Development of a decentralized nonlinear controller for a class of uncertain polynomial interconnected systems: Application for a large scale power system

Abstract

This paper presents a new robust decentralized control of nonlinear interconnected systems, which is applied and validated on a large scale power system. Our work is performed in three steps. Firstly, we have developed the polynomial description of the nonlinear uncertain and interconnected system using odd Kronecker power of state vectors, which is an easy-manipulation model for such complex systems. Then we applied Lyapunov’s direct method of stability analysis, associated with a quadratic function, in order to determine a sufficient condition for global asymptotic stability by applying a nonlinear, decentralized and optimal polynomial control. Finally, we carried out a simulation study on a nonlinear uncertain power system with three interconnected machines. We considered different cases of perturbations on its state variables as well as different cases of fault locations. We prove via advanced simulations, the effectiveness of the proposed control technique which is able to mitigate the successive amplitudes of the oscillations, to limit the control actions and to enhance the power system transient stability.

Keywords

Decentralized control interconnected power systems Kronecker product optimal control polynomial systems robust control

Introduction

Modeling, stability analysis, and control of nonlinear and complex systems is one of the most important problems that have attracted the attention of researchers (Benhadj, 1995; Bouzaouache and Benhadj, 2008; Elloumi and Benhadj, 2014; Elloumi et al., 2016; Lakshmikantham et al., 1991; Loussifi et al., 2016; Ortega et al., 1998; Phulpin et al., 2011). The resolution of such problems should take into account several factors. Firstly, the complex systems are generally characterized by several interconnections of many subsystems that are geographically distributed, thereby the stabilization of such large scale interconnected systems is generally based on decentralized schemes (Elloumi et al., 2013). The main objectives of decentralized control are to find some feedback laws for adapting the interactions from the other subsystems where no state information is transferred. In other words, each subsystem is controlled using only its locally available state and should obviously guarantee the stability of the global interconnected system. The advantage of decentralized control designs is to reduce complexity and therefore allow the control implementation to be more feasible.

Secondly, these processes are characterized by complex nonlinear dynamic models with varying parameters and can be subjected to important perturbations. The application of a decentralized robust control is then required to guarantee system stability under model parametric perturbations and different system configurations.

Decentralized robust control problems of large scale interconnected systems have been widely studied in the literature. In this context, several design approaches have been proposed and successfully applied to improve the transient stability, we can cite the feedback linearization technique (Wang et al., 1997; Wang and Hill, 2000), Hamiltonian techniques (Hill and Wang, 2000; Maschke et al., 2000; Wang et al., 2003; Xi et al., 2002), H_∞ synthesis (Tlili, 2017), guaranteed cost control approach (Rtibi et al., 2015), and sliding-mode control (Huerta et al., 2010; Yan et al., 2004). However, these researchers were limited to some special classes of nonlinear systems because there is no common method to investigate stability of general large scale interconnected systems, as it is known in the case of linear systems. Therefore, researchers in this direction are still actively in progress, aiming to develop and apply some advanced approaches for the study and the control of nonlinear interconnected systems.

The polynomial model constitutes an important class of nonlinear systems. It presents the advantage to permit the use of the Kronecker product and the power of matrices and vectors (Brewer, 1978; Graham, 1981) allowing important algebraic manipulations as shown in Bouafoura et al. (2011) and Rotella and Dauphin-Tanguy (1988). This class is able to describe the dynamic behavior of a large set of processes like electrical machines, power systems and robot manipulators (Elloumi and Benhadj Braiek, 2012). Moreover, since analytical nonlinear function can be approximated by a polynomial expansion, then the polynomial class allows to approach any nonlinear system.

Based on considerations cited above, this work aims mainly to provide a new design of a decentralized robust nonlinear controller for the stability of uncertain polynomial large scale interconnected systems. The description of these systems using Kronecker product and the use of the quadratic Lyapunov function have allowed the definition of sufficient conditions for the global asymptotic stability of the system equilibrium.

The second part of our contribution concerns the implementation of the developed approach on a large scale interconnected power systems. In fact, power system stability has been recognized as an important problem for secure system operation; so we aim in this paper to improve system transient stability, considering different fault locations and different cases of perturbations on its state variables.

The subsequent sections of this work are organized as follows. In section 2, the studied system is presented. Section 3 focuses on the presentation of decentralized robust control law and the determination of modified Algebraic Riccati Equation (ARE). Section 4 is devoted to the development of the decentralized nonlinear robust control and the establishment of some sufficient conditions for global stability of a nonlinear polynomial system. Some simulation results of the studied multimachine power system are presented in section 5. Finally, conclusions are provided in section 6.

Description of the studied system

We consider the class of nonlinear systems, composed of the interconnection of n subsystems, and for which the r order polynomial development is composed only by odd Kronecker power of vectors, that is, $r = 2 s - 1, s \in N$ (Elloumi and Benhadj Braiek, 2012)

\begin{matrix} \begin{matrix} {\overset{\cdot}{X}}_{i} = f_{i} (X_{i}) + B_{i} U_{i} + g_{i} (X_{1}, X_{2}, \dots, X_{n}) \\ = \sum_{k = 0}^{s - 1} (A_{i, 2 k + 1}) X_{i}^{[2 k + 1]} + B_{i} U_{i} + \sum_{\binom{j = 1}{j \neq i}}^{n} \sum_{k = 1}^{r} \sum_{s = 1}^{k} (G_{ij}^{k, s}) X_{i}^{[k - s]} \otimes X_{j}^{[s]} \end{matrix} \\ i = 1, 2, \dots, n \end{matrix}

(1)

where

$X_{i} \in R^{n_{i}}$ are the state vectors of the $i^{th}$ subsystem,

$A_{i, 2 k + 1} \in R^{n_{i} \times n_{i}^{2 k + 1}}$ is the characteristic matrix of app-ropriate dimension of the $i^{th}$ subsystem,

$B_{i} \in R^{n_{i} \times m_{i}}$ is the control vector of the of the $i^{th}$ subsystem,

$G_{ij}^{k, s}$ is the interconnection matrix of appropriate dimension of the $i^{th}$ subsystem,

${X_{i}}^{[k]} \in R^{n_{i}^{k}}$ is the power of order k of the vector X, as defined Kronecker,

$U_{i} (t) \in R^{m_{i}}$ the control of the $i^{th}$ subsystem.

The overall interconnected system is described by the following compact form

\begin{matrix} \overset{\cdot}{X} = A_{1} X^{[1]} + A_{3} X^{[3]} + \dots A_{r} X^{[r]} + BU \\ \overset{\cdot}{X} = \sum_{j = 0}^{s - 1} A_{2 j + 1} X^{[2 j + 1]} + BU \\ r = 2 s - 1, s \in N . \end{matrix}

(2)

where

$X^{T} = [X_{1}^{T}, X_{2}^{T}, \dots, X_{n}^{T}], X \in R^{N}, N = \sum_{i = 1}^{n} n_{i},$

$A_{2 j + 1} = diag (A_{i, 2 k + 1}) + G$ , with G expressed from $G_{ij}^{k, s}$ , and $A_{2 j + 1} \in R^{N \times N^{2 j + 1}}$ ,

$B = diag [B_{i}]$ ,

$U^{T} = [U_{1}^{T}, U_{2}^{T}, \dots, U_{n}^{T}]$ .

