Advanced Sliding Mode Design for Optimal Automatic Generation Control in Multi-Area Multi-Source Power System Considering HVDC Link

Abstract

The multi-area multi-source power system (MAMSPS), which uses a variety of power sources including gas, hydro, thermal, and renewable energy, has recently been implemented to balance the growing demand for electricity and the overall capacity for power generation. In this paper, an integral sliding mode control with a single-phase technique (ISMCSP) is applied to two areas, with each area including gas–wind–thermal power systems with HVDC system. Firstly, a two-area gas–wind–thermal power system with HVDC (TAGWTPSH) is the first model in this scheme to consider the parameter uncertainties of a MAMSPS. Secondly, sliding mode design law with a single-phase technique is introduced to alleviate chattering and oscillation problems. Then, power system stability is ensured by the Lyapunov control theory based on the new LMIs technique. Thirdly, the ISMCSP’s effectiveness in a MAMSPS is also assessed under random load patterns and parameter variations regarding settling time and over-/undershoot. The ISMCSP was created to alter the fundamental sliding mode control, and therefore the suggested approach performs better than recently published approaches. This is demonstrated by the frequency overshoot deviation value in frequency deviations: 0.7 × 10⁻³ to 2.8 × 10⁻³ for the TAGWTPSH with the suggested ISMCSP. In the last case, for random changes in load from −0.4 to +0.5 p. u, the proposed ISMCSP method still stabilizes the frequency of the areas meeting the standard requirements for AGC.

1. Introduction

Maintaining equilibrium between the electrical load and total net generated power is the primary goal of power plant (PP) control. The frequency must be within the allowable range of the power system; this issue is very clearly regulated. PP frequency is affected by a change between load demand and total net generation capacity. Sudden load changes will make automatic generation control (AGC) increasingly complicated. An AGC regulator makes the total generating capacity balance with the load. The AGC of a PP was subjected to classical control techniques by researchers [1]. In the transfer function, the PP is modeled and represented to achieve the classical scheme. Additionally, control gains are employed in PI, PD, and PID standard controllers to handle system stabilization by adjusting the parameters of the PP and AGC following measurements of frequency error and load disturbance. In industry, PID, PD, and PI techniques are commonly employed for the AGC of perturbed PPs. As the daily demand for electricity rises, PPs get bigger and more complex, and they have a taller shape, which increases the complexity of the PPs’ AGC.

Large power plants (PPs) with many generating sets in each area, like multi-area power plants (MAPPs), frequently face the following issues: nonlinearities, random load disturbance, subsystem parameter deviation, extended frequency transient time delay, and area control error (ACE). Therefore, these characteristics affect the AGC of MAPPs, which must be considered for MAPP modeling. Nevertheless, many of the PP models that are currently in use are ill-suited to handle the aforementioned features because the AGC can only monitor two inputs. By expressing the PP modeling of power systems in state space, control engineers have recently solved the previously mentioned problem and made it possible to use modern control techniques for multiple inputs. The AGC of MAPPs has been studied using several different methods in the interim. These include adaptive techniques, observer schemes, optimal control (also known as particle swarm optimization, or PSO), intelligent control (also known as fuzzy logic, etc.), and more. These approaches are combined with traditional PI and PID. In [2], the proportional–integral (PI) approach for the AGC of two areas following an abrupt frequency change was applied using fuzzy logic techniques. The suggested method of control was updated online based on demand for loads using fuzzy logic rules and PI parameters. The improved particle swarm optimization algorithm control was successfully used to track unknown parameters for the MAPP AGC in combination with the classical PI method. The results showed that this approach produces better results than the classical control method [3]. The observer scheme was used when it was difficult to retrieve some of the MAPP parameters. For the AGC of the original MAPP, the variables related to the system state are estimated using the reconstruction of a PP that was originally intended for use with the Luenberger observer [4]. It was evaluated against the standard Luenberger observer to assess the AGC of the single-area PS to confirm the observer–PI’s advantage over it [4]. Practically speaking, the AGC needs to be extremely resistant to significant disturbances. As a result, the AGC of MAPPs developed variable structure control (VSC).

The sliding mode control (SMC) plan is the most often used VSC method. These designs adhere to sliding surface choice structures, switching laws, and control laws, which require that variables be brought to the sliding surface and stay there for a finite amount of time [5,6,7,8,9]. The SMC is indispensable because of its resilience and ability to withstand significant disturbances. The AGC of MAPPs has previously been investigated in SMC [10,11,12,13,14,15,16,17,18,19,20]. The MAPP SMC has been used in conjunction with other techniques over time to address the AGC problem. For MAPP AGC, a unique adaptive strategy based on SMC was applied in response to step load disruption [21]. However, in a power system with an independent source of power for each area, the observer-based SMC was utilized for the AGC of a MAPP in circumstances where it was difficult to obtain certain system state variables [22]. For the MAPP AGC, an integral output feedback control based on SMC was created to stabilize the system under specific disturbances [23]. The single-phase SMC was created to alter the basic SMC to provide unmatched durability for the AGC of MAPPs. This allows for fast and stable system parameters in sliding surfaces along various trajectories. Recently, it was determined that the AGC of a MAPP during step load and randomized load disturbance should adopt single-phase SMC via an observer [24]. The AGC with random load disruption of the New England 39 bus system also employed this technique [24]. The aforementioned SMC techniques were used to calculate the AGC of MAPPs that comprised thermal plants or hydro alone in all regions. In reality, a multi-area multi-source power plant (MAMSPP) consists of multiple generators in each area, encompassing nuclear, hydro, gas, thermal, and additional sources. Examining the AGC of MAMSPPs has not been the subject of many studies. An implementation of a generator-based PID-structured AGC concept controlled the AGC of the two-area thermal power plants (TATPPs) by using a bacterial foraging algorithm (BFA) in response to a step demand shift [25]. To enhance the TATPP dynamic under load disturbance, a unique Teaching Learning Optimization (TLBO) methodology for AGC with a two-degrees-of-freedom of PID (2-DOF PID) controller was additionally established for LFC [26]. To design PID-structured controllers for the Optimized Generation Control (OGC) methodology, which connects two control zones of diverse-source power systems, a new Artificial Intelligence (AI) method called the Jaya algorithm was also used [27].

