Invertibility of Nonlinear Differential-Algebraic-Equation Subsystems with Application to Power Systems

Abstract

For nonlinear differential-algebraic-equation subsystems, whose index is one and interconnection input is locally measurable, the problem of invertibility is discussed and the results are applied to the power systems component decentralized control. The inverse systems’ definitions for such a class of differential-algebraic-equation subsystems are put forward. A recursive algorithm is proposed to judge whether the controlled systems are invertible. Then physically feasible α-order integral right inverse systems are constructed, with which the composite systems are linearizaed and decoupled. Finally, decentralized excitation and valve coordinative control for one synchronous generator within multimachine power systems are studied and the simulation results based on MATLAB demonstrate the effectiveness of the control scheme proposed in this paper.

1. Introduction

A number of physical systems such as power systems, economic systems, and constrained robot systems are mathematically described by differential-algebraic-equation (DAE) systems [1]. Various concepts, theories, and methods of ordinary-differential-equation (ODE) systems are extended and great progress has been made for DAE systems [2–6]. In [2], for linear DAE systems with input saturation, the composite nonlinear feedback control problem was considered through introducing the generalized Lyapunov function. In [3], the state-feedback stabilization is considered for nonlinear discrete DAE large-scale control systems using Lyapunov matrix equation. 𝐻_∞ Observer was designed for a class of continuous time nonlinear DAE systems in [4], where necessary and sufficient condition for observer existence was established under the worst conditions. In [5], the traditional Kalman filter was improved and a recursive state estimation method is presented for nonlinear DAE systems. In [6], the output feedback compensation problem was considered by coupling the design of controller and observer instead of separation principle.

In order to provide a measure of the difference between DAE systems and ODE systems, the notion of differential index is commonly used, which corresponds to the minimum number of differentiations of the algebraic equations required to obtain equivalent ODE systems [7]. Among nonlinear DAE systems, the systems of index one represent an important class of physical systems such as power systems and electric circuits. In [8], the Lyapunov method of nonlinear ODE systems was extended to nonlinear DAE systems of index one and the sufficient conditions of stability are presented. In [9], still for such a class of nonlinear DAE systems, an explicit constructing method was given and the state space order-reduction realization was achieved with application to power systems control.

In most existing results, the controlled systems are treated as isolated ones. However, in many practical applications, controlled systems are subsystems within large-scale systems and there exist mutual influence and constraint between the controlled systems and the rest of the large-scale systems. Typically, a so-called “power systems component structural model” put forward in [10, 11] for power systems component decentralized control problem just falls into this category. At the same time, it should be noted that DAE subsystems control problems are common not only for power systems but also for large-scale systems of others areas [12]. As far as the authors know, the research about nonlinear DAE subsystems is seldom found.

Among various nonlinear control methods, the invertibility of systems plays an important role in linearization and decoupling of general nonlinear systems (not restricted to affine form) [13]. The research on invertibility of nonlinear DAE systems can be traced to [14, 15], where the invertibility of continuous and discrete DAE systems was discussed, respectively.

In this paper, for nonlinear DAE systems whose index is one and interconnection inputs are locally measurable and bounded, the invertibility is studied and the results are applied to power systems component decentralized control. The structure of this paper is as follows. Firstly, the definitions of unit right inverse systems and 𝛼-order integral right inverse systems are given. Secondly, a recursive algorithm is proposed with which to determine whether the controlled nonlinear DAE subsystems are invertible. If the controlled systems are invertible, physically realizable 𝛼-order integral right inverse systems are constructed through state-feedback and dynamic compensation, with which the decoupling and linearization of the composite systems are both achieved so that various linear control methods and theories can be applied. At last, decentralized excitation and valve coordinative controller are designed for one synchronous generator set within multimachine power systems. The simulation is conducted based on MATLAB and the simulation results illustrate the effectiveness of the proposed scheme in this paper.