We suppose now, that the considered interconnected system presents some parametric fluctuations, then it can be described by the following uncertain expression

\begin{matrix} {\overset{\cdot}{X}}_{i} = f_{i} (X_{i}) + (B_{i} + Δ B_{i}) U_{i} + g_{i} (X_{1}, X_{2}, \dots, X_{n}) \\ i = 1, 2, \dots, n \end{matrix}

(3)

with

f_{i} (X_{i}) = \sum_{k = 0}^{s - 1} (A_{i, 2 k + 1} + Δ A_{i, 2 k + 1}) X_{i}^{[2 k + 1]}

(4)

and

g_{i} (X_{1}, \dots, X_{n}) = \sum_{\binom{j = 1}{j \neq i}}^{n} \sum_{k = 1}^{r} \sum_{s = 1}^{k} (G_{ij}^{k, s} + Δ G_{ij}^{k, s}) X_{i}^{[k - s]} \otimes X_{j}^{[s]}

(5)

where $Δ A_{i, 2 k + 1} \in R^{n_{i} \times n_{i}^{2 k + 1}}$ , $Δ B_{i} \in R^{n_{i} \times m_{i}}$ and $Δ G_{ij}^{k, s}$ are, respectively, the terms of uncertainties on the characteristic matrix, the control vector and the system interconnection term. These uncertainties are assumed to be of the following structure

\begin{matrix} Δ A_{i, 2 k + 1} = L_{i, 2 k + 1} F_{i, 2 k + 1} (t) E_{1 i, 2 k + 1} \\ Δ B_{i} = L_{i} F_{i} (t) E_{2 i} \\ Δ G_{ij}^{k, s} = L_{ij}^{k, s} F_{ij} {(t)}^{k, s} E_{ij}^{k, s} \end{matrix}

(6)

where $L_{i, 2 k + 1}, E_{1 i, 2 k + 1}, E_{2 i}, L_{i}, L_{ij}^{k, s}$ and $E_{ij}^{k, s}$ are known constant real matrices of appropriate dimensions.

$F_{i} (t) \in R^{p_{i} \times q_{i}}, F_{i, 2 k + 1} (t) \in R^{p_{i} \times {q_{i}}^{2 k + 1}}$ and ${F_{ij}}^{k, s} (t) \in R^{p_{ij} \times q_{ij}}$ are unknown matrix functions with Lebesgue measurable elements and satisfying

\begin{matrix} F_{i}^{T} (t) F_{i} (t) \leq I_{q_{i}} \\ F_{i, 2 k + 1} {(t)}^{T} . F_{i, 2 k + 1} (t) ⩽ I_{{q_{i}}^{2 k + 1}} \\ {(F_{ij}^{k, s})}^{T} (t) . F_{ij}^{k, s} (t) ⩽ I_{q_{ij}} \end{matrix}

(7)

where $I \in R^{l \times l}$ denotes the identity matrix.

The corresponding overall uncertain interconnected system is then described by the following compact form

\begin{matrix} \overset{\cdot}{X} = (A_{1} + Δ A G_{1}) X^{[1]} + (A_{3} + Δ A G_{3}) X^{[3]} + \dots \\ + (A_{r} + Δ A G_{r}) X^{[r]} + (B + Δ B) U \\ \overset{\cdot}{X} = \sum_{j = 0}^{s - 1} (A_{2 j + 1} + Δ A G_{2 j + 1}) X^{[2 j + 1]} + (B + Δ B) U \\ r = 2 s - 1, s \in N \end{matrix}

(8)

where $Δ A G_{2 j + 1} = diag (Δ A_{i, 2 k + 1}) + Δ G$ , with $Δ G$ expressed from $Δ G_{ij}^{k, s}$ , and $Δ A G_{2 j + 1} \in R^{N \times N^{2 j + 1}}$ , and $Δ B = diag [Δ B_{i}]$ .

The proposed polynomial decentralized robust control law

In this section, we propose a nonlinear decentralized control structure of the polynomial uncertain system. The local decentralized control law is expressed by the following polynomial form

\begin{matrix} U_{i} = - (K_{i 1} X_{i} + K_{i 3} X_{i}^{[3]} + K_{i 5} X_{i}^{[5]} + \dots + K_{ir} X_{i}^{[r]}) \\ i = 1, 2, \dots, n \end{matrix}

(9)

This corresponds to an overall control law given by

\begin{matrix} U = {(U_{1} \dots U_{n})}^{T} \\ = - (K_{1} X \begin{matrix} + K_{3} X^{[3]} + K_{5} X^{[5]} + \dots + K_{r} X^{[r]}) \end{matrix} \\ = - \sum_{j = 0}^{s - 1} K_{2 j + 1} X^{[2 j + 1]}, r = 2 s - 1 \end{matrix}

(10)

with

$K_{1} = diag (K_{i 1}), i = 1 \dots, n,$

$K_{2 j + 1}, j = 1 \dots, s - 1,$ are expressed from $K_{i, 2 j + 1},$

$K_{i, 2 j + 1}, i = 1 \dots, n$ and $j = 1 \dots, s - 1$ are given by

{\begin{matrix} a) K_{i 1} = R_{i}^{- 1} B_{i}^{T} P_{i} \\ b) M_{j, j + 1} (K_{i, 2 j + 1}) = (R_{i}^{- 1} B_{i}^{T} P_{i}) \otimes I_{n_{i}^{j}} \end{matrix}

(11)

where the mat-function $M_{j, j + 1}$ is defined in Appendix A, $R_{i} \in R^{m_{i} \times m_{i}}$ is a positive definite matrix, and $P_{i} (i = 1, \dots, n)$ is the symmetric positive definite matrix solution of a Riccati equation that should take into account the system uncertainties.

The determination of this modified Riccati equation is based on the consideration of decoupled uncertain subsystems defined by the following state equations

{\overset{\cdot}{X}}_{i} = [A_{i} + Δ A_{i}] X_{i} + [B_{i} + Δ B_{i}] U_{i}, i = 1, 2, \dots, n

(12)

with the associated modified quadratic criterion

J_{i} = \frac{1}{2} \int_{0}^{\infty} e^{2 α t} (X_{i}^{T} Q_{i} X_{i} + U_{i}^{T} R_{i} U_{i}) dt

(13)

with $α ⩾ 0$ and $Q_{i} \in R^{n_{i} \times n_{i}}$ a semi positive definite matrix.

