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Sliding mode controller design via delaydependent \(H_{\infty }\) stabilization criterion for load frequency regulation
Protection and Control of Modern Power Systems volume 8, Article number: 49 (2023)
Abstract
This work presents a control approach based on slidingmodecontrol (SMC) to design robust \(H_{\infty }\) state feedback controllers for load frequency regulation of delayed interconnected power system (IPS) with parametric uncertainties. Considering both state feedback control strategy and delayed feedback control strategy, two SMC laws are proposed. The proposed control laws are designed to improve the stability and disturbance rejection performance of delayed IPS, while stabilization criteria in the form of linear matrix inequality are derived by choosing a Lyapunov–Krasovskii functional. An artificial timedelay is incorporated in the control law design of the delayed feedback control structure to enhance the controller performance. A numerical example is considered to study the control performance of the proposed controllers and simulation results are provided to observe the dynamic response of the IPS.
1 Introduction
Stable power system operation needs stability in frequency because frequency reflects the status of the real power balance between power supply and load demand [1, 2]. Alternators and other power devices deviate from stable operating condition due to fluctuation in frequency, which can lead to unstable and unsafe operation of the entire power system [3, 4]. Disturbance in load demand is the main cause of originating frequency and voltage oscillations in power system [5, 6]. Therefore, load frequency control (LFC) is an essential need of power system for its smooth operation [7, 8]. Using modern technologies, simple power system areas are interconnected through tielines to create an interconnected power system (IPS) [9, 10]. The main objective of a well designed IPS is to regulate frequency variation within tolerable limits during abnormal condition of IPS operation, i.e., load variation or sudden change in load demand [6, 11, 12].
In the last few decades, many traditional control techniques have been proposed to solve LFC problem in power system. ProportionalIntegral (PI) control method [13] is first proposed to solve LFC problem, while decentralized robust PI controller based on Kharitonov’s theorem is designed for LFC of a multiarea IPS in [14]. Various control schemes are designed for solving LFC problem of uncertain IPS, such as robust control scheme based on Riccatiequation [15] and adaptive control scheme with system parameter variation [16]. An active disturbance rejection control approach is proposed to develop a robust decentralized LFC algorithm for a threearea IPS in [17], while in [18], poleplacement method based variable structure controller is proposed for LFC of IPS.
A finite time is always needed to measure control signals, compute control action and actuate the plant. This finite time requirement in every closedloop system is termed as timedelay. Stability of closedloop systems is affected by this delay because of its destabilizing nature [8]. Timedelay appears in IPS due to two main reasons, i.e.: (i) time needed for frequency and tieline power measurement; and (ii) time taken in signal transmission from the remote terminal unit to the control centre and from the control centre to the generating unit [19]. Presence of timedelay in IPS sometimes leads to system instability and performance degradation [20, 21]. Therefore, timedelay should be included properly in LFC design of IPS to solve frequency deviation problem. Different control schemes have been used to design controllers for solving the LFC problem of timedelayed IPS. A decentralized robust PI controller is designed for LFC of power system with delay in [22], while state feedback control strategy based on linear matrix inequality (LMI) is adopted for LFC of IPS with delay in [23]. An iterative LMI based robust LFC algorithm is developed in [24] for timedelay IPS, while Lyapunov theory based delaydependent stability criterion is developed in [25] for LFC by considering constant delay and timevarying delay in IPS. In [26], a delaydependent robust PIDtype controller is designed to regulate frequency of a threearea IPS with communication delay. Considering the existence of timedelay in the local PItype LFC signals of an IPS, delaydependent stability criteria are developed for the delayed LFC scheme in [27]. Although the above mentioned literatures based on delayed LFC design have well considered the effect of timedelay in LFC scheme, no specific method has been implemented to improve the performance of LFC scheme which is degraded due to the existence of timedelay.
Stabilizing nature of timedelay in system dynamics is an interesting aspect studied in [28, 29]. A finite valued known timedelay is introduced intentionally in the control law design to improve the stability of a delayed system in [8]. A few research results are available in literatures where timedelay is introduced in control design for improving the control performance of multiarea delayed IPS. In [19], a known finite delay is used to design \(H_{\infty }\) twoterm frequency controller for enhancing the control performance of a twoarea IPS considering communication delays, while tolerable delay margin of IPS is enhanced using timedelay in a robust \(H_{\infty }\) frequency controller design in [21]. The timedelay used purposefully in controller design is termed as artificial delay. Besides the above discussed control approaches, sliding mode control (SMC) approach is very effective to solve the LFC problem by designing robust state feedback controller for timedelayed multiarea IPS. SMC approach is popular for its insensitivity to parameters variation, excellent control performance and finitetime convergence [6, 30]. In [31], a nonlinear sliding mode controller is designed for load frequency regulation of multiarea power system with timevarying delay. SMC scheme for LFC is considered for windintegrated delayed IPS in [32], and SMC scheme is proposed to design a robust state feedback controller for LFC of multiarea power system with timedelay in [6, 33]. In [34], a decentralised robust load frequency controller is proposed for interconnected timedelay power system using sliding mode technique, while in [35], an eventtriggered SMC design is proposed for LFC of power systems with sensor faults and communication delay. In [36], an adaptive delaydependent sliding mode faulttolerant LFC design is proposed for nonlinear power system with unknown timevarying state and input delays, whereas a dynamic integral SMC based LFC scheme is proposed for an interconnected delayed power system in [37].
Based on the above study, it is clearly observed that \(H_{\infty }\) control and SMC approaches are mostly used for the LFC design of IPS with timedelay and parametric uncertainty. The reason behind for the application of SMC approach on LFC design of delayed IPS is that SMC based controllers are very insensitive to parametric variation [37]. Similarly, the popularity of \(H_{\infty }\) control is widespread for its disturbance rejection performance and robustness against uncertainties [2, 19]. As timedelay and parametric uncertainty are two inevitable phenomena in LFC of IPS and they have negative impact on system stability, the effects of these phenomena on LFC should be accounted properly during LFC design for IPS. To neutralize the destabilizing effect of the existence of timedelay in a system, purposeful use of a known finite timedelay may be considered while designing the control method [38]. In view of the above discussions, to obtain the benefit of both SMC and \(H_{\infty }\) control methods, sliding mode controller design based on delaydependent \(H_{\infty }\) stabilization criterion is proposed for load frequency regulation of IPS considering the existence of timedelay and parametric uncertainty in this paper. The main contributions of this paper are described as follows:

Two new SMC laws are proposed considering two different controller structures, including state feedback controller and delayed feedback controller designing strategies, for load frequency regulation of delayed IPS with parametric uncertainty. Chattering is a major issue in SMC approach because it degrades control performance [33]. This paper takes care of the chattering issue of SMC approach to ensure that the controller output signals are free from chattering.