2. System Description and Problem Formulation

We consider general nonlinear DAE subsystems as follows: 𝑥̇_𝑖=𝑓_𝑖𝑥_𝑖,𝑧_𝑖,𝑢_𝑖,𝑣_𝑖,𝑔_𝑖𝑥_𝑖,𝑧_𝑖,𝑢_𝑖,𝑣_𝑖𝑦=0,_𝑖=ℎ_𝑖𝑥_𝑖,𝑧_𝑖,𝑢_𝑖,𝑣_𝑖,𝑖=1,…,𝑁, (1) where 𝑥_𝑖∈𝑁^𝑖₀⊂𝑅^𝑛𝑖,𝑧_𝑖∈𝑀^𝑖₀⊂𝑅^𝑙𝑖,𝑢_𝑖∈𝐿^𝑖₀⊂𝑅^𝑚𝑖,𝑦_𝑖∈𝐾^𝑖₀⊂𝑅^𝑚𝑖,𝑣_𝑖∈𝑆^𝑖₀⊂𝑅^𝑠𝑖 are differential variable, algebraic variable, manipulated input, controlled output, and interconnection input, respectively. 𝑣_𝑖 reflects the influence of the rest of the large-scale systems on controlled systems (1). 𝑓_𝑖∈𝑅^𝑛𝑖,𝑔_𝑖∈𝑅^𝑙𝑖,ℎ_𝑖∈𝑅^𝑚𝑖 are smooth vector fields. For the sake of simplicity, we will omit the subscript 𝑖 of (1) in the remainder of this paper. Let 𝑋₀=(𝑥₀,𝑢₀,𝑧₀,𝑦₀,𝑣₀)∈𝑈₀ is compatible initial conditions, that is, 𝑔(𝑥₀,𝑧₀,𝑢₀,𝑣₀)=0 where 𝑈₀=[𝑁₀,𝑀₀,𝐿₀,𝐾₀,𝑆₀] [8].

Throughout this paper, the following basic hypotheses are made for (1).

(H1) The Jacobian matrix of 𝑔(𝑥,𝑧,𝑢,𝑣) with respect to 𝑧 has constant full rank on 𝑈₀: rank𝜕𝑔𝜕𝑧=𝑙,∀𝑥,𝑧,𝑢,𝑦,𝑣∈𝑈₀, (2) that is, (1) is of index one.

(H2) The interconnection input 𝑣 and its sufficient order derivatives are locally measurable and bounded.

Remark 1. Above basic hypotheses hold for power systems components under normal operating conditions.
In sequel we will give inverse systems definition of nonlinear DAE subsystems (1), including unit right inverse systems and 𝛼-order integral right inverse systems.
The controlled output 𝑦(𝑡) of nonlinear DAE subsystems (1) is determined together by the interconnection input 𝑣, manipulated input 𝑢(𝑡) and initial conditions. From the functional point of view, nonlinear DAE subsystems (1) can be regarded as an operator (marked by 𝜃) which maps the manipulated input 𝑢(𝑡), and interconnection input 𝑣(𝑡) to control output 𝑦(𝑡), that is, 𝑦=𝜃(𝑢,𝑣). (3)

Definition 2. Suppose there exist systems Σ which bear input-output mapping relationship: 𝑦=𝜃(𝑟,𝑣), where the input 𝑟(𝑡)=(𝑟₁,…,𝑟_𝑚)^𝑇 is a smooth vector. For nonlinear DAE subsystems (1), if 𝑢(𝑡)=𝑦(𝑡), we have 𝑦(𝑡)=𝑟(𝑡), and then the systems Σ are the unit right inverse systems of nonlinear DAE subsystems (1). We call systems (1) are invertible.

Definition 3. Suppose there exist systems Σ_𝛼 which bear input-output mapping relationship: 𝑦=𝜃_𝛼(𝜑,𝑣) where the input 𝜑(𝑡)=(𝜑₁,…,𝜑_𝑚)^𝑇=𝑟^(𝛼)=(𝑟^(𝛼1)₁,…,𝑟^(𝛼𝑚)_𝑚)^𝑇 is a continuous vector. For nonlinear DAE subsystems (1), if 𝑢(𝑡)=𝑦(𝑡), we have 𝑦^(𝛼)=𝜑 where 𝑦^(𝛼)=(𝑦^(𝛼1)₁,…,𝑦^(𝛼𝑚)_𝑚)^𝑇 (i.e., 𝑦_𝑖^(𝛼𝑖)=𝜑_𝑖,𝑖=1,…,𝑚), and then the systems Σ are the unit right inverse systems of nonlinear DAE subsystems (1). We call systems (1) are invertible.

The aim of this paper is to study the invertibility of nonlinear DAE subsystems (1) satisfying (H1) and (H2). The linearization and decoupling of the composite systems can be achieved if the controlled systems (1) are invertible. As a result, various linear control theorems and methods can be applied.

3. Recursive Algorithm and Invertibility of Nonlinear DAE Subsystems

We will give a recursive algorithm, with which to determine the invertibility of nonlinear DAE subsystems (1).

