Mathematical Problems in Engineering

Volume 2015, Article ID 415281, 11 pages

http://dx.doi.org/10.1155/2015/415281

## Multiobject Holographic Feedback Control of Differential Algebraic System with Application to Power System

^{1}College of Electrical Engineering, Guangxi University, Nanning, Guangxi 530005, China^{2}School of Automobile Engineering, Linyi University, Linyi, Shandong 276005, China

Received 13 October 2014; Accepted 2 May 2015

Academic Editor: Hari M. Srivastava

Copyright © 2015 Lanmei Cong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A multiobject holographic feedback (MOHF) control method for studying the nonlinear differential algebraic (NDA) system is proposed. In this method, the nonlinear control law is designed in a homeomorphous linear space by means of constructing the multiobject equations (MOEq) which is in accord with Brunovsky normal form. The objective functions of MOEq are considered to be the errors between the output functions and their references. The relative degree for algebraic system is defined that is key to connecting the nonlinear and the linear control laws. Pole assignment method is addressed for the stability domain of this MOHF control. Since there is no any approximation, the MOHF control is effective in governing the dynamic performance stably both to the small and major disturbance. The application in single machine infinite system (SMIS) shows that this approach is effective in the improvement of stable and transient stability for power system on the disturbance of active power or three-phase short circuit fault.

#### 1. Introduction

Power system shows strong nonlinear properties, especially in the action of large disturbance such as three-phase short circuit fault. Many effective control theories and methods have been proposed for this nonlinear system in literatures, such as differential geometry theory [1, 2], Lyapunov energy function [3, 4] and Hamiltonian function [5, 6], machine learning, and neural network [7–10]. Feedback linearization based on differential geometry theory is effective in the stability control of the nonlinear power system [2]. Different from the approximate linearization, Differential geometry method designs the nonlinear control law by constructing the mapping from the nonlinear space to the homeomorphous linear without approximation. Tan and Wang in [11] designed the adaptive excitation and phase shifter controller based on third-order generator model. Kennedy et al. in [12] designed a nonlinear excitation controller based on a form of state feedback linearization using the geometric approach. With the work of Sastry and Isidori [13], the adaptive control of “minimum-phase” nonlinear system was studied by state feedback exactly linearization and several initial results were derived.

Developing the differential geometry theory, Li et al. in [14] proposed the Multi-index Nonlinear Control (MINC) method. Their research demonstrated that it was not necessary for the system to satisfy the condition of completely exact linearization, which is usually too strict condition to satisfy for most of systems. Li et al. also addressed that the part exact linearization made the selection of output function more flexible to adapt the control requirement of the controlled system, which made MINC method get more satisfied control performance. Li et al. [14] also proposed an effective and practical method of determining output function that was linear combination of state vectors. The precondition of MINC method adopting part exact linearization is that the zero-dynamic must be stable, which is possibly difficult to prove for general system. For solving this problem, Liu et al. in [15] proposed the multiobject holographic feedback (MOHF) control method. This method designed the nonlinear feedback law in linear space by constructing multiobjective equations that satisfy Brunovsky normal form [16]. Furthermore, it is unnecessary to prove the stability of zero-dynamic. These research works are based on the ordinary differential equation (ODE) model.

For most of the complicated power system, it is insufficient that ODE describes wholly the dynamic activity, since the connecting nodes of power system have to be formulated in algebraic equations. The general power system, hence, is usually modeled as differential algebraic equation (DAE).