A novel sliding surface with the incorporation of artificial delay is defined to obtain the proposed SMC law based on delayed feedback controller designing strategy. An important feature of the proposed SMC law based on delayed feedback controller designing strategy is that a finite known delay is introduced judiciously in the proposed sliding mode LFC structure to improve closedloop IPS performance which is degraded due to the existence of timedelay in uncontrolled IPS.

New delaydependent \(H_{\infty }\) stabilization criteria in LMI framework for the sliding mode dynamics of IPS are derived by using a simple Lyapunov–Krasovskii (LK) functional approach. These stabilization criteria satisfy parametric uncertainty and timedelay existence of the closedloop IPS, while the gains of the proposed sliding mode controllers are computed by solving these stabilization criteria. The inclusion of \(H_{\infty }\) criterion into the stabilization condition improves disturbance rejection performance.

A wellknown numerical example of a twoarea IPS with timedelay and parametric uncertainty is considered to study the performance of the proposed load frequency regulation method. The performance of the proposed frequency regulation method is compared with some existing methods [19, 21] to show its superiority.
Notation: Throughout this paper, the superscript ‘T’ stands for matrix transposition, the superscript ‘\(1\)’ stands for matrix inverse. For any arbitrary matrix B and two symmetric matrices A and C, \(\left[ \begin{matrix} A &{} B \\ * &{} C \\ \end{matrix} \right]\) denotes a symmetric matrix, where \(*\) represents \(B^{T}\).
2 Modeling of delayed interconnected power system (IPS)
A twoarea IPS with timedelay in each control area is considered in this paper to analyse the LFC problem. The LFC model of the twoarea IPS with timedelay in each control area is presented in Fig. 1 [19].
From Fig. 1, the dynamic model of the twoarea IPS with timedelay can be described as [21]:
with \(\Delta {{P}_{12}}=\Delta {{P}_{21}}\), \(i,j=1,2,\,i\ne j\).
The above dynamic model can be represented in statespace form as [2]:
where the state vector
and load disturbance vector
\(w(t)={{\left[ \begin{matrix} \Delta P_{d1}(t) &{} \Delta P_{d2}(t) \\ \end{matrix} \right] }^{T}}\).
Here, \(\Delta P_{d1}(t)\) and \(\Delta P_{d2}(t)\) are norm bounded, and satisfy \(\left\ \Delta {{P}_{d1}}(t) \right\ <{{b}_{1}}\) and \(\left\ \Delta {{P}_{d2}}(t) \right\ <{{b}_{2}}\). \({{b}_{1}}\) and \({{b}_{2}}\) are positive constants, while \({{\tau }_{1}}\) and \({{\tau }_{2}}\) are time delays of area 1 and area 2, respectively. The system parameters are given for \(e=1,2\) as:
Considering parametric uncertainties in system matrices of (6), one may write:
where \(\hat{A}(t)\), \({{\hat{A}}_{d1}}(t)\) and \({{\hat{A}}_{d2}}(t)\) are uncertain state matrices, i.e., \(\hat{A}(t)=A+\Delta A(t)\), \({{\hat{A}}_{d1}}(t)={{A}_{d1}}+\Delta {{A}_{d1}}(t)\) and \({{\hat{A}}_{d2}}(t)={{A}_{d2}}+\Delta {{A}_{d2}}(t)\). \(\Delta A(t)\), \(\Delta {{A}_{d1}}(t)\) and \(\Delta {{A}_{d2}}(t)\) are uncertainties with the system matrices A, \({{A}_{d1}}\) and \({{A}_{d2}}\), respectively. The uncertainty in system matrices can be considered to be bounded with norm, and can be described as:
where H, \({{E}_{1}}\), \({{E}_{2}}\), and \({{E}_{3}}\) are constant matrices with appropriate dimensions. F(t) is a timevarying matrix, which satisfies \({{F}^{T}}(t)F(t)\le I\).
3 Controller designing strategy
This paper aims to design load frequency controllers based on sliding mode control for delayed IPS, whereas they satisfy the \(H_{\infty }\) criterion, defined as [2]:
where \(\gamma\) is the \(H_{\infty }\) performance indicator, which indicates rejection of load disturbance. Minimization of \(\gamma\) is required to obtain the minimal effect of load variation in the IPS performance.
The controller designing strategy proposed for load frequency regulation of the twoarea IPS with timedelay is shown in Fig. 2. As shown, state vector (x(t)) of the twoarea delayed IPS acts as input to the proposed sliding mode controller, and the controller outputs (\(u_{1}\) and \(u_{2}\)) act as control inputs to the twoarea delayed IPS, provided that the controller gains are based on \(H_{\infty }\) performance condition (10). Some steps are required to compute \(H_{\infty }\) performancebased gains of the sliding mode controller, which are described as follows.

Step1 Selection of switching surface, which is a function of the state vector of delayed IPS.

Step2 Differentiate the selected switching surface and obtain an equivalent controller by equating this differentiation to zero.

Step3 Obtain the closedloop sliding mode dynamics of delayed IPS by using the equivalent controller in the statespace form of delayed IPS.

Step4 Develop delaydependent \({{H}_{\infty }}\) stabilization criterion in LMI framework for the closedloop sliding mode dynamics of delayed IPS.