From (2) we know that (𝜕𝑔/𝜕𝑧)⁻¹ exist on 𝑈₀. We use the following operator 𝐸_𝜉(⋅): 𝐸_𝜉(𝐹)≜𝜕𝐹𝜕𝜉_{0=𝑔(𝑥,𝑧,𝑢,𝑣)}=𝜕𝐹−𝜕𝜉𝜕𝐹𝜕𝑧𝜕𝑔𝜕𝑧⁻¹𝜕𝑔𝜕𝜉 (4) to denote the Jacobian matrix of vector function 𝐹(𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘),𝑣,…,𝑣^(𝑘)) with respect to some variable 𝜉∈(𝑥,𝑢,𝑣) under the algebraic constraint 0=𝑔(𝑥,𝑧,𝑢,𝑣). For the limit of space, only the procedure of step 𝑘 is presented.

Step 𝑘(𝑘=1,2,…). Suppose that, until to step 𝑘, we can get a sequence of nonnegative integers 𝛾₀,…,𝛾_𝑘−2,𝛾_𝑘−1 and the distribution Δ_𝑘−1 with 𝑋₀^𝑘−1 is the regular point. Around 𝑋₀^𝑘−1 we have rank𝐸_𝑢(ℎ_𝑘−1)=𝛾_𝑘−1 where ℎ_𝑘−1=^𝐻𝑘−2ℎ_𝑘−1. Obviously 𝛾_𝑘−1≥𝛾_𝑘−2, and let ℎ_𝑘−1,1 denote 𝛾_𝑘−1−𝛾_𝑘−2 rows chosen from ℎ_𝑘−1 such that rank𝐸_𝑢(ℎ^𝐻𝑘−2_𝑘−1,1)=𝛾_𝑘−1. Let 𝐻_𝑘−1=ℎ^𝐻𝑘−2_𝑘−1,1 and ℎ_𝑘−1,2 denote the rest rows of ℎ_𝑘−1. Then there exist neighborhood 𝑈_𝑘=[𝑁_𝑘,𝑀_𝑘,𝐿_𝑘,𝐾_𝑘,𝑆_𝑘]⊆𝑈_𝑘−1 and smooth mapping 𝜆_𝑘−1(𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘−1),𝑣,…,𝑣^(𝑘−1)) such that 𝐸_𝑢ℎ_𝑘−1,2=𝜆_𝑘−1(⋅)𝐸_𝑢𝐻_𝑘−1, (5) where 𝑁_𝑘=𝑥∣𝑥∈𝑁_𝑘−1,rank𝐸_𝑢ℎ_𝑘−1=𝛾_𝑘−1𝑀_𝑘=𝑧∣𝑧∈𝑀_𝑘−1,rank𝐸_𝑢ℎ_𝑘−1=𝛾_𝑘−1𝐿_𝑘=𝑢∣𝑢∈𝐿_𝑘−1,rank𝐸_𝑢ℎ_𝑘−1=𝛾_𝑘−1𝐾_𝑘=𝑦,…,𝑦^(𝑘−1)∣𝑦,…,𝑦^(𝑘−1)∈𝐾_𝑘−1,rank𝐸_𝑢ℎ_𝑘−1=𝛾_𝑘−1𝑆_𝑘=𝑣,…,𝑣^(𝑘−1)∣𝑣,…,𝑣^(𝑘−1)∈𝑆_𝑘−1,rank𝐸_𝑢ℎ_𝑘−1=𝛾_𝑘−1. (6)