The sole goal of these activities, which were completed without taking into account the influence of variables and interconnection defects in the system situation matrix, was the AGC of the TAGWTPSH despite load disruptions. To sum up, putting these algorithms into practice is difficult and time-consuming. Therefore, we proposed using an ISMCSP to study the impacts of parameter uncertainty, load disturbances, and changes in subsystem parameters on the AGC of the TAGWTPSH. Compared to modeling the TAGWTPSH and accounting for the impact of connectivity and parameter uncertainties on the scenario matrix, implementing this method is less difficult and demanding. This is the first time that the AGC of the TAGWTPSH was validated using the ISMCSP in comparison to the recently established conventional methodologies. The following is a list of this work’s principal contributions:

By mandating that all system status directions begin at the surface in the initial period, achieve the surface, and remain there indefinitely, the ISMCSP is intended to modify the basic SMC in a way that greatly increases its resistance to disturbances. Unlike a fundamental SMC, which depends on reaching time, this guarantees that the particulars of the PP and the surface for sliding are equivalent.

Comparing the suggested approach to other recently developed AGC approaches in [25,26,27,28,29], the simulation results further confirm the suggested approach’s applicability for the AGC of MAMSPPs. The TAGWTPSH model does take into account the effects of a load trouble subsystem parameter change or the uncertainty of parameters in the current state matrix as in the connected matrix.

2. Methods and Materials

2.1. The Mathematical Representation of the Related Multi-Area Power Plant

A PP block diagram is shown in this section. Power system dynamic models are typically nonlinear. An MAMSPP comprises various generating sets in each region. Nevertheless, the AGC of MAMSPPs does not apply to nuclear plants, since they are mentioned, due to their basic load system [25,26,27,28,29]. Consequently, we consider a TAGWTPS in each area, as shown in Figure 1. The area control error, or ACE, is the power error that results from the linear combination of the power errors in the link network and the frequency variation errors across the system. The TAGWTPS model, which takes into account the connection matrix and load disturbance, is constructed using the following methodology:

$$\begin{array}{l}\Delta {\dot{f}}_1=-{\displaystyle \frac{1}{T_{P{S}_1}}}\Delta f_1+{\displaystyle \frac{K_{P{S}_1}{\alpha }_{11}}{T_{P{S}_1}}}\Delta P_{P{T}_1}+{\displaystyle \frac{K_{P{S}_1}{\alpha }_{12}}{T_{P{S}_i}}}\Delta P_{G{W}_1}+{\displaystyle \frac{K_{P{S}_1}{\alpha }_{13}}{T_{P{S}_1}}}\Delta P_{G{G}_1}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{K_{P{S}_1}}{T_{P{S}_1}}}\Delta P_{ti{e}_{12}}\\ -{\displaystyle \frac{K_{P{S}_1}}{T_{P{S}_1}}}\Delta P_{D_1}-{\displaystyle \frac{K_{P{S}_1}}{T_{P{S}_1}}}\Delta P_{D{C}_1}\end{array}$$

(1)

$$\Delta {\dot{P}}_{P{T}_1}=-{\displaystyle \frac{1}{T_{T_1}}}\Delta P_{P{T}_1}+{\displaystyle \frac{1}{T_{T_1}}}\Delta P_{G{T}_1}$$

(2)

$$\Delta {\dot{P}}_{G{T}_1}=-{\displaystyle \frac{K_{R_1}}{T_{S{G}_1}{R}_{11}}}\Delta f_1-{\displaystyle \frac{1}{T_{R_1}}}\Delta P_{G{T}_1}+\left({\displaystyle \frac{1}{T_{R_1}}}-{\displaystyle \frac{K_{R_1}}{T_{S{G}_1}}}\right)\Delta X_{E{T}_1}+{\displaystyle \frac{K_{R_1}}{T_{S{G}_1}}}\Delta AC{E}_1+{\displaystyle \frac{K_{R_1}}{T_{S{G}_1}}}{U}_{11}$$

(3)

$$\Delta {\dot{X}}_{E{T}_1}=-{\displaystyle \frac{1}{T_{S{G}_1}{R}_{11}}}\Delta f_1-{\displaystyle \frac{1}{T_{S{G}_1}}}\Delta X_{E{T}_1}+{\displaystyle \frac{1}{T_{S{G}_1}}}\Delta AC{E}_1+{\displaystyle \frac{1}{T_{S{G}_1}}}{U}_{11}$$

