i-\Delta f_j\right)-w_{2i}\hfill \\ \Delta {\dot{P}}_{{\mathrm{v}}_i}=A_{{31}_i}\Delta f_i+B_{3_i}{u}_i\left(t\right)+A_{{33}_i}\Delta P_{{\mathrm{v}}_i}+F_{{35}_i}{w}_{3i}.\hfill \end{array}\right.$$

(8)

Define

$$\begin{array}{c}\hfill {\tilde{w}}_i=\mathrm{col}\left[\begin{array}{c}\Delta P_{\mathrm{wind}},w_{1i},\sum _{j=1,j\ne i}^N{T}_{ij}\Delta f_j,w_{2i},w_{3i}\end{array}\right],w_{3i}=\mathrm{col}\left[\begin{array}{c}{w}_{3i,1},w_{3i,2},\dots ,w_{3i,n}\end{array}\right]\end{array}$$

we can obtain that

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A_i{x}_i\left(t\right)+B_i{u}_i\left(t\right)+A_{{\tau }_i}{x}_i(t-{\tau }_i\left(t\right))+F_i{\tilde{w}}_i\left(t\right)\hfill \\ y_i\left(t\right)=C_i{x}_i\left(t\right),\hfill \end{array}\right.$$

(9)

where

$$\begin{array}{cc}\hfill & F_i=\left[\begin{array}{ccccc}0& -\frac{1}{M_i}& 0& -\frac{1}{M_i}& 0\\ {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}\\ {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& {\mathbf{0}}_{n\times 1}& F_{35i}\\ 0& 0& -2\pi & -1& 0\\ \frac{1}{T_{{\mathrm{w}}_i}}& 0& 0& 0& 0\end{array}\right],F_{{35}_i}=\mathrm{col}\left[\begin{array}{c}\frac{1}{T_{{\mathrm{g}}_{1i}}},\dots ,\frac{1}{T_{{\mathrm{g}}_{ni}}}\end{array}\right].\hfill \end{array}$$

2.2. Non-Fragile Discrete PID LFC Scheme

Different areas in interconnected LFC wind power systems are often associated with digital devices, which send discrete data to their neighboring areas. Define $T>0$ as the sampling interval. Then, one can obtain a discrete state-space model of multi-area as

$$\left\{\begin{array}{c}{x}_i(k+T)={\overline{A}}_i{x}_i\left(k\right)+{\overline{B}}_i{u}_i\left(k\right)+{\overline{A}}_{{\tau }_i}{x}_i(k-{\tau }_i\left(k\right))+{\overline{F}}_i{w}_i\left(k\right)\hfill \\ y_i\left(k\right)={\overline{C}}_i{x}_i\left(k\right)\hfill \\ x_i\left(l\right)={\phi }_i\left(l\right),l=-{\tau }_M,-{\tau }_M+1,\dots ,0,\hfill \end{array}\right.$$

(10)

where ${\overline{A}}_i=e^{A_{i}T}$, ${\overline{B}}_i={\int }_0^T{e}^{A_{i}t}{B}_{i}dt$, ${\overline{A}}_{{\tau }_i}={\int }_0^T{e}^{A_{i}t}{A}_{{\tau }_i}dt$, ${\overline{F}}_i={\int }_0^T{e}^{A_{i}t}{F}_{i}dt$, ${\overline{C}}_i=C_i$. ${\tau }_i\left(k\right)$ denotes the discrete time delay of the ith area.

Define $x\left(k\right)=\mathrm{col}[x_1\left(k\right),\dots ,x_N\left(k\right)],u\left(k\right)=\mathrm{col}[u_1\left(k\right),\dots ,u_N\left(k\right)],y\left(k\right)=\mathrm{col}[y_1\left(k\right),\dots ,y_N\left(k\right)]$, and $w\left(k\right)=\mathrm{col}[{\tilde{w}}_1\left(k\right),\dots ,{\tilde{w}}_N\left(k\right)]$. Then, a state-space representation of multi-area power systems can be expressed as

$$\left\{\begin{array}{c}x(k+1)=\overline{A}x\left(k\right)+\overline{B}u\left(k\right)+{\overline{A}}_{\tau }x(k-\tau \left(k\right))+\overline{F}w\left(k\right)\hfill \\ y\left(k\right)=\overline{C}x\left(k\right)\hfill \\ x\left(l\right)=\phi \left(l\right),l=-{\tau }_M,-{\tau }_M+1,\dots ,0,\hfill \end{array}\right.$$

(11)

where $\overline{A}=\mathrm{diag}\{{\overline{A}}_1,\dots ,{\overline{A}}_N\}$, $\overline{B}=\mathrm{diag}\{{\overline{B}}_1,\dots ,{\overline{B}}_N\}$, $\overline{C}=\mathrm{diag}\{{\overline{C}}_1,\dots ,{\overline{C}}_N\}$, ${\overline{A}}_{\tau }=\mathrm{diag}\{{\overline{A}}_{{\tau }_1},\dots ,{\overline{A}}_{{\tau }_N}\}$, $\overline{F}=\mathrm{diag}\{{\overline{F}}_1,\dots ,{\overline{F}}_N\}$. $\tau \left(k\right)=\max \left\{{\tau }_i\left(k\right)\right\}(i=1,2,...,N)$, and $\phi \left(l\right)(l=-{\tau }_M,-{\tau }_M+1,\dots ,0)$ are the initial conditions.