Consider the following variable changes

{\begin{matrix} X_{i α} = e^{α t} X_{i} \\ U_{i α} = e^{α t} U_{i} \end{matrix}

(14)

Based on the expression (14), equation (12) becomes

{\overset{\cdot}{X}}_{i α} = A_{i α} X_{i α} + B U_{i α}

(15)

with

A_{i α} = A_{i} + Δ A_{i} + α I_{n}, and B = B_{i} + Δ B_{i}

(16)

then

{\overset{\cdot}{X}}_{i α} = (A_{i} + Δ A_{i} + α I_{n}) e^{α t} X_{i} + (B_{i} + Δ B_{i}) e^{α t} U_{i}

(17)

and

U_{i} = - K_{i} X_{i}, i = 1, 2, \dots, n

(18)

with

K_{i} = R_{i}^{- 1} B_{i}^{T} P_{i}

(19)

where $P_{i} (i = 1, 2, \dots, n)$ is the symmetric positive definite matrix, which is the solution of the following Algebraic Riccati Equation (ARE)

A_{i α}^{T} P_{i} + P_{i} A_{i α} - P_{i} B {R_{i}}^{- 1} B^{T} P_{i} + Q_{i} = 0

(20)

Then using (20), we obtain the following modified Algebraic Riccati Equation that depends on the uncertainty matrices

\begin{matrix} {[A_{i} + Δ A_{i}]}^{T} P_{i} + P_{i} [A_{i} + Δ A_{i}] - P_{i} B_{i} R_{i}^{- 1} B_{i}^{T} P_{i} - P_{i} B_{i} R_{i}^{- 1} Δ B_{i}^{T} P_{i} - P_{i} Δ B_{i} R_{i}^{- 1} B_{i}^{T} P_{i} \\ - P_{i} Δ B_{i} R_{i}^{- 1} Δ B_{i}^{T} P_{i} + 2 α P_{i} + Q_{i} = 0 \end{matrix}

(21)

Stability study

In this section, we present our contribution concerning the development of a sufficient condition to guarantee the global asymptotic stability of the uncertain interconnected polynomial system (8) with the decentralized polynomial control law (10).

This stability study is based on the Lyapunov direct method. Let the Lyapunov function V to be defined by the following quadratic form

V (X) = X^{T} P X

(22)

where $P = diag [P_{i}]$ is an $(n \times n)$ positive definite symetric matrix. The global asymptotic stability of the equilibrium state $(X = 0)$ of system (8) is ensured when the time derivative $\overset{\cdot}{V} (X)$ of $V (X)$ is negative definite for all $X \in R^{N}$ . The derivative of equation (22) gives

\overset{\cdot}{V} (X) = {\overset{\cdot}{X}}^{T} P X + X^{T} P \overset{\cdot}{X}

(23)

Using equations (8) and (10), the development of $\overset{\cdot}{V} (X)$ becomes

\begin{matrix} \overset{\cdot}{V} (X) = X^{T} P \sum_{j = 0}^{s - 1} [\begin{matrix} (A_{2 j + 1} + Δ A G_{2 j + 1}) - (B + Δ B) K_{2 j + 1} \end{matrix}] X^{[2 j + 1]} \\ + \sum_{j = 0}^{s - 1} [\begin{matrix} X^{{[2 j + 1]}^{T}} ((A_{2 j + 1} + Δ A G_{2 j + 1})^{T} - K_{2 j + 1}^{T} {(B + Δ B)}^{T}) \end{matrix}] PX \\ = 2 X^{T} P \sum_{j = 0}^{s - 1} [\begin{matrix} (A_{2 j + 1} + Δ A G_{2 j + 1}) - (B + Δ B) K_{2 j + 1}) \end{matrix}] X^{[2 j + 1]} \end{matrix}

(24)

By applying the ‘vec’ function property given by (62) in Appendix A, relation (24) can be written as

\begin{matrix} \dot{V} (X) = 2 \sum_{j = 0}^{s - 1} [v e c (P (A_{2 j + 1} + Δ A G_{2 j + 1}) - P (B + Δ B) K_{2 j + 1} {)]}^{T} X^{[2 j + 1]} \\ = 2 \sum_{j = 0}^{s - 1} X^{{[j + 1]}^{T}} M a t_{(N^{j + 1}, N^{j + 1})} [v e c (P (A_{2 j + 1} + Δ A G_{2 j + 1}) \\ - P (B + Δ B) K_{2 j + 1})] X^{[j + 1]} \end{matrix}

(25)

Now, when applying (63), (25) becomes

\begin{matrix} Ma t_{(N^{j + 1}, N^{j + 1})} [\begin{matrix} vec (P (A_{2 j + 1} + Δ A G_{2 j + 1}) - P (B + Δ B) K_{2 j + 1}) \end{matrix}] = \\ {\bar{U}}_{N^{j}, N} (P \otimes I_{N^{j}}) M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1}) - (B + Δ B) K_{2 j + 1}) \end{matrix}

(26)

This result and the following equation

\forall i, j \in N; {\bar{U}}_{N^{i} \times N^{j}} X^{[i + j]} = X^{[i + j]}

(27)

imply

\begin{matrix} \overset{\cdot}{V} (X) = 2 \sum_{j = 0}^{s - 1} X^{{[j + 1]}^{T}} Ma t_{(N^{j + 1}, N^{j + 1})} [vec (P (A_{2 j + 1} + Δ A G_{2 j + 1}) - P (B + Δ B) K_{2 j + 1})] X^{[j + 1]} \\ = 2 \sum_{j = 0}^{s - 1} {X^{[j + 1]}}^{T} {\bar{U}}_{N^{J}, N} (P \otimes I_{N^{J}}) M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1}) - (B + Δ B) K_{2 j + 1}) X^{[j + 1]} \\ = 2 \sum_{j = 0}^{s - 1} {X^{[j + 1]}}^{T} (P \otimes I_{N^{J}}) M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1}) - (B + Δ B) K_{2 j + 1}) X^{[j + 1]} \\ = 2 \sum_{j = 0}^{s - 1} {X^{[j + 1]}}^{T} (P \otimes I_{N^{J}}) [(M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) - M_{j, j + 1} ((B + Δ B) K_{2 j + 1})] X^{[j + 1]} \\ = 2 \sum_{j = 0}^{s - 1} X^{{[j + 1]}^{T}} (P \otimes I_{N^{J}}) [(M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) - (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N_{j}}] X^{[j + 1]} \\ = X^{T} [P (A_{1} + Δ A G_{1}) + {(A_{1} + Δ A G_{1})}^{T} P - P (B + Δ B) K_{1} - K_{1}^{T} {(B + Δ B)}^{T} P] X + 2 \sum_{j = 1}^{s - 1} X^{{[j + 1]}^{T}} \\ [(P \otimes I_{N^{J}}) (M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) - (P \otimes I_{N_{j}}) (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N_{j}}] X^{[j + 1]} \\ = X^{T} [P (A_{1} + Δ A G_{1}) + {(A_{1} + Δ A G_{1})}^{T} P - P (B + Δ B) K_{1} - K_{1}^{T} {(B + Δ B)}^{T} P] X + 2 \sum_{j = 1}^{s - 1} X^{{[j + 1]}^{T}} \\ [(P \otimes I_{N^{J}}) (M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) - (P (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N_{j}}] X^{[j + 1]} \end{matrix}