(4)

$$\Delta {\dot{P}}_{G{W}_1}=-\Delta P_{G{W}_1}+K_{W3}\Delta P_{R{W}_1}$$

(5)

$$\begin{array}{l}\Delta {\dot{P}}_{R{W}_1}=-{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}{R}_{12}}}\Delta f_1-\Delta P_{R{W}_1}+\left(1-{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}\right)\Delta X_{E{W}_1}+{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}\Delta AC{E}_1\\ +{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}{U}_{12}\end{array}$$

(6)

$$\Delta {\dot{X}}_{E{W}_1}=-{\displaystyle \frac{1}{T_{W_2}{R}_{12}}}\Delta f_1-{\displaystyle \frac{1}{T_{W_2}}}\Delta X_{E{W}_1}+{\displaystyle \frac{1}{T_{W_2}}}\Delta AC{E}_1+{\displaystyle \frac{1}{T_{W_2}}}{U}_{12}$$

(7)

$$\Delta {\dot{P}}_{G{G}_1}=-{\displaystyle \frac{1}{T_{C{D}_1}}}\Delta P_{G{G}_1}+{\displaystyle \frac{1}{T_{C{D}_1}}}\Delta P_{R{G}_1}$$

(8)

$$\begin{array}{l}\Delta {\dot{P}}_{R{G}_1}=+{\displaystyle \frac{X_{G_i}{T}_{C{R}_1}}{R_{13}{b}_{G_1}{Y}_{G_1}{T}_{F_1}}}\Delta f_1-{\displaystyle \frac{1}{T_{F_1}}}\Delta P_{R{G}_1}+\left({\displaystyle \frac{1}{T_{F_1}}}+{\displaystyle \frac{T_{C{R}_1}}{Y_{G_1}{T}_{F_1}}}\right)\Delta X_{V{G}_1}\\ \left(\hspace{0.17em}{\displaystyle \frac{c_{G_1}{X}_{G_1}{T}_{C{R}_1}}{b_{G_1}{Y}_{G_1}{T}_{F_1}}}-{\displaystyle \frac{T_{C{R}_1}}{Y_{G_1}{T}_{F_1}}}\right)\Delta X_{E{G}_1}-{\displaystyle \frac{X_{G_1}{T}_{C{R}_1}}{b_{G_1}{Y}_{G_1}{T}_{F_1}}}\Delta AC{E}_1-{\displaystyle \frac{X_{G_1}{T}_{C{R}_1}}{b_{G_1}{Y}_{G_1}{T}_{F_1}}}{U}_{13}\end{array}$$

(9)

$$\begin{array}{l}\Delta {\dot{P}}_{V{G}_1}=-{\displaystyle \frac{X_{G_1}}{R_{13}{b}_{G_1}{Y}_{G_1}}}\Delta f_1-{\displaystyle \frac{1}{Y_{G_1}}}\Delta P_{V{G}_1}+\left({\displaystyle \frac{1}{Y_{G_1}}}-{\displaystyle \frac{c_{G_1}{X}_{G_1}}{b_{G_1}{Y}_{G_1}}}\right)\Delta X_{E{G}_1}+{\displaystyle \frac{X_{G_1}}{b_{G_1}{Y}_{G_1}}}\Delta AC{E}_1\\ +{\displaystyle \frac{X_{G_1}}{b_{G_1}{Y}_{G_1}}}{U}_{13}\end{array}$$

(10)

$$\Delta {\dot{X}}_{E{G}_1}=-{\displaystyle \frac{1}{b_{G_1}{R}_{13}}}\Delta f_1-{\displaystyle \frac{c_{G_1}}{b_{G_1}}}\Delta X_{E{G}_1}+{\displaystyle \frac{1}{b_{G_1}}}\Delta AC{E}_1+{\displaystyle \frac{1}{b_{G_1}}}{U}_{13}$$

(11)

$$\Delta {\dot{P}}_{D{C}_1}=+{\displaystyle \frac{K_{D{C}_1}}{T_{D{C}_1}}}\Delta f_1-{\displaystyle \frac{1}{T_{D{C}_1}}}\Delta P_{D{C}_1}$$

(12)

$$\begin{array}{l}\Delta {\dot{f}}_2=-{\displaystyle \frac{1}{T_{P{S}_2}}}\Delta f_2+{\displaystyle \frac{K_{P{S}_2}{\alpha }_{21}}{T_{P{S}_2}}}\Delta P_{P{T}_2}+{\displaystyle \frac{K_{P{S}_2}{\alpha }_{22}}{T_{P{S}_2}}}\Delta P_{G{W}_2}+{\displaystyle \frac{K_{P{S}_2}{\alpha }_{23}}{T_{P{S}_2}}}\Delta P_{G{G}_2}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{a_{12}{K}_{P{S}_2}}{T_{P{S}_1}}}\Delta P_{ti{e}_{12}}\\ -{\displaystyle \frac{K_{P{S}_2}}{T_{P{S}_1}}}\Delta P_{D_2}-{\displaystyle \frac{K_{P{S}_2}}{T_{P{S}_1}}}\Delta P_{D{C}_2}\end{array}$$