Remark 4. If 𝐸_𝑢(ℎ𝑘−1,2)=0, then we only need to set 𝜆_𝑘−1(⋅)=0.
Let ℎ_𝑘=ℎ_𝑘𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘),𝑣,…,𝑣^(𝑘)=𝐸_𝑥ℎ_𝑘−1,2−𝜆_𝑘−1(⋅)𝐸_𝑥𝐻_𝑘−1𝑥̇+^𝑘−1_𝑖=0𝜕ℎ_𝑘−1,2𝜕𝑦^(𝑖)−𝜆_𝑘−1(⋅)𝜕𝐻_𝑘−1𝜕𝑦^(𝑖)𝑦^(𝑖+1)+𝐸_𝑣ℎ_𝑘−1,2−𝜆_𝑘−1(⋅)𝐸_𝑣𝐻_𝑘−1𝑣̇+^𝑘−1_𝑖=1𝜕ℎ_𝑘−1,2𝜕𝑣^(𝑖)−𝜆_𝑘−1(⋅)𝜕𝐻_𝑘−1𝜕𝑣^(𝑖)𝑣^(𝑖+1) (7) and define recursively ℎ_𝑘𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘),𝑣,…,𝑣^(𝑘)=𝐻_𝑘−1ℎ_𝑘=0. (8) Let 𝑈_𝑘=[𝑁_𝑘,𝑀_𝑘,𝐿_𝑘,𝐾_𝑘,𝑆_𝑘], where 𝐾_𝑘=[𝐾_𝑘,𝐾_𝑘],𝑆_𝑘=[𝑆_𝑘,𝑆_𝑘], and 𝐾_𝑘,𝑆_𝑘 are the value region of 𝑦^(𝑘) and 𝑣^(𝑘), respectively, when 𝑥∈𝑁_𝑘−1,𝑧∈𝑀_𝑘−1,𝑢∈𝐿_𝑘−1,(𝑦,…,𝑦^(𝑘−1))∈𝐾_𝑘−1,(𝑣,…,𝑣^(𝑘−1))∈𝑆_𝑘−1. Accordingly we get the point 𝑋^𝑘₀=(𝑥₀,𝑧₀,𝑢₀,𝑦₀,…,𝑦₀^(𝑘),𝑣₀,…,𝑣₀^(𝑘)) and let Δ_𝑘 denote the distribution generated by the row vectors of 𝐸_𝑢(ℎ_𝑘). If 𝑋^𝑘₀ is the regular point of Δ_𝑘, that is, the matrix 𝐸_𝑢(ℎ_𝑘) has constant rank 𝛾_𝑘 at some neighborhood of 𝑋^𝑘₀. If 𝛾_𝑘=𝑚, then algorithm stops; if 𝛾_𝑘<𝑚, then algorithm enters next step.
In the above construction we produce a sequence of nonnegative integers 𝛾₁,𝛾₂,… which satisfy 0≤𝛾₁≤𝛾₂≤𝛾₃≤⋯≤𝑚.

Definition 5. The relative order 𝜌 for the nonlinear DAE subsystems (1) is the least positive integer 𝑘 such that 𝛾_𝑘=𝑚 or 𝜌=∞ if 𝛾_𝑘<𝑚 for all 𝑘=1,2,….

The main results of this paper can be concluded into the following theorems.

Theorem 6. Consider the nonlinear DAE subsystems (1) with relative order 𝜃. If 𝜃<∞, then there exist unit right inverse systems of nonlinear DAE subsystems (1).

