Robust Stabilizer Rewritten

7/31/2019 Robust Stabilizer Rewritten

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DESIGN OF NONLINEAR CONTROLLER FOR SINGLE MACHINE

CONNECTED TO AN INFINITE BUS SYSTEM

Assignment 2

Final Year Project 2012

External Advisor: Dr. Arshad Habib Malik

Organization: KINPOE (KANUPP)

Internal Advisor: Sir Raja Masood Larik

Group Members:

Abdul MajeedSyed Danish-ullah

Osama Idris

M. Salman Younus

NED UNIVERSITY OF ENGINEERING AND TECHNOLOGY


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Table Of Contents:

ABSTRACT 3

LIST OF SYMBOLS 4

1. INTRODUCTION 5

2. POWER SYSTEM MODEL 5

3. DEFINITIONS 7

4. LINEARIZATION METHOD 8

5. SLIDING MODE CONTROL 10


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Abstract:

This paper presents a new method for designing nonlinear power system stabilizer to improve

transient and dynamic stability. In these studies, a 4th order nonlinear model of synchronous

generator has been used. First the existence of a diffeomorphism Transformation for nonlinear

power system has been proved and then, the nonlinear model has been transformed to a canonical

form using input- output linearization method. Finally, the linearized model is controlled using

sliding mode technique. At these procedures, the stability of zero dynamics has been attained. The

simulation results on a single machine connected to an infinite bus show the good performance of

stabilizer.


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List of symbols:

EfExcitation voltage

E'd , E'q. d- and q-axis components of the transient internal EMF

id , iq d and q axis component of armature current.

H Inertia coefficient

Te

, Tm

Electro mechanical air gap torque and mechanical torque supplied by the

prime mover.

xd , x'd , xq , x'q d and q axis synchronous and transient reactance of the generator.

, s Rotor speed and the synchronous speed.

= - s Rotor speed deviation

Rotor angle with respect to the synchronous reference axis defined by P-V Bus

Ra Armature Resistance

T'do , T'qo Open circuit d- and q-axis transient time constants


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1. Introduction:In the recent decade, the complexity and size of power systems have increased remarkably. So, power

engineers have had to face some major problems such as dynamic stability.

Traditional designs of power systems assumed a mainly linear nature for the system. The stabilization

was therefore insured by linear regulators such as PSS and AVR. These compensators are based on classical

control algorithms, which assume a linearized model of the power system about an equilibrium point [5]. But

as mentioned, the trend is for power systems to be operated closer to limits and outside the linear region.

So the classical regulators can hardly maintain the synchronism and the stability of the system.

Therefore the use of nonlinear controllers has become crucial.

Nonlinear control theory based on differential geometry has provided powerful tools for symmetric

synthesis of nonlinear feedback systems [2], [8]. Most of available methods linearized the input-output

response of a nonlinear system by using a state feedback, or linearized exactly the state equations using a

coordinate transformation and a state feedback [9],[10],[11],[12]. The feedback linearization techniques

were first explored in power system application in [16] and later in [17]. The main object in these studies

was to enhance the systems stability. Since the nonlinear model used in these studies was a reduced

third order model of the machine, no internal dynamics were left to worry about [6],[13]. After that, the

domain of these studies was developed. The approach uses a feedback- linearizing controller to regulate

both the terminal voltage at the generating plant and the rotor angle.

In this paper a single machine-infinite bus system has been considered. The model according to thedesign method developed in this paper, the system is modeled in the form of an affine nonlinear system

instead of linear one at a certain equilibrium point. The main goal of this paper is to investigate the

application of a nonlinear model of a power system in order to improve both its stability and damping

properties even under large and sudden disturbances. One of the major limitations of the input-output

linearization approach is that the nonlinear system must have a minimum phase property, which has been

attained in our approach. Finally, the robustness of the approach is improved using a sliding version of

the controller.

2. Power system modelIn this section, its assumed that the external network connecting the generator terminals to the infinite bus

is a linear two- port network. This includes any complex representation of the external system with several

Transmission lines, Transformers and Loads. The loads are assumed to be of constant impedance. A typical

network is shown in figure (1). For simulation, a forth order model of synchronous generator is used

(damping neglected):

(1)

, -

(2)


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[ ] (3)

[ ( ) ] (4)The Electrical torque, Te is expressed as:

( ) (5)The non state variables , can be obtained from the stator algebraic and networkequations:

( )

(6)Where,

co in (7) co in

(ZR +jZI) is the input impedance of the external network viewed from the generator terminals with

the infinite bus shorted. ((h1+jh2) is the voltage gain at the generator terminals with armature open-

circuited.

By choosing the following state variables:

(8)The non-linear equations (1-4) can be put into the form:

(9)Where:

(10) And


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(11)

Therefore, we have a power system model with general state equation in the form of an affine

nonlinear system.

Figure 1: Typical n/w diagram

3. Definitions:In this part, we first introduce some mathematical tools from differential geometry and topology to

limit the conceptual and notational complexity. We discuss these tools directly in the context of

nonlinear dynamic systems.