Step5 Obtain \(H_{\infty }\) performancebased gains of sliding mode controller from the solution of delaydependent \({{H}_{\infty }}\) stabilization criterion.
After obtaining the \(H_{\infty }\) performancebased controller gains, sliding mode control law is designed by using these gains and system state vector for load frequency regulation of IPS with timedelay.
Subsequently, it proceeds to design the proposed \({{H}_{\infty }}\) performancebased sliding mode controller by using both state feedback controller designing strategy and delayed feedback controller designing strategy.
3.1 Robust \({{H}_{\infty }}\) state feedback sliding mode controller designing strategy
3.1.1 Selection of switching surface
A switching surface can be selected considering delay in IPS as [6]:
where J and K are two constant matrices, and J is selected such that the matrix JB becomes nonsingular. Time derivative of (11) is given as:
When delayed IPS trajectory reaches its sliding mode, the conditions of \({\sigma }(t)=0\) and \({\dot{\sigma }}(t)=0\) are satisfied by the switching function. Substituting (8) in (12), one can obtain:
The following equivalent controller can be obtained from (13) by considering \({\dot{\sigma }}(t)=0\):
The closedloop system of delayed IPS is obtained by substituting (14) in (8) as:
where matrix \(\hat{D}=DB{{(JB)}^{1}}JD\). System (15) without parametric uncertainty can be written as:
Remark 1
The equivalent controller structure (14) contains an inverse matrix of \({{(JB)}^{1}}\). It is well known that a singular matrix can not be inverted, and thus, matrix JB must be a nonsingular matrix. Therefore, care must be taken while selecting matrix J for the construction of the switching surface (11).
3.1.2 Robust \({{H}_{\infty }}\) stabilization criterion
Theorem 1
Closedloop IPS (15) satisfies the \(H_{\infty }\) criterion \(\left\ {{T}_{wy}} \right\ \le \gamma\) and \(\gamma >0\), if there exist positive definite matrices \(\hat{P}\), \({{\hat{Q}}_{1}}\), \({{\hat{Q}}_{2}}\), \({{\hat{R}}_{{{\tau }_{1}}}}\), \({{\hat{R}}_{{{\tau }_{2}}}}\) and matrix G, such that an LMI holds as follows:
where
and n is the dimension of x(t).
The corresponding robust \({{H}_{\infty }}\) controller gain matrix is obtained as \(K=G{{\left( {{Y}^{T}} \right) }^{1}}\).
Proof
For investigating the stability of (15), the LK functional [39] is considered as:
where
Differentiating (18) yields:
Referring to Lemma 1 of [40], the integral terms in (19) may be replaced with inequalities as:
Stabilization condition requires information regarding system dynamics. Therefore, the zerovalued quadratic formulation of IPS dynamics (15) is considered to incorporate IPS dynamics in the stabilization criterion instead of replacing \(\dot{x}(t)\) in (20) directly by using (15), i.e.:
where \(\xi \left( t \right) ={{\left[ \begin{matrix} {{x}^{T}}(t) &{} {{x}^{T}}(t{{\tau }_{1}}) &{} {{x}^{T}}(t{{\tau }_{2}}) &{} {{{\dot{x}}}^{T}}(t) \\ \end{matrix} \right] }^{T}}\), and \(S={{\left[ \begin{matrix} S_{1}^{T} &{} S_{2}^{T} &{} S_{3}^{T} &{} S_{4}^{T} \\ \end{matrix} \right] }^{T}}\). \(S_1\), \(S_2\), \(S_3\) and \(S_4\) are proper dimensional arbitrary matrices. The above zero term (21) is able to fulfill the requirement of involving system states in stability condition. One can rewrite (21) by splitting certain and uncertain terms as:
The certain terms in (22) can be written as:
where \({\bar{\Theta }}={{\left[ {{{\bar{\Theta }} }_{lm}} \right] }_{l,m=1,2,3,4}}\),
The second term in the right hand side (RHS) of (23) can be expressed as [41]:
Substituting (24) into (23) yields:
By following Lemma 2 of [40], the uncertain terms of (22) may be represented as:
where
Substituting (25) and (26) into (22) yields:
Addition of (20) and (27) gives:
where \({\bar{\Psi }} ={{\left[ {{{\bar{\Psi }} }_{lm}} \right] }_{l,m=1,2,3,4}},\)
A cost function is obtained from (10) for the investigation of \({{H}_{\infty }}\) criterion, as:
Closedloop IPS (15) satisfies the criterion (10) only when \({{J}_{yw}}\le 0\). For initial condition \(V(0)=0\) and since \(V(\infty )\ge 0\), one can write:
Substituting (28) into (30), one obtains:
where \({\bar{\Omega }} ={\bar{\Psi }} +\sum \limits _{j=1}^{3}{{{\Xi }_{j}}}+{{\gamma }^{2}}{\bar{\theta }} {{{\bar{\theta }} }^{T}}+\hat{C}{{\hat{C}}^{T}}\) and \(\hat{C}={{\left[ \begin{matrix} C &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] }^{T}}\).
The condition \({{J}_{yw}}\le 0\) is satisfied, if there is:
Now, using Schur complement [41] in inequality (32), we can write:
where \({{{\bar{\Phi }} }_{1}}={{\left[ \begin{matrix} {{E}_{1}} &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}\), \({{{\bar{\Phi }} }_{2}}={{\left[ \begin{matrix} {{0}_{1n\times 1n}} &{} {{E}_{2}} &{} {{0}_{1n\times 2n}} \\ \end{matrix} \right] }^{T}}\), \({{{\bar{\Phi }} }_{3}}={{\left[ \begin{matrix} {{0}_{1n\times 2n}} &{} {{E}_{3}} &{} {{0}_{1n\times 1n}} \\ \end{matrix} \right] }^{T}}\), \({{{\bar{\Phi }} }_{4}}={{\left[ \begin{matrix} {{\hat{D}}^{T}}S_{1}^{T} &{} {{\hat{D}}^{T}}S_{2}^{T} &{} {{\hat{D}}^{T}}S_{3}^{T} &{} {{\hat{D}}^{T}}S_{4}^{T} \\ \end{matrix} \right] }^{T}}\), \({{{\bar{\Phi }} }_{5}}={{\left[ \begin{matrix} C &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}\).
Considering \({{S}_{1}}={{S}_{2}}={{S}_{3}}={{S}_{4}}=S\), pre and postmultiplying with \(diag\left\{ \begin{matrix} {{S}^{1}} &{} {{S}^{1}} &{} {{S}^{1}} &{} {{S}^{1}} &{} I &{} I &{} I &{} I &{} I \\ \end{matrix} \right\}\) and its transpose in (33), and finally adopting variables change as \({{S}^{1}}=Y\), \(K{{\left( {{S}^{1}} \right) }^{T}}=K{{Y}^{T}}=G\), \({{S}^{1}}{{Q}_{1}}{{\left( {{S}^{1}} \right) }^{T}}={{\hat{Q}}_{1}}\), \({{S}^{1}}{{Q}_{2}}{{\left( {{S}^{1}} \right) }^{T}}={{\hat{Q}}_{2}}\), \({{S}^{1}}{{R}_{{{\tau }_{1}}}}{{\left( {{S}^{1}} \right) }^{T}}={{\hat{R}}_{{{\tau }_{1}}}}\), \({{S}^{1}}{{R}_{{{\tau }_{2}}}}{{\left( {{S}^{1}} \right) }^{T}}={{\hat{R}}_{{{\tau }_{2}}}}\),
\({{S}^{1}}P{{\left( {{S}^{1}} \right) }^{T}}=\hat{P}\), one obtains (17). Hence, proof of Theorem 1 is completed. \(\square\)
Remark 2
For a specific value of \(\gamma\), \({{H}_{\infty }}\) controller gain matrix K may be obtained from the feasible solution of LMI (17). As per (10), the minimum value of \(\gamma\) should be utilized for computing controller gains to have minimal disturbance effect on system response. The minimum \(\gamma\) value can be obtained by minimizing \({{\gamma }^{2}}\) of LMI (17) and K can then be easily obtained from the solution of LMI (17) by using this \(\gamma\) value. But, this process results in high value of controller gains [19]. From Theorem 1, it is known that variables G and Y of LMI (17) are involved in the calculation of K. So, minimization of \(\left\ G \right\\) and \(\left\ {{Y}^{1}} \right\\) can result in the values of K within limits. Based on the analysis, following LMI optimization problem is designed, and solution of this optimization problem gives minimum \(\gamma\) as well as stabilizing gain K.
LMI Optimization Problem 1:
Min \({\gamma }^{2}+{\bar{g}}+{\bar{y}}\) Subject to (17), \(\left[ \begin{matrix} {\bar{g}}I &{} G \\ * &{} I \\ \end{matrix} \right] >0\) and \(\left[ \begin{matrix} Y &{} I \\ * &{} {\bar{y}}I \\ \end{matrix} \right] >0.\)
where \({\bar{y}}\) and \({\bar{g}}\) denote the matrix norms \(\left\ {{Y}^{1}} \right\\) and \(\left\ G \right\\), respectively.
Remark 3
The LK functional defined in (18) has double integral terms whereas the LK functionals chosen in [6, 21] have single integral terms. Use of multiple integral terms in LK functional reduces conservatism of delaydependent stabilization criterion. On the other hand, it increases the computational burden. To reduce computational burden as well as conservatism of stabilization criterion, LK functional defined in this paper is restricted to double integral terms.