Several typical methods are described in literatures for studying the differential algebraic system and its application in power systems. Kurina and März in [17] addressed the linear quadratic optimal control problems with constraints described by general linear DAEs with variable coefficients. The optimal feedback control matrix can be obtained by suitably formulating a Riccati DAE system, which is similar to the classical example in which the constraints are described by explicit ODE system. Tsolas et al. in [18] presented the preserving structure model of general power system and suggested that the DA model could be replaced locally by an ODE description. They developed further some geometric characterizations of the region of attraction for a stable equilibrium point. The transient stability was studied in [19, 20] based on preserving structure model and the total energy function were shaped. Wang and Chen [21] presented the differential geometric method that used the preserved structure model for DA systems. The Lyapunov function was proposed in [22] for the stability research of multimachine power systems. The energy function presented in [23] as well as [18] showed some properties of Lyapunov function. Kautsky et al. [24] described the numerical method for determining robust solutions of pole assignment by state feedback. The robust pole assignment problem is also discussed in [24]. One may see the eigenstructure assignment approach of the state output feedback pole placement for input output in [25]. The pole assignment problems were proposed by [26–28] with projection and deflation methods together with optimization approach, respectively.

This paper explores the MOHF control described by the differential algebraic model. As an example of application in power system, the feedback law is designed for single machine infinite system. The structure is organized as follows. Section 2 presents a model of general SIMO differential algebraic system. Section 3 defines the derivative and relative degree for differential algebraic system. Section 4 proposes the MOHF design method and pole assignment approach for this design. Section 5 designs a practical power system based on the preserving structure as an application of the MOHF theory. Section 6 gives the conclusions.

#### 2. Differential Algebraic Model

Consider the single-input multiout differential algebraic power system: The first differential equation describes the behaviors of dynamic components of power system. The second algebraic equation shows the relations of the voltage, active power, and (or) reactive power flows among the nodes. is state vector. is algebraic variable. is the input of the plant. is the output function which is decided by control performance. , , and are the following mapped functions:

The design in this paper is based on the following assumptions.

*Assumption 1. *, , , and are smooth manifolds.

*Assumption 2. *, , and satisfy the compatible initial conditions, which are if the input there exist , , and on the equilibrium point of (1). Without loss of generality, we assume that .

*Assumption 3. *The Jacobi matrix of with respect to is nonsingular.

#### 3. Relative Degree of the Differential Algebraic System

The definition of the relative degree for ordinary differential equation system was given by Sastry and Isidori in [13]. We redefine it here in order to design differential algebraic system. Firstly, a special matrix is given aswhere is an unit matrix of . This matrix exists in the condition of Assumption 3 holding.

The definition of derivative is formulated as follows.

*Definition 4. *The product of the Jacobi matrix of output function with respect to and , the matrix , and the vector field are defined as derivative of with respect to in differential algebraic system; that is The higher derivatives are then defined asCorrespondingly, the derivative of the () to is defined asAn immediate definition of the relative degree for the differential algebraic systems is yielded as the following.

*Definition 5. *The DA system of (1) is said to have relative degree to the output function , if and only if the following conditions holds:With these definitions of derivative and relative degree, we can establish the homeomorphous mapping between nonlinear space and linear in the MOHF control design of next section.

#### 4. MOHF Control Design

In this section, we describe the MOHF design method in detail. We establish, firstly, the adaptive model and then discuss the rules of choosing objective function. In the following section we propose the structure of multiobjective equation and design feedback control law. The problem of pole assignment is solved in Section 4.4.

##### 4.1. Adaptive Model

Consider the reference model:

The adaptive model is derived from (9) and (1):where

The reference adaptive output function of plant (1) is followed aswhere is the output of (9) and also the reference output of plant (1).

##### 4.2. Objective Functions

Objective functions are the variables that need to be constrained in control. These functions are considered as the formwhereand and are the th component of output function in (12) and its reference trajectory.

In this MOHF control, the output functions (12) are determined for the purpose of getting satisfied control object. They include not only the practical outputs of the system but also other variables related to control properties. We consider them as the linear combination of state vectors and algebraic variables, which meanswhere , andare the constant matrices decided by the dynamic and stability properties of plant.