Then, with the fluctuation in the gain matrices considered, a non-fragile LFC controller is designed as

$$\begin{array}{c}\hfill u\left(k\right)=(K_{\mathrm{P}}+\alpha \left(k\right)\Delta K_{\mathrm{P}})y\left(k\right)+(K_{\mathrm{I}}+\beta \left(k\right)\Delta K_{\mathrm{I}})\sum _{l=k-m}^{k-1}y\left(l\right)+(K_{\mathrm{D}}+\lambda \left(k\right)\Delta K_{\mathrm{D}})(y\left(k\right)-y(k-1)),\end{array}$$

(12)

where the change in proportion deviation can be calculated by $\Delta K_{\mathrm{P}}$, the change in integral deviation can be calculated by $\Delta K_{\mathrm{I}}$, and $\Delta K_{\mathrm{D}}$ denotes the change in differential deviation. m is a known scalar representing the length of the time-windows, and it is assumed that $m\ge {\tau }_M$ in this paper. The mathematical characters $\alpha \left(k\right)\in \left\{0,1\right\}$, $\beta \left(k\right)\in \left\{0,1\right\}$, $\lambda \left(k\right)\in \left\{0,1\right\}$ are assumed to be known as

$$\begin{array}{c}\hfill \mathbb{E}\left\{\varpi \left(k\right)\right\}=\varpi ,\mathbb{E}\left\{{(\varpi \left(k\right)-\varpi )}^2\right\}=\varpi (1-\varpi )(\varpi \in \{\alpha ,\beta ,\lambda \}).\end{array}$$

(13)

Define $\chi \left(k\right)=\mathrm{col}\left[\begin{array}{c}x(k-1),x(k-2),\dots ,x(k-m)\end{array}\right]$, it follows from (11) that a discrete model can be expressed as

$$\left\{\begin{array}{c}x(k+1)=({\mathcal{A}}_1+\tilde{\alpha }\left(k\right){\mathcal{A}}_2+\tilde{\lambda }\left(k\right){\mathcal{A}}_3)x\left(k\right)+({\mathcal{B}}_1+\tilde{\beta }\left(k\right){\mathcal{B}}_2-\tilde{\lambda }\left(k\right){\mathcal{B}}_3)\chi \left(k\right)\hfill \\ +{\overline{A}}_{\tau }x(k-\tau \left(k\right))+\overline{F}w\left(k\right)\hfill \\ y\left(k\right)=\overline{C}x\left(k\right)\hfill \\ x\left(l\right)=\phi \left(l\right),l=-{\tau }_M,-{\tau }_M+1,\dots ,0,\hfill \end{array}\right.$$

(14)

where

$$\begin{array}{cc}\hfill & {\mathcal{A}}_1=\overline{A}+\overline{B}(K_{\mathrm{D}}+K_{\mathrm{P}}+\overline{\lambda }\Delta K_{\mathrm{D}}+\overline{\alpha }\Delta K_{\mathrm{P}})\overline{C},{\mathcal{A}}_2=\overline{B}\Delta K_{\mathrm{P}}\overline{C},{\mathcal{A}}_3=\overline{B}\Delta K_{\mathrm{D}}\overline{C},\tilde{\lambda }\left(k\right)=\lambda \left(k\right)-\overline{\lambda }\left(k\right)\hfill \\ \hfill & {\mathcal{B}}_2=\left[\begin{array}{c}\overline{B}\Delta K_{\mathrm{I}}\overline{C},\overline{B}\Delta K_{\mathrm{I}}\overline{C},\dots ,\overline{B}\Delta K_{\mathrm{I}}\overline{C}\end{array}\right],{\mathcal{B}}_3=\left[\begin{array}{c}\overline{B}\Delta K_{\mathrm{D}}\overline{C},0,\dots ,0\end{array}\right],\tilde{\beta }\left(k\right)=\beta \left(k\right)-\overline{\beta }\left(k\right)\hfill \\ \hfill & {\mathcal{B}}_1=\left[\begin{array}{c}\overline{B}(K_{\mathrm{I}}+\overline{\beta }\Delta K_{\mathrm{I}}-(K_{\mathrm{D}}+\overline{\lambda }\Delta K_{\mathrm{D}}))\overline{C},\overline{B}{K}_{\mathrm{I}}\overline{C},\dots ,\overline{B}{K}_{\mathrm{I}}\overline{C}\end{array}\right],\tilde{\alpha }\left(k\right)=\alpha \left(k\right)-\overline{\alpha }\left(k\right)\hfill \\ \hfill & \left[\begin{array}{c}\Delta K_{\mathrm{P}},\Delta K_{\mathrm{I}},\Delta K_{\mathrm{D}}\end{array}\right]=\mathsf{\Xi }ϝ\left(k\right)\left[\begin{array}{c}{\beth }_1,{\beth }_2,{\beth }_3\end{array}\right].\hfill \end{array}$$