(28)

Since $A_{1} = A + G$ , where $A = diag (A_{i 1})$ , and G expressed from $G_{ij}^{k, s}$ , and suppose further that $Δ A G_{1} = Δ A + Δ G$ , with $Δ A = diag (Δ A_{i 1})$ , and $Δ G$ expressed from $Δ G_{ij}^{k, s}$ , we use the modified algebraic Riccati equation (21). It follows that inequality (28) is equivalent to

\begin{matrix} \overset{\cdot}{V} (X) = - X^{T} [Q + PB R^{- 1} B^{T} P - P Δ {BR}^{- 1} Δ B^{T} P + 2 α P \\ - P (G + Δ G) - (G^{T} + Δ G^{T}) P] X \\ + 2 \sum_{j = 1}^{s - 1} X^{T} [(P \otimes I_{N^{J}}) (M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) \\ - (P (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N_{j}}] X^{[j + 1]} \end{matrix}

(29)

where $Q = diag [Q_{i}], R^{- 1} = diag [R_{i}^{- 1}]$ .

Then (29) can be written as

\overset{\cdot}{V} = - Z^{T} M Z

(30)

Z = {(\begin{matrix} X \\ ⋮ \\ X^{[j + 1]} \end{matrix})}^{T}

(31)

M = (\begin{matrix} M_{1} & O \\ ⋱ \\ O & 2 M_{2 j + 1} \end{matrix})

(32)

where

\begin{matrix} M_{1} = Q + PB R^{- 1} B^{T} P - P Δ B R^{- 1} Δ B^{T} P + 2 α P \\ - (PG + G^{T} P) - (P Δ G + Δ G^{T} P) \end{matrix}

(33)

\begin{matrix} M_{2 j + 1} = (P \otimes I_{N^{J}}) (M_{j, j + 1} ((A_{2 j + 1} + Δ A G_{2 j + 1})) \\ - (P (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N^{j}} \end{matrix}

(34)

To ensure the asymptotic stability of system (8) with the decentralized control law $(10)$ , $\overset{\cdot}{V} (X)$ should be negative definite, which is equivalent to M positive definite, that means $M_{1}$ is a positive definite matrix and $M_{2 j + 1}$ is a semi positive definite matrix. So the main problem with the analysis of robust stability of uncertain system is to find the upper bound norm for uncertainty matrices $Δ B$ , $Δ G$ , $Δ A G_{2 j + 1}$ , then by exploiting this new upper bound norm of uncertainties matrices, we will obtain new practical sufficient conditions.

The uncertainty matrix $Δ B$ has a majorant matrix such that

\begin{matrix} Δ B ⩽ ‖ Δ B ‖ I \\ = ‖ (\begin{matrix} Δ B_{1} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & Δ B_{n} \end{matrix}) ‖ I \\ ⩽ ‖ (\begin{matrix} ‖ L_{1} F_{1} E_{21} ‖ & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & ‖ L_{n} F_{n} E_{2 n} ‖ \end{matrix}) ‖ I \end{matrix}

(35)

Using equation (6) and hypotheses (7), inequality (35) becomes

\begin{matrix} Δ B ⩽ μ I \end{matrix}

(36)

where

μ = ‖ (\begin{matrix} ‖ L_{1} ‖ ‖ E_{21} ‖ & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & ‖ L_{n} ‖ ‖ E_{2 n} ‖ \end{matrix}) ‖

(37)

So the uncertain part of $Δ B$ in equality (33) verifies

P Δ B R^{- 1} Δ B^{T} P ⩽ P μ^{2} ‖ R^{- 1} ‖ P

(38)

On the other hand, the uncertain part of $Δ G$ can be bounded by

\begin{matrix} P Δ G ⩽ ‖ P Δ G ‖ I \\ ⩽ ‖ P ‖ ‖ Δ G ‖ I \\ = ‖ P ‖ ‖ (\begin{matrix} 0 & Δ G_{12}^{11} & \dots & Δ G_{1 n}^{11} \\ Δ G_{21}^{11} & 0 & \dots & Δ G_{2 n}^{11} \\ \dots & \dots & 0 & \dots \\ Δ G_{n 1}^{11} & Δ G_{n 2}^{11} & \dots & 0 \end{matrix}) ‖ I \\ = ‖ P ‖ ‖ (\begin{matrix} 0 & ‖ Δ G_{12}^{11} ‖ & \dots & ‖ Δ G_{1 n}^{11} ‖ \\ ‖ Δ G_{21}^{11} ‖ & 0 & \dots & ‖ Δ G_{2 n}^{11} ‖ \\ \dots & \dots & 0 & \dots \\ ‖ Δ G_{n 1}^{11} ‖ & ‖ Δ G_{n 2}^{11} ‖ & \dots & 0 \end{matrix}) ‖ I \end{matrix}

(39)

From (6) and (7), it follows

P Δ G = ‖ P ‖ ‖ (\begin{matrix} 0 & ‖ L_{12}^{11} ‖ ‖ E_{12}^{11} ‖ & \dots & ‖ L_{1 n}^{11} ‖ ‖ E_{1 n}^{11} ‖ \\ ‖ L_{21}^{11} ‖ ‖ E_{21}^{11} ‖ & 0 & \dots & ‖ L_{2 n}^{11} ‖ ‖ E_{2 n}^{11} ‖ \\ \dots & \dots & 0 & \dots \\ ‖ L_{n 1}^{11} ‖ ‖ E_{n 1}^{11} ‖ & ‖ L_{n 2}^{11} ‖ ‖ E_{n 2}^{11} ‖ & \dots & 0 \end{matrix}) ‖ I

(40)

then

P Δ G ⩽ ‖ P ‖ β I

(41)

where

β = ‖ (\begin{matrix} 0 & ‖ L_{12}^{11} ‖ ‖ E_{12}^{11} ‖ & \dots & ‖ L_{1 n}^{11} ‖ ‖ E_{1 n}^{11} ‖ \\ ‖ L_{21}^{11} ‖ ‖ E_{21}^{11} ‖ & 0 & \dots & ‖ L_{2 n}^{11} ‖ ‖ E_{2 n}^{11} ‖ \\ \dots & \dots & 0 & \dots \\ ‖ L_{n 1}^{11} ‖ ‖ E_{n 1}^{11} ‖ & ‖ L_{n 2}^{11} ‖ ‖ E_{n 2}^{11} ‖ & \dots & 0 \end{matrix}) ‖

(42)

Consequently, it becomes

P Δ G + Δ G^{T} P ⩽ 2 ‖ P ‖ β I

(43)

By substituting the results of equalities (38) and (43) above in equation (33), we conclude that $M_{1}$ is bounded as follows

\begin{matrix} M_{1} \geq Q + 2 α P + P B R^{- 1} B^{T} P - P μ^{2} ‖ R^{- 1} ‖ P - (P G + G^{T} P) - 2 ‖ P ‖ β I \end{matrix}

(44)

We note that for $M_{1}$ to be positive, it is necessary that equality (44) should be verified.