The following corollary presents \({{H}_{\infty }}\) performance based criterion for stabilization of delayed IPS (16).
Corollary 1
Closedloop IPS (16) satisfies \(H_{\infty }\) criterion \(\left\ {{T}_{wy}} \right\ \le \gamma\) and \(\gamma >0\), for positive definite matrices \(\hat{P}\), \({{\hat{Q}}_{1}}\), \({{\hat{Q}}_{2}}\), \({{\hat{R}}_{{{\tau }_{1}}}}\), \({{\hat{R}}_{{{\tau }_{2}}}}\) and arbitrary matrix G, such that an LMI holds as follows:
where\({\tilde{\Lambda }}=\left[ \begin{matrix} {{{\tilde{\Lambda }} }_{11}} &{} {{{\tilde{\Lambda }} }_{12}} &{} {{{\tilde{\Lambda }} }_{13}} &{} {{{\tilde{\Lambda }} }_{14}} \\ * &{} {{{\tilde{\Lambda }} }_{22}} &{} {{{\tilde{\Lambda }} }_{23}} &{} {{{\tilde{\Lambda }} }_{24}} \\ * &{} * &{} {{{\tilde{\Lambda }} }_{33}} &{} {{{\tilde{\Lambda }} }_{34}} \\ * &{} * &{} * &{} {{{\tilde{\Lambda }} }_{44}} \\ \end{matrix} \right]\),
The corresponding \({{H}_{\infty }}\) controller gain matrix is obtained as \(K=G{{\left( {{Y}^{T}} \right) }^{1}}\).
Proof
One can prove Corollary 1 by adopting similar procedures as in proof of Theorem 1 without considering uncertainties of IPS parameters. \(\square\)
3.1.3 Sliding mode control law based on state feedback control strategy
Theorem 2
Switching control law satisfying reaching condition \({{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0\) is given by:
where \(\zeta\) is a constant having positive value close to zero.
Proof
Proof of Theorem 2 is given in Appendix. \(\square\)
Remark 4
Signum function of switching surface is generally used in sliding mode controller structure to drive system trajectory into the predefined sliding surface [6, 31, 32]. However, such controller has chattering issue, and therefore, use of signum function is avoided in design of sliding mode controller (35) to reduce the chattering effect.
3.2 Robust \({{H}_{\infty }}\) delayed feedback sliding mode controller designing strategy
3.2.1 Selection of switching surface
The switching surface is selected by including artificial delay to improve the performance of closedloop IPS, as:
where \({K}_{h}\) is a constant matrix, and h is the artificial delay chosen by the control designer. Derivative of (36) gives:
Substituting (8) into (37) yields:
Considering \({\dot{\sigma }}(t)=0\), the equivalent controller is derived from (38) as:
The sliding mode closedloop delayed IPS is obtained by substituting (39) into (8) as:
where matrix \(\hat{D}=DB{{(JB)}^{1}}JD\). System (40) without parametric uncertainty can be written as:
3.2.2 Robust \({{H}_{\infty }}\) stabilization criterion
Theorem 3
Closedloop IPS (40) satisfies \(H_{\infty }\) criterion \(\left\ {{T}_{wy}} \right\ \le \gamma\) and \(\gamma >0\), for positive definite matrices \(\hat{P}\), \({{\hat{Q}}_{1}}\), \({{\hat{Q}}_{2}}\), \({{\hat{Q}}_{h}}\), \({{\hat{R}}_{{{\tau }_{1}}}}\), \({{\hat{R}}_{{{\tau }_{2}}}}\), \({{\hat{R}}_{h}}\) and arbitrary matrices G and V, such that an LMI holds as follows:
where \({\hat{\Omega }} =\left[ \begin{matrix} {{{\hat{\Omega }} }_{11}} &{} {{{\hat{\Omega }} }_{12}} &{} {{{\hat{\Omega }} }_{13}} &{} {{{\hat{\Omega }} }_{14}} &{} {{{\hat{\Omega }} }_{15}} \\ * &{} {{{\hat{\Omega }} }_{22}} &{} {{{\hat{\Omega }} }_{23}} &{} {{{\hat{\Omega }} }_{24}} &{} {{{\hat{\Omega }} }_{25}} \\ * &{} * &{} {{{\hat{\Omega }} }_{33}} &{} {{{\hat{\Omega }} }_{34}} &{} {{{\hat{\Omega }} }_{35}} \\ * &{} * &{} * &{} {{{\hat{\Omega }} }_{44}} &{} {{{\hat{\Omega }} }_{45}} \\ * &{} * &{} * &{} * &{} {{{\hat{\Omega }} }_{55}} \\ \end{matrix} \right] ,\)
The corresponding robust \({{H}_{\infty }}\) controller gain matrices are obtained as \(K=G{{\left( {{Y}^{T}} \right) }^{1}}\) and \({{K}_{h}}=V{{\left( {{Y}^{T}} \right) }^{1}}\).
Proof
To investigate the stability of (40), the LK functional is considered as:
where
and
Next, by implementing similar steps as in proof of Theorem 1, one obtains (42). This completes the proof of Theorem 3. \(\square\)
LMI Optimization Problem 2:
Min \({\gamma }^{2}+{\bar{g}}+{\bar{v}}+{\bar{y}}\)
Subject to (42), \(\left[ \begin{matrix} {\bar{g}}I &{} G \\ * &{} I \\ \end{matrix} \right] >0\), \(\left[ \begin{matrix} {\bar{v}}I &{} V \\ * &{} I \\ \end{matrix} \right] >0\) and \(\left[ \begin{matrix} Y &{} I \\ * &{} {\bar{y}}I \\ \end{matrix} \right] >0.\)
where \({\bar{v}}\) represents the norm of matrix \(\left\ V \right\\). Stabilizing gains K and \(K_h\) can be calculated by solving the above LMI optimization problem.
\({{H}_{\infty }}\) performance based stabilization criterion for (41) can be deduced from Theorem 3. The following corollary presents the criterion for (41).
Corollary 2
Closedloop IPS (41) satisfies \(H_{\infty }\) criterion \(\left\ {{T}_{wy}} \right\ \le \gamma\) and \(\gamma >0\), for positive definite matrices \(\hat{P}\), \({{\hat{Q}}_{1}}\), \({{\hat{Q}}_{2}}\), \({{\hat{Q}}_{h}}\), \({{\hat{R}}_{{{\tau }_{1}}}}\), \({{\hat{R}}_{{{\tau }_{2}}}}\), \({{\hat{R}}_{h}}\) and arbitrary matrices G and V, such that an LMI holds as follows:
where \({\tilde{\varphi }} =\left[ \begin{matrix} {{{\tilde{\varphi }} }_{11}} &{} {{{\tilde{\varphi }} }_{12}} &{} {{{\tilde{\varphi }} }_{13}} &{} {{{\tilde{\varphi }} }_{14}} &{} {{{\tilde{\varphi }} }_{15}} \\ * &{} {{{\tilde{\varphi }} }_{22}} &{} {{{\tilde{\varphi }} }_{23}} &{} {{{\tilde{\varphi }} }_{24}} &{} {{{\tilde{\varphi }} }_{25}} \\ * &{} * &{} {{{\tilde{\varphi }} }_{33}} &{} {{{\tilde{\varphi }} }_{34}} &{} {{{\tilde{\varphi }} }_{35}} \\ * &{} * &{} * &{} {{{\tilde{\varphi }} }_{44}} &{} {{{\tilde{\varphi }} }_{45}} \\ * &{} * &{} * &{} * &{} {{{\tilde{\varphi }} }_{55}} \\ \end{matrix} \right]\),
The corresponding \({{H}_{\infty }}\) controller gain matrices are obtained as \(K=G{{\left( {{Y}^{T}} \right) }^{1}}\) and \({{K}_{h}}=V{{\left( {{Y}^{T}} \right) }^{1}}\).
Proof
One may follow Theorem 3 to prove Corollary 2. \(\square\)
3.2.3 Sliding mode control law based on delayed feedback control strategy
Theorem 4
Switching control law satisfying reaching condition \({{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0\) is given by:
Proof
Proof of Theorem 4 is similar to that of Theorem 2, and thus no further description is given here. \(\square\)
Remark 5
The performance of the proposed SMC scheme may be improved by selecting a suitable h value which can improve the transient stability of a delayed system. It can also be used to improve the dynamic response of a system with timedelay.
4 Results analysis
4.1 Numerical example
A wellknown numerical example of an IPS with two areas is considered for analyzing the performance of the proposed SMC schemes. Parameters of the IPS are given in Table 1 [19, 21].
The uncertainties in IPS parameters are assumed as \(E_{1}=E_{2}=E_{3}=H=0.001I\) as described in (9). Timedelays of area1 and area2 are considered to be fixed as \({\tau }_{1}=0.1s\) and \({\tau }_{2}=0.2s\), respectively.
For the \({{H}_{\infty }}\) state feedback sliding mode controller (35) and \({{H}_{\infty }}\) delayed feedback sliding mode controller (45), the design parameters \(\zeta\) and \(b_{i}\) are set as 0.001 and 1, respectively, whereas matrix J is selected as:
Besides \(\zeta\), \(b_{i}\) and J, controller gain matrix K is required to design controller (35), and an artificial delay (h) and controller gains (K and \(K_{h}\)) are needed to design controller (45). h is tuned at 0.86s, while LMI Optimization Problem 1 and Problem 2 are solved by using mincx solver of LMI control Toolbox in MATLAB to obtain \({{H}_{\infty }}\) performance index \(\gamma\) and gains of controllers (35) and (45), respectively. \(\gamma\) for controller (35) is computed as 1.4942 and the corresponding gain matrix K is presented in (46). Similarly, \(\gamma\) for controller (45) is computed as 2.6686 and the corresponding gain matrices K and \(K_{h}\) are presented in (47) and (48). Replacing (17) with (34) in LMI Optimization Problem 1 and (42) with (44) in LMI Optimization Problem 2, \({{H}_{\infty }}\) performance index and gains of controllers (35) and (45) are computed neglecting the uncertainties of IPS parameters. At this condition, \(\gamma\) is obtained as 1.4938 and K is computed as (49) for controller (35). Similarly, for controller (45), \(\gamma\) is obtained as 2.6679, and K and \(K_{h}\) are computed as (50) and (51).