where $\mathsf{\Xi }$ and ${\beth }_{\iota }$ ( $\iota =1,2,3$) are the constant matrices, and $ϝ\left(k\right)$ is an indeterminate matrix constrained by $ϝ\left(k\right){ϝ}^T\left(k\right)\le I$. The discrete time delay $\tau \left(k\right)$ satisfies ${\tau }_m\le \tau \left(k\right)\le {\tau }_M(k\in {\mathbb{N}}^{+})$. ${\tau }_m$ represents the lower bound of $\tau \left(k\right)$, and ${\tau }_M$ represents the upper bound of $\tau \left(k\right)$, which are all known scalars.

Remark 1.

Multiple generation sets have been introduced into the interconnected wind power system to adjust the turbine input power, they decide the share of the load demands. In the reconstructed model, a reasonable power resource allocation scheme can balance the load requirements of the demand side and the supply side, and new information signals representing the power flow mobility between regions have been proposed, which can improve the efficiency and reliability of frequency regulation.

Remark 2.

Notice that in the above discrete-time controller, three PID controller gain matrices, $K_{\mathrm{P}}$, $K_{\mathrm{I}}$, and $K_{\mathrm{D}}$, are included. Current information and the trend in the information of outputs are well utilized with the introduction of the proportional terms $K_{\mathrm{P}}$ and differential terms $K_{\mathrm{D}}$. Additionally, history measurements are fully used in the integral term $K_{\mathrm{I}}$. In particular, compared with the continuous-time controller proposed in [36], a limited but adjusted length of time window is also designed in integral terms of discrete PID to relieve the computational burden.

Remark 3.

Further, non-fragile control schemes are taken into account in the discrete PID controller in (12). $\alpha \left(k\right)$, $\beta \left(k\right)$, and $\lambda \left(k\right)$ describe the occurring status of the perturbation. When $\alpha \left(k\right)$, $\beta \left(k\right)$, and $\lambda \left(k\right)=0$, it means that there is no proportion deviation. When $\alpha \left(k\right)$, $\beta \left(k\right)$, and $\lambda \left(k\right)=1$, it means that there are multiplicative uncertainties being considered, which better explains the possible gain changes in the implementation process.

Lemma 1

([37]). Given matrices $Y>0$, D, and E with the appropriate dimensions, then

$$\begin{array}{c}\hfill Y+DF\left(k\right)E+E^{\mathrm{T}}{F}^{\mathrm{T}}\left(k\right)D^{\mathrm{T}}<0\end{array}$$

(15)

for all $F\left(k\right)$ satisfying $F^{\mathrm{T}}\left(k\right)F\left(k\right)\le I$, if and only if there exists a scalar $\xi D{D}^{\mathrm{T}}+{\xi }^{-1}{E}^{\mathrm{T}}E<0$.

3. Results

Based on the model (14) proposed above, this section established dissipativity conditions in Theorem 1. Then, the control gain matrices based on a non-fragile discrete PID control scheme are designed in Theorem 2. This study aims to achieve an analysis of multi-area wind power systems, save limited network resources, and guarantee dissipative control. In the following, the definition of dissipativity is given.

Definition 1

([30]). System (14) is said to be strictly $(U_1,U_2,U_3)$-γ-dissipative if, for any $\gamma >0$, such that

$$\begin{array}{c}\hfill \sum _{k=0}^{k^{*}}r(w\left(k\right),y\left(k\right))\ge \gamma \sum _{k=0}^{k^{*}}{w}^{\mathrm{T}}\left(k\right)w\left(k\right),\forall k^{*}\ge 0\end{array}$$

(16)

holds under the zero initial condition. The energy supply rate function $r\left(w\right(k),y(k\left)\right)$ is defined as follows.

$$\begin{array}{c}\hfill r(w\left(k\right),y\left(k\right))=y^{\mathrm{T}}\left(k\right)U_{1}y\left(k\right)+2y^{\mathrm{T}}\left(k\right)U_{2}w\left(k\right)+w^{\mathrm{T}}\left(k\right)U_{3}w\left(k\right),\end{array}$$

(17)

where $U_1$, $U_2$, and $U_3$ are real matrices with $U_1=U_1^{\mathrm{T}}$, $U_3=U_3^{\mathrm{T}}$, and $U_1\le 0$.