*Remark 6. *The choice of objective function acts importantly in MOHF control. It decides the effect of control and the complexity of feedback law. Two factors fall into main considerations. The first, also the most important, is the relations between objective functions and control performance. The second is the practicability of the designed control law, since the excessively complicated law is probably difficult to carry out and maybe declines the robustness. Based on both of the considerations, it is of high priority to let the relative degree holdin determining the objective functions. Compared with higher , this can make feedback law more simple and hence easy to be used in practical system. For most of system, (17) is not a hard condition if the objective functions are arranged appropriately. Our following design is based on this condition satisfied.

##### 4.3. Design of MOHF Control Law

In this section, we use the objective functions in (13) to construct the multiobjective equation that is in agreement with Brunovsky normal form [16] and design the nonlinear feedback law in linear space. The multiobjective equation is formed aswhereare Brunovsky constant matrices.

This system (18) is called the extra system. The feedback law is called extra system control law. It is designed as the following form according to linear optimal theory

The feedback matrices can be yielded by the next pole assignment.

The control law of system (1) is accordingly called as the inner system control law, which is bridged with by (5)–(7). From (18) we haveand from (20), the inner feedback control law is obtained:

*Remark 7. *This MOHF method uses the extra control law to govern the objective functions to their equilibriums. The homeomorphous transformations of (21)-(22) transfer this law to equivalent input of nonlinear system to constrain the nonlinear dynamic oscillation. Since this transformation from nonlinear linear space to linear holds all the state information without approximation, this approach may enhance the dynamic stability when is designed appropriately.

The feedback and output constant matrix are designed by the following pole assignment.

##### 4.4. Pole Assignment

The linear reference system of (1) is derived from the Taylor expansion on The linear expansion of output function (2) iswhere , , , , are Jacobian matrices.

For the stability control of power system, the aim of control is to make the system reinstate to its original equilibrium or achieve a new equilibrium after the disturbance deleted. So we consider that the reference trajectories of (23)–(25) are equilibrium; that isSubstituting (24) into (23) and (25), the linear adaptive model is as follows:which is expressed concisely as

Selecting the nonsingular linear transformation matrix , let ; then (29) are transformed to the following controllable normal formwhere

From (29), (30), and (31) we have that

Assume that (30)-(31) hold condition (17), we get the feedback law

Substituting (34) into (30) we getwhereThe characteristic polynomial of (31) is

*Remark 8. *Equation (27) describes the linear asymptotical system of plant (1) in the neighborhood of hyperbolic singular point. According to Hartman-Grobman theory, the above pole assignment to the linear system can achieve locally stability control of (1).

*Remark 9. *This MOHF design method supplies an inner adaptive reference track that is the equilibrium state. In case that the feedback law constrains the objective functions to its reference tracks, the stability of the system is held.

#### 5. Application

As an example of application, we designed MOHF law for single machine infinite system (SMIS) based on a DA model.

The DA model of the SMIS iswhere is the state vector of the generator and the algebraic constraint variable. is the rotor angle, is the rotor speed, and is the transient internal voltage of the generator. and are direct-axis and quadrature-axis synchronous reactance of the generator, respectively. is direct-axis transient reactance. ,, and , where, is the transformer reactance and the transmission line reactance. is output terminal voltage of the generator and is the voltage at infinite bus. is the time constant of the field-winding with open-circuit armature winding. is the constant of damper winding and the inertia time constant of the rotor.

Compared with (1), the matrices , , and of (38) are

The output functions and their reference are considered asThe objective function is as follows:

Construct the multiobjective equation in accordance with the Brunovsky form as

The feedback lawwhere and the constants , , and are designed according to pole assignment.

From (43) (44), we get another expression of

From (39)-(40), substituting (5)~(7) in (46), we have thatwhere and ( is the input of the SIMS in the stable state). From (46) we haveFrom of (40) we have thatThe is gotten by the substitution (49) into (4) Combining (48)–(50) with (5)–(7), the and are as follows:

The nonlinear adaptive feedback control law is derived as*Simulation and Discussions*. The constants of (51)-(52) are given as ,, ,, , and. The initial running conditions are considered as ,, and .

Figure 1 shows the simulation results when the disturbance of the active power steps up 20% with the constants , , and ; for MOHF.