Now, we focus on determining a minorant to $M_{2 j + 1}$ , given by expression (34), starting from

Δ A G_{2 j + 1} = diag (Δ A_{i, 2 j + 1}) + Δ G_{2 j + 1}, j = 1, \dots, s - 1

(45)

where

\begin{matrix} diag (Δ A_{i, 2 j + 1}) = (\begin{matrix} Δ A_{1, 2 j + 1} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & Δ A_{n, 2 j + 1} \end{matrix}) \\ \leq ‖ diag (Δ A_{i, 2 j + 1}) ‖ I \\ \leq ‖ (\begin{matrix} ‖ L_{1, 2 j + 1} ‖ ‖ E_{1, 2 j + 1} ‖ & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & ‖ L_{n, 2 j + 1} ‖ ‖ E_{n, 2 j + 1} ‖ \end{matrix}) ‖ I \\ = γ I \end{matrix}

(46)

where $γ = max (diag (‖ L_{i, 2 j + 1} ‖ ‖ E_{i, 2 j + 1} ‖))$ for $i = 1, 2, \dots, n$ and $j = 1, \dots, s - 1$ , in the same way the uncertainties parameters $Δ G_{2 j + 1}, j = 1, \dots, s - 1$ can be bounded as follows

Δ G_{2 j + 1} ⩽ σ . I

(47)

where $σ$ is the norm of a matrix expressed in terms of $L_{ij}^{k, s}$ and $E_{ij}^{k, s}$ .

Substitute the conditions aforementioned in (46) and (47) into (45) yields

Δ A G_{2 j + 1} ⩽ (γ + σ) I

(48)

Then applying the linear ’mat’ function, defined in Appendex A, inequality (48) becomes

\begin{matrix} M_{j, j + 1} (Δ A G_{2 j + 1}) ⩽ M_{j, j + 1} ((γ + σ) I), \\ - M_{j, j + 1} ((γ + σ) I) ⩽ M_{j, j + 1} (Δ A G_{2 j + 1}) ⩽ M_{j, j + 1} ((γ + σ) I) \\ M_{j, j + 1} (A_{2 j + 1}) + M_{j, j + 1} (Δ A G_{2 j + 1}) ⩾ M_{j, j + 1} (A_{2 j + 1}) - M_{j, j + 1} ((γ + σ) I) \end{matrix}

(49)

On the other hand, we have

\begin{matrix} (P (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N^{j}} = PB R^{- 1} B^{T} P \otimes I_{N^{j}} \\ + P Δ B R^{- 1} B^{T} P \otimes I_{N^{j}} \end{matrix}

(50)

now let

\begin{matrix} Δ B R^{- 1} B^{T} ⩽ ‖ Δ B R^{- 1} B^{T} ‖ I ⩽ ‖ Δ B ‖ ‖ R^{- 1} B^{T} ‖ I \\ = μ ‖ R^{- 1} B^{T} ‖ I \end{matrix}

(51)

then

P Δ B R^{- 1} B^{T} P \otimes I_{N^{j}} ⩽ P μ ‖ R^{- 1} B^{T} ‖ P \otimes I_{N^{j}}

(52)

and

(P (B + Δ B) R^{- 1} B^{T} P) \otimes I_{N^{j}} ⩽ P B R^{- 1} B^{T} P \otimes I_{N^{j}} + P μ ‖ R^{- 1} B^{T} ‖ P \otimes I_{N^{j}}

(53)

Combining (53) with (49) allows

\begin{matrix} M_{2 j + 1} ⩾ (P \otimes I_{N^{j}}) [M_{j, j + 1} (A_{2 j + 1}) - M_{j, j + 1} ((γ + σ) I)] \\ - [P (B R^{- 1} B^{T} + μ ‖ R^{- 1} B^{T} ‖ I) P] \otimes I_{N^{j}} \end{matrix}

(54)

Consequently, for $M_{2 j + 1}$ to be semi-positive defined, (54) should be verified. We can now state the following result:

Theorem 1: The decentralized robust control laws (9) (or (10)) assures a global asymptotic stability for a class of large scale interconnected uncertain polynomial system (8) if there exists $(n_{i} \times n_{i})$ positive definite matrices $Q_{i}$ , $i = 1, \dots, n$ and $α \in R$ such that $M_{1}$ of expression $(33)$ is positive definite and $M_{2 j + 1}$ for $j \geq 1$ of expression (34) are semi-positive definite matrices.

Application of the proposed control to a multimachine power system

In this section, we consider the application of the above method, presented in section 4, to the synthesis of a decentralized robust control, applied to an uncertain three-machine power system model (Figure 1) (Elloumi and Benhadj Braiek, 2002; Wang et al., 1998), characterized by the parameters (Table 1) (Elloumi and Benhadj Braiek, 2002).

Table 1.

Three-machine system parameters.

Parameters	Machine 1	Machine 2	Machine 3
$x_{d} (pu)$	1.863	2.36	2.36
$x_{d'} (pu)$	0.257	0.319	0.319
$x_{T} (pu)$	0.129	0.11	0.11
$x_{ad} (pu)$	1.712	0.712	0.712
$T_{d 0}' (pu)$	6.9	7.96	7.96
$H (s)$	4	5.1	5.1
$D (pu)$	5	3	3
$T_{m} (s)$	0.35	0.35	0.35
$T_{e} (s)$	0.1	0.1	0.1
R	0.05	0.05	0.05
$K_{m}$	1	1	1
$K_{e}$	1	1	1
$x_{12} (pu)$	0.55
$x_{13} (pu)$	0.53
$x_{23} (pu)$	0.6
$ω_{0} (rad / s)$	314.159

Multimachine power system modelization

A three-machine power system controlled by the steam valve opening, can be described with the interconnection of three subsystems as follows (Wang et al., 1998)

\begin{matrix} {\overset{\cdot}{X}}_{i} (t) = A_{i} X_{i} (t) + B_{i} U_{i} (t) + \sum_{j = 1, j \neq i}^{3} p_{ij} G_{ij} g_{ij} (X_{i}, X_{j}) \\ i = 1, 2, 3 \end{matrix}

(55)

where $X_{i} \in R^{n_{i}}$ is the state vector defined by

X_{i} (t)^{T} = [Δ δ_{i} (t) ω_{i} (t) Δ P_{m_{i}} (t) Δ X_{e_{i} (t)}],

with

\begin{matrix} Δ δ_{i} (t) = δ_{i} (t) - δ_{i_{0}} \\ Δ P_{m_{i}} (t) = P_{m_{i}} (t) - P_{{m_{i}}_{0}} \\ Δ X_{e_{i}} (t) = X_{e_{i}} (t) - X_{e_{i 0}} \end{matrix}

$δ_{i_{0}}, P_{{m_{i}}_{0}}$ and $X_{e_{i_{0}}}$ are the initial values of $δ_{i} (t)$ , $P_{mi} (t)$ and $X_{ei} (t)$ respectively.