(13)

$$\Delta {\dot{P}}_{P{T}_2}=-{\displaystyle \frac{1}{T_{T_2}}}\Delta P_{P{T}_2}+{\displaystyle \frac{1}{T_{T_2}}}\Delta P_{G{T}_2}$$

(14)

$$\Delta {\dot{P}}_{G{T}_2}=-{\displaystyle \frac{K_{R_2}}{T_{S{G}_2}{R}_{21}}}\Delta f_2-{\displaystyle \frac{1}{T_{R_2}}}\Delta P_{G{T}_2}+\left({\displaystyle \frac{1}{T_{R_2}}}-{\displaystyle \frac{K_{R_2}}{T_{S{G}_2}}}\right)\Delta X_{E{T}_2}+{\displaystyle \frac{K_{R_2}}{T_{S{G}_2}}}\Delta AC{E}_2+{\displaystyle \frac{K_{R_2}}{T_{S{G}_2}}}{U}_{21}$$

(15)

$$\Delta {\dot{X}}_{E{T}_2}=-{\displaystyle \frac{1}{T_{S{G}_2}{R}_{21}}}\Delta f_2-{\displaystyle \frac{1}{T_{S{G}_2}}}\Delta X_{E{T}_2}+{\displaystyle \frac{1}{T_{S{G}_2}}}\Delta AC{E}_2+{\displaystyle \frac{1}{T_{S{G}_2}}}{U}_{21}$$

(16)

$$\Delta {\dot{P}}_{G{W}_2}=-\Delta P_{G{W}_2}+K_{W3}\Delta P_{R{W}_2}$$

(17)

$$\begin{array}{l}\Delta {\dot{P}}_{R{W}_2}=-{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}{R}_{21}}}\Delta f_2-\Delta P_{R{W}_2}+\left(1-{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}\right)\Delta X_{E{W}_2}+{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}\Delta AC{E}_2\\ +{\displaystyle \frac{K_{W_2}{T}_{W1}}{T_{W_2}}}{U}_{22}\end{array}$$

(18)

$$\Delta {\dot{X}}_{E{W}_2}=-{\displaystyle \frac{1}{T_{W_2}{R}_{21}}}\Delta f_2-{\displaystyle \frac{1}{T_{W_2}}}\Delta X_{E{W}_2}+{\displaystyle \frac{1}{T_{W_2}}}\Delta AC{E}_2+{\displaystyle \frac{1}{T_{W_2}}}{U}_{22}$$

(19)

$$\Delta {\dot{P}}_{G{G}_2}=-{\displaystyle \frac{1}{T_{C{D}_2}}}\Delta P_{G{G}_2}+{\displaystyle \frac{1}{T_{C{D}_2}}}\Delta P_{R{G}_2}$$

(20)

$$\begin{array}{l}\Delta {\dot{P}}_{R{G}_2}=+{\displaystyle \frac{X_{G_2}{T}_{C{R}_2}}{R_{23}{b}_{G_2}{Y}_{G_2}{T}_{F_2}}}\Delta f_2-{\displaystyle \frac{1}{T_{F_2}}}\Delta P_{R{G}_2}+\left({\displaystyle \frac{1}{T_{F_2}}}+{\displaystyle \frac{T_{C{R}_2}}{Y_{G_1}{T}_{F_2}}}\right)\Delta X_{V{G}_2}\\ \left(\hspace{0.17em}+{\displaystyle \frac{c_{G_2}{X}_{G_2}{T}_{C{R}_2}}{b_{G_2}{Y}_{G_2}{T}_{F_2}}}-{\displaystyle \frac{T_{C{R}_2}}{Y_{G_1}{T}_{F_2}}}\right)\Delta X_{E{G}_2}-{\displaystyle \frac{X_{G_2}{T}_{C{R}_2}}{b_{G_2}{Y}_{G_2}{T}_{F_2}}}\Delta AC{E}_2-{\displaystyle \frac{X_{G_2}{T}_{C{R}_2}}{b_{G_2}{Y}_{G_2}{T}_{F_2}}}{U}_{23}\end{array}$$

(21)

$$\begin{array}{l}\Delta {\dot{P}}_{V{G}_2}=-{\displaystyle \frac{X_{G_2}}{R_{23}{b}_{G_2}{Y}_{G_2}}}\Delta f_2-{\displaystyle \frac{1}{Y_{G_2}}}\Delta P_{V{G}_2}+\left({\displaystyle \frac{1}{Y_{G_1}}}-{\displaystyle \frac{c_{G_2}{X}_{G_2}}{b_{G_2}{Y}_{G_2}}}\right)\Delta X_{E{G}_2}\\ +{\displaystyle \frac{X_{G_2}}{b_{G_2}{Y}_{G_2}}}\Delta AC{E}_2+{\displaystyle \frac{X_{G_2}}{b_{G_2}{Y}_{G_2}}}{U}_{23}\end{array}$$

(22)

$$\Delta {\dot{X}}_{E{G}_2}=-{\displaystyle \frac{1}{b_{G_2}{R}_{23}}}\Delta f_2-{\displaystyle \frac{c_{G_2}}{b_{G_2}}}\Delta X_{E{G}_2}+{\displaystyle \frac{1}{b_{G_2}}}\Delta AC{E}_2+{\displaystyle \frac{1}{b_{G_2}}}{U}_{23}$$