Proof. According to the recursive algorithm, at step 𝜃 we can get the following equation: ℎ_𝜃𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝜃),𝑣,…,𝑣^(𝜃)=0 (9) with rank(𝐸_𝑢(ℎ_𝜃))=𝛾_𝜃=𝑚. By the virtue of the Implicit Function Theorem, (9) determines a unique solution about 𝑢: 𝑢=ℎ_𝜃⁻¹𝑥,𝑧,𝑦,…,𝑦^(𝜃),𝑣,…,𝑣^(𝜃). (10) Denote 𝑣_𝑘=(𝑣^𝑇,…,(𝑣^(𝑘))^𝑇)^𝑇,𝑘=0,1,…,𝜃 and construct the following systems: 𝑥̇=𝑓𝑥,𝑧,ℎ_𝜃⁻¹𝑥,𝑧,𝑟,…,𝑟^(𝜃),𝑣_𝜃,𝑣,𝑥𝑡₀,𝑧𝑡₀=𝑥𝑡₀,𝑧𝑡₀0=𝑔𝑥,𝑧,ℎ_𝜃⁻¹𝑥,𝑧,𝑟,…,𝑟^(𝜃),𝑣_𝜃,𝑣,𝑦=ℎ_𝜃⁻¹𝑥,𝑧,𝑟,…,𝑟^(𝜃),𝑣_𝜃, (11) where 𝑥∈𝑅^𝑛,𝑧∈𝑅^𝑙,𝑟∈𝑅^𝑚 and 𝑦∈𝑅^𝑚, are differential variables, algebraic variables, manipulated input, and controlled output, respectively. Systems (11) are unit right inverse systems of nonlinear DAE subsystems (1), and conclusion can be proved as follows.
By virtue of the recursive algorithm, there exists a row transmit matrix Π_𝑘=^{𝐼𝑟𝑘−100}Π_𝑘10Π_𝑘2,𝑘=1,…,𝜃−1 where Π_𝑘1∈𝑅^{(𝛾𝑘−𝛾𝑘−1)×(𝑚−𝛾𝑘−1)} and Π_𝑘2∈𝑅^{(𝑚−𝛾𝑘)×(𝑚−𝛾𝑘−1)} such that ^𝐻ℎ^𝑘−1_𝑘,1=Π_𝑘⋅ℎ_𝑘, that is, 𝐻_𝑘=𝐻_𝑘−1ℎ_𝑘,1=𝐻_𝑘−1Π_𝑘1𝑑ℎ𝑑𝑡^𝑘−1,2−𝜆_𝑘𝑑𝐻𝑑𝑡^𝑘−1,ℎ_𝑘,2=Π_𝑘2𝑑ℎ𝑑𝑡^𝑘−1,2−𝜆_𝑘𝑑𝐻𝑑𝑡^𝑘−1. (12) For ℎ₀ especially, we have ^𝐻ℎ⁰₀=Π₀₁Π₀₂ℎ₀, where Π₀₁∈𝑅^𝑟0×𝑚 and Π_𝑘2∈𝑅^{(𝑚−𝛾0)×𝑚}.
When the manipulated input of nonlinear DAE subsystems (1) is set to 𝑢(𝑡)=𝑦(𝑡), the states (𝑥(𝑡),𝑧(𝑡)) satisfy 𝑥̇=𝑓𝑥,𝑧,𝑦,𝑣,0=𝑔𝑥,𝑧,𝑦,𝑣. (13) From (11) we have 𝑥̇=𝑓(𝑥,𝑧,𝑦,𝑣),0=𝑔(𝑥,𝑧,𝑦,𝑣) as well as (𝑥(𝑡₀),𝑧(𝑡₀))=(𝑥(𝑡₀),𝑧(𝑡₀)); thus, we have (𝑥,𝑧)=(𝑥,𝑧).
Define 𝑦₀=Π_0,1𝑦,𝑦₀=Π_0,2𝑦 and let 𝑦_𝑘=Π_𝑘,1𝑦_𝑘−1,𝑦_𝑘=Π_𝑘,2𝑦_𝑘−1,𝑘=1,…,𝜃−1. It can be proved that if the manipulated input 𝑟(𝑡)∈𝐶^𝑛(𝑅) of systems (11) satisfies ℎ_𝛼(𝑥,𝑧,𝑦,𝑟,…,𝑟^(𝛼),𝑣_𝜃)=0 and 𝐻_𝑘𝑥,𝑧,𝑢,𝑟,…,𝑟^(𝑘),𝑣_𝑘=𝐻_𝑘𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘),𝑣_𝑘,𝑘=1,…,𝜃−1 (14) as well as initial conditions satisfying ℎ_𝑘,2𝑥𝑡₀,𝑧𝑡₀,𝑢𝑡₀,𝑟𝑡₀,…,𝑟^(𝑘)𝑡₀,𝑣_𝑘𝑡₀=ℎ_𝑘,2𝑥𝑡₀,𝑧𝑡₀,𝑢𝑡₀,𝑦𝑡₀𝑦,…,_(𝑘)𝑡₀,𝑣_𝑘𝑡₀, (15) then we have 𝑦(𝑡)=𝑟(𝑡) where 𝑦(𝑡) is the controlled output of nonlinear DAE subsystems (1). From Definition 2, systems (11) are unit right inverse systems of nonlinear DAE subsystems (1). This completes the proof.

Since (𝑥,𝑧)=(𝑥,𝑧), unit right inverse systems (11) can be realized by directly feedbacking states (𝑥,𝑧) of nonlinear DAE subsystems (1). The composite systems will be linearized and decoupled when (11) is series connected before (1).

It should be noted that there exist differential operator in above realization (11), which is difficult to be realized in physics. For this we have the following theorem.

Theorem 7. Consider the nonlinear DAE subsystems (1) with relative order 𝜃. If 𝜃<∞, then there exist 𝛼-order integral right inverse systems of nonlinear DAE subsystems (1).