$U_{i} (t) \in R^{m_{i}}$ is the control of the $i^{th}$ subsystem.

A_{i} = (\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & - \frac{D_{i}}{2 H_{i}} & \frac{ω_{0}}{2 H_{i}} & 0 \\ 0 & 0 & - \frac{1}{T_{mi}} & \frac{K_{mi}}{T_{mi}} \\ 0 & - \frac{K_{ei}}{T_{ei} R_{i} ω_{0}} & 0 & - \frac{1}{T_{ei}} \end{matrix})

B_{i} = (\begin{matrix} 0 \\ 0 \\ 0 \\ \frac{1}{T_{ei}} \end{matrix}), G_{ij} = (\begin{matrix} 0 \\ \frac{ω_{0} E_{qi}' E_{qj}' B_{ij}}{2 H_{i}} \\ 0 \\ 0 \end{matrix})

g_{ij} (t) = \sin (δ_{i} (t) - δ_{j} (t)) - \sin (δ_{i 0} - δ_{j 0})

with

$p_{ij}$ a constant of either 1 or 0 ( if 0 means that $j^{th}$ has no connection with $i^{th}$ );

$δ_{i}$ the rotor angle for $i^{th}$ machine, in radian;

$ω_{i}$ the relative speed for $i^{th}$ machine, in $radian / s;$

$P_{mi}$ the mechanical power for $i^{th}$ machine, in pu;

$X_{ei}$ the steam valve opening for $i^{th}$ machine, in pu;

$H_{i}$ the inertia constant for $i^{th}$ machine, in pu;

$D_{i}$ the damping coefficient for $i^{th}$ machine, in pu;

$T_{m_{i}}$ the time constant for $i^{th}$ machine’s turbine, in second;

$K_{m_{i}}$ the gain of $i^{th}$ machine turbine;

$T_{e_{i}}$ the time constant of $i^{th}$ machine speed governor, in second;

$R_{i}$ the regulation constant of $i^{th}$ machine, in pu;

$B_{ij}$ the nodal susceptance between $i^{th}$ and $j^{th}$ , in pu;

$ω_{0}$ the synchronous machine speed, in $radian / s$ ;

$E'_{qi}$ the internal transient voltage for $i^{th}$ machine, in pu, which is assumed to be constant;

$E'_{qj}$ the internal transient voltage for $j^{th}$ machine, in pu, which is assumed to be constant;

$x_{di}$ the direct axis reactance of the $i^{th}$ generator, in pu;

$x'_{di}$ the direct axis transient of the $i^{th}$ generator, in pu;

$x_{Ti}$ the transformer reactance, in pu;

$x_{adi}$ the mutual reactance between the excitation coil and the stator coil, in pu;

$T'_{d 0 i}$ the direct axis transient short-circuit time constant, in second;

$x_{ij}$ the transmission line reactance between the $i^{th}$ and $j^{th}$ the generators, in pu.

Polynomial model of the electrical system

The multimachine power system model (55) can be transformed to the nonlinear analytic model (3), after a usual limited development of the ‘sinus’ function. Then the power system model can be represented by a third order truncated polynomial form, which can be considered sufficient to reproduce the nonlinear analytical model; that is, for $n = 3$ (three-machine), $r = 2 s - 1 = 3$ , then $s = 2$ .

We obtain the following form

\begin{matrix} {\overset{\cdot}{X}}_{i} = (A_{i, 1} + Δ A_{i, 1}) X_{i} + (A_{i, 3} + Δ A_{i, 3}) {X_{i}}^{[3]} + (B_{i} + Δ B_{i}) U_{i} \\ + \sum_{\binom{j = 1}{j \neq i}}^{3} \sum_{k = 1}^{3} \sum_{s = 1}^{k} (G_{ij}^{k, s} + {Δ G}_{ij}^{k, s}) X_{i}^{[k - s]} \otimes X_{j}^{[s]}, i = 1, 2, 3 \end{matrix}

(56)

with

A_{i 1} {\begin{matrix} A_{1, 1} (1, 2) = 1, A_{1, 1} (2, 2) = - 0.625, \\ A_{1, 1} (4, 2) = - 0.637, A_{1, 1} (3, 2) = 39.27, \\ A_{1, 1} (3, 3) = - 2.857, A_{1, 1} (3, 4) = 2.857, \\ A_{1, 1} (4, 4) = - 10, \\ A_{2, 1} (1, 2) = 1, A_{2, 1} (2, 2) = - 0.3, \\ A_{2, 1} (4, 2) = - 0.637, \\ A_{2, 1} (3, 2) = 30.8, A_{2, 1} (3, 3) = - 2.857, \\ A_{2, 1} (3, 4) = 2.857, A_{2, 1} (4, 4) = - 10, \\ A_{3, 1} (1, 2) = 1, A_{3, 1} (2, 2) = - 0.3, \\ A_{3, 1} (4, 2) = - 0.637, \\ A_{3, 1} (3, 2) = 30.8, A_{3, 1} (3, 3) = - 2.857, \\ A_{3, 1} (3, 4) = 2.857, A_{3, 1} (4, 4) = - 10, \\ A_{1, 1} = A_{2, 1} = A_{3, 1} = O_{4 \times 4} for the other terms . \\ of these matrices . \end{matrix}

A_{i 3} {\begin{matrix} A_{1, 3} (2, 1) = 9.16, A_{2, 3} (2, 1) = 7.7, \\ A_{3, 3} (2, 1) = 8.43, \\ A_{1, 3} = A_{2, 3} = A_{3, 3} = O_{4 \times 64} for the \\ other terms of these matrices . \end{matrix}

B_{i} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 10 \end{matrix}]; i = 1, 2, 3

G_{ij}^{1, 1} {\begin{matrix} G_{12}^{1, 1} (2, 1) = 27.49, G_{21}^{1, 1} (2, 1) = 23.1, \\ G_{31}^{1, 1} (2, 1) = 23.1, \\ G_{13}^{1, 1} (2, 1) = 27.49, G_{23}^{1, 1} (2, 1) = 27.49, \\ G_{32}^{1, 1} (2, 1) = 27.49, \\ G_{12}^{1, 1} = G_{21}^{1, 1} = G_{31}^{1, 1} = G_{13}^{1, 1} = G_{23}^{1, 1} = G_{32}^{1, 1} = O_{4 \times 4} \\ for the other terms of these matrices . \end{matrix}