Based on the discrete-type Lyapunov–Krasovskii functional, sufficient conditions for the control of system (14) are derived in the following theorem for given control feedback gains $K_{\mathrm{P}}$, $K_{\mathrm{I}}$, and $K_{\mathrm{D}}$.

Theorem 1.

For a given integer $\gamma >0$, $\overline{\alpha }>0$, $\overline{\lambda }>0$, $\overline{\beta }>0$, $\check{\alpha }>0$, $\check{\lambda }>0$, $\check{\beta }>0$, ${\tau }_M>0$, ${\tau }_m>0$, ${\epsilon }_1$, ${\epsilon }_2$, system (14) is strictly $(U_1,U_2,U_3)$-γ-dissipative, if there exist $P>0$, $Q_1>0$, $Q_2>0$, $Q_3>0$, $R>0$ and ${\beth }_1$, ${\beth }_2$, ${\beth }_3$, Ξ of appropriate dimensions satisfying

$$\begin{array}{c}\hfill {\Pi }_1=\left[\begin{array}{cc}{\Pi }_1^{11}& *\\ {\Pi }_1^{21}& {\Pi }_1^{22}\end{array}\right]<0,\end{array}$$

(18)

$$\begin{array}{c}\hfill {\Pi }_2=\left[\begin{array}{cc}{\Pi }_2^{11}& *\\ {\Pi }_2^{21}& {\Pi }_2^{22}\end{array}\right]<0,\end{array}$$

(19)

where

$$\begin{array}{cc}\hfill & {\Pi }_1^{22}=-\mathrm{diag}{\left\{\mathrm{P}\right\}}_6,{\Pi }_2^{22}=-(1+1/{\epsilon }_2)P,\check{\lambda }=\overline{\lambda }(1-\overline{\lambda }),{\Pi }_2^{21}=P\overline{F}\hfill \\ \hfill & \varsigma =(2n+3)\times N,{\Pi }_2^{11}=\gamma I-U_3-1/{\epsilon }_2{U}_2,Q=\mathrm{diag}\{Q_1,Q_2,\dots ,Q_m\}\hfill \\ \hfill & \check{\alpha }=\overline{\alpha }(1-\overline{\alpha }),\check{\beta }=\overline{\beta }(1-\overline{\beta }),\mathcal{P}=-P+\sum _{n=1}^m{Q}_n+({\tau }_M-{\tau }_m+1)R\hfill \\ \hfill & {\Pi }_1^{11}=\left[\begin{array}{ccc}\mathcal{P}-{\epsilon }_2{\overline{C}}^T{U}_2\overline{C}-{\overline{C}}^T{U}_1\overline{C}& {\mathbf{0}}_{\varsigma \times \varsigma }& {\mathbf{0}}_{\varsigma \times \varsigma }\\ {\mathbf{0}}_{\varsigma \times \varsigma }& -Q& {\mathbf{0}}_{\varsigma \times \varsigma }\\ {\mathbf{0}}_{\varsigma \times \varsigma }& {\mathbf{0}}_{\varsigma \times \varsigma }& -R\end{array}\right],{\Pi }_1^{21}=\left[\begin{array}{ccc}P{\mathcal{A}}_1& P{\mathcal{B}}_1& P{\overline{A}}_{\tau }\\ \sqrt{{\epsilon }_1}P{\mathcal{A}}_1& \sqrt{{\epsilon }_1}P{\mathcal{B}}_1& \sqrt{{\epsilon }_1}P{\overline{A}}_{\tau }\\ \sqrt{\check{\alpha }}P{\mathcal{A}}_2& {\mathbf{0}}_{\varsigma \times \varsigma }& {\mathbf{0}}_{\varsigma \times \varsigma }\\ \sqrt{\check{\lambda }}P{\mathcal{A}}_3& {\mathbf{0}}_{\varsigma \times \varsigma }& {\mathbf{0}}_{\varsigma \times \varsigma }\\ {\mathbf{0}}_{\varsigma \times \varsigma }& \sqrt{\check{\beta }}P{\mathcal{B}}_2& {\mathbf{0}}_{\varsigma \times \varsigma }\\ {\mathbf{0}}_{\varsigma \times \varsigma }& \sqrt{\check{\lambda }}P{\mathcal{B}}_3& {\mathbf{0}}_{\varsigma \times \varsigma }\end{array}\right].\hfill \end{array}$$

 Proof

(Proof of Theorem 1). Please see Appendix A. □

Remark 4.