Proof. Replace 𝑦 of (10) with 𝑟: 𝑢=ℎ_𝜃⁻¹𝑥,𝑧,𝑟,…,𝑟^(𝜃),𝑣,…,𝑣^(𝜃). (16) Let the highest and lowest order derivative of 𝑟_𝑖(𝑖=1,…,𝑚) are 𝑟^(𝛽𝑖)_𝑖 and 𝑟^(𝛼𝑖)_𝑖, respectively. Define 𝜑=𝜑₁,…,𝜑_𝑚^𝑇=𝑟^(𝛼1)₁,…,𝑟^(𝛼𝑚)_𝑚^𝑇,𝜉_𝑖=𝑟^(𝛽𝑖)_𝑖,𝑟^(𝛽𝑖_𝑖⁺¹⁾,…,𝑟^(𝛼𝑖_𝑖^−1)𝑇,𝑖=1,…,𝑚 (17) and construct the following systems Σ_𝛼∶𝜉̇_𝑖=𝐴_𝑖𝜉_𝑖+𝐵_𝑖𝜑_𝑖,𝑖=1,…,𝑚𝑦=ℎ_𝛼⁻¹𝑥,𝑧,𝜉,𝜑,𝑣_𝛼, (18) where 𝐴_𝑖=001⋱⋱⋱1_{(𝛼𝑖−𝛽𝑖)×(𝛼𝑖−𝛽𝑖)}𝐵_𝑖=0⋮01_{(𝛼𝑖−𝛽𝑖)×1},𝜉=𝜉^𝑇₁,…,𝜉^𝑇_𝑚^𝑇,𝑣_𝜃=𝑣^𝑇,…,(𝑣^(𝜃))^𝑇𝑇. (19) If 𝑟(𝑡)∈𝐶^𝜃(𝑅) of (18) satisfies ℎ_𝜃𝑥,𝑧,𝑢,𝑟,…,𝑟^(𝜃),𝑣_𝜃𝐻=0,_𝑘𝑥,𝑧,𝑢,𝑟,…,𝑟^(𝑘),𝑣_𝑘=𝐻_𝑘𝑥,𝑧,𝑢,𝑦,…,𝑦^(𝑘),𝑣_𝑘,𝑘=1,…,𝜃−1. (20) Meanwhile initial conditions satisfy ℎ_𝑘,2𝑥𝑡₀,𝑧𝑡₀,𝑢𝑡₀,𝑟𝑡₀,…,𝑟^(𝑘)𝑡₀,𝑣_𝑘𝑡₀=ℎ_𝑘,2𝑥𝑡₀,𝑧𝑡₀,𝑢𝑡₀,𝑦𝑡₀𝑦,…,_(𝑘)𝑡₀,𝑣_𝑘𝑡₀ (21) when 𝜑=(𝑟^(𝛼1)₁,…,𝑟^(𝛼𝑚)_𝑚)^𝑇, 𝜉_𝑖(𝑡₀)=(𝑟^(𝛽𝑖)_𝑖(𝑡₀),…,𝑟^(𝛼𝑖_𝑖⁻¹⁾(𝑡₀))^𝑇(𝑖=1,…,𝑚), we have output of nonlinear DAE subsystems satisfies 𝑦^(𝛼𝑖)_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑖=1,…,𝑚 (where 𝜑 is new control input to be designed). By Definition 3, systems (18) are 𝛼-order integral right inverse systems realized by state-feedback and dynamic compensation. This completes the proof.

There are two possible difficulties existing in realization of inverse systems: The one hand, the proposed method depends on the exact model of the controlled systems and sensitive to the perturbation of parameters or variation of structure. On the other hand, analysis solution of manipulated control may be difficult to be obtained. Inspired by [16], we may use the excellent approximation ability of Neural Networks to overcome the imprecise model and unknown analysis solution of manipulated control. This will be the next work we will undertake.

4. Decentralized Excitation and Valve Coordinative Control of Synchronous Generator

Excitation and valve coordinative control is studied for one synchronous generator based on the scheme proposed in this paper. For the sake of simplicity, the subscript 𝑖 is still omitted.

The mathematical model of synchronous generator excitation and valve coordinative control is described by the following two-input two-output nonlinear DAE subsystems [10]: 𝛿̇=𝑓₁(⋅)=𝜔−𝜔₀𝜔̇=𝑓₂(⋅)=𝜔₀𝐻𝑃_𝐻+𝐶_𝑀𝐿𝑃_𝑚0−𝐷𝜔₀𝜔−𝜔₀−𝐸^′_𝑞+𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑𝐼_𝑞𝐸̇^′_𝑞=𝑓₃(⋅)=1𝑇^′_𝑑0𝐸_𝑓−𝐸^′_𝑞−𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑𝑃̇_𝐻=𝑓₄(⋅)=1𝑇_𝐻Σ−𝑃_𝐻+𝐶_𝐻𝑃_𝑚0+𝐶_𝐻𝑈_𝑐𝑔(𝑥,𝑧,𝑢,𝑣)=0, (22) where 𝑔(𝑥,𝑧,𝑢,𝑣)=𝑃_𝑡−𝐸^′_𝑞+𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑𝐼_𝑞+𝑟_𝑎𝐼²_𝑑+𝐼²_𝑞𝜃_𝑈𝑥−𝛿+arc𝑐𝑡𝑔^𝑞𝐼_𝑞−𝑟_𝑎𝐼_𝑑𝐸^′_𝑞−𝑥^′_𝑑𝐼_𝑑−𝑟_𝑎𝐼_𝑞𝐼_𝑡−𝐼²_𝑑+𝐼²_𝑞𝑄_𝑡−𝐸^′_𝑞𝐼_𝑑+𝑥_𝑞𝐼²_𝑞+𝑥^′_𝑑𝐼²_𝑑, (23) where the differential variables 𝑥=(𝛿,𝜔,𝐸^′_𝑞,𝑃_𝐻)^𝑇 are relative power angle between G1 and G4, rotate speed deviation of G1, 𝑞-axis transient potential, and the high pressure mechanical power, respectively, the algebraic variables 𝑧=(𝑃_𝑡,𝜃_𝑈,𝐼_𝑑,𝐼_𝑞)^𝑇 are active power, the angle of voltage, the 𝑑-axis current, and the 𝑞-axis current, respectively, and the interconnection input 𝑣=(𝐼_𝑡,𝑄_𝑡)^𝑇 are the generator stator current and the reactive power, respectively. The manipulated input 𝑢=(𝐸_𝑓,𝑈_𝑐)^𝑇 are induction electromotive force and the governor position, respectively. The controlled output 𝑦 is chosen as voltage and rotate speed deviation, respectively: (⋅)=𝑉𝑦=ℎ^𝑡𝜔=𝑃²_𝑡+𝑄²_𝑡𝐼_𝑡𝜔. (24) The others are the systems parameters.