G_{i j, i \neq j}^{k, s \neq 1, 1} {\begin{array}{l} G_{12}^{3, 1} (2, 1) = - 13.74, G_{21}^{3, 1} (2, 1) = - 11.55, \\ G_{31}^{3, 1} (2, 1) = - 11.55, G_{13}^{3, 1} (2, 1) = 13.74, \\ G_{23}^{3, 1} (2, 1) = - 11.55, G_{32}^{3, 1} (2, 1) = - 13.74, \\ G_{12}^{3, 2} (2, 1) = 13.74, G_{21}^{3, 2} (2, 1) = 11.55, \\ G_{32}^{3, 2} (2, 1) = 13.74, G_{13}^{3, 2} (2, 1) = 13.74, \\ G_{23}^{3, 2} (2, 1) = 11.55, G_{32}^{3, 2} (2, 1) = - 13.74, \\ G_{12}^{3, 3} (2, 1) = - 4.58, G_{21}^{3, 3} (2, 1) = - 3.85, \\ G_{31}^{3, 3} (2, 1) = - 3.85, G_{13}^{3, 3} (2, 1) = - 4.58, \\ G_{23}^{3, 3} (2, 1) = - 3.85, G_{32}^{3, 3} (2, 1) = - 4.58, \\ G_{12}^{3.1} = G_{21}^{3, 1} = G_{31}^{3, 1} = G_{13}^{3, 1} = G_{23}^{3, 1} \\ = G_{32}^{3, 1} = G_{12}^{3, 2} = G_{21}^{3, 2} = G_{31}^{3, 2} \\ = G_{13}^{3, 2} = G_{23}^{3, 2} = G_{32}^{3, 2} = G_{12}^{3, 3} \\ = G_{21}^{3, 3} = G_{31}^{3, 3} = G_{13}^{3, 3} = G_{23}^{3, 3} \\ = G_{32}^{3, 3} = O_{4 \times 64} for the other terms of these matrices . \end{array}

The uncertainties matrices are chosen such that

Δ A_{i 1} {\begin{matrix} Δ A_{1, 1} (3, 3) = - 0.635, \\ Δ A_{1, 1} (3, 4) = 0.635, \\ Δ A_{2, 1} (3, 3) = - 0.635, \\ Δ A_{2, 1} (3, 4) = 0.635, \\ Δ A_{3, 1} (3, 3) = 0, \\ Δ A_{3, 1} (3, 4) = 0, \\ Δ A_{1, 1} = Δ A_{2, 1} \\ = Δ A_{3, 1} = O_{4 \times 4} \\ for the other terms of these matrices . \end{matrix} A_{i 3} {\begin{matrix} Δ A_{1, 3} (2, 1) = 4.5650, \\ Δ A_{2, 3} (2, 1) = - 0.7317, \\ Δ A_{3, 3} (2, 1) = 0, \\ Δ A_{1, 3} = Δ A_{2, 3} = Δ A_{3, 3} = O_{4 \times 64} \\ for the other terms of these matrices . \end{matrix}

Δ G_{i j, i \neq j}^{k, s \neq 1, 1} {\begin{array}{l} Δ G_{12}^{3, 1} (1, 1) = - 13.74, Δ G_{13}^{3, 1} (1, 1) = - 11.55, \\ Δ G_{12}^{3, 2} (1, 1) = - 11.55, Δ G_{13}^{3, 2} (1, 1) = 13.74, \\ Δ G_{12}^{3, 3} (1, 1) = - 11.55, Δ G_{13}^{3, 3} (1, 1) = - 13.74, \\ Δ G_{21}^{3, 3} (1, 1) = 13.74, Δ G_{21}^{3, 2} (1, 1) = 11.55, \\ Δ G_{21}^{3, 1} (1, 1) = 13.74, Δ G_{23}^{3, 1} (1, 1) = 13.74, \\ Δ G_{23}^{3, 2} (1, 1) = 11.55, Δ G_{23}^{3, 3} (1, 1) = - 13.74, \\ Δ G_{31}^{3, 3} (1, 1) = - 4.58, Δ G_{32}^{3, 3} (1, 1) = - 3.85, \\ Δ G_{31}^{3, 2} (1, 1) = - 3.85, Δ G_{32}^{3, 2} (1, 1) = - 4.58, \\ Δ G_{31}^{3, 1} (1, 1) = - 3.85, Δ G_{32}^{3, 1} (1, 1) = - 4.58, \\ Δ G_{12}^{3.1} = Δ G_{13}^{3, 1} = Δ G_{12}^{3, 2} = Δ G_{13}^{3, 2} = Δ G_{12}^{3, 3} \\ = Δ G_{13}^{3, 3} = Δ G_{21}^{3, 3} = Δ G_{21}^{3, 2} = Δ G_{21}^{3, 1} \\ = Δ G_{23}^{3, 1} = Δ G_{23}^{3, 2} = Δ G_{32}^{3, 3} = Δ G_{31}^{3, 3} \\ = Δ G_{32}^{3, 3} = Δ G_{31}^{3, 2} = Δ G_{32}^{3, 2} = Δ G_{31}^{3, 1} \\ = Δ G_{32}^{3, 1} = O_{4 \times 64} for the other terms of these matrices . \end{array}

The global interconnected system can be defined by the following polynomial form

\overset{\cdot}{X} = [A_{1} + Δ A G_{1}] X + [A_{3} + Δ A G_{3}] X^{[3]} + [B + Δ B] U

(57)

with

X^{T} = [X_{1}^{T}, X_{2}^{T}, \dots, X_{n}^{T}], B = diag [B_{i}],

U^{T} = [U_{1}^{T}, U_{2}^{T}, U_{3}^{T}],

A_{1} (12 \times 12) = (\begin{matrix} A_{11} & G_{12}^{1, 1} & G_{13}^{1, 1} \\ G_{21}^{1, 1} & A_{21} & G_{23}^{1, 1} \\ G_{31}^{1, 1} & G_{32}^{1, 1} & A_{31}, \end{matrix})

$A_{1} = (A_{1} + G_{1}), A_{1} = diag [A_{i 1}],$ $G_{1}$ is the interconnection matrix of the first order.