In this paper, the discrete-type Lyapunov–Krasovskii functional is a mathematical function that assigns a scalar value to each state, which provides a systematic approach to analyze the stability of discrete-time wind power systems with higher computational efficiency. It is constructed based on the system dynamics and positive definiteness. The functional plays a crucial role in analyzing the stability and designing non-fragile discrete PID controllers for restructured wind power systems. We can see from the proof of Theorem 1 that the stability condition was derived based on the calculation of the forward difference $\Delta V\left(k\right)$, which deserves some comments. To bring the information of the delay into the final result, the Lyapunov–Krasovskii functional is chosen to be $V_3\left(k\right)$, and the calculation of $\Delta V_3\left(k\right)$ led to ${\sum }_{\rho =k-\tau (k+1)+1}^{k-1}{x}^{\mathrm{T}}\left(\rho \right)Rx\left(\rho \right)$, which was additionally introduced and gave rise to possible conservativeness. We have enlarged it to ${\sum }_{\rho =k-\tau (k+1)+1}^{k-{\tau }_m}{x}^{\mathrm{T}}\left(\rho \right)Rx\left(\rho \right)$ and ${\sum }_{\rho =k-{\tau }_m+1}^{k-1}{x}^{\mathrm{T}}\left(\rho \right)Rx\left(\rho \right)$ to reduce possible conservativeness.

Remark 5.

It is worth mentioning that a strict $(U_1,U_2,U_3)-\gamma$ dissipative analysis problem is more general than some other problems. When $U_1=-I$, $U_2=0$, and $U_3=2\gamma I$, the $(U_1,U_2,U_3)$ dissipative properties degenerate into $H_{\infty }$ properties strictly, which is shown by a dissipativity analysis. The dissipatiivity analysis in this paper is less conservative than $H_{\infty }$ adopted for LFC of power system by most other studies.

Theorem 2.

For a given integer $\gamma >0$, $\overline{\alpha }>0$, $\overline{\lambda }>0$, $\overline{\beta }>0$, $\check{\alpha }>0$, $\check{\lambda }>0$, $\check{\beta }>0$, ${\tau }_M>0$, ${\tau }_m>0$, ${\epsilon }_1$, ${\epsilon }_2$, system (14) is strictly $(U_1,U_2,U_3)$-γ-dissipative, if there exist $\widehat{P}>0$, ${\widehat{Q}}_1>0$, ${\widehat{Q}}_2>0$, ${\widehat{Q}}_3>0$, $\widehat{R}>0$ and ${\beth }_1$, ${\beth }_2$, ${\beth }_3$, Ξ of appropriate dimensions satisfying

$$\begin{array}{c}\hfill {\Pi }_3=\left[\begin{array}{cc}{\Pi }_3^{11}& *\\ {\Pi }_3^{21}& {\Pi }_3^{22}\end{array}\right]<0,\end{array}$$

(20)

$$\begin{array}{c}\hfill {\tilde{\Pi }}_2=\left[\begin{array}{cc}{\tilde{\Pi }}_2^{11}& *\\ {\tilde{\Pi }}_2^{21}& {\tilde{\Pi }}_2^{22}\end{array}\right]<0,\end{array}$$

(21)