(23)

$$\Delta {\dot{P}}_{D{C}_2}=+{\displaystyle \frac{K_{D{C}_2}}{T_{D{C}_2}}}\Delta f_2-{\displaystyle \frac{1}{T_{D{C}_2}}}\Delta P_{D{C}_2}$$

(24)

$$\Delta {\dot{P}}_{ti{e}_{12}}=2\pi T_{12}(\Delta f_1-\Delta f_2)$$

(25)

$$\Delta A\dot{C}{E}_1=B_1\Delta f_1+\Delta P_{ti{e}_{12}}$$

(26)

$$\Delta A\dot{C}{E}_2=B_2\Delta f_2+a_{12}\Delta P_{ti{e}_{12}}$$

(27)

where $U_{11}$, $U_{12}$, $U_{13}$, $U_{21}$, $U_{22}$, and $U_{23}$ denote the control signal; $B_1$ and $B_2$ are the frequency bias parameters; $\Delta X_{E{T}_1}$ and $\Delta X_{E{T}_2}$ denote the change in valve position; $\Delta P_{P{T}_1}$ and $\Delta P_{P{T}_2}$ are the change in thermal turbine speed changer position (p.u.MW); $\Delta P_{G{G}_1}$ and $\Delta P_{G{G}_2}$ are the change in gas turbine speed changer position; $\Delta P_{G{T}_1}$ and $\Delta P_{G{T}_2}$ show the change in the high pressure of the thermal turbine power; $\Delta P_{G{W}_1}$ and $\Delta P_{G{W}_1}$ present the wind turbine power output; $\Delta f_1$ and $\Delta f_2$ denote the incremental change in frequency of each control area (Hz); $\Delta P_{D_1}$ and $\Delta P_{D_2}$ display total incremental charge in the local load; and $\Delta P_{ti{e}_{12}}$ shows the incremental change in actual tie-line power flow from control areas 1 to 2 (p.u.MW); and $AC{E}_1$ and $AC{E}_2$ denote the area control error of the first and second areas. After defining the previously mentioned PP parameters, the dynamics equation is utilized to ascertain the PP state space structure. Equation (28) below can be used to write and express the matrix form that is displayed in the dynamic equations in Equations (1)–(27), the two areas of thermal power plants that are depicted in Figure 1.

$$\dot{z}\left(t\right)=Az(t)+Bu(t)+F\Delta P_d(t)$$

(28)

The practical, interconnected TAGWTPSH’s changing sources of load are continually affected by changes in working points. This component can be conceptualized as parameter uncertainties. The general stateless model of the matrix form is shown as follows: $A;B;F$. The TAGWTPSH’s state space form is denoted by Equation (28). The system parameters and state system variables are given in the Appendix B.

The system matrices with nominal values are represented by A and B, and the lumped uncertainty is represented by $w_i(z_i,t)$.

$$w(z,t)=F\Delta P_d(t)$$

(29)

Assumption 1 

([23,24]). It is assumed that the load disturbance $w(z,t)$ is bounded, such that the differential of $\dot{w}(z,t)$ is bounded so that $‖w(z,t)‖\le \gamma$ and $‖\dot{w}(z,t)‖\le \partial$. The elements in this vector space, which is a matrix norm, are denoted by $‖.‖$

Lemma 1 

([23,24]). The following matrix inequality is true for every scalar if $X$ and$Y$ are real matrices of the proper dimension:

$$Z^{T}T+T^{T}Z\le \mu Z^{T}Z+{\mu }^{-1}{T}^{T}T.$$

Lemma 2 

([23]). For a given inequality

$$\left[\begin{array}{cc}P(z)& \mathrm{\Gamma }(z)\\ {\mathrm{\Gamma }}^T(z)& \mathrm{{\rm Y}}(z)\end{array}\right]>0$$

(30)

then $\mathrm{{\rm Y}}(z)>0$ and $P(z)-\mathrm{\Gamma }(z){\mathrm{{\rm Y}}}^{-1}(z){\mathrm{\Gamma }}^T(z)>0$, in which $P(z)=P^T(z)$, $\mathrm{{\rm Y}}(z)={\mathrm{{\rm Y}}}^T(z)$, and $\mathrm{\Gamma }(z)$ are dependent-affinitive on z, which holds for all $\epsilon >0$

2.2. Single-Phase Sliding Surface

In order to guarantee that sliding variables start at the surface at a specific time and that the SMC is independent of reaching time, we choose a single-phase sliding surface. As a result, the single-phase surface is shown as

$$\Im [z(t)]=\Re z(t)-{\displaystyle \underset{0}{\overset{t}{\int }}\Re (A-BT)z(\tau )d\tau }-\Re z(0)e^{-\alpha t}$$

(31)

where $\Re$ is a constant matrix, and $\Re$ is constructed to warrant that $\Re B$ is an invertible matrix. $T$ is the design matrix, and matrix $T$ is chosen by polar assignment so that the eigenvalues of the matrix $(A-BT)$ are always negative.