Let h :

be a smooth scalar function, andf:

be a smooth vector field on

, Then

the lie derivative of h with respect tofis a scalar function defined by:

(12)Thus, the lie derivative Lfh is simply the directional derivative of h along the direction of the vector f

(projection ofh in the direction of f).If we assume:

(13)

, - (14)Then we have:

(15)Repeated lie derivatives can be defined recursively:

(16)


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4. Linearization MethodAs discussed in part (1), The basic approach of input-output linearization is simply to differentiate

the input function y repeatedly until the input u appears, and then design control signal, u. Using

the procedure which is mentioned in [2],[8], The nonlinear equation (1-4) can be linearized.

We assume the output signal of the system as:

(17)The successive derivatives of y can be calculated as follows:

*+ (18)

, - (19) ( ) (20)Please notice, that the relative degree of system is three (r =3), but the order of main model is four,

consequently the system has a hidden or internal dynamic. This means that one of state equations

is related to the internal dynamics.

In order to check the existence of a diffeomorphism, we must prove that g(x) is involutive, but we

know that any vector with constant values is involutive, hence this condition is satisfied. On the

other hand, if the relative degree of system in region is equals to r then the gradients in region are linearly independent, where: (21)This means 1,, r can be chosen as new state variables and (x) will be the new state vector. The

new variables can be defined as follows:

y hx

x

y f x (22) y fLet us show that nr =j (=1) more variable j exists to complete the new state variable

transformation. To have diffeomorphism, we simply need that gradient vector j can be founduch ha i j are linearly independen in he neighbourhood From he Frobenioutheorem, since we have only one vector field g (which is involutive), this guaranties the

existence of n-

r linear independen gradien funcion j

jn-r).

At the final step, from the result of the Frobenious Theorem, we mustelec uch ha:


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(23)Hence,

(24)

Since only g3 is not equal to zero we must have:

(25)Thi repreen ha mu be independen of x3 For aifying hi condiion we choe simply as:

x4 (26)Therefore the diffeomorphism will be: , -Or:

, - (27)Since is not singular, hence, exists. This diffeomorphism can transform the non-linear model to a normal form as:

b u

(28)

Where, (29) hx (30)

And the hidden dynamic is represented by:

(31)With output y: (32)It can be seen that the 4th state variable is not directly accessible; therefore, the zero dynamics

of the stabilizer should be examined in order to guarantee the overall stability of the system.

The zero dynamics of a system is defined to be the internal dynamics of the system when the

system output is kept constantly to zero by (28-31). The zero dynamics is established in the

following way:


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Let , by this procedure, from eq (31) it is obvious that the zero dynamics ofthe system is asymptotically stable.

Consequently, at the final step after a long procedure can be calculated by means of (1-7).Then by separating the final equation,

. The control signal, u, will be given as below:

(33)And we can reach the linear equation:

(34)Therefore, by this procedure, the non-linear term is cancelled.

5. Sliding Mode ControlAs discussed in the last part, by means of a diffeomorphism, we can transform the nonlinear

model to a canonical form. One of the difficulties with input-output linearization is that, the

approach relies on exact cancellation of nonlinear terms in order to make linear input-output

relation. Due to normal error and uncertainties in the model of nonlinear system, thecancellaion won be exac To have a robu conrol deign many mehod have beenproposed like sliding mode or adaptive control.

The sliding mode control (SMC) is a robust nonlinear control strategy employing feedback of a

discontinuous signal. The discontinuous feedback control method is chosen so that system

trajectory is directed towards the sliding hyper plane when the sliding mode occurs, The

system state will remain on it forever and the system behaves as an equivalent system with

desired dynamics which is governed by the sliding hyper plane equation.

The control design procedure consists of two steps:

1- A feedback control law to verify sliding condition with considering the uncertainty.

2- Discontinuous control law is suitably smoothed to achieve an optimal trade-off between

control bandwidth and tracking precision.

Please notice that this control procedure is discontinuous, which leads to control chattering.

Chattering is undesirable in practice because it involves high control activity and further may

excite high frequency dynamics.

The control objective is to choose control signal, v, to make Y(t) track Yd(t) (desired output) in

the presence of boundable modeling errors, where :

, - is the external state. The tracking error vector can be defined as:


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, - (35)The fir ep in liding mode conrol i o define he wiching urface Se with:

(36)

Where is a positive constant that will interpret the place of poles in closed loop system andr=3. Therefore:

(37)If the control signal can be chosen such that:

(38)

Then the system will hit the switching surface S=0, and slide along this surface. Further,

satisfying Eq. (38) guarantees that error asymptotically approaches zero.

From (38) we have:

(39)For smoothing the switching law in a neighborhood close to sliding surface, we can use

saturation function. Therefore:

(40)By substituting final equation in the main model, the control signal of the nonlinear system will

be obtained.

Robust Stabilizer Rewritten

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