where

$$\begin{array}{cc}\hfill & {\Pi }_3^{11}=\left[\begin{array}{cc}{\Phi }_1& *\\ {\Phi }_2& {\Phi }_3\end{array}\right],{\Phi }_5=\left[\begin{array}{ccc}{\mathbf{0}}_{N\times \varsigma }& {\gimel }_{2}X& {\gimel }_{2}X\\ {\mathbf{0}}_{N\times \varsigma }& {\gimel }_{3}X& {\mathbf{0}}_{N\times \varsigma }\end{array}\right],{\Phi }_2^{31}=\left[\begin{array}{c}{(-U_1)}^{\frac{1}{2}}\overline{C}\\ {(-U_2)}^{\frac{1}{2}}\overline{C}\end{array}\right],\widehat{\mathsf{\Xi }}=\overline{B}\mathsf{\Xi }{\mathsf{\Xi }}^{\mathrm{T}}{\overline{B}}^{\mathrm{T}}\hfill \\ \hfill & {\gimel }_1={\beth }_1\overline{C},{\Pi }_3^{22}=\mathrm{diag}{\{-\eta {\mathit{I}}_{N\times N}\}}_6,{\Phi }_1(3,3)=-\widehat{R},{\tilde{\Pi }}_2^{22}=-(1+1/{\epsilon }_2)\widehat{P},{\gimel }_2={\beth }_2\overline{C}\hfill \\ \hfill & {\Phi }_1(1,1)=-\widehat{P}+\sum _{n=1}^m{\widehat{Q}}_n+({\tau }_M-{\tau }_m+1)\widehat{R},{\Phi }_2^{13}={\left[\begin{array}{cc}{\overline{A}}_{\tau }X& \sqrt{{\epsilon }_1}{\overline{A}}_{\tau }X\end{array}\right]}^{\mathrm{T}},{\gimel }_3={\beth }_3\overline{C}\hfill \\ \hfill & {\Phi }_3=\mathrm{diag}\{{\Phi }_3^1,-\widehat{P},-\widehat{P},-\widehat{P},-\widehat{P},{\widehat{U}}_1\},{\tilde{\Pi }}_2^{21}=\overline{F},{\widehat{U}}_1={U_1}^{-1},{\Phi }_1(2,2)=-\widehat{Q}\hfill \\ \hfill & {\Phi }_3^1=\left[\begin{array}{cc}-\widehat{P}+{\overline{\alpha }}^2\widehat{\mathsf{\Xi }}+{\overline{\lambda }}^2\widehat{\mathsf{\Xi }}& \sqrt{{\epsilon }_1}({\overline{\alpha }}^2\widehat{\mathsf{\Xi }}+{\overline{\lambda }}^2\widehat{\mathsf{\Xi }})\\ \sqrt{{\epsilon }_1}({\overline{\alpha }}^2\widehat{\mathsf{\Xi }}+{\overline{\lambda }}^2\widehat{\mathsf{\Xi }})& -\widehat{P}+{\epsilon }_1({\overline{\alpha }}^2\widehat{\mathsf{\Xi }}+{\overline{\lambda }}^2\widehat{\mathsf{\Xi }})\end{array}\right],{\Pi }_3^{21}=\left[\begin{array}{cc}{\Phi }_4& {\mathbf{0}}_{4N\times 3\varsigma }\\ {\Phi }_5& {\Phi }_6\end{array}\right],{\tilde{\Pi }}_2^{11}=\gamma {\mathit{I}}_{N\times N}-U_3\hfill \\ \hfill & {\Phi }_6=\left[\begin{array}{cccc}{\gimel }_{2}X& {\gimel }_{2}X& {\gimel }_{2}X& {\mathbf{0}}_{N\times \varsigma }\\ {\gimel }_{3}X& {\mathbf{0}}_{N\times \varsigma }& {\mathbf{0}}_{N\times \varsigma }& {\mathbf{0}}_{N\times \varsigma }\end{array}\right],{\Phi }_4=\left[\begin{array}{cc}{\gimel }_{3}X& {\gimel }_{2}X\\ {\gimel }_{1}X& -{\gimel }_{3}X\\ {\gimel }_{1}X& {\mathbf{0}}_{N\times \varsigma }\\ {\gimel }_{3}X& {\mathbf{0}}_{N\times \varsigma }\end{array}\right],{\Phi }_2=\left[\begin{array}{ccc}{\Phi }_2^{11}& {\mathbf{0}}_{2\varsigma \times 2\varsigma }& {\Phi }_2^{13}\\ {\mathbf{0}}_{4\times \varsigma }& {\mathbf{0}}_{4\times 2\varsigma }& {\mathbf{0}}_{4\times \varsigma }\\ {\Phi }_2^{31}& {\mathbf{0}}_{N\times \varsigma }& {\mathbf{0}}_{N\times \varsigma }\end{array}\right]\hfill \\ \hfill & {\Phi }_2^{11}=\left[\begin{array}{cc}\overline{A}X+\overline{B}{Y}_{\mathrm{D}}+\overline{B}{Y}_{\mathrm{P}}& \overline{B}\mathcal{Y}\\ \sqrt{{\epsilon }_1}(\overline{A}X+\overline{B}{Y}_{\mathrm{D}}+\overline{B}{Y}_{\mathrm{P}})& \sqrt{{\epsilon }_1}\overline{B}\mathcal{Y}\end{array}\right],\widehat{Q}=\mathrm{diag}\{{\widehat{Q}}_1,{\widehat{Q}}_2,\dots ,{\widehat{Q}}_m\}.\hfill \end{array}$$

In addition, the PID control feedback gains are given by

$$\begin{array}{c}\hfill K_{\mathrm{P}}=N_{\mathrm{p}}{M_1}^{-1},K_{\mathrm{I}}=N_{\mathrm{I}}{M_1}^{-1},K_{\mathrm{D}}=N_{\mathrm{D}}{M_1}^{-1}.\end{array}$$

(22)

 Proof

(Proof of Theorem 2). Please see Appendix B. □

4. Illustrative Examples

In this section, the effectiveness of the model proposed in the above section are demonstrated in two parts. In Section 4.1, the effectiveness of the proposed control scheme is considered to be tested by two comparative simulation examples. In Section 4.2, the robustness of the proposed control scheme is validated by using various load disturbances.

4.1. Effectiveness of Non-Fragile Discrete PID LFC

Frequency deviation and tie-line power are key indicators of the system’s response to load changes and generation imbalances. By analyzing the frequency deviation and tie-line power under different operating conditions and disturbances, we can assess the performance of the load frequency control. The goal is to keep the frequency and net power interchanges with neighboring areas at the scheduled values, typically around the nominal value. If the frequency deviations and tie-line power are small and quickly return to the nominal value after disturbances, it indicates effective load frequency control.