The SMC now has no reaching time thanks to the addition of the term $\xi z(0)e^{-\alpha t}$. We then create the control law. We start by differentiating the sliding surface $\Im [z(t)]$ concerning time, which leads to the following equation:

$$\Im [z]=\Re [Az+Bu+w(z,t)]-\Re (A-BT)z+\alpha \Re z(0)e^{-\alpha t}$$

(32)

So, this makes $\Im [z(t)]=\dot{\Im }[z(t)]=0$.

From Equation (32), $u_i^{eq}(t)$ is the equivalent control input, which is given as

$$u^{eq}(t)=-{(\Re B)}^{-1}[\Re Az+\Re w(z,t)-\Re (A-BT)z_i(t)+\alpha \Re z(0)e^{-\alpha t}]$$

(33)

Prior to designing the control rule, we study the sliding surface TAGWTPSH asymptotic stability. We do so by changing the value of $u_i^{eq}$ in Equation (28) and simplifying as follows:

$$\begin{array}{l}\dot{z}=Az-B{(EB)}^{-1}[\Re w(z,t)+\Re BTz(t)+\alpha \Re z(0)e^{-\alpha t}]+w(z,t)\\ =(A-BT)z+\hspace{0.17em}[I-B_i{(\Re B_i)}^{-1}\Re ]w(z,t)-B{(\Re B)}^{-1}\Re \alpha \Re z(0)e^{-\alpha t}\\ =(A-BT)z+\hspace{0.17em}\mathrm{\Phi }w(z,t)+\mathrm{\Theta }z(0)e^{-\alpha t}\end{array}$$

(34)

where $\mathrm{\Phi }=[I-B{(\Re B)}^{-1}\Re ]$ and $\mathrm{\Theta }=-B{(\Re B)}^{-1}\alpha \Re$. Next, we examine the TAGWTPSH stability theoretically. We start by representing the system state matrix as a linear matrix inequality (LMI) accompanied by a basic theorem.

Theorem 1. 

Equation (34) is asymptotically stable for the TAGWTPSH if and only if it embraces a symmetric positive matrix $\mathrm{{\rm Y}}$ and positive scalars $q$, $\phi$, and $\beta$ such that the following LMIs satisfy it:

$$\left[\begin{array}{cccc}\mathrm{\Xi }& 0& \mathrm{{\rm Y}}\mathrm{\Phi }& \mathrm{{\rm Y}}\mathrm{\Theta }\\ 0& -{\pi }^{-1}\xi {\mathrm{{\rm Y}}}^{-1}& 0& 0\\ {\mathrm{\Phi }}^T\mathrm{{\rm Y}}& 0& -\chi & 0\\ {\mathrm{\Theta }}^T\mathrm{{\rm Y}}& 0& 0& -\zeta \end{array}\right]<0$$

(35)

where $\mathrm{\Xi }=[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)+\kappa \mathrm{{\rm Y}}]$

The system with the LMI is a differential equation; thus, we use the Lyapunov stability concept, whose function is to examine the stability of the system Equation (18).

$$V=z^T(t)\hspace{0.17em}\mathrm{{\rm Y}}z(t)$$

(36)

where ${\mathrm{{\rm Y}}}_i>0$ satisfies (35). If we acquire the first-time derivative of Equation (36) and simplify, we arrive at

$$\begin{array}{l}\dot{V}=z^T(t)[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)]z(t)+w^T(z,t){\mathrm{\Phi }}^T\mathrm{{\rm Y}}z(t)\\ +z^T(t)\mathrm{{\rm Y}}\mathrm{\Phi }w(z,t)+e^{-\alpha t}{z}^T(0){\mathrm{\Theta }}^T\mathrm{{\rm Y}}z(t)+z^T(t)\mathrm{{\rm Y}}\mathrm{\Theta }z(0)e^{-\alpha t}\end{array}$$

(37)

When we apply Lemma 1 to Equation (37) and further simplify, we obtain

$$\begin{array}{l}\dot{V}\le z^T(t)[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)]z(t)+{\chi }^{-1}{z}^T(t)\mathrm{{\rm Y}}\mathrm{\Phi }{\mathrm{\Phi }}^T\mathrm{{\rm Y}}z(t)\\ +\chi w^T(z,t)w(z,t)+{\zeta }^{-1}{z}^T(t)\mathrm{{\rm Y}}\mathrm{\Theta }{\mathrm{\Theta }}^T\mathrm{{\rm Y}}z(t)+\zeta e^{-\alpha t}{z}^T(0)z(0)e^{-\alpha t}\end{array}$$

(38)

If we also apply Lemma 1 to Equation (38) and simplify, we obtain

$$\begin{array}{l}\dot{V}\le z^T(t)[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)]z(t)+\xi z^T(t)\mathrm{{\rm Y}}z(t)+{\beta }^{-1}{z}^T(t)\mathrm{{\rm Y}}z(t)\\ +{\chi }^{-1}{z}^T(t)\mathrm{{\rm Y}}\mathrm{\Phi }{\mathrm{\Phi }}^T\mathrm{{\rm Y}}z(t)+\chi w^T(z,t)w(z,t)+{\zeta }^{-1}{z}^T(t)\mathrm{{\rm Y}}\mathrm{\Theta }{\mathrm{\Theta }}^T\mathrm{{\rm Y}}z(t)+\zeta e^{-\alpha t}{z}^T(0)z(0)e^{-\alpha t}\end{array}$$