The Jacobian matrix of 𝑔(𝑥,𝑧,𝑢,𝑣) with respect to 𝑧 is 𝜕𝑔=𝑥𝜕𝑧10_𝑞−𝑥^′_𝑑𝐼_𝑞+2𝑟_𝑎𝐼_𝑑𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑+2𝑟_𝑎𝐼_𝑞01−11+𝐴²𝜕𝐴𝜕𝐼_𝑑−11+𝐴²𝜕𝐴𝜕𝐼_𝑞𝐼00^𝑑𝐼_𝑡−𝐼_𝑞𝐼_𝑡00−𝐸^′_𝑞+2𝑥^′_𝑑𝐼_𝑑2𝑥_𝑞𝐼_𝑞≜10𝑎₁₃𝑎₁₄01𝑎₂₃𝑎₂₄00𝑎₃₃𝑎₃₄00𝑎₄₃𝑎₄₄, (25) where 𝐴=(𝑥_𝑞𝐼_𝑞−𝑟_𝑎𝐼_𝑑)/(𝐸^′_𝑞−𝑥^′_𝑑𝐼_𝑑−𝑟_𝑎𝐼_𝑞). It can be verified that under the normal operating condition, the following equation det(𝜕𝑔/𝜕𝑧)=−(1/𝐼_𝑡)(2𝑥_𝑞𝐼_𝑑𝐼_𝑞+𝐸^′_𝑞𝐼_𝑞−2𝑥^′_𝑑𝐼_𝑑𝐼_𝑞)≠0 holds, that is, matrix 𝜕𝑔/𝜕𝑧 is of full rank and generator is of index one.

From (22) and (23), we can get the differential equation of 𝑧 as follows: 𝑧̇=−𝜕𝑔𝜕𝑧⁻¹𝜕𝑔𝜕𝑥𝑓−𝜕𝑔𝜕𝑧⁻¹𝜕𝑔𝜕𝑣𝑣̇, (26) where 𝜕𝑔𝜕𝑧⁻¹=10𝑏₁𝑐₁01𝑏₂𝑐₂00𝑏₃𝑐₃00−𝑎⁻¹₃₃𝑎₄₃1,𝑏₁=−𝑎⁻¹₃₃𝑎₁₃−𝑎⁻¹₃₃𝑎₄₃𝑐₁,𝑏₂=−𝑎⁻¹₃₃𝑎₂₃−𝑎⁻¹₃₃𝑎₄₃𝑐₂,𝑏₃=1−𝑎⁻¹₃₃𝑎₄₃𝑐₃,𝑐₁=−𝑐₄⁻¹𝑎₁₄+𝑎⁻¹₃₃𝑐₄⁻¹𝑎₁₃𝑎₃₄,𝑐₂=−𝑐₄⁻¹𝑎₂₄+𝑎⁻¹₃₃𝑐₄⁻¹𝑎₂₃𝑎₃₄,𝑐₃=𝑐₄⁻¹𝑎₃₄,𝑐₄=𝑎₄₄−𝑎⁻¹₃₃𝑎₃₄𝑎₄₃. (27) The Jacobian matrix of 𝑔 with respect to 𝑥 and interconnection input 𝑣 are 𝜕𝑔=𝜕𝑥00−𝐼_𝑞0−10−11+𝐴²𝜕𝐴𝜕𝐸^′_𝑞0000000−𝐼_𝑑0,𝜕𝑔𝜕𝑣=.00001001 (28)