In this section, $\overline{\alpha }=0.7$, $\overline{\beta }=0.6$, and $\overline{\lambda }=0.5$, which are the average success probabilities of the non-fragile items, and the trajectories of $\alpha ,\beta ,$ and $\lambda$ are shown in Figure 2. The time length of the integral loop in the discrete PID controller (12) is taken as $m=3$, the lower bound of the discrete time delay is taken as ${\tau }_m=1$, and the upper bound is taken as ${\tau }_M=3$.

Example 1

([38]). The parameters of the model (14) in two areas are listed in

Table 2. Parameters of two-area LFC scheme and each including two governors.

Parameters	${\mathit{T}}_{\mathbf{t}}$	${\mathit{T}}_{\mathbf{g}}$	R	D	$\mathit{\beta }$	M	$\mathit{\alpha }$
Area 1	3.0	1.0	0.05	0.1884	21.0	0.1667	1.0
Area 2	4.0	1.7	0.05	0.1884	21.5	0.2084	1.0

.

The initial states in this example for two areas are defined as $x(-3)=x(-2)=x(-1)=x\left(0\right)={\left[\begin{array}{ccccc}-4& 4& -5& 5& 2\\ 1& 1& 2& -2& 4\end{array}\right]}^{\mathrm{T}}$, and the fluctuation in the gain matrices of the non-fragile item in the control scheme of two areas are given as

$$\begin{array}{c}\hfill \Delta K_{\mathrm{P}}=\left[\begin{array}{cc}-1.8400& -0.0400\\ -0.8750& -1.1250\end{array}\right],\Delta K_{\mathrm{I}}=\left[\begin{array}{cc}-0.2400& -0.4400\\ -0.3750& -0.6250\end{array}\right],\Delta K_{\mathrm{D}}=\left[\begin{array}{cc}-0.4800& -0.1000\\ -0.6250& -0.1875\end{array}\right].\end{array}$$

By solving LMI, the solution to matrix inequalities (A15) and (20) in Theorem 2 is computed. Then, the control gains of (14) for two areas are outlined as follows:

$$\begin{array}{c}\hfill K_{\mathrm{P}}=\left[\begin{array}{cc}0.4770& -0.0050\\ 0.0008& 0.4638\end{array}\right],K_{\mathrm{I}}=\left[\begin{array}{cc}0.0005& -0.0013\\ -0.0012& 0.0009\end{array}\right],K_{\mathrm{D}}=\left[\begin{array}{cc}0.4770& -0.0050\\ 0.0008& 0.4638\end{array}\right].\end{array}$$

In order to better demonstrate the effectiveness of the proposed non-fragile discrete PID in this paper, the responses of the system (14) are shown in Figure 3 and Figure 4 for both the cases considering and not considering uncertainties. Additionally, Figure 5 and Figure 6 describe the control inputs for the two operating conditions mentioned above. From Figure 3 and Figure 4, it can be observed that the proposed non-fragile discrete PID control scheme operates stably and exhibits better control performance compared to other control schemes.

Example 2

([38]). The parameters of the proposed model (14) in three areas are listed in

Table 3. Parameters of the three-area LFC scheme and each including two governors.

Parameters	(k-i: kth Governor in ith Area)
	1-1	2-1	1-2	2-2	1-3	2-3
$T_{\mathrm{t}}$	3.2	3	3	3.2	3.1	3.4
$T_{\mathrm{g}}$	0.6	0.8	0.6	0.7	0.8	0.6
R	2.4	2.5	2.5	2.7	2.8	2.4
$\alpha$	0.5	0.5	0.5	0.5	0.6	0.4
Parameters	Areas
	1		2		3
M	0.1667		0.2084		0.1600
D	0.1884		0.1884		0.1780
$\beta$	0.4250		0.3966		0.3522
$T_{ij}$	$T_{12}$ = 0.2450		$T_{13}$ = 0.212		$T_{23}$ = 0.11

.