(39)

The following equation is attainable given Assumption 1:

$$\begin{array}{l}\dot{V}\le z^T(t)[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)+\kappa \mathrm{{\rm Y}}+{\chi }^{-1}\mathrm{{\rm Y}}\mathrm{\Phi }{\mathrm{\Phi }}^T\mathrm{{\rm Y}}\\ +{\zeta }^{-1}\mathrm{{\rm Y}}\mathrm{\Theta }{\mathrm{\Theta }}^T\mathrm{{\rm Y}}]z(t)+\vartheta +\zeta e^{-\alpha t}{z}^T(0)z(0)e^{-\alpha t}\end{array}$$

(40)

where $\vartheta =\chi {\gamma }^2$ and $\kappa =\xi +{\displaystyle \frac{N-1}{\beta }}$.

From Lemma 2 and LMI (35), we obtain

$$\Lambda \hspace{0.17em}=-[{(A-BT)}^T\mathrm{{\rm Y}}+\mathrm{{\rm Y}}(A-BT)+\kappa \mathrm{{\rm Y}}+{\chi }^{-1}\mathrm{{\rm Y}}\mathrm{\Phi }{\mathrm{\Phi }}^T\mathrm{{\rm Y}}+{\zeta }^{-1}\mathrm{{\rm Y}}\mathrm{\Theta }{\mathrm{\Theta }}^T\mathrm{{\rm Y}}+]>0$$

(41)

Based on Equations (40) and (41), we can infer

$$\dot{V}[z(t)]\le -{\lambda }_{\min }(\Lambda ){‖z(t)‖}^2+\upsilon +\zeta e^{-\alpha t}{z}^T(0)z(0)e^{-\alpha t}$$

(42)

where ${\upsilon }_i$ is the constant value and ${\lambda }_{\min }(\Lambda )>0$. The term $\zeta e^{-\alpha t}{z}^T(0)z(0)e^{-\alpha t}$ in Equation (42) will converge to zero as the time approaches infinity. So, $\dot{V}<\hspace{0.17em}0$ is achieved with $‖z(t)‖>\sqrt{{\displaystyle \frac{\upsilon }{{\lambda }_{\min }(\Lambda )}}}$.

Consequently, the TAGWTPSH (34) is asymptotically stable.

2.3. Integral Sliding Mode Control with Single-Phase Design

Two control strategies are often used for AGC in the local control regions of the TAGWTPSH: decentralized and centralized. As a result, the control law and Equation (31) are used to build the centralized sliding mode control:

$$u(t)=-{(\Re B)}^{-1}[‖\Re ‖\gamma +‖\Re ‖‖B‖‖T‖‖z‖+\alpha ‖\Re ‖‖z(0)‖e^{-\alpha t}+\overline{\epsilon }]{\displaystyle \frac{\Im [z(t)]}{‖\Im [z(t)]‖}}$$

(43)

Equation (26) of the control rule guarantees that every state variable remains within the all-time single-phase sliding surface. We theoretically examine the sliding variables’ reachability once again in order to support this claim. Our first theorem is established.

Theorem 2. 

We take the TAGWTPSH (7) into account using the centralized SMC in (26). The single-phase sliding surface is then used to preserve the trajectory of each system state. Once more, we utilize Lyapunov theory, and the function becomes.

$$\overline{V}(t)=‖\Im [z(t)]‖$$

(44)

Hence, by simplifying and differentiating Equation (44), we obtain

$$\dot{\overline{V}}={\displaystyle \frac{{\Im }_i^T[z(t)]}{‖\Im [z(t)]‖}}\{\Re Bu+\Re w(z,t)+\Re BTz+\alpha Ez(0)e^{-\alpha t}\}$$

(45)

From Equation (45) and inequality properties $‖AB‖\le ‖A‖‖B‖$, one can infer

$$\dot{\overline{V}}\le ‖\Re ‖‖w(z,t)‖+‖\Re ‖‖B‖‖T‖‖z‖+\alpha ‖\Re ‖‖z(0)‖e^{-\alpha t}+{\displaystyle \frac{{\Im }^T[z(t)]}{‖\Im [z_i(t)]‖}}\Re Bu(t)$$

(46)

Assuming 1, we are able to show that

$$\dot{\overline{V}}\le ‖\Re ‖\gamma +‖\Re ‖‖B‖‖T_i‖‖z‖+\alpha ‖\Re ‖‖z(0)‖e^{-\alpha t}+{\displaystyle \frac{{\Im }^T[z(t)]}{‖\Im [z(t)]‖}}\Re Bu(t)$$

(47)

Equation (47) is substituted with the suggested control technique, Equation (43):

$$\dot{\overline{V}}\le -{\overline{\epsilon }}_i$$

(48)

The system state trajectories of the TAGWTPSH in Equation (28) are in the integral sliding mode control with single-phase technique, according to the stated inequality criteria.

Remark 1. 

Lyapunov stability theory and theoretical analysis of the asymptotic stability of the TAGWTPSH serve as the foundation for the novel LMI technique, which focuses on the parametric stability of the system close to an equilibrium point.

Remark 2. 

The sliding variable is particularly resilient against other SMCs addressed in the w since the term $\Re z(0)e^{-\alpha t}$, which is a function of time, demonstrates that the sliding variable’s reachability to the surface does not depend on reaching time.