According to the recursive algorithm, the relative degree of (22) is 2 and the solution of 𝑢 can be derived as follows: 𝐸_𝑓=𝑇^′_𝑑0−𝐼_𝑞−𝑐₁𝐼_𝑑×1𝑃_𝑡1𝐼_𝑡𝑃²_𝑡+𝑄²_𝑡𝐼²_𝑡𝑦̇₁+𝐼̇_𝑡𝑃²_𝑡+𝑄²_𝑡−𝑄_𝑡𝑄̇_𝑡+𝑏₁𝐼̇_𝑡+𝑐₁𝑄̇_𝑡+𝐸^′_𝑞+𝑥_𝑑−𝑥^′_𝑑𝐼_𝑑,𝑈_𝑐=𝜔₀𝐶_𝐻𝐻𝑇_𝐻Σ×𝑦̈₂−𝜔₀𝐻𝑇_𝐻Σ−𝑃_𝐻+𝐶_𝐻𝑃_𝑚0−𝐷𝜔₀𝑓₂−𝐼_𝑞𝑓₃−𝑥_𝑞−𝑥^′_𝑑𝑐₃𝐼_𝑑𝐼_𝑞𝑓₃+𝑥_𝑞−𝑥^′_𝑑𝑏₃𝐼_𝑞𝐼̇_𝑡+𝑥_𝑞−𝑥^′_𝑑𝑐₃𝐼_𝑞𝑄̇_𝑡−𝐸^′_𝑞𝐼_𝑑𝑓₃+𝑎⁻¹₃₃𝑎₄₃𝐸^′_𝑞𝐼̇_𝑡+𝐸^′_𝑞𝑄̇_𝑡−𝑥_𝑞−𝑥^′_𝑑𝐼²_𝑑𝑓₃+𝑎⁻¹₃₃𝑎₄₃𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑𝐼̇_𝑖+𝑥_𝑞−𝑥^′_𝑑𝐼_𝑑𝑄̇_𝑡. (29) According to (18), the (1,2)-order integral right inverse systems for synchronous generator (22) can be constructed, with which the decoupling and linearization of the composed systems can be achieved.

The simulation is conducted based on a two-area four-machine power systems (as shown in Figure 1).

Figure 1 

Two-area four-machine power system.

The parameters of each generator and transformer are the same and the other parameters can be found in [17]. Systems operating condition is as follows: at first systems operate under double circuit stable state, then a three-phase symmetrical earth fault happens at 𝑘 point at line 7-8 on 0.5 second and the ground reactance of the failure point is 0.0001 pu.The fault is cut on 0.65 second and the systems return to original operating condition. Generators G1, G3 both adopted inverse excitation and valve controller and the closed-loop 𝜑₁,𝜑₂ adopted 𝑃𝐼𝐷 controller where 𝜑₁ only adopts proportion part equal to 10 and 𝜑₂ adopt proportion part equal to 30 and differential part equal to 5. The other generators adopt traditional linear controller. The simulation is based-on MATLAB and the results are shown in Figure 2.

(a) Relative power angle between G1 and G4

(b) Relative power angle between G3 and G4

(c) Load bus voltage of G1

(d) Relative power angle between G1 and G2

(e) Load bus voltage of G3

Figure 2 

Simulation results of G1 and G3.

As shown in Figure 2, when generators G1 and G3 both equipped excitation and valve coordinative inverse controller, both interarea oscillation and area oscillation are improved dramatically.

With noting that 𝑥_𝑑,𝑥_𝑞 cannot be measured online. To overcome this difficulty, we can adopt the method proposed in [16] to replace the 𝑥_𝑑,𝑥_𝑞 with local measured signals.

5. Conclusion

In this paper, the problem of invertibility for a special class of nonlinear DAE subsystems is studied. The definitions of inverse systems for nonlinear DAE subsystems are put forward. Then a recursive algorithm is given, with which to determine the invertibility of nonlinear DAE subsystems. Physically realizable right inverse systems are constructed through state-feedback and dynamic compensation, with which the decoupling and linearization of composite systems are achieved so that various linear control theorems and methods can be applied. Not restricted to power systems components decentralized control, the result of this paper is also meaningful to decentralized control of other areas of large-scale systems.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61004001, 61104103, 51177019), Natural Science Foundation of Jiangsu Province of China (BK2011826), Jiangsu Government Scholarship for Overseas Studies, Open Project of Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University (MCCSE2012A07), and State Grid Corporation of China, Major Projects on Planning and Operation Control of Large Scale Grid (SGCC-MPLG022-2012).