In Table 2, a comparative study of h and \(\gamma\) values is presented. Larger h and smaller \(\gamma\) are obtained by using the proposed control approach in comparison to the other existing values [19, 21]. A larger value of \(\sigma\) indicates better delay tolerability of the closedloop IPS [2], while a smaller value of \(\gamma\) indicates a better reduction of disturbance effect on the system performance [8]. As the existence of timedelay in LFC scheme of IPS is an inevitable phenomenon, improvement in delay tolerability improves the stability of the IPS. As load disturbance is the main cause of load frequency deviation in IPS, the reduction of disturbance effect on system performance results in better stability of IPS.
Next, simulation studies of the proposed LFC schemes are performed in MATLAB/Simulink using the above mentioned parameters of the twoarea delayed IPS and the \({{H}_{\infty }}\) performancebased sliding mode controllers. The \({{H}_{\infty }}\) state feedback sliding mode LFC model is shown in Fig. 3. As seen, the LFC model consists of the twoarea delayed IPS and the proposed \({{H}_{\infty }}\) state feedback sliding mode controller. In Fig. 3, \(w_{1}\) and \(w_{2}\) are the disturbances of area1 and area2, respectively, while \({\sigma _{1}}(t)\) and \({\sigma _{2}}(t)\) are the sliding mode switching surface functions for area1 and area2, respectively. \(u_{1}\) and \(u_{2}\) are the respective \({{H}_{\infty }}\) state feedback SMC terms to area1 and area2. \(B_1\) and \(D_1\) are the first columns of the matrices B and D, respectively, while \(B_2\) and \(D_2\) are the second columns of the matrices B and D, respectively. \(J_1\) and \(J_2\) are the first and second rows of the matrix J, while \(K_1\) and \(K_2\) are the first and second rows of the controller gain matrix K, respectively.
Similarly, the \({{H}_{\infty }}\) delayed feedback sliding mode LFC model is shown in Fig. 4. As shown, the LFC model consists of the twoarea delayed IPS and the proposed \({{H}_{\infty }}\) delayed feedback sliding mode controller. In Fig. 4, \(w_{1}\) and \(w_{2}\) are the disturbances of area1 and area2, respectively, \({\sigma _{1}}(t)\) and \({\sigma _{2}}(t)\) are the sliding mode switching surface functions for area1 and area2, respectively, and \(u_{1}\) and \(u_{2}\) are the \({{H}_{\infty }}\) delayed feedback SMC terms to area1 and area2, respectively. \(B_1\) and \(D_1\) are the first columns of the matrices B and D, respectively, while \(B_2\) and \(D_2\) are the second columns of the matrices B and D, respectively. \(J_1\) and \(J_2\) are the first and second rows of the matrix J. \(K_1\) and \({K}_{h1}\) are the first rows of the controller gain matrices K and \(K_{h}\), respectively, while \(K_2\) and \(K_{h2}\) are the second rows of the controller gain matrices K and \(K_{h}\), respectively.
4.2 Simulation results
The twoarea closedloop delayed power system without parametric uncertainty [i.e., statespace model of twoarea IPS (6) with SMC laws (35) and (45)], and the twoarea closedloop delayed power system with parametric uncertainty [i.e., statespace model of twoarea IPS (8) with SMC laws (35) and (45)], are simulated in MATLAB/Simulink platform. Simulation studies on LFC schemes (i.e., state feedback \({{H}_{\infty }}\) SMC and delayed feedback \({{H}_{\infty }}\) SMC) are analyzed by considering step load disturbances and random load disturbances in both areas of the IPS.
In [19], a twoterm \({{H}_{\infty }}\) LFC scheme based on delayed feedback controller designing strategy is proposed for a twoarea IPS with timedelay. Moreover, in [21], delayed feedback \({{H}_{\infty }}\) LFC scheme is proposed for a twoarea IPS with timedelay and parametric uncertainty. By using the twoarea IPS parameters shown in Table 1, simulation studies of the schemes in [19] and [21] are performed in MATLAB/Simulink for both step load disturbances and random load disturbances in the two areas of the IPS. The simulation results of the proposed LFC schemes and the LFC schemes of [19] and [21] are then compared.
4.2.1 Stepload disturbance
Stepload changes in control area1 and control area2 are set as \(\Delta P_{d1}=0.1\) p.u. and \(\Delta P_{d2}=0.2\) p.u., respectively. Simulation results on frequency deviations and control signals of the closedloop IPS without parametric uncertainty are shown in Figs. 5 and 6, respectively. In comparison, the frequency deviations and control signals of the closedloop IPS with parametric uncertainty are shown in Figs. 7 and 8, respectively.
It is observed from Figs. 5 and 7 that the frequency deviations of the twoarea delayed IPS with and without parametric uncertainty are converging to zero within 15s by adopting the proposed delayed feedback \({{H}_{\infty }}\) SMC law (45). In comparison, by adopting the proposed state feedback \({{H}_{\infty }}\) SMC law (35) or the control approaches of [19, 21] (i.e., delayed feedback \({{H}_{\infty }}\) control approaches), the frequency deviations of the twoarea delayed IPS with and without parametric uncertainty take almost 30s for converging to zero. Furthermore, the observation of these frequency response curves in between 25 to 30 s is interesting, during which the frequency deviations of the twoarea delayed IPS with and without parametric uncertainty are almost zero by using the proposed delayed feedback \({{H}_{\infty }}\) SMC law (45). The damping of frequency deviation of the twoarea delayed IPS without parametric uncertainty by using the proposed state feedback \({{H}_{\infty }}\) SMC law (35) is not much better than those using the control approaches of [19, 21] (see Fig. 5), whereas, the damping of frequency deviation of the twoarea delayed IPS with parametric uncertainty by using the proposed state feedback \({{H}_{\infty }}\) SMC law (35) is better than that of using the control approaches of [19, 21] (see Fig. 7). Therefore, by comparing the frequency response curves of Figs. 5 and 7 from 25s to 30s, one can observe that the proposed control approaches perform better than the existing control approaches for the consideration of parametric uncertainty in delayed IPS. There is not much difference in the respective frequency response curves of delayed IPS with and without parametric uncertainty by using the proposed approaches. This study shows that the proposed \({{H}_{\infty }}\) performancebased sliding mode controllers are not sensitive to parametric variation.
One can observe from Fig. 6 that the convergence of control signals generated by the proposed delayed feedback \({{H}_{\infty }}\) SMC law (45) for the twoarea delayed IPS without parametric uncertainty is faster than the control signals generated by the proposed state feedback \({{H}_{\infty }}\) SMC law (35) or the control approaches of [19, 21]. Also, From Fig. 8, one can observe that with consideration of parametric uncertainty in the delayed IPS, the convergence of control signals generated by both the proposed control approaches is faster than those generated by the control approaches of [19, 21]. Moreover, it can be observed from Figs. 6 and 8 that the control signals generated by the proposed methods are free from chattering. There are significant differences in the convergence values of control signals produced by the proposed \({{H}_{\infty }}\) SMC laws (35) and (45), due to the difference in the structure of \({{H}_{\infty }}\) SMC laws (35) and (45). The \({{H}_{\infty }}\) SMC law (45) contains a delayed state feedback term, which is not present in the structure of the \({{H}_{\infty }}\) SMC law (35).