3. Results and Discussion

This section examines three simulated scenarios for an MAMSPP with multiple power sources under various load disturbances, taking into account simulation parameter uncertainties, in order to demonstrate the effectiveness of the proposed ISMCSP control strategy. The MAMSPP’s nominal values are taken from [1]. The results of the suggested approach are contrasted with those of the most recent techniques published in the literature. The output response’s peak value, settling time, and tie-line power (TLP) were all used to assess performance. The parameters for the simulation are in the Appendix A.

3.1. Scenario 1

With nominal power system characteristics equal to [1], the proposed ISMCSP for the AGC approach was applied to an MAMSPP with numerous power sources under comparatively high step load disturbances. Areas 1 and 2 both saw an abrupt 1% p.u. step load increase. Figure 2 and Figure 3 show the TAGWTPSH’s frequency deviations (FDs) in Hertz. In results 2, 3, and 4, the frequency deviations (FDs)for areas 1 and 2 were 0.0028 and 0.0007 in (Hz), respectively, and the tie-line power (TLP) could quickly converge to zero for the ISMCSP (Figure 4). Areas 1 and 2 had corresponding frequency stabilization times of 1.3 and 1.1 s; these results demonstrate the superiority of the proposed technique. The data show that the ISMCSP controller quickly smooths the oscillations compared to the New Approach in [1].

Table 1. Determining the comparison between time $T_s(s)$ and maximum overshoot MO (Hz) under scenario 1.

Measurement	The Suggested Approach		The New Approach in [1]
	Ts (s)	MO (Hz)	Ts (s)	MO (Hz)
$\Delta f_1$	1.3	0.0028	8.5	0.0134
$\Delta f_2$	1.1	0.0007	5.5	0.013

displays the outcomes.

3.2. Scenario 2

For scenario 2, the simulated response of the ISMCSP approach was compared with [1], and Figure 5, Figure 6 and Figure 7 display the frequency deviation in area 1, area 2, and tie-line power flow between the two areas, respectively. The system performance was assessed in each controlled area having a lumped disruption, consisting of a 3% step load variation for areas 1 and 2, in addition to system parameter uncertainties to assess the resilience of the proposed ISMCSP under actual operating conditions. Using the same parameters as in [1], the TAGWTPSH parameters were in a steady state for a duration of 0 ≤ t ≤ 4 s. The frequency stabilization times for areas 1 and 2 were 3 and 4 s, respectively. It is clear from the waveforms previously discussed that the frequency error and ACE both rapidly approached zero. Areas 1 and 2 in Figure 5 and Figure 6 had frequency overshoot deviations (FOD) of 0.0019 and 0.002 in (Hz), respectively. These outcomes are consistent with the frequency needs of the power system. The comparison results are presented in

Table 2. Determining the comparison between time $T_s(s)$ and maximum overshoot MO (Hz) under scenario 2.

Measurement	The Suggested Approach		The New Approach in [1]
	Ts (s)	MO (Hz)	Ts (s)	MO (Hz)
$\Delta f_1$	3	0.0019	5.5	0.04
$\Delta f_2$	4	0.002	5.6	0.027

.

3.3. Scenario 3

This subsection provides evidence of the effectiveness of the recommended control strategy in handling random disturbances. Figure 8 illustrates the random step load in areas 1 and 2 in scenario 3. The disturbance loads for this simulation were created using random perturbations of 50 s. The suggested method’s performance was assessed using TAGWTPSH simulation results. In areas 1 and 2, the random step load was taken into account. The frequency deviation in areas 1 and 2 and the tie-line power flow between the two areas for scenario 3 are shown in Figure 9, Figure 10 and Figure 11. These figures demonstrate how well the suggested ISMCSP controller handles frequency deviation and tie-line power variation for erratic load variations. The ISMCSP controller rapidly stabilized, outperforming the controllers of the New Approach in [1].

From the simulation results, larger load variation leads to longer overshoot and settling times. The overshoot and settling times are shorter under the proposed control scheme compared to the New Approach [1] scenario, proving the superiority of the proposed control method. In conclusion, the proposed method shows better performance in the scenarios.

4. Conclusions

Our aim in this study is to manage the load frequency of an MAMSPP by developing sliding mode control for the first time for an ISMCSP. The model used to evaluate the viability of the constructed ISMCSP is called the two-area thermal power plant, or TAGWTPSH. The unpredictability of the scenario and the associated parameters are also taken into consideration by the TAGWTPSH model. TAGWTPSH stability is provided by this novel linear matrix inequality. The areas where the ISMCSP performs better are contrasted with simulation results from recently mentioned controllers. The proposed ISMCSP outperforms the recently described methods in terms of TAGWTPSH performance. The fact that the ISMCSP is immune to random demand disruptions, subsystem variable changes, and the unpredictability of variables in the structure and connected states further demonstrates its frequency resilience. Consequently, the suggested ISMCSP greatly enhances the AGC of MAMSPPs. Additionally, the proposed approach is extended to a two-area power system with multiple sources and HVDC. The simulation study indicates that the proposed ISMCSP strategy outperforms a number of recently proposed strategies and could be a very promising method for handling more difficult engineering optimization problems in future research.