The initial states in this example for three areas are defined as $x(-3)=x(-2)=x(-1)=x\left(0\right)={\left[\begin{array}{ccccccc}4& 4& -5& 5& 2& 2& 2\\ 5& -4& -5& 5& 2& 2& 2\\ -4& 4& 4& 5& 2& 3& -2\end{array}\right]}^{\mathrm{T}}$; the fluctuation in the gain matrices of the non-fragile item in the control scheme of three areas are given as

$$\begin{array}{cc}\hfill & \Delta K_{\mathrm{P}}=\left[\begin{array}{ccc}-0.8800& -0.0100& -0.6400\\ 0.0900& -1.8650& -0.2500\\ -0.0100& -0.2100& -2.8500\end{array}\right],\Delta K_{\mathrm{I}}=\left[\begin{array}{ccc}0.020& 0.370& 0.210\\ 0.315& 0.035& 0.230\\ 0.260& 0.360& 0.020\end{array}\right]\hfill \\ \hfill & \Delta K_{\mathrm{D}}=\left[\begin{array}{ccc}-0.8800& -0.0750& -0.3800\\ 0.0900& -1.8875& -0.2500\\ -0.0100& -0.2750& -1.9100\end{array}\right].\hfill \end{array}$$

Control gains of (12) for three areas are outlined as follows

$$\begin{array}{cc}\hfill & K_{\mathrm{P}}=\left[\begin{array}{ccc}0.9923& -0.0846& -0.1897\\ -0.0032& 0.9285& -0.1846\\ -0.0168& -0.1236& 0.9407\end{array}\right],K_{\mathrm{I}}=\left[\begin{array}{ccc}0.0135& -0.0057& -0.0015\\ -0.0015& 0.0100& 0.0022\\ -0.0001& 0.0020& 0.0055\end{array}\right]\hfill \\ \hfill & K_{\mathrm{D}}=\left[\begin{array}{ccc}0.9644& 0.0534& 0.1757\\ -0.0046& 1.0155& 0.1810\\ 0.0119& 0.1164& 1.0255\end{array}\right].\hfill \end{array}$$

In this case study, the effectiveness of the non-fragile control scheme was validated through an uncertainty analysis. The simulation results, as shown in Figure 7 and Figure 8, depict the responses of the system (14) in three regions under both uncertain and certain conditions. The input trajectories of the controllers are illustrated in Figure 9 and Figure 10. It is evident that under the design conditions of the non-fragile discrete PID, the system tends to stabilize and exhibits improved performance.

4.2. Case Study of Wind Power System

Robustness assessment involves analyzing the performance of load frequency control under various operating conditions, parameter uncertainties, and disturbances. The control strategy demonstrates resilience and effectiveness in handling uncertainties and disturbances while maintaining system stability and desired frequency response. This section tests the system responses for different scenarios, including possible different bilateral contracts (GPM), wind power system is as shown in the Figure 11.

Case 1: As the first test scenario, the following step load disturbances were applied to two areas: $\Delta P_{{\mathrm{L}}_1}\left(k\right)=100$ MW and $\Delta P_{{\mathrm{L}}_2}\left(k\right)=60$ MW. From $k=0$ to $k=200$, no disturbances were present in the system, and, at $k=200$, a disturbance satisfying the previous condition was introduced. The GPM is given as follows:

$$\begin{array}{c}\hfill GPM=\left[\begin{array}{ccc}0.5& 0& 0\\ 0.25& 0& 0\\ 0& 0.5& 0\\ 0& 0.25& 0\\ 0& 0& 0.5\\ 0& 0& 0.25\end{array}\right].\end{array}$$

As can be seen from Figure 12, the system finally reached stability under the conditions of Case 1.

Case 2: Generally, load disturbances in wind power systems are uncertain. Therefore, it is necessary to test the effectiveness of the controller considering randomly varying load disturbances. For simulation, the uncertainty of load disturbance for each region is calculated by using: $P_{\mathrm{L}}\left(k\right)=R\cos \left(2\pi k\right)$, where R is a random number satisfying $R<1$. The random load disturbance is shown in Figure 12, and it is applied in the system from $k=600$. Given the signing rules between Discos and the available Gencos in other areas as the following GPM:

$$\begin{array}{c}\hfill GPM=\left[\begin{array}{ccc}0.25& 0.25& 0\\ 0.5& 0& 0\\ 0& 0.25& 0.75\\ 0.25& 0.25& 0\\ 0& 0.25& 0\\ 0& 0& 0.25\end{array}\right].\end{array}$$

The simulation results confirm that the output signal eventually tends to zero, thus demonstrating the stability of the proposed system under random load disturbance.

The output signal of the area controlled can be seen in Figure 13, which indicates that the proposed control scheme can effectively achieve stability under various load fluctuations. The simulation results establish the efficacy of the suggested controller in moderating frequency deviation and tie-line power change in the restructured environment.

5. Conclusions

This paper has designed an LFC for reconstructed power systems containing multiple generation sets. Firstly, a restructured wind power system model has been introduced, which incorporated competition among market participants and provided a favorable environment for the integration of wind power into the grid to maintain grid reliability. Then, in order to relieve the computation burden of the system, a discrete PID control scheme has been designed. Moreover, considering the fluctuation in the gain matrices, the non-fragile model has been adopted to describe this phenomenon. Finally, two cases of wind power systems have been given to demonstrate the effectiveness of the proposed control approach under sufficient conditions to satisfy dissipativity. The proposed control scheme has also been verified by numerical examples.