A_{3} {\begin{matrix} A_{3} (1, 1) = A_{13}, A_{3} (1, 65) = G_{12}^{31}, \\ A_{3} (1, 129) = G_{13}^{3, 1}, A_{3} (1, 257) = G_{12}^{3, 2}, \\ A_{3} (1, 513) = G_{13}^{3, 2}, A_{3} (1, 833) = G_{12}^{3, 3}, \\ A_{3} (1, 1665) = G_{13}^{3, 3}, A_{3} (5, 1) = G_{21}^{3, 3}, \\ A_{3} (5, 577) = G_{21}^{3, 2}, A_{3} (5, 769) = G_{21}^{3, 1}, \\ A_{3} (5, 833) = A_{23}, A_{3} (5, 897) = G_{23}^{31}, \\ A_{3} (5, 1089) = G_{23}^{3, 2}, A_{3} (5, 1665) = G_{23}^{3, 3}, \\ A_{3} (9, 1) = G_{31}^{3, 3}, A_{3} (9, 833) = G_{32}^{3, 3}, \\ A_{3} (9, 1153) = G_{31}^{3, 2}, A_{3} (9, 1409) = G_{32}^{3, 2}, \\ A_{3} (9, 1537) = G_{31}^{3, 1}, A_{3} (9, 1601) = G_{22}^{31}, \\ A_{3} (9, 1665) = A_{33}, \\ A_{3} (i, j) = 0 for the other values of 1 ⩽ i ⩽ 12 and 1 ⩽ j ⩽ 728 \end{matrix}

$Δ A G_{1} (12 \times 12) = (Δ A_{1} + Δ G_{1}),$ with $Δ A_{1} = d i a g (Δ A_{11}, Δ A_{21}, Δ A_{31})$ and

Δ G_{1} = (\begin{matrix} 0 & Δ G_{12}^{11} & Δ G_{13}^{11} \\ Δ G_{21}^{11} & 0 & Δ G_{23}^{11} \\ Δ G_{31}^{11} & Δ G_{32}^{11} & 0 \end{matrix})

Δ A G_{3} (12 \times 1728) = (Δ A_{3} + Δ G_{3})

with $Δ A_{3} = diag (Δ A_{13}, Δ A_{23}, Δ A_{33}),$ and $Δ G_{3}$ expressed from $Δ G_{ij, i \neq j}^{k, s \neq 1, 1}$

Δ B_{1} = (\begin{matrix} Δ B_{1} & 0 & 0 \\ 0 & Δ B_{2} & 0 \\ 0 & 0 & Δ B_{3} \end{matrix})

Calculation of the control law

The decentralized control laws, for the three machines are expressed in the following form

{\begin{matrix} U_{1} = - K_{11} X_{1} - K_{13} X_{1}^{[3]} \\ U_{2} = - K_{21} X_{2} - K_{23} X_{2}^{[3]} \\ U_{3} = - K_{31} X_{3} - K_{33} X_{3}^{[3]} \end{matrix}

(58)

which corresponds to the overall control law

U = (\begin{matrix} U_{1} \\ U_{2} \\ U_{3} \end{matrix}) = - K_{1} X - K_{3} X^{[3]}

(59)

where $K_{1} = diag (K_{11}, K_{21}, K_{31})$ corresponds to the linear part of the control, and $K_{3}$ is expressed in terms of $K_{13}, K_{23}, K_{33}$ . To calculate the gains $K_{1}$ and $K_{3}$ the matrices $R_{i}$ , $Q_{i}$ for $i = 1, 2, 3,$ and the coefficient $α$ , are chosen, via a trial method, such that

\begin{matrix} α = 0 \\ R_{i} = 2 \\ Q_{i} = diag {0.001, 0.001, 0.01, 0.01} \end{matrix}

The gains $K_{i 1}$ and $K_{i 3}$ are given by

$K_{i 1} = R_{i}^{- 1} B_{i}^{T} P_{i}, i = 1, 2, 3$ where $P_{i}$ is the solution of algebraic Riccati equation (21).

$M_{1, 2} (K_{i 3}) = (R_{i}^{- 1} B_{i}^{T} P_{i}) \otimes I_{4}, i = 1, 2, 3$

We obtain

K_{11} = [216.2867, 37.1001, 88.7956, 8.4863]

K_{21} = [273.2567, 47.9028, 89.3301, 8.5082]

K_{31} = [273.2567, 47.9028, 89.3301, 8.5082]

It can be easily checked that the matrix

M = (\begin{matrix} M_{1} & O \\ O & 2 M_{3} \end{matrix})

is positive definite, where $M_{1}$ and $M_{3}$ are given in $(33)$ and $(34)$ . So, the decentralized control law (9) stabilizes asymptotically the overall interconncected power system.

Simulation results

In this section, the dynamic performances of the controllers are evaluated. Simulation studies are carried out under different contingencies. The following cases are considered.

Effect of small disturbance on the dynamic performance of the system: Perturbations of state variables

The performances of the designed control are shown in Figures (2) and (3), on which the evolution are simulated of the system state variables $Δ δ_{i} (t), ω_{i} (t)$ , $Δ P_{m_{i}} (t), Δ X_{e_{i} (t)}$ and the corresponding control inputs, when a perturbation has occurred on the rotor angle variation of the first machine.

The curves of Figures (4) and (5) show, respectively, the evolution of the three machine state variables when a perturbation has occurred on the relative speed variation of the second machine, and the corresponding decentralized control signals.

The simulation results presented in these figures demonstrate clearly the ability of the decentralized nonlinear control to neutralize rapidly the effect of the occurred perturbations by enhancing rapidly the transient stability of the multimachine power system.

Effect of severe disturbance on the dynamic performance of the system: Perturbations of state variables with temporary three-phase short-circuit fault

We consider a symmetrical three-phase short-circuit fault as in Wang et al. (1998) and Wang and Hill (2000), which is assumed to be on the transmission line between the first and the second machine at 1s and is cut off at 1.1s. The original system is restored after the fault clearance. Let $λ$ denote the fraction of the transmission line to the left of the fault. The persistent disturbance of 10% is used in the simulation.

Case 1: The fault location is $λ = 0.2$ , the corresponding closed loop system responses are shown in Figure 6.

Case 2: The fault location is $λ = 0.8$ , the corresponding closed loop system responses are shown in Figure 7.

Case 3: The fault location is $λ = 0.5$ , the corresponding closed loop system responses are shown in Figure 8.

From the simulation results shown in these figures, it can be seen that the robust decentralized control law is able to damp the oscillations of the state variables and to enhance transient stability of the multimachine power system despite the different cases of perturbations and fault locations. So the proposed nonlinear decentralized control ensures an effective stability for the global system.

Conclusions

In this paper, we have developed and validated a new decentralized control approach of uncertain polynomial interconnected systems. The studied systems are described by a polynomial model with odd Kronecker power of state vectors. The nonlinear decentralized control law, which was described by a polynomial form, was able to guarantee the asymptotic stability of the overall interconnected system when some sufficient conditions were verified. The numerical simulation results have confirmed the validity and the feasibility of the proposed approach that was able to damp the system oscillations, and to enhance the power system transient stability, despite the high nonlinear interconnection between generators, and the different perturbations that would occur on the system state variables.

Although the proposed control structure seems to be strongly nonlinear and complex, it succeeds to resolve the problem of robust decentralized control of nonlinear interconnected process. Due to the continuous huge progress of the computer technology, the proposed nonlinear control structure can be easily implemented on real processes such as electrical power systems.

Footnotes

Appendix A

We recall here the useful mathematical notations and properties used in this paper concerning the Kronecker tensor product.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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