The above analysis of the simulation results indicates that the performance of the proposed \({{H}_{\infty }}\) SMC approaches is better than the performance of the existing \({{H}_{\infty }}\) control approaches [19, 21] in terms of robustness against parametric uncertainty, fast convergence and disturbance rejection performance. Furthermore, better load frequency regulation of delayed IPS is obtained with the proposed delayed feedback \({{H}_{\infty }}\) SMC approach than the proposed state feedback \({{H}_{\infty }}\) SMC approach. This means that artificial delay incorporation in SMC structure improves stability of delayed power system.
4.2.2 Randomload disturbance
Randomload disturbances applied to control area1 and control area2 are shown in Fig. 9. Frequency deviations and control signals of the closedloop delayed IPS without parametric uncertainty are depicted in Figs. 10 and 11, respectively. Considering uncertainty in the closedloop delayed IPS parameters, frequency deviations and control signals are depicted in Figs. 12 and 13, respectively.
It is observed from Figs. 10 and 12, the proposed control schemes reduce frequency deviations of the delayed IPS satisfactorily for randomload changes. Damping of frequency deviations by using the delayed feedback \(H_{\infty }\) SMC strategy (45) is better than that of state feedback \(H_{\infty }\) SMC strategy (35). Moreover, from Figs. 11 and 13, one can observe that the control signals generated by the delayed feedback \(H_{\infty }\) SMC strategy (45) have faster convergence rate than those by the state feedback \(H_{\infty }\) SMC strategy (35). This observation shows that incorporation of artificial delay in SMC structure improves the stability of IPS with timedelay. The control signals [i.e., \(u_{1}(t)\) and \(u_{2}(t)\)] generated by the proposed \(H_{\infty }\) SMC approaches for the delayed IPS areas are chattering free. Hence, \(H_{\infty }\) criterion based sliding mode controllers (35) and (45) give chattering free control signals. Furthermore, one can observe from these simulation results (i.e., Figs. 10, 11, 12 and 13) that the performance of the proposed control approaches is better than the existing control approaches [19, 21] in terms of frequency regulation and control signal convergence of the closedloop delayed IPS with randomload change in each control area.
The objective of any control design for load frequency regulation of IPS is to maintain any load frequency oscillation within acceptable range, which in power system is defined by two main levels such as statutory limit and operational limit [4]. Frequency oscillation of \(\pm 0.5\) Hz and \(\pm 0.2\) Hz are acceptable in statutory and operational limits, respectively [8]. From the frequency response curves shown in Figs. 5, 7, 10, and 12, it is clearly observed that frequency oscillations in each area of the closedloop delayed IPS with and without parametric uncertainty are within the range of \(\pm 0.2\) Hz. This observation indicates that the load frequency oscillation of delayed IPS with and without parametric uncertainty is damped satisfactorily by using the proposed \(H_{\infty }\) performancebased sliding mode controllers. Hence, application of the proposed sliding mode \(H_{\infty }\) LFC designs based on the proposed state feedback control and delayed feedback control strategies can make IPS operation safe, secure and stable.
\(H_{\infty }\) performancebased state feedback sliding mode controller (35) and \(H_{\infty }\) performancebased delayed feedback sliding mode controller (45) take state variables of the statespace model (6) as their inputs and generate control signals for the twoarea delayed IPS. Performance of controller (35) depends on the selection of suitable values of controller parameters \({b}_{1}\), \({{b}_{2}}\), J and K. Similarly, performance of controller (45) depends on the selection of suitable values of controller parameters \({b}_{1}\), \({{b}_{2}}\), h, J, K and \(K_{h}\). The values of \({b}_{1}\) and \({b}_{2}\) are selected according to the disturbances of area1 and area2, respectively, while the value of h is tuned by considering the values of timedelays appeared in area1 and area2. Selection of matrix J depends on the input matrix B of the statespace model (6), while computation of K and \(K_{h}\) needs the values of \(\tau _{1}\), \(\tau _{2}\), h, and matrices A, \(A_{d1}\), \(A_{d2}\), B, C, D of the statespace model (6) and (2). The matrices A, \(A_{d1}\), \(A_{d2}\), B, C and D are obtained by using the values of the twoarea IPS parameters given in Table 1. In real situation, the number of control areas of an IPS is not limited to two, and there may be a variety of generation technologies in each control area. For such situation, the number of state variables of the IPS (i.e., number of inputs to controller) and number of control inputs to IPS (i.e., number of controller outputs) may increase, which increase the dimensions of matrices A, \(A_{d1}\), \(A_{d2}\), B, C, D and J. Hence, dimensions of controller gain matrices K and \(K_{h}\) are increased and new values of K and \(K_{h}\) are computed. The above discussed changes in power system and controller parameters may change the control performance in real situations.
Conventional control design for load frequency regulation of IPS consists of an integral controller or a PI controller in each control area of the IPS to control the frequency oscillation of the respective control area [19]. Integral controllers and PI controllers are preferred to design conventional LFC schemes due to their simple structure and availability of various tuning methods [21], and when used in each control area of IPS for load frequency regulation, they are known as local controllers. For high and random load demand change as shown in Fig. 9, the local controllers may take a long time (up to several minutes) to bring the frequency oscillation to its tolerable limit. To handle such situation, the proposed sliding mode \(H_{\infty }\) LFC designs based on state feedback or delayed feedback control strategies can be considered, since in the proposed LFC designs, local controllers are considered as integral parts of the IPS while an \(H_{\infty }\) sliding mode controller is used to solve LFC problem of such IPS which already has local controllers. By observing the results analyzed in this paper, it is evident that the application of the proposed \(H_{\infty }\) performancebased sliding mode LFC designs can bring the frequency oscillation of IPS to its tolerable limit within a minute.
5 Conclusions
Two types of sliding mode controller designs are proposed in this paper by considering two different strategies, such as state feedback control strategy and delayed feedback control strategy, for power frequency regulation of delayed IPS with parametric uncertainty. In the delayed feedback sliding mode controller design, an artificial delay is incorporated intentionally to obtain better dynamic performance of the closedloop delayed IPS. To obtain minimal disturbance effect on the performance of closedloop systems (delayed IPS with state feedback sliding mode controller and with delayed feedback sliding mode controller), two delaydependent \(H_{\infty }\) criteria (Theorems 1 and 3) in LMI framework are developed separately using two separate LK functionals for the stabilization of the two closedloop IPSs with timedelay and parametric uncertainty. Stabilizing controller gains of these two types of sliding mode controllers are computed from the solution of these developed \(H_{\infty }\) criteria. These proposed LFC methods are tested for a numerical example of a twoarea power system. Results of the test system show that these control methods can damp frequency deviations adequately for both step and random load changes in each control area. The findings of this paper are summarized as follows:

The proposed \({{H}_{\infty }}\) performancebased sliding mode controller designs for load frequency regulation of delayed IPS with parametric uncertainty are not affected by chattering. So, its control performance is improved.

Damping of frequency oscillation is significantly improved by using the delayed feedback \({{H}_{\infty }}\) SMC approach in comparison to the state feedback \({{H}_{\infty }}\) SMC approach, which indicates that incorporation of an artificial delay in SMC design improves the stability of IPS with timedelay.

For the high and random load changes in each area of the delayed IPS (see Fig. 9), the frequency oscillations are within the operational limit (\(\pm 0.2\) Hz) by adopting the proposed \({{H}_{\infty }}\) SMC approaches (see Figs. 10 and 12). This observation shows the applicability of the proposed control approaches for making the operation of IPS stable, secure and safe.

The performance of the proposed delayed feedback \({{H}_{\infty }}\) sliding mode LFC approach is superior to those of the existing delayed feedback \({{H}_{\infty }}\) LFC approaches [19, 21] in terms of robustness against parametric variation, fast convergence and disturbance attenuation performance.
The study may be extended to analyzing the load frequency regulation of IPS considering a variety of generation technologies in each control area. Controller design for load frequency regulation of timedelay IPS with parametric uncertainty may also be proposed by using fractional order sliding mode controller in place of the integral sliding mode controller.
Availability of data and materials
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Abbreviations
 SMC:

Sliding mode control
 IPS:

Interconnected power system
 LFC:

Load frequency control
 PI:

Proportionalintegral
 PID:

Proportionalintegralderivative
 LMI:

Linear matrix inequality
 LK:

Lyapunov–Krasovskii
 ACE:

Area control error
 \(\Delta {{P}_{vi}}\) :

Governor valve position deviation of area i
 \(\Delta {{P}_{mi}}\) :

Mechanical output power deviation of area i
 \(\Delta {{f}_{i}}\) :

Frequency deviation of area i
 \(\Delta {{P}_{ij}}\) :

Tieline power deviation of area i and area j
 \(\Delta {{P}_{di}}\) :

Load disturbance of area i
 \({{T}_{gi}}\) :

Governor time constant of area i
 \({{T}_{pi}}\) :

Power system time constant of area i
 \({{T}_{chi}}\) :

Turbine time constant of area i
 \({{T}_{1}}\) :

Stiffness coefficient between area 1 and area 2
 \({{k}_{pi}}\) :

Power system gain of area i
 \({{k}_{i}}\) :

Gain of local integral controller of area i
 \(\Delta {{E}_{i}}\) :

Output of local integral controller in area i
 \(B_{i}\) :

Frequency bias parameter of area i
 \({{R}_{i}}\) :

Speed droop of area i
 \(\tau _{i}\) :

Timedelay in control area i
 h :

Artificial delay
 \(u_{i}\) :

Control input to area i
 K :

Controller gains correspond to present states
 \(K_{h}\) :

Controller gains correspond to delayed states
 \(\gamma\) :

\(H_{\infty }\) performance index
 I :

Identity matrix with proper dimension
 \({{R}^{1}}\) :

Inverse of matrix R
 \({{R}^{T}}\) :

Transpose of matrix R
 *:

Symmetric terms of a matrix
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Appendix A Proof of Theorem 2
Appendix A Proof of Theorem 2
Consider the Lyapunov function as:
Derivative of (52) is given by:
Using (13), one can write (53) as:
Substituting (35) in (54), one obtains:
Equation (55) can be expressed as:
The above equation may be further solved as:
Thus, control law (35) ensures \({{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0\).
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Pradhan, S.K., Das, D.K. Sliding mode controller design via delaydependent \(H_{\infty }\) stabilization criterion for load frequency regulation. Prot Control Mod Power Syst 8, 49 (2023). https://doi.org/10.1186/s4160102300322w
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DOI: https://doi.org/10.1186/s